# Tangent Ind-Categories

Geoff Vooys

###### Abstract.

In this paper we show that if $\operatorname{\mathscr{C}}$ is a tangent
category then the Ind-category
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ is a tangent category as well
with a tangent structure which locally looks like the tangent structure on
$\operatorname{\mathscr{C}}$. Afterwards we give a pseudolimit description of
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{/X}$ when
$\operatorname{\mathscr{C}}$ admits finite products, show that the
$\operatorname{Ind}$-tangent category of a representable tangent category
remains representable (in the sense that it has a microlinear object), and we
characterize the differential bundles in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ when
$\operatorname{\mathscr{C}}$ is a Cartesian differential category. Finally we
compute the $\operatorname{Ind}$-tangent category for the categories
$\mathbf{CAlg}_{A}$ of commutative $A$-algebras,
$\operatorname{\mathbf{Sch}}_{/S}$ of schemes over a base scheme $S$,
$\operatorname{\mathnormal{A}-\mathbf{Poly}}$ (the Cartesian differential
category of $A$-valued polynomials), and
$\operatorname{\operatorname{\mathbb{R}}-\mathbf{Smooth}}$ (the Cartesian
differential category of Euclidean spaces). In particular, during the
computation of $\operatorname{Ind}(\operatorname{\mathbf{Sch}}_{/S})$ we give
a definition of what it means to have a formal tangent scheme over a base
scheme $S$.

###### Key words and phrases:

Tangent category, $\operatorname{Ind}$-category,
$\operatorname{Ind}$-completion, Cartesian differential category, formal
scheme

###### 1991 Mathematics Subject Classification:

Primary 18F40

###### Contents

  1. 1 Introduction
    1. 1.1 Results and Structure of the Paper
  2. 2 A Review of Tangent Categories
  3. 3 Tangent Ind-Categories
  4. 4 Properties of Ind-Tangent Categories
    1. 4.1 A Pseudolimit Result
    2. 4.2 Representable Ind-Tangent Categories
    3. 4.3 Differential Bundles and Ind-Tangent Categories
  5. 5 Ind-Tangent Category Examples
    1. 5.1 The Tangent Structure on Commutative $A$-Algebras
    2. 5.2 The Zariski Tangent Structure on Formal Schemes
    3. 5.3 The Ind-Tangent Category of the CDC of Polynomials
    4. 5.4 The Ind-Tangent Category of the CDC of Smooth Maps

##  1\. Introduction

This paper is a study of two different categorical geometric concepts and the
ways in which they interact with each other. On one hand, we have the
categorification of tangent bundles and differential geometry (as discovered
by Rosický in [17] and later rediscovered by Cockett and Crutwell in [3]) and
the categorical approach to differential reasoning by the semantics of the
tangent bundle functor itself; on the other hand we have
$\operatorname{Ind}$-categories (as discovered by Grothendieck in [1] in its
relation to presheaf toposes the mechanics involved in taking filtered
colimits) which give an inductive cocompletion of a category and allow for
well-behaved “at the limit” geoemtric arguments. In particular,
$\operatorname{Ind}$-completions and $\operatorname{Ind}$-arguemnts often
carry local, nearly infinitesimal, information and behave as though they are
completions of spaces in certain adic topologies. Because of the geometric
similarities between studying infinitesimal neighborhoods of successive
tangent vectors and their geometry together with the completions of a space in
an adic topology involving tangent vectors, in this paper we examine how these
categorical structures interact. Let us first recall in more detail how
tangent categories are incarnated.

On one hand, tangent categories are a categorical generalization of the
tangent bundle of a (smooth) manifold and provide a category-theoretic setting
in which to do differential geometry. Originally discovered by Rosický in [17]
and later rediscovered and refined by Cockett and Crutwell in [3], tangent
categories have given connections between differential geometry, higher
category theory, (de Rham) cohomology, linear logic, programming, machine
learning, and have become ubiquitous in both theoretical computer science and
pure mathematics. These have been studied in detail in many of these different
contexts, and in [7] we even see some interactions of enriched tangent
structures. Moreover, work of Crutwell and Lemay (cf. [5]) has recently shown
that the category $\operatorname{\mathbf{Sch}}_{/S}$ of schemes over a base
scheme $S$ admits a tangent structure as well.

On the other hand, ind-categories arise as the free filtered cocompletion of a
category $\operatorname{\mathscr{C}}$ and were first studied in detail by
Grothendieck in [1]. These categories are important in algebraic geoemtry, as
by [19] there is a close relationship with the categories
$\operatorname{\mathbf{FSch}}_{/S}$ and
$\operatorname{Ind}(\operatorname{\mathbf{Sch}}_{/S})$ where
$\operatorname{\mathbf{FSch}}_{/S}$ is the category of formal schemes over
$S$. This means in particular that ind-categories can be thought of as formal
completions of objects along some closed subobject. We then can think of each
object in $\operatorname{Ind}(\operatorname{\mathscr{C}})$, by way of formal
analogy, as an object which is (adically) completed along some closed
subobject and hence admits all formal infinitesimal neighborhoods along that
subobject. Our intuition as we proceed is to study these formal infinitesimal
neighborhoods in these objects when they _also_ have a notion of tangent in
the following sense: if we have infinitesimal neighborhoods and we have
tangent vectors, then we should expect to be able to complete the tangent
space to allow infinitesimal neighborhoods of the tangent vectors themselves.
Note that this sits in contrast to the fact that if we take the free
cocompletion of the category $\operatorname{\mathscr{C}}$ (as modeled, for
instance, by the presheaf topos
$[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]$),
we cannot expect
$[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]$
to be a tangent category even when $\operatorname{\mathscr{C}}$ is — cf. [3]
for details. The difference here is that our infinitesimal neighborhoods are
sufficiently well behaved (via their filtrations) whereas the free
cocompletion has colimits which are too wild to witness and interact well with
the categorical limits that describe tangent-theoretic information.

###  1.1. Results and Structure of the Paper

We begin this paper with a quick review of tangent categories, their
morphisms, and the category (and $2$-category) of tangent categories.
Afterwards we familiarize ourselves with the ind-construction and get to know
the $\operatorname{Ind}$ pseudofunctor
$\operatorname{Ind}:\operatorname{\mathfrak{Cat}}\to\operatorname{\mathfrak{Cat}}$.
We then study the interaction of the tangent structures as they pass through
the $\operatorname{Ind}$ pseudofunctor and lift to the ind-case. This
culminates in our first big result, which is Theorem 3.27.

######  Theorem 1.1 (cf. Theorem 3.27).

Let $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ be a tangent
category. Then the category
$(\operatorname{Ind}(\operatorname{\mathscr{C}}),\operatorname{Ind}(\operatorname{\mathbb{T}}))$
is a tangent category where $\operatorname{Ind}(\operatorname{\mathbb{T}})$ is
the tangent structure

$\operatorname{Ind}(\operatorname{\mathbb{T}}):=(\operatorname{Ind}(T),\hat{p},\operatorname{Ind}(0),\hat{+},\hat{\ell},\hat{c})$

where $\operatorname{Ind}(T)$ is the indicization of $T$ induced by
Proposition 3.7 and $\hat{p},\operatorname{Ind}(0),\hat{+},\hat{\ell},$ and
$\hat{c}$ are the natural transformations constructed in Lemmas 3.15, 3.19,
3.21, and Lemma 3.23, respectively.

After establishing this we show a functoriality result in Theorem 3.28 (which,
as a corollary, allows us to deduce that
$\operatorname{Ind}:\operatorname{\mathfrak{Cat}}\to\operatorname{\mathfrak{Cat}}$
restricts to a pseudofunctor
$\operatorname{Ind}:\operatorname{\mathfrak{Tan}}\to\operatorname{\mathfrak{Tan}}$).

######  Theorem 1.2 (cf. Theorem 3.28).

Let
$(F,\alpha):(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})\to(\operatorname{\mathscr{D}},\operatorname{\mathbb{S}})$
be a morphism of tangent categories. Then the induced map
$(\operatorname{Ind}(F),\hat{\alpha}):(\operatorname{Ind}(\operatorname{\mathscr{C}}),\operatorname{Ind}(\operatorname{\mathbb{T}}))\to(\operatorname{Ind}(\operatorname{\mathscr{D}}),\operatorname{Ind}(\operatorname{\mathbb{S}}))$
is a morphism of tangent categories where $\hat{\alpha}$ is the natural
transformation:

${\operatorname{Ind}(\operatorname{\mathscr{C}})}$${\operatorname{Ind}(\operatorname{\mathscr{D}})}$$\scriptstyle{\operatorname{Ind}(F)\circ\operatorname{Ind}(T)}$$\scriptstyle{\operatorname{Ind}(F\circ
T)}$$\scriptstyle{\operatorname{Ind}(S\circ
F)}$$\scriptstyle{\operatorname{Ind}(S)\circ\operatorname{Ind}(F)}$$\scriptstyle{\phi_{T,F}}$$\scriptstyle{\operatorname{Ind}(\alpha)}$$\scriptstyle{\phi_{F,S}^{-1}}$

Furthermore, $(\operatorname{Ind}(F),\hat{\alpha})$ is a strong tangent
morphism if and only if $(F,\alpha)$ is a strong tangent morphism.

We also show how the $\operatorname{Ind}$ construction interacts with the
forgetful functor
$\operatorname{Forget}:\operatorname{\mathfrak{Tan}}\to\operatorname{\mathfrak{Cat}}$
and the free tangent functor
$\operatorname{Free}:\operatorname{\mathfrak{Cat}}\to\operatorname{\mathfrak{Tan}}$.
More explicitly we show that the diagram

${\operatorname{\mathfrak{Tan}}}$${\operatorname{\mathfrak{Tan}}}$${\operatorname{\mathfrak{Cat}}}$${\operatorname{\mathfrak{Cat}}}$$\scriptstyle{\operatorname{Ind}}$$\scriptstyle{\operatorname{Forget}}$$\scriptstyle{\operatorname{Forget}}$$\scriptstyle{\operatorname{Ind}}$

commutes strictly but in the other case that there is a pseudonatural
transformation

${\operatorname{\mathfrak{Cat}}}$${\operatorname{\mathfrak{Cat}}}$${\operatorname{\mathfrak{Tan}}}$${\operatorname{\mathfrak{Tan}}}$$\scriptstyle{\operatorname{Free}}$$\scriptstyle{\operatorname{Ind}}$$\scriptstyle{\operatorname{Free}}$$\scriptstyle{\operatorname{Ind}}$$\scriptstyle{\alpha}$

which cannot be a pseudonatural equivalence. In particular, we show that
$\operatorname{Ind}$ does not commute with $\operatorname{Free}$, even up to
pseudonatural equivalence.

After presenting these theorems we show how the $\operatorname{Ind}$-tangent
structure interacts with various constructions which are important in the
tangent category literature. In particular, we show that the
$\operatorname{Ind}$-tangent category of a representable tangent category
remains representable and then determine when, for a Cartesian differential
category $\operatorname{\mathscr{C}}$, an object in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ is a differential bundle over
the terminal object $\top_{\operatorname{Ind}(\operatorname{\mathscr{C}})}$.
As a corollary we give a necessary and sufficient condition for recognizing
when, for a Cartesian differential category $\operatorname{\mathscr{C}}$,
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ is not a Cartesian
differential category.

######  Proposition 1.3 (cf. Proposition 4.4).

Let $\operatorname{\mathscr{C}}$ be a Cartesian closed tangent category. If
$\operatorname{\mathscr{C}}$ is representable then there is an object
$\underline{{D}}$ in $\operatorname{Ind}(\operatorname{\mathscr{C}})$ for
which $\operatorname{Ind}(T)\cong[\underline{{D}},-]$.

######  Proposition 1.4 (cf. Proposition 4.23).

Let $\operatorname{\mathscr{C}}$ be an $A$-linear Cartesian differential
category for a commutative rig $A$. Then
$\operatorname{\mathscr{C}}=\operatorname{\mathscr{C}}_{\operatorname{\mathnormal{D}\operatorname{lin}}}$
if and only if every object in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ is a differential bundle over
the terminal object $\top_{\operatorname{Ind}(\operatorname{\mathscr{C}})}$.

We close this paper with a review of the tangent structure on the categories
$\mathbf{CAlg}_{A}$ of commutative algebras over a commutative rig $A$;
$\operatorname{\mathbf{Sch}}_{/S}$ over a base scheme $S$; the $A$-linear
Cartesian differential category $\operatorname{\mathnormal{A}-\mathbf{Poly}}$
of polynomials over, again, a commutative rig $A$; and then the Cartesian
differential category
$\operatorname{\operatorname{\mathbb{R}}-\mathbf{Smooth}}$ of Euclidean spaces
and smooth maps between them. Of particular interest are a definition of a
formal tangent scheme (together with an explicit calculation) together with
the following characterizations of the differential objects in
$\operatorname{Ind}(\operatorname{\mathnormal{A}-\mathbf{Poly}})$ and
$\operatorname{Ind}(\operatorname{\operatorname{\mathbb{R}}-\mathbf{Smooth}})$.

######  Theorem 1.5 (cf. Theorem 5.10).

The category of differential objects in
$\operatorname{Ind}(\operatorname{\mathnormal{A}-\mathbf{Poly}})$ together
with linear bundle maps between them is equivalent to
$\operatorname{Ind}(\operatorname{\operatorname{\mathnormal{A}-\mathbf{Mod}}_{f.d.}})$.
In particular, there is an equivalence of categories of
$\mathbf{Diff}(\operatorname{Ind}(\operatorname{\mathnormal{A}-\mathbf{Poly}}))$
and the category $\operatorname{\mathnormal{A}-\mathbf{Mod}}$ of $A$-modules.

######  Theorem 1.6 (cf. Theorem 5.12).

The category of differential objects in
$\operatorname{Ind}(\operatorname{\operatorname{\mathbb{R}}-\mathbf{Smooth}})$
together with linear bundle maps between them is equivalent to
$\operatorname{Ind}(\operatorname{\operatorname{\mathbb{R}}-\mathbf{Vec}_{f.d.}})$.
In particular, there is an equivalence of categories between
$\mathbf{Diff}(\operatorname{Ind}(\operatorname{\operatorname{\mathbb{R}}-\mathbf{Smooth}}))$
and the category $\operatorname{\operatorname{\mathbb{R}}-\mathbf{Vec}}$ of
$\operatorname{\mathbb{R}}$ vector spaces.

### Acknowledgments

I would like to thank JS Lemay for many interesting conversations regarding
this material and suggesting many of the directions in which to examine the
$\operatorname{Ind}$-construction and the $\operatorname{Ind}$-tangent
structure111Much of the material in Sections 4.2, 4.3, 5.1, 5.3, and 5.4 was
examined and developed following our conversations.. I’d also like to thank
JS, Rick Blute, and Dorette Pronk for reading early drafts of this paper and
providing helpful suggestions. Additionally I’d like to thank Rory Lucyshyn-
Wright for interesting conversations regarding the
$\operatorname{Ind}$-pseudofunctor. Some of this work was presented at the
2023 Foundational Methods in Computer Science conference, and I’m grateful to
the organizers for the hospitality and intellectually stimulating environment
which allowed me to produce Sections 4 – 5 of this paper.

##  2\. A Review of Tangent Categories

Let us recall what it means for a category to be a tangent category following
R. Cockett and G. Crutwell in [3]. These categories are an abstraction of what
it means to have a category and tangent bundle functor, together with all the
relations and structure we require/expect of such a bundle object. In what
follows below, if $\alpha:FX\to X$ is a natural transformation of an
endofunctor $F:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{C}}$, we
write $F_{2}X$ for the pullback

$\textstyle{F_{2}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\textstyle{FX\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{X}}$$\textstyle{FX\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{X}}$$\textstyle{X}$

if it exists in $\operatorname{\mathscr{C}}$.

######  Definition 2.1 ([3, Definition 2.3]).

A category $\operatorname{\mathscr{C}}$ has a tangent structure
$\operatorname{\mathbb{T}}=(T,p,0,+,\ell,c)$ when the following six axioms
hold:

  1. (1)

$T:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{C}}$ is a functor
equipped with a natural transformation
$p:T\Rightarrow\operatorname{id}_{\operatorname{\mathscr{C}}}$ such that for
every object $X$ in $\operatorname{\mathscr{C}}$ all pullback powers of
$p_{X}:TX\to X$ exist and for all $n\in\operatorname{\mathbb{N}}$ the functors
$T^{n}$ preserve these pullback powers.

  2. (2)

There are natural transformations $+:T_{2}\Rightarrow T$ and
$0:\operatorname{id}_{\operatorname{\mathscr{C}}}\to T$ for which each map
$p_{X}:TX\to X$ describes an additive bundle in $\operatorname{\mathscr{C}}$,
i.e., $p:TX\to X$ is an internal commutative monoid in
$\operatorname{\mathscr{C}}_{/X}$ with addition and unit given by $+$ and $0$,
respectively.

  3. (3)

There is a natural transformation $\ell:T\Rightarrow T^{2}$ such that for any
$X\in\operatorname{\mathscr{C}}_{0}$, the squares

${TX}$${T^{2}X}$${TX\times_{X}TX}$${T^{2}X\times_{TX}T^{2}X}$${X}$${TX}$${X}$${TX}$${TX}$${T^{2}X}$${TX}$${T^{2}X}$$\scriptstyle{\ell_{X}}$$\scriptstyle{p_{X}}$$\scriptstyle{Tp_{X}}$$\scriptstyle{+_{X}}$$\scriptstyle{\langle\ell_{X}\circ\pi_{1},\ell_{X}\circ\pi_{2}\rangle}$$\scriptstyle{+_{TX}}$$\scriptstyle{0_{X}}$$\scriptstyle{0_{X}}$$\scriptstyle{0_{TX}}$$\scriptstyle{0_{X}}$$\scriptstyle{\ell_{X}}$$\scriptstyle{\ell_{X}}$

all commute, i.e., $(\ell_{X},0_{X})$ is a morphism of bundles in
$\operatorname{\mathscr{C}}$ (cf. [3, Definition 2.2]).

  4. (4)

There is a natural transformation $c:T^{2}\Rightarrow T^{2}$ such that for all
$X\in\operatorname{\mathscr{C}}_{0}$ the squares

${T^{2}X}$${T^{2}X}$${T^{2}X\times_{TX}T^{2}X}$${T^{2}X\times_{TX}T^{2}X}$${TX}$${TX}$${TX}$${TX}$${T^{2}X}$${T^{2}X}$${T^{2}X}$${T^{2}X}$$\scriptstyle{c_{X}}$$\scriptstyle{Tp_{X}}$$\scriptstyle{p_{TX}}$$\scriptstyle{(T\ast+)_{X}}$$\scriptstyle{\langle
c_{X}\circ\pi_{1},c_{X}\circ\pi_{2}\rangle}$$\scriptstyle{(+\ast
T)_{X}}$$\scriptstyle{(T\ast
0)_{X}}$$\scriptstyle{\operatorname{id}_{TX}}$$\scriptstyle{(0\ast
T)_{X}}$$\scriptstyle{\operatorname{id}_{TX}}$$\scriptstyle{c_{X}}$$\scriptstyle{c_{X}}$

commute, i.e., $(c_{X},\operatorname{id}_{TX})$ describes a bundle morphism in
$\operatorname{\mathscr{C}}$

  5. (5)

We have the equations
$c^{2}=\operatorname{id}_{\operatorname{\mathscr{C}}},c\circ\ell=\ell$, and
the diagrams

${T}$${T^{2}}$${T^{3}}$${T^{3}}$${T^{3}}$${T^{2}}$${T^{3}}$${T^{3}}$${T^{3}}$${T^{3}}$$\scriptstyle{\ell}$$\scriptstyle{\ell}$$\scriptstyle{T\ast\ell}$$\scriptstyle{c\ast
T}$$\scriptstyle{T\ast c}$$\scriptstyle{c\ast T}$$\scriptstyle{c\ast
T}$$\scriptstyle{\ell\ast T}$$\scriptstyle{T\ast c}$$\scriptstyle{c\ast
T}$${T^{2}}$${T^{3}}$${T^{3}}$${T^{2}}$${T^{3}}$$\scriptstyle{c}$$\scriptstyle{\ell\ast
T}$$\scriptstyle{T\ast c}$$\scriptstyle{c\ast T}$$\scriptstyle{T\ast\ell}$

commute in the functor category
$[\operatorname{\mathscr{C}},\operatorname{\mathscr{C}}]$ where $F\ast\alpha$
and $\alpha\ast F$ denote the left and right whiskering of a natural
transformation $\alpha$ by a functor $F$, respectively.

  6. (6)

The diagram

${T_{2}X}$${T^{2}X}$${TX}$$\scriptstyle{(T\ast+)_{X}\circ\langle\ell\circ\pi_{1},0_{TX}\circ\pi_{2}\rangle}$$\scriptstyle{T(p_{X})}$$\scriptstyle{0_{X}\circ
p_{X}\circ p_{TX}}$

is an equalizer in $\operatorname{\mathscr{C}}$.

Finally, a category $\operatorname{\mathscr{C}}$ is said to be a tangent
category if it admits a tangent structure $\operatorname{\mathbb{T}}$.

######  Remark 2.2.

In [3], the axioms above are named and are given intuitive explanations. While
we will not recall this here in complete detail, we will at least recall the
stated content of each axiom:

  * •

Axiom 1 names $T$ the tangent functor of the tangent structure
$\operatorname{\mathbb{T}}$ on $\operatorname{\mathscr{C}}$ and $p$ the bundle
map.

  * •

Axiom 2 describes the object $TX\to X$ as an additive tangent bundle in
$\operatorname{\mathscr{C}}$.

  * •

Axiom 3 names $\ell$ as the vertical lift which lifts a tangent to a tangent
of a tangent.

  * •

Axiom 4 names $c$ as the canonical flip which looks like the interchange of
mixed partial derivatives.

  * •

Axiom 5 gives the coherence relations $\ell$ and $c$ must satisfy.

  * •

Axiom 6 describes the universality of the vertical lift.

Note also that when we say that $\operatorname{\mathscr{C}}$ is a tangent
category, we really mean that
$(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ is a category
equipped with a specific tangent structure $\operatorname{\mathbb{T}}$ and
have simply left the explicit mention of the tangent structure
$\operatorname{\mathbb{T}}$ out. In fact, this is an abuse of notation; it is
possible for a category to have multiple distinct tangent structures so we
will only say $\operatorname{\mathscr{C}}$ is a tangent category to mean that
$(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ is a category
equipped with a specific (potentially unspecified) tangent structure.

Equally as important to tangent categories are the morphisms of tangent
categories. These come in two flavours: one, which is lax, and another which
is strong. While we will usually work with strong tangent morphisms, it is
important for Theorem 3.28 to have the full definition.

######  Definition 2.3 ([3, Definition 2.7]).

Let
$(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})=(\operatorname{\mathscr{C}},T,p,0,+,\ell,c)$
and
$(\operatorname{\mathscr{D}},\operatorname{\mathbb{S}})=(\operatorname{\mathscr{D}},S,q,0^{\prime},\oplus,\ell^{\prime},c^{\prime})$
be tangent categories. A morphism of tangent categories is a pair
$(F,\alpha):(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})\to(\operatorname{\mathscr{D}},\operatorname{\mathbb{S}})$
where $F:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{D}}$ is a functor
and $\alpha$ is a natural transformation

${\operatorname{\mathscr{C}}}$${\operatorname{\mathscr{D}}}$$\scriptstyle{F\circ
T}$$\scriptstyle{S\circ F}$$\scriptstyle{\alpha}$

for which the diagrams of functors and natural transformations

${F\circ T}$${S\circ F}$${F}$$\scriptstyle{\alpha}$$\scriptstyle{F\ast
p}$$\scriptstyle{q\ast F}$${F}$${F\circ T}$${S\circ F}$$\scriptstyle{F\ast
0}$$\scriptstyle{0^{\prime}\ast F}$$\scriptstyle{\alpha}$${F\circ
T_{2}}$${S_{2}\circ F}$${F\circ T}$${S\circ
F}$$\scriptstyle{F\ast\alpha_{2}}$$\scriptstyle{F\ast+}$$\scriptstyle{\oplus\ast
F}$$\scriptstyle{\alpha}$ ${F\circ T}$${S\circ F}$${F\circ T^{2}}$${S^{2}\circ
F}$$\scriptstyle{\alpha}$$\scriptstyle{F\ast\ell}$$\scriptstyle{\ell^{\prime}\ast
F}$$\scriptstyle{(S\ast\alpha)\circ(\alpha\ast T)}$${F\circ
T^{2}}$${S^{2}\circ F}$${F\circ T^{2}}$${S^{2}\circ
F}$$\scriptstyle{(S\ast\alpha)\circ(\alpha\ast T)}$$\scriptstyle{F\ast
c}$$\scriptstyle{c^{\prime}\ast F}$$\scriptstyle{(S\ast\alpha)\circ(\alpha\ast
T)}$

commute. Additionally, we say that the morphism $(F,\alpha)$ is strong if
$\alpha$ is a natural isomorphism and if $F$ preserves the equalizers and
pullbacks of the tangent structure
$(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$.

To finish up this section we give a short description of the $1$ and
$2$-categories of tangent categories, tangent morphisms, and their natural
transformations.

######  Definition 2.4.

The category $\operatorname{\mathbf{Tan}}$ is defined as follows:

  * •

Objects: Tangent categories
$(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$.

  * •

Morphisms: Tangent morphisms
$(F,\alpha):(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})\to(\operatorname{\mathscr{D}},\operatorname{\mathbb{S}})$.

  * •

Composition: The composition of two tangent morphisms
$(F,\alpha):(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})\to(\operatorname{\mathscr{D}},\operatorname{\mathbb{S}})$
and
$(G,\beta):(\operatorname{\mathscr{D}},\operatorname{\mathbb{S}})\to(\operatorname{\mathscr{E}},\operatorname{\mathbb{R}})$
is defined to be the pair

$\big{(}G\circ F,(\beta\ast F)\circ(G\ast\alpha)\big{)}.$

  * •

Identities: The identity on a tangent category is
$(\operatorname{id}_{\operatorname{\mathscr{C}}},e):(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})\to(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$
where $e$ is the natural transformation witnessing the equality
$\operatorname{id}_{\operatorname{\mathscr{C}}}\circ
T=T\circ\operatorname{id}_{\operatorname{\mathscr{C}}}$.

######  Definition 2.5.

The $2$-category $\operatorname{\mathfrak{Tan}}$ is defined by:

  * •

Zero-and-one-morphisms: As in $\operatorname{\mathbf{Tan}}$.

  * •

$2$-morphisms: A $2$-morphism as in the displayed diagram

${(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})}$${(\operatorname{\mathscr{D}},\operatorname{\mathbb{S}})}$$\scriptstyle{(F,\alpha)}$$\scriptstyle{(G,\beta)}$$\scriptstyle{\rho}$

is a natural transformation $\rho:F\Rightarrow
G:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{D}}$ for which the
diagram

${F\circ T}$${S\circ F}$${G\circ T}$${S\circ
G}$$\scriptstyle{\alpha}$$\scriptstyle{\rho\ast
T}$$\scriptstyle{S\ast\rho}$$\scriptstyle{\beta}$

of functors and natural transformations above commutes. Said natural
transformations are called tangent transformations

  * •

Vertical and horizontal composition: As in $\operatorname{\mathfrak{Cat}}$.

  * •

Vertical composition identities: As in $\operatorname{\mathfrak{Cat}}$.

There are two canonical forgetful functors,
$\operatorname{\mathbf{Tan}}\to\operatorname{\mathbf{Cat}}$ and
$\operatorname{\mathfrak{Tan}}\to\operatorname{\mathfrak{Cat}}$, which are
defined by
$(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})\mapsto\operatorname{\mathscr{C}}$
on $0$-morphisms (objects), $(F,\alpha)\mapsto F$ on $1$-morphisms, and by
$\rho\mapsto\rho$ on $2$-morphisms. Ultimately our main two theorems in this
paper (cf. Theorems 3.27 and 3.28) can be summaried as showing that the
$\operatorname{Ind}$-pseudofunctor defined on $\operatorname{\mathfrak{Cat}}$
can be restricted/extended to a pseudofunctor
$\operatorname{Ind}:\operatorname{\mathfrak{Tan}}\to\operatorname{\mathfrak{Tan}}$
and that the diagram

${\operatorname{\mathfrak{Tan}}}$${\operatorname{\mathfrak{Tan}}}$${\operatorname{\mathfrak{Cat}}}$${\operatorname{\mathfrak{Cat}}}$$\scriptstyle{\operatorname{Forget}}$$\scriptstyle{\operatorname{Ind}}$$\scriptstyle{\operatorname{Forget}}$$\scriptstyle{\operatorname{Ind}}$

commutes strictly; cf. Corollary 3.29 below.

##  3\. Tangent Ind-Categories

In this section we examine tangent structures on ind-completions of tangent
categories. The general intuition here is that if we have a category of
geometric objects in which we can form tangent bundles, when we take formal
completions of these objects along closed subobjects we can make a formal
tangent object along these completions. As an application, we will show and
discuss the Zariski tangent structure on the category of formal schemes. We
begin this journey with a recollection of basic properties of the ind-
category.

######  Remark 3.1.

We have chosen to follow the more explicit and hands-on definition of tangent
categories for the majority this section (as opposed to the Weil algebra
approach) for use later in describing equivariant tangent ind-categories and
introducing the tangent structure on equivariant formal schemes. While this
does mean there is more explicit combinatorial work to be done, it has the
benefit of making the theory more explicit and amenable to calculations in
practice. That being said, we will review Weil algebras briefly before
discussing the free tangent functor.

If $\operatorname{\mathscr{C}}$ is a category, the ind-category
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ is the free filtered formal
cocompletion of $\operatorname{\mathscr{C}}$. The construction and
presentation of $\operatorname{Ind}(\operatorname{\mathscr{C}})$ we use is
essentially the one in [1, Section I.8.2]; an alternative but equivalent
presentation is in [12, Chapter 6]. Where necessary in what follows, we write
$\operatorname{\mathbf{y}}:\operatorname{\mathscr{C}}\to[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]$
for the Yoneda embedding of $\operatorname{\mathscr{C}}$. We also will
generally omit size-related issues in our discussion, but warn here that we
should be careful to work with categories which are
$\operatorname{\mathscr{U}}$-small for some fixed Grothendieck universe
$\operatorname{\mathscr{U}}$ in order to avoid issues such as having large
hom-sets. That being said, we will not generally worry about size-related
issues in this paper.

######  Definition 3.2.

If $\operatorname{\mathscr{C}}$ is a locally small category, its presheaf-
valued ind-category
$\operatorname{Ind}_{\mathbf{PSh}}(\operatorname{\mathscr{C}})$ is defined as
follows:

  * •

Objects: Presheaves
$P\in[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]$
for which there is an isomorphism of functors

$P\cong\lim_{\begin{subarray}{c}\longrightarrow\\\ i\in
I\end{subarray}}\operatorname{\mathscr{C}}(-,X_{i})=\lim_{\begin{subarray}{c}\longrightarrow\\\
i\in I\end{subarray}}\operatorname{\mathbf{y}}X_{i}$

where $I$ is a filtered category.

  * •

Morphisms: For any objects
$P,Q\in\operatorname{Ind}(\operatorname{\mathscr{C}})_{0}$, we define

$\operatorname{Ind}(\operatorname{\mathscr{C}})(P,Q):=[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}](P,Q).$

  * •

Composition and identities: As in
$[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]$.

We will call the objects of
$\operatorname{Ind}_{\mathbf{PSh}}(\operatorname{\mathscr{C}})$ ind-presheaves
of $\operatorname{\mathscr{C}}$.

In [12] the category
$\operatorname{Ind}_{\mathbf{PSh}}(\operatorname{\mathscr{C}})$ is taken as
the definition of the ind-category of $\operatorname{\mathscr{C}}$. We will
take an approach more like the one used in [1] where we think of
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ as the category of filtered
colimits by keeping track of the filtered diagram in
$\operatorname{\mathscr{C}}$ which defines the colimit. The only difficult
part of this more “representation agnostic” view is that it is more difficult
to define hom-sets. However, from the presheaf description we have above and
the co(Yoneda Lemma) we get natural isomorphisms which tell us how to define
the $\operatorname{Ind}$-hom sets. Fix two $\operatorname{Ind}$-presheaves $P$
and $Q$ over $\operatorname{\mathscr{C}}$ and write each as a filtered colimit

$P\cong\lim_{\begin{subarray}{c}\longrightarrow\\\ i\in
I\end{subarray}}\operatorname{\mathbf{y}}(X_{i})$

and

$Q\cong\lim_{\begin{subarray}{c}\longrightarrow\\\ j\in
J\end{subarray}}\operatorname{\mathbf{y}}(X_{j}).$

We then have by definition, the co(Yoneda Lemma), the Yoneda Lemma, and the
fact that (co)limits in
$[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]$
are representable (cf. [1, Equation I.8.2.5.1])

$\displaystyle\operatorname{Ind}_{\mathbf{PSh}}(\operatorname{\mathscr{C}})(P,Q)$
$\displaystyle\cong[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]\left(\lim_{\begin{subarray}{c}\longrightarrow\\\
i\in
I\end{subarray}}\operatorname{\mathbf{y}}X_{i},\lim_{\begin{subarray}{c}\longrightarrow\\\
j\in
J\end{subarray}}\operatorname{\mathbf{y}}Y_{j}\right)\cong\lim_{\begin{subarray}{c}\longleftarrow\\\
i\in
I\end{subarray}}[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]\left(\operatorname{\mathbf{y}}X_{i},\lim_{\begin{subarray}{c}\longrightarrow\\\
j\in J\end{subarray}}\operatorname{\mathbf{y}}Y_{j}\right)$
$\displaystyle\cong\lim_{\begin{subarray}{c}\longleftarrow\\\ i\in
I\end{subarray}}\left(\lim_{\begin{subarray}{c}\longrightarrow\\\ j\in
J\end{subarray}}[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}](\operatorname{\mathbf{y}}X_{i},\operatorname{\mathbf{y}}Y_{j})\right)\cong\lim_{\begin{subarray}{c}\longleftarrow\\\
i\in I\end{subarray}}\left(\lim_{\begin{subarray}{c}\longrightarrow\\\ j\in
J\end{subarray}}\operatorname{\mathscr{C}}(X_{i},Y_{j})\right).$

In particular, this leads us to a definition of hom-sets in a “representation
agnostic” formulation of $\operatorname{Ind}(\operatorname{\mathscr{C}})$.

######  Definition 3.3 ([1, Section I.8.2]).

If $\operatorname{\mathscr{C}}$ is a category then we define an ind-object of
$\operatorname{\mathscr{C}}$ to be a functor
$F:I\to\operatorname{\mathscr{C}}$ where $I$ is a filtered category. If no
confusion is likely to arise, we will write ind-objects as systems
$\underline{{X}}=(X_{i})_{i\in I}=(X_{i})$ where $X_{i}:=F(i)$ for all $i\in
I_{0}$ and leave the transition morphisms $F(f),f\in I_{1}$ and even the
functor $F$ implicit.

######  Definition 3.4 ([1, Section I.8.2]).

The $\operatorname{Ind}$-category
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ of a category
$\operatorname{\mathscr{C}}$ is defined as follows:

  * •

Objects: Ind-objects $\underline{{X}}$ of $\operatorname{\mathscr{C}}$.

  * •

Morphisms: We define the hom-sets
$\operatorname{Ind}(\operatorname{\mathscr{C}})(\underline{{X}},\underline{{Y}})$
for $\underline{{X}}=(X_{i})_{i\in I}$ and $\underline{{Y}}=(Y_{j})_{j\in J}$
via

$\operatorname{Ind}(\operatorname{\mathscr{C}})(\underline{{X}},\underline{{Y}}):=\lim_{\begin{subarray}{c}\longleftarrow\\\
i\in I\end{subarray}}\left(\lim_{\begin{subarray}{c}\longrightarrow\\\ j\in
J\end{subarray}}\operatorname{\mathscr{C}}(X_{i},Y_{j})\right).$

  * •

Composition: Induced by degree-wise composition after taking the filtered
limit of the filtered colimit of maps $X_{i}\to Y_{j}$.

  * •

Identities: $\operatorname{id}_{(X_{i})}$ is the identity transformation for
the functor $\underline{{X}}:I\to\operatorname{\mathscr{C}}$.

######  Remark 3.5.

While the definition of hom-sets above looks complicated in general, defining
morphisms between ind-objects that have the same indexing category is
straightforward. In the case $\underline{{X}}=(X_{i})_{i\in I}$ and
$\underline{{Y}}=(Y_{i})_{i\in I}$ then a morphism
$\rho:\underline{{X}}\to\underline{{Y}}$ is given by a collection of maps
$\rho_{i}:X_{i}\to Y_{i}$ for all $i\in I$ which are compatible with the
transition morphisms of each functor. In particular, $\rho$ is a natural
transformation

${I}$${\operatorname{\mathscr{C}}}$$\scriptstyle{F}$$\scriptstyle{G}$$\scriptstyle{\rho}$

when $F$ and $G$ are the functors representing $\underline{{X}}$ and
$\underline{{Y}}$, respectively.

When working with $\operatorname{Ind}(\operatorname{\mathscr{C}})$ it is
frequently helpful to represent each object as a presheaf (and in fact we will
use this technique below); we will thus follow [1] and define the functor
$L:\operatorname{Ind}(\operatorname{\mathscr{C}})\to[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]$
as follows:

  * •

For any ind-object $\underline{{X}}=(X_{i})_{i\in I}$ define
$L(\underline{{X}})$ to be the presheaf

$L(\underline{{X}})=\lim_{\begin{subarray}{c}\longrightarrow\\\ i\in
I\end{subarray}}\operatorname{\mathbf{y}}X_{i}.$

  * •

For any morphism $\rho:\underline{{X}}\to\underline{{Y}}$ in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ we define $L(\rho)$ to be the
image of $\rho$ under the natural isomorphism

$\operatorname{Ind}(\operatorname{\mathscr{C}})(\underline{{X}},\underline{{Y}})=\lim_{\begin{subarray}{c}\longleftarrow\\\
i\in I\end{subarray}}\left(\lim_{\begin{subarray}{c}\longrightarrow\\\ j\in
J\end{subarray}}\operatorname{\mathscr{C}}(X_{i},Y_{j})\right)\cong[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]\left(\lim_{\begin{subarray}{c}\longrightarrow\\\
i\in
I\end{subarray}}\operatorname{\mathbf{y}}X_{i},\lim_{\begin{subarray}{c}\longrightarrow\\\
j\in
J\end{subarray}}\operatorname{\mathbf{y}}Y_{j}\right)=[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}](L\underline{{X}},L\underline{{Y}}).$

######  Proposition 3.6 ([1, Section I.8.2.4; cf. Line I.8.2.4.8]).

The functor
$L:\operatorname{Ind}(\operatorname{\mathscr{C}})\to[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]$
is fully faithful, exact, and has essential image equal to the category
$\operatorname{Ind}_{\mathbf{PSh}}(\operatorname{\mathscr{C}})$. In
particular,
$\operatorname{Ind}(\operatorname{\mathscr{C}})\simeq\operatorname{Ind}_{\mathbf{PSh}}(\operatorname{\mathscr{C}})$.

This means that when checking to see if certain objects are isomorphic in the
ind-category, where constructions are often more straightforward to perform
(there is less worry about making sure you haven’t done something weird and
misused the Yoneda Lemma), and then check for isomorphisms through the functor
$L$.222Because $L$ is fully faithful, it is in particular conservative so
isomorphisms $L\underline{{X}}\cong L\underline{{Y}}$ come uniquely from
isomorphisms $\underline{{X}}\cong\underline{{Y}}$. We now recall and discuss
the construction of ind-functors so that we can then build ind-natural
transformations and introduce the ind-pseudofunctor.

######  Proposition 3.7 ([1, Section I.8.6.1]).

If $F:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{D}}$ is a functor
then there is a functor
$\operatorname{Ind}(F):\operatorname{Ind}(\operatorname{\mathscr{C}})\to\operatorname{Ind}(\operatorname{\mathscr{D}})$
for which given a composable pair of functors
$\operatorname{\mathscr{C}}\xrightarrow{F}\operatorname{\mathscr{D}}\xrightarrow{G}\operatorname{\mathscr{E}}$
there is a natural isomorphism

$\operatorname{Ind}(G\circ
F)\cong\operatorname{Ind}(G)\circ\operatorname{Ind}(F).$

######  Remark 3.8.

The construction of the functor
$\operatorname{Ind}(F):\operatorname{Ind}(\operatorname{\mathscr{C}})\to\operatorname{Ind}(\operatorname{\mathscr{D}})$
is important for our applications, so we will describe it here in the two main
ways of working with the category
$\operatorname{Ind}(\operatorname{\mathscr{C}})$: the reduced
$\underline{{X}}=(X_{i})$ language and the more formal language
$\underline{{X}}:I\to\operatorname{\mathscr{C}}$. In the first case, the
functor $\operatorname{Ind}(F)(\underline{{X}})$ acts by

$\operatorname{Ind}(F)(\underline{{X}}):=(FX_{i}).$

In the second case we find that

$\operatorname{Ind}(F)(\underline{{X}})=F\circ\underline{{X}}:I\to\operatorname{\mathscr{D}}.$

The assignment on morphisms is more complicated to describe. If we have a
morphism

$\rho\in\operatorname{Ind}(\operatorname{\mathscr{C}})((X_{i})_{i\in
I},(Y_{j})_{j\in J})$

then $\operatorname{Ind}(F)(\rho)$ is induced by the assignment

$\lim_{I}\left(\operatorname*{colim}_{J}\rho_{ij}\right)\mapsto\lim_{I}\left(\operatorname*{colim}_{J}F(\rho_{ij})\right)$

However, there is a clean description in case
$\underline{{X}},\underline{{Y}}:I\to\operatorname{\mathscr{C}}$ are functors
defined on the same indexing category. In this case a morphism
$\rho:\underline{{X}}\to\underline{{Y}}$ is exactly a natural transformation

${I}$${\operatorname{\mathscr{C}}}$$\scriptstyle{\underline{{X}}}$$\scriptstyle{\underline{{Y}}}$$\scriptstyle{\rho}$

and so we define $\operatorname{Ind}(F)$ algebraically by

$\operatorname{Ind}(F)(\rho_{i})=(F\rho_{i})$

and, in the categorical perspective,

$\operatorname{Ind}(F)(\rho)=F\ast\rho.$

######  Corollary 3.9.

For any category $\operatorname{\mathscr{C}}$ there is an equality
$\operatorname{Ind}(\operatorname{id}_{\operatorname{\mathscr{C}}})=\operatorname{id}_{\operatorname{Ind}(\operatorname{\mathscr{C}})}$.

###### Proof.

The verification that
$\operatorname{id}_{\operatorname{Ind}(\operatorname{\mathscr{C}})}(\underline{{X}})=\underline{{X}}=\operatorname{Ind}(\operatorname{id}_{\operatorname{\mathscr{C}}})(\underline{{X}})$
for any object $\underline{{X}}$ is trivial and omitted. For the case of
morphisms note that if
$\lim(\operatorname*{colim}\rho_{ij}):(X_{i})\to(Y_{j})$ is any morphism then
$\operatorname{Ind}(\operatorname{id}_{\operatorname{\mathscr{C}}})$ acts on
$\lim(\operatorname*{colim}\rho_{ij})$ via

$\lim(\operatorname*{colim}\rho_{ij})\mapsto\lim\bigg{(}\operatorname*{colim}\big{(}\operatorname{id}_{\operatorname{\mathscr{C}}}(\rho_{ij})\big{)}\bigg{)}=\lim(\operatorname*{colim}(\rho_{ij})),$

which is exactly the identity assignment. ∎

######  Remark 3.10.

As we proceed in this paper, for any composable pair of functors
$\operatorname{\mathscr{C}}\xrightarrow{F}\operatorname{\mathscr{D}}\xrightarrow{G}\operatorname{\mathscr{E}}$
we write $\phi_{F,G}$ for the compositor natural isomorphism:

${\operatorname{Ind}(\operatorname{\mathscr{C}})}$${\operatorname{Ind}(\operatorname{\mathscr{E}})}$$\scriptstyle{\operatorname{Ind}(G)\circ\operatorname{Ind}(F)}$$\scriptstyle{\operatorname{Ind}(G\circ
F)}$$\scriptstyle{\phi_{F,G}}$$\scriptstyle{\cong}$

While in general we do only have a natural isomorphism
$\operatorname{Ind}(G\circ
F)\cong\operatorname{Ind}(G)\circ\operatorname{Ind}(F)$, the lemma we prove
below (for use later when we prove that $\hat{\ell}$ and $\hat{c}$ are
components of bundle maps) shows that these compositor natural isomorphisms
are actually relatively well-behaved in the sense that the action of
$\operatorname{Ind}(G)\circ\operatorname{Ind}(F)$ and
$\operatorname{Ind}(G\circ F)$ on certain classes of morphisms are the same.

######  Lemma 3.11.

Let $F:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{D}}$ and
$G:\operatorname{\mathscr{D}}\to\operatorname{\mathscr{E}}$ be functors. If
$\underline{{X}},\underline{{Y}}:I\to\operatorname{\mathscr{C}}$ are two
objects in $\operatorname{Ind}(\operatorname{\mathscr{C}})$ with
${\rho}=(\rho_{i})_{i\in I}:\underline{{X}}\to\underline{{Y}}$ an ind-
morphism, then

$\big{(}\operatorname{Ind}(G)\circ\operatorname{Ind}(F)\big{)}(\underline{{\rho}})=\operatorname{Ind}(G\circ
F)(\underline{{\rho}}).$

###### Proof.

Because both $\underline{{X}}$ and $\underline{{Y}}$ have the same indexing
category, it follows from Remark 3.8

$\operatorname{Ind}(G\circ F)(\underline{{\rho}})=(G\circ F)\ast\rho$

Similarly, we find that

$\operatorname{Ind}(F)({\rho})=F\ast\rho=\iota_{F}\ast\rho$

where $\iota_{F}$ is the identity natural transformation on $F$. Then

$\operatorname{Ind}(G)\big{(}F\rho_{i}\big{)}_{i\in
I}=G\ast(\iota_{F}\ast\rho)=\iota_{G}\ast(\iota_{F}\ast\rho)=(\iota_{G}\ast\iota_{F})\ast\rho=(\iota_{G\circ
F})\ast\rho.$

Thus we conclude that

$\operatorname{Ind}(G\circ F)(\rho_{i})=(G\circ F)\ast\rho=\iota_{G\circ
F}\ast\rho=\iota_{G}\ast(\iota_{F}\ast\rho)=\operatorname{Ind}(G)\big{(}\operatorname{Ind}(F)(\rho)\big{)},$

as was desired. ∎

######  Proposition 3.12.

Let $\operatorname{\mathscr{C}},\operatorname{\mathscr{D}}$ be categories with
functors $F,G:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{D}}$ and let
$\alpha:F\Rightarrow G$ be a natural transformation. Then there is a natural
transformation

$\operatorname{Ind}(\alpha):\operatorname{Ind}(F)\Rightarrow\operatorname{Ind}(G).$

###### Proof.

Fix
$\underline{{X}}=(X_{i})\in\operatorname{Ind}(\operatorname{\mathscr{C}})_{0}$.
We define
$\operatorname{Ind}(\alpha)_{\underline{{X}}}:\operatorname{Ind}(F)(X_{i})\to\operatorname{Ind}(G)(X_{i})$
by setting (cf. Remark 3.5)

$\operatorname{Ind}(\alpha)_{\underline{{X}}}:=(\alpha_{X_{i}})_{i\in I}.$

To check that this is a natural transformation fix a map $\rho:(X_{i})_{i\in
I}\to(Y_{j})_{j\in J}$ so that the sequences $(\rho_{ij})$ are a family of
morphisms compatible with the transition morphisms of $\underline{{X}}$ and
$\underline{{Y}}$. In particular, for any $f:i\to i^{\prime}$ in $I$ and
$g:j\to j^{\prime}$ in $J$ the square

$\textstyle{X_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{ij}}$$\scriptstyle{f_{\ast}}$$\textstyle{Y_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{\ast}}$$\textstyle{X_{i^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{i^{\prime}j^{\prime}}}$$\textstyle{Y_{j^{\prime}}}$

commutes. To prove naturality, it suffices to show that for any any $f:i\to
i^{\prime}$ in $I$ and for any $g:j\to j^{\prime}$ in $J$, the cube

${FX_{i}}$${GX_{i}}$${FY_{j}}$${GY_{j}}$${FX_{i^{\prime}}}$${GX_{i^{\prime}}}$${FY_{j^{\prime}}}$${GY_{j^{\prime}}}$$\scriptstyle{F\rho_{ij}}$$\scriptstyle{\alpha_{X_{i}}}$$\scriptstyle{F(f_{\ast})}$$\scriptstyle{G(f_{\ast})}$$\scriptstyle{G\rho_{ij}}$$\scriptstyle{G(g_{\ast})}$$\scriptstyle{F(\rho_{i^{\prime}j^{\prime}})}$$\scriptstyle{\alpha_{X_{i^{\prime}}}}$$\scriptstyle{G\rho_{i^{\prime}j^{\prime}}}$$\scriptstyle{\alpha_{Y_{j^{\prime}}}}$$\scriptstyle{\alpha_{Y_{j}}}$$\scriptstyle{F(g_{\ast})}$

commutes; note that we are writing $f_{\ast}$ for the image of $f:i\to
i^{\prime}$ as a transition morphism of $\underline{{X}}$ and similarly for
$g_{\ast}$ as a transtion morphism of $\underline{{Y}}$. However, note that
the squares

${FX_{i}}$${GX_{i}}$${FX_{i^{\prime}}}$${GX_{i^{\prime}}}$$\scriptstyle{\alpha_{X_{i}}}$$\scriptstyle{F(f_{\ast})}$$\scriptstyle{G(f_{\ast})}$$\scriptstyle{\alpha_{X_{i^{\prime}}}}$
${FY_{j}}$${GY_{j}}$${FY_{j^{\prime}}}$${GY_{j^{\prime}}}$$\scriptstyle{\alpha_{Y_{j}}}$$\scriptstyle{F(g_{\ast})}$$\scriptstyle{G(g_{\ast})}$$\scriptstyle{\alpha_{Y_{j}}}$

commute by the naturality of $\alpha$. Similarly, the squares

${FX_{i}}$${FY_{j}}$${GX_{i}}$${GY_{j}}$$\scriptstyle{F\rho_{ij}}$$\scriptstyle{\alpha_{X_{i}}}$$\scriptstyle{\alpha_{Y_{j}}}$$\scriptstyle{G\rho_{ij}}$
${FX_{i^{\prime}}}$${FY_{j^{\prime}}}$${GX_{i^{\prime}}}$${GY_{j^{\prime}}}$$\scriptstyle{F(\rho_{i^{\prime}j^{\prime}})}$$\scriptstyle{\alpha_{X_{i^{\prime}}}}$$\scriptstyle{\alpha_{Y_{j}^{\prime}}}$$\scriptstyle{G(\rho_{i^{\prime}j^{\prime}})}$

commute by assumption of the $\rho_{ij}$ being compatible with the transition
morphisms and the functoriality of $F$ and $G$. Thus the cube commutes, which
shows that $\operatorname{Ind}(\alpha)$ is indeed natural after taking the
filtered limit of the filtered colimit of maps. ∎

######  Corollary 3.13.

Given a two-cell of categories

${\operatorname{\mathscr{C}}}$${\operatorname{\mathscr{D}}}$$\scriptstyle{F}$$\scriptstyle{G}$$\scriptstyle{H}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$

we have

$\operatorname{Ind}(\beta\circ\alpha)=\operatorname{Ind}(\beta)\circ\operatorname{Ind}(\alpha).$

###### Proof.

Let $\underline{{X}}=(X_{i})_{i\in I}$ be an object of
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{0}$. We then calculate from
Proposition 3.12 that

$\operatorname{Ind}(\beta\circ\alpha)_{\underline{{X}}}=\left((\beta\circ\alpha)_{X_{i}}\right)_{i\in
I}=(\beta_{X_{i}}\circ\alpha_{X_{i}})_{i\in I}=(\beta_{X_{i}})_{i\in
I}\circ(\alpha_{X_{i}})_{i\in
I}=\operatorname{Ind}(\beta)_{\underline{{X}}}\circ\operatorname{Ind}(\alpha)_{\underline{{X}}}.$

∎

We now need to know how the $\operatorname{Ind}$-assignment interacts with
horizontal composition of natural transformations in order to verify that it
is pseudofunctorial in $\operatorname{\mathfrak{Cat}}$. Recall that
pseudofuctoriality declares immediately that if we have a $2$-cell

${\operatorname{\mathscr{C}}}$${\operatorname{\mathscr{D}}}$${\operatorname{\mathscr{E}}}$$\scriptstyle{F}$$\scriptstyle{G}$$\scriptstyle{\alpha}$$\scriptstyle{H}$$\scriptstyle{K}$$\scriptstyle{\beta}$

then we have the identity

$\operatorname{Ind}(\beta)\ast\operatorname{Ind}(\alpha)=\phi_{G,K}^{-1}\circ\operatorname{Ind}(\beta\ast\alpha)\circ\phi_{F,H};$

in particular, establishing the above identity is equivalent to establishing
pseudofunctoriality on $2$-morphisms.

######  Lemma 3.14.

The $\operatorname{Ind}$-assignment is pseudofunctorial on $2$-morphisms in
the sense that if we have natural transformations $\alpha:F\Rightarrow
G:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{D}}$ and if
$\beta:H\Rightarrow K:\operatorname{\mathscr{D}}\to\operatorname{\mathscr{E}}$
then

$\operatorname{Ind}(\beta\ast\alpha)=\phi_{G,K}^{-1}\circ\operatorname{Ind}(\beta)\ast\operatorname{Ind}(\alpha)\circ\phi_{F,H}.$

###### Proof.

This is a straightforward verifiction using Lemma 3.11 and the observation
that every object and morphism in sight in the definitions of
$\operatorname{Ind}(\beta\ast\alpha)$ and
$\operatorname{Ind}(\beta)\ast\operatorname{Ind}(\alpha)$ involve only the
same indexing categories. ∎

We now will endeavour to show that if
$(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ is a tangent category
then the ind-category $\operatorname{Ind}(\operatorname{\mathscr{C}})$
naturally inherits an ind-tangent structure which we will call
$\operatorname{Ind}(\operatorname{\mathbb{T}})$ from
$\operatorname{\mathbb{T}}$. We begin this study by first showing the
existence of the natural transformations $\operatorname{Ind}(p)$ and
$\operatorname{Ind}(0)$; constructing these is straightforward, while the
natural transformations $\hat{+},\hat{\ell}$, and $\hat{c}$ take a little more
work to define for technical $2$-categorical reasons: the $2$-functor
$\operatorname{Ind}:\operatorname{\mathfrak{Cat}}\to\operatorname{\mathfrak{Cat}}$
is a pseudofunctor and not strictly functorial.

######  Lemma 3.15.

If $\operatorname{\mathscr{C}}$ is a tangent category then there are ind-
bundle and ind-zero natural transformations
$\operatorname{Ind}(p):\operatorname{Ind}(T)\Rightarrow\operatorname{id}_{\operatorname{Ind}(\operatorname{\mathscr{C}})}$
and
$\operatorname{Ind}(0):\operatorname{id}_{\operatorname{Ind}(\operatorname{\mathscr{C}})}\Rightarrow\operatorname{Ind}(T)$.

###### Proof.

Apply Corollary 3.9 and Proposition 3.12 to the natural transformations
$p:T\Rightarrow\operatorname{id}_{\operatorname{\mathscr{C}}}$ and
$0:\operatorname{id}_{\operatorname{\mathscr{C}}}\Rightarrow T$. ∎

Because every tangent category $\operatorname{\mathscr{C}}$ admits (finite)
pullbacks against the bundle maps $p_{X}:TX\to X$ for every object
$X\in\operatorname{\mathscr{C}}_{0}$, it follows from [1, Proposition
I.8.9.1.c] that $\operatorname{Ind}(\operatorname{\mathscr{C}})$ admits finite
pullbacks agains the ind-bundle maps
$\operatorname{Ind}(p)_{\underline{{X}}}:\operatorname{Ind}(T)\underline{{X}}\to\underline{{X}}$
for every
$\underline{{X}}\in\operatorname{Ind}(\operatorname{\mathscr{C}})_{0}$. In
order to have our tangent structure
$\operatorname{Ind}(\operatorname{\mathbb{T}})$ be essentially given as the
indicization of the tangent structure $\operatorname{\mathbb{T}}$, we nee now
must establish a natural isomorphism (and hence also the existence of)
$\operatorname{Ind}(T){}_{\operatorname{Ind}(p)}\times_{\operatorname{Ind}(p)}\operatorname{Ind}(T)=\operatorname{Ind}(T)_{2}\cong\operatorname{Ind}(T_{2})$.
Afterwards, we’ll establish that compositional powers of
$\operatorname{Ind}(T)$ preserve and commute with these pullback powers before
constructiong the ind-bundle addition and proving that
$\operatorname{Ind}(p)_{\underline{{X}}}$ is a commutative monoid in
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\underline{{X}}}$.

######  Proposition 3.16.

If $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ is a tangent
category then there is a natural isomorphism of functors
$\operatorname{Ind}(T_{2})\cong\operatorname{Ind}(T)_{2}:\operatorname{Ind}(\operatorname{\mathscr{C}})\to\operatorname{Ind}(\operatorname{\mathscr{C}})$.

###### Proof.

First that the pullback powers
$\operatorname{Ind}(T_{n})\underline{{X}}=\operatorname{Ind}(T)\underline{{X}}\times_{\underline{{X}}}\cdots\times_{\underline{{X}}}\operatorname{Ind}(T)\underline{{X}}$
follows from the fact that for any objects
$\underline{{X}}=(X_{i}),\underline{{Y}}=(Y_{i}),\underline{{Z}}=(Z_{i})$ the
object $\underline{{W}}=(X_{i}\times_{Z_{i}}Y_{i})$ represents the pullback
$\underline{{X}}\times_{\underline{{Z}}}\underline{{Y}}$ in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$; the verification of this
fact is routine and straightforward.

We use the presheaf realization functor
$L:\operatorname{Ind}(\operatorname{\mathscr{C}})\to[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]$
to prove the remainder of proposition, i.e., that
$\operatorname{Ind}(T)_{2}\cong\operatorname{Ind}(T_{2})$. On objects we have
that on one hand

$\operatorname{Ind}(T)_{2}\underline{{X}}=\operatorname{Ind}(T)\underline{{X}}\times_{\underline{{X}}}\operatorname{Ind}(T)\underline{{X}}=(TX_{i})\times_{(X_{i})}(TX_{i})$

while on the other hand

$\operatorname{Ind}(T_{2})\underline{{X}}=(T_{2}X_{i})=(TX_{i}\times_{X_{i}}TX_{i}).$

Taking the image under $L$ and using that filtered colimits commute with
finite limits in
$[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]$
we find

$\displaystyle L\big{(}\operatorname{Ind}(T_{2})\underline{{X}}\big{)}$
$\displaystyle=\lim_{\begin{subarray}{c}\longrightarrow\\\ i\in
I\end{subarray}}\operatorname{\mathbf{y}}(TX_{i}\times_{X_{i}}TX_{i})\cong\lim_{\begin{subarray}{c}\longrightarrow\\\
i\in
I\end{subarray}}\left(\operatorname{\mathbf{y}}TX_{i}\times_{\operatorname{\mathbf{y}}X_{i}}\operatorname{\mathbf{y}}TX_{i}\right)\cong\lim_{\begin{subarray}{c}\longrightarrow\\\
i\in
I\end{subarray}}\operatorname{\mathbf{y}}TX_{i}\times_{\operatorname*{colim}\operatorname{\mathbf{y}}X_{i}}\lim_{\begin{subarray}{c}\longrightarrow\\\
i\in I\end{subarray}}\operatorname{\mathbf{y}}TX_{i}$ $\displaystyle\cong
L(TX_{i})\times_{LX_{i}}L(TX_{i})\cong
L\left(\operatorname{Ind}(T)\underline{{X}}\times_{\underline{{X}}}\operatorname{Ind}(T)\underline{{X}}\right)=L\left(\operatorname{Ind}(T)_{2}\underline{{X}}\right).$

From the fact that $L$ is conservative, we conclude that
$\operatorname{Ind}(T)_{2}\underline{{X}}\cong\operatorname{Ind}(T_{2})\underline{{X}}$;
that this isomorphism is natural in $\underline{{X}}$ because it is induced by
the universal property of the limit together with limit preservation
isomorphisms. ∎

######  Corollary 3.17.

For any tangent category
$(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ and any
$m\in\operatorname{\mathbb{N}}$ there is an isomorphism
$\operatorname{Ind}(T)_{m}\cong\operatorname{Ind}(T_{m})$.

###### Proof.

This follows mutatis mutandis to the proof of Proposition 3.16. ∎

######  Corollary 3.18.

Let $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ be a tangent
category and let $m,n\in\operatorname{\mathbb{N}}$. Then there is a natural
isomorphism of functors

$\operatorname{Ind}(T)^{n}\circ\operatorname{Ind}(T)_{m}\cong\operatorname{Ind}(T)_{m}\circ\operatorname{Ind}(T)^{n}.$

###### Proof.

Since $\operatorname{\mathscr{C}}$ is a tangent category, we have natural a
natural isomorphism $T^{n}\circ T_{m}\cong T_{m}\circ T^{n}$. Apply
Proposition 3.12 to this natural isomorphism to get a natural isomorphism
$\operatorname{Ind}(T^{n}\circ T_{m})\cong\operatorname{Ind}(T_{m}\circ
T^{n})$. Finally, conjugating by the to the natural isomorphism by the
$\operatorname{Ind}$-compositor
$\phi_{f,g}:\operatorname{Ind}(f)\circ\operatorname{Ind}(g)\to\operatorname{Ind}(g\circ
f)$ multiple times together together with using Corollary 3.17 gives

$\displaystyle\operatorname{Ind}(T)^{n}\circ\operatorname{Ind}(T)_{m}$
$\displaystyle\cong\operatorname{Ind}(T^{n})\circ\operatorname{Ind}(T_{m})\cong\operatorname{Ind}(T^{n}\circ
T_{m})\cong\operatorname{Ind}(T_{m}\circ
T^{n})\cong\operatorname{Ind}(T_{m})\circ\operatorname{Ind}(T^{n})$
$\displaystyle\cong\operatorname{Ind}(T)_{m}\circ\operatorname{Ind}(T)^{n}.$

∎

Let us now prove the existence of the ind-addition natural transformation
$\operatorname{Ind}(+)$ and prove that for any
$\underline{{X}}\in\operatorname{Ind}(\operatorname{\mathscr{C}})_{0}$,
$\operatorname{Ind}(p)_{\underline{{X}}}$ is a commutative monoid in
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\underline{{X}}}$.

######  Lemma 3.19.

Let $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ be a tangent
category. Then there is an ind-addition natural transformation

$\hat{(}+):\operatorname{Ind}(T)_{2}\Rightarrow\operatorname{Ind}(T).$

###### Proof.

We define $\operatorname{Ind}(+)$ to be the natural transformation displayed
below

$\textstyle{\operatorname{Ind}(T)_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\operatorname{Ind}(T_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ind}(+)}$$\textstyle{\operatorname{Ind}(T)}$

where $\operatorname{Ind}(+)$ is the indicization of the natural
transformation $+:T_{2}\Rightarrow T$ of Proposition 3.12. ∎

######  Proposition 3.20.

For any tangent category
$(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ and any
$\underline{{X}}\in\operatorname{Ind}(\operatorname{\mathscr{C}})_{0}$, the
object
$\hat{(}p)_{\underline{{X}}}:\operatorname{Ind}(T)\underline{{X}}\to\underline{{X}}$
is a commutative monoid in
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\underline{{X}}}$ with unit
$\operatorname{Ind}(0)_{\underline{{X}}}$ and addition $\hat{+}$.

###### Proof.

We must show that in
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\underline{{X}}}$ the
diagrams

$\textstyle{\operatorname{Ind}(T)\underline{{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{\operatorname{Ind}(T)\underline{{X}}\times_{\underline{{X}}}\underline{{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{\operatorname{Ind}(T)\underline{{X}}}\times\operatorname{Ind}(0)_{\underline{{X}}}}$$\textstyle{\operatorname{Ind}(T)_{2}\underline{{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{+}_{\underline{{X}}}}$$\textstyle{\underline{{X}}\times_{\underline{{X}}}\operatorname{Ind}(T)\underline{{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ind}(0)_{\underline{{X}}}\times\operatorname{id}_{\operatorname{Ind}(T)\underline{{X}}}}$$\textstyle{\operatorname{Ind}(T)_{2}\underline{{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{+}_{\underline{{X}}}}$$\textstyle{\operatorname{Ind}(T)\underline{{X}}}$
$\textstyle{\operatorname{Ind}(T)_{2}\underline{{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{\hat{+}_{\underline{{X}}}}$$\textstyle{\operatorname{Ind}(T)_{2}\underline{{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{+}_{\underline{{X}}}}$$\textstyle{\operatorname{Ind}(T)\underline{{X}}}$
${(\operatorname{Ind}(T)_{2}\underline{{X}})\times_{\underline{{X}}}\operatorname{Ind}(T)\underline{{X}}}$${\operatorname{Ind}(T)_{2}\underline{{X}}}$${\operatorname{Ind}(T)\underline{{X}}\times(\operatorname{Ind}(T)_{2}\underline{{X}})}$${\operatorname{Ind}(T)\underline{{X}}}$${\operatorname{Ind}(T)_{2}\underline{{X}}}$$\scriptstyle{\hat{+}_{\underline{{X}}}\times\operatorname{id}_{\operatorname{Ind}(T)\underline{{X}}}}$$\scriptstyle{\cong}$$\scriptstyle{\hat{+}_{\underline{{X}}}}$$\scriptstyle{\operatorname{id}_{\operatorname{Ind}(T)\underline{{X}}}\times\hat{+}_{\underline{{X}}}}$$\scriptstyle{\hat{+}_{\underline{{X}}}}$

commute. To prove the commutativity of these diagrams we note that
$\operatorname{Ind}(T)_{2}\underline{{X}}$ carries the natural isomorphism
$\operatorname{Ind}(T)_{2}\cong\operatorname{Ind}(T_{2})$ which mediates
between the pullback in $\operatorname{Ind}(\operatorname{\mathscr{C}})$ and
the ind-object given by the pullbacks $(TX_{i}\times_{X_{i}}TX_{i})$ and the
map $\hat{+}_{\underline{{X}}}$ is defined by first using this mediating
isomorphism before acting on the ind-object $(TX_{i}\times_{X_{i}}TX_{i})$. As
such, it follows by naturality to verify each of the diagrams above on the
corresponding incarnation of ind-objects whose components are all given
$i$-locally as the pullback of objects in $\operatorname{\mathscr{C}}$. More
explicitly, to verify the first diagram it suffices to show that the diagram

$\textstyle{(TX_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{(TX_{i}\times_{X_{i}}X_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{TX_{i}}\times
0_{X_{i}}}$$\textstyle{(T_{2}X_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+_{X_{i}}}$$\textstyle{(X_{i}\times_{X_{i}}TX_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0_{X_{i}}\times\operatorname{id}_{TX_{i}}}$$\textstyle{(T_{2}X_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+_{X_{i}}}$$\textstyle{(TX_{i})}$

commutes in $\operatorname{Ind}(\operatorname{\mathscr{C}})$. However, because
$TX_{i}$ is a bundle over $X_{i}$ for all $i\in I$, so the diagram

$\textstyle{TX_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{TX_{i}\times_{X_{i}}X_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{TX_{i}}\times
0_{X_{i}}}$$\textstyle{T_{2}X_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+_{X_{i}}}$$\textstyle{X_{i}\times_{X_{i}}TX_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0_{X_{i}}\times\operatorname{id}_{TX_{i}}}$$\textstyle{T_{2}X_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+_{X_{i}}}$$\textstyle{TX_{i}}$

for each $i\in I$. Thus by taking the image of the diagram under the functor
$L$ it follows that the diagram of ind-presheaves commutes and so via the fact
that $L$ is fully faithful, it follows that the diagram commutes in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ as well. The commutativity of
the other diagrams is verified mutatis mutandis to this diagrams, and so are
omitted. ∎

We now build the ind-vertical lift
$\operatorname{Ind}(\ell):\operatorname{Ind}(T)\Rightarrow\operatorname{Ind}(T)^{2}$
and prove that it induces a bundle morphism in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$.

######  Lemma 3.21.

If $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ is a tangent
category then there is an ind-vertical lift transformation
$\hat{\ell}:\operatorname{Ind}(T)\Rightarrow\operatorname{Ind}(T)^{2}$.

###### Proof.

We define $\hat{\ell}$ as in the diagram of functors and natural
transformations

$\textstyle{\operatorname{Ind}(T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ind}{\ell}}$$\textstyle{\operatorname{Ind}(T^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{T,T}}$$\textstyle{\operatorname{Ind}(T)^{2}}$

where
$\operatorname{Ind}{\ell}:\operatorname{Ind}(T)\Rightarrow\operatorname{Ind}(T^{2})$
is the transformation induced by applying Proposition 3.12 and
$\phi_{T,T}:\operatorname{Ind}(T^{2})\cong\operatorname{Ind}(T)^{2}$ is the
compositor isomorphism. ∎

######  Proposition 3.22.

If $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ is a tangent
category then for any object $\underline{{X}}$ of
$\operatorname{Ind}(\operatorname{\mathscr{C}})$, the pair of morphisms
$(\hat{\ell}_{\underline{{X}}},\operatorname{Ind}(0)_{\underline{{X}}})$
describes a morphism of bundles in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$.

###### Proof.

We must prove that the diagrams

${\operatorname{Ind}(T)\underline{{X}}}$${\operatorname{Ind}(T)^{2}\underline{{X}}}$${\underline{{X}}}$${\operatorname{Ind}(T)\underline{{X}}}$$\scriptstyle{\hat{\ell}_{\underline{{X}}}}$$\scriptstyle{\operatorname{Ind}(p)_{\underline{{X}}}}$$\scriptstyle{(\operatorname{Ind}(T)\ast\operatorname{Ind}(p))_{\underline{{X}}}}$$\scriptstyle{\operatorname{Ind}(0)_{\underline{{X}}}}$
${\operatorname{Ind}(T)\underline{{X}}\times_{\underline{{X}}}\operatorname{Ind}(T)\underline{{X}}}$${\operatorname{Ind}(T)^{2}\underline{{X}}\times_{\operatorname{Ind}(T)\underline{{X}}}\operatorname{Ind}(T)^{2}\underline{{X}}}$${\operatorname{Ind}(T)\underline{{X}}}$${\operatorname{Ind}(T)^{2}\underline{{X}}}$$\scriptstyle{\langle\hat{\ell}_{\underline{{X}}}\circ\pi_{1},\hat{\ell}_{\underline{{X}}}\circ\pi_{2}\rangle}$$\scriptstyle{\hat{+}_{\underline{{X}}}}$$\scriptstyle{(\hat{+}\ast\operatorname{Ind}(T))_{\underline{{X}}}}$$\scriptstyle{\hat{\ell}_{\underline{{X}}}}$
${\underline{{X}}}$${\operatorname{Ind}(T)\underline{{X}}}$${\operatorname{Ind}(T)\underline{{X}}}$${\operatorname{Ind}(T)^{2}\underline{{X}}}$$\scriptstyle{\operatorname{Ind}(0)_{\underline{{X}}}}$$\scriptstyle{\operatorname{Ind}(0)_{\underline{{X}}}}$$\scriptstyle{(\operatorname{Ind}(0)\ast\operatorname{Ind}(T))_{\underline{{X}}}}$$\scriptstyle{\hat{\ell}_{\underline{{X}}}}$

commute in $\operatorname{Ind}(\operatorname{\mathscr{C}})$. Write
$\underline{{X}}=(X_{i})_{i\in I}$ and consider that the first diagram has
bottom edge calculated by

$\operatorname{Ind}(0)_{\underline{{X}}}\circ\operatorname{Ind}(p)_{\underline{{X}}}=(0_{X_{i}})\circ(p_{X_{i}})=(0_{X_{i}}\circ
p_{X_{i}})=((T\ast p)_{X_{i}}\circ\ell_{X_{i}})$

because $\operatorname{\mathscr{C}}$ is a tangent category. Alternatively, by
Lemma 3.21 we find that the upper half of the diagram is calculated by

$\big{(}\operatorname{Ind}(T)\ast\operatorname{Ind}(p)\big{)}_{\underline{{X}}}\circ\hat{\ell}_{X_{i}}=\big{(}(T\ast
p)_{X_{i}}\big{)}\circ(\ell_{X_{i}})=\big{(}(T\ast
p)_{X_{i}}\circ\ell_{X_{i}}\big{)}=\operatorname{Ind}(0)_{\underline{{X}}}\circ\operatorname{Ind}(p)_{\underline{{X}}},$

so the first diagram indeed commutes. The third diagram is verified to commute
similarly, so it suffices to show now that the second diagram commutes. For
this we note that on one hand

$\hat{\ell}_{\underline{{X}}}\circ\hat{+}_{\underline{{X}}}=(\ell_{X_{i}})_{i\in
I}\circ(0_{X_{i}})_{i\in I}=(\ell_{X_{i}}\circ 0_{X_{i}})_{i\in
I}=\big{(}(+\ast
T)_{X_{i}}\circ\langle\ell_{X_{i}}\circ\pi_{1,i},\ell_{X_{i}}\circ\pi_{2,i}\rangle\big{)}_{i\in
I}$

because $\operatorname{\mathscr{C}}$ is a tangent category. Now on the other
hand we note that

$\displaystyle\big{(}\hat{+}\ast\operatorname{Ind}(T)\big{)}_{\underline{{X}}}\circ\left\langle\hat{\ell}_{\underline{{X}}}\circ\pi_{1},\hat{\ell}_{\underline{{X}}}\circ\pi_{2}\right\rangle$
$\displaystyle=\left((+\ast
T)_{X_{i}}\right)\circ\left\langle(\ell_{X_{i}})\circ(\pi_{1,i}),\ell_{X_{i}}\circ(\pi_{2,i})\right\rangle$
$\displaystyle=\big{(}(+\ast
T)_{X_{i}}\big{)}\circ\left\langle(\ell_{X_{i}}\circ\pi_{1,i}),(\ell_{X_{i}}\circ\pi_{2,i})\right\rangle$
$\displaystyle=\big{(}(+\ast
T)_{X_{i}}\big{)}\circ\big{(}\langle\ell_{X_{i}}\circ\pi_{1,i},\ell_{X_{i}}\circ\pi_{2,i}\rangle\big{)}$
$\displaystyle=\left((+\ast
T)_{X_{i}}\circ\langle\ell_{X_{i}}\circ\pi_{1,i},\ell_{X_{i}}\circ\pi_{2,i}\rangle\right)$

so it follows that the diagram indeed commutes. ∎

We now provide the existence of the ind-canonical flip. This amounts to being
able to commute partial derivative operators within formal $2$-jets of our
tangent object $\operatorname{Ind}(T)\underline{{X}}\to\underline{{X}}$.
Afterwards we will prove that $\hat{c}$ is one of the components of a bundle
morphism.

######  Lemma 3.23.

Let $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ be a tangent
category. Then there is an ind-canonical flip natural transformation

$\hat{c}:\operatorname{Ind}(T)^{2}\Rightarrow\operatorname{Ind}(T)^{2}.$

###### Proof.

We define the ind-canonical flip via the diagram

${\operatorname{Ind}(T)^{2}}$${\operatorname{Ind}(T^{2})}$${\operatorname{Ind}(T)^{2}}$${\operatorname{Ind}(T^{2})}$$\scriptstyle{\hat{c}}$$\scriptstyle{\phi_{T,T}}$$\scriptstyle{\operatorname{Ind}{c}}$$\scriptstyle{\phi_{T,T}^{-1}}$

where $\phi_{T,T}$ is the compositor isomorphism
$\theta:\operatorname{Ind}(T)^{2}\xRightarrow{\cong}\operatorname{Ind}(T^{2})$
and $\operatorname{Ind}{c}$ is the indicization of the canonical flip
$c:T^{2}\Rightarrow T^{2}$ asserted by Proposition 3.12. ∎

######  Proposition 3.24.

If $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ is a tangent
category then for any object $\underline{{X}}$ of
$\operatorname{Ind}(\operatorname{\mathscr{C}})$, the pair of morphisms
$(\operatorname{id}_{\operatorname{Ind}(T)\underline{{X}}},\hat{c}_{\underline{{X}}})$
describe a bundle morphism.

###### Proof.

Following Definition 2.1, we must show that the diagrams

${\operatorname{Ind}(T)^{2}{\underline{{X}}}}$${\operatorname{Ind}(T)^{2}{\underline{{X}}}}$${\operatorname{Ind}(T){\underline{{X}}}}$${\operatorname{Ind}(T){\underline{{X}}}}$$\scriptstyle{\hat{c}_{\underline{{X}}}}$$\scriptstyle{(\operatorname{Ind}(T)\ast\operatorname{Ind}(p))_{\underline{{X}}}}$$\scriptstyle{(\operatorname{Ind}(p)\ast\operatorname{Ind}(T))_{{\underline{{X}}}}}$
${\operatorname{Ind}(T)^{2}{\underline{{X}}}\times_{\operatorname{Ind}(T){\underline{{X}}}}\operatorname{Ind}(T)^{2}{\underline{{X}}}}$${\operatorname{Ind}(T)^{2}{\underline{{X}}}\times_{\operatorname{Ind}(T){\underline{{X}}}}\operatorname{Ind}(T)^{2}{\underline{{X}}}}$${\operatorname{Ind}(T)^{2}{\underline{{X}}}}$${\operatorname{Ind}(T)^{2}{\underline{{X}}}}$$\scriptstyle{(\operatorname{Ind}(T)\ast\hat{+})_{{\underline{{X}}}}}$$\scriptstyle{\langle\hat{c}_{\underline{{X}}}\circ\pi_{1},\hat{c}_{\underline{{X}}}\circ\pi_{2}\rangle}$$\scriptstyle{(\hat{+}\ast\operatorname{Ind}(T))_{\underline{{X}}}}$$\scriptstyle{\hat{c}_{\underline{{X}}}}$
${\operatorname{Ind}(T){\underline{{X}}}}$${\operatorname{Ind}(T){\underline{{X}}}}$${\operatorname{Ind}(T)^{2}{\underline{{X}}}}$${\operatorname{Ind}(T)^{2}{\underline{{X}}}}$$\scriptstyle{(\operatorname{Ind}(T)\ast
0)_{\underline{{X}}}}$$\scriptstyle{(0\ast\operatorname{Ind}(T))_{\underline{{X}}}}$$\scriptstyle{\hat{c}_{\underline{{X}}}}$

commute. The first and third diagrams are established similarly, so we need
only establish the commutativity of the first and second diagrams to prove the
proposition. Let $\underline{{X}}=(X_{i})_{i\in I}=(X_{i})$ be an object of
$\operatorname{Ind}(\operatorname{\mathscr{C}})$. For this we begin by
establishing the commutativity of the first digaram. Note that on one hand

$\big{(}\operatorname{Ind}(T)\ast\operatorname{Ind}(p)\big{)}_{\underline{{X}}}=\big{(}(T\ast
p)_{X_{i}}\big{)}=\big{(}(p\ast T)_{X_{i}}\circ c_{X_{i}}\big{)}$

because $\operatorname{\mathscr{C}}$ is a tangent category. On the other hand
we calculate that

$\displaystyle\big{(}\operatorname{Ind}(p)\ast\operatorname{Ind}(T)\big{)}_{\underline{{X}}}\circ\hat{c}_{\underline{{X}}}$
$\displaystyle=\big{(}(p\ast
T)_{X_{i}}\big{)}\circ\theta^{-1}_{\underline{{X}}}\circ\underline{{c}}_{\underline{{X}}}\circ\theta_{\underline{{X}}}=\big{(}(p\ast
T)_{X_{i}}\big{)}\circ\theta^{-1}_{\underline{{X}}}\circ(c_{X_{i}})\circ\theta_{\underline{{X}}}$
$\displaystyle=\big{(}(p\ast
T)_{X_{i}}\big{)}\circ(c_{X_{i}})\circ\theta^{-1}_{\underline{{X}}}\circ\theta_{\underline{{X}}}=\big{(}(p\ast
T)_{X_{i}}\big{)}\circ(c_{X_{i}})$ $\displaystyle=\big{(}(p\ast
T)_{X_{i}}\circ c_{X_{i}}\big{)},$

which shows that the first diagram indeed commutes.

To show the commutativity of the second diagram we note that on one hand a
routine check shows

$\displaystyle\hat{c}_{\underline{{X}}}\circ\big{(}\operatorname{Ind}(T)\ast\hat{+}\big{)}_{\underline{{X}}}$
$\displaystyle=\theta_{\underline{{X}}}^{-1}\circ(c_{X_{i}})\circ\big{(}(T\ast+)_{X_{i}}\big{)}\circ\theta_{\underline{{X}}}.$

On the other hand

$\displaystyle\big{(}\hat{+}\ast
T\big{)}_{\underline{{X}}}\circ\left\langle\hat{c}_{\underline{{X}}}\circ\pi_{1},\hat{c}_{\underline{{X}}}\circ\pi_{2}\right\rangle$
$\displaystyle=\big{(}\hat{+}\ast
T\big{)}_{\underline{{X}}}\circ\left\langle\theta^{-1}_{\underline{{X}}}\circ(c_{X_{i}})\circ\theta_{\underline{{X}}}\circ\pi_{1},\theta^{-1}_{\underline{{X}}}\circ(c_{X_{i}})\circ\theta_{\underline{{X}}}\circ\pi_{2}\right\rangle$
$\displaystyle=\theta_{\underline{{X}}}^{-1}\circ\big{(}(+\ast
T)_{X_{i}}\big{)}\circ\left\langle(c_{X_{i}})\circ\theta_{\underline{{X}}}\circ\pi_{1},(c_{X_{i}})\circ\theta_{\underline{{X}}}\circ\pi_{2}\right\rangle$
$\displaystyle=\theta_{\underline{{X}}}^{-1}\circ\big{(}(+\ast
T)_{X_{i}}\big{)}\circ\left(\big{\langle}(c_{X_{i}})\circ\pi_{1,i},(c_{X_{i}})\circ\pi_{2,i}\big{\rangle}\right)\circ\theta_{\underline{{X}}}$
$\displaystyle=\theta_{\underline{{X}}}^{-1}\circ\left((p\ast
T)_{X_{i}}\circ\left\langle(c_{X_{i}})\circ\pi_{1,i},(c_{X_{i}})\circ\pi_{2,i}\right\rangle\right)\circ\theta_{\underline{{X}}}$
$\displaystyle=\theta^{-1}_{\underline{{X}}}\circ\big{(}c_{X_{i}}\circ(T\ast+)_{X_{i}}\big{)}\circ\theta_{\underline{{X}}}=\theta_{\underline{{X}}}^{-1}\circ(c_{X_{i}})\circ\big{(}(T\ast+)_{X_{i}}\big{)}\circ\theta_{\underline{{X}}},$

which shows that the secon diagram commutes. ∎

From here we show the coherences that $\hat{c}$ and $\hat{\ell}$ satisfy.

######  Lemma 3.25.

Let $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ be a tangent
category. The ind-canonical flip
$\hat{c}:\operatorname{Ind}(T)^{2}\Rightarrow\operatorname{Ind}(T)^{2}$ is an
involution, $\hat{c}\circ\hat{\ell}=\hat{\ell}$, and the diagrams

${\operatorname{Ind}(T)}$${\operatorname{Ind}(T)^{2}}$${\operatorname{Ind}(T)^{2}}$${\operatorname{Ind}(T)^{3}}$$\scriptstyle{\hat{\ell}}$$\scriptstyle{\hat{\ell}}$$\scriptstyle{\operatorname{Ind}(T)\ast\hat{\ell}}$$\scriptstyle{\hat{\ell}\ast\operatorname{Ind}(T)}$
${\operatorname{Ind}(T)^{3}}$${\operatorname{Ind}(T)^{3}}$${\operatorname{Ind}(T)^{3}}$${\operatorname{Ind}(T)^{3}}$${\operatorname{Ind}(T)^{3}}$${\operatorname{Ind}(T)^{3}}$$\scriptstyle{\hat{c}\ast\operatorname{Ind}(T)}$$\scriptstyle{\operatorname{Ind}(T)\ast\hat{c}}$$\scriptstyle{\hat{c}\ast\operatorname{Ind}(T)}$$\scriptstyle{\hat{c}\ast\operatorname{Ind}(T)}$$\scriptstyle{\operatorname{Ind}(T)\ast\hat{c}}$$\scriptstyle{\hat{c}\ast\operatorname{Ind}(T)}$
${\operatorname{Ind}(T)^{2}}$${\operatorname{Ind}(T)^{3}}$${\operatorname{Ind}(T)^{3}}$${\operatorname{Ind}(T)^{2}}$${\operatorname{Ind}(T)^{3}}$$\scriptstyle{\hat{c}}$$\scriptstyle{\hat{\ell}\ast\operatorname{Ind}(T)}$$\scriptstyle{\operatorname{Ind}(T)\ast\hat{c}}$$\scriptstyle{\hat{c}\ast\operatorname{Ind}(T)}$$\scriptstyle{\operatorname{Ind}(T)\ast\hat{\ell}}$

commute.

###### Proof.

That $\hat{c}$ is an involution follows from the calculation, for any
$\underline{{X}}=(X_{i})\in\operatorname{Ind}(\operatorname{\mathscr{C}})_{0}$,

$\hat{c}^{2}=\theta^{-1}\circ(c_{X_{i}})\circ\theta\circ\theta^{-1}\circ(c_{X_{i}})\circ\theta`=\theta^{-1}\circ(c_{X_{i}})^{2}\circ\theta=\theta^{-1}\circ\operatorname{id}_{\operatorname{Ind}(T^{2})\underline{{X}}}\circ\theta=\operatorname{id}_{\operatorname{Ind}(T)^{2}\underline{{X}}}.$

The second identity follows from the calculation

$\displaystyle\hat{c}_{\underline{{X}}}\circ\hat{\ell}_{\underline{{X}}}$
$\displaystyle=\theta^{-1}_{\underline{{X}}}\circ(c_{X_{i}})\circ\theta_{\underline{{X}}}\circ\theta^{-1}_{\underline{{X}}}\circ(\ell_{X_{i}})=\theta_{\underline{{X}}}^{-1}\circ(c_{X_{i}})\circ(\ell_{X_{i}})=\theta_{\underline{{X}}}^{-1}\circ(c_{X_{i}}\circ\ell_{X_{i}})$
$\displaystyle=\theta_{\underline{{X}}}^{-1}\circ(\ell_{X_{i}})=\hat{\ell}_{\underline{{X}}}.$

Finally the verification of the commuting diagrams is tedious but
straightforward check using the naturality of the isomorphisms
$\operatorname{Ind}(T^{n})\cong\operatorname{Ind}(T)^{n}$ together with the
identities satisfied by the canonical flip and vertical lift in
$\operatorname{\mathscr{C}}$. ∎

As the last necessary ingredient in showing that
$(\operatorname{Ind}(\operatorname{\mathscr{C}}),\operatorname{Ind}(\operatorname{\mathbb{T}}))$,
we will prove the universality of the ind-vertical lift. This proof will rely
on ind-presheaves
$\operatorname{Ind}_{\mathbf{PSh}}(\operatorname{\mathscr{C}})$ and the
functor
$L:\operatorname{Ind}(\operatorname{\mathscr{C}})\to[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]$
as it is convenient when proving properties about limits and colimits that
need not hold on the nose, i.e., those properties which need only hold up to
isomorphism.

######  Proposition 3.26.

Let $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ be a tangent
category and let
$\underline{{X}}\in\operatorname{Ind}(\operatorname{\mathscr{C}})_{0}$. Then
the diagram

${\operatorname{Ind}(T)_{2}\underline{{X}}}$${\operatorname{Ind}(T)^{2}\underline{{X}}}$${\operatorname{Ind}(T)\underline{{X}}}$$\scriptstyle{\big{(}\operatorname{Ind}(T)\ast\hat{+}\big{)}_{\underline{{X}}}\circ\left\langle\hat{\ell}\circ\pi_{1},\big{(}\operatorname{Ind}(0)\ast
T\big{)}_{\underline{{X}}}\circ\pi_{2}\right\rangle}$$\scriptstyle{\big{(}\operatorname{Ind}(T)\ast\operatorname{Ind}(p)\big{)}_{\underline{{X}}}}$$\scriptstyle{\operatorname{Ind}(0)_{\underline{{X}}}\circ\operatorname{Ind}(p)_{\underline{{X}}}\circ\big{(}\operatorname{Ind}(p)\ast\operatorname{Ind}(T)\big{)}_{\underline{{X}}}}$

is an equalizer in $\operatorname{Ind}(\operatorname{\mathscr{C}})$.

###### Proof.

Because being an equalizer is true along isomorphic objects in
$\operatorname{\mathscr{C}}$, stable under equivalence of categories, and in
light of Proposition 3.6, it suffices to prove that the diagram

${L\operatorname{Ind}(T)_{2}\underline{{X}}}$${L\operatorname{Ind}(T)^{2}\underline{{X}}}$${L\operatorname{Ind}(T)\underline{{X}}}$$\scriptstyle{L\big{(}\operatorname{Ind}(T)\ast\hat{+}\big{)}_{\underline{{X}}}\circ
L\left\langle\hat{\ell}\circ\pi_{1},\big{(}\operatorname{Ind}(0)\ast
T\big{)}_{\underline{{X}}}\circ\pi_{2}\right\rangle}$$\scriptstyle{L\big{(}\operatorname{Ind}(T)\ast\operatorname{Ind}(p)\big{)}_{\underline{{X}}}}$$\scriptstyle{L\operatorname{Ind}(0)_{\underline{{X}}}\circ
L\operatorname{Ind}(p)_{\underline{{X}}}\circ
L\big{(}\operatorname{Ind}(p)\ast\operatorname{Ind}(T)\big{)}_{\underline{{X}}}}$

is an equalizer in
$[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]$.
For this we calculate that the diagram above is is isomorphic to the diagram

${L\operatorname{Ind}(T)\underline{{X}}\times_{L\underline{{X}}}L\operatorname{Ind}(T)\underline{{X}}}$${L\operatorname{Ind}(T)^{2}\underline{{X}}}$${L\operatorname{Ind}(T)\underline{{X}}}$

where the horizontal morphisms are the untangling of the $L$ morphisms defined
above. By the fact that filtered colimits commute with finite limits in the
presheaf topos
$[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]$,
we find that the diagram above is isomorphic to the diagram of presheaves:

${\lim\limits_{\begin{subarray}{c}\longrightarrow\\\ i\in
I\end{subarray}}\operatorname{\mathscr{C}}(-,T_{2}X_{i})}$${\lim\limits_{\begin{subarray}{c}\longrightarrow\\\
i\in
I\end{subarray}}\operatorname{\mathscr{C}}(-,T^{2}X_{i})}$${\lim\limits_{\begin{subarray}{c}\longrightarrow\\\
i\in I\end{subarray}}\operatorname{\mathscr{C}}(-,TX_{i})}$

Finally, each $i$-indexed component of the diagram above,

${\operatorname{\mathscr{C}}(-,T_{2}X_{i})}$${\operatorname{\mathscr{C}}(-,T^{2}X_{i})}$${\operatorname{\mathscr{C}}(-,TX_{i})}$

is an equalizer by the fact that the Yoneda Lemma is continuous and the fact
that $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ is a tangent
category. This implies in turn that the diagram

${\lim\limits_{\begin{subarray}{c}\longrightarrow\\\ i\in
I\end{subarray}}\operatorname{\mathscr{C}}(-,T_{2}X_{i})}$${\lim\limits_{\begin{subarray}{c}\longrightarrow\\\
i\in
I\end{subarray}}\operatorname{\mathscr{C}}(-,T^{2}X_{i})}$${\lim\limits_{\begin{subarray}{c}\longrightarrow\\\
i\in I\end{subarray}}\operatorname{\mathscr{C}}(-,TX_{i})}$

and hence

${L\operatorname{Ind}(T)_{2}\underline{{X}}}$${L\operatorname{Ind}(T)^{2}\underline{{X}}}$${L\operatorname{Ind}(T)\underline{{X}}}$$\scriptstyle{L\big{(}\operatorname{Ind}(T)\ast\hat{+}\big{)}_{\underline{{X}}}\circ
L\left\langle\hat{\ell}\circ\pi_{1},\big{(}\operatorname{Ind}(0)\ast
T\big{)}_{\underline{{X}}}\circ\pi_{2}\right\rangle}$$\scriptstyle{L\big{(}\operatorname{Ind}(T)\ast\operatorname{Ind}(p)\big{)}_{\underline{{X}}}}$$\scriptstyle{L\operatorname{Ind}(0)_{\underline{{X}}}\circ
L\operatorname{Ind}(p)_{\underline{{X}}}\circ
L\big{(}\operatorname{Ind}(p)\ast\operatorname{Ind}(T)\big{)}_{\underline{{X}}}}$

are both equalizers in
$[\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathbf{Set}}]$.
Finally, appealing Proposition to 3.6 proves the result. ∎

We now have the ingredients to show that
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ is a tangent category
whenever $\operatorname{\mathscr{C}}$ is a tangent category.

######  Theorem 3.27.

Let $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ be a tangent
category. Then the category
$(\operatorname{Ind}(\operatorname{\mathscr{C}}),\operatorname{Ind}(\operatorname{\mathbb{T}}))$
is a tangent category where $\operatorname{Ind}(\operatorname{\mathbb{T}})$ is
the tangent structure

$\operatorname{Ind}(\operatorname{\mathbb{T}}):=(\operatorname{Ind}(T),\operatorname{Ind}(p),\operatorname{Ind}(0),\hat{+},\hat{\ell},\hat{c})$

where $\operatorname{Ind}(T)$ is the indicization of $T$ induced by
Proposition 3.7 and
$\operatorname{Ind}(p),\operatorname{Ind}(0),\hat{+},\hat{\ell},$ and
$\hat{c}$ are the natural transformations constructed in Lemmas 3.15, 3.19,
3.21, and Lemma 3.23, respectively.

###### Proof.

We now estalish that
$(\operatorname{Ind}(\operatorname{\mathscr{C}}),\operatorname{Ind}(\operatorname{\mathbb{T}}))$
satisfies the axioms of Definition 2.1:

  1. (1)

That $\operatorname{Ind}(\operatorname{\mathscr{C}})$ admits all tangent
pullbacks folloes from [1, Proposition I.8.9.1.c] and that
$\operatorname{Ind}(T)$ and its compositional powers preserve the pullback
powers $\operatorname{Ind}(T)_{m}$ is Corollary 3.18.

  2. (2)

The ind-zero transformation $\operatorname{Ind}(0)$ is constructed in Lemma
3.15 and the ind-addition transformation is constructed in Lemma 3.19. That
each object
$\operatorname{Ind}(p)_{\underline{{X}}}:\operatorname{Ind}(T)\underline{{X}}\to\underline{{X}}$
is a commutative monoid in
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\underline{{X}}}$ is
Proposition 3.20.

  3. (3)

The ind-vertical lift $\hat{\ell}$ exists by Lemma 3.21 and the pair
$(\hat{\ell}_{\underline{{X}}},\operatorname{Ind}(0)_{\underline{{X}}})$ is a
bundle morphism for any $\underline{{X}}$ in
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{0}$ by Proposition 3.22.

  4. (4)

The existence of the ind-canonical flip follows from Lemma 3.23 while the fact
that
$(\hat{c}_{\underline{{X}}},\operatorname{id}_{\operatorname{Ind}(T)\underline{{X}}})$
is a bundle map for any $\underline{{X}}$ in
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{0}$ follows from Proposition
3.24.

  5. (5)

The coherences between the ind-canonical flip and ind-vertical lift are given
in Lemma 3.25.

  6. (6)

The universality of the ind-vertical lift is proved in Proposition 3.26.

Because each axiom of Definition 2.1 is satisfied it follows that
$(\operatorname{Ind}(\operatorname{\mathscr{C}}),\operatorname{Ind}(\operatorname{\mathbb{T}}))$
is a tangent category. ∎

We now show the functoriality of the $\operatorname{Ind}$-construction on
tangent morphisms.

######  Theorem 3.28.

Let
$(F,\alpha):(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})\to(\operatorname{\mathscr{D}},\operatorname{\mathbb{S}})$
be a morphism of tangent categories. Then the induced map
$(\operatorname{Ind}(F),\hat{\alpha}):(\operatorname{Ind}(\operatorname{\mathscr{C}}),\operatorname{Ind}(\operatorname{\mathbb{T}}))\to(\operatorname{Ind}(\operatorname{\mathscr{D}}),\operatorname{Ind}(\operatorname{\mathbb{S}}))$
is a morphism of tangent categories where $\hat{\alpha}$ is the natural
transformation:

${\operatorname{Ind}(\operatorname{\mathscr{C}})}$${\operatorname{Ind}(\operatorname{\mathscr{D}})}$$\scriptstyle{\operatorname{Ind}(F)\circ\operatorname{Ind}(T)}$$\scriptstyle{\operatorname{Ind}(F\circ
T)}$$\scriptstyle{\operatorname{Ind}(S\circ
F)}$$\scriptstyle{\operatorname{Ind}(S)\circ\operatorname{Ind}(F)}$$\scriptstyle{\phi_{T,F}}$$\scriptstyle{\operatorname{Ind}(\alpha)}$$\scriptstyle{\phi_{F,S}^{-1}}$

Furthermore, $(\operatorname{Ind}(F),\hat{\alpha})$ is a strong tangent
morphism if and only if $(F,\alpha)$ is a strong tangent morphism.

###### Proof.

The fact that $(\operatorname{Ind}(F),\hat{\alpha})$ is a tangent morphism is
a straightforward but tedious $2$-categorical verification; we illustrate the
first such verification and omit the rest. To establish the first diagram of
functors and natural transformations we simply paste the commuting triangles
as in the diagram below

${\operatorname{Ind}(F)\circ\operatorname{Ind}(T)}$${\operatorname{Ind}(F\circ
T)}$${\operatorname{Ind}(S\circ
F)}$${\operatorname{Ind}(S)\circ\operatorname{Ind}(F)}$${\operatorname{Ind}(F)}$$\scriptstyle{\phi_{F,T}}$$\scriptstyle{\operatorname{Ind}(F)\ast\operatorname{Ind}(p)}$$\scriptstyle{\operatorname{Ind}(F\ast
p)}$$\scriptstyle{\operatorname{Ind}(\alpha)}$$\scriptstyle{\phi_{F,S}^{-1}}$$\scriptstyle{\operatorname{Ind}(q\ast
F)}$$\scriptstyle{\operatorname{Ind}(q)\ast\operatorname{Ind}(p)}$

to establish that

${\operatorname{Ind}(F)\circ\operatorname{Ind}(T)}$${\operatorname{Ind}(S)\circ\operatorname{Ind}(F)}$${\operatorname{Ind}(F)}$$\scriptstyle{\hat{\alpha}}$$\scriptstyle{\operatorname{Ind}(F)\ast\operatorname{Ind}(p)}$$\scriptstyle{\operatorname{Ind}(q)\ast\operatorname{Ind}(F)}$

commutes. The remaining four axioms are verified and established similarly and
so are omitted.

For the final claim regarding detecting when $(F,\alpha)$ is strong:

$\implies:$ assume that $(\operatorname{Ind}(F),\hat{\alpha})$ is a strong
tangent morphism. Then it is strong on all constant objects
$\underline{{X}}=(X)$ for $X\in\operatorname{\mathscr{C}}_{0}$; as such it
follows that $\alpha_{X}$ is an isomorphism for every
$X\in\operatorname{\mathscr{C}}_{0}$. That $F$ preserves tangent pullbacks and
equalizers is shown similarly.

$\impliedby:$ Assume $\alpha$ is a natural isomorphism so that
$\operatorname{Ind}(\alpha)$ and
$\hat{\alpha}=\phi_{F,S}^{-1}\circ\operatorname{Ind}(\alpha)\circ\phi_{F,S}$
are isomorphisms as well. Similarly, if $F$ preserves tangent pullbacks and
equalizers, so does $\operatorname{Ind}(F)$. ∎

######  Corollary 3.29.

The diagram

${\operatorname{\mathfrak{Tan}}}$${\operatorname{\mathfrak{Tan}}}$${\operatorname{\mathfrak{Cat}}}$${\operatorname{\mathfrak{Cat}}}$$\scriptstyle{\operatorname{Ind}}$$\scriptstyle{\operatorname{Forget}}$$\scriptstyle{\operatorname{Forget}}$$\scriptstyle{\operatorname{Ind}}$

commutes strictly.

###### Proof.

This is immediate after unwinding the definitions. ∎

We now close this section by proving that while the $\operatorname{Ind}$
pseudofunctor does not commute (even up to isomorphism) with the free tangent
functor, there is a pseudonatural transformation:

${\operatorname{\mathfrak{Cat}}}$${\operatorname{\mathfrak{Cat}}}$${\operatorname{\mathfrak{Tan}}}$${\operatorname{\mathfrak{Tan}}}$$\scriptstyle{\operatorname{Ind}}$$\scriptstyle{\operatorname{Free}}$$\scriptstyle{\operatorname{Free}}$$\scriptstyle{\operatorname{Ind}}$$\scriptstyle{\alpha}$

To do this, however, we necessitate a discussion of what it means to be a free
tangent functor, i.e., we need to at least give a short description of the
$2$-functor
$\operatorname{Free}:\operatorname{\mathfrak{Cat}}\to\operatorname{\mathfrak{Tan}}$.

In [14] P. Leung showed that the category $\operatorname{\mathbf{Weil}}_{1}$
of Weil algebras constitute the free tangent structure in a very precise sense
(we will recall and see some of this later, but one of the main results in
[14] is that tangent structures on a category $\operatorname{\mathscr{C}}$ are
equivalent to Weil actegory structures on $\operatorname{\mathscr{C}}$). While
we will get to know this category in more detail later (by comparing and
contrasting the exposition of [14] and what is presented in B. MacAdam’s
thesis [15]), for now we just need to know the following result: for any
category $\operatorname{\mathscr{C}}$, the free tangent category over
$\operatorname{\mathscr{C}}$ is
$\operatorname{Free}(\operatorname{\mathscr{C}}):=\operatorname{\mathbf{Weil}}_{1}\times\operatorname{\mathscr{C}}$.
We will briefly recall the definition of $\operatorname{\mathbf{Weil}}_{1}$,
however.

######  Definition 3.30 ([14, Definition 4.2]).

The category $\operatorname{\mathbf{Weil}}_{1}$ is defined as follows:

  * •

Objects: For each $n\in\operatorname{\mathbb{N}}$, define the following crigs:

$W^{n}:=\begin{cases}\operatorname{\mathbb{N}}&\text{if}\,n=0;\\\
\operatorname{\mathbb{N}}[x_{1},\cdots,x_{n}]/(x_{i}x_{j}:1\leq i,j\leq
n)&\text{if}\,n\geq 1.\end{cases}$

Then the objects of $\operatorname{\mathbf{Weil}}_{1}$ are the closure of
$\\{W^{n}\;|\;n\in\operatorname{\mathbb{N}}\\}$ under finite coproducts
$\otimes_{\operatorname{\mathbb{N}}}$ of crigs.

  * •

Morphisms: Morphisms $\varphi:A\to B$ of crigs for which the canonical maps

${A}$${B}$${\operatorname{\mathbb{N}}}$$\scriptstyle{\pi_{\operatorname{\mathfrak{m}}_{A}}}$$\scriptstyle{\varphi}$$\scriptstyle{\pi_{\operatorname{\mathfrak{m}}_{B}}}$

commute; note that in each case $\operatorname{\mathfrak{m}}_{A}$ and
$\operatorname{\mathfrak{m}}_{B}$ are the unique maximal ideals of $A$ and
$B$, respectively.

######  Theorem 3.31 ([14]; [15, Observation 4.2.5]).

For any category $\operatorname{\mathscr{C}}$, there is a free tangent
category given by
$\operatorname{\mathbf{Weil}}_{1}\times\operatorname{\mathscr{C}}$.

Because the product construction is $2$-functorial we find immediately that
there is a functor
$\operatorname{Free}:\operatorname{\mathfrak{Cat}}\to\operatorname{\mathfrak{Tan}}$
which sends a category $\operatorname{\mathscr{C}}$ to the free tangent
category over itself:
$\operatorname{Free}(\operatorname{\mathscr{C}}):=\operatorname{\mathbf{Weil}}_{1}\times\operatorname{\mathscr{C}}$.
It acts on $1$-morphisms by
$\operatorname{Free}(f):=\operatorname{id}_{\operatorname{\mathbf{Weil}}_{1}}\times
f$ and on $2$-morphisms by
$\operatorname{Free}(\alpha):=\iota_{\operatorname{id}_{\operatorname{\mathbf{Weil}}_{1}}}\times\alpha$.
To construct our pseudonatural transformation

${\operatorname{\mathfrak{Cat}}}$${\operatorname{\mathfrak{Cat}}}$${\operatorname{\mathfrak{Tan}}}$${\operatorname{\mathfrak{Tan}}}$$\scriptstyle{\operatorname{Ind}}$$\scriptstyle{\operatorname{Free}}$$\scriptstyle{\operatorname{Free}}$$\scriptstyle{\operatorname{Ind}}$$\scriptstyle{\alpha}$

we compute that on one hand

$\operatorname{Ind}(\operatorname{Free}(\operatorname{\mathscr{C}}))=\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1}\times\operatorname{\mathscr{C}})$

while on the other hand

$\operatorname{Free}(\operatorname{Ind}(\operatorname{\mathscr{C}}))=\operatorname{\mathbf{Weil}}_{1}\times\operatorname{Ind}(\operatorname{\mathscr{C}}).$

To construct our pseudonatural transformation $\alpha$ we first note that by
[12, Proposition 6.1.12] for any category $\operatorname{\mathscr{C}}$ there
is a natural equivalence of categories

$\operatorname{Ind}(\operatorname{Free}(\operatorname{\mathscr{C}}))=\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1}\times\operatorname{\mathscr{C}})\simeq\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1})\times\operatorname{Ind}(\operatorname{\mathscr{C}}).$

Now consider the embedding
$\operatorname{incl}_{\operatorname{\mathbf{Weil}}_{1}}:\operatorname{\mathbf{Weil}}_{1}\to\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1})$
which sends an object $X$ to the functor333Recall that $\mathbbm{1}$ is the
terminal category, i.e., the category with one object (which we label as
$\ast$) and one morphism (namely $\operatorname{id}_{\ast}$).
$\underline{{X}}:\mathbbm{1}\to\operatorname{\mathscr{C}}$ given by
$\ast\mapsto X,\operatorname{id}_{\ast}\mapsto\operatorname{id}_{X}$ and sends
a morphism $f:X\to Y$ to the corresponding natural transformation
$\underline{{f}}:\underline{{X}}\to\underline{{Y}}$. We then define
$\alpha_{\operatorname{\mathscr{C}}}$ to be the composite:

${\operatorname{\mathbf{Weil}}_{1}\times\operatorname{Ind}(\operatorname{\mathscr{C}})}$${\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1})\times\operatorname{Ind}(\operatorname{\mathscr{C}})}$${\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1}\times\operatorname{\mathscr{C}})}$$\scriptstyle{\operatorname{incl}_{\operatorname{\mathbf{Weil}}_{1}}\times\operatorname{id}_{\operatorname{Ind}(\operatorname{\mathscr{C}})}}$$\scriptstyle{\simeq}$

To see how to define the witness transformations let
$F:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{D}}$ be a functor. We
then get the pasting diagram

${\operatorname{\mathbf{Weil}}_{1}\times\operatorname{Ind}(\operatorname{\mathscr{C}})}$${\operatorname{\mathbf{Weil}}_{1}\times\operatorname{Ind}(\operatorname{\mathscr{D}})}$${\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1})\times\operatorname{Ind}(\operatorname{\mathscr{C}})}$${\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1})\times\operatorname{Ind}(\operatorname{\mathscr{D}})}$${\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1}\times\operatorname{\mathscr{C}})}$${\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1}\times\operatorname{\mathscr{D}})}$$\scriptstyle{\alpha_{\operatorname{\mathscr{C}}}}$$\scriptstyle{\operatorname{id}_{\operatorname{\mathbf{Weil}}_{1}}\times\operatorname{Ind}(F)}$$\scriptstyle{\operatorname{incl}_{\operatorname{\mathbf{Weil}}_{1}}\times\operatorname{id}}$$\scriptstyle{\operatorname{incl}_{\operatorname{\mathbf{Weil}}_{1}}\times\operatorname{id}}$$\scriptstyle{\alpha_{\operatorname{\mathscr{D}}}}$$\scriptstyle{\operatorname{id}_{\operatorname{Ind}(\operatorname{\mathbf{Weil}})_{1}}\times\operatorname{Ind}(F)}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1}\times
F)}$$\scriptstyle{\cong}$

where the natural isomorphism in the bottom-most square is induced from the
natural equivalence of [12, Proposition 6.2.11]. Note this pastes to the
invertible $2$-cell

${(\operatorname{Free}\circ\operatorname{Ind})(\operatorname{\mathscr{C}})}$${(\operatorname{Free}\circ\operatorname{Ind})(\operatorname{\mathscr{D}})}$${(\operatorname{Ind}\circ\operatorname{Free})(\operatorname{\mathscr{C}})}$${(\operatorname{Ind}\circ\operatorname{Free})(\operatorname{\mathscr{D}})}$$\scriptstyle{(\operatorname{Free}\circ\operatorname{Ind})(F)}$$\scriptstyle{\alpha_{\operatorname{\mathscr{C}}}}$$\scriptstyle{\alpha_{\operatorname{\mathscr{D}}}}$$\scriptstyle{(\operatorname{Ind}\circ\operatorname{Free})(F)}$$\scriptstyle{\cong}$

which we define to be our witness transformation $\alpha_{f}$. From here the
pseudonaturality of $\alpha$ is routine but straightforward to check: the
compatibility with the compositors for $\operatorname{Free}(-)$ and
$\operatorname{Ind}(-)$ more or less comes down to the fact that the
equivalence
$\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1}\times\operatorname{\mathscr{C}})\simeq\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1})\times\operatorname{Ind}(\operatorname{\mathscr{C}})$
is natural and the action of $\operatorname{Ind}$ on
$\operatorname{\mathfrak{Cat}}$ occurs only in the right-handed variable and
does not interact with the $\operatorname{\mathbf{Weil}}_{1}$-variable
otherwise. This leads to the following proposition.

######  Proposition 3.32.

There is a pseudonatural transformation

${\operatorname{\mathfrak{Cat}}}$${\operatorname{\mathfrak{Cat}}}$${\operatorname{\mathfrak{Tan}}}$${\operatorname{\mathfrak{Tan}}}$$\scriptstyle{\operatorname{Ind}}$$\scriptstyle{\operatorname{Free}}$$\scriptstyle{\operatorname{Free}}$$\scriptstyle{\operatorname{Ind}}$$\scriptstyle{\alpha}$

where the transition functors are given by

${\operatorname{\mathbf{Weil}}_{1}\times\operatorname{Ind}(\operatorname{\mathscr{C}})}$${\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1})\times\operatorname{Ind}(\operatorname{\mathscr{C}})}$${\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1}\times\operatorname{\mathscr{C}})}$$\scriptstyle{\operatorname{incl}_{\operatorname{\mathbf{Weil}}_{1}}\times\operatorname{id}_{\operatorname{Ind}(\operatorname{\mathscr{C}})}}$$\scriptstyle{\simeq}$

######  Remark 3.33.

Note that since
$\operatorname{\mathbf{Weil}}_{1}\not\simeq\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1})$444Since
$\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1})$ admits all small
filtered colimits, we only need to demonstrate that there is a filtered
colimit which $\operatorname{\mathbf{Weil}}_{1}$ does not admit. However,
$\operatorname{\mathbf{Weil}}_{1}$ does not have infinite colimits as the
countably infinite coproduct
$X=\bigotimes_{n\in\operatorname{\mathbb{N}}}\operatorname{\mathbb{N}}[x_{n}]/(x_{n}^{2})\cong\operatorname{\mathbb{N}}[x_{n}:n\in\operatorname{\mathbb{N}}]/(x_{n}^{2}:n\in\operatorname{\mathbb{N}})$
is not an object in $\operatorname{\mathbf{Weil}}_{1}$. As such we get that
$\operatorname{\mathbf{Weil}}_{1}\not\simeq\operatorname{Ind}(\operatorname{\mathbf{Weil}}_{1})$.
it is impossible for $\alpha$ to be taken to be an invertible $2$-cell as
$\alpha_{\operatorname{\mathscr{C}}}$ is an equivalence if and only if
$\operatorname{\mathscr{C}}=\varnothing$ is the empty category.

##  4\. Properties of Ind-Tangent Categories

In this section we examine how the $\operatorname{Ind}$-construction interacts
with various important and standard constructions in the tangent category
world and in algebraic geometry. We in particular give a pseudolimit
characterization of the slice category
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{X}$ where $X=(X_{i})$ is an
$\operatorname{Ind}$-object, we exampine the $\operatorname{Ind}$-tangent
category of a representable tangent category, how $\operatorname{Ind}$
interacts with differential objects (and more generally how
$\operatorname{Ind}$ interacts with differential bundles), and finally how
$\operatorname{Ind}$ interacts with Cartesian differential categories. To do
these investigations we need some basic structural results regarding the
$\operatorname{Ind}$-pseudofunctor and its preservation of adjoints. Before
proceeding, however, let us recall the definition of a representable tangent
category.

###  4.1. A Pseudolimit Result

In this short subsection we show that there is a sensible characterization of
the slice category $\operatorname{Ind}(\operatorname{\mathscr{C}})_{/(X_{i})}$
(at least when $\operatorname{\mathscr{C}}$ has finite pullbacks) in terms of
a pseudolimit over the diagram of categories
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{/X_{i}}$ with morphism
functors induced by the pullback functors. While we do not explicitly use this
in this article, I anticipate that this will have future uses when studying
the tangent category of formal schemes explicitly and also in certain
functional-analytic characterizations (cf. Subsection 5.4 below) involving
slice tangent categories.

######  Proposition 4.1.

Let $\operatorname{\mathscr{C}}$ be a finitely complete category and let
$\operatorname{\mathfrak{X}}=(X_{i})$ be an object in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$. Consider the pseudofunctor
$G:I^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$ given by:

  * •

The object assignment sends $i\in I_{0}$ to the category
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{/X_{i}}$.

  * •

The morphism assignment sends $\varphi:i\to i^{\prime}$ to the pullback
functor
$\tilde{\varphi}^{\ast}:\operatorname{Ind}(\operatorname{\mathscr{C}})_{/X_{i^{\prime}}}\to\operatorname{Ind}(\operatorname{\mathscr{C}})_{X_{i}}$,
i.e., the functor which sends an object $Z\to X_{i^{\prime}}$ to a chosen
pullback

${Z\times_{X_{i^{\prime}}}X_{i}}$${Z}$${X_{i}}$${X_{i^{\prime}}}$$\scriptstyle{\tilde{\varphi}}$

where $\tilde{\varphi}:X_{i}\to X_{i^{\prime}}$ is the structure map induced
from $\operatorname{\mathfrak{X}}$.

  * •

The compositors are induced by the natural isomorpshims of pullbacks.

Then
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\operatorname{\mathfrak{X}}}$
is the pseudolimit of the diagram in $\operatorname{\mathfrak{Cat}}$ induced
by $F$.

###### Proof.

Begin by noting that $\operatorname{\mathfrak{X}}$ is the filtered colimit of
the $X_{i}$ in $\operatorname{Ind}(\operatorname{\mathscr{C}})$; this follows
from the fact that under the equivalence
$\operatorname{Ind}(\operatorname{\mathscr{C}})\simeq\operatorname{Ind}_{\mathbf{PSh}}(\operatorname{\mathscr{C}})$,
$\operatorname{\mathfrak{X}}$ is identified with the presheaf

$\operatorname*{colim}_{i\in I}\operatorname{\mathbf{y}}(X_{i})$

in $\operatorname{Ind}_{\mathbf{PSh}}(\operatorname{\mathscr{C}})$ and
equivalences preserve colimits. Consequently we write
$\alpha_{i}:X_{i}\to\operatorname{\mathfrak{X}}$ for the colimit structure
maps for all $i\in I$.

We now will establish that any diagram factors through
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\operatorname{\mathfrak{X}}}$.
Let $\operatorname{\mathscr{D}}$ be a category such that there are functors
$F_{i}:\operatorname{\mathscr{D}}\to\operatorname{\mathscr{C}}_{/X_{i}}$ for
all $i\in I_{0}$ such that for any $\varphi:i\to i^{\prime}$ in $I$, there is
an invertible $2$-cell

${\operatorname{Ind}(\operatorname{\mathscr{C}})_{/X_{i^{\prime}}}}$${\operatorname{\mathscr{D}}}$${\operatorname{Ind}(\operatorname{\mathscr{C}})_{/X_{i}}}$$\scriptstyle{\tilde{\varphi}^{\ast}}$$\scriptstyle{F_{i^{\prime}}}$$\scriptstyle{F_{i}}$$\scriptstyle{\rho_{\varphi}}$$\scriptstyle{\cong}$

Now observe that the natural isomorphism $\rho$ implies that in
$\operatorname{\mathscr{C}}$ there is an isomorphism of functors

$F_{i}(-)\cong(\tilde{\varphi}^{\ast}\circ
F_{i^{\prime}})(-)=F_{i^{\prime}}(-)\times_{X_{i^{\prime}}}X_{i}$

where the pullback is taken as in the diagram:

${F_{i^{\prime}}(-)\times_{X_{i^{\prime}}}X_{i}}$${F_{i^{\prime}}(-)}$${X_{i}}$${X_{i^{\prime}}}$$\scriptstyle{p_{0}}$$\scriptstyle{p_{1}}$$\scriptstyle{\tilde{\varphi}}$

Consequently taking the first projection of the pullback above and pre-
composing with the isomorphism $\rho$ gives a structure natural transformation

${F_{i}}$${F_{i^{\prime}}\times_{X_{i^{\prime}}}X_{i}}$${F_{i^{\prime}}}$$\scriptstyle{\cong}$$\scriptstyle{\hat{\varphi}}$$\scriptstyle{p_{0}}$

Taking the colimit of the $\hat{\varphi}$ for every object
$d\in\operatorname{\mathscr{D}}_{0}$, which exists in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ because $I$ is a filtered
category, produces a family of objects which we define by

$F(d):=\operatorname*{colim}_{i\in I}F_{i}(d).$

The universal property of the colimit allows us to extend this to a functor
$\tilde{F}:\operatorname{\mathscr{D}}\to\operatorname{Ind}(\operatorname{\mathscr{C}})$;
we claim that this extension naturally factors through the forgetful functor
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\operatorname{\mathfrak{X}}}\to\operatorname{Ind}(\operatorname{\mathscr{C}})$.
To see this, however, we simply use the universal property of the colimits in
sight to produce the commuting

${F(d)}$${F_{i}(d)}$${F_{i^{\prime}}(d)}$${X_{i}}$${X_{i^{\prime}}}$${\operatorname{\mathfrak{X}}}$$\scriptstyle{\hat{\varphi}_{d}}$$\scriptstyle{\beta_{i}}$$\scriptstyle{\beta_{i^{\prime}}}$$\scriptstyle{\alpha_{i}}$$\scriptstyle{\tilde{\varphi}}$$\scriptstyle{\alpha_{i^{\prime}}}$$\scriptstyle{\exists!\theta}$

diagram with $\theta$ defined by the universal property of $F(d)$. This gives
us the functor
$F:\operatorname{\mathscr{D}}\to\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\operatorname{\mathfrak{X}}}$.

We now claim that this functor $\theta$ allows us to factor the $2$-cells
$\rho_{\varphi}$ in the sense that there is a pasting diagram

${\operatorname{Ind}(\operatorname{\mathscr{C}})_{/X_{i^{\prime}}}}$${\operatorname{\mathscr{D}}}$${\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\operatorname{\mathfrak{X}}}}$${\operatorname{Ind}(\operatorname{\mathscr{C}})_{/X_{i}}}$$\scriptstyle{\tilde{\varphi}^{\ast}}$$\scriptstyle{F_{i^{\prime}}}$$\scriptstyle{F_{i}}$$\scriptstyle{\theta}$$\scriptstyle{\alpha_{i^{\prime}}^{\ast}}$$\scriptstyle{\alpha_{i}^{\ast}}$$\scriptstyle{\gamma}$$\scriptstyle{\cong}$$\scriptstyle{\delta}$$\scriptstyle{\cong}$$\scriptstyle{\phi_{\widetilde{\varphi},\alpha_{i^{\prime}}}}$$\scriptstyle{\cong}$

which pastes to $\rho_{\varphi}$. To see this we first will show the existence
of $\gamma$. To this end note that on one hand, for any
$d\in\operatorname{\mathscr{D}}_{0}$ we have

$(\alpha_{i}^{\ast}\circ\theta)(d)=F(d)\times_{\operatorname{\mathfrak{X}}}X_{i}.$

Now note that $X_{i}$ as a constant object is isomorphic in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ to the $I$-indexed object
$\underline{{X_{i^{\prime}}}}:I\to\operatorname{\mathscr{C}}$ where
$\underline{{X_{i^{\prime}}}}(x):=X_{i^{\prime}}$ for all objects and
$\underline{{X_{i^{\prime}}}}(\varphi)=\operatorname{id}_{X_{i^{\prime}}}$ for
all morphisms. Under this representation we get that
$X_{i^{\prime}}\cong\operatorname*{colim}_{I}\underline{{X_{i^{\prime}}}}(i)$
and so

$(\alpha_{i^{\prime}}^{\ast}\circ\theta)(d)=F(d)\times_{\operatorname{\mathfrak{X}}}X_{i}\cong
F(d)\times_{\operatorname{\mathfrak{X}}}\left(\operatorname*{colim}_{i\in
I}\underline{{X_{i^{\prime}}}}(i)\right)\cong\left(\operatorname*{colim}_{I}F_{i}(d)\right)\times_{\operatorname*{colim}_{i\in
I}X_{i}}\left(\operatorname*{colim}_{i\in
I}\underline{{X_{i^{\prime}}}}(i)\right).$

Using now that each filtered colimit in sight is taken over the same (filtered
indexing category and that filtered colimits commute naturally with pullbacks
in $\operatorname{Ind}(\operatorname{\mathscr{C}})$ we find

$\displaystyle(\alpha_{i^{\prime}}^{\ast}\circ\theta)(d)$
$\displaystyle\cong\left(\operatorname*{colim}_{I}F_{i}(d)\right)\times_{\operatorname*{colim}_{i\in
I}X_{i}}\left(\operatorname*{colim}_{i\in
I}\underline{{X_{i^{\prime}}}}(i)\right)\cong\operatorname*{colim}_{i\in
I}\left(F_{i}(d)\times_{X_{i}}\underline{{X_{i^{\prime}}}}(i)\right)=\operatorname*{colim}_{i\in
I}\left(F_{i}(d)\times_{X_{i}}X_{i^{\prime}}\right)$

Note that since there is a natural isomorphism $F_{i}(-)\cong
F_{i^{\prime}}(-)\times_{X_{i^{\prime}}}X_{i}$ whenever there is a morphism
$\tilde{\varphi}:i\to i^{\prime}$ in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$. Using that $I$ is filtered
and that we can compute the colimit after applying the $d$-component of this
natural isomorphism allows us to deduce that

$\operatorname*{colim}_{i\in
I}\left(F_{i}(d)\times_{X_{i}}X_{i^{\prime}}\right)\cong\operatorname*{colim}_{i\in
I}\left(\big{(}F_{i^{\prime}}(d)\times_{X_{i^{\prime}}}X_{i}\big{)}\times_{X_{i}}X_{i^{\prime}}\right)\cong\operatorname*{colim}_{i\in
I}\left(F_{i^{\prime}}(d)\right)\cong F_{i^{\prime}}(d)$

because the final colimit is taken in a variable which is not present in the
object $F_{i^{\prime}}(d)$. Note that each isomorphism presented is natural;
defining $\gamma$ to be the composite of these gives our desired $2$-cell:

${\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\operatorname{\mathfrak{X}}}}$${\operatorname{\mathscr{D}}}$${\operatorname{Ind}(\operatorname{\mathscr{C}})_{/X_{i^{\prime}}}}$$\scriptstyle{\alpha_{i^{\prime}}^{\ast}}$$\scriptstyle{\theta}$$\scriptstyle{F_{i^{\prime}}}$$\scriptstyle{\gamma}$$\scriptstyle{\cong}$

Establishing the existence of $\delta$ is similar to establishing that of
$\gamma$ and omitted. Finally the existence of the $2$-cell follows
immediately from the fact that
$\operatorname{\mathfrak{X}}\cong\operatorname*{colim}_{I}X_{i}$. That the
resulting pasting diagram pastes to $\rho_{\varphi}$ is also an extremely
tedious but straightforward argument that uses the fact that $\gamma$ and
$\delta$ involve $\rho_{\varphi}$ in establishing the natural isomorphism and
is also omitted.

We now finally establish the remaining property of
$\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\operatorname{\mathfrak{X}}}$
as a pseudolimit of the diagram of shape $G$. That is, assume that we have two
functors
$F,F^{\prime}:\operatorname{\mathscr{D}}\to\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\operatorname{\mathfrak{X}}}$
such that for all $i\in I_{0}$ there is a $2$-cell

${\operatorname{\mathscr{D}}}$${\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\operatorname{\mathfrak{X}}}}$${\operatorname{Ind}(\operatorname{\mathscr{C}})_{/\operatorname{\mathfrak{X}}}}$${\operatorname{Ind}(\operatorname{\mathscr{C}})_{/X_{i}}}$$\scriptstyle{F}$$\scriptstyle{F^{\prime}}$$\scriptstyle{\alpha_{i}^{\ast}}$$\scriptstyle{\alpha_{i}^{\ast}}$$\scriptstyle{\gamma_{i}}$

for which the diagram of functors and natural transformations, for any
$\varphi:i\to i^{\prime}$ in $I$,

${\tilde{\varphi}^{\ast}\circ\alpha_{i^{\prime}}^{\ast}\circ
F}$${\tilde{\varphi}^{\ast}\circ\alpha_{i^{\prime}}^{\ast}\circ
F^{\prime}}$${\alpha_{i}^{\ast}}$${\alpha_{i}^{\ast}\circ
F^{\prime}}$$\scriptstyle{\tilde{\varphi}^{\ast}\ast\gamma_{i^{\prime}}}$$\scriptstyle{\phi_{\tilde{\varphi},\alpha_{i^{\prime}}}\ast
F}$$\scriptstyle{\phi_{\tilde{\varphi},\alpha_{i^{\prime}}}\ast
F^{\prime}}$$\scriptstyle{\alpha_{i}^{\ast}\ast\gamma_{i}}$

commutes. We must establish that there is a unique $2$-cell $\gamma:F\to
F^{\prime}$ making $\alpha_{i}^{\ast}\ast\gamma=\gamma_{i}$ for all $i\in
I_{0}$. However, note that $\alpha_{i}^{\ast}\circ
F=F(-)\times_{\operatorname{\mathfrak{X}}}X_{i}$ and $\alpha_{i}^{\ast}\circ
F^{\prime}=F^{\prime}(-)\times_{\operatorname{\mathfrak{X}}}X_{i}$ for all
$i\in I_{0}$ and the conditions on the naturality of $\gamma_{i}$ and the
coherences regarding the interchange of the $\gamma_{i}$ with the pullbacks
$\tilde{\varphi}^{\ast}$ tell us that we have a uniquely determined natural
transformation

$\operatorname*{colim}_{I}\left(F(-)\times_{\operatorname{\mathfrak{X}}}X_{i}\right)\xrightarrow{\operatorname*{colim}_{I}(\gamma_{i})}\operatorname*{colim}_{I}\left(F^{\prime}(-)\times_{\operatorname{\mathfrak{X}}}X_{i}\right).$

Using the natural isomorphisms
$\operatorname*{colim}_{I}(F(-)\times_{\operatorname{\mathfrak{X}}}X_{i})\cong
F(-)$ and
$\operatorname*{colim}_{I}(F^{\prime}(-)\times_{\operatorname{\mathfrak{X}}}X_{i})$
induced by the fact that filtered colimits commute with pullbacks in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ gives our desired natural
transformation:

${\operatorname*{colim}_{I}\left(F(-)\times_{\operatorname{\mathfrak{X}}}X_{i}\right)}$${F(-)}$${\operatorname*{colim}_{I}\left(F^{\prime}(-)\times_{\operatorname{\mathfrak{X}}}X_{i}\right)}$${F^{\prime}(-)}$$\scriptstyle{\operatorname*{colim}_{I}(\gamma_{i})}$$\scriptstyle{\cong}$$\scriptstyle{\gamma}$$\scriptstyle{\cong}$

Finally that this satisfies $\alpha_{i}^{\ast}\ast\gamma=\gamma_{i}$ is
immediate by construction. ∎

###  4.2. Representable Ind-Tangent Categories

######  Definition 4.2 ([7, Definition 3]).

Let $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ be a Cartesian
closed tangent category. We say that $\operatorname{\mathscr{C}}$ is a
representable tangent category if there exists an object $D$ of
$\operatorname{\mathscr{C}}$ for which there is a natural isomorphism
$T(-)\cong[D,-]$.

Because of this we will first explain how the
$\operatorname{Ind}$-construction preserves adjoints.

######  Proposition 4.3.

Let $F\dashv G:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{D}}$ be an
adjunction. Then there is an adjunction
$\operatorname{Ind}(F)\dashv\operatorname{Ind}(G):\operatorname{Ind}(\operatorname{\mathscr{C}})\to\operatorname{Ind}(\operatorname{\mathscr{D}})$.

###### Proof.

Let $\underline{{X}}=(X_{i})_{i\in I}$ be an object of
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ and let
$\underline{{Y}}=(Y_{j})_{j\in J}$ be an object of
$\operatorname{Ind}(\operatorname{\mathscr{D}})$. Then because $F\dashv G$,
$\operatorname{\mathscr{C}}(X_{i},GY_{j})\cong\operatorname{\mathscr{D}}(FX_{i},Y_{j})$
for all $i\in I_{0}$ and all $j\in J_{0}$. Then we get that

$\displaystyle\operatorname{Ind}(\operatorname{\mathscr{C}})(\underline{{X}},\operatorname{Ind}(G)\underline{{Y}})$
$\displaystyle=\operatorname{Ind}(\operatorname{\mathscr{C}})((X_{i}),(GY_{j}))=\lim_{\begin{subarray}{c}\longleftarrow\\\
i\in I\end{subarray}}\left(\lim_{\begin{subarray}{c}\longrightarrow\\\ j\in
J\end{subarray}}\operatorname{\mathscr{C}}(X_{i},GY_{j})\right)$
$\displaystyle\cong\lim_{\begin{subarray}{c}\longleftarrow\\\ i\in
I\end{subarray}}\left(\lim_{\begin{subarray}{c}\longrightarrow\\\ j\in
J\end{subarray}}\operatorname{\mathscr{D}}(FX_{i},Y_{j})\right)=\operatorname{Ind}(\operatorname{\mathscr{D}})((FX_{i}),(Y_{j}))=\operatorname{Ind}(\operatorname{\mathscr{D}})(\operatorname{Ind}(F)\underline{{X}},\underline{{Y}}).$

Thus
$\operatorname{Ind}(F)\dashv\operatorname{Ind}(G):\operatorname{Ind}(\operatorname{\mathscr{C}})\to\operatorname{Ind}(\operatorname{\mathscr{D}})$.
∎

We can then use this to expand internal hom functors on Cartesian closed
categories to the $\operatorname{Ind}$-category at least when the co-
representing object $\underline{{X}}$ is constant, i.e., when
$\underline{{X}}=(X)_{\ast\in\\{\ast\\}}$.

######  Proposition 4.4.

Let $\operatorname{\mathscr{C}}$ be a for which there is an object
$X\in\operatorname{\mathscr{C}}_{0}$ such that $\operatorname{\mathscr{C}}$
admits a product functor $(-)\times
X:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{C}}$ and there is an
internal hom functor
$[X,-]:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{C}}$ with
$(-)\times X\dashv[X,-]$. Then if $\underline{{X}}=(X)_{\ast\in\\{\ast\\}}$ is
the constant object at $X$ in $\operatorname{Ind}(\operatorname{\mathscr{C}})$
the internal hom functor
$[\underline{{X}},-]:\operatorname{Ind}(\operatorname{\mathscr{C}})\to\operatorname{Ind}(\operatorname{\mathscr{C}})$
exists and is right adjoint to the product functor $(-)\times\underline{{X}}$.

###### Proof.

That $\operatorname{Ind}((-)\times X)\dashv\operatorname{Ind}([X,-])$ is a
consequence of Proposition 4.3. Because there is a natural isomorphism of
functors

$\operatorname{Ind}((-)\times X)\cong(-)\times\underline{{X}}$

we get that $(-)\times\underline{{X}}\dashv\operatorname{Ind}([X,-])$ and so
the internal hom exists. The final statement about the functor
$[\underline{{X}},-]$ being isomorphic to $\operatorname{Ind}([X,-])$ follows
from the fact that for any $(Y_{i})_{i\in I}$ and any $(Z_{j})_{j\in J}$ in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$,

$\displaystyle\operatorname{Ind}(\operatorname{\mathscr{C}})\big{(}(Y_{i}),\left[\underline{{X}},(Z_{j})\right]\big{)}$
$\displaystyle=\lim_{\begin{subarray}{c}\longleftarrow\\\ i\in
I\end{subarray}}\left(\lim_{\begin{subarray}{c}\longrightarrow\\\ j\in
J\end{subarray}}\operatorname{\mathscr{C}}(Y_{i},[X,Z_{j}])\right)\cong\lim_{\begin{subarray}{c}\longleftarrow\\\
i\in I\end{subarray}}\left(\lim_{\begin{subarray}{c}\longrightarrow\\\ j\in
J\end{subarray}}\operatorname{\mathscr{C}}(Y_{i}\times X,Z_{j})\right)$
$\displaystyle=\operatorname{Ind}(\operatorname{\mathscr{C}})\left((Y_{j})\times\underline{{X}},(Z_{j})\right)$

which gives the desired isomorphism
$[\underline{{X}},-]\cong\operatorname{Ind}([X,-]).$ ∎

######  Proposition 4.5.

Let $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ be a
representable tangent category. Then there is an object $\underline{{D}}$ in
the $\operatorname{Ind}$-tangent category
$(\operatorname{Ind}(\operatorname{\mathscr{C}}),\operatorname{Ind}(\operatorname{\mathbb{T}}))$
for which $\operatorname{Ind}(T)\cong[\underline{{D}},-]$.

###### Proof.

Since $\operatorname{\mathscr{C}}$ is a representable tangent category,
$\operatorname{\mathscr{C}}$ is Cartesian closed and there is an object
$D\in\operatorname{\mathscr{C}}_{0}$ for which $[D,-]\cong T$. It then follows
from Proposition 4.4 that the internal hom functor
$\operatorname{Ind}([D,-])\cong[\underline{{D}},-]$ exists. A routine
calculation shows

$[\underline{{D}},-]\cong\operatorname{Ind}([D,-])\cong\operatorname{Ind}(T)$

which in turn proves the proposition. ∎

######  Remark 4.6.

Proposition 4.5 technically holds in more general (formal) situations than
merely for Cartesian closed categories $\operatorname{\mathscr{C}}$. If there
were a tangent category
$(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ with an object
$D\in\operatorname{\mathscr{C}}_{0}$ for which there is an internal hom
functor and natural isomorphism $[D,-]\cong T$, then there is a natural
isomorphism $[\underline{{D}},-]\cong\operatorname{Ind}(T)$ as well. I am
unaware, however, of tangent categories
$(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ with this property
that are not Cartesian closed555And in fact I would be very interested in
seeing such an example which arises “in nature” as it were..

###  4.3. Differential Bundles and Ind-Tangent Categories

One of the most important structures we can encounter in the theory of tangent
cateogries is that of differential bundles. These describe objects $E$ over
some base object $M$ together with a structure map $q:E\to M$ that make $E$
look “locally like” they are given by tangent bundles. While these are of
course related to vector bundles in algebraic geometry and differential
geometry, they are also related to linear logic and allow a clean description
of Cartesian differential categories (cf. [4, Section 3.5]) In this subsection
we will study how the $\operatorname{Ind}$-construction translates
pseudofunctors and describe the cases where we can say that an object
$\underline{{E}}\to\underline{{M}}$ in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ are “locally” determined by
differential bundles in $\operatorname{\mathscr{C}}$. Consequently we
necessitate a short review of differential bundles and differential objects in
tangent categories.

######  Definition 4.7 ([3, Definition 2.1, 2.2]).

Let $\operatorname{\mathscr{C}}$ be a category. An additive bundle in
$\operatorname{\mathscr{C}}$ is a morphism $p:A\to M$ such that $p:A\to M$ is
a commutative monoid in $\operatorname{\mathscr{C}}_{/M}$.

A morphism of additive bundles $(p:A\to M,\alpha:A\times_{M}A\to A,\zeta:M\to
A)$ and $(q:B\to N,\mu:B\times_{N}B\to B,\eta:N\to B)$ is a pair of maps
$f:A\to B$ and $g:M\to N$ such that the diagrams

${A}$${B}$${M}$${N}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{g}$
${A\times_{M}A}$${B\times_{N}B}$${A}$${B}$$\scriptstyle{\langle
f\circ\pi_{0},f\circ\pi_{1}\rangle}$$\scriptstyle{\alpha}$$\scriptstyle{\mu}$$\scriptstyle{f}$
${M}$${N}$${A}$${B}$$\scriptstyle{g}$$\scriptstyle{\zeta}$$\scriptstyle{\eta}$$\scriptstyle{f}$

commute.

######  Definition 4.8 ([4, Definition 2.3]).

Let $(\operatorname{\mathscr{C}},\operatorname{\mathbb{T}})$ be a tangent
category. A differential bundle in $\operatorname{\mathscr{C}}$ is a quadruple
$q=(q,\sigma,\zeta,\lambda)$ where:

  * •

$q:E\to M$ is a morphism which is a commutative monoid internal to
$\operatorname{\mathscr{C}}_{/M}$ with addition map $\sigma:E_{2}\to E$ and
unit map $\zeta:M\to E$;

  * •

$\lambda:E\to TE$ is a morphism called the lift of $q$;

  * •

For any $n\in\operatorname{\mathbb{N}}$, the pullback powers
$E_{n}=E\times_{M}\cdots\times_{M}E$ exist in $\operatorname{\mathscr{C}}$ for
all $n\in\operatorname{\mathbb{N}}$ and for each
$m\in\operatorname{\mathbb{N}}$ the functor $T^{m}$ preserves these pullbacks;

  * •

The pair $(\lambda,0):(q,\sigma,\zeta,\lambda)\to(Tq,T\sigma,T\zeta,T\lambda)$
is an additive bundle map;

  * •

The pair $(\lambda,\zeta):(q,\sigma,\zeta,\lambda)\to(p,+,0,\ell)$ is an
additive bundle morphism;

  * •

The lift $\lambda$ is universal in the sense that if $\mu$ is the morphism

${E_{2}}$${TE}$${T(E_{2})}$$\scriptstyle{\mu}$$\scriptstyle{\langle\lambda\circ\pi_{0},0\circ\pi_{1}\rangle}$$\scriptstyle{T\sigma}$

then the diagram below is a pullback which is preserved by $T^{n}$ for all
$n\in\operatorname{\mathbb{N}}$:

${E_{2}}$${TE}$${M}$${TM}$$\scriptstyle{\mu}$$\scriptstyle{q\circ\pi_{1}}$$\scriptstyle{Tq}$$\scriptstyle{0}$

  * •

The equation $\ell\circ\lambda=T(\lambda)\circ\lambda$ holds.

######  Definition 4.9 ([4, Definition 2.3]).

A morphism of differential bundles from $(q:E\to M,\sigma,\zeta,\lambda)$ to
$(q^{\prime}:F\to N,\mu,\eta,\rho)$ is a pair of morphisms $f:E\to F$ and
$g:M\to N$ such that the diagram

${E}$${F}$${M}$${N}$$\scriptstyle{f}$$\scriptstyle{q}$$\scriptstyle{q^{\prime}}$$\scriptstyle{g}$

commutes. Additionally, we say that the morphism $(f,g)$ is linear if it
preserves the lift in the sense that the diagram

${E}$${F}$${T(E)}$${T(F)}$$\scriptstyle{f}$$\scriptstyle{\lambda}$$\scriptstyle{\rho}$$\scriptstyle{Tf}$

commutes.

######  Remark 4.10.

In the definition of a morphism of differential bundles, the pair $(f,g)$ need
not be an additive bundle map, i.e., $(f,g)$ need not preserve the monoidal
operations of $q:A\to M$. However, by [4, Proposition 2.16] it does follow
that if the bundle map is _linear_ then it is also an additive bundle map.

We now show the first preservation result of the subsection: namely that two
ind-objects indexed by the same filtered category which are locally
differential bundles with linear transition maps give rise to an ind-
differential bundle.

######  Proposition 4.11.

Let $I$ be a filtered index category and let $\operatorname{\mathscr{C}}$ be a
tangent category. If $\underline{{E}}=(E_{i})_{i\in I}$ and
$\underline{{M}}=(M_{i})_{i\in I}$ are $\operatorname{Ind}$-objects such that:

  * •

For each $i\in I_{0}$, $E_{i}$ and $M_{i}$ there is a differential bundle
$(q_{i}:E_{i}\to M_{i},\sigma_{i}:E_{i}\times_{M_{i}}E_{i}\to
E_{i},\zeta_{i}:M_{i}\to E_{i},\lambda_{i}:E_{i}\to TE_{i})$;

  * •

For each map $\varphi:i\to i^{\prime}$ in $I$, if $\tilde{\varphi}:E_{i}\to
E_{i^{\prime}}$ and $\widehat{\varphi}:M_{i}\to M_{i^{\prime}}$ denote the
corresponding structure maps in $\underline{{E}}$ and $\underline{{M}}$ then
$(\tilde{\varphi},\widehat{\varphi}):(q_{i},\sigma_{i},\zeta_{i},\lambda_{i})\to(q_{i^{\prime}},\sigma_{i^{\prime}},\zeta_{i^{\prime}},\lambda_{i^{\prime}})$
is a linear morphism of differential bundles;

then there is a differential bundle
$(\underline{{q}},\underline{{\sigma}},\underline{{\zeta}},\underline{{\lambda}})$
in $\operatorname{Ind}(\operatorname{\mathscr{C}})$.

###### Proof.

First note that the condition that there be a bundle map $q_{i}:E_{i}\to
M_{i}$ for all $i\in I$ and that the maps
$(\tilde{\varphi},\widehat{\varphi})$ being a linear differential bundle map
mean that the diagrams

${E_{i}}$${E_{i^{\prime}}}$${M_{i}}$${M_{i^{\prime}}}$$\scriptstyle{\tilde{\varphi}}$$\scriptstyle{q_{i}}$$\scriptstyle{q_{i^{\prime}}}$$\scriptstyle{\widehat{\varphi}}$
${E_{i}}$${E_{i^{\prime}}}$${TE_{i}}$${TE_{i^{\prime}}}$$\scriptstyle{\tilde{\varphi}}$$\scriptstyle{\lambda_{i}}$$\scriptstyle{\lambda_{i^{\prime}}}$$\scriptstyle{T\tilde{\varphi}}$

all commute for every morphism $\varphi:i\to i^{\prime}$ in $I$. This allows
us to deduce that there are maps $(q_{i})_{i\in
I}=:\underline{{E}}\to\underline{{M}}$ and $(\lambda_{i})_{i\in
I}=:\underline{{\lambda}}:\underline{{E}}\to\operatorname{Ind}(T)\underline{{E}}$
in $\operatorname{Ind}(\operatorname{\mathscr{C}})$. Similarly, by [4,
Proposition 2.16] it follows that each pair
$(\tilde{\varphi},\widehat{\varphi})$ is an additive morphism of bundles which
implies that the diagrams

${E_{i}\times_{M_{i}}E_{i}}$${E_{i^{\prime}}\times_{M_{i^{\prime}}}E_{i^{\prime}}}$${E_{i}}$${E_{i^{\prime}}}$$\scriptstyle{\langle\tilde{\varphi}\circ\pi_{0},\tilde{\varphi}\circ\pi_{1}\rangle}$$\scriptstyle{\sigma_{i}}$$\scriptstyle{\sigma_{i^{\prime}}}$$\scriptstyle{\tilde{\varphi}}$
${M_{i}}$${M_{i^{\prime}}}$${E_{i}}$${E_{i^{\prime}}}$$\scriptstyle{\widehat{\varphi}}$$\scriptstyle{\zeta_{i}}$$\scriptstyle{\zeta_{i^{\prime}}}$$\scriptstyle{\tilde{\varphi}}$

commute. Since
$\underline{{E}}\times_{\underline{{M}}}\underline{{E}}=(E_{i}\times_{M_{i}}E_{i})_{i\in
I}$, this allows us to deduce that the maps $(\sigma_{i})_{i\in
I}=:\underline{{\sigma}}:\underline{{E}}\times_{\underline{{M}}}\underline{{E}}\to\underline{{E}}$
and $(\zeta_{i})_{i\in
I}=:\underline{{\zeta}}:\underline{{M}}\to\underline{{E}}$ exist in
$\operatorname{Ind}(\operatorname{\mathscr{C}})$ as well. This defines our
quadruple
$(\underline{{q}},\underline{{\sigma}},\underline{{\zeta}},\underline{{\lambda}})$
which we will now show is a differential bundle in an axiom-by-axiom
verification.

First note that the pullbacks
$\underline{{E}}_{n}:=\underline{{E}}\times_{\underline{{M}}}\cdots\times_{\underline{{M}}}\underline{{E}}$
exist by an argument mutatis mutandis to the argument which showed why tangent
pullbacks exist in Proposition 3.16, i.e., the object
$\underline{{X}}=(E_{i}\times_{M_{i}}\times_{M_{i}}\cdots\times_{M_{i}}E_{i})_{i\in
I}$ represents the pullback $\underline{{E}}_{n}$. The argument as to why the
compositional powers $\operatorname{Ind}(T)^{m}$ preserve these pullbacks is
similar to the argument why $\operatorname{Ind}(T)^{m}$ preserves tangent
pullbacks given in Corollary 3.18.

To show that
$(\underline{{\lambda}},\operatorname{Ind}(0)_{\underline{{M}}}):(\underline{{q}},\underline{{\sigma}},\underline{{\zeta}},\underline{{\lambda}})\to(\operatorname{Ind}(T)\underline{{q}},\operatorname{Ind}(T)(\underline{{\sigma}}),\operatorname{Ind}(T)(\underline{{\zeta}}),\operatorname{Ind}(T)(\underline{{\lambda}}))$
is an additive bundle morphism we need to show that the diagrams

${\underline{{E}}}$${\operatorname{Ind}(T)\underline{{E}}}$${\underline{{M}}}$${\operatorname{Ind}(T)\underline{{M}}}$$\scriptstyle{\underline{{\lambda}}}$$\scriptstyle{\underline{{q}}}$$\scriptstyle{\operatorname{Ind}(T)\underline{{q}}}$$\scriptstyle{\operatorname{Ind}(0)_{\underline{{M}}}}$
${\underline{{E}}\times_{\underline{{M}}}\underline{{E}}}$${\operatorname{Ind}(T)\underline{{E}}\times_{\operatorname{Ind}(T)\underline{{M}}}\operatorname{Ind}(T)\underline{{E}}}$${\underline{{E}}}$${\operatorname{Ind}(T)\underline{{E}}}$$\scriptstyle{\langle\underline{{\lambda}}\circ\pi_{0},\underline{{\lambda}}\circ\pi_{1}\rangle}$$\scriptstyle{\underline{{\sigma}}}$$\scriptstyle{\operatorname{Ind}(T)\underline{{\sigma}}}$$\scriptstyle{\underline{{\lambda}}}$
${\underline{{M}}}$${\operatorname{Ind}(T)\underline{{M}}}$${\underline{{E}}}$${\operatorname{Ind}(T)\underline{{E}}}$$\scriptstyle{\operatorname{Ind}(0)_{\underline{{M}}}}$$\scriptstyle{\underline{{\zeta}}}$$\scriptstyle{\operatorname{Ind}(T)\underline{{\zeta}}}$$\scriptstyle{\underline{{\lambda}}}$

all commute. However, as each of the diagrams

${{E}_{i}}$${TE_{i}}$${M_{i}}$${TM_{i}}$$\scriptstyle{\underline{{\lambda}}}$$\scriptstyle{\underline{{q}}}$$\scriptstyle{Tq_{i}}$$\scriptstyle{0_{{M}_{i}}}$
${E_{i}\times_{M_{i}}E_{i}}$${TE_{i}\times_{TM_{i}}TE_{i}}$${E_{i}}$${TE_{i}}$$\scriptstyle{\langle\lambda_{i}\circ\pi_{0},\lambda_{i}\circ\pi_{1}\rangle}$$\scriptstyle{\sigma_{i}}$$\scriptstyle{T\sigma_{i}}$$\scriptstyle{\lambda_{i}}$
${M_{i}}$${TM_{i}}$${E_{i}}$${TE_{i}}$$\scriptstyle{0_{M_{i}}}$$\scriptstyle{{\zeta}_{i}}$$\scriptstyle{T{\zeta}_{i}}$$\scriptstyle{{\lambda}_{i}}$

commutes for each $i\in I_{0}$, and each given pullback represents the
$i$-component of the corresponding $\operatorname{Ind}$-object it follows that
the diagrams between $\operatorname{Ind}$-objects commute. Thus
$(\underline{{\lambda}},\operatorname{Ind}(0)_{\underline{{M}}})$ is an
additive bundle map. Establishing that
$(\underline{{\lambda}},\underline{{\zeta}})$ is an additive bundle morphism
is done similarly and hence omitted.

For the universality of the lift $\underline{{\lambda}}$ we note that each of
the maps in the desired pullback are defined $I$-locally. Consequently we can
deduce that the pullbacks exist and are preserved by all powers
$\operatorname{Ind}(T)^{m}$ follows similarly to our verification of the
existence and preservation of $\underline{{E}}_{n}$.