††thanks: The authors acknowledge with gratitude the support of an Australian
Research Council Discovery Project DP190102432. The first author is a JSPS
Overseas Research Fellow.

18D20, 18N10 s.fujii.math<EMAIL_ADDRESS>
# The oplax limit of an enriched category

Soichiro Fujii and Stephen Lack  School of Mathematical and Physical Sciences,
Macquarie University, NSW 2109, Australia

(2022)

###### Abstract

We show that 2-categories of the form $\mathscr{B}\mbox{-}\mathbf{Cat}$ are
closed under slicing, provided that we allow $\mathscr{B}$ to range over
bicategories (rather than, say, monoidal categories). That is, for any
$\mathscr{B}$-category $\mathbb{X}$, we define a bicategory
$\mathscr{B}/\mathbb{X}$ such that
$\mathscr{B}\mbox{-}\mathbf{Cat}/\mathbb{X}\cong(\mathscr{B}/\mathbb{X})\mbox{-}\mathbf{Cat}$.
The bicategory $\mathscr{B}/\mathbb{X}$ is characterized as the oplax limit of
$\mathbb{X}$, regarded as a lax functor from a chaotic category to
$\mathscr{B}$, in the 2-category $\mathbf{BICAT}$ of bicategories, lax
functors and icons. We prove this conceptually, through limit-preservation
properties of the 2-functor $\mathbf{BICAT}\to 2\mbox{-}\mathbf{CAT}$ which
maps each bicategory $\mathscr{B}$ to the 2-category
$\mathscr{B}\mbox{-}\mathbf{Cat}$. When $\mathscr{B}$ satisfies a mild local
completeness condition, we also show that the isomorphism
$\mathscr{B}\mbox{-}\mathbf{Cat}/\mathbb{X}\cong(\mathscr{B}/\mathbb{X})\mbox{-}\mathbf{Cat}$
restricts to a correspondence between fibrations in
$\mathscr{B}\mbox{-}\mathbf{Cat}$ over $\mathbb{X}$ on the one hand, and
$\mathscr{B}/\mathbb{X}$-categories admitting certain powers on the other.

###### keywords:

Enriched categories, bicategories

##  1 Introduction

It is well-known that for any monoidal category $\mathscr{V}$ and monoid
$M=(M,e\colon I\to M,$ $m\colon M\otimes M\to M)$ therein, the slice category
$\mathscr{V}/M$ has a canonical monoidal structure; the unit is $e$ and the
monoidal product of objects $(s\colon S\to M)$ and $(t\colon T\to M)$ is

$S\otimes T$$M\otimes M$$M$.$s\otimes t$$m$

Moreover, there is a canonical isomorphism of categories

$\mathbf{Mon}(\mathscr{V}/M)\cong\mathbf{Mon}(\mathscr{V})/M.$

This paper originated from a natural generalization of this, replacing the
notion of monoid in $\mathscr{V}$ by that of $\mathscr{V}$-category. That is,
for any $\mathscr{V}$-category $\mathbb{X}$, there is an appropriate “base”
$\mathscr{V}/\mathbb{X}$ admitting a canonical isomorphism of 2-categories

${(\mathscr{V}/\mathbb{X})}\mathchar
45\relax\mathbf{Cat}\cong{\mathscr{V}}\mathchar
45\relax\mathbf{Cat}/\mathbb{X}.$ (1)

Here, the “base” $\mathscr{V}/\mathbb{X}$ is in general not a monoidal
category but a bicategory. Enriched category theory over bicategories is
developed in, e.g., [BCSW83, Str05]. We recall that, for a bicategory
$\mathscr{B}$, a $\mathscr{B}$-category $\mathbb{X}$ is given by

  * •

a set $\mathrm{ob}(\mathbb{X})$;

  * •

a function $|-|\colon\mathrm{ob}(\mathbb{X})\to\mathrm{ob}(\mathscr{B})$
($|x|$ is called the _extent_ of $x$);

  * •

for each $x,x^{\prime}\in\mathrm{ob}(\mathbb{X})$, a 1-cell
$\mathbb{X}(x,x^{\prime})\colon|x|\to|x^{\prime}|$ in $\mathscr{B}$;

  * •

for each $x\in\mathrm{ob}(\mathbb{X})$, a 2-cell

$|x|$$|x|$$1_{|x|}$$\mathbb{X}(x,x)$$j_{x}$

in $\mathscr{B}$, where $1_{|x|}$ is the identity 1-cell on $|x|$; and

  * •

for each $x,x^{\prime},x^{\prime\prime}\in\mathrm{ob}(\mathbb{X})$, a 2-cell

$|x|$$|x^{\prime}|$$|x^{\prime\prime}|$$\mathbb{X}(x,x^{\prime})$$\mathbb{X}(x^{\prime},x^{\prime\prime})$$\mathbb{X}(x,x^{\prime\prime})$$M_{x,x^{\prime},x^{\prime\prime}}$

in $\mathscr{B}$,

subject to the associativity and identity laws, generalizing the usual axioms
for a category.

Since the isomorphism (1) already forces us to consider enrichment over
bicategories, it is natural to wonder whether there is a generalization of the
isomorphism involving a bicategory $\mathscr{B}$ in place of the monoidal
category $\mathscr{V}$. Indeed this turns out to be the case: for any
bicategory $\mathscr{B}$ and $\mathscr{B}$-category $\mathbb{X}$, there is a
bicategory $\mathscr{B}/\mathbb{X}$ with a canonical isomorphism of
2-categories ${(\mathscr{B}/\mathbb{X})}\mathchar
45\relax\mathbf{Cat}\cong{\mathscr{B}}\mathchar
45\relax\mathbf{Cat}/\mathbb{X}$. Thus 2-categories of the form
${\mathscr{B}}\mathchar 45\relax\mathbf{Cat}$ are closed under slicing,
provided that we allow $\mathscr{B}$ to range over bicategories.

The construction of $\mathscr{B}/\mathbb{X}$ is simple enough to carry out at
this point; see also Remark 4.8 for a more abstract point of view. We set
$\mathrm{ob}(\mathscr{B}/\mathbb{X})=\mathrm{ob}(\mathbb{X})$ and, for each
$x,x^{\prime}\in\mathrm{ob}(\mathscr{B}/\mathbb{X})$, the hom-category
$(\mathscr{B}/\mathbb{X})(x,x^{\prime})$ is the slice category
$\mathscr{B}(|x|,|x^{\prime}|)/\mathbb{X}(x,x^{\prime})$. The identity 1-cell
at $x$ is $j_{x}$, and the composite of 1-cells $(s\colon
S\to\mathbb{X}(x,x^{\prime}))\colon x\to x^{\prime}$ and $(t\colon
T\to\mathbb{X}(x^{\prime},x^{\prime\prime}))\colon x^{\prime}\to
x^{\prime\prime}$ is the pasting composite

$|x|$$|x^{\prime}|$$|x^{\prime\prime}|$.$S$$T$$\mathbb{X}(x,x^{\prime})$$\mathbb{X}(x^{\prime},x^{\prime\prime})$$\mathbb{X}(x,x^{\prime\prime})$$M_{x,x^{\prime},x^{\prime\prime}}$$s$$t$

Of course, when both $\mathscr{B}$ and $\mathbb{X}$ have only one object, the
construction of $\mathscr{B}/\mathbb{X}$ reduces to that of the slice of a
monoidal category over a monoid.

This observation allows one to view (enriched) functors as (enriched)
categories, and suggests new perspectives even on notions which are not
directly related to enrichment. For example, for any ($\mathbf{Set}$-)category
$\mathbb{X}$, there is a bicategory $\mathbf{Set}/\mathbb{X}$ with an
isomorphism ${(\mathbf{Set}/\mathbb{X})}\mathchar
45\relax\mathbf{Cat}\cong\mathbf{Cat}/\mathbb{X}$. So we can view functors
into $\mathbb{X}$ as enriched categories (see Example 4.6 below and [Gar14]
for a related construction), and we may potentially interpret properties of
functors via enriched categorical terms. Indeed, we shall show that a functor
$\mathbb{Y}\to\mathbb{X}$ is a Grothendieck fibration if and only if the
corresponding $\mathbf{Set}/\mathbb{X}$-category $\overline{\mathbb{Y}}$ has
powers by a certain class of 1-cells in $\mathbf{Set}/\mathbb{X}$, as well as
a $\mathscr{B}$-enriched version of this result.

The notation $\mathscr{B}/\mathbb{X}$ is justified by its characterization as
the oplax limit of a 1-cell in a suitable 2-category. To explain this, recall
that a $\mathscr{B}$-category $\mathbb{X}$ can be given equivalently as a lax
functor $\mathbb{X}\colon X_{c}\to\mathscr{B}$, where $X_{c}$ is the chaotic
category with the same set of objects as $\mathbb{X}$.111Lax functors of this
form were studied by Bénabou [Bén67] under the name polyad; for the connection
with enriched categories see [Str05]. So we can view the
$\mathscr{B}$-category $\mathbb{X}$ as a 1-cell in the 2-category
$\mathbf{BICAT}$ of bicategories, lax functors and icons [Lac10]. The
bicategory $\mathscr{B}/\mathbb{X}$ is the oplax limit of this 1-cell in
$\mathbf{BICAT}$:

$\mathscr{B}/\mathbb{X}$$X_{c}$$\mathscr{B}$.$\mathbb{X}$

(Although $\mathbf{BICAT}$ is not complete, it does have oplax limits of
1-cells [Lac05, LS12].) This generalizes the characterization of the slice
monoidal category $\mathscr{V}/M$ as the oplax limit of the monoid $M$ in
$\mathscr{V}$, regarded as a lax monoidal functor from the terminal monoidal
category to $\mathscr{V}$, in the 2-category of monoidal categories, lax
monoidal functors and monoidal natural transformations.

In this paper, we study properties of the 2-functor
$\mathrm{Enr}\colon\mathbf{BICAT}\to 2\mathchar 45\relax\mathbf{CAT}$ mapping
each bicategory $\mathscr{B}$ to the 2-category ${\mathscr{B}}\mathchar
45\relax\mathbf{Cat}$, in order to understand the isomorphism
${(\mathscr{B}/\mathbb{X})}\mathchar
45\relax\mathbf{Cat}\cong{\mathscr{B}}\mathchar
45\relax\mathbf{Cat}/\mathbb{X}$ conceptually, as well as to establish further
closure properties of 2-categories of the form ${\mathscr{B}}\mathchar
45\relax\mathbf{Cat}$. To this end, it is useful to factorize $\mathrm{Enr}$
as

$\mathbf{BICAT}$$2\mathchar
45\relax\mathbf{CAT}/\mathrm{Enr}(\mathbf{1})$$2\mathchar
45\relax\mathbf{CAT}$,$\mathrm{Enr}_{\mathbf{1}}$$\mathrm{Enr}$forgetful

where $\mathbf{1}$ is the terminal bicategory. The 2-functor
$\mathrm{Enr}_{\mathbf{1}}$ maps each bicategory $\mathscr{B}$ to
${\mathscr{B}}\mathchar 45\relax\mathbf{Cat}$ equipped with the 2-functor
$\mathrm{Enr}(!)\colon{\mathscr{B}}\mathchar
45\relax\mathbf{Cat}\to\mathrm{Enr}(\mathbf{1})$ induced from the unique lax
functor $!\colon\mathscr{B}\to\mathbf{1}$. The underlying category of
$\mathrm{Enr}(\mathbf{1})$ is $\mathbf{Set}$, and $\mathrm{Enr}(!)$ can be
regarded as $\mathrm{ob}(-)$, mapping each $\mathscr{B}$-category $\mathbb{X}$
to its set of objects $\mathrm{ob}(\mathbb{X})$. (Although $\mathrm{Enr}$ is
usually denoted simply as ${(-)}\mathchar 45\relax\mathbf{Cat}$, we adopted
the current notation in order to avoid the potentially misleading expression
${\mathbf{1}}\mathchar 45\relax\mathbf{Cat}$.)

In our main theorem (Theorem 2.1), we show that
$\mathrm{Enr}_{\mathbf{1}}\colon\mathbf{BICAT}\to 2\mathchar
45\relax\mathbf{CAT}/\mathrm{Enr}(\mathbf{1})$ preserves _any_ limit which
happens to exist in $\mathbf{BICAT}$. This implies that $\mathrm{Enr}$
preserves any limit which happens to exist in $\mathbf{BICAT}$ _and_ is
created by the forgetful 2-functor $2\mathchar
45\relax\mathbf{CAT}/\mathrm{Enr}(\mathbf{1})\to 2\mathchar
45\relax\mathbf{CAT}$. In ordinary category theory, the limits created by the
forgetful functors from slice categories are precisely the connected limits.
In Section 3 we generalize this to 2-categories (or in fact to
$\mathscr{V}$-categories where $\mathscr{V}$ is any complete and cocomplete
cartesian closed category), introducing the class of $\mathbf{Cat}$-connected
limits with several characterizations. So $\mathrm{Enr}\colon\mathbf{BICAT}\to
2\mathchar 45\relax\mathbf{CAT}$ preserves any $\mathbf{Cat}$-connected limit
which happens to exist in $\mathbf{BICAT}$. This includes Eilenberg–Moore
objects of comonads, for example. Although oplax limits of 1-cells are not
$\mathbf{Cat}$-connected, the isomorphism ${(\mathscr{B}/\mathbb{X})}\mathchar
45\relax\mathbf{Cat}\cong{\mathscr{B}}\mathchar
45\relax\mathbf{Cat}/\mathbb{X}$ is explained via the limit-preservation
property of $\mathrm{Enr}$ and a 2-categorical argument in Section 4.

Finally, in Section 5, we investigate (internal) fibrations in the 2-category
${\mathscr{B}}\mathchar 45\relax\mathbf{Cat}$ of $\mathscr{B}$-categories.
Specifically, we show that (assuming a mild local completeness condition on
$\mathscr{B}$) a $\mathscr{B}$-functor $\mathbb{Y}\to\mathbb{X}$ is a
fibration in ${\mathscr{B}}\mathchar 45\relax\mathbf{Cat}$ if and only if the
corresponding $\mathscr{B}/\mathbb{X}$-category $\overline{\mathbb{Y}}$ admits
certain powers.

##  2 The limit-preservation theorem

The issue of size is not essential in this paper, but for the sake of
definiteness we make the following conventions. We fix two Grothendieck
universes $\mathcal{U}_{0}$ and $\mathcal{U}_{1}$ with
$\mathcal{U}_{0}\in\mathcal{U}_{1}$. Sets, categories, etc. in
$\mathcal{U}_{0}$ and $\mathcal{U}_{1}$ are called _small_ and _large_
respectively.

Let $\mathbf{BICAT}$ be the 2-category of large bicategories, lax functors and
icons [Lac10, Theorem 3.2], and $2\mathchar 45\relax\mathbf{CAT}$ be the
2-category of large 2-categories, 2-functors and 2-natural transformations. We
have a 2-functor $\mathrm{Enr}\colon\mathbf{BICAT}\to 2\mathchar
45\relax\mathbf{CAT}$ sending each bicategory $\mathscr{B}$ to the 2-category
${\mathscr{B}}\mathchar 45\relax\mathbf{Cat}$ of all small
$\mathscr{B}$-categories, $\mathscr{B}$-functors and $\mathscr{B}$-natural
transformations. The bicategory $\mathbf{1}$ with a single 2-cell is the
terminal object of $\mathbf{BICAT}$, and hence $\mathrm{Enr}$ induces the
2-functor $\mathrm{Enr}_{\mathbf{1}}\colon\mathbf{BICAT}\to 2\mathchar
45\relax\mathbf{CAT}/\mathrm{Enr}(\mathbf{1})$, where $2\mathchar
45\relax\mathbf{CAT}/\mathrm{Enr}(\mathbf{1})$ denotes the (strict) slice
2-category of $2\mathchar 45\relax\mathbf{CAT}$ over
$\mathrm{Enr}(\mathbf{1})\in 2\mathchar 45\relax\mathbf{CAT}$. The 2-category
$\mathrm{Enr}(\mathbf{1})$ is the locally chaotic 2-category whose underlying
category is $\mathbf{Set}$. More precisely, the objects of
$\mathrm{Enr}(\mathbf{1})$ can be identified with the small sets, and for each
pair of small sets $X$ and $Y$ we have
$\mathrm{Enr}(\mathbf{1})(X,Y)=\mathbf{Set}(X,Y)_{c}$, where $(-)_{c}$ appears
in the string of adjunctions

$\mathbf{SET}$$\mathbf{CAT}_{0}$.$\pi_{0}$$(-)_{d}$$\mathrm{ob}$$(-)_{c}$$\vdash$$\vdash$$\vdash$
(2)

Here, $\mathbf{SET}$ and $\mathbf{CAT}_{0}$ denote the categories of large
sets and of large categories respectively. The (finite-product-preserving)
functors in (2) induce 2-adjunctions

$\mathbf{CAT}$$2\mathchar
45\relax\mathbf{CAT}$.$(\pi_{0})_{\ast}$$(-)_{ld}$$(-)_{0}$$(-)_{lc}$$\vdash$$\vdash$$\vdash$

So we shall write the 2-category $\mathrm{Enr}(\mathbf{1})$ as
$\mathbf{Set}_{lc}$.

Explicitly, the 2-functor $\mathrm{Enr}_{\mathbf{1}}\colon\mathbf{BICAT}\to
2\mathchar 45\relax\mathbf{CAT}/\mathbf{Set}_{lc}$ maps each bicategory
$\mathscr{B}$ to the 2-category ${\mathscr{B}}\mathchar 45\relax\mathbf{Cat}$
equipped with the 2-functor $\mathrm{ob}(-)\colon{\mathscr{B}}\mathchar
45\relax\mathbf{Cat}\to\mathbf{Set}_{lc}$ which extracts the set of objects of
a $\mathscr{B}$-category.

In the following theorem, by a weighted (2-)limit we mean a 2-limit weighted
by a 2-functor $F\colon\mathscr{D}\to\mathbf{CAT}$, where $\mathscr{D}$ is a
large 2-category and $\mathbf{CAT}$ is the 2-category of large categories.

######  Theorem 2.1.

The 2-functor $\mathrm{Enr}_{\mathbf{1}}\colon\mathbf{BICAT}\to 2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}$ preserves all weighted limits which
happen to exist in $\mathbf{BICAT}$.

######  Proof 2.2.

We shall show the following.

  1. (a)

The set $\mathcal{G}$ of all objects of $2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}$ of the form
$(\mathbf{2_{2}}\to\mathbf{Set}_{lc})$, where $\mathbf{2_{2}}$ denotes the
free 2-category on a single 2-cell, is a strong generator of the 2-category
$2\mathchar 45\relax\mathbf{CAT}/\mathbf{Set}_{lc}$.

  2. (b)

For each object $A\in\mathcal{G}$, the 2-functor $2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}(A,\mathrm{Enr}_{\mathbf{1}}(-))\colon\mathbf{BICAT}\to\mathbf{CAT}$
is a 2-limit of representable 2-functors, and hence preserves all weighted
limits which happen to exist in $\mathbf{BICAT}$.

From these, the main claim follows. Indeed, let $\mathscr{D}$ be a large
2-category, $F\colon\mathscr{D}\to\mathbf{CAT}$ be a 2-functor (the weight)
and $S\colon\mathscr{D}\to\mathbf{BICAT}$ be a 2-functor such that the
weighted limit $\\{F,S\\}$ exists in $\mathbf{BICAT}$. Then the weighted limit
$\\{F,\mathrm{Enr}_{\mathbf{1}}\circ S\\}$ exists in $2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}$, because $2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}$ has all (large) weighted limits. We
have a comparison 1-cell
$M\colon\mathrm{Enr}_{\mathbf{1}}\\{F,S\\}\to\\{F,\mathrm{Enr}_{\mathbf{1}}\circ
S\\}$ in $2\mathchar 45\relax\mathbf{CAT}/\mathbf{Set}_{lc}$. Now for each
$A\in\mathcal{G}$, the functor

$2\mathchar 45\relax\mathbf{CAT}/\mathbf{Set}_{lc}(A,M)\colon 2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}(A,\mathrm{Enr}_{\mathbf{1}}\\{F,S\\})\to
2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}(A,\\{F,\mathrm{Enr}_{\mathbf{1}}\circ
S\\})$

is an isomorphism by (b), from which we conclude that $M$ is an isomorphism by
(a).

$\mathcal{G}$ is a strong generator of $2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}$ because, given any 1-cell
$T\colon(\mathscr{X}\to\mathbf{Set}_{lc})\to(\mathscr{Y}\to\mathbf{Set}_{lc})$,
i.e., a 2-functor $T\colon\mathscr{X}\to\mathscr{Y}$ between 2-categories
$\mathscr{X}$ and $\mathscr{Y}$ over $\mathbf{Set}_{lc}$, the condition that
$2\mathchar 45\relax\mathbf{CAT}/\mathbf{Set}_{lc}(A,T)$ is an isomorphism for
all $A\in\mathcal{G}$ means that $T$ is bijective on 2-cells.

To show (b), observe that a 2-functor $\mathbf{2_{2}}\to\mathbf{Set}_{lc}$
corresponds to a parallel pair of functions $f_{0},f_{1}\colon X\to Y$. Such a
2-functor can be seen as an object of $2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}$. Given
$((f_{0},f_{1})\colon\mathbf{2_{2}}\to\mathbf{Set}_{lc})$ where
$f_{0},f_{1}\colon X\to Y$, first define the category
$X_{c}\mathbin{\star}X_{c}$ by

$\mathrm{ob}(X_{c}\mathbin{\star}X_{c})=\\{(0,x)\mid x\in X\\}\cup\\{(1,x)\mid
x\in X\\}$

and for each pair of objects $(i,x)$ and $(i^{\prime},x^{\prime})$,

$(X_{c}\mathbin{\star}X_{c})((i,x),(i^{\prime},x^{\prime}))=\begin{cases}\\{\ast\\}&\text{if
}i\leq i^{\prime};\\\ \emptyset&\text{otherwise}.\end{cases}$

We regard $X_{c}\mathbin{\star}X_{c}$ as a bicategory as well. We have
functors

$\nabla=[\mathrm{id},\mathrm{id}]\colon X_{c}\mathbin{\star}X_{c}\to
X_{c}\qquad\text{and}\qquad[f_{0},f_{1}]\colon X_{c}\mathbin{\star}X_{c}\to
Y_{c};$

these can also be regarded as lax functors, i.e., morphisms in
$\mathbf{BICAT}$. The 2-functor

$2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}((f_{0},f_{1}),\mathrm{Enr}_{\mathbf{1}}(-))\colon\mathbf{BICAT}\to\mathbf{CAT}$

is the comma object (in $[\mathbf{BICAT},\mathbf{CAT}]$) as in

$2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}((f_{0},f_{1}),\mathrm{Enr}_{\mathbf{1}}(-))$$\mathbf{BICAT}(Y_{c},-)$$\mathbf{BICAT}(X_{c},-)$$\mathbf{BICAT}(X_{c}\mathbin{\star}X_{c},-)$.$\mathbf{BICAT}(\nabla,-)$$\mathbf{BICAT}([f_{0},f_{1}],-)$

Indeed, for any bicategory $\mathscr{B}\in\mathbf{BICAT}$, an object of the
comma category of the functors $\mathbf{BICAT}([f_{0},f_{1}],\mathscr{B})$ and
$\mathbf{BICAT}(\nabla,\mathscr{B})$ consists of lax functors
$\mathbb{C}\colon X_{c}\to\mathscr{B}$ and $\mathbb{D}\colon
Y_{c}\to\mathscr{B}$ together with an icon

$X_{c}\mathbin{\star}X_{c}$$Y_{c}$$X_{c}$$\mathscr{B}$.$\nabla$$\mathbb{C}$$[f_{0},f_{1}]$$\mathbb{D}$$\alpha$

This corresponds to $\mathscr{B}$-categories $\mathbb{C}$ and $\mathbb{D}$
with $\mathrm{ob}(\mathbb{C})=X$ and $\mathrm{ob}(\mathbb{D})=Y$ such that
$|x|_{\mathbb{C}}=|f_{i}(x)|_{\mathbb{D}}$ for all $x\in X$ and
$i\in\\{0,1\\}$, together with a 2-cell
$\alpha_{(i,x),(i^{\prime},x^{\prime})}\colon\mathbb{C}(x,x^{\prime})\to\mathbb{D}(f_{i}(x),f_{i^{\prime}}(x^{\prime}))$
in $\mathscr{B}$ for each $(i,x),(i^{\prime},x^{\prime})\in
X_{c}\mathbin{\star}X_{c}$ with $i\leq i^{\prime}$, satisfying some equations.
These latter data in turn correspond to $\mathscr{B}$-functors
$F_{0}\colon\mathbb{C}\to\mathbb{D}$ and $F_{1}\colon\mathbb{C}\to\mathbb{D}$
(with $\mathrm{ob}(F_{i})=f_{i}$) together with a $\mathscr{B}$-natural
transformation $\alpha\colon F_{0}\to F_{1}$. (We record in Lemma 2.3 below an
observation which is useful for the verification.)

This gives a bijective correspondence on objects of $2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}((f_{0},f_{1}),\mathrm{Enr}_{\mathbf{1}}(\mathscr{B}))$
and the comma category of $\mathbf{BICAT}([f_{0},f_{1}],\mathscr{B})$ and
$\mathbf{BICAT}(\nabla,\mathscr{B})$, which routinely extends to an
isomorphism of categories natural in $\mathscr{B}$.

######  Lemma 2.3.

Let $\mathscr{B}$ be a bicategory, $\mathbb{C},\mathbb{D}$ be
$\mathscr{B}$-categories and $T,S\colon\mathbb{C}\to\mathbb{D}$ be
$\mathscr{B}$-functors. To give a $\mathscr{B}$-natural transformation
$\alpha\colon T\to S$, i.e., a family of 2-cells

$|x|$$|x|$$1_{|x|}$$\mathbb{D}(Tx,Sx)$$\alpha_{x}$

in $\mathscr{B}$ for each $x\in\mathbb{C}$, satisfying the naturality axiom:
for each $x,x^{\prime}\in\mathbb{C}$,

$\mathbb{C}(x,x^{\prime})$$1_{|x^{\prime}|}.\mathbb{C}(x,x^{\prime})$$\mathbb{C}(x,x^{\prime}).1_{|x|}$$\mathbb{D}(Tx^{\prime},Sx^{\prime}).\mathbb{D}(Tx,Tx^{\prime})$$\mathbb{D}(Sx,Sx^{\prime}).\mathbb{D}(Tx,Sx)$$\mathbb{D}(Tx,Sx^{\prime})$$\cong$$\alpha_{x^{\prime}}.T_{x,x^{\prime}}$$M^{\mathbb{D}}_{Tx,Tx^{\prime},Sx^{\prime}}$$\cong$$S_{x,x^{\prime}}.\alpha_{x}$$M^{\mathbb{D}}_{Tx,Sx,Sx^{\prime}}$
(3)

commutes, is equivalent to give a family of 2-cells

$|x|$$|x^{\prime}|$$\mathbb{C}(x,x^{\prime})$$\mathbb{D}(Tx,Sx^{\prime})$$\alpha_{x,x^{\prime}}$

in $\mathscr{B}$ for each $x,x^{\prime}\in\mathbb{C}$, such that for each
$x,x^{\prime},x^{\prime\prime}\in\mathbb{C}$,

$\mathbb{C}(x^{\prime},x^{\prime\prime}).\mathbb{C}(x,x^{\prime})$$\mathbb{D}(Sx^{\prime},Sx^{\prime\prime}).\mathbb{D}(Tx,Sx^{\prime})$$\mathbb{C}(x,x^{\prime\prime})$$\mathbb{D}(Tx,Sx^{\prime\prime})$$M^{\mathbb{C}}_{x,x^{\prime},x^{\prime\prime}}$$\alpha_{x,x^{\prime\prime}}$$S_{x^{\prime},x^{\prime\prime}}.\alpha_{x,x^{\prime}}$$M^{\mathbb{D}}_{Tx,Sx^{\prime},Sx^{\prime\prime}}$$\mathbb{C}(x^{\prime},x^{\prime\prime}).\mathbb{C}(x,x^{\prime})$$\mathbb{D}(Tx^{\prime},Sx^{\prime\prime}).\mathbb{D}(Tx,Tx^{\prime})$$\mathbb{C}(x,x^{\prime\prime})$$\mathbb{D}(Tx,Sx^{\prime\prime})$$M^{\mathbb{C}}_{x,x^{\prime},x^{\prime\prime}}$$\alpha_{x,x^{\prime\prime}}$$\alpha_{x^{\prime},x^{\prime\prime}}.T_{x,x^{\prime}}$$M^{\mathbb{D}}_{Tx,Tx^{\prime},Sx^{\prime\prime}}$
and

commute; the correspondence is given by mapping $(\alpha_{x})$ to
$(\alpha_{x,x^{\prime}})$ whose component at $(x,x^{\prime})$ is the composite
(3).

As observed in [Lac10, Section 6.2], the 2-category $\mathbf{BICAT}$ can be
seen as the 2-category of strict algebras, lax morphisms, and algebra 2-cells
for a 2-monad $T$ on a certain locally presentable 2-category of
$\mathbf{CAT}$-enriched graphs, and so by [Lac05] has oplax limits,
Eilenberg–Moore objects of comonads, and limits of diagrams containing only
strict morphisms; this last class includes in particular products and powers.
It also has various other sorts of limits where certain parts of the diagram
are required to be pseudofunctors. For a more precise characterization see
[LS12].

The case of oplax limits of 1-cells is our motivating example, and is
formalized in Section 4, specifically in Theorem 4.4. The case of
Eilenberg–Moore objects of comonads is treated in Example 3.13. As a final
example, we consider products. In this case, Theorem 2.1 says that, for
bicategories $\mathscr{B}$ and $\mathscr{C}$, the diagram

${(\mathscr{B}\times\mathscr{C})}\mathchar
45\relax\mathbf{Cat}$${\mathscr{C}}\mathchar
45\relax\mathbf{Cat}$${\mathscr{B}}\mathchar
45\relax\mathbf{Cat}$$\mathbf{Set}_{lc}$$\mathrm{ob}$$\mathrm{ob}$

is a pullback of 2-categories. In particular, to give a
$\mathscr{B}\times\mathscr{C}$-category is equivalent to giving a
$\mathscr{B}$-category and a $\mathscr{C}$-category with the same set of
objects.

######  Remark 2.4.

It is possible to remove any size-related conditions on the notion of weighted
limit in Theorem 2.1. That is, for any (possibly larger than “large”)
2-category $\mathscr{D}$ and a weight
$F\colon\mathscr{D}\to\mathbf{CAT}^{\prime}$, where $\mathbf{CAT}^{\prime}$ is
a 2-category of categories in a universe containing $\mathcal{U}_{1}$,
$\mathrm{Enr}_{\mathbf{1}}$ preserves all $F$-weighted limits which happen to
exist in $\mathbf{BICAT}$. Indeed, let $S\colon\mathscr{D}\to\mathbf{BICAT}$
be a 2-functor such that $\\{F,S\\}$ exists in $\mathbf{BICAT}$. Then,
although a priori we do not know if $\\{F,\mathrm{Enr}_{\mathbf{1}}\circ S\\}$
exists in $2\mathchar 45\relax\mathbf{CAT}/\mathbf{Set}_{lc}$ or not, we can
certainly consider a large enough variant $2\mathchar
45\relax\mathbf{CAT}^{\prime}/\mathbf{Set}_{lc}$ in which it does. Then by the
above discussion we have
$\mathrm{Enr}_{\mathbf{1}}\\{F,S\\}\cong\\{F,\mathrm{Enr}_{\mathbf{1}}\circ
S\\}$ in $2\mathchar 45\relax\mathbf{CAT}^{\prime}/\mathbf{Set}_{lc}$. Since
the fully faithful 2-functor $2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}\to 2\mathchar
45\relax\mathbf{CAT}^{\prime}/\mathbf{Set}_{lc}$ reflects limits, and
$\mathrm{Enr}_{\mathbf{1}}$ does land in $2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}$, we see that the limit
$\\{F,\mathrm{Enr}_{\mathbf{1}}\circ S\\}$ actually exists in $2\mathchar
45\relax\mathbf{CAT}/\mathbf{Set}_{lc}$.

##  3 Weighted limits created by forgetful 2-functors
$\mathscr{K}/A\to\mathscr{K}$

Theorem 2.1 implies that the 2-functor $\mathrm{Enr}\colon\mathbf{BICAT}\to
2\mathchar 45\relax\mathbf{CAT}$ preserves all weighted limits created by the
forgetful 2-functor $2\mathchar 45\relax\mathbf{CAT}/\mathbf{Set}_{lc}\to
2\mathchar 45\relax\mathbf{CAT}$. We now investigate these. In the following,
$1$ and $\mathbf{1}$ denote the terminal category and the terminal 2-category,
respectively.

######  Theorem 3.1.

Let $\mathscr{D}$ be a small 2-category and
$F\colon\mathscr{D}\to\mathbf{Cat}$ be a 2-functor.222As $\mathbf{BICAT}$ and
$2\mathchar 45\relax\mathbf{CAT}/\mathbf{Set}_{lc}$ are not
$\mathbf{Cat}$-enriched but rather $\mathbf{CAT}$-enriched, perhaps we should
have considered $F\colon\mathscr{D}\to\mathbf{CAT}$ with $\mathscr{D}$ large.
However, since the issue of size is irrelevant, we prefer to retain the more
familiar notation. Then the following are equivalent.

  1. 1.

All $F$-weighted limits are created by the forgetful 2-functor
$\mathscr{K}/A\to\mathscr{K}$ for any locally small 2-category $\mathscr{K}$
and $A\in\mathscr{K}$.

  2. 2.

All $F$-weighted limits commute with copowers in $\mathbf{Cat}$. In other
words, $F$-weighted limits are preserved by the 2-functor
$X\times(-)\colon\mathbf{Cat}\to\mathbf{Cat}$ for any $X\in\mathbf{Cat}$.

  3. 3.

The $F$-weighted limit of the unique 2-functor $\mathscr{D}\to\mathbf{1}$ is
preserved by any 2-functor $\mathbf{1}\to\mathbf{Cat}$:
$X\cong[\mathscr{D},\mathbf{Cat}](F,\Delta X)$ for any $X\in\mathbf{Cat}$.

  4. 4.

The $F$-weighted limit of the unique 2-functor $\mathscr{D}\to\mathbf{1}$ is
preserved by any 2-functor $\mathbf{1}\to\mathscr{K}$: $A\cong\\{F,\Delta
A\\}$ for any locally small 2-category $\mathscr{K}$ and $A\in\mathscr{K}$.

  5. 5.

$F\ast(-)\colon[\mathscr{D}^{\mathrm{op}},\mathbf{Cat}]\to\mathbf{Cat}$
preserves the terminal object. In other words, the $F$-weighted colimit of
${\Delta}1\colon\mathscr{D}^{\mathrm{op}}\to\mathbf{Cat}$ is the terminal
category: $F\ast{\Delta}1\cong 1$.

  6. 6.

The (conical) colimit of $F$ is the terminal category: $\Delta 1\ast F\cong
1$.

######  Proof 3.2.

[(1)$\implies$(2)] For any $X\in\mathbf{Cat}$, copower by $X$ is given by
$X\times(-)\colon\mathbf{Cat}\to\mathbf{Cat}$, which is the composite of the
right adjoint 2-functor $X\times(-)\colon\mathbf{Cat}\to\mathbf{Cat}/X$ and
the forgetful 2-functor $\mathbf{Cat}/X\to\mathbf{Cat}$.

[(2)$\implies$(3)] Note that we have $1\cong\\{F,\Delta 1\\}$ in
$\mathbf{Cat}$. Since $X\times(-)\colon\mathbf{Cat}\to\mathbf{Cat}$ preserves
the $F$-weighted limit $\\{F,\Delta 1\\}$, we have $X\cong\\{F,\Delta X\\}$.

[(3)$\implies$(4)] For any $B\in\mathscr{K}$ we have
$\mathscr{K}(B,A)\cong[\mathscr{D},\mathbf{Cat}](F,\Delta\mathscr{K}(B,A))$.
This shows that $A\in\mathscr{K}$ is the weighted limit $\\{F,\Delta A\\}$.

[(4)$\implies$(1)] Let $T\colon\mathscr{D}\to\mathscr{K}/A$ be a 2-functor,
with the corresponding cone

$\mathscr{D}$$\mathbf{1}$$\mathscr{K}$.$S$$A$$\gamma$

In particular, $S$ is the composite of $T$ and the forgetful 2-functor
$\mathscr{K}/A\to\mathscr{K}$. Suppose that the weighted limit $\\{F,S\\}$
exists in $\mathscr{K}$. We have a 1-cell
$\\{F,\gamma\\}\colon\\{F,S\\}\to\\{F,\Delta A\\}\cong A$ in $\mathscr{K}$. We
claim that the object $(\\{F,\gamma\\}\colon\\{F,S\\}\to A)\in\mathscr{K}/A$
is the limit $\\{F,T\\}$ in $\mathscr{K}/A$. For any $(p\colon B\to
A)\in\mathscr{K}/A$, the hom category
$(\mathscr{K}/A)((B,p),(\\{F,S\\},\\{F,\gamma\\}))$ is given by the equalizer

$(\mathscr{K}/A)((B,p),(\\{F,S\\},\\{F,\gamma\\}))$$\mathscr{K}(B,\\{F,S\\})$$\mathscr{K}(B,A)$,$\mathscr{K}(B,\\{F,\gamma\\})$$\Delta
p$

which is easily seen to be canonically isomorphic to
$[\mathscr{D},\mathbf{Cat}](F,(\mathscr{K}/A)((B,p),T-))$.

[(4)$\implies$(5)] Applying (4) to
$1\colon\mathbf{1}\to\mathbf{Cat}^{\mathrm{op}}$, we obtain
$F\ast{\Delta}1\cong 1$ in $\mathbf{Cat}$.

[(5)$\implies$(3)] For any $X\in\mathbf{Cat}$, we have

$X\cong[1,X]\cong[F\ast{\Delta}1,X]\cong[\mathscr{D},\mathbf{Cat}](F,[{\Delta}1(-),X])\cong[\mathscr{D},\mathbf{Cat}](F,{\Delta}X).$

[(5)$\iff$(6)] By $F\ast{\Delta}1\cong{\Delta}1\ast F$.

A 2-functor $F\colon\mathscr{D}\to\mathbf{Cat}$ is called _$\mathbf{Cat}$
-connected_ if $F$ satisfies the equivalent conditions of Theorem 3.1.
Similarly, a weighted limit is _$\mathbf{Cat}$ -connected_ if its weight is
so. Note that $F\colon\mathscr{D}\to\mathbf{Cat}$ is connected if and only if
$[\mathscr{D},\mathbf{Cat}](F,\Delta X)\cong X$ for $X=1+1$, which implies the
same condition for any discrete category $X$; on the other hand it is
$\mathbf{Cat}$-connected if this holds for all categories $X$.

######  Remark 3.3.

Theorem 3.1 can be proved more generally for categories enriched over a
complete and cocomplete cartesian closed category $\mathscr{V}$ in place of
$\mathbf{Cat}$, indeed the proof carries over essentially word-for-word upon
replacing each instance of $\mathbf{Cat}$ by $\mathscr{V}$.

We now give a few simple results about $\mathbf{Cat}$-connected weights in
order to clarify the scope of the notion.

######  Proposition 3.4.

If $\mathscr{D}$ has a terminal object, then
$F\colon\mathscr{D}\to\mathbf{Cat}$ is $\mathbf{Cat}$-connected if and only if
$F$ preserves the terminal object.

######  Proof 3.5.

If $\mathscr{D}$ has a terminal object $1$ then the colimit of $F$ is $F(1)$.

######  Proposition 3.6.

Let $\mathscr{C}$ be an ordinary category, and
$G\colon\mathscr{C}\to\mathbf{Set}$ a functor. This determines a 2-functor
$G_{d}\colon\mathscr{C}_{ld}\to\mathbf{Cat}$, where now $\mathscr{C}_{ld}$ is
regarded as a locally discrete 2-category. This $G_{d}$ sends an object $C$ to
the discrete category $G(C)_{d}$ with object-set $G(C)$. Then $G_{d}$ is
$\mathbf{Cat}$-connected if and only if the corresponding $G$ is connected.

######  Proof 3.7.

Since the functor $(-)_{d}\colon\mathbf{Set}\to\mathbf{Cat}_{0}$ preserves
colimits, $\mathrm{colim}(G_{d})=\mathrm{colim}(G)_{d}$.

######  Proposition 3.8.

If $\mathscr{D}_{0}$ is connected then $\Delta
1\colon\mathscr{D}\to\mathbf{Cat}$ is $\mathbf{Cat}$-connected.

######  Proof 3.9.

The colimit of $\Delta 1\colon\mathscr{D}\to\mathbf{Cat}$ is a quotient of the
colimit of $\Delta 1\colon\mathscr{D}_{0}\to\mathbf{Cat}$. The terminal
category $1$ has no proper quotients.

######  Example 3.10.

Equifiers are $\mathbf{Cat}$-connected: here it is easiest to verify directly
that equifiers in $\mathbf{Cat}$ commute with copowers. Similarly, one
verifies that Eilenberg–Moore objects of monads and of comonads are
$\mathbf{Cat}$-connected. Equalizers and pullbacks are
$\mathbf{Cat}$-connected by Proposition 3.8.

######  Example 3.11.

Products are not $\mathbf{Cat}$-connected: they are not even connected. Powers
by a category $X$ are limits weighted by $X\colon\mathbf{1}\to\mathbf{Cat}$;
since the colimit of such a weight is just $X$, powers by $X$ are
$\mathbf{Cat}$-connected if and only if $X=1$. Inserters, comma objects and
oplax limits of 1-cells are not $\mathbf{Cat}$-connected: in particular they
are not preserved by the 2-functor $\mathbb{N}\colon\mathbf{1}\to\mathbf{Cat}$
which picks out the monoid $\mathbb{N}$ of natural numbers. More generally,
inserters are not preserved by $X\colon\mathbf{1}\to\mathbf{Cat}$ if $X$ has a
non-identity endomorphism, while comma objects and oplax limits of 1-cells are
not preserved by $X\colon\mathbf{1}\to\mathbf{Cat}$ unless $X$ is discrete.

As an immediate consequence of Theorems 2.1 and 3.1, we have:

######  Corollary 3.12.

The 2-functor $\mathrm{Enr}\colon\mathbf{BICAT}\to 2\mathchar
45\relax\mathbf{CAT}$ preserves all $\mathbf{Cat}$-connected limits which
happen to exist in $\mathbf{BICAT}$.

######  Example 3.13.

Eilenberg–Moore objects of comonads are $\mathbf{Cat}$-connected, and exist in
$\mathbf{BICAT}$ by the results of [Lac05, LS12], thus they are preserved by
$\mathrm{Enr}$. In more detail, a comonad $G$ in $\mathbf{BICAT}$ on a
bicategory $\mathscr{B}$ consists of a comonad $G=G_{a,b}$ on each hom-
category $\mathscr{B}(a,b)$, together with 2-cells $G_{2}\colon Gg.Gf\to
G(gf)$ for all $f\colon a\to b$ and $g\colon b\to c$, and 2-cells $G_{0}\colon
1_{Ga}\to G1_{a}$ for all objects $a$, subject to various conditions, which
say that the $G_{a,b}$, the $G_{2}$ and the $G_{0}$ can be assembled into an
identity-on-objects lax functor $\mathscr{B}\to\mathscr{B}$, in such a way
that the counits and comultiplications for the comonads become icons. The
Eilenberg–Moore object $\mathscr{B}^{G}$ is the bicategory with the same
objects as $\mathscr{B}$, and with hom-category $\mathscr{B}^{G}(a,b)$ given
by the Eilenberg–Moore category $\mathscr{B}(a,b)^{G_{a,b}}$ of $G_{a,b}$.
Corollary 3.12 then says that ${\mathscr{B}^{G}}\mathchar
45\relax\mathbf{Cat}$ is the Eilenberg–Moore 2-category for the induced
2-comonad on ${\mathscr{B}}\mathchar 45\relax\mathbf{Cat}$.

##  4 Oplax limits and fibrations

A 1-cell $f\colon A\to B$ in a 2-category $\mathscr{K}$ is called a fibration,
when $\mathscr{K}(C,f)\colon\mathscr{K}(C,A)\to\mathscr{K}(C,B)$ is a
Grothendieck fibration for each $C\in\mathscr{K}$, and

$\mathscr{K}(C,A)$$\mathscr{K}(D,A)$$\mathscr{K}(C,B)$$\mathscr{K}(D,B)$$\mathscr{K}(c,A)$$\mathscr{K}(D,f)$$\mathscr{K}(C,f)$$\mathscr{K}(c,B)$

is a morphism of fibrations for each $c\colon C\to D$ in $\mathscr{K}$, in the
sense that $\mathscr{K}(c,A)$ sends cartesian morphisms to cartesian
morphisms.

As explained by Street [Str74], this notion can be reformulated if the
2-category $\mathscr{K}$ has oplax limits of 1-cells, as we shall henceforth
suppose. Recall that the oplax limit of a 1-cell $f\colon A\to B$ in
$\mathscr{K}$ is the universal diagram

$B/f$$A$$B$$u_{f}$$v_{f}$$f$$\lambda_{f}$

wherein we often drop the subscripts $f$ unless multiple oplax limits are
being used.

If $\mathscr{K}=\mathbf{Cat}$, then these oplax limits are comma categories,
as the notation suggests. On the other hand, we have:

######  Example 4.1.

Let $X$ be a small set, seen as a chaotic bicategory $X_{c}$ (that is,
$(X_{c})_{ld}$ or equivalently $(X_{c})_{lc}$). To give an $X_{c}$-enriched
category is just to give a set of objects with a map into $X$. Similar
calculations involving $X_{c}$-enriched functors and natural transformations
show that the diagram

${X_{c}}\mathchar
45\relax\mathbf{Cat}$$\mathbf{1}$$\mathbf{Set}_{lc}$$\mathrm{ob}$$X$

is an oplax limit in $2\mathchar 45\relax\mathbf{CAT}$; in other words, the
2-category ${X_{c}}\mathchar 45\relax\mathbf{Cat}$ is isomorphic to the slice
2-category $\mathbf{Set}_{lc}/X$.

The fibrations in $\mathscr{K}$ with codomain $B$ can be understood in terms
of a 2-monad $T_{B}$ on $\mathscr{K}/B$ whose underlying 2-functor maps
$f\colon A\to B$ to $v_{f}\colon B/f\to B$; the component at $f\colon A\to B$
of its unit is the unique map $d=d_{f}\colon A\to B/f$ with $ud=1_{A}$,
$vd=f$, and $\lambda d$ equal to the identity 2-cell on $f$. This 2-monad is
lax idempotent (has the “Kock–Zöberlein property”), and so an object $f\colon
A\to B$ of $\mathscr{K}/B$ has a pseudo $T_{B}$-algebra structure if and only
if $d\colon(A,f)\to(B/f,v_{f})$ has a right adjoint in $\mathscr{K}/B$; and
this in turn is the case if and only if $f$ is a fibration [Str74].

Likewise, the strict $T_{B}$-algebras are the split fibrations in
$\mathscr{K}$: those $f\colon A\to B$ for which each
$\mathscr{K}(C,f)\colon\mathscr{K}(C,A)\to\mathscr{K}(C,B)$ is a split
fibration, and each $\mathscr{K}(c,A)\colon$
$\mathscr{K}(C,A)\to\mathscr{K}(D,A)$ preserves the chosen cartesian lifts.

In particular, $v\colon B/f\to B$ is a split fibration for any $f\colon A\to
B$, and $d$ exhibits $v\colon B/f\to B$ as the free (split) fibration on $f$.
Thus if $p\colon E\to B$ is a fibration, and $g\colon A\to E$ defines a
morphism $(A,f)\to(E,p)$ in $\mathscr{K}/B$, there is an essentially unique
morphism of fibrations $r\colon(B/f,v)\to(E,p)$ extending $g$. (If $q\colon
F\to B$ and $p\colon E\to B$ are fibrations in $\mathscr{K}$ with the common
codomain $B$, then a 1-cell $r\colon(F,q)\to(E,p)$ in $\mathscr{K}/B$ is a
_morphism of fibrations_ if for each $C\in\mathscr{K}$, $\mathscr{K}(C,r)$ is
a morphism of fibrations, i.e., preserves cartesian morphisms.)

In general, oplax limits of 1-cells are not preserved by the projection
$\mathscr{K}/B\to\mathscr{K}$, but to some extent fibrations can be used to
remedy this, as the following result shows.

######  Proposition 4.2.

Let $p\colon A\to B$ be a fibration in $\mathscr{K}$, and consider a morphism
$g$ in $\mathscr{K}/B$ into $p$, and the (essentially unique) induced morphism
$r$ of fibrations, as below

$C$$A$$B$$g$$pg$$p$ $B/pg$$A$$B$.$r$$v_{pg}$$p$

Then the oplax limit of $g$ in $\mathscr{K}$ is the oplax limit of $r$ in
$\mathscr{K}/B$.

######  Proof 4.3.

Without loss of generality we may take $\mathscr{K}=\mathbf{Cat}$. An object
of the oplax limit of $g$ consists of objects $a\in A$, $c\in C$, and a
morphism $\alpha\colon a\to gc$.

An object of $B/pg$ consists of objects $b\in B$, $c\in C$, and a morphism
$\beta\colon b\to pgc$; and $r$ sends this to the domain of the cartesian
lifting $\overline{\beta}\colon\beta^{*}gc\to gc$. An object of the oplax
limit of $r$ in $\mathscr{K}/B$ consists of objects $(b,c,\beta)\in B/pg$,
$a\in A$, and a morphism $\alpha^{\prime}\colon a\to\beta^{*}{gc}$ with
$p(\alpha^{\prime})$ an identity. But by the fibration property of $p$, to
give such an $\alpha^{\prime}$ is equivalently to give $\alpha\colon a\to gc$
with $p\alpha=\beta$.

This gives the desired bijection between the objects of the two oplax limits,
and this extends to a bijection between morphisms.

We can use this to prove the following key result, already alluded to in the
introduction.

######  Theorem 4.4.

Let $\mathscr{B}$ be a bicategory and $\mathbb{X}$ a $\mathscr{B}$-category.
Then the slice 2-category ${\mathscr{B}}\mathchar
45\relax\mathbf{Cat}/\mathbb{X}$ has the form
${(\mathscr{B}/\mathbb{X})}\mathchar 45\relax\mathbf{Cat}$ for a bicategory
$\mathscr{B}/\mathbb{X}$.

######  Proof 4.5.

If we regard $\mathbb{X}$ as a lax functor $\mathbb{X}\colon
X_{c}\to\mathscr{B}$, where $X=\mathrm{ob}(\mathbb{X})$, we may take its oplax
limit

$\mathscr{B}/\mathbb{X}$$X_{c}$$\mathscr{B}$$\mathbb{X}$

in $\mathbf{BICAT}$. Explicitly,
$\mathrm{ob}(\mathscr{B}/\mathbb{X})=\mathrm{ob}(\mathbb{X})=X$, while the hom
$(\mathscr{B}/\mathbb{X})(x,x^{\prime})$ is given by the slice category
$\mathscr{B}(|x|,|x^{\prime}|)/\mathbb{X}(x,x^{\prime})$ for all
$x,x^{\prime}\in X$.

It follows by Theorem 2.1 that ${(\mathscr{B}/\mathbb{X})}\mathchar
45\relax\mathbf{Cat}$ is the oplax limit

${(\mathscr{B}/\mathbb{X})}\mathchar 45\relax\mathbf{Cat}$${X_{c}}\mathchar
45\relax\mathbf{Cat}$${\mathscr{B}}\mathchar
45\relax\mathbf{Cat}$$\mathbf{Set}_{lc}$$\mathrm{Enr}(\mathbb{X})$$\mathrm{ob}$$\mathrm{ob}$

in $2\mathchar 45\relax\mathbf{CAT}/\mathbf{Set}_{lc}$.

Now by Example 4.1, $\mathrm{ob}(-)\colon{X_{c}}\mathchar
45\relax\mathbf{Cat}\to\mathbf{Set}_{lc}$ is the free fibration on
$X\colon\mathbf{1}\to\mathbf{Set}_{lc}$, while $\mathrm{Enr}(\mathbb{X})$ is
the morphism of fibrations induced by
$\mathbb{X}\colon\mathbf{1}\to{\mathscr{B}}\mathchar 45\relax\mathbf{Cat}$,
and so by Proposition 4.2 the diagram

${(\mathscr{B}/\mathbb{X})}\mathchar
45\relax\mathbf{Cat}$$\mathbf{1}$${\mathscr{B}}\mathchar
45\relax\mathbf{Cat}$$\mathbb{X}$

is an oplax limit in $2\mathchar 45\relax\mathbf{CAT}$. But this says
precisely that ${(\mathscr{B}/\mathbb{X})}\mathchar
45\relax\mathbf{Cat}\cong{\mathscr{B}}\mathchar
45\relax\mathbf{Cat}/\mathbb{X}$.

######  Example 4.6.

In particular, when $\mathscr{B}$ is the cartesian monoidal category
$\mathbf{Set}$ regarded as a one-object bicategory, we have for each
($\mathbf{Set}$-)category $\mathbb{X}$ the bicategory
$\mathbf{Set}/\mathbb{X}$ whose set of objects is $\mathrm{ob}(\mathbb{X})$
and whose hom-category $(\mathbf{Set}/\mathbb{X})(x,x^{\prime})$ is the slice
category $\mathbf{Set}/\mathbb{X}(x,x^{\prime})$. Each functor
$F\colon\mathbb{Y}\to\mathbb{X}$ corresponds to a
$\mathbf{Set}/\mathbb{X}$-category $\overline{\mathbb{Y}}$ given as follows:
the objects of $\overline{\mathbb{Y}}$ are the same as those of $\mathbb{Y}$,
the extent of $y$ in $\overline{\mathbb{Y}}$ is $Fy$, and the hom
$\overline{\mathbb{Y}}(y,y^{\prime})$ is
$F_{y,y^{\prime}}\colon\mathbb{Y}(y,y^{\prime})\to\mathbb{X}(Fy,Fy^{\prime})$.
Note that since
$\mathbf{Set}/\mathbb{X}(x,x^{\prime})\simeq\mathbf{Set}^{\mathbb{X}(x,x^{\prime})}$,
$\mathbf{Set}/\mathbb{X}$ is (biequivalent to) the free local cocompletion of
$\mathbb{X}$ regarded as a locally discrete bicategory, as pointed out to us
by Ross Street.

A variant of $\mathbf{Set}/\mathbb{X}$ is the free quantaloid
$\mathcal{P}\mathbb{X}$ over $\mathbb{X}$. Specifically,
$\mathcal{P}\mathbb{X}$ is also a bicategory with the same objects as
$\mathbb{X}$, but whose hom-category $(\mathcal{P}\mathbb{X})(x,x^{\prime})$
is the powerset lattice $\mathcal{P}(\mathbb{X}(x,x^{\prime}))$, which is
equivalent to the full subcategory of the slice category
$\mathbf{Set}/\mathbb{X}(x,x^{\prime})$ consisting of the injections to
$\mathbb{X}(x,x^{\prime})$. Accordingly, the
$\mathcal{P}\mathbb{X}$-categories correspond to the _faithful_ functors
$\mathbb{Y}\to\mathbb{X}$ [Gar14, Proposition 3.5].

######  Example 4.7.

Let $\mathscr{B}$ be a bicategory with all right liftings. Then for each
$b\in\mathscr{B}$, we have a $\mathscr{B}$-category $\mathbb{B}_{b}$ whose
objects are the 1-cells $f\colon x\to b$ in $\mathscr{B}$ with codomain $b$,
with extent $|(x,f)|=x$, and whose hom $\mathbb{B}_{b}((x,f),(y,g))\colon x\to
y$ is the right lifting of $f$ along $g$:

$x$$y$$b$.$\mathbb{B}_{b}((x,f),(y,g))$$f$$g$

(See [GP97, Section 2] for the dual construction.) Given a
$\mathscr{B}$-category $\mathbb{X}$, the $\mathscr{B}$-functors
$\mathbb{X}\to\mathbb{B}_{b}$ correspond to the $\mathscr{B}$-presheaves on
$\mathbb{X}$ with extent $b$. Hence if we consider the bicategory
$\mathscr{B}/\mathbb{B}_{b}$, then a $\mathscr{B}/\mathbb{B}_{b}$-category can
be identified with a $\mathscr{B}$-category equipped with a
$\mathscr{B}$-presheaf with extent $b$.

By the universality of right liftings, the bicategory
$\mathscr{B}/\mathbb{B}_{b}$ is canonically isomorphic to the lax slice
bicategory $\mathscr{B}{\sslash}b$: this has 1-cells with codomain $b$ as
objects, and diagrams of the form

$x$$y$$b$$f$$g$

as 1-cells from $f$ to $g$. Unlike $\mathbb{B}_{b}$, this lax slice bicategory
$\mathscr{B}{\sslash}b$ can be defined even when $\mathscr{B}$ does not have
right liftings, and it is true in general that a
$\mathscr{B}{\sslash}b$-category corresponds to a $\mathscr{B}$-category
equipped with a $\mathscr{B}$-presheaf with extent $b$. (For a general
bicategory $\mathscr{B}$, the notion of $\mathscr{B}$-presheaf can be defined
in terms of actions; see [Str05] for a definition of the more general notion
of module.)

######  Remark 4.8.

The bicategory $\mathscr{B}/\mathbb{X}$ can be obtained from $\mathbb{X}$ via
a change-of-base process for bicategories enriched in a tricategory. Since the
theory of tricategory-enriched bicategories, let alone change-of-base for
them, has not really been developed in detail, we mention this only in
passing.

We regard $\mathscr{B}$ as a tricategory with no non-identity 3-cells, and we
regard the cartesian monoidal 2-category $\mathbf{Cat}$ as a one-object
tricategory $\Sigma(\mathbf{Cat})$. There is a lax morphism of tricategories
$\Theta\colon\mathscr{B}\to\Sigma(\mathbf{Cat})$ sending each object
$b\in\mathscr{B}$ to the unique object of $\Sigma(\mathbf{Cat})$, and sending
a 1-cell $f\colon b\to b^{\prime}$ in $\mathscr{B}$ to the category
$B(b,b^{\prime})/f$. Composition with $\Theta$ then sends each
$\mathscr{B}$-enriched bicategory to a $\Sigma(\mathbf{Cat})$-enriched
bicategory. Since $\mathscr{B}$ has no non-identity 3-cells, a
$\mathscr{B}$-enriched bicategory is just a $\mathscr{B}$-enriched category;
on the other hand, a $\Sigma(\mathbf{Cat})$-enriched bicategory is just a
bicategory in the ordinary sense. Applying this to the $\mathscr{B}$-category
$\mathbb{X}$ gives the bicategory $\mathscr{B}/\mathbb{X}$.

##  5 Variation through enrichment

In the paper [BCSW83], the authors showed how fibrations with codomain
$\mathbb{X}$ can be seen as certain categories enriched over a bicategory
$\mathscr{W}(\mathbb{X})$ depending on the category $\mathbb{X}$. In this
section we give a result of the same type, although it differs in several
important respects. The bicategory we use is $\mathbf{Set}/\mathbb{X}$ (see
Example 4.6), which is like $\mathscr{W}(\mathbb{X})$ in having as objects the
objects of $\mathbb{X}$: see Remark 5.1 below for the relationship between the
two bicategories. Then we show that fibrations over $\mathbb{X}$ can be
identified with $\mathbf{Set}/\mathbb{X}$-categories which have certain
powers.

######  Remark 5.1.

Given objects $x,x^{\prime}\in\mathbb{X}$, a 1-cell in
$\mathscr{W}(\mathbb{X})$ from $x$ to $x^{\prime}$ consists of a presheaf $E$
on $\mathbb{X}$ equipped with maps to $\mathbb{X}(-,x)$ and
$\mathbb{X}(-,x^{\prime})$; in other words, it consists of a span of
presheaves from $\mathbb{X}(-,x)$ to $\mathbb{X}(-,x^{\prime})$. Now a 1-cell
$S\to\mathbb{X}(x,x^{\prime})$ in $\mathbf{Set}/\mathbb{X}$ from $x$ to
$x^{\prime}$ determines, via Yoneda, a map
$S\cdot\mathbb{X}(-,x)\to\mathbb{X}(-,x^{\prime})$ of presheaves, where
$S\cdot\mathbb{X}(-,x)$ denotes the copower of $\mathbb{X}(-,x)$ by $S$: the
coproduct of $S$ copies of $\mathbb{X}(-,x)$. On the other hand there is the
codiagonal $S\cdot\mathbb{X}(-,x)\to\mathbb{X}(-,x)$, and so we obtain a span

$\mathbb{X}(-,x)$$\mathbb{X}(-,x^{\prime})$$S\cdot\mathbb{X}(-,x)$

of presheaves; that is, a 1-cell in $\mathscr{W}(\mathbb{X})$ from $x$ to
$x^{\prime}$. This defines the 1-cell part of a homomorphism of bicategories
$\mathbf{Set}/\mathbb{X}\to\mathscr{W}(\mathbb{X})$ which is the identity on
objects and locally fully faithful. Just as we characterize fibrations over
$\mathbb{X}$ as $\mathbf{Set}/\mathbb{X}$-categories with certain limits, so
in [BCSW83] these fibrations are seen as $\mathscr{W}(\mathbb{X})$-categories
with certain limits; one key difference is that in the case of
$\mathscr{W}(\mathbb{X})$ the limits in question are absolute.

In fact we work not just with fibrations of ordinary categories, but rather
fibrations in the 2-category ${\mathscr{B}}\mathchar 45\relax\mathbf{Cat}$ of
$\mathscr{B}$-enriched categories, as in Section 4. We have seen in Theorem
4.4 that, for a $\mathscr{B}$-category $\mathbb{X}$, $\mathscr{B}$-functors
with codomain $\mathbb{X}$ correspond to $\mathscr{B}/\mathbb{X}$-enriched
categories. We shall see in this section that a $\mathscr{B}$-functor
$F\colon\mathbb{Y}\to\mathbb{X}$ is a fibration in ${\mathscr{B}}\mathchar
45\relax\mathbf{Cat}$ if and only if the corresponding
$\mathscr{B}/\mathbb{X}$-category $\overline{\mathbb{Y}}$ has certain powers.

First, however, we give an elementary characterization of fibrations in
${\mathscr{B}}\mathchar 45\relax\mathbf{Cat}$. To do this, we start with the
fact that every $\mathscr{B}$-category $\mathbb{X}$ has an underlying ordinary
category $\mathbb{X}_{0}$ with the same objects; a morphism $x\to x^{\prime}$
in $\mathbb{X}_{0}$ can exist only if $x$ and $x^{\prime}$ have the same
extent ($|x|=|x^{\prime}|$), in which case it amounts to a 2-cell
$1_{|x|}\to\mathbb{X}(x,x^{\prime})$ in $\mathscr{B}$. We shall sometimes
refer to such morphisms in $\mathbb{X}_{0}$ simply as morphisms in
$\mathbb{X}$. The assignment $\mathbb{X}\mapsto\mathbb{X}_{0}$ is the object-
part of a 2-functor ${\mathscr{B}}\mathchar
45\relax\mathbf{Cat}\to\mathbf{Cat}$,333In fact this arises via change-of-base
with respect to a lax functor from $\mathscr{B}$ to the cartesian monoidal
category $\mathbf{Set}$, seen as a one-object bicategory. The lax functor
sends each object $b$ to this unique object; it sends a 1-cell $f\colon b\to
c$ to the set $\mathscr{B}(b,c)(1_{b},f)$ if $b=c$ and the empty set
otherwise; with the evident action on 2-cells. while the assignment
$(x,x^{\prime})\mapsto\mathbb{X}_{0}(x,x^{\prime})$ is the object-part of a
functor $\mathbb{X}_{0}^{\mathrm{op}}\times\mathbb{X}_{0}\to\mathbf{Set}$. In
particular, if $f\colon x^{\prime}\to x^{\prime\prime}$ is a morphism in
$\mathbb{X}$ and $x$ is an object, there is an induced 2-cell
$\mathbb{X}(x,f)\colon\mathbb{X}(x,x^{\prime})\to\mathbb{X}(x,x^{\prime\prime})$
defined by pasting $f\colon
1_{|x^{\prime}|}\to\mathbb{X}(x^{\prime},x^{\prime\prime})$ together with the
composition 2-cell
$M_{x,x^{\prime},x^{\prime\prime}}\colon\mathbb{X}(x^{\prime},x^{\prime\prime}).\mathbb{X}(x,x^{\prime})\to\mathbb{X}(x,x^{\prime\prime})$.

######  Definition 5.2.

Let $F\colon\mathbb{Y}\to\mathbb{X}$ be a $\mathscr{B}$-functor. A morphism
$h\colon y^{\prime}\to y$ in $\mathbb{Y}_{0}$ is said to be cartesian with
respect to $F$ if the square

$\mathbb{Y}(z,y^{\prime})$$\mathbb{Y}(z,y)$$\mathbb{X}(Fz,Fy^{\prime})$$\mathbb{X}(Fz,Fy)$$\mathbb{Y}(z,h)$$F_{z,y}$$F_{z,y^{\prime}}$$\mathbb{X}(Fz,Fh)$

is a pullback in $\mathscr{B}(|z|,|y|)$ for all objects $z$ in $\mathbb{Y}$.

This says in particular that $h$ is cartesian with respect to the ordinary
functor $F_{0}\colon\mathbb{Y}_{0}\to\mathbb{X}_{0}$, but in general is
stronger than this.

######  Proposition 5.3.

Suppose that the bicategory $\mathscr{B}$ has pullbacks in each hom-category
$\mathscr{B}(a,b)$. A $\mathscr{B}$-functor $F\colon\mathbb{Y}\to\mathbb{X}$
is a fibration in ${\mathscr{B}}\mathchar 45\relax\mathbf{Cat}$ if and only
if, for each object $y\in\mathbb{Y}$ and each morphism $g\colon x\to Fy$ in
$\mathbb{X}$ there is a cartesian morphism $\overline{g}\colon g^{*}y\to y$ in
$\mathbb{Y}$ with $F\overline{g}=g$.

######  Proof 5.4.

The pullbacks in the hom-categories of $\mathscr{B}$ can be used to construct
oplax limits in ${\mathscr{B}}\mathchar 45\relax\mathbf{Cat}$, as we shall now
show. Let $F\colon\mathbb{Y}\to\mathbb{X}$ be a $\mathscr{B}$-functor; then
the oplax limit $\mathbb{L}=\mathbb{X}/F$ has:

  * •

objects given by pairs $(g,y)$, with $y\in\mathbb{Y}$ and $g\colon x\to Fy$ in
$\mathbb{X}_{0}$

  * •

the extent of $(g,y)$ equal to the extent of $y$ (which is also the extent of
$x$)

  * •

homs given by pullbacks as in

$\mathbb{L}((g^{\prime},y^{\prime}),(g,y))$$\mathbb{Y}(y^{\prime},y)$$\mathbb{X}(x^{\prime},x)$$\mathbb{X}(Fy^{\prime},Fy)$$\mathbb{X}(x^{\prime},Fy)$$U_{(g^{\prime},y^{\prime}),(g,y)}$$F_{y^{\prime},y}$$\mathbb{X}(g^{\prime},Fy)$$V_{(g^{\prime},y^{\prime}),(g,y)}$$\mathbb{X}(x^{\prime},g)$

  * •

projections $V\colon\mathbb{L}\to\mathbb{X}$ and
$U\colon\mathbb{L}\to\mathbb{Y}$ sending an object $(g,y)$ to $x$ and to $y$,
and defined on homs as in the diagram above.

The diagonal $\mathscr{B}$-functor $D\colon\mathbb{Y}\to\mathbb{L}$ sends an
object $z\in\mathbb{Y}$ to $(1_{Fz},z)\in\mathbb{L}$. Taking
$(g^{\prime},y^{\prime})=Dz$ in the above diagram gives a pullback

$\mathbb{L}(Dz,(g,y))$$\mathbb{Y}(z,y)$$\mathbb{X}(Fz,x)$$\mathbb{X}(Fz,Fy)$.$U$$F_{z,y}$$V$$\mathbb{X}(Fz,g)$

Now $F$ is a fibration just when $D$ has a right adjoint in
${\mathscr{B}}\mathchar 45\relax\mathbf{Cat}/\mathbb{X}$. Such an adjoint is
given on objects by a lifting of $g\colon Fx\to y$ to some $\overline{g}\colon
g^{*}y\to y$, and the universal property says that this lifting is cartesian.

We now turn to the characterization of fibrations of $\mathscr{B}$-categories
in terms of $\mathscr{B}/\mathbb{X}$-categories. First recall that if
$\mathscr{W}$ is a bicategory and $\mathbb{Z}$ is a $\mathscr{W}$-category
then powers in $\mathbb{Z}$ involve an object $y$ of $\mathbb{Z}$ and a 1-cell
$v\colon x\to|y|$ in $\mathscr{W}$ with codomain the extent of $y$. The power
of $y$ by $v$ consists of an object $v\pitchfork y$ of $\mathbb{Z}$ with
extent $|v\pitchfork y|=x$, together with a 2-cell

$|v\pitchfork y|$$|y|$$v$$\mathbb{Z}(v\pitchfork y,y)$$\eta$

such that for all $z\in\mathbb{Z}$ and all

$|z|$$|v\pitchfork y|$$|y|$$b$$\mathbb{Z}(z,y)$$v$$\alpha$

there exists a unique $\gamma$ making the pasting composite

$|z|$$|v\pitchfork y|$$|y|$$b$$\mathbb{Z}(z,v\pitchfork
y)$$\mathbb{Z}(z,y)$$v$$\mathbb{Z}(v\pitchfork y,y)$$M$$\gamma$$\eta$

equal to $\alpha$. (In other words, the pasting of $\eta$ and $M$ exhibits
$\mathbb{Z}(z,v\pitchfork y)$ as the right lifting of $\mathbb{Z}(z,y)$ along
$v$.)

We consider this in the case where $\mathscr{W}=\mathscr{B}/\mathbb{X}$ and
$\mathbb{Z}$ is the $\mathscr{B}/\mathbb{X}$-category $\overline{\mathbb{Y}}$
corresponding to a $\mathscr{B}$-functor $F\colon\mathbb{Y}\to\mathbb{X}$.
Then an object $y$ of $\overline{\mathbb{Y}}$ is just an object of
$\mathbb{Y}$, and the extent of $y$, as an object of $\mathscr{B}/\mathbb{X}$,
is the object $Fy$ of $\mathbb{X}$. A general 1-cell $x\to Fy$ in
$\mathscr{B}/\mathbb{X}$ has the form

$|x|$$|Fy|$,$v$$\mathbb{X}(x,Fy)$$w$

but we shall only consider the special case where $|x|=|Fy|$ and $v=1_{|x|}$,
so that in fact we are dealing with a morphism $w\colon x\to Fy$ in
$\mathbb{X}_{0}$. In general, we call a 1-cell $(w\colon
v\to\mathbb{X}(x,x^{\prime}))\colon x\to x^{\prime}$ in
$\mathscr{B}/\mathbb{X}$ a singleton 1-cell if $|x|=|x^{\prime}|$ and
$v=1_{|x|}$. Note that the category $\mathbb{X}_{0}$ can be regarded as a sub-
bicategory of $\mathscr{B}/\mathbb{X}$ whose 1-cells are the singleton
1-cells. When $\mathscr{B}=\mathbf{Set}$, a 1-cell $x\to x^{\prime}$ in
$\mathbf{Set}/\mathbb{X}$ corresponds to a set $v$ equipped with a function
$w\colon v\to\mathbb{X}(x,x^{\prime})$; in this case, the singleton 1-cells in
$\mathbf{Set}/\mathbb{X}$ can be identified with those 1-cells with $v$ a
singleton, whence the name singleton.

A power of $y$ by $w\colon 1\to\mathbb{X}(x,Fy)$ then consists of an object
$w\pitchfork y$ of $\mathbb{Y}$ with $F(w\pitchfork y)$ $=x$ together with a
morphism $\overline{w}\colon w\pitchfork y\to y$ in $\mathbb{Y}_{0}$ with
$F\overline{w}=w$ — that is, a lifting $\overline{w}$ of $w$ — subject to the
universal property stating that for all $z\in\mathbb{Y}$, $b\colon|z|\to|y|$,
$\alpha$, and $\beta$ making

$\leavevmode\hbox to135.65pt{\vbox
to127.36pt{\pgfpicture\makeatletter\hbox{\hskip
10.87811pt\lower-33.66876pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.5451pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$|z|$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{49.04793pt}{82.85828pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$|x|$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{106.18028pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$|y|$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}{
{}{}{}}{{{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}}
}{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}} }{{}{}}{{}}
{}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}{}{{}}{}{}{}{}{}{{}}{}{}{}{}{{}}\pgfsys@moveto{0.5501pt}{8.533pt}\pgfsys@curveto{1.85481pt}{41.39047pt}{16.4072pt}{64.43256pt}{45.16086pt}{79.54536pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.88518}{0.46526}{-0.46526}{0.88518}{45.33789pt}{79.6384pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.0692pt}{54.24968pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{$b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}{
{}{}{}}{{{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}}
}{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}} }{{}{}}{{}}
{}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}{}{{}}{}{}{}{}{}{{}}{}{}{}{}{{}}\pgfsys@moveto{68.29611pt}{82.02849pt}\pgfsys@curveto{100.63289pt}{71.70746pt}{119.30312pt}{42.17065pt}{114.80717pt}{8.92938pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.13403}{-0.99098}{0.99098}{-0.13403}{114.78038pt}{8.7312pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{108.89018pt}{57.55746pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}{
{}{}{}}{{{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}}
}{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}} }{{}{}}{{}}
{}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}{}{}{}{}{{}}\pgfsys@moveto{11.07811pt}{6.38712pt}\pgfsys@curveto{41.99379pt}{24.25826pt}{71.71255pt}{24.27461pt}{102.30078pt}{6.6363pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.86629}{-0.49954}{0.49954}{0.86629}{102.47403pt}{6.53639pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@rect{42.7466pt}{12.96974pt}{28.21783pt}{13.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{46.0796pt}{18.05275pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\mathbb{Y}(z,y)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}}{}{ {}{}{}}{ {}{}{}}{{{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}}
}{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}} }{{}{}}{{}}
{}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}{}{{}}{}{}{}{}{}{{}}{}{}{}{}{{}}\pgfsys@moveto{11.07811pt}{-6.38712pt}\pgfsys@curveto{41.99379pt}{-24.25826pt}{71.71255pt}{-24.27461pt}{102.30078pt}{-6.6363pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.86629}{0.49954}{-0.49954}{0.86629}{102.47403pt}{-6.53639pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{40.60599pt}{-28.58575pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\mathbb{X}(Fz,Fy)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{} {}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.33748pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{56.90552pt}{58.28644pt}\pgfsys@lineto{56.90552pt}{39.64835pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{1.53749pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{56.90552pt}{58.28644pt}\pgfsys@lineto{56.90552pt}{39.64835pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{56.90552pt}{39.64835pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{60.43852pt}{46.86266pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{$\alpha$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.33748pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{56.90552pt}{9.91684pt}\pgfsys@lineto{56.90552pt}{-8.72125pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{1.53749pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{56.90552pt}{9.91684pt}\pgfsys@lineto{56.90552pt}{-8.72125pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{56.90552pt}{-8.72125pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{60.43852pt}{-1.58861pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{$F_{z,y}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad=\qquad\leavevmode\hbox
to137.04pt{\vbox to127.61pt{\pgfpicture\makeatletter\hbox{\hskip
10.87811pt\lower-33.91696pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.5451pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$|z|$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{49.04793pt}{82.85828pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$|x|$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{104.7914pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$|y|,$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}{
{}{}{}}{{{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}}
}{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}} }{{}{}}{{}}
{}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}{}{{}}{}{}{}{}{}{{}}{}{}{}{}{{}}\pgfsys@moveto{0.5501pt}{8.533pt}\pgfsys@curveto{1.85481pt}{41.39047pt}{16.4072pt}{64.43256pt}{45.16086pt}{79.54536pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.88518}{0.46526}{-0.46526}{0.88518}{45.33789pt}{79.6384pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.0692pt}{54.24968pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{$b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}{
{}{}{}}{{{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}}
}{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}} }{{}{}}{{}}
{}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}{}{}{}{}{{}}\pgfsys@moveto{11.07811pt}{5.47244pt}\pgfsys@curveto{40.2821pt}{20.74126pt}{54.9765pt}{43.89948pt}{56.3387pt}{76.42564pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.04184}{0.99911}{-0.99911}{0.04184}{56.34705pt}{76.62544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@rect{13.27467pt}{11.59961pt}{34.00903pt}{13.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.60768pt}{16.68262pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\mathbb{X}(Fz,x)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}}{}{ {}{}{}}{ {}{}{}}{{{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}}
}{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}} }{{}{}}{{}}
{}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}{}{{}}{}{}{}{}{}{{}}{}{}{}{}{{}}\pgfsys@moveto{11.07811pt}{-6.38712pt}\pgfsys@curveto{41.4854pt}{-24.32715pt}{70.51819pt}{-24.60066pt}{100.91013pt}{-7.43398pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.8707}{0.4918}{-0.4918}{0.8707}{101.08426pt}{-7.33562pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{39.79385pt}{-28.83395pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\mathbb{X}(Fz,Fy)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}}{}{ {}{}{}}{ {}{}{}}{{{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}}
}{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}} }{{}{}}{{}}
{}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}{}{{}}{}{}{}{}{}{{}}{}{}{}{}{{}}\pgfsys@moveto{68.29611pt}{79.73146pt}\pgfsys@curveto{97.40384pt}{64.43256pt}{111.95622pt}{41.39047pt}{113.24509pt}{8.93268pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.03967}{-0.9992}{0.9992}{0.03967}{113.253pt}{8.73285pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{104.73764pt}{54.24968pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}{
{}{}{}}{{{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}}
}{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}} }{{}{}}{{}}
{}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}{}{}{}{}{{}}\pgfsys@moveto{57.45561pt}{76.82527pt}\pgfsys@curveto{58.48027pt}{44.3725pt}{72.64117pt}{21.53995pt}{100.90585pt}{6.38982pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.88136}{-0.47243}{0.47243}{0.88136}{101.08212pt}{6.29533pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@rect{65.42091pt}{12.31038pt}{34.12894pt}{13.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{68.75392pt}{17.39339pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\mathbb{X}(x,Fy)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{} {}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.33748pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{56.90552pt}{18.45276pt}\pgfsys@lineto{56.90552pt}{-3.03079pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{1.53749pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{56.90552pt}{18.45276pt}\pgfsys@lineto{56.90552pt}{-3.03079pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{56.90552pt}{-3.03079pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{60.43852pt}{4.72153pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{$M$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.33748pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{28.45276pt}{52.59598pt}\pgfsys@lineto{28.45276pt}{33.95787pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{1.53749pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{28.45276pt}{52.59598pt}\pgfsys@lineto{28.45276pt}{33.95787pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{28.45276pt}{33.95787pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.98576pt}{40.92914pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{$\beta$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.33748pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{85.35828pt}{52.59598pt}\pgfsys@lineto{85.35828pt}{33.95787pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{1.53749pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{85.35828pt}{52.59598pt}\pgfsys@lineto{85.35828pt}{33.95787pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{85.35828pt}{33.95787pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{88.89128pt}{41.1722pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{$w$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

there exists a unique $\gamma$ making the pasting composites

$|z|$$|w\pitchfork y|$$|y|$$b$$1$$\mathbb{Y}(z,w\pitchfork
y)$$\mathbb{X}(Fz,x)$$\gamma$$F_{z,w\pitchfork y}$ $|z|$$|w\pitchfork
y|$$|y|$$b$$\mathbb{Y}(z,v\pitchfork
y)$$\mathbb{Y}(z,y)$$1$$\mathbb{Y}(w\pitchfork y,y)$$M$$\gamma$$\overline{w}$

equal respectively to $\beta$ and $\alpha$. But this says exactly that if the
exterior of the diagram

$\mathbb{Y}(z,w\pitchfork
y)$$\mathbb{Y}(z,y)$$\mathbb{X}(Fz,x)$$\mathbb{X}(Fz,Fy)$$b$$\mathbb{Y}(z,\overline{w})$$F_{z,y}$$F_{z,w\pitchfork
y}$$\mathbb{X}(Fz,w)$$\alpha$$\beta$$\gamma$

in $\mathscr{B}(|z|,|y|)$ commutes, then there is a unique $\gamma$ making the
triangular regions commute; in other words, that the internal square is a
pullback. This in turn says that $\overline{w}$ is a cartesian lifting of $w$.
This now proves:

######  Proposition 5.5.

Let $\mathscr{B}$ be a bicategory in which each hom-category has pullbacks. A
$\mathscr{B}$-functor $F\colon\mathbb{Y}\to\mathbb{X}$ is a fibration if and
only if the corresponding $\mathscr{B}/\mathbb{X}$-category
$\overline{\mathbb{Y}}$ has powers by morphisms in $\mathbb{X}_{0}$; that is,
powers by singleton 1-cells.

We conclude by strengthening this correspondence to an isomorphism between
suitable 2-categories. Let $\mathscr{W}$ be a bicategory and
$H\colon\mathbb{Z}\to\mathbb{Z}^{\prime}$ a $\mathscr{W}$-functor. Suppose
that the power $v\pitchfork y$ of $y\in\mathbb{Z}$ by $v\colon x\to|y|$ exists
in $\mathbb{Z}$, with the associated 2-cell $\eta\colon
v\to\mathbb{Z}(v\pitchfork y,y)$. Then $H$ is said to _preserve_ the power
$v\pitchfork y$ if the 2-cell $H_{v\pitchfork y,y}\circ\eta\colon
v\to\mathbb{Z}^{\prime}(H(v\pitchfork y),Hy)$ exhibits $H(v\pitchfork y)$ as
the power $v\pitchfork Hy$ in $\mathbb{Z}^{\prime}$.

######  Theorem 5.6.

Let $\mathscr{B}$ be a bicategory in which each hom-category has pullbacks.
The canonical isomorphism of 2-categories ${(\mathscr{B}/\mathbb{X})}\mathchar
45\relax\mathbf{Cat}\cong{\mathscr{B}}\mathchar
45\relax\mathbf{Cat}/\mathbb{X}$ restricts to an isomorphism between the
locally full sub-2-category of ${(\mathscr{B}/\mathbb{X})}\mathchar
45\relax\mathbf{Cat}$ having

  * •

the $\mathscr{B}/\mathbb{X}$-categories with powers by singleton 1-cells as
objects, and

  * •

the $\mathscr{B}/\mathbb{X}$-functors preserving these powers as 1-cells,

and the locally full sub-2-category of ${\mathscr{B}}\mathchar
45\relax\mathbf{Cat}/\mathbb{X}$ having

  * •

the fibrations to $\mathbb{X}$ as objects, and

  * •

the fibration morphisms as 1-cells.

## References

  * [BCSW83] Renato Betti, Aurelio Carboni, Ross Street, and Robert Walters.  Variation through enrichment.  J. Pure Appl. Algebra, 29(2):109–127, 1983.
  * [Bén67] Jean Bénabou.  Introduction to bicategories.  In Reports of the Midwest Category Seminar, pages 1–77. Springer, Berlin, 1967.
  * [Gar14] Richard Garner.  Topological functors as total categories.  Theory Appl. Categ., 29(15):406–422, 2014.
  * [GP97] R. Gordon and A. J. Power.  Enrichment through variation.  J. Pure Appl. Algebra, 120(2):167–185, 1997.
  * [Lac05] Stephen Lack.  Limits for lax morphisms.  Appl. Categ. Structures, 13(3):189–203, 2005.
  * [Lac10] Stephen Lack.  Icons.  Appl. Categ. Structures, 18(3):289–307, 2010.
  * [LS12] Stephen Lack and Michael Shulman.  Enhanced 2-categories and limits for lax morphisms.  Adv. Math., 229(1):294–356, 2012.
  * [Str74] Ross Street.  Fibrations and Yoneda’s lemma in a $2$-category.  In Category Seminar (Proc. Sem., Sydney, 1972/1973), Lecture Notes in Math., Vol. 420, pages 104–133. Springer, Berlin, 1974.
  * [Str05] Ross Street.  Enriched categories and cohomology.  Repr. Theory Appl. Categ., (14):1–18, 2005.  Reprinted from Quaestiones Math. 6 (1983), no. 1-3, 265–283, with new commentary by the author.