$1_{\upharpoonleft\\!\\!{\;\mathfrak{sing}\,\,\upharpoonleft\\!\\!{\;C\;}\\!\\!\downharpoonright\;}\\!\\!\downharpoonright}\leftrightsquigarrow_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright}{}\upharpoonleft\\!\\!{\eta^{\prime}_{C}}\\!\\!\downharpoonright\epsilon^{\prime}_{\upharpoonleft\\!\\!{\;C\;}\\!\\!\downharpoonright}$

natural in $\mathscr{R}$-objects $C$, where $\eta^{\prime}$ and
$\epsilon^{\prime}$ denote the unit and counit of the adjunction
$\upharpoonleft\\!\\!{-}\\!\\!\downharpoonright\dashv\mathfrak{sing}\,$ and
$\mathscr{R}$ is the full subcategory $\hat{\square}$ consisting of those
cubical sets whose atomic cubical sets are all isomorphic to representables.

The idea of the proof is to lift a stream map
$f:\upharpoonleft\\!\\!{\square[1]^{n}}\\!\\!\downharpoonright\rightarrow\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright$
along a stream map $\upharpoonleft\\!\\!{C^{*}_{f;k}\rightarrow
C}\\!\\!\downharpoonright$ up to natural
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright_{*}$-homotopy based
on a $2^{k}$-fold subdivision of $\square[1]^{n}$ to a stream admitting a
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright$-convexity
structure natural in $f$ and, in a suitable sense, $k$.

###### Proof.

Let $S=\mathfrak{sing}\,\upharpoonleft\\!\\!{-}\\!\\!\downharpoonright$. Let
$f$ denote a
$(\upharpoonleft\\!\\!{\square[-]}\\!\\!\downharpoonright/\upharpoonleft\\!\\!{\mathfrak{sd}_{3}C}\\!\\!\downharpoonright)$-object.
Define $n_{f}$ by

$f:\upharpoonleft\\!\\!{\square[1]^{n_{f}}}\\!\\!\downharpoonright\rightarrow\upharpoonleft\\!\\!{\mathfrak{sd}_{3}C}\\!\\!\downharpoonright.$

construction of lift $f_{k}$ of
$\upharpoonleft\\!\\!{\epsilon_{C}f}\\!\\!\downharpoonright$: Let $k\gg 0$.
Let $\mathcal{I}_{f;k}$ denote the poset, natural in $f$, of all Boolean
intervals in $\mathfrak{sd}_{2}^{k}[1]^{n_{f}}$ ordered under inclusion. Let
$I$ denote a $\mathcal{I}_{f;k}$-object. The image of each closed cell
$|\square[I]|$ under $f\varphi_{\theta;k}$ lies in an open star of a vertex by
$|\mathfrak{sd}_{2}^{k}\square[1]^{n_{f}}|$. Thus the image of each closed
cell under $|\epsilon_{C}|f\varphi_{\theta;k}$ lies in a closed cell [Lemma
3.20]. Let $C_{f;I}=\mathfrak{supp}_{|-|}\left(f|\square[I]|,C\right)$. Define
$C^{*}_{f;k}$ as the $\hat{\square}$-colimit

$C^{*}_{f;k}=\mathrm{colim}_{I}\left(C_{f;I}\otimes\square[I]\right),$

an $\mathscr{R}$-object because it is a finite iterated pushout of inclusions
of $\mathscr{R}$-objects.

Define $\mathscr{R}$-morphisms $\theta_{f;k}$ and $\rho_{f;k}$, natural in
$f$, by commutative diagrams

${C_{f;I}\otimes\square[I]}$${C_{f;I}}$${C^{*}_{f;k}}$${C}$$\scriptstyle{C_{f;I}\otimes(\square[I\rightarrow\star])}$$\scriptstyle{\theta_{f;k}}$
${C_{f;I}\otimes\square[I]}$${\square[I]}$${C^{*}_{f;k}}$${\mathfrak{sd}_{2}^{k}[1]^{n_{f}}}$$\scriptstyle{(C_{f;I}\rightarrow\star)\otimes\square[I]}$$\scriptstyle{\rho_{f;k}}$

There exists a unique dotted stream map $f_{k}$, natural in $f$ by uniqueness,
making the following diagram, in which $f_{I}$ is a suitable restriction and
corestriction of the composite of the bottom row, commute:

(8)
${\upharpoonleft\\!\\!{\square[I]}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{C_{f;I}}\\!\\!\downharpoonright\times\upharpoonleft\\!\\!{\square[I]}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{C^{*}_{f;k}}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{\mathfrak{sd}_{2}^{k}\square[1]^{n_{f}}}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{\square[1]^{n_{f}}}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{\mathfrak{sd}_{3}C}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright,}$$\scriptstyle{f_{I}\times
1_{\upharpoonleft\\!\\!{\;\square[I]\;}\\!\\!\downharpoonright}}$$\scriptstyle{\upharpoonleft\\!\\!{\;\square[I\hookrightarrow\mathfrak{sd}_{2}^{k}[1]^{n_{f}}]\;}\\!\\!\downharpoonright}$$\scriptstyle{\upharpoonleft\\!\\!{\;\theta_{f;k}\;}\\!\\!\downharpoonright}$$\scriptstyle{\varphi_{\square[1]^{n_{f}};2^{k}}}$$\scriptstyle{f_{k}}$$\scriptstyle{f}$$\scriptstyle{\upharpoonleft\\!\\!{\;\epsilon_{C}\;}\\!\\!\downharpoonright}$

convexity structure on
$\upharpoonleft\\!\\!{C^{*}_{f;k}}\\!\\!\downharpoonright$: Let $\pi_{f;k;1}$
and $\pi_{f;k;2}$ denote the respective projections

$\pi_{f;k;1},\pi_{f;k;2}:\upharpoonleft\\!\\!{C^{*}_{f;k}}\\!\\!\downharpoonright^{2}\rightarrow\upharpoonleft\\!\\!{C^{*}_{f;k}}\\!\\!\downharpoonright$

onto first and second factors. Define $s_{f;k}$ by the commutative diagram

(9)
${\upharpoonleft\\!\\!{C^{*}_{f;k}}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{C^{*}_{f;k}}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{\mathfrak{sd}_{2}^{k}\square[1]^{n_{f}}}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{C^{*}_{f;k}}\\!\\!\downharpoonright}$$\scriptstyle{s_{f;k}}$$\scriptstyle{f_{k}}$

natural in $f$. For each $x\in|C^{*}_{f;k}|$, $s_{f;k}(x)$ and $x$ both lie in
the same closed cell in $|C^{*}_{f;k}|$, the directed realization of an atomic
subpresheaf of $C^{*}_{f;k}$ and hence the directed realization of a
representable up to isomorphism by our assumption on $C$. Thus there exists a
$\upharpoonleft\\!\\!{\mathfrak{d}_{1}}\\!\\!\downharpoonright$-homotopy
$s_{f;k}\leftrightsquigarrow_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright}{}1_{\upharpoonleft\\!\\!{\;C^{*}_{f;k}\;}\\!\\!\downharpoonright}$
natural in $f$ [Lemma 4.5]. The stream maps
$\pi_{f;k;1}s_{f;k},\pi_{f;k;2}s_{f;k}$ both naturally factor through
$\upharpoonleft\\!\\!{\square[1]^{n_{f}}}\\!\\!\downharpoonright$. Thus there
exists a
$\upharpoonleft\\!\\!{\mathfrak{d}_{1}}\\!\\!\downharpoonright$-homotopy
$\pi_{f;k;1}s_{f;k}\leftrightsquigarrow_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright}{}\pi_{f;k;2}s_{f;k}$
natural in $f$ [Lemma 4.5]. Concatenating the
$\upharpoonleft\\!\\!{\mathfrak{d}_{1}}\\!\\!\downharpoonright$-homotopies

$\pi_{f;k;1}\leftrightsquigarrow_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright}{}\pi_{f;k;1}s_{f;k}\leftrightsquigarrow_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright}{}\pi_{f;k;2}s_{f;k}\leftrightsquigarrow_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright}{}\pi_{f;k;2}$

yields a $\mathfrak{d}_{3}$-homotopy
$h^{*}:_{f;k}:\pi_{f;k;1}\leftrightsquigarrow_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright}{}\pi_{f;k;2}$
natural in $f$.

constructing the requisite directed homotopy: Consider the solid arrows in the
diagram

(10)
${\upharpoonleft\\!\\!{S\upharpoonleft\\!\\!{C_{f;k}^{*}}\\!\\!\downharpoonright}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{S\upharpoonleft\\!\\!{C^{*}_{f;k}}\\!\\!\downharpoonright}\\!\\!\downharpoonright^{2}}$${\upharpoonleft\\!\\!{\square[1]^{n_{f}}}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{S\upharpoonleft\\!\\!{C^{*}_{f;k}}\\!\\!\downharpoonright}\\!\\!\downharpoonright^{\upharpoonleft\\!\\!{\;\mathfrak{d}_{3}[1]\;}\\!\\!\downharpoonright}}$${\upharpoonleft\\!\\!{S\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{S\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright}\\!\\!\downharpoonright^{2}}$${\upharpoonleft\\!\\!{S\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright}\\!\\!\downharpoonright^{\upharpoonleft\\!\\!{\;\mathfrak{d}_{3}[1]\;}\\!\\!\downharpoonright}}$$\scriptstyle{1_{\upharpoonleft\\!\\!{\;S\upharpoonleft\\!\\!{\;C^{*}_{f;k}\;}\\!\\!\downharpoonright\;}\\!\\!\downharpoonright}\times\epsilon^{\prime}_{S\upharpoonleft\\!\\!{\;C^{*}_{f;k}\;}\\!\\!\downharpoonright}\upharpoonleft\\!\\!{\;\eta^{\prime}_{C^{*}_{f;k}}\;}\\!\\!\downharpoonright}$$\scriptstyle{S\upharpoonleft\\!\\!{\;\Lambda^{*}_{f;k}\;}\\!\\!\downharpoonright}$$\scriptstyle{Sh^{*}_{f;k}}$$\scriptstyle{S\upharpoonleft\\!\\!{\;\Lambda^{*}_{f;k}\;}\\!\\!\downharpoonright^{\otimes
2}}$$\scriptstyle{\upharpoonleft\\!\\!{\;\mathfrak{adj}(f)\;}\\!\\!\downharpoonright}$$\scriptstyle{\mathfrak{adj}(f_{k}\varphi^{-1}_{\square[1]^{n_{f}};2^{k}})}$$\scriptstyle{{S\upharpoonleft\\!\\!{\;\Lambda^{*}_{\theta;o}\;}\\!\\!\downharpoonright^{{\upharpoonleft\\!\\!{\;\mathfrak{d}_{1}[1]\;}\\!\\!\downharpoonright}}}}$$\scriptstyle{1_{\upharpoonleft\\!\\!{\;S\upharpoonleft\\!\\!{\;C_{\theta}\;}\\!\\!\downharpoonright\;}\\!\\!\downharpoonright}\times\epsilon^{\prime}_{S\upharpoonleft\\!\\!{\;C\;}\\!\\!\downharpoonright}\upharpoonleft\\!\\!{\;\eta^{\prime}_{C}\;}\\!\\!\downharpoonright}$

The top triangle commutes by construction of $h^{*}_{f;k}$ and the left
triangle commutes up to
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright_{*}$-homotopy.
Therefore there exist remaining dotted stream maps making the entire diagram
commute up to
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright_{*}$-homotopy.

invariance of $h_{f;k}$ in $k$: The top horizontal stream maps in diagrams

${\upharpoonleft\\!\\!{\square[C_{f;I}]}\\!\\!\downharpoonright\times\upharpoonleft\\!\\!{\square[I]}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{\square[C_{f;I^{\prime}}]}\\!\\!\downharpoonright\times\upharpoonleft\\!\\!{\square[I^{\prime}]}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{C^{*}_{f;i}}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{C^{*}_{f;i+1}}\\!\\!\downharpoonright}$$\scriptstyle{\upharpoonleft\\!\\!{\;C_{f;I}\hookrightarrow
C_{f;I^{\prime}}\;}\\!\\!\downharpoonright\otimes\varphi_{\square[I^{\prime}];2}\upharpoonleft\\!\\!{\;\square[I\hookrightarrow\mathfrak{sd}_{2}\square[I^{\prime}]]\;}\\!\\!\downharpoonright}$$\scriptstyle{e_{f;i}}$

where $I$ denotes a $\mathcal{I}_{f;i+1}$-object and
$I^{\prime}=\mathfrak{supp}_{\mathfrak{sd}_{2}}(I,\mathfrak{sd}_{2}^{k}[1]^{n_{f}})$,
induce dotted stream maps $e_{f;k}$ natural in $f$. The stream map $e_{f;k}$
define the $k$-indexed components of natural transformations between diagrams
in (8), (9), and (10) from the case $k=i\gg 0$ to the case $k=i+1\gg 0$
natural in $f$. In particular, $h_{f;k}$ is independent of $k$ and therefore
defines the desired
$\upharpoonleft\\!\\!{\mathfrak{d}_{4}}\\!\\!\downharpoonright$-homotopy. ∎

Define $\mathfrak{h}$ by the commutative diagram

${\square_{1}}$${\mathrm{\bf DiTop}}$${\square}$${\mathrm{\bf
Top}}$$\scriptstyle{\mathfrak{h}}$$\scriptstyle{|-|}$

In other words, an $\mathfrak{h}$-homotopy is exactly a classical homotopy
through stream maps. An $\mathfrak{h}$-homotopy is sometimes referred to in
the literature as an h-homotopy (eg. [46]) or a dihomotopy [20]. On one hand,
$\mathfrak{h}$-homotopy is an equivalence relation because of natural
isomorphisms
$\mathfrak{h}_{0}\cong\mathfrak{h}_{1}\cong\mathfrak{h}_{2}\cdots$. On the
other hand,
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright$-homotopy is not an
equivalence relation on the hom-set $\mathrm{\bf DiTop}(X,Y)$ unless every
dipath in $Y$ is reversible. Every left
$\upharpoonleft\\!\\!{\mathfrak{d}_{n}}\\!\\!\downharpoonright$-homotopy
determines a left $\mathfrak{h}$-homotopy by the existence of a natural
transformation
$\mathfrak{h}\rightarrow\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright_{n}$.
It turns out that $f\leftrightsquigarrow_{\mathfrak{h}}g$ if and only if
$f\leftrightsquigarrow_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright}{}g$
for a pair of stream maps $f,g$ from a compact stream to a directed
realization of a cubical set by a straightforward adaptation of a proof under
the usual definition of the cube category [44]; however we do not examine
$\mathfrak{h}$-homotopy theory in this paper.

###  4.3. Cubical

Define $\pi_{0}C$ as the coequalizer

$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C(\delta_{-})}$$\scriptstyle{C(\delta_{+})}$$\textstyle{C_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{0}C}$

natural in cubical sets $C$.

######  Example 4.9.

We can make the natural identification $[A,B]_{\mathfrak{d}}=\pi_{0}B^{A}$.

Unravelling the definition, $\pi_{0}\Omega^{n}(C,v)$ is the set of equivalence
classes of cubical functions $\square[1]^{n}\rightarrow C$ sending
$\partial\square[1]^{n}$ to the subcubical set of $C$ generated by a vertex
$v$, where two such cubical functions $\alpha,\beta$ are equivalent if
$\alpha\leftrightsquigarrow_{\mathfrak{d}}\beta$ relative
$\partial\square[1]^{n}\hookrightarrow\square[1]^{n}$.

####  4.3.1. Classical

For each small category ${\small\bigcirc}$, $\hat{{\small\bigcirc}}$-morphism
is a classical weak equivalence if it is a Thomason equivalence after applying
${\small\bigcirc}/-:\hat{\square}\rightarrow\mathrm{\bf Cat}$. The classical
weak equivalences and monos form the respective weak equivalences and
cofibrations of a test model structure on $\hat{\square}$. [Proposition C.1,
Theorem 1.4.3, [12]]. Call the localization of $\hat{\square}$ by its
classical weak equivalences the classical homotopy category of
$\hat{\square}$. Call the fibrant cubical sets in the test model structure
simply fibrant. The fundamental groupoid $\Pi_{1}$ is a classical homotopy
invariant in the sense of the following proposition, whose proof is given at
the end of §4.5.1.

######  Proposition 4.10.

For each classical weak equivalence $\psi:A\rightarrow B$ of cubical sets,

$\Pi_{1}\psi:\Pi_{1}A\rightarrow\Pi_{1}B$

is a categorical equivalence.

As a consequence, cubical nerves of small groupoids are fibrant (cf.
Proposition 3.36.)

######  Corollary 4.11.

For each small groupoid $\mathcal{G}$, $\mathfrak{ner}\,\mathcal{G}$ is
fibrant.

###### Proof.

Consider the solid functors in the left of the diagrams

${A}$${\mathfrak{ner}\,\mathcal{G}}$${B}$$\scriptstyle{\psi}$$\scriptstyle{\mathfrak{adj}(\phi)}$
${\Pi_{1}A}$${\mathcal{G}}$${\Pi_{1}B}$$\scriptstyle{\cong}$$\scriptstyle{\Pi_{1}\psi}$$\scriptstyle{\phi}$

Suppose $A\hookrightarrow B$ is an acyclic cofibration in the test model
structure. There exists a dotted functor $\phi$ making the entire right
diagram commute by $\Pi_{1}(A\hookrightarrow B)$ a faithful equivalence of
small categories. Therefore there exists exists a dotted functor making the
left diagram commute. ∎

Classical weak equivalences and monos form the respective weak equivalences
and cofibrations of a model structure on presheaves over the minimal variant
of $\square$. In this model structure, the set of inclusions (2) generate the
acyclic cofibrations [12, §8.4.34]. It therefore follows that each inclusion
(2) is an acyclic cofibrations in the test model structure on $\hat{\square}$.
For each fibrant cubical set $C$ having vertex $v$, let

$\pi_{n}(C,v)=\pi_{0}\Omega^{n}(C,v).$

The set $\pi_{n+1}(C,v)$ naturally admits the extra structure of a group whose
operations come from the right lifting properties of the fibration
$C\rightarrow\star$ against (2). The groups $\pi_{1}(C,v),\pi_{2}(C,v),\ldots$
are analogous to combinatorial homotopy groups on Kan simplicial sets [41].

######  Example 4.12.

For each group $G$, $\pi_{n}(\mathfrak{ner}\,G,\star)=\begin{cases}G&n=1\\\
0&n\neq 1\end{cases}$.

Write $H^{1}(C;\pi)$ for classical cubical $1$-cohomology

$H^{1}(C;\pi)=[C,\mathfrak{ner}\,\pi]_{\mathfrak{d}}=\pi_{0}(\mathfrak{ner}\,\pi)^{C},$

an Abelian group natural in Abelian groups $\pi$ and cubical sets $C$.
Classical cubical $1$-cohomology sends classical weak equivalences to
isomorphisms by $\mathfrak{ner}\,\pi$ fibrant [Corollary 4.11]. The higher
cubical cohomology groups are obtained by generalizing the cubical nerve of a
(discrete cubical) Abelian group $\pi$ to a suitable iterated fibrant cubical
delooping construction $W^{n}\pi$ (cf. [41].)

####  4.3.2. Directed

Just as classical cubical homotopy theory is the $\mathfrak{d}$-homotopy
theory of fibrant cubical sets, we can take directed cubical homotopy to mean
the $\mathfrak{d}$-homotopy theory of cubcats. This cubical directed theory
extends classical cubical homotopy theory by the following proposition, whose
proof is given in §4.5.2.

######  Proposition 4.13.

For each fibrant cubical set $B$, there exists a monic cubical function

$B\rightarrow C$

and retraction $\rho:C\rightarrow B$ such that
$1_{C}\leftrightsquigarrow_{\mathfrak{d}}(B\hookrightarrow C)\rho$.

We can generalize $\pi_{n}$ as follows. For each cubcat $C$ having vertex $v$,
let

$\tau_{n}(C,v)=\pi_{0}\Omega^{n}(C,v).$

The set $\tau_{n+1}(C,v)$ admits the extra structure of a monoid whose
products are induced by $\infty$-fold compositions on $C$ compatible with the
extension of $C$ to $\blacksquare^{\mathrm{op}}$.

######  Example 4.14.

For each monoid $M$, $\tau_{n}(\mathfrak{ner}\,M,\star)=\begin{cases}M&n=1\\\
0&n\neq 1\end{cases}$.

Extend first classical cohomology to a first directed $1$-cohomology

$H^{1}(C;\tau)=[C,\mathfrak{ner}\,\tau]_{\mathfrak{d}}=\pi_{0}(\mathfrak{ner}\,\tau)^{C},$

a commutative monoid natural in commutative monoids $\tau$ and cubical sets
$C$.

######  Example 4.15.

For a cubical model $\square[1]/\partial\square[1]$ of the circle,

$H^{1}(\square[1]/\partial\square[1];\tau)=[\mathbb{N},\tau]_{\mathrm{T}_{1}\mathfrak{d}}={\raisebox{1.99997pt}{$\tau$}\left/\raisebox{-1.99997pt}{$\equiv$}\right.}$

where $\equiv$ is the smallest monoid congruence on $\tau$ equating two
elements if they coincide after adding a common element to both of them. This
congruence $\equiv$ is trivial precisely when $\tau$ is cancellative.

Group-completion induces a monoid homomorphism

$H^{1}(C;\tau\rightarrow\tau[\tau]^{-1}):H^{1}(C;\tau)\rightarrow
H^{1}(C;\tau[\tau]^{-1})$

from directed cohomology to classical cohomology, natural in commutative
monoid coefficients $\tau$. Directed $1$-cohomology and this natural
comparison homomorphism generalize to higher $n>1$ by representing
$H^{n}(-;\tau)$ with a suitable iterated delooping construction $W^{n}\tau$ on
the (discrete cubical) commutative monoid $\tau$.

###  4.4. Algebraic

We recall three homotopy theories on the category $\mathrm{\bf Cat}$ of small
categories and functors between them, in order of increasing refinement. All
three of these homotopy theories coincide on the full subcategory $\mathrm{\bf
Gpd}$ of small groupoids.

####  4.4.1. Classical

The class of Thomason weak equivalences is the smallest retract-closed class
$\mathscr{W}$ of $\mathrm{\bf Cat}$-morphisms having the 2-out-of-3 property
and containing all terminal functors such that a functor
$\alpha:\mathcal{X}\rightarrow\mathcal{Y}$ lies in $\mathscr{W}$ whenever the
induced functor $\beta\alpha/o\rightarrow\beta/o$ lies in $\mathscr{W}$ for
each functor $\beta:\mathcal{Y}\rightarrow\mathcal{Z}$ and
$\mathcal{Z}$-object $o$ [11, Theorem 2.2.11]. The localization of
$\mathrm{\bf Cat}$ by the Thomason weak equivalences exists [65] and will be
referred to as the classical homotopy category of $\mathrm{\bf Cat}$.

######  Example 4.16.

A sufficient and intrinsic condition for a $\mathrm{\bf Cat}$-morphism

$\zeta:\mathcal{X}\rightarrow\mathcal{Y}$

to be a Thomason weak equivalence is if $o/\zeta$ has a terminal object for
each $\mathcal{X}$-object $o$ by Quillen’s Theorem A.

It is difficult to give a complete characterization of the Thomason weak
equivalences that is at once explicit and intrinsic, at least without
reference to the simplex category $\Delta$ (cf. [16].) We write $h(\mathrm{\bf
Cat})$ and $h(\mathrm{\bf Gpd})$ for the respective localizations of
$\mathrm{\bf Cat}$ and $\mathrm{\bf Gpd}$ by their Thomason weak equivalences.
Thomason weak equivalences can be defined more generally for $n$-fold functors
between $n$-fold categories. These weak equivalences, part of Thomason model
structures on categories of $n$-fold small categories for each $n=1,2,\ldots$
[22], model classical homotopy theory in terms of strict (higher) categorical
structure.

####  4.4.2. Directed

Let $\mathrm{T}_{1}\mathfrak{d}_{n}$ denote the interval object

$\mathrm{T}_{1}\mathfrak{d}_{n}=\mathrm{T}_{1}(\mathfrak{d}_{n})=(\mathrm{T}_{1}\mathfrak{d})_{n}:\square_{1}\rightarrow\mathrm{\bf
Cat}.$

In particular, $\mathrm{T}_{1}\mathfrak{d}$ is the cannonical interval object

$\mathrm{T}_{1}\mathfrak{d}:\square_{1}\hookrightarrow\mathrm{\bf Cat}.$

The homotopy theory in which weak equivalences are the
$(\mathrm{T}_{1}\mathfrak{d})_{*}$-equivalences [56], as well a slightly
weaker homotopy theory [36] in which homotopy is defined by a single path
object in terms of $\mathfrak{d}_{1},\mathfrak{d}_{2},\ldots$, have been
studied previously. The $(\mathrm{T}_{1}\mathfrak{d})_{*}$-equivalences, while
not the weak equivalences of a model structure, are the weak equivalences of a
$\Lambda$-cofibration category [36] structure on $\mathrm{\bf Cat}$. While
each $(\mathrm{T}_{1}\mathfrak{d})_{*}$-equivalence is a Thomason weak
equivalence, not each Thomason weak equivalence is a
$(\mathrm{T}_{1}\mathfrak{d})_{*}$-equivalence. Write $d(\mathrm{\bf Cat})$
for the quotient of $\mathrm{\bf Cat}$ by the congruence relation
$\leftrightsquigarrow_{\mathrm{T}_{1}\mathfrak{d}}$.

######  Example 4.17.

For parallel $\mathrm{\bf Cat}$-morphisms $\alpha,\beta$, a
$\mathrm{T}_{1}\mathfrak{d}$-homotopy

$\alpha\leadsto\beta$

is exactly a natural transformation $\alpha\rightarrow\beta$. In particular, a
(left or right) adjoint in $\mathrm{\bf Cat}$ is a
$\mathrm{T}_{1}\mathfrak{d}_{1}$-equivalence.

######  Example 4.18.

Consider a functor of small categories

$F:\mathcal{X}\rightarrow\mathcal{Y}.$

The functor $F$ is sometimes referred to as a future equivalence [31] if $F$
is a $\mathrm{T}_{1}\mathfrak{d}$-equivalence and a past equivalence [31] if
$F$ is a $(\mathrm{T}_{1}\mathfrak{d})^{\mathrm{op}}$-equivalence. Future
equivalences and past equivalences preserve certain properties of interest in
state space analyses, such as terminal objects and initial objects
respectively [27].

####  4.4.3. Categorical

There exist natural isomorphisms

$\Pi_{1}\mathfrak{d}\cong\Pi_{1}(\mathfrak{d}_{n})=(\Pi_{1}\mathfrak{d})_{n},\quad
n=0,1,2,\ldots$

A categorical equivalence between small categories is exactly a
$\Pi_{1}\mathfrak{d}$-equivalence. Every categorical equivalence is a
$\mathrm{T}_{1}\mathfrak{d}$-equivalence because localization defines a
natural transformation
$\mathrm{T}_{1}\mathfrak{d}\rightarrow\Pi_{1}\mathfrak{d}$.

###  4.5. Comparisons

The different homotopy theories can be compared. The classical homotopy
theories of small categories, cubical sets, simplicial sets, and topological
spaces are all equivalent to one another. The directed homotopy theories of
cubical sets and streams are equivalent to one another, with the directed
homotopy theory of small categories acting as a special case.

####  4.5.1. Classical

We can compare different classical homotopy theories.

######  Proposition 4.19.

Topological realization defines the left map of a Quillen equivalence

$|-|:\hat{\square}\leftrightarrows\mathrm{\bf Top}$

between $\hat{\square}$ equipped with its test model structure and
$\mathrm{\bf Top}$ equipped with its q-model structure.

A simple consequence is a cubical description of homotopy groups.

######  Corollary 4.20.

For each $n$, the function

$\tau_{n}(C,v)\rightarrow\tau_{n}(|C|,|v|)$

induced from the unit of the adjunction with left adjoint
$|-|:\hat{\square}\rightarrow\mathrm{\bf Top}$ is bijective, and in particular
is a group isomorphism in the case $n>0$, for all fibrant cubical sets $C$ and
vertices $v\in C_{0}$.

Previously established equivalences [[11, Theorem 2.2.11], [60, §II.3,
§VI.3.3.1]] and a categorical equivalence between classical homotopy
categories of simplicial sets and cubical sets in the sense of this paper
[Proposition C.4] imply the following.

######  Corollary 4.21.

The functor $\mathfrak{ner}\,$ induces a categorical equivalence

$h(\mathrm{\bf Cat})\simeq h(\hat{\square}).$

###### proof of Proposition 4.10.

Consider a cubical function $\psi:A\rightarrow B$. The diagram

${\Pi_{1}A}$${\Pi_{1}B}$${\Pi_{1}A}$${\Pi_{1}|B|}$$\scriptstyle{\Pi_{1}\psi}$$\scriptstyle{\Pi_{1}|\psi|}$$\scriptstyle{\Pi_{1}|\psi|}$

in which $\Pi_{1}$ in the bottom row denotes the fundamental groupoid of a
topological space and the vertical arrows are inclusions of fundamental
groupoids induced by topological realization, commutes. The vertical arrows
are categorical equivalences because every point in a topological realization
is path-connected to a vertex. Thus if $\psi$ is a classical weak equivalence,
$|\psi|$ is a classical homotopy equivalence [Proposition 4.19], hence
$\Pi_{1}|A|\rightarrow\Pi_{1}|B|$ is a categorical equivalence, and hence
$\Pi_{1}\psi:\Pi_{1}A\rightarrow\Pi_{1}B$ is a categorical equivalence. ∎

####  4.5.2. Directed

We now give the main results.

######  Theorem 4.22.

There exist
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright_{*}$-homotopies

$\upharpoonleft\\!\\!{\eta_{C}}\\!\\!\downharpoonright\epsilon_{\upharpoonleft\\!\\!{\;C\;}\\!\\!\downharpoonright}\leftrightsquigarrow_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright_{*}}1_{\upharpoonleft\\!\\!{\;\mathfrak{sing}\,\,\upharpoonleft\\!\\!{\;C\;}\\!\\!\downharpoonright\;}\\!\\!\downharpoonright}$

natural in cubical sets $C$.

###### Proof.

Write $S$ for $\mathfrak{sing}\,$. Write $\eta^{\prime},\epsilon^{\prime}$ for
the respective unit and counit of
$\upharpoonleft\\!\\!{-}\\!\\!\downharpoonright\dashv\mathfrak{sing}\,$. Let
$\mathscr{R}$ denote the full subcategory of $\hat{\square}$ consisting of
cubical sets whose atomic subpresheaves are all isomorphic to representables.

Consider the solid stream maps in the diagram

${\upharpoonleft\\!\\!{S\upharpoonleft\\!\\!{C_{\theta;o}^{*}}\\!\\!\downharpoonright}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{S\upharpoonleft\\!\\!{C_{\theta;o}^{*}}\\!\\!\downharpoonright}\\!\\!\downharpoonright^{2}}$${\upharpoonleft\\!\\!{\square[1]^{n_{\theta}}}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{S\upharpoonleft\\!\\!{C^{*}_{\theta;o}}\\!\\!\downharpoonright}\\!\\!\downharpoonright^{\upharpoonleft\\!\\!{\;\mathfrak{d}_{3}[1]\;}\\!\\!\downharpoonright}}$${\upharpoonleft\\!\\!{S\upharpoonleft\\!\\!{C_{\theta}}\\!\\!\downharpoonright}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{S\upharpoonleft\\!\\!{C_{\theta}}\\!\\!\downharpoonright}\\!\\!\downharpoonright^{2}}$${\upharpoonleft\\!\\!{S\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright}\\!\\!\downharpoonright^{\upharpoonleft\\!\\!{\;\mathfrak{d}_{3}[1]\;}\\!\\!\downharpoonright}}$$\scriptstyle{1_{\upharpoonleft\\!\\!{\;S\upharpoonleft\\!\\!{\;C^{*}_{\theta;o}\;}\\!\\!\downharpoonright\;}\\!\\!\downharpoonright}\times\epsilon^{\prime}_{S\upharpoonleft\\!\\!{\;C^{*}_{\theta;o}\;}\\!\\!\downharpoonright}\upharpoonleft\\!\\!{\;\eta^{\prime}_{C^{*}_{\theta;o}}\;}\\!\\!\downharpoonright}$$\scriptstyle{S\upharpoonleft\\!\\!{\;\Lambda^{*}_{\theta;o}\;}\\!\\!\downharpoonright}$$\scriptstyle{S\upharpoonleft\\!\\!{\;\Lambda^{*}_{\theta;o}\;}\\!\\!\downharpoonright^{\otimes
2}}$$\scriptstyle{\upharpoonleft\\!\\!{\;\theta\;}\\!\\!\downharpoonright}$$\scriptstyle{{S\upharpoonleft\\!\\!{\;\Lambda^{*}_{\theta;o}\;}\\!\\!\downharpoonright^{{\upharpoonleft\\!\\!{\;\mathfrak{d}_{1}[1]\;}\\!\\!\downharpoonright}}}}$$\scriptstyle{1_{\upharpoonleft\\!\\!{\;S\upharpoonleft\\!\\!{\;C_{\theta}\;}\\!\\!\downharpoonright\;}\\!\\!\downharpoonright}\times\epsilon^{\prime}_{S\upharpoonleft\\!\\!{\;C\;}\\!\\!\downharpoonright}\upharpoonleft\\!\\!{\;\eta^{\prime}_{C_{\theta}}\;}\\!\\!\downharpoonright}$

Let $\theta$ denote a
$(\square/S\upharpoonleft\\!\\!{-}\\!\\!\downharpoonright)$-object. For $k\gg
0$,
$\mathfrak{adj}(\theta)\varphi_{\square[1]^{n_{\theta}};2^{k}}:\upharpoonleft\\!\\!{\mathfrak{sd}_{2^{k}}\square[1]^{n_{\theta}}}\\!\\!\downharpoonright$
maps each closed cell into the open start of a vertex by
$\mathbb{I}^{n_{\theta}}$ compact. Thus there exists an
$(\mathscr{R}/C_{\theta})$-object
$\Lambda^{*}_{\theta;o}:C^{*}_{\theta;o}\rightarrow C_{\theta}$ and
$\mathfrak{d}_{*}$-homotopy, natural in $\theta$ and object $o$ in a small
cofiltered category, and cubical function $\theta^{*}$, up to which the left
triangle commutes [Lemmas 3.23 and 4.6]. There exists a dotted stream map and
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright_{*}$-homotopy, both
natural in $\theta$ and $o$, up to which the top triangle commutes [Lemma
4.8]. Thus there exist the remaining dotted stream maps from
$\upharpoonleft\\!\\!{\square[1]^{n_{\theta}}}\\!\\!\downharpoonright$,
natural in $\theta$ and $o$ and
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright_{*}$-homotopy, up
to which the bottom triangle commutes. Thus there exists the lower left
diagonal dotted stream map, natural in cubical sets $C_{\theta}$, making the
entire diagram commute up to
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright_{*}$-homotopy. ∎

######  Corollary 4.23.

There exist $\mathfrak{d}_{*}$-homotopies

$\eta_{\mathfrak{sing}\,\,X}\mathfrak{sing}\,\epsilon_{\upharpoonleft\\!\\!{\;\mathfrak{sing}\,\,X\;}\\!\\!\downharpoonright}\leftrightsquigarrow_{\mathfrak{d}_{*}}1_{\mathfrak{sing}\,\upharpoonleft\\!\\!{\;\mathfrak{sing}\,\,X\;}\\!\\!\downharpoonright}$

natural in streams $X$.

###### Proof.

Let $S=\mathfrak{sing}\,$. Let $X$ denote a stream. The stream map
$\epsilon_{\upharpoonleft\\!\\!{\;SX\;}\\!\\!\downharpoonright}$ is a
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright_{*}$-equivalence
[Theorem 4.22] and hence the cubical function
$S\epsilon_{\upharpoonleft\\!\\!{\;SX\;}\\!\\!\downharpoonright}$ is a
$\mathfrak{d}_{*}$-equivalence because $S$ sends
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright_{*}$-equivalences
to $\mathfrak{d}_{*}$-equivalences. The
$(\hat{\square}/\leftrightsquigarrow_{\mathfrak{d}})$-isomorphism represented
by $S\epsilon_{\upharpoonleft\\!\\!{\;SX\;}\\!\\!\downharpoonright}$ admits a
section and hence inverse represented by $\eta_{SX}$. ∎

A simple application is a proof of Proposition 4.13.

###### proof of Proposition 4.13.

Let $\eta$ denote the unit of
$\upharpoonleft\\!\\!{-}\\!\\!\downharpoonright\dashv\mathfrak{sing}\,$. Let
$C=\mathfrak{sing}\,\upharpoonleft\\!\\!{B}\\!\\!\downharpoonright$. The
stream map $\upharpoonleft\\!\\!{\eta_{B}}\\!\\!\downharpoonright$ is a
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright_{*}$-equivalence
[Theorem 4.22], hence $|\eta_{B}|$ is a $|\mathfrak{d}|$-equivalence, hence
$\eta_{B}$ is an acyclic cofibration from a fibrant object in the test model
structure on $\hat{\square}$, and hence $\eta_{B}$ admits a retraction
$\mu_{B}$. Then $\mathfrak{sing}\,\mu_{B}$ represents a retraction and hence
unique inverse to the isomorphism in
$\hat{\square}/\leftrightsquigarrow_{\mathfrak{d}}$ [Corollary 4.23]
represented by $\mathfrak{sing}\,\eta_{B}$. It therefore follows that
$\mathfrak{sing}\,\upharpoonleft\\!\\!{\eta_{B}\mu_{B}}\\!\\!\downharpoonright$
is $\leftrightsquigarrow_{\mathfrak{d}}$-equivalent to
$1_{\mathfrak{sing}\,\upharpoonleft\\!\\!{\;C\;}\\!\\!\downharpoonright}$.
Thus the diagram

${C}$${C}$${\mathfrak{sing}\,\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright}$${\mathfrak{sing}\,\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright}$$\scriptstyle{\eta_{B}\mu_{B}}$$\scriptstyle{\eta_{C}}$$\scriptstyle{\eta_{C}}$$\scriptstyle{1_{\mathfrak{sing}\,\upharpoonleft\\!\\!{\;C\;}\\!\\!\downharpoonright}}$

commutes up to $\leftrightsquigarrow_{\mathfrak{d}}$. Therefore the top
horizontal arrow is $\leftrightsquigarrow_{\mathfrak{d}}$-equivalent to
$1_{C}$ because the vertical arrows are $\mathfrak{d}$-equivalences [Corollary
4.23]. ∎

######  Corollary 4.24.

The following are equivalent for a ${\mathscr{G}}$-stream map $f$.

  1. (1)

$\mathfrak{sing}\,\,f$ is a $\mathfrak{d}^{{\mathscr{G}}}_{*}$-equivalence

  2. (2)

$\upharpoonleft\\!\\!{\mathfrak{sing}\,f}\\!\\!\downharpoonright$ is a
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright^{{\mathscr{G}}}$-equivalence

###### Proof.

Let $\eta$ denote the unit of the adjunction
$\upharpoonleft\\!\\!{-}\\!\\!\downharpoonright^{{\mathscr{G}}}\dashv\mathfrak{sing}\,^{{\mathscr{G}}}$.

If (1) then (2) by
$\upharpoonleft\\!\\!{-}\\!\\!\downharpoonright^{{\mathscr{G}}}$ sending
$\mathfrak{d}^{{\mathscr{G}}}_{*}$-equivalences to
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright^{{\mathscr{G}}}_{*}$-equivalences.

Suppose (2). Take $f:X\rightarrow Y$. In the commutative diagram

${\mathfrak{sing}\,\,X}$${\mathfrak{sing}\,\,Y}$${\mathfrak{sing}\,\,\upharpoonleft\\!\\!{\mathfrak{sing}\,\,X}\\!\\!\downharpoonright}$${\mathfrak{sing}\,\,\upharpoonleft\\!\\!{\mathfrak{sing}\,\,Y}\\!\\!\downharpoonright,}$$\scriptstyle{\mathfrak{sing}\,\,f}$$\scriptstyle{\eta_{\mathfrak{sing}\,\,X}}$$\scriptstyle{\eta_{\mathfrak{sing}\,\,Y}}$$\scriptstyle{\mathfrak{sing}\,\,\upharpoonleft\\!\\!{\;f\;}\\!\\!\downharpoonright}$

the bottom horizontal arrow is a
$\mathfrak{d}^{{\mathscr{G}}}_{*}$-equivalence because $\mathfrak{sing}\,$
sends
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright^{{\mathscr{G}}}_{*}$-equivalences
to $\mathfrak{d}^{{\mathscr{G}}}_{*}$-equivalences, the vertical arrows are
$\mathfrak{d}^{{\mathscr{G}}}_{*}$-equivalences [Corollary 4.23], and hence
the top horizontal arrow is also a
$\mathfrak{d}^{{\mathscr{G}}}_{*}$-equivalence. ∎

######  Corollary 4.25.

Fix a ${\mathscr{G}}$-cubical function $\psi:A\rightarrow B$. The following
are equivalent.

  1. (1)

$\mathfrak{sing}\,\upharpoonleft\\!\\!{\psi}\\!\\!\downharpoonright$ is a
$\mathfrak{d}^{{\mathscr{G}}}$-equivalence

  2. (2)

$\upharpoonleft\\!\\!{\psi}\\!\\!\downharpoonright$ is a
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright^{{\mathscr{G}}}$-equivalence

  3. (3)

$[\psi,C]_{\mathfrak{d}^{{\mathscr{G}}}}$ is a bijection for each cubcat $C$

If $A$ and $B$ are both ${\mathscr{G}}$-cubcats, then the above conditions are
equivalent to $\psi$ being a $\mathfrak{d}^{{\mathscr{G}}}_{*}$-equivalence.

###### Proof.

Let $\epsilon$ denote the counit of
$\upharpoonleft\\!\\!{-}\\!\\!\downharpoonright^{{\mathscr{G}}}\dashv\mathfrak{sing}\,^{{\mathscr{G}}}$.

Take $\psi:A\rightarrow B$. If (2) then (1) because
$\mathfrak{sing}\,^{{\mathscr{G}}}$ sends
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright_{*}$-equivalences
to $\mathfrak{d}_{*}$-equivalences.

Suppose (1). In the commutative diagram

${\upharpoonleft\\!\\!{A}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{B}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{\mathfrak{sing}\,\,\upharpoonleft\\!\\!{A}\\!\\!\downharpoonright}\\!\\!\downharpoonright}$${\upharpoonleft\\!\\!{\mathfrak{sing}\,\,\upharpoonleft\\!\\!{B}\\!\\!\downharpoonright}\\!\\!\downharpoonright}$$\scriptstyle{\upharpoonleft\\!\\!{\;\psi\;}\\!\\!\downharpoonright^{{\mathscr{G}}}}$$\scriptstyle{\epsilon_{\upharpoonleft\\!\\!{\;A\;}\\!\\!\downharpoonright}}$$\scriptstyle{\upharpoonleft\\!\\!{\;\mathfrak{sing}\,\,\upharpoonleft\\!\\!{\;\psi\;}\\!\\!\downharpoonright\;}\\!\\!\downharpoonright}$$\scriptstyle{\epsilon_{\upharpoonleft\\!\\!{\;B\;}\\!\\!\downharpoonright}}$

the bottom horizontal arrow is a
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright^{{\mathscr{G}}}_{*}$-equivalence
because $\upharpoonleft\\!\\!{-}\\!\\!\downharpoonright$ sends
$\mathfrak{d}^{{\mathscr{G}}}_{*}$-equivalences to
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright^{{\mathscr{G}}}_{*}$-equivalences,
the vertical arrows are
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright^{{\mathscr{G}}}_{*}$-equivalences
[Theorem 4.22], and hence the top arrow is also a
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright^{{\mathscr{G}}}_{*}$-equivalence.
Hence (2).

Suppose (2). Then
$[\upharpoonleft\\!\\!{\psi}\\!\\!\downharpoonright,X]_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright^{{\mathscr{G}}}}=[\psi,\mathfrak{sing}\,\,X]_{\mathfrak{d}^{{\mathscr{G}}}}$
is a bijection for each ${\mathscr{G}}$-stream $X$ by the Yoneda Lemma. For
each ${\mathscr{G}}$-cubcat $C$,
$\eta_{C}:C\rightarrow\mathfrak{sing}\,\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright$
is a $\mathfrak{d}^{{\mathscr{G}}}_{*}$-equivalence [Lemmas 3.37 and 4.6].
Hence for each ${\mathscr{G}}$-cubcat $C$,
$[\psi,C]_{\mathfrak{d}^{{\mathscr{G}}}}=[\psi,\mathfrak{sing}\,\,\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright]_{\mathfrak{d}^{{\mathscr{G}}}}$
is a bijection. Hence (3).

Suppose (3). Then
$[\psi,\mathfrak{sing}\,\,X]_{\mathfrak{d}^{{\mathscr{G}}}}=[\upharpoonleft\\!\\!{\psi}\\!\\!\downharpoonright,X]_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright^{{\mathscr{G}}}}$
is a bijection for each stream $X$. Hence $\psi$ represents an isomorphism in
$\mathrm{\bf
DiTop}/\leftrightsquigarrow_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright^{{\mathscr{G}}}}$
by the Yoneda Lemma. Hence (2).

If $\psi$ is a $\mathfrak{d}^{{\mathscr{G}}}_{*}$-equivalence, then (3)
because
$[-,C]_{\mathfrak{d}^{{\mathscr{G}}}}:(\hat{\square}^{{\mathscr{G}}}/\leftrightsquigarrow_{\mathfrak{d}^{{\mathscr{G}}}})\rightarrow\mathrm{\bf
Set}$, a functor, preserves isomorphisms. Suppose $A$ and $B$ are both
${\mathscr{G}}$-cubcats and (3). Then $\psi$ is a
$\mathfrak{d}^{{\mathscr{G}}}_{*}$-equivalence by the Yoneda Lemma. ∎

######  Corollary 4.26.

The following are equivalent for a ${\mathscr{G}}$-functor $\zeta$ of
${\mathscr{G}}$-categories.

  1. (1)

$\zeta$ is a $(\mathrm{T}_{1}\mathfrak{d})^{{\mathscr{G}}}_{*}$-equivalence

  2. (2)

$\mathfrak{ner}\,\zeta$ is a $\mathfrak{d}^{{\mathscr{G}}}_{*}$-equivalence

  3. (3)

$\upharpoonleft\\!\\!{\mathfrak{ner}\,\zeta}\\!\\!\downharpoonright$ is a
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright^{{\mathscr{G}}}_{*}$-equivalence

###### Proof.

If (3) then $[\mathfrak{ner}\,\zeta,C]_{\mathfrak{d}^{{\mathscr{G}}}}$ is a
bijection for each ${\mathscr{G}}$ cubcat $C$ and hence (2) [Corollary 4.25]
because cubical nerves are cubcats [Proposition 3.36]. If (2) then (1) because
$\mathrm{T}_{1}^{{\mathscr{G}}}$ sends
$\mathfrak{d}^{{\mathscr{G}}}_{*}$-equivalences to
$(\mathrm{T}_{1}\mathfrak{d})^{{\mathscr{G}}}_{*}$-equivalences. If (1) then
(3) because
$\upharpoonleft\\!\\!{\mathfrak{ner}\,-}\\!\\!\downharpoonright^{{\mathscr{G}}}$
sends $(\mathrm{T}_{1}\mathfrak{d})^{{\mathscr{G}}}_{*}$-equivalences to
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright^{{\mathscr{G}}}_{*}$-equivalences.
∎

Our main result, when specialized for the case ${\mathscr{G}}=\star$ of
trivial diagrams, is a directed analogue of the classical Quillen equivalence
between cubical sets and topological spaces. Recall that a class $\mathscr{W}$
of morphisms in a category $\mathscr{X}$ for which the localization
$\mathscr{X}[\mathscr{W}^{-1}]$ exists is saturated if it coincides with the
isomorphisms in the localization $\mathscr{X}[\mathscr{W}^{-1}]$ of
$\mathscr{X}$ by $\mathscr{W}$.

######  Corollary 4.27.

There exist dotted localizations in the diagram

${\mathrm{\bf
Cat}^{{\mathscr{G}}}}$${\hat{\square}^{{\mathscr{G}}}}$${\mathrm{\bf
DiTop}^{{\mathscr{G}}}}$${d\left(\mathrm{\bf
Cat}^{{\mathscr{G}}}\right)}$${d\left(\hat{\square}^{{\mathscr{G}}}\right)}$${d\left(\mathrm{\bf
DiTop}^{{\mathscr{G}}}\right)}$$\scriptstyle{\mathfrak{ner}\,}$$\scriptstyle{\upharpoonleft\\!\\!{\;-\;}\\!\\!\downharpoonright}$$\scriptstyle{\simeq}$

by the following respective saturated classes of morphisms: the
$\mathrm{T}_{1}{}\mathfrak{d}^{{\mathscr{G}}}$-equivalences; those
${\mathscr{G}}$-cubical functions $\psi$ for which
$\upharpoonleft\\!\\!{\psi}\\!\\!\downharpoonright^{{\mathscr{G}}}$ is a
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright^{{\mathscr{G}}}$-equivalence;
and those ${\mathscr{G}}$-stream maps $f$ for which
$\mathfrak{sing}\,^{{\mathscr{G}}}\,f$ is a
$\mathfrak{d}^{{\mathscr{G}}}$-equivalence. There exist dotted horizontal
functors making the entire diagram commute up to natural isomorphism with the
left dotted horizontal functor a fully faithful embedding and the right dotted
horizontal functor an adjoint categorical equivalence. A
${\mathscr{G}}$-cubical function $\psi$ represents an isomorphism in
$d(\hat{\square}^{{\mathscr{G}}})$ if and only if
$[\psi,C]_{\mathfrak{d}^{{\mathscr{G}}}}$ is a bijection for all cubcats $C$.

###### Proof.

Let $\hat{d}(\hat{\square}^{{\mathscr{G}}})$ and $\hat{d}(\mathrm{\bf
DiTop}^{{\mathscr{G}}})$ denote the quotient categories

$\hat{d}(\hat{\square}^{{\mathscr{G}}})={\raisebox{1.99997pt}{$\hat{\square}^{{\mathscr{G}}}$}\left/\raisebox{-1.99997pt}{$\leftrightsquigarrow_{\mathfrak{d}}$}\right.}\quad\hat{d}(\mathrm{\bf
DiTop}^{{\mathscr{G}}})={\raisebox{1.99997pt}{$\mathrm{\bf
DiTop}^{{\mathscr{G}}}$}\left/\raisebox{-1.99997pt}{$\leftrightsquigarrow_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright}{}$}\right.}$

Write $S$ for $\mathfrak{sing}\,$.

The comonad $Q$ on $\hat{d}(\mathrm{\bf DiTop}^{{\mathscr{G}}})$ induced by
the comonad $\upharpoonleft\\!\\!{\mathfrak{sing}\,-}\\!\\!\downharpoonright$
is idempotent [Theorem 4.22]. Thus we can define $d(\mathrm{\bf
DiTop}^{{\mathscr{G}}})$ to be the localization
$\mathscr{X}[\mathscr{W}^{-1}]$ of a saturated class $\mathscr{W}$ where
$\mathscr{X}=\hat{d}(\mathrm{\bf DiTop}^{{\mathscr{G}}})$ and $\mathscr{W}$ is
the saturated class of all $Q$-isomorphisms, or equivalently
$\mathscr{X}=\mathrm{\bf DiTop}^{{\mathscr{G}}}$ and $\mathscr{W}$ is the
class of all ${\mathscr{G}}$-stream maps $f$ for which
$\upharpoonleft\\!\\!{\mathfrak{sing}\,^{{\mathscr{G}}}f}\\!\\!\downharpoonright^{{\mathscr{G}}}$
represents a $\hat{d}(\mathrm{\bf DiTop}^{{\mathscr{G}}})$-isomorphism or
equivalently those ${\mathscr{G}}$-stream maps $f$ for which $Sf$ are
$\mathfrak{d}^{{\mathscr{G}}}_{*}$-equivalences [Corollary 4.24].

The monad $R$ on $\hat{d}(\mathrm{\bf DiTop}^{{\mathscr{G}}})$ induced by the
monad $\mathfrak{sing}\,\upharpoonleft\\!\\!{-}\\!\\!\downharpoonright$ is
idempotent [Corollary 4.23]. Thus we can take
$d(\hat{\square}^{{\mathscr{G}}})=\mathscr{X}[\mathscr{W}^{-1}]$ where
$\mathscr{X}=\hat{d}(\hat{\square}^{{\mathscr{G}}})$ and $\mathscr{W}$ is the
saturated class of all $R$-isomorphisms, or equivalently
$\mathscr{X}=\hat{\square}^{{\mathscr{G}}}$ and $\mathscr{W}$ consists of the
${\mathscr{G}}$-cubical functions $\psi$ for which
$\mathfrak{sing}\,^{{\mathscr{G}}}\upharpoonleft\\!\\!{\psi}\\!\\!\downharpoonright^{{\mathscr{G}}}$
represents a $\hat{d}(\hat{\square}^{{\mathscr{G}}})$-isomorphism, or
equivalently $\mathscr{X}=\hat{\square}^{{\mathscr{G}}}$ and $\mathscr{W}$
consists of the ${\mathscr{G}}$-cubical functions $\psi$ for which
$\upharpoonleft\\!\\!{\psi}\\!\\!\downharpoonright^{{\mathscr{G}}}$ are
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright^{{\mathscr{G}}}_{*}$-equivalences
[Corollary 4.25].

The adjunction
$\upharpoonleft\\!\\!{-}\\!\\!\downharpoonright^{{\mathscr{G}}}\dashv\mathfrak{sing}\,^{{\mathscr{G}}}$
induces an adjunction $d(\hat{\square}^{{\mathscr{G}}})\leftrightarrows
d(\mathrm{\bf DiTop}^{{\mathscr{G}}})$, a categorical equivalence because the
unit and counit are natural isomorphisms [Theorem 4.22 and Corollary 4.23].
The fully faithful embedding $\mathfrak{ner}\,^{{\mathscr{G}}}$ passes to a
fully faithful embedding $d(\mathrm{\bf Cat}^{{\mathscr{G}}})\hookrightarrow
d(\hat{\square}^{{\mathscr{G}}})$ [Corollary 4.26]. ∎

A consequence is an equivalence between combinatorial and topological
constructions of directed homotopy monoids.

######  Corollary 4.28.

The unit of the adjunction
$\upharpoonleft\\!\\!{-}\\!\\!\downharpoonright\dashv\mathfrak{sing}$ induces
a bijection

$\tau_{n}\left(C,v\right)\cong\tau_{n}\left(\mathfrak{sing}\,\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright,v\right)\quad
n=0,1,2,\ldots$

which, for the case $n>0$, is a monoid isomorphism, for each cubical set $C$
with vertex $v\in C_{0}$.

######  Example 4.29.

For each monoid $M$,

$\tau_{n}\left(\upharpoonleft\\!\\!{\mathfrak{ner}\,M}\\!\\!\downharpoonright,\star\right)=\tau_{n}\left(\mathfrak{ner}\,M,\star\right)=\begin{cases}M&n=1\\\
0&n\neq 1\end{cases}$

follows because $\Omega^{n}(\mathfrak{ner}\,M)$ is the discrete cubical set
$M$ for $n=1$ and $\star$ otherwise, even though the underlying classifying
space $|\mathfrak{ner}\,M|$ can realize any classical weak type [53].

######  Corollary 4.30.

For each cubcat $C$, $\upharpoonleft\\!\\!{-}\\!\\!\downharpoonright$ induces
a natural bijection

$\pi_{0}B^{(-)}\cong[-,\upharpoonleft\\!\\!{B}\\!\\!\downharpoonright]_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright}:\hat{\square}^{\mathrm{op}}\rightarrow\mathrm{\bf
Set}.$

A special case is the equivalence between cubical directed $1$-cohomology and
singular directed $1$-cohomology. Write $H^{1}_{\mathfrak{sing}\,}(X;\tau)$
for the commutative monoid
$[\mathfrak{sing}\,\,X,\mathfrak{ner}\,\,\tau]_{\mathfrak{d}}$ natural in
streams $X$ and commutative monoids $\tau$. This equivalence can be extended
to higher degrees for iterated delooping constructions $W^{n}\tau$ (cf. [41])
on discrete commutative monoids $\tau$ that extend $\mathfrak{ner}\,\tau$ for
the case $n=1$ and define cubcats for all $n$.

######  Corollary 4.31.

There exists an isomorphism

$H^{1}(\eta_{C}:C\rightarrow\mathfrak{sing}\,\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright;\tau):H^{1}_{\mathfrak{sing}\,}(\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright;\tau)\cong
H^{1}(C;\tau)$

of commutative monoids, natural in cubical sets $C$ and commutative monoids
$\tau$.

###### Proof.

In the commutative diagram

${{[\mathfrak{sing}\,\,\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright,\mathfrak{ner}\,\tau]_{\mathfrak{d}}}}$${{[C,\mathfrak{ner}\,\tau]}}$${{[\upharpoonleft\\!\\!{\mathfrak{sing}\,\,\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright}\\!\\!\downharpoonright,\upharpoonleft\\!\\!{\mathfrak{ner}\,\tau}\\!\\!\downharpoonright]_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright}}}$${{[\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright,\upharpoonleft\\!\\!{\mathfrak{ner}\,\tau}\\!\\!\downharpoonright]_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright}}}$$\scriptstyle{[\eta_{C},\mathfrak{ner}\,\tau]_{\mathfrak{d}}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{[\upharpoonleft\\!\\!{\;\eta_{C}\;}\\!\\!\downharpoonright,\upharpoonleft\\!\\!{\;\mathfrak{ner}\,\tau\;}\\!\\!\downharpoonright]}$

in which the vertical arrows are induced by
$\upharpoonleft\\!\\!{-}\\!\\!\downharpoonright$, the vertical arrows are
bijections [Corollary 4.30] by $\mathfrak{ner}\,\,\tau$ a cubcat [Proposition
3.36], the bottom horizontal arrow is a bijection by
$\upharpoonleft\\!\\!{\eta_{C}}\\!\\!\downharpoonright$ a
$\upharpoonleft\\!\\!{\mathfrak{d}}\\!\\!\downharpoonright$-equivalence
[Corollary 4.27], and hence the top horizontal arrow is also a bijection. ∎

######  Corollary 4.32.

For each cancellative commutative monoid $\tau$,

$H^{1}_{\mathfrak{sing}\,}(\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright;\tau)=\mathrm{\bf
Cat}(\mathrm{T}_{1}C,\tau)$

if the cubical set $C$ has a unique vertex.

###### Proof.

We have the following identities

$\displaystyle
H^{1}_{\mathfrak{sing}\,}(\upharpoonleft\\!\\!{C}\\!\\!\downharpoonright;\tau)$
$\displaystyle=[K,\upharpoonleft\\!\\!{\mathfrak{ner}\,\tau}\\!\\!\downharpoonright]_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright}$
$\displaystyle=[C,\mathfrak{ner}\,\tau]_{\mathfrak{d}}$
$\displaystyle=[\mathrm{T}_{1}C,\tau]_{\mathrm{T}_{1}\mathfrak{d}}$
$\displaystyle=\mathrm{\bf Cat}(\mathrm{T}_{1}C,\tau),$

with the last line by cancellativity. ∎

Some easy calculations follow for the unique two closed and connected
$(1+1)$-spacetimes that arise as directed realizations of (finite) cubical
sets.

######  Example 4.33.

Fix a cancellative commutative monoid $\tau$. Then

$H^{1}(T;\tau)=\tau^{2},$

where $T$ is the unique underlying stream of a time-oriented Lorentzian torus
[Figure 2], by identifying $T$ with the directed realization of
$(\square[1]/\partial\square[1])^{\otimes 2}$, whose fundamental category is
$\mathbb{N}^{2}$.

######  Example 4.34.

Fix a cancellative commutative monoid $\tau$. Then

$H^{1}(K;\tau)=\tau\times_{2\tau}\tau,$

where $K$ is the unique underlying stream of a time-oriented Lorentzian Klein
bottle [Figure 2], by identifying $K$ with the directed realization of the
quotient $C$ of $\square[1]^{2}$ by the smallest equivalence relation
identifying $\delta_{\pm 1;2}$ with $\delta_{\mp 2;2}$ and calculating
$\mathrm{T}_{1}C$ to be the monoid $\langle x,y\;|\;x^{2}=y^{2}\rangle$.

Small categories are insufficient for modelling all directed homotopy types.

######  Example 4.35.

For each $n>1$ and small category $\mathcal{X}$, every stream map

$\upharpoonleft\\!\\!{{\square[1]^{n}}/{\partial\square[1]^{n}}}\\!\\!\downharpoonright\rightarrow\upharpoonleft\\!\\!{\mathfrak{ner}\,\mathcal{X}}\\!\\!\downharpoonright$

is
$\leftrightsquigarrow_{\upharpoonleft\\!\\!{\;\mathfrak{d}\;}\\!\\!\downharpoonright}{}$-homotopic
to a constant stream map. It therefore follows that higher directed spheres
$\upharpoonleft\\!\\!{{\square[1]^{n}}/{\partial\square[1]^{n}}}\\!\\!\downharpoonright$
do not have the h-homotopy, much less d-homotopy type, type of directed
realizations of cubical nerves of small categories. Intuitively, the cubical
model $\square[1]^{n}/\partial\square[1]^{n}$ of a directed sphere presents a
cubcat freely generated by a single $n$-cell between a single vertex. In fact,
these higher directed spheres likely do not have the h-homotopy type of
directed realizations of cubical models of $(1,\infty)$-categories [9, 17].
Thus directed homotopy types encode higher categories, albeit up to directed
homotopy, more general than $(1,\infty)$-categories (cf. [18]).

The theorem in §1 follows from Corollary 4.27, Example 4.35, and equivalent
formulations of the classical homotopy category.

##  5\. Conclusion

Much early work in directed homotopy theory went into generalizing categorical
equivalences between groupoids to notions of equivalences between small
categories that preserve computational behavior of interest (eg. [21, 26, 27]
and [Example 4.18]). These directed equivalences are stronger than
$\mathrm{T}_{1}\mathfrak{d}_{*}$-equivalences but weaker than categorical
equivalences. Unfortunately, these directed equivalences have poor formal
properties compared to $\mathrm{T}_{1}\mathfrak{d}_{*}$-equivalences;
$\mathrm{\bf Cat}$ admits a localization with respect to the latter but not
the former. The typical application was to capture the behavior of executions
in a concurrent program having a directed state space $X$ by computing a
minimal model of $\mathrm{T}_{1}\mathfrak{sing}\,X$ with respect to the
relevant notion of directed equivalence. It is in this sense that many
original applications of directed homotopy were $1$-categorical in nature,
albeit up to generalizations of $1$-categorical equivalence.

It is also in this sense that later applications have often [23, 62] been
$(1,\infty)$-categorical in nature. For example, more subtle computational
behavior of executions in a concurrent program having directed state space $X$
appears in the properties of Moore path categories on $X$, topological
categories in which the morphisms form spaces of directed paths on $X$. In
fact, directed state spaces $X$ have sometimes been defined as topological
categories of some sort [23]. Moore path categories on directed spaces in the
applications have minimal models with tractable descriptions [62] and are in
fact conjectured to model all $(1,\infty)$ categories [18]. Unfortunately, the
class of stream maps preserving the relevant $(1,\infty)$-categories of
interest also have poor formal properties compared to stream maps $f$ for
which $\mathfrak{sing}\,f$ are $\mathfrak{d}_{*}$-equivalences; $\mathrm{\bf
DiTop}$ admits a localization with respect to the latter but likely not the
former.

Recent years have seen computations modelled abstractly by homotopy types. A
(dependently) typed higher order programming language for reversible
computations has been shown to admit semantics in $\infty$-groupoids (fibered
over other $\infty$-groupoids) [3]. Objects represent states, $1$-morphisms
represent reversible executions, and higher order morphisms represent
reversible transformations of those executions, or equivalently, concurrent
executions of sequential computations. Since $\infty$-equivalences between
$\infty$-groupoids ignore differences like subdivisions, state space reduction
is built into the very syntax of the language. This language, higher order,
can thus be used to reason efficiently about computations expressed in the
same language. The recent literature has seen extensions of (dependent) type
theory to synthetic theories of (fibered) higher categories [50, 64]. These
more expressive languages model irreversible computations [50] because the
morphisms in higher categories need not be invertible.

Ideally, (dependent) type theory can be alternatively extended so that edges
in (fibered) cubcats represent computations, higher cubes in (fibered) cubcats
represent higher order transformations, and directed homotopy invariance is
built into the syntax of the language (cf. [58]). Such a language ought to
share both some of the efficiency of automated reasoning within dependent type
theory as well as some of the expressiveness of synthetic higher category
theory.

##  6\. Acknowledgements

This work was supported by AFOSR grant FA9550-16-1-0212. The author is
grateful to Robert Ghrist for conceiving of and producing the visualizations
of conal manifolds behind Figure 2. The author would like to thank Emily
Rudman for pointing out some simplifications in earlier proofs of Lemmas 3.8
and 3.9.

##  Appendix A Lattices

A lattice $L$ is modular if for all $x,y,z\in L$,

$(x\wedge_{L}y)\vee_{L}(x\wedge_{L}z)=((x\wedge_{L}y)\vee_{L}z)\wedge_{L}x.$

######  Example A.1.

Distributive lattices are modular.

The following Diamond Isomorphism Theorem characterizes modular lattices.

######  Diamond Isomorphism Theorem.

The following are equivalent for a lattice $L$.

  1. (1)

$L$ is modular.

  2. (2)

For each $x,y\in L$, the rules $x\vee_{L}-$ and $y\wedge_{L}-$ define
respective bijections $[x\wedge_{L}y,y]\cong[x,x\vee_{L}y]$ and
$[x,x\vee_{L}y]\cong[x\wedge_{L}y,y]$, where $[x^{\prime},y^{\prime}]$ denotes
the smallest interval in $L$ containing $x^{\prime}$ as its minimum and
$y^{\prime}$ as its maximum.

######  Theorem, [49].

The following are equivalent for a finite lattice $L$.

  1. (1)

$L$ is distributive

  2. (2)

For all $x,y,z\in L$ with $y,z$ either both immediate successors to $x$ or
both immediate predecessors to $x$ in $L$, $\\{y\wedge_{L}z,y\vee_{L}z,y,z\\}$
is a Boolean interval in $L$.

  3. (3)

The smallest interval in $L$ containing Boolean intervals $I,J$ in $L$ with
$\max\,I=\max\,J$ or $\min\,I=\min\,J$ is also Boolean.

######  Lemma A.2.

For Boolean intervals $I,J$ in a finite distributive lattice $L$, the images

$I\vee_{L}J,I\wedge_{L}J$

of $I\times J$ under $\vee_{L},\wedge_{L}$ are Boolean intervals in $L$.

###### Proof.

The intervals $I\vee_{L}\min\,J,J\vee_{L}\min\,I$ are Boolean by the Diamond
Isomorphism for Modular Lattices and hence $I\vee_{L}J$ is Boolean [Theorem,
[49]]. Thus $I\wedge_{L}J$ is also Boolean by duality. ∎

While every finite poset, including every finite lattice, is a colimit of its
($1$-dimensional) Boolean intervals in the category of posets and monotone
functions, not every finite lattice is such a colimit in the category
$\mathrm{\bf Cat}$.

######  Lemma A.3.

Every finite distributive lattice is a $\mathrm{\bf Cat}$-colimit of its
Boolean intervals.

###### Proof.

Consider a finite distributive lattice $L$. Let $\mathcal{X}$ be the
$\mathrm{\bf Cat}$-colimit

$\mathcal{X}=\mathrm{colim}_{I\rightarrow L}I$

over the Boolean intervals $I$ in $L$. The object sets of $\mathcal{X}$ and
$L$ coincide. There exists a relation $x\leqslant_{L}y$ if and only if there
exists a $\mathcal{X}$-morphism $x\rightarrow y$ because $\mathcal{X}$ admits
as generators relations of the form $x\leqslant_{L}y$ with $y$ an immediate
successor to $x$ in $L$. Consider parallel $\mathcal{X}$-morphisms
$\alpha,\beta:x\rightarrow y$. It thus suffices to show

$\alpha=\beta.$

We induct on the length $k$ of a maximal chain in $L$ having minimum $x$ and
maximum $y$. In the base case $k=1$, $\alpha,\beta$ are identities and hence
$\alpha=\beta$. Inductively assume that $\alpha=\beta$ when there exists a
maximal chain in $L$ having minimum $x$ and maximum $y$ with length less than
$k$. Consider the case $k>1$. Then $\alpha,\beta$ both factor as composites
$x\rightarrow a\rightarrow y$ and $x\rightarrow b\rightarrow y$ with $a$ and
$b$ both immediate successors to $x$ in $L$. Then $\alpha$ and $\beta$ are
choices of dotted monotone function respectively making the left and bottom
triangles commute in

${a}$${a\vee_{L}b}$${y}$${x}$${b,}$

a diagram in $\mathcal{X}$. There exists a dotted morphism making the top and
right triangles commute by the inductive hypothesis. The outer square, whose
elements form a Boolean interval in $L$ [Theorem, [49]], commutes in
$\mathcal{X}$. Thus $\alpha=\beta$. ∎

We can now give a proof of Lemma 3.12.

###### proof of Lemma 3.12.

Suppose (1). Let $I$ be a Boolean interval in $L$. The restriction of $\phi$
to $I$ corestricts to a surjection $\phi_{I}:I\rightarrow J_{I}$ with $J_{I}$
a Boolean interval in $M$ because $\phi$ preserves Boolean intervals. The
function $\phi_{I}:I\rightarrow J_{I}$, surjective by construction, is a
lattice homomorphism by $I\hookrightarrow L$ and $J\hookrightarrow M$ both
inclusions of sublattices into lattices. Thus (2).

Suppose (2). Consider $x,y\in L$. It suffices to show that

(11) $\phi(x\vee_{L}y)=\phi(x)\vee_{M}\phi(y).$

by double induction on the minimal lengths $m,n$ of maximal chains in $L$
having as their extrema $x\wedge_{L}y$ and, respectively, $x$ and $y$. For
then $\phi$ preserves binary suprema, hence also binary infima by duality, and
hence $\phi$ is a lattice homomorphism, mapping Boolean intervals onto Boolean
intervals by (2).

In the case $m=1$, $x\wedge_{L}y=x$, hence $x\vee_{L}y=y$, hence
$\phi(x)\leqslant_{M}\phi(x\vee_{L}y)=\phi(y)$, and consequently (11). The
case $n=1$ follows by symmetry.

Consider the case $m=n=2$. Then $x,y,x\wedge_{L}y,x\vee_{L}y$ form the
elements of a Boolean interval $I$ in $L$ [Theorem, [49]]. Then the
restriction of $\phi$ to $I$ corestricts to a Boolean interval $J_{I}$ in $M$.
It therefore follows from (2) and the preservation of finite non-empty suprema
and infima by $I\hookrightarrow L$ and $J_{I}\hookrightarrow M$ that

$\phi(x\vee_{L}y)=\phi(x\vee_{I}y)=\phi(x)\vee_{J_{I}}\phi(y)=\phi(x)\vee_{M}\phi(y).$

Consider the case $m\leqslant 2$. Suppose $n>2$. Then there exists an
immediate successor $y^{\prime}\neq y$ to $x\wedge_{L}y$ such that
$y^{\prime}\leqslant_{L}y$. Then
$y\wedge_{L}(x\vee_{L}y^{\prime})=(x\wedge_{L}y)\vee_{L}y^{\prime}=y^{\prime}$
by $L$ distributive and hence the length of a maximal chain in $L$ having as
its extrema $y\wedge_{L}(x\vee_{L}y^{\prime})$ and $y$ is strictly less than
$n$. And $x\wedge_{L}y^{\prime}=x\wedge_{L}y$ and hence the length of a
maximal chain in $L$ having as its extrema $x\wedge_{L}y^{\prime}$ and
$y^{\prime}$ is $m=2$. Inductively assume
$\phi(x\vee_{L}y^{\prime})=\phi(x)\vee_{L}\phi(y^{\prime})$ and
$\phi(y\vee_{L}(x\vee_{L}y^{\prime}))=\phi(y)\vee_{M}\phi(x\vee_{L}y^{\prime})$.
It therefore follows that
$\phi(x\vee_{L}y)=\phi(x\vee_{L}y^{\prime}\vee_{L}y)=\phi(x\vee_{L}y^{\prime})\vee_{M}\phi(y)=\phi(x)\vee_{M}\phi(y^{\prime})\vee_{M}\phi(y)=\phi(x)\vee_{M}\phi(y)$.
Then (11) follows from induction on $n$ for the case $m\leqslant 2$. Thus (11)
holds whenever $\min(m,n)\leqslant 2$ by symmetry.

Consider the general case. To show (11), it suffices to take the case $m>2$.
Then there exists an immediate successor $x^{\prime}\neq x$ to $x\wedge_{L}y$
such that $x^{\prime}\leqslant_{L}x$. Then
$x\wedge_{L}(x^{\prime}\vee_{L}y)=(x\wedge_{L}y)\vee_{L}x^{\prime}=x^{\prime}$
by $L$ distributive and hence the length of a maximal chain in $L$ having as
its extrema $x\wedge_{L}(x^{\prime}\vee_{L}y)$ and $x$ is strictly less than
$m$. And $x^{\prime}\wedge_{L}y=x\wedge_{L}y$ and hence the length of a
maximal chain from $x^{\prime}\wedge_{L}y$ to $x$ is $2$. Inductively assume
$\phi(x\vee_{L}(x^{\prime}\vee_{L}y))=\phi(x)\vee_{M}\phi(x^{\prime}\vee_{L}y)$.
Then
$\phi(x\vee_{L}y)=\phi(x\vee_{L}x^{\prime}\vee_{L}y)=\phi(x)\vee_{M}\phi(x\vee_{L}y^{\prime})=\phi(x)\vee_{M}\phi(x^{\prime})\vee_{M}\phi(y)=\phi(x)\vee_{M}\phi(y)$.
Hence (11). ∎

Besides the previous observations, the preservation of fully faithful
embeddings by $\mathrm{\bf Cat}$-pushouts[66] is used in the following proof
of Proposition 3.13.

###### proof of Proposition 3.13.

Let $F_{k}$ denote the bottom left Kan extension. Uniqueness follows by the
right vertical arrow an inclusion. To show existence, it suffices to show
$F_{k}$ preserves $\mathrm{\bf Dis}$-objects and $\mathrm{\bf Dis}$-morphisms.

$F_{k}$ preserves $\mathrm{\bf Dis}$-objects: Let $I$ denote a Boolean
interval in $L$. Inclusions of the forms $(I\hookrightarrow
L)^{[k]}:I^{[k]}\rightarrow L^{[k]}$ are fully faithful embeddings. It follows
that the natural functor $F_{k}L\rightarrow L^{[k]}$, an iterated pushout of
inclusions of the form $(I\rightarrow L)^{[k]}$ [Lemma A.3], is a full and
faithful embedding and hence can henceforth be regarded as an inclusion of
posets. In other words, we can identify $\mathfrak{sd}_{k+1}L$ with the poset
of all monotone functions $[k]\rightarrow L$ which corestrict to Boolean
intervals in $L$, with partial order $\leqslant_{\mathfrak{sd}_{k+1}L}$
defined by $\alpha\leqslant_{\mathfrak{sd}_{k+1}L}\beta$ if and only if
$\alpha(i)\leqslant_{L}\beta(i)$ for each $0\leqslant i\leqslant k$.

Consider $\alpha,\beta\in\mathfrak{sd}_{k+1}L$. The monotone functions
$\alpha\vee_{L}\beta$ and $\alpha\wedge_{L}\beta$ corestrict to Boolean
intervals in $L$ [Lemma A.2]. Thus $F_{k}L$ is a sublattice of the finite
distributive lattice $L^{[k]}$ and hence finite distributive.

$F_{k}$ preserves $\mathrm{\bf Dis}$-morphisms: Consider a general
$\mathrm{\bf Dis}$-morphism $\phi:L\rightarrow M$. To show that $F_{k}\phi$ is
a $\mathrm{\bf Dis}$-morphism, it suffices to take the case $\phi$ a
$\square$-morphism [Lemma 3.12]. Then $\phi$ is an iterated a Cartesian
monoidal product in $\mathrm{\bf Cat}$ of $\delta_{\pm},\sigma$. Then
$\phi^{[k]}$ is an iterated Cartesian monoidal product in $\mathrm{\bf Cat}$
of $\delta_{\pm}^{[k]}$ and $\sigma^{[k]}$ by $(-)^{[k]}$ a right adjoint and
hence product-preserving. The functions $\delta_{\pm}^{[k]}$ and
$\sigma^{[k]}$ are monotone functions to or from a terminal object and hence
$\mathrm{\bf Dis}$-morphisms. Hence $\phi$ is a $\mathrm{\bf Dis}$-morphism.

last claim: In order to show that the natural transformation $F_{m}\rightarrow
F_{n}$ induced from $\phi$ component-wise corestricts to the desired natural
transformation $\mathfrak{sd}_{m+1}\rightarrow\mathfrak{sd}_{n+1}$, it
suffices to show that $J^{\phi}$ is a $\mathrm{\bf Dis}$-morphism for each
$\square$-object $J$ [Lemma 3.12]. It therefore suffices to take the case
$J=[1]$ because $(-)^{\phi}$ is a Cartesian monoidal natural transformation
$\mathrm{\bf Cat}^{\mathrm{op}}\rightarrow\mathrm{\bf Cat}$. In that case,
non-singleton Boolean intervals in $J^{[m]}=[m+1]$ and $J^{[n]}=[n+1]$ are
intervals between elements and their immediate successors. Consider a non-
singleton Boolean interval $I$ in $J^{[m]}=[1]^{[m]}$. Let $\zeta_{-}=\min\,I$
and $\zeta_{+}=\max\,I$. Then there exists $0\leqslant j\leqslant m$ such that
$\zeta_{-}(i)=\zeta_{-}(i+1)=\zeta_{+}(i)=0$ for all $i<j$, $\zeta_{+}(i)=1$
for all $i\geqslant j$, and $\zeta_{-}(i)=1$ for all $i>j$. The preimage of
$j$ under $\phi$ is either a singleton or empty by $\phi$ injective.

In the case that the preimage is empty, then $\zeta_{-}\phi=\zeta_{+}\phi$.

In the case that the preimage contains the unique element $j^{*}$, then
$\phi(i)<j$ for all $i<j^{*}$, $\phi(i)\geqslant j$ for all $i\geqslant
j^{*}$, and consequently $\zeta_{-}\phi(i)=\zeta_{-}\phi(i+1)=\zeta_{+}(i)=0$
for all $i<j^{*}$, $\zeta_{+}\phi(i)=1$ for all $i\geqslant j^{*}$,
$\zeta_{-}\phi(i)=1$ for all $i>j^{*}$, and consequently $\zeta_{+}\phi$ is an
immediate successor to $\zeta_{-}\phi$ in $[1]^{[n]}$.

In either case, $\\{\phi\zeta_{-},\phi\zeta_{+}\\}$ is a Boolean interval in
$[1]^{[n]}$.

Thus $J^{\phi}$ is a $\mathrm{\bf Dis}$-morphism. ∎

##  Appendix B Pro-objects

We recall a characterization of the data of pro-morphisms as follows.

######  Lemma B.1.

Fix a category $\mathscr{X}$. Consider the following data:

  1. (1)

cofiltered diagrams $X:\mathcal{X}\rightarrow\mathscr{X}$ and
$Y:\mathcal{Y}\rightarrow\mathscr{X}$.

  2. (2)

choices of $\mathcal{X}$-object $x_{y}$ and $\mathscr{X}$-morphism
$\zeta_{y}:X(x_{y})\rightarrow Y(y)$ for each choice $y$ of
$\mathcal{Y}$-object such that for each $\mathcal{Y}$-morphism
$\upsilon:y_{1}\rightarrow y_{2}$, there exist $\mathcal{X}$-morphisms
$\chi_{1}:x\rightarrow x_{y_{1}}$ and $\chi_{2}:x\rightarrow x_{y_{2}}$

Suppose that for each $\mathcal{Y}$-morphism $\upsilon:y_{1}\rightarrow
y_{2}$, there exist $\mathcal{X}$-morphisms $\chi_{1}:x\rightarrow x_{y_{1}}$
and $\chi_{2}:x\rightarrow x_{y_{2}}$ such that the left of the diagrams below
commutes. Then there exists a unique $(\mathrm{\bf pro}\mathchar
45\relax\mathscr{X})$-morphism $\zeta:\lim\,X\rightarrow\lim\,Y$ such that the
following diagram, in which the vertical arrows are canonically defined,
commutes for each $\mathcal{Y}$-object $y$.

${X(x)}$${X(x_{y_{1}})}$${Y(y_{1})}$${X(x_{y_{2}})}$${Y(y_{2})}$${Y(y_{2})}$$\scriptstyle{X(\chi_{1})}$$\scriptstyle{X(\chi_{2})}$$\scriptstyle{\zeta_{y_{1}}}$$\scriptstyle{Y(\upsilon)}$$\scriptstyle{\zeta_{y_{2}}}$$\scriptstyle{1_{Y(y_{2})}}$
${\lim\,X}$${\lim\,Y}$${X(x_{y})}$${Y(y),}$$\scriptstyle{\zeta}$$\scriptstyle{\zeta_{y}}$

##  Appendix C Test Categories

A test model structure on a presheaf category $\hat{{\small\bigcirc}}$ is a
model structure on $\hat{{\small\bigcirc}}$ in which the cofibrations are the
monos and the weak equivalences are those $\hat{{\small\bigcirc}}$-morphisms
$\psi:A\rightarrow B$ for which ${\small\bigcirc}/\psi$ are Thomason weak
equivalences. A small category ${\small\bigcirc}$ is a test category if there
is a Thomason weak equivalence ${\small\bigcirc}\rightarrow\star$ and
$\hat{{\small\bigcirc}}$ admits a test model structure [Theorem 1.4.3, [12]].
The reader is referred elsewhere [12] for details. Test categories can be
recognized by the following criteria.

######  Proposition, p.86 44(d), [35].

A small category ${\small\bigcirc}$ is a test category if there exist functors

$\zeta:{\small\bigcirc}\rightarrow\mathrm{\bf
Cat},\quad\mathfrak{i}:\mathscr{I}\rightarrow\hat{{\small\bigcirc}},$

with $\mathfrak{i}$ an interval object in $\hat{{\small\bigcirc}}$, satisfying
the following:

  1. (1)

${\small\bigcirc}\rightarrow\star$ is a Thomason weak equivalence

  2. (2)

$\zeta(o)$ has a terminal object for each ${\small\bigcirc}$-object $o$

  3. (3)

the equalizer of
${\small\bigcirc}[\mathfrak{i}(\delta_{-})],{\small\bigcirc}[\mathfrak{i}(\delta_{+})]$
is initial in $\hat{{\small\bigcirc}}$

  4. (4)

$({\small\bigcirc}/\mathfrak{i}([1]))\rightarrow\star$ is a Thomason weak
equivalence

  5. (5)

there exists a natural transformation
$\zeta_{!}\mathfrak{i}\rightarrow(\square_{1}\hookrightarrow\mathrm{\bf
Cat})$, where $\zeta_{!}$ denotes the left Kan extension of $\zeta$ along the
Yoneda embedding ${\small\bigcirc}[-]$.

In abstract homotopical parlance [35], conditions (1), (4) require that
${\small\bigcirc},\mathfrak{i}$ are aspherical and condition (3) requires that
$\mathfrak{i}$ be separated.

######  Proposition C.1.

Consider a small category ${\small\bigcirc}$ contained in a chain

$\square\subset{\small\bigcirc}\subset\mathrm{\bf Cat}$

of subcategories, with $\square$ wide in ${\small\bigcirc}$. Then
${\small\bigcirc}$ is a test category. In particular, $\square$ is a test
category.

###### Proof.

Let $\zeta$ denote inclusion ${\small\bigcirc}\hookrightarrow\mathrm{\bf
Cat}$. Let $\mathfrak{i}$ be the interval object
${\small\bigcirc}[-](\mathscr{I}\hookrightarrow{\small\bigcirc})$ in
$\hat{{\small\bigcirc}}$.

The functor ${\small\bigcirc}\rightarrow\star$ is a Thomason weak equivalence
by $[0]$ a terminal object in $\square$, $\mathrm{\bf Cat}$, and hence
${\small\bigcirc}$.

Each small category $\zeta([1]^{n})=[1]^{n}$ has terminal object
$(1,\cdots,1)$ for each ${\small\bigcirc}$-object $[1]^{n}$.

The equalizer of $\mathfrak{i}(\delta_{-})$ and $\mathfrak{d}(\delta_{+})$,
whose restriction to a cubical set is the initial cubical set because it
defines the empty equalizer of $\mathfrak{d}(\delta_{-})$ and
$\mathfrak{d}(\delta_{+})$, is the initial presheaf on
$\hat{{\small\bigcirc}}$.

The category $({\small\bigcirc}/{\small\bigcirc}[1])$ has final object
$1_{{\small\bigcirc}[1]}:{\small\bigcirc}[1]\rightarrow{\small\bigcirc}[1]$ by
$\square$ wide in ${\small\bigcirc}$ and therefore admits a Thomason weak
equivalence to $\star$.

There exists natural isomorphisms
$\zeta_{!}\mathfrak{i}\cong(\square\hookrightarrow\mathrm{\bf
Cat})_{!}\mathfrak{d}\cong\mathrm{T}_{1}\mathfrak{d}:\square_{1}\hookrightarrow\mathrm{\bf
Cat}$.

The desired conclusion follows [Proposition, p.86 44(d), [35]]. ∎

######  Lemma C.2.

There exists a $\mathrm{T}_{1}\mathfrak{d}_{1}$-equivalence, natural in
cubical sets $C$, of the form

$(\Delta/(\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}C))\simeq({\small\bigcirc}/C)$

for each small subcategory ${\small\bigcirc}\subset\mathrm{\bf Cat}$
containing $\square$ as a wide subcategory such that each
${\small\bigcirc}$-morphism admits a factorization, unique up to isomorphism,
into a surjective ${\small\bigcirc}$-morphism followed by an injective
${\small\bigcirc}$-morphism and the poset of subobjects of every
${\small\bigcirc}$-object is a lattice.

###### Proof.

Let $\bigstar$ denote one of $\Delta,{\small\bigcirc}$. It is possible to
define an endofunctor

$E_{P}:({\small\bigcirc}/P)\rightarrow({\small\bigcirc}/P),$

natural in $\hat{\bigstar}$-objects $P$, naturally sending each
$(\bigstar/P)$-object $\theta$ to the terminal $(\bigstar/P)$-object having
the same image as the $\hat{\bigstar}$-morphism $\theta$ because each
$\bigstar$-morphism admits a unique factorization up to isomorphism into a
surjection followed by an injection. Then $E_{P}$ is pointed, uniquely and
hence naturally in $\hat{\bigstar}$-objects $P$. Thus there exists a
$\mathrm{T}_{1}\mathfrak{d}$-homotopy
$1_{\bigstar/P}\leadsto_{\mathrm{T}_{1}\mathfrak{d}}E_{P}$.

Define functors $F_{C},G_{C}$, natural in $\hat{{\small\bigcirc}}$-objects
$C$, of the forms

$F_{C}:(\Delta/(\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}C))\rightarrow({\small\bigcirc}/C)\quad
G_{C}:({\small\bigcirc}/C)\rightarrow\Delta/(\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}C)$

as follows. We can take the $({\small\bigcirc}/C)$-object $F_{C}\psi$, natural
in $(\Delta/(\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}C))$-objects $\psi$, to
be terminal among all $({\small\bigcirc}/C)$-objects $\theta$ with
${\mathfrak{im}}\,\psi\subset{\mathfrak{im}}\,\mathfrak{tri}\;\\!\\!\,\theta$
because the poset of subobjects of each ${\small\bigcirc}$-object is a finite
and hence complete lattice. The
$(\Delta/(\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}C))$-object $G_{C}\theta$,
natural in $({\small\bigcirc}/C)$-objects
$\theta:{\small\bigcirc}[1]^{n}\rightarrow C$, is defined by the commutative
diagram

${\Delta[1]}$${\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}C}$${\mathfrak{ner}^{\Delta}[1]^{n}}$${\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}{\small\bigcirc}[1]^{n}}$$\scriptstyle{\prod_{i=1}^{n}1_{\Delta[1]}}$$\scriptstyle{G_{C}(\theta)}$$\scriptstyle{\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}\theta}$

Dotted simplicial nerves of extrema-preserving monotone functions
$[1]\rightarrow[n]$ make

${\Delta[1]}$${\mathfrak{ner}^{\Delta}[1]^{n}}$${\Delta[n]}$${\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}C}$$\scriptstyle{G_{C}F_{C}(\psi)}$$\scriptstyle{F_{C}(\psi)}$$\scriptstyle{E_{\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}C}(\psi)}$

commute and therefore define the components of a natural transformation or
equivalently a $\mathrm{T}_{1}\mathfrak{d}$-homotopy
$G_{C}F_{C}\leadsto_{\mathrm{T}_{1}\mathfrak{d}}E_{\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}C}$.
Concatenating this $\mathrm{T}_{1}\mathfrak{d}$-homotopy with
$1_{{\small\bigcirc}/\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}C}\leadsto_{\mathrm{T}_{1}\mathfrak{d}}E_{\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}C}$
and the $\mathrm{T}_{1}\mathfrak{d}$-homotopy
$1_{{\small\bigcirc}/C}\leadsto_{\mathrm{T}_{1}\mathfrak{d}}E_{C}=F_{C}G_{C}$
with a constant $\mathrm{T}_{1}\mathfrak{d}$-homotopy yield the desired
$\mathrm{T}_{1}\mathfrak{d}_{1}$-homotopies. ∎

######  Lemma C.3.

There exists a $\mathrm{T}_{1}\mathfrak{d}_{2}$-equivalence, natural in
simplicial sets $S$, of the form

$(\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}{\small\bigcirc}[-]/S)\simeq(\Delta/S)$

for each small subcategory ${\small\bigcirc}\subset\mathrm{\bf Cat}$
containing $\square$ as a wide subcategory.

###### Proof.

Define functors $F_{S},G_{S}$, natural in simplicial sets $S$, of the forms

$F_{S}:(\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}{\small\bigcirc}[-]/S)\rightarrow(\Delta/S)\quad
G_{S}:(\Delta/S)\rightarrow(\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}{\small\bigcirc}[-]/S)$

by natural commutative diagrams of the following forms:

${\Delta[1]}$${\Delta[1]^{n}}$${S}$${\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}{\small\bigcirc}[1]^{n}}$$\scriptstyle{F_{S}(\theta)}$$\scriptstyle{\prod_{i=1}^{n}1_{\Delta[1]}}$$\scriptstyle{\theta}$
${\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}{\small\bigcirc}[1]^{n}}$${\mathfrak{ner}^{\Delta}[n]}$${S}$${\Delta[n]}$$\scriptstyle{G_{S}(\psi)}$$\scriptstyle{\mathfrak{ner}^{\Delta}(x\mapsto\sum_{i}x_{i})}$$\scriptstyle{\psi}$

Let $\mathfrak{diag}_{[1]}=1_{[1]}^{\times_{\mathrm{\bf
Cat}}n}:[1]\rightarrow[1]^{n}$ and
$\delta_{\raisebox{0.4pt}{\scalebox{0.6}{+}}\raisebox{0.4pt}{\scalebox{0.6}{+}}}=\delta_{+1}\cdots\delta_{+1}:[1]\rightarrow[1]^{n}$.
For all $x\in[1]$,

$\displaystyle\mathfrak{diag}_{[1]}(x)$ $\displaystyle=(x,\ldots,x)$
$\displaystyle\leqslant_{[1]^{n}}(1,\ldots,1,x)$
$\displaystyle=\delta_{\raisebox{0.4pt}{\scalebox{0.6}{+}}\raisebox{0.4pt}{\scalebox{0.6}{+}}}(x)$

Therefore the function $\phi:[1]^{2}\rightarrow[1]^{n}$ characterized by

$\phi\delta_{-1}=\mathfrak{diag}_{[1]}\quad\phi\delta_{+}=\delta_{\raisebox{0.4pt}{\scalebox{0.6}{+}}\raisebox{0.4pt}{\scalebox{0.6}{+}}}$

is monotone. Hence there exists a dotted simplicial nerve of $\phi$ making

${\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}{\small\bigcirc}[1]}$${\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}{\small\bigcirc}[1]^{2}}$${\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}{\small\bigcirc}[1]}$${\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}{\small\bigcirc}[1]^{n}}$${\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}{\small\bigcirc}[1]^{n}}$${S}$$\scriptstyle{G_{S}F_{S}(\psi)}$$\scriptstyle{\mathfrak{ner}^{\Delta}\Delta_{[1]}}$$\scriptstyle{\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}{\small\bigcirc}[\delta_{-}]}$$\scriptstyle{\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}{\small\bigcirc}[\delta_{+}]}$$\scriptstyle{\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}{\small\bigcirc}[\delta_{\raisebox{0.4pt}{\scalebox{0.6}{+}}\raisebox{0.4pt}{\scalebox{0.6}{+}}}]}$$\scriptstyle{\psi}$$\scriptstyle{\psi}$

commute. The arrows in the top row define the components of $3$
$\mathrm{T}_{1}\mathfrak{d}$-homotopies, which, when concatenated with a
constant $\mathrm{T}_{1}\mathfrak{d}$-homotopy, yields a
$\mathrm{T}_{1}\mathfrak{d}_{2}$-homotopy
$G_{S}F_{S}\leftrightsquigarrow_{\mathrm{T}_{1}\mathfrak{d}}1_{{\small\bigcirc}/S}$

Dotted simplicial nerves of extrema-preserving monotone functions
$[1]\rightarrow[n]$ make

${\Delta[1]}$${\mathfrak{ner}^{\Delta}[1]^{n}}$${\Delta[n]}$${S}$$\scriptstyle{\mathfrak{ner}^{\Delta}(\prod_{i=1}^{n}1_{[1]})}$$\scriptstyle{F_{S}G_{S}(\psi)}$$\scriptstyle{G_{S}(\psi)}$$\scriptstyle{\mathfrak{ner}^{\Delta}(x\mapsto\sum_{i}x_{i})}$$\scriptstyle{\psi}$

commute and hence define the components of natural transformations
$F_{S}G_{S}\rightarrow 1_{(\Delta/S)}$ or equivalently
$\mathrm{T}_{1}\mathfrak{d}$-homotopies
$F_{S}G_{S}\leadsto_{\mathrm{T}_{1}\mathfrak{d}}1_{(\Delta/S)}$. ∎

Fix a subcategory ${\small\bigcirc}$ of $\mathrm{\bf Cat}$ defining a test
category. Even though there exists a zig-zag of Quillen equivalences between
$\hat{{\small\bigcirc}}$ and $\hat{\Delta}$, it is not necessarily the case
that triangulation directly defines a Quillen equivalence between them.

######  Proposition C.4.

Triangulation defines the left map of a Quillen equivalence

$\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}:\hat{{\small\bigcirc}}\leftrightarrows\hat{\Delta}$

between presheaf categories equipped with test model structures, where
${\small\bigcirc}$ is a subcategory of $\mathrm{\bf Cat}$ containing $\square$
as a wide subcategory such that each ${\small\bigcirc}$-morphism admits a
factorization, unique up to isomorphism, into a surjective
${\small\bigcirc}$-morphism followed by an injective
${\small\bigcirc}$-morphism and the poset of subobjects of each
${\small\bigcirc}$-object is a complete lattice.

###### Proof.

Both $\hat{\Delta}$ and $\hat{{\small\bigcirc}}$ admit test model structures
[Proposition C.1]. There exist dotted Thomason weak equivalences making the
left [Lemma C.2] and right [Lemma C.3] diagrams below commute for maps $\psi$
of presheaves, where
$\mathfrak{qua}\;\\!\\!\vdash\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}$:

${{\small\bigcirc}/A}$${{\small\bigcirc}/B}$${\Delta/(\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}A)}$${\Delta/(\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}B)}$$\scriptstyle{{\small\bigcirc}/\psi}$$\scriptstyle{\Delta/(\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}\psi)}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$
${\Delta/A}$${\Delta/B}$${{\small\bigcirc}/(\mathfrak{qua}\;\\!\\!\,A)}$${{\small\bigcirc}/(\mathfrak{qua}\;\\!\\!\,B)}$$\scriptstyle{{\small\bigcirc}/\psi}$$\scriptstyle{{\small\bigcirc}/(\mathfrak{qua}\;\\!\\!\,\psi)}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

In each of the commutative diagrams, the top horizontal arrow is a classical
weak equivalence if and only if the bottom horizontal arrow is a classical
weak equivalence. Thus $\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}$ and its
right adjoint both preserve and reflect weak equivalences in test model
structures. Additionally $\mathfrak{tri}\;\\!\\!_{{\small\bigcirc}}$ preserves
monos, cofibrations. ∎

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