# Homotopy invariants in small categories

I. Carcacía-Campos, E. Macías-Virgós and D. Mosquera-Lois  CITMAga
Universidade de Santiago de Compostela, 15782-Spain
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Tanaka introduced a notion of Lusternik Schnirelmann category, denoted
$\mathop{\mathrm{ccat}}\mathcal{C}$, of a small category $\mathcal{C}$. Among
other properties, he proved an analog of Varadarajan’s theorem for fibrations,
relating the LS-categories of the total space, the base and the fiber.

In this paper we recall the notion of homotopic distance $\mathrm{cD}(F,G)$
between two functors $F,G\colon\mathcal{C}\to\mathcal{D}$, later introduced by
us, which has
$\mathop{\mathrm{ccat}}\mathcal{C}=\mathrm{cD}(\mathrm{id}_{\mathcal{C}},\bullet)$
as a particular case. We consider another particular case, the distance
$\mathrm{cD}(p_{1},p_{2})$ between the two projections
$p_{1},p_{2}\colon\mathcal{C}\times\mathcal{C}\to\mathcal{C}$, which we call
the categorical complexity of the small category $\mathcal{C}$. Moreover, we
define the higher categorical complexity of a small category and we show that
it can be characterized as a higher distance.

We prove the main properties of those invariants. As a final result we prove a
Varadarajan’s theorem for the homotopic distance for Grothendieck bi-
fibrations between small categories.

## Introduction

In the thirties, Lusternik and Schnirelmann introduced a homotopic invariant
of manifolds which bounded from the below the number of critical points of any
smooth function defined on them (see [4]). For a topological space $X$, this
homotopic invariant is known as the Lusternik-Schnirelmann category of $X$,
denoted $\mathop{\mathrm{cat}}(X)$.

Twenty years ago, M. Farber introduced the topological complexity of a
topological space ([6]). It is a homotopic invariant which measures the
difficulty in finding a motion planning algorithm on the space under
consideration.

Both, the Lusternik-Schnirelmann category and the topological complexity are
instances of a more general homotopic invariant called homotopic distance
between maps, introduced by us in [13]. It provides a way to relate and extend
several other homotopic invariants (see [14]).

Recently, it has grown the interest in developing topological and homotopic
invariants in the setting of small categories. For example, this is the case
of the Euler characteristic by Leinster ([2, 11]) or the integration with
respect to the Euler-Poincaré characteristic by Tanaka ([22]). The definition
of homotopy between functors by M-J. Lee ([9]) made it possible for Tanaka
([21]) to extend the Lusternik-Schnirelmann category to the setting of small
categories. Later, two of the authors extended the notion of homotopic
distance to the context of categories ([12]).

In this work, we continue the study begun in [12]. First, we introduce a novel
homotopic invariant for small categories: a notion of higher homotopic
distance. Second, we study fibrations between small categories culminating
with a result in the spirit of Varadarajan’s theorem ([23]) for the homotopic
distance for Grothendieck bi-fibrations between small categories (Theorem
5.1), which can be seen both as a generalization of [21, Theorem 4.5] and as
an extension of our previous work ([13]) to the context of categories.

The paper is organized as follows. Section 1 is devoted to fixing notation
while recalling some preliminaries about homotopies and homotopic distance in
the setting of small categories. Moreover, the novel notion of higher
categorical distance is introduced (Definition 1.15) and it is proved to be a
particular case of a homotopic distance (Theorem 1.16). In Section 2 bi-
fibrations are presented in this setting from scratch. The exposition is
intended to be self-contained so no previous knowledge from the reader is
necessary on the topic. Section 4 addresses the equivalence of fibers whose
base objects are connected by an arrow (Theorem 4.1). In Section 5, we state
and prove the main result (Theorem 5.1).

##  1\. Categorical distance between functors

We will assume that all categories are small unless stated otherwise. If
$\mathcal{E}$ is a category, we also denote by $\mathcal{E}$ its set of
objects, and by $\mathcal{E}(e_{1},e_{2})$ the set of arrows between the
objects $e_{1},e_{2}\in\mathcal{E}$. If $P\colon\mathcal{E}\to\mathcal{B}$ is
a functor, we denote by $Pe$ and $P\phi$ the image of the object $e$ and the
arrow $\phi$, respectively.

###  1.1. Homotopies between functors

We recall the notion of homotopy between functors introduced by Lee ([9, 10]).

######  Definition 1.1.

The interval category $\mathcal{I}_{m}$ of length $m\geq 0$ consists of $m+1$
objects with zigzag arrrows,

$0\longrightarrow 1\longleftarrow
2\longrightarrow\cdots\longrightarrow(\longleftarrow)m.$

Given two small categories $\mathcal{C}$ and $\mathcal{D}$ we denote its
product by $\mathcal{C}\times\mathcal{D}$. Recall that the objects of
$\mathcal{C}\times\mathcal{D}$ are pairs of objects in $\mathcal{C}$ and
objects in $\mathcal{D}$, and its arrows are products of arrows in
$\mathcal{C}$ and arrows in $\mathcal{D}$.

######  Definition 1.2.

Let $F,G\colon\mathcal{C}\to\mathcal{D}$ be two functors between small
categories. We say that $F$ and $G$ are homotopic, denoted by $F\simeq G$, if,
for some $m\geq 0$, there exists a functor
$H\colon\mathcal{C}\times\mathcal{I}_{m}\rightarrow\mathcal{D}$, called a
homotopy (of length $m$), such that $H_{0}=F$ and $H_{m}=G$.

Alternatively, the notion of homotopy between functors can be defined as
follows.

######  Proposition 1.3.

The functors $F,G\colon\mathcal{C}\to\mathcal{D}$ are homotopic if and only if
there is a finite sequence of functors
$F_{0},\dots,F_{m}\colon\mathcal{C}\to\mathcal{D}$, with $F_{0}=F$ and
$F_{m}=G$, such that for each $i\in\\{0,\ldots,m-1\\}$ there is a natural
transformation either between $F_{i}$ and $F_{i+1}$ or between $F_{i+1}$ and
$F_{i}$.

That both definitions are equivalent follows from the following Lemma.

######  Lemma 1.4.

There is a natural transformation $\Phi\colon F\Rightarrow G$ if and only if
there exists a homotopy
$H\colon\mathcal{C}\times\mathcal{I}_{1}\to\mathcal{D}$ such that $H_{0}=F$
and $H_{1}=G$.

###### Proof.

For an object, we define $H(c,0)=Fc$, $H(c,1)=Gc$. For an arrow $f\colon c\to
c^{\prime}$ we define $H(f\times\mathrm{id}_{0})=Ff$,
$H(f\times\mathrm{id}_{1})=Gf$ and, for the only arrow $s\colon 0\to 1$ in
$\mathcal{I}_{1}$, we define

$H(c\times s)=\Phi_{c}\colon Fc\to Gc.$

∎

The homotopy relation defined above is an equivalence relation. Also, it
behaves well with respect to compositions, that is, if $F\simeq F^{\prime}$
and $G\simeq G^{\prime}$, then $F\circ G\simeq F^{\prime}\circ G^{\prime}$
whenever $F\circ F^{\prime}$ and $G\circ G^{\prime}$ make sense.

###  1.2. Categorical distance between functors

We introduce now the categorical homotopic distance between functors (see
[12]).

######  Definition 1.5.

Let $\mathcal{C}$ be a small category, a family $\\{{\mathcal{U}}_{i}\\}_{i\in
I}$ of subcategories of $\mathcal{C}$ if a geometric cover of $\mathcal{C}$ if
for every chain of arrows

$c_{0}\xrightarrow{\alpha_{1}}\cdots\xrightarrow{\alpha_{m}}c_{m}$

in $\mathcal{C}$ there is some $i\in I$ such that the chain lies in
${\mathcal{U}}_{i}$.

There are at least three reasons for adopting this definition: first, just
covering objects would leave the arrows between different subcategories
uncovered; second, an arbitrary covering of the arrows would leave the
compositions of arrows between different subcategories uncovered; third,
geometric coverings correspond to coverings of the classifying space of the
category, thus making easy to compare the categorical distance with the
topological distance of the classifying space.

######  Definition 1.6.

Let $F,G:\mathcal{C}\rightarrow\mathcal{D}$ be two functors between small
categories. The homotopical distance $\mathrm{cD}(F,G)$ beween $F$ and $G$ is
the least positive integer $n\geq 0$ such that there is geometric cover
$\\{{\mathcal{U}}_{0},...,{\mathcal{U}}_{n}\\}$ of $\mathcal{C}$ such that
$F|_{{\mathcal{U}}_{i}}\simeq G|_{{\mathcal{U}}_{i}}$ for every $0\leq i\leq
n$.

If there is no such cover we define $\mathrm{cD}(F,G)=\infty$.

We call $\\{{\mathcal{U}}_{0},...,{\mathcal{U}}_{n}\\}$ a geometric cover by
homotopy domains for $F$ and $G$.

###  1.3. Properties

The following properties are obvious:

  1. (1)

$\mathrm{cD}(F,G)=\mathrm{cD}(G,F)$.

  2. (2)

$\mathrm{cD}(G,F)=0$ if and only if $F\simeq G$.

  3. (3)

If $F\simeq\widehat{F}$ and $G\simeq\widehat{G}$ then
$\mathrm{cD}(F,G)=\mathrm{cD}(\widehat{F},\widehat{G})$.

We finish the section by introducing a notion of higher homotopic distance in
the setting of small categories.

######  Definition 1.7.

Let $\mathcal{C}$ and $\mathcal{D}$ be two small categories and let
$\\{F_{i}\\}_{i=1}^{n}:\mathcal{C}\rightarrow\mathcal{D}$ be a finite set of
functors between them. The higher homotopical distance
$\mathrm{cD}(F_{1},...,F_{n})$ is the least integer $m\geq 0$ such that there
is a geometric cover $\\{{\mathcal{U}}_{0},...,{\mathcal{U}}_{m}\\}$ that
satisfies $F_{i}|_{{\mathcal{U}}_{k}}\simeq F_{j}|_{{\mathcal{U}}_{k}}$ for
every $i,j\in\\{1,...,n\\}$ and $0\leq k\leq m$.

If there is no such cover we define $\mathrm{cD}(F_{1},...,F_{n})=\infty$.

###  1.4. LS-Category

The following notion of “categorical LS-category” is due to Tanaka [21].

A subcategory $\mathcal{U}$ of a small category $\mathcal{C}$ is called
$0$-categorical if the inclusion functor
$\iota\colon\mathcal{U}\rightarrow\mathcal{C}$ is homotopic to a constant
functor. The geometric cover $\\{{\mathcal{U}}_{0},...,{\mathcal{U}}_{m}\\}$
of the small category $\mathcal{C}$ is called categorical if every
${\mathcal{U}}_{i}$ is $0$-categorical.

######  Definition 1.8.

Let $\mathcal{C}$ be a small category. We define the (normalized) Lusternik-
Schnirelmann category of $\mathcal{C}$, denoted by
$\mathop{\mathrm{ccat}}(\mathcal{C})$, as the least integer $n\geq 0$ such
that there is a categorical cover of $\mathcal{C}$.

If there is no such cover we define $\mathop{\mathrm{ccat}}(\mathcal{C})$ as
$\infty$.

We state a result from [13] and provide a proof since we will make use of it
in the proof of Theorem 5.1.

######  Proposition 1.9.

For every connected small category $\mathcal{C}$ we have that

$\mathop{\mathrm{ccat}}(\mathcal{C})=\mathrm{cD}(\mathrm{id}_{\mathcal{C}},\bullet)$

where $\bullet$ is any constant functor.

###### Proof.

Let $\\{{\mathcal{U}}_{0},\dots,{\mathcal{U}}_{n}\\}$ be a geometric cover
satisfying that $n=\mathop{\mathrm{ccat}}(\mathcal{C})$ and
$\iota_{{\mathcal{U}}_{i}}\simeq\bullet_{i}$ for some constant functors
$\bullet_{i}$ and every $i\in\\{0,...,n\\}$. We have to prove that the
constant functors $\bullet_{i}$ and $\bullet$ are homotopic. But this follows
because there is some arrow which connectes the objects $\bullet_{i}$ and
$\bullet$, and thsi arrow defines a natural transformation between the
corresponding constant functors.

It is now evident that
$\mathrm{id}|_{U_{i}}=\iota_{{\mathcal{U}}_{i}}\simeq\bullet$ so
$\\{{\mathcal{U}}_{0},\dots,{\mathcal{U}}_{n}\\}$ is a homotopy domain for
$\mathrm{id}_{\mathcal{C}}$ and $\bullet$. So we have proved that
$\mathop{\mathrm{ccat}}(\mathcal{C})\geq\mathrm{cD}(\mathrm{id}_{\mathcal{C}},\bullet)$.

Alternatively if ${\mathcal{U}}_{0},\dots,{\mathcal{U}}_{n}$ are homotopy
domains for $\mathrm{id}_{\mathcal{C}}$ and $\bullet$ it follows that
$\mathrm{id}|_{{\mathcal{U}}_{i}}=\iota_{{\mathcal{U}}_{i}}\simeq\bullet$, so
$\mathrm{cD}(\mathrm{id}_{\mathcal{C}},\bullet)\geq\mathop{\mathrm{ccat}}(\mathcal{C})$.
∎

######  Corollary 1.10.

The small category $\mathcal{C}$ is contractible if and only if
$\mathop{\mathrm{ccat}}(\mathcal{C})=0$.

######  Example 1.11.

If the small category $\mathcal{C}$ admits finite products, then $\mathcal{C}$
is contractible.

###### Proof.

Fix an object $c_{0}$ in $\mathcal{C}$ and define the functor
$c_{0}\times-\colon\mathcal{C}\to\mathcal{C}$ which sends the object $c$ into
$c_{0}\times c$, and the morphism $f\colon c_{1}\to c_{2}$ into
$\mathrm{id}_{c_{0}}\times f$. Denote by
$c_{0}\colon\mathcal{C}\to\mathcal{C}$ the constant functor.

Then there are natural transformations (that is, homotopies)

$\mathrm{id}_{\mathcal{C}}\Leftarrow c_{0}\times-\Rightarrow c_{0}$

given by the two projections $p_{1}\colon c_{0}\times c\to c_{0}$ and
$p_{2}\colon c_{0}\times c\to c$. ∎

Let $\mathcal{C}$ be a category and let $c_{0}$ be an object in $\mathcal{C}$.
There are two functors
$i_{1},i_{2}\colon\mathcal{C}\rightarrow\mathcal{C}\times\mathcal{C}$ defined
as follows:

  * •

For every object $c$ in $\mathcal{C}$ we have that $i_{1}(c)=(c,c_{0})$ e
$i_{2}(c)=i_{2}(c_{0},c)$.

  * •

Let $f$ be a morphism in $\mathcal{C}$,then $i_{1}(f)=(f,\mathrm{id}_{c_{0}})$
e $i_{2}(f)=(\mathrm{id}_{c_{0}},f)$.

######  Proposition 1.12 ([12, Proposition 6]).

Let $\mathcal{C}$ be a small category and $C$ an objet in $\mathcal{C}$. We
claim that:

$\mathop{\mathrm{ccat}}(\mathcal{C})=\mathrm{cD}(i_{1},i_{2}).$

###  1.5. Categorical complexity

If $\mathcal{C}$ is a small category, we define the diagonal functor
$\Delta\colon\mathcal{C}\rightarrow\mathcal{C}\times\mathcal{C}$ as the
functor that takes every object $c$ to $\Delta(c)=(c,c)$ and that takes every
morphism $f$ to $\Delta(f)=(f,f)$.

######  Definition 1.13.

  1. (1)

We say that the subcategory $\mathcal{U}$ of $\mathcal{C}\times\mathcal{C}$ is
a Farber subcategory if there is a functor
$F\colon\mathcal{U}\rightarrow\mathcal{C}$ such that $\Delta\circ
F\simeq\iota_{U}$.

  2. (2)

The (normalized) categorical complexity of $\mathcal{C}$, denoted by
$\mathop{\mathrm{cTC}}(\mathcal{C})$ is the least integer $n\geq 0$ such that
there is a geometric cover $\\{{\mathcal{U}}_{0},\dots,{\mathcal{U}}_{n}\\}$
of $\mathcal{C}\times\mathcal{C}$ by Farber subcategories.

If there is no such cover we define $\mathop{\mathrm{cTC}}(\mathcal{C})$ as
$\infty$.

######  Proposition 1.14 ([12, Theorem 1]).

Let $\mathcal{C}$ be a small category category, we have that

$\mathop{\mathrm{cTC}}(\mathcal{C})=\mathrm{cD}(p_{1},p_{2}),$

where $p_{1},p_{2}\colon\mathcal{C}\times\mathcal{C}\to\mathcal{C}$ are the
projections.

###  1.6. Higher categorical complexity

We sketch now a definition of a higher topological complexity, analogous to
that existing in the topological setting ([18]).

Let $\mathcal{C}$ be a small category. We denote by $\mathcal{C}^{n}$ the
product $\mathcal{C}\times\stackrel{{\scriptstyle
n}}{{\cdots}}\times\mathcal{C}$. The higher $n$-diagonal functor
$\Delta_{n}\colon\mathcal{C}\rightarrow\mathcal{C}^{n}$ is the functor that
takes every object $X$ in $\mathcal{C}$ to $\Delta_{n}X=(X,...,X)$ and that
takes every morphism $f$ to $\Delta_{n}f=(f,...,f)$.

######  Definition 1.15.

  1. (1)

We say that the subcategory $\iota\colon\Omega\hookrightarrow\mathcal{C}^{n}$
is a higher $n$-Farber subcategory if there is a functor
$F\colon\Omega\rightarrow\mathcal{C}$ wich is a right homotopy inverse of
$\Delta_{n}$, that is, $\Delta_{n}\circ F\simeq i_{\Omega}$.

  2. (2)

The (normalized) higher $n$-categorical complexity of $\mathcal{C}$, denoted
by $\mathop{\mathrm{cTC}}_{n}(\mathcal{C})$ is the least integer number $n\geq
0$ such that there is a geometric cover $\\{\Omega_{0},...,\Omega_{n}\\}$ of
$\mathcal{C}^{n}$ by higher Farber subcategories.

The following result guarantees that higher categorical complexity can be seen
as a homotopic distance.

######  Theorem 1.16.

The higher topological complexity equals the higher categorical distance
between the projections $p_{i}\colon\mathcal{C}^{n}\to\mathcal{C}$, $1\leq
i\leq n$, that is,

$\mathop{\mathrm{cTC}}\nolimits_{n}(\mathcal{C})=\mathrm{cD}(p_{1},...,p_{n}).$

###### Proof.

( $\leq$) Let $\mathrm{cD}(p_{1},\dots,p_{n})=m$. Let $\\{U_{0},...,U_{m}\\}$
be a geometric cover of $\mathcal{C}^{n}$ by homotopy domains, that is,
subcategories $U_{k}$ such that $p_{i}|_{U_{k}}\simeq p_{j}|_{U_{k}}$ for all
$i,j\in\\{0,\dots,n\\}$. We will see that they are also higher Farber
subcategories.

Indeed, for every $U_{k}$ we have that the first projection $p_{1}\colon
U_{k}\to\mathcal{C}$ verifies $\Delta_{n}\circ p_{1}=\iota_{U_{k}}$. To check
that, for every $i\in\\{1,...,n\\}$ consider the homotopy $H_{i}\colon
U_{k}\times\mathcal{I}_{m_{i}}\rightarrow\mathcal{C}$ between $p_{1}|_{U_{i}}$
and $p_{i}|_{U_{i}}$. We can normalize all the homotopies by taking
$m=\max\\{m_{i}\\}_{i=0}^{n}$ and extending $H_{i}\colon
U_{i}\times\mathcal{I}_{m}\rightarrow\mathcal{C}$ by identities if $j\geq
m_{i}$. Now we can define a new homotopy

$G:U_{k}\times\mathcal{I}_{m}\rightarrow\mathcal{C}$

as

$G(c_{1},c_{2},\dots,c_{n},j)=(H_{1}(c_{1},\dots,c_{n}),H_{2}(c_{1},...,c_{n},j),...,H_{n}(c_{1},...,c_{n},j).$

Let us check that $G$ is a homotopy between $\Delta\circ p_{1}$ and the
inclusion $\iota_{U_{k}}$:

$\displaystyle\quad G(c_{1},\dots,c_{n},0)$
$\displaystyle=(H_{1}(c_{1},\dots,c_{n},0),\dots,H_{n}(c_{1},\dots,c_{n},0))$
$\displaystyle=(p_{1}(c_{1},\dots,c_{n}),\dots,p_{1}(c_{1},\dots,c_{n}))$
$\displaystyle=(c_{1},\dots,c_{1})$
$\displaystyle=\Delta_{n}p_{1}(c_{1},\dots,c_{n}),$

and

$\displaystyle\quad G(c_{1},c_{2},\dots,c_{n},m)$
$\displaystyle=G(H_{1}(c_{1},\dots,c_{n},m),\dots,H_{n}(c_{1}\dots,c_{n},m))$
$\displaystyle=(p_{1}(c_{1},c_{2},\dots,c_{n}),\dots,p_{n}(c_{1},...,c_{n}))$
$\displaystyle=(c_{1},c_{2},\dots,c_{n}).$

This shows that $\mathop{\mathrm{cTC}}\nolimits_{n}(\mathcal{C})\leq m$.

($\geq$) Now, let $\mathop{\mathrm{cTC}}\nolimits_{n}(\mathcal{C})=m$. Let
$\\{\Omega_{0},...,\Omega_{m}\\}$ be a geometric cover of $\mathcal{C}^{n}$ by
Farber subcategories. Let us fix some index $k\in\\{0,...,n\\}$ and let us see
that $p_{i}|_{U_{k}}\simeq p_{j}|_{U_{k}}$ for every $i,j\in\\{1,...,n\\}$.

Since $U_{k}$ is a Farber subcategory, there is a functor $F\colon
U_{k}\rightarrow\mathcal{C}$ such that $\Delta\circ F\simeq\iota_{U_{k}}$.
Hence, there is a homotopy $H\colon
U_{k}\times\mathcal{I}_{m}\rightarrow\mathcal{C}^{n}$ such that
$H_{0}=\Delta_{n}\circ F$ and $H_{m}=\iota_{U_{k}}$. We take the functor
$K\colon U_{k}\times\mathcal{I}_{2m}\rightarrow\mathcal{C}$ given by

$K(c_{1},\dotsc_{n},l)=\begin{cases}p_{i}\circ H(c_{1},...,c_{n},m-l)&\text{
if }\quad 0\leq l\leq m\\\ p_{j}\circ H(c_{1},...,c_{n},l-m)&\text{if}\quad
m\leq l\leq 2m\\\ \end{cases}$

The functor $K$ is well-defined because, for $l=m$:

$\displaystyle\quad p_{i}\circ H(X_{1},...,X_{n},0)$ $\displaystyle=$
$\displaystyle p_{i}\circ\Delta\circ F(c_{1},...,c_{n})$ $\displaystyle=$
$\displaystyle p_{i}(F(c_{1},\dots,c_{n}),\dots,F(c_{1},\dots,X_{n}))$
$\displaystyle=$ $\displaystyle F(c_{1},\dots,c_{n})$ $\displaystyle=$
$\displaystyle p_{j}\circ\Delta\circ F(c_{1},...,c_{n})$ $\displaystyle=$
$\displaystyle p_{j}\circ H(X_{1},...,X_{n},0).$

Moreover $K$ is a homotopy between $p_{i}$ and $p_{j}$:

$\displaystyle K(c_{1},\dots,c_{n},0)=$ $\displaystyle p_{i}\circ
H(c_{1},\dots,c_{n},m)$ $\displaystyle=$ $\displaystyle
p_{i}\circ\iota_{U_{k}}(c1,\dots,c_{n})=p_{i}(c_{1},\dots,c_{n}),$

and

$\displaystyle K(c_{1},\dots,c_{n},2m)=$ $\displaystyle p_{j}\circ
H(c_{1},\dots,c_{n},m)$ $\displaystyle=$ $\displaystyle p_{j}\circ
i_{U_{k}}(c1,\dots,c_{n})=p_{j}(c_{1},\dots,c_{n}).$

We have proven that $\mathrm{cD}(p_{1},...,p_{n})\leq m$. ∎

###  1.7. Properties

We recall several basic properties of the categorical distance. They serve to
give short proofs of many results.

######  Proposition 1.17 ([12, Propositions 7 and 8]).

  1. (1)

Let $\mathcal{C}$, $D$ and $\mathcal{B}$ be three small categories and let
$F,G\colon\mathcal{C}\rightarrow\mathcal{D}$ and
$H:\mathfrak{D}\rightarrow\mathcal{B}$ be three functors. We have that:

$\mathrm{cD}(H\circ F,H\circ G)\leq\mathrm{cD}(F,G).$

  2. (2)

Analogously, let $\mathcal{C}$, $\mathcal{D}$ and $\mathcal{B}$ be three small
categories and let $F,G\colon\mathcal{C}\rightarrow\mathcal{D}$ and
$H:\mathcal{B}\rightarrow\mathcal{C}$ be three functors. We have that:

$\mathrm{cD}(H\circ F,H\circ G)\leq\mathrm{cD}(F,G).$

######  Corollary 1.18.

Let $\mathcal{C}$ be a connected small category. Then we have that

$\mathop{\mathrm{ccat}}(\mathcal{C})\leq\mathop{\mathrm{cTC}}(\mathcal{C}).$

###### Proof.

$\mathop{\mathrm{ccat}}(\mathcal{C})=\mathrm{cD}(\mathrm{id}_{\mathcal{C}},C_{0})=\mathrm{cD}(p_{1}\circ
i_{1},p_{2}\circ
i_{1})\leq\mathrm{cD}(p_{1},p_{2})=\mathop{\mathrm{cTC}}(\mathcal{C}).$

∎

In what follows we note that category and categorical complexity are in some
sense dual and extreme cases of categorical distance.

######  Corollary 1.19.

Let $F,G\colon\mathcal{C}\rightarrow\mathcal{D}$ be two functors between small
categories, Then we have that

$\mathrm{cD}(F,G)\leq\mathop{\mathrm{cTC}}(\mathcal{D}).$

###### Proof.

We define the functor
$(F,G)\colon\mathcal{C}\rightarrow\mathcal{D}\times\mathcal{D}$ such that
$p_{1}\circ(F,G)=F$ and $p_{2}\circ(F,G)=G$. Therefore we have that

$\mathrm{cD}(F,G)=cD(p_{1}\circ(F,G),p_{2}\circ(F,G))\leq\mathrm{cD}(p_{1},p_{2})=\mathop{\mathrm{cTC}}(\mathcal{D}).$

∎

######  Proposition 1.20 ([12, Theorem 2]).

Let $F,G\colon\mathcal{C}\rightarrow\mathcal{D}$ be two functors between small
categories, Then

$\mathrm{cD}(F,G)\leq\mathop{\mathrm{ccat}}(\mathcal{C}).$

##  2\. Bi-fibrations

We recall the notions of cartesian morphism and Grothendieck fibration. We
follow the references [5], [8, Appendix A], [20], [24] and [25, Chapter 12],
as well as [21, Section 4].

###  2.1. Cartesian arrows

Let $P\colon\mathcal{E}\to\mathcal{B}$ be a functor.

######  Definition 2.1.

The morphism $\phi\in\mathcal{E}(e_{1},e_{2})$ is cartesian (with respect to
$P$) if, for every arrow $\beta\in\mathcal{E}(e,e_{2})$ and every arrow
$\overline{\alpha}\in\mathcal{B}(Pe,Pe_{1})$ such that
$P\phi\circ\overline{\alpha}=P\beta$, there exists a unique arrow
$\alpha\in\mathcal{E}(e,e_{1})$ such that $\phi\circ\alpha=\beta$ and
$P\alpha=\overline{\alpha}$ (see Diagram (1)).

(1)
${e_{1}}$${e}$${e_{2}}$$\scriptstyle{\phi}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$
${Pe_{1}}$${Pe}$${Pe_{2}}$$\scriptstyle{P\phi}$$\scriptstyle{\overline{\alpha}}$$\scriptstyle{P\beta}$

The name cartesian seems to have its origin in the following characterization,
whose proof is left to the reader:

######  Proposition 2.2.

The morphism $\phi$ is cartesian if and only if the following commutative
square in the category of Sets is a pullback:

${\mathcal{E}(e,e_{1})}$${\mathcal{E}(e,e_{2})}$${\mathcal{B}(Pe,Pe_{1})\ \
}$${\ \
\mathcal{B}(Pe,Pe_{2})}$$\scriptstyle{P(-)}$$\scriptstyle{\phi\circ-}$$\scriptstyle{P(-)}$$\scriptstyle{P(\phi)\circ-}$

######  Example 2.3.

Let $P\colon\mathcal{C}\to\mathcal{C}$ the identity functor. Then any arrow in
$\mathcal{C}$ is cartesian. This is obvious because $P\alpha=\alpha$.

######  Example 2.4.

Let $P\colon\mathcal{C}\to\bullet$ be the constant functor. An arrow $\phi$ in
$\mathcal{C}$ is $P$-cartesian if and only if $\phi$ is an isomorphism.

Let us check it. From the diagram

${e_{1}}$${e_{2}}$${e_{2}}$$\scriptstyle{\phi}$$\scriptstyle{\alpha}$$\scriptstyle{\mathrm{id}_{e_{2}}}$
${\bullet}$${\bullet}$${\bullet}$$\scriptstyle{\mathrm{id}_{\bullet}}$$\scriptstyle{\mathrm{id}_{\bullet}}$$\scriptstyle{\mathrm{id}_{\bullet}}$

we obtain an arrow such that $\phi\alpha=\mathrm{id}_{e_{2}}$. Now,
$\phi(\alpha\phi)=\phi$ means that we have the diagram

${e_{1}}$${e_{1}}$${e_{2}}$$\scriptstyle{\phi}$$\scriptstyle{\alpha\phi}$$\scriptstyle{\phi}$
${\ast}$${\ast}$${\ast}$$\scriptstyle{\mathrm{id}_{\ast}}$$\scriptstyle{\mathrm{id}_{\ast}}$$\scriptstyle{\mathrm{id}_{\ast}}$

and by unicity it must be $\alpha\phi=\mathrm{id}_{e_{1}}$. So the morphism
$\phi$ is an isomorphism.

The converse is immediate.

######  Example 2.5.

Posets can be viewed as small categories, with an arrow existing between two
objects if and only if $c_{1}\leq c_{2}$. Functors between posets are the
order preserving maps $P\colon\mathcal{E}\to\mathcal{B}$. For such a functor,
a map $e_{1}\leq e_{2}$ is cartesian if and only if it verifies the following
condition: if $e\leq e_{2}$ and $Pe\leq e_{1}$ then $e\leq e_{1}$.

This condition ressembles the definition of a down beat point [1]. In fact.
down beat points are cartesian for any functor.

For a study of fibrations in the setting of posets we refer the reader to [3].

###  2.2. Fibrations

######  Definition 2.6.

The functor $P\colon\mathcal{E}\to\mathcal{B}$ is a fibration if for any arrow
$\overline{\phi}\in\mathcal{B}(b_{1},b_{2})$ and any object
$e_{2}\in\mathcal{E}$ such that $Pe_{2}=b_{2}$, there exists a cartesian arrow
$\phi\in\mathcal{E}(e_{1},e_{2})$ such that $P\phi=\overline{\phi}$.

The arrow $\phi$ is called a cartesian lift of $\overline{\phi}$ with codomain
$e_{2}$.

######  Definition 2.7.

If $P\colon\mathcal{E}\to\mathcal{B}$ is a fibration, we define the fiber of
$P$ over $b\in\mathcal{B}$ as the subcategory of $\mathcal{E}$ with objects
$e\in E$ such that $Pe=b$, and with arrows $\nu\in\mathcal{E}(e_{1},e_{2})$
such that $P\nu=\mathrm{id}_{b}$. These arrows are called vertical arrows.

The cartesian lift of a given $\overline{\phi}\colon b_{1}\to b_{2}$ with a
given codomain $e_{2}$ is unique, up to a unique vertical arrow. This follows
from the unicity in the definition of cartesian arrow.

By using the axiom of choice, we can take a particular lift, which will be
denoted by

$\mathrm{Cart}(\overline{\phi},e_{2})\colon\bar{\phi}^{*}e_{2}\to e_{2}.$

This particular choice defines a functor
${\bar{\phi}}^{*}\colon\mathcal{E}_{b_{2}}\to\mathcal{E}_{b_{1}}$, where the
image of a vertical arrow $\nu\in\mathcal{E}_{b_{2}}$ is given by the unique
arrow ${\overline{\phi}}^{*}\nu$ making the following diagram commute:

${\overline{\phi}^{*}e_{2}}$${e_{2}}$${\overline{\phi}^{*}e_{2}^{\prime}\ \
}$${\ \
e_{2}^{\prime}.}$$\scriptstyle{\overline{\phi}^{*}\nu}$$\scriptstyle{\mathrm{Cart}(\overline{\phi},e_{2})}$$\scriptstyle{\nu}$$\scriptstyle{\mathrm{Cart}(\overline{\phi},e_{2}^{\prime})}$

It corresponds to the diagram

${{\overline{\phi}}^{*}e_{2}^{\prime}}$${{\overline{\phi}}^{*}e_{2}}$${e_{2}}$${e_{2}}$$\scriptstyle{\mathrm{Cart}}$$\scriptstyle{\nu}$
${b_{1}}$${b_{1}}$${b_{2}}$${b_{2}}$$\scriptstyle{\phi}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\phi}$$\scriptstyle{\mathrm{id}}$

The functoriality of ${\overline{\phi}}^{*}$ follows again from unicity. It is
called the pullback functor.

######  Example 2.8.

Every group $G$ can be considered as a category with one object, where the
arrows are the elements $g\in G$, and the composition is given by the
operation in $G$. A group homomorphism $F\colon H\to G$ can be considered as a
functor. An arrow in $H$ is always cartesian; hence $F$ is a fibration if and
only if is surjective. The fiber is the kernel.

######  Example 2.9.

Let $\mathcal{C}^{\mathcal{I}_{1}}$ be the category whose objects are the
arrows in $\mathcal{C}$, and whose arrows are the commutative squares. The
codomain functor $P\colon\mathcal{C}^{\mathcal{I}_{1}}\to\mathcal{C}$ is a
fibration if and only if $\mathcal{C}$ has enough pullbacks. The cartesian
arrows are the pullbacks in $\mathcal{C}$. It is called the fundamental
fibration.

We introduce another example of a fibration in the setting of locally small
categories.

######  Example 2.10.

Let $U\colon\textbf{{Top }}\to\textbf{{Sets }}$ be the forgetful functor from
topological spaces to sets. It associates to each topological space $X$ its
underlying set $UX$, and to each continuous map the set map itself. It is a
fibration. The $U$-cartesian maps are the continuous maps $f\colon e_{1}\to
e_{2}$ such that the topology on $e_{1}$ is the initial topology, that is, the
smallest (coarsest) topology making $f$ continuous.

###  2.3. Op-cartesian arrows and op-fibrations

Let
$P^{\mathrm{op}}\colon\mathcal{E}^{\mathrm{op}}\to\mathcal{B}^{\mathrm{op}}$
be the opposite functor of $P\colon\mathcal{E}\to\mathcal{B}$.

######  Definition 2.11.

The arrow $\varphi\colon e_{1}\to e_{2}$ in $\mathcal{E}$ is op-cartesian for
the functor $P$ if the opposite arrow $\varphi^{\mathrm{op}}$ in
$\mathcal{E}^{\mathrm{op}}$ is cartesian for $P^{\mathrm{op}}$. Explicitly,
that means that for any given $\beta\in\mathcal{E}(e_{1},e)$ and any given
$\overline{\alpha}\in\mathcal{B}(Pe_{2},Pe)$ such that $\overline{\alpha}\circ
P\varphi=P\beta$, there exists a unique $\alpha\in\mathcal{E}(e_{2},e)$ such
that $\alpha\circ\varphi=\beta$ and $P\alpha=\overline{\alpha}$, as in the
following diagram:

${e_{1}}$${e}$${e_{2}}$$\scriptstyle{\varphi}$$\scriptstyle{\beta}$$\scriptstyle{\alpha}$
${Pe_{1}}$${Pe}$${Pe_{2}}$$\scriptstyle{P\varphi}$$\scriptstyle{P\beta}$$\scriptstyle{\overline{\alpha}}$

######  Definition 2.12.

A functor $P\colon\ E\to\mathcal{B}$ is an op-fibration if for any map
$\overline{\varphi}\colon b_{1}\to b_{2}$ in $\mathcal{B}$, and for any object
$e_{1}$ in $\mathcal{E}$ with $Pe_{1}=b_{1}$, there exists an op-cartesian
arrow $\varphi\colon b_{1}\to b_{2}$ such that $P\varphi=\overline{\varphi}$.

Again, this op-cartesian lifting is unique up to a unique vertical
isomorphism. Then we can choose some particular lifting

$\mathrm{opCart}(\overline{\varphi},e_{1})\colon
e_{1}\to\overline{\varphi}_{*}e_{1}$

so defining a functor

$\overline{\varphi}_{*}\colon\mathcal{E}_{b_{1}}\to\mathcal{E}_{b_{2}}$

bewteen the fibers.

######  Example 2.13.

The codomain functor $P\colon\mathcal{C}^{\mathcal{I}_{1}}\to\mathcal{C}$ of
Example 2.9 is always an op-fibration.

######  Example 2.14.

Analogously, the domain functor
$P\colon\mathcal{C}^{\mathcal{I}_{1}}\to\mathcal{C}$ is an op-fibration if and
only if $\mathcal{C}$ has enough push-outs.

######  Definition 2.15.

We say that the functor $P\colon\mathcal{E}\to\mathcal{B}$ is a bi-fibration
if it is both a fibration and an op-fibration.

We prefer this terminology instead of “fibration” and “cofibration”, see [16]
for a discussion.

######  Example 2.16.

The forgetful functor $U\colon\textbf{{Top }}\to\textbf{{Sets }}$ from
topological spaces to sets is an op-fibration. Op-cartesian maps are the
continuous maps $e_{1}\to e_{2}$ where $e_{2}$ has the quotient topology.

######  Example 2.17.

Let $\varphi\colon K\to K^{\prime}$ be a simplicial map of simplicial
complexes. Let
$\mathcal{X}(\varphi)\colon\mathcal{X}(K)\to\mathcal{X}(K^{\prime})$ be the
corresponding map between the associated face posets. A poset can be
considered as a small category with the obvioius arrows. Then the functor
$\varphi$ is a fibration, but it is not an op-fibration ([25, Example 13.5]).

######  Example 2.18.

Take a base category $\mathcal{B}$ and another category $\mathcal{C}$, then
the first projection $P_{1}:\mathcal{B}\times\mathcal{C}\rightarrow\mathrm{B}$
is a bi-fibration. In fact, given an arrow $\overline{\phi}\colon b_{1}\to
b_{2}$ in $\mathcal{C}$, and an object $(b_{2},c)$ in
$\mathcal{B}\times\mathcal{C}$, the morphism
$\phi=\overline{\phi}\times\mathrm{id}_{c}\colon(b_{1},c)\to(b_{2},c)$ is a
cartesian lifting. In order to prove it we only must take into account the
diagram:

${{(b_{1},c)}}$${{(b,c^{\prime})}}$${{(b_{2},c)}}$$\scriptstyle{\phi}$$\scriptstyle{w\times
v}$$\scriptstyle{u\times v}$
${b_{1}}$${b}$${b_{2}}$$\scriptstyle{\overline{\phi}}$$\scriptstyle{w}$$\scriptstyle{u}$

Analogously, $P_{1}$ is an op-fibration.

###  2.4. Further examples

######  Example 2.19.

Let $\mathcal{E}$ be the category with objects the integers $n\in\mathbb{Z}$
and a unique morphism $m\rightarrow n$ if and only if $m\leq n$.

$\cdots\to-1\to 0\to 1\to\cdots;$

and let $\mathcal{B}$ be the monoid of the natural numbers $n\geq 0$ as a
category.

${\bullet}$$\scriptstyle{n\in\mathbb{N}}$

The functor $P\colon\mathcal{E}\to\mathcal{B}$ with $P(n)=\bullet$ and
$P(m\rightarrow n)=n-m$ is a fibration. In fact, let us consider
$n\colon\bullet\rightarrow\bullet$ a morphism in $\mathcal{B}$ and $m$ any
object in $\mathcal{E}$ ; then the morphism $m-n\rightarrow m$ covers $n$.
Moreover it is $P$-cartesian as indicated in the following commutative
diagrams:

${m-n}$${l}$${m}$
${\bullet}$${\bullet}$${\bullet}$$\scriptstyle{n}$$\scriptstyle{m-l-n}$$\scriptstyle{m-l}$

Furthermore, $P$ is an op-fibration by a similar argument.

######  Example 2.20.

Let $\mathcal{E}$ be the category with objects the integer numbers
$n\in\mathbb{Z}$, and ziz-zag arrows:

$\cdots\longleftarrow-2\longrightarrow-1\longleftarrow 0\longrightarrow
1\longleftarrow 2\longrightarrow\cdots$

and let $\mathcal{B}$ be the following category

${\overline{0}}$${\overline{2}}$${\overline{3}}$${\overline{1}}$

It is easy to show that the functor $P(n)=\overline{n}\mod 4$ is a fibration
and an op-fibration.

##  3\. Lifting properties

The lifting properties of Grothendieck fibrations were considered by Gray in
[7].

Recall that we denote by $\mathcal{I}_{n}$ the $n$-chain category generated by
the following diagram:

$0\longrightarrow 1\longrightarrow\cdots\longrightarrow n.$

In particular, $\mathcal{I}_{1}$ denotes the category
$0\stackrel{{\scriptstyle s}}{{\longrightarrow}}1$ consisting of two objects
and one non-identity morphism $s$.

Gray proved that the functor $P\colon\mathcal{E}\to\mathcal{B}$ is a fibration
if and only if homotopies have cartesian liftings. Since we represent a
natural transformation ending at $P\circ G$ as a functor
$\eta\colon\mathcal{C}\times\mathcal{I}_{1}\to\mathcal{B}$ where
$\eta_{1}=P\circ G$, we can state the following, which improves [21,
Prop.4.2]:

######  Proposition 3.1.

The functor $P\colon\mathcal{E}\to\mathcal{B}$ is a fibration if and only for
any category $\mathcal{C}$ and for any functor
$H\colon\mathcal{C}\times\mathcal{I}_{1}\to\mathcal{B}$ such that
$H_{1}=H\circ i_{1}=P\circ G$ there exists a functor
$\widetilde{H}\colon\mathcal{C}\times\mathcal{I}_{1}\to\mathcal{E}$ such that
$P\circ\widetilde{H}=H$ and $\widetilde{H}_{1}=\widetilde{H}\circ i_{1}=G$, as
in the following diagram:

${\mathcal{C}}$${\mathcal{E}}$${\mathcal{C}\times\mathcal{I}_{1}}$${\mathcal{B}}$$\scriptstyle{i_{1}}$$\scriptstyle{G}$$\scriptstyle{P}$$\scriptstyle{\widetilde{H}}$$\scriptstyle{H}$

and moreover $\tilde{H}(f\times s)$ is a cartesian arrow for $P$ for any arrow
$f$ in $\mathcal{C}$.

Here, $i_{1}\colon\mathcal{C}\to\mathcal{C}\times\mathcal{I}_{1}$ is the
functor sending the object $c$ into $(c,1)$, and the arrow $f\colon c_{1}\to
c_{2}$ into $f\times\mathrm{id}_{1}$, and $s$ is the only arrow $0\to 1$ in
$\mathcal{I}_{1}$.

###### Proof.

Obviously we must define $\widetilde{H}_{1}=G$. The problem is how to define
$\widetilde{H}_{0}$ in order to have a functor. To do that we have to use the
definition of fibered category. As we know, if we have an arrow
$\overline{\phi}$ in $\mathcal{B}$ and we have an object $e_{2}$ in the fiber
of the codomain $b_{2}$, we can lift the arrow into a cartesian morphism
$\mathrm{Cart}(\overline{\phi},e_{2})\colon{\overline{\phi}}^{*}e_{2}\to
e_{2}$. In particular, we can choose
$\mathrm{Cart}(\mathrm{id}_{b},e)=\mathrm{id}_{e}$.

Now, if we have the arrow $H(\mathrm{id}_{c}\times s)\colon H(c,0)\rightarrow
H(c,1)$ in $\mathcal{B}$ and the object $G(X)\in\mathcal{E}_{H(c,1)}$, there
is a cartesian morphism

$\mathrm{Cart}H(\mathrm{id}_{c}\times s))\colon H(\mathrm{id}_{c}\times
s)^{*}G(c)\rightarrow G(c),$

so we define

$\widetilde{H}(c,0)=H(\mathrm{id}_{c}\times s)^{*}G(c)$

and

$\widetilde{H}(\mathrm{id}_{c}\times s)=\mathrm{Cart}H(\mathrm{id}_{c}\times
s).$

Finally, for every morphism $f\colon c_{1}\rightarrow c_{2}$ in $\mathcal{C}$
we define $\widetilde{H}(f\times s)$ as the unique cartesian arrow filling the
diagram

${{\widetilde{H}(c_{1},0)}}$${{\widetilde{H}(c_{1},1)=Gc_{1}}}$${{\widetilde{H}(c_{2},0)}}$${{\widetilde{H}(c_{2},1)=Gc_{2}}}$$\scriptstyle{\widetilde{H}(\mathrm{id}_{c_{1}}\times
s)}$$\scriptstyle{\widetilde{H}(f\times\mathrm{id}_{1})=Gf}$$\scriptstyle{\widetilde{H}(\mathrm{id}_{c_{2}}\times
s)}$
${{H(c_{1},0)}}$${{H(c_{1},1)}}$${{H(c_{2},0)}}$${{H(c_{2},1)}}$$\scriptstyle{H(f\times\mathrm{id}_{0})}$$\scriptstyle{H(\mathrm{id}_{c_{1}}\times
s)}$$\scriptstyle{H(f\times\mathrm{id}_{1})}$$\scriptstyle{H(\mathrm{id}_{c_{2}}\times
s)}$

For the converse statement, let $\overline{\phi}\colon b_{1}\to b_{2}$ in
$\mathrm{B}$ be an arrow and consider the diagram

${\bullet}$${\mathcal{E}}$${\bullet\times\mathcal{I}_{1}}$${\mathcal{B}}$$\scriptstyle{i_{1}}$$\scriptstyle{G}$$\scriptstyle{P}$$\scriptstyle{\widetilde{H}}$$\scriptstyle{H}$

where $g(\bullet)=e_{2}$ and $H(\mathrm{id}_{\bullet}\times
s)=\overline{\phi}$. Then the map
$\phi={\tilde{H}}(\mathrm{id}_{\bullet}\times s)$ is cartesian and verifies
that $P\phi=\overline{\phi}$. This ends the proof.∎

Analogously, we have the lifting propery for op-fibrations, where the left
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Taking in account this and the previous proposition we have the following
general lifting property.

######  Corollary 3.2.

Let $P:\mathcal{E}\rightarrow\mathcal{B}$ be a bi-fibration. Then the lifting
propery holds for any chain category $\mathcal{I}_{n}$:

${\mathcal{C}}$${\mathcal{E}}$${\mathcal{C}\times\mathcal{I}_{n}}$${\mathcal{B}}$$\scriptstyle{G}$$\scriptstyle{i_{0}}$$\scriptstyle{P}$$\scriptstyle{\widetilde{H}}$$\scriptstyle{H}$

##  4\. Homotopic fiber

The next result is a crucial one. It proves that in a bi-fibration, two
objects $b_{1},b_{2}\in\mathcal{B}$ in the base which are connected by an
arrow $\overline{\varphi}\colon b_{1}\to b_{2}$, have fibers which are
homotopically equivalent as small categories.

######  Theorem 4.1 ([21, Proposition 4.4]).

Let $P:\mathcal{E}\rightarrow\mathcal{B}$ be a bi-fibration. If $b_{1}$ and
$b_{2}$ are two objects in $\mathrm{B}$ such that there is a morphism $u\colon
b_{1}\to b_{2}$, then there is a categorical equivalence between the fibers
$\mathcal{E}_{b_{1}}$ and $\mathcal{E}_{b_{2}}$.

###### Proof.

Remember that in a bi-fibration we have both the pushforward functor
$u_{*}\colon\mathcal{E}_{b_{1}}\to\mathcal{E}_{b_{2}}$ and the pullback
functor $u^{*}\colon\mathcal{E}_{b_{2}}\to\mathcal{E}_{b_{1}}$.

We defined $u_{*}$ as follows: for every object $e_{1}$ in the fiber
$\mathcal{E}_{b_{1}}$ there is a unique object $u_{*}{e_{1}}$ in
$\mathcal{E}_{b_{2}}$ such that $\mathrm{opCart}(u,e_{1})\colon e_{1}\to
u_{*}e_{1}$ is the unique op-cartesian lift of $u$.

Alternatively, for every $e_{2}$ in $\mathcal{E}_{b_{2}}$ there is a unique
object $u^{*}e_{2}\in\mathcal{E}_{b_{1}}$ such that
$\mathrm{Cart}(u,e_{2})\colon u^{*}e_{2}\to e_{2}$ is the unique cartesian
lift of $u$.

Now in order to show tha $u_{*}$ and $u^{*}$ induce a categorical equivalence
it is enough to show that for every object $e_{2}$ in $\mathcal{E}_{b_{2}}$
there are natural arrows $\alpha$ and $\beta$ as in the following diagram:

${e_{2}}$${u^{*}e_{2}}$${u_{*}(u^{*}e_{2})}$$\scriptstyle{\alpha}$$\scriptstyle{\mathrm{Cart}(u,e_{2})}$$\scriptstyle{\mathrm{opCart}(u,u^{*}e_{2})}$$\scriptstyle{\beta}$
${b_{2}}$${b_{1}}$${b_{2}}$$\scriptstyle{\mathrm{id}_{b_{2}}}$$\scriptstyle{u}$$\scriptstyle{u}$

and that their compositions are the identities because
$\mathrm{Cart}(u,e_{2})$ and $\mathrm{opCart}(u,e_{1})$ are a cartesian and
op-cartesian morphism (respectively). Thus, we have a natural isomorphism
between the functors $\mathrm{id}_{\mathcal{E}_{b_{2}}}$ and $u_{*}\circ
u^{*}$.

Analogously we have another natural isomorphism between
$\mathrm{id}_{\mathcal{E}_{b_{1}}}$ and $u^{*}\circ u^{*}$ using the
followings diagrams:

${e_{1}}$${u_{*}e_{1}}$${u^{*}(u_{*}e_{1})}$$\scriptstyle{\mathrm{opCart}(u,e_{1})}$$\scriptstyle{\gamma}$$\scriptstyle{\mathrm{Cart}(u,u_{*}e_{1})}$$\scriptstyle{\phi}$
${b_{1}}$${b_{2}}$${b_{1}}$$\scriptstyle{u}$$\scriptstyle{\mathrm{id}_{b_{1}}}$$\scriptstyle{u}$

∎

A consequence of the latter theorem is that we can speak about the homotopic
invariants of “the fiber” of a bi-fibration.

######  Remark 1.

In the proof of Theorem 4.1 we implicitly assumed that both fibers
$\mathcal{E}_{b_{1}}$ and $\mathcal{E}_{b_{2}}$ are non empty. This might not
be true. However, the Theorem is still true, because for a bi-fibration, if
two objects $b_{1}$, $b_{2}$ in the base are connected by an arrow, say
$\overline{\phi}\colon b_{1}\to b_{2}$, then either
$\mathcal{E}_{b_{1}}=\emptyset=\mathcal{E}_{b_{2}}$ or
$\mathcal{E}_{b_{1}}\neq\emptyset\neq\mathcal{E}_{b_{2}}$ simultaneously.

In fact, assume first that $P$ is only a fibration. If
$\mathcal{E}_{b_{2}}=\emptyset$ the pullback functor is not defined, because
we have no codomain $e_{2}$ to lift the arrow $\overline{\phi}$. Since a
fibration $P\colon\mathcal{E}\to\mathcal{B}$ may not be surjective-on-objects
(see Example 4.2), it may happen that $\mathcal{E}_{b_{2}}$ is empty, while
$\mathcal{E}_{b_{1}}$ is not.

However, for a fibration, if $\mathcal{E}_{b_{2}}\neq\emptyset$ then
$\mathcal{E}_{b_{1}}\neq\emptyset$ too, because to each codomain
$e_{2}\in\mathcal{E}_{b_{2}}$ we associate the domain
${\overline{\phi}}^{*}e_{2}$ of
$\mathrm{Cart}(\phi,e_{2})\in\mathcal{E}_{b_{1}}$.

Dually, for an op-fibration, if $\mathcal{E}_{b_{1}}\neq\emptyset$ then
$\mathcal{E}_{b_{2}}\neq\emptyset$, because to each domain $e_{1}$ we
associate the codomain ${\overline{\phi}}_{*}e_{1}$ of
$\mathrm{opCart}(\overline{\phi},e_{1})$.

As a consequence, in a bi-fibration there are no arrows connecting objects in
the image with objects outside the image, or conversely.

######  Example 4.2.

A fibration which is not surjective-on-objects is for instance the constant
functor $P\colon\mathcal{I}_{1}\to\mathcal{I}_{1}$ with $P(0)=P(1)=0$, and
$P(s)=\mathrm{id}_{0}$. We have $\mathcal{E}_{0}=\mathcal{I}_{1}$ while
$\mathcal{E}_{1}=\emptyset$. This shows that $P$ is not an op-fibration. In
fact, given the arrow $s\colon 0\to 1$ and the domain $1\in\mathcal{E}_{0}$
there is no op-cartesian lift of $s$.

Notice that the arrow $s\colon 0\to 1$ can not be lifted to any cartesian
arrow, but this does not contradict the definition of fibration (there are no
possible codomains).

######  Example 4.3.

The following functor $P:\mathcal{E}\rightarrow\mathcal{B}=\mathcal{I}_{1}$ is
a bi-fibration, but the fibers over $0$ and $1$ are not isomorphic. However,
they are homotopically equivalent, as stated in Theorem 4.1.

Let $\mathcal{E}$ be the category generated by the following diagram:

${\bar{1}}$${0}$${1}$$\scriptstyle{f}$$\scriptstyle{s}$$\scriptstyle{g}$

where $f\circ g=\mathrm{id}_{1}$ and $g\circ f=\mathrm{id}_{\bar{1}}$. Let
$\mathcal{B}=\mathcal{I}_{1}$ the category with only one arrow
$0\xrightarrow{s}1$.

The functor $P$ is defined as $P(0)=0$, $P(1)=P(\bar{1})=1$, $P(s)=s$ and
$P(f)=P(g)=\mathrm{id}_{1}$. It is a bi-fibration because the arrows
$\mathrm{id}_{0}$, $\mathrm{id}_{1}$, $\mathrm{id}_{\bar{1}}$, $s$ and $g\circ
s$ are cartesian and op-cartesian. The fiber over $0$ is the discrete category
with one object $0$ and the fiber over $1$ is the following category:

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It is contractible by the natural transformation $\alpha$ between the identity
and the constant functor $\bar{1}$ given by $\alpha(1)=g$,
$\alpha(\bar{1})=\mathrm{id}_{\bar{1}}$. In fact, we have

${1}$${\bar{1}}$${\bar{1}}$${\bar{1}}$$\scriptstyle{g}$$\scriptstyle{\alpha(1)=g}$$\scriptstyle{\mathrm{id}_{\bar{1}}}$$\scriptstyle{\alpha(\bar{1})=\mathrm{id}_{\bar{1}}}$
${\bar{1}}$${\bar{1}}$${1}$${\bar{1}}$$\scriptstyle{f}$$\scriptstyle{\alpha(\bar{1})=\mathrm{id}_{\bar{1}}}$$\scriptstyle{\mathrm{id}_{\bar{1}}}$$\scriptstyle{\alpha(1)=g}$

##  5\. Varadarajan’s theorem

In the classical LS-category theory of topological spaces, Varadarajan [23]
proved a formula relating the categories of the fiber, the base and the total
space for a Hurewicz fibration. We generalized this result for the homotopic
distance, in [13, Theorem 6.1].

In [21, Theorem 4.5], Tanaka proved an analogous result for a bi-fibration
$P\colon\mathcal{E}\to\mathcal{B}$ with fiber $\mathcal{F}$ between small
categories, with path-connected base, namely

(2)
$\mathop{\mathrm{ccat}}\mathcal{E}+1\leq(\mathop{\mathrm{ccat}}\mathcal{B}+1)\cdot(\mathop{\mathrm{ccat}}\mathcal{F}+1).$

Recall that by Corollary 4.1, if the base category $\mathcal{B}$ of a bi-
fibration $P:\mathcal{E}\rightarrow\mathcal{B}$ is path-connected, then every
two fibers are homotopy equivalent. We will refer to any of them as (the
homotopy type of) “the fiber $\mathcal{F}$ of the bi-fibration”.

We will both extend our [13, Theorem 6.1] to the context of bi-fibrations of
small categories and generalize Tanaka’s result, as follows.

######  Theorem 5.1.

Let $(F,\overline{F})$ and $(G,\overline{G})$ be two morphisms between the bi-
fibrations $P\colon\mathcal{E}\rightarrow\mathcal{B}$ and
$P^{\prime}\colon\mathcal{E}^{\prime}\rightarrow\mathcal{B}^{\prime}$. Let
$\mathcal{B}$ and $\mathcal{B}^{\prime}$ be path-connected. Let $b$ be an
object in $\mathcal{B}$ such that $F(b)=G(b)=b^{\prime}$ and let
$F_{b},G_{b}\colon\mathcal{E}_{b}\rightarrow\mathcal{E}^{\prime}_{b^{\prime}}$
be the induced functors between the fibers. Then

$\mathrm{cD}(F,G)+1\leq(\mathrm{cD}(F_{b},G_{b})+1)\cdot(\mathop{\mathrm{ccat}}(\mathcal{B})+1).$

###### Proof.

We assume that $\mathrm{cD}(F_{b},G_{b})$ and
$\mathop{\mathrm{ccat}}\mathcal{B}$ are both finite, otherwise the result is
trivial.

Let $\mathop{\mathrm{ccat}}(\mathcal{B})=m$, with $\\{U_{0},...,U_{m}\\}$ a
categorical cover of $\mathcal{B}$, and let $\mathrm{cD}(F_{b},G_{b})=n$, with
$\\{V_{0},...,V_{n}\\}$ a covering of $\mathcal{E}_{b}$ by homotopy domains
for $F_{b}$ and $G_{b}$.

For every $i\in\\{0,...,m\\}$ we have a homotopy $C^{i}\colon
U_{i}\times\mathcal{I}_{k_{i}}\to\mathcal{B}$ between the inclusion
$U_{i}\hookrightarrow\mathcal{B}$ and a constant functor $\bullet_{i}$ for
some object $\ast_{i}\in\mathcal{B}$. Since $\mathcal{B}$ is connected we can
assume that all $\bullet_{i}$ is the same object $\ast$ for all $i$ (see the
proof of Proposition 1.9).

Let $P^{-1}(U_{i})$ be the subcategory of $\mathcal{E}$ whose objects are the
objects $e\in\mathcal{E}$ with $Pe\in U_{i}$, and whose arrows are the arrows
$\alpha\in\mathcal{E}(e_{1},e_{2})$ such that $P\alpha$ is an arrow in
$U_{i}$.

By the homotopy lifting property applied to the following diagram:

${P^{-1}(U)}$${\mathcal{E}}$${P^{-1}(U)\times\mathcal{I}_{k_{i}}}$${U\times\mathcal{I}_{k_{i}}}$${\mathcal{B}}$$\scriptstyle{i_{0}}$$\scriptstyle{P}$$\scriptstyle{\widetilde{C}^{i}}$$\scriptstyle{P\times\mathrm{id}}$$\scriptstyle{C^{i}}$

we have a homotopy
$\widetilde{C}^{i}:U_{i}\times\mathcal{I}_{k_{i}}\to\mathcal{E}$ such that
${\widetilde{C}^{i}}_{0}$ is the inclusion
$P^{-1}(U_{i})\hookrightarrow\mathcal{E}$ and ${\widetilde{C}^{i}}_{k_{i}}$
lies inside the fiber $\mathcal{E}_{b}$.

For each $i\in\\{1,\dots,m\\}$ and $j\in\\{1,\dots,n\\}$ we define

$W_{i,j}=P^{-1}(U_{i})\cap({\widetilde{C}^{i}}_{k_{i}})^{-1}(V_{j}).$

We claim that $\\{W_{i,j}\\}_{0\leq i\leq m,0\leq j\leq n}$ is a geometric
cover of $\mathcal{E}$ such that each $W_{i,j}$ is a homotopy domain for $F$
and $G$.

(1) $\\{W_{i,j}\\}$ is a geometric cover of $\mathcal{E}$.

Let

$\mathcal{C}\colon C_{1}\longrightarrow
C_{2}\longrightarrow\cdots\longrightarrow C_{l}$

be a chain in $\mathcal{E}$. Then we obtain the chain

$P(\mathcal{C})\colon P(C_{1})\longrightarrow
P(C_{2})\longrightarrow\cdots\longrightarrow P(C_{l})$

in $\mathcal{B}$. Since $\\{U_{0},...,U_{m}\\}$ is a geometric cover of
$\mathcal{B}$, there is some $i$ such that the chain $P\mathcal{C})$ lies in
$U_{i}$, so the chain $\mathcal{C}$ lies in $P^{-1}(U_{i})$. Moreover, the
functor ${\widetilde{C}^{i}}_{k_{i}}$ is defined in $\mathcal{C}$, hence we
have a new chain ${\widetilde{C}}^{i}_{k_{i}}(\mathcal{C})$ that lies in the
fiber $\mathcal{E}_{b}$. Now, we know that $\\{V_{j}\\}$ is a geometric cover
of $F_{b}$, so ${\widetilde{C}}^{i}_{k_{i}}(\mathcal{C})$ lies in some
$V_{j}$. We conclude that $\mathcal{C}$ is in $W_{i,j}$.

(2) Each $W_{i,j}$ is a homotopy domain for $F$ and $G$.

For the sake of simplicity we change the notations as follows: $U=U_{i}$,
$\widetilde{C}=\widetilde{C}^{i}$, $k=k_{i}$, $V=V_{j}$.

Let $K\colon V\times\mathcal{I}_{l}\rightarrow\mathcal{E}_{b}$ be a homotopy
between $F|_{V}$ and $G|_{V}$ and let
$\iota\colon\mathcal{E}_{b}\hookrightarrow\mathcal{E}$ be the inclusion of the
fiber into the total category $\mathcal{E}$. The homotopy that we need is the
functor

$H:W_{i,j}\times\mathcal{I}_{k+l+k}\rightarrow\mathcal{E}^{\prime}$

given by

$H(c,n)=\begin{cases}F\widetilde{C}(c,n)&\text{if}\quad 0\leq n\leq k\cr\iota
K(\widetilde{C}_{k}c,n-k)&\text{if}\quad k\leq n\leq k+l\cr
G\widetilde{C}(c,k+l+k-n)&\text{if}\quad k+l\leq n\leq k+l+k\cr\end{cases}.$

It only remains to check that $H$ is well defined and that it is the wanted
homotopy:

$\displaystyle H(c,0)=$ $\displaystyle F\widetilde{C}(c,0)=Fc\text{ since
}\widetilde{C}_{0}\text{ is the inclusion},$ $\displaystyle\iota
K(\widetilde{C}_{k}c,0)=$ $\displaystyle\iota
F(\widetilde{C}_{k}c)=F\widetilde{C}(c,k),$ $\displaystyle\iota
K(\widetilde{C}_{k}c,l)=$ $\displaystyle\iota
G\widetilde{C}_{k}c=G\widetilde{C}(c,k),$ $\displaystyle H(c,k+l+k)=$
$\displaystyle G\widetilde{C}(c,0)=Gc\text{ since }\widetilde{C}_{0}\text{ is
the inclusion}.$

This proves that $\mathrm{cD}(F,G)\leq m+n$, as stated. ∎

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