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 Leonard Guetta committed Jun 13, 2020 1 \chapter{Homotopy and homology type of free $2$-categories}  Leonard Guetta committed Jun 25, 2020 2 3 4 \section{Preliminaries : the case of free $1$-categories} In this section, we review some homotopical results concerning free ($1$-)categories that will be of great help in the sequel. \begin{paragr}  Leonard Guetta committed Jul 02, 2020 5 6 7 8 9 10 11  A \emph{reflexive graph} $G$ consists of the data of two sets $G_0$ and $G_1$ together with \begin{itemize} \item a source'' map $\src : G_1 \to G_0$, \item a target'' map $\trgt : G_1 \to G_0$, \item a unit'' map $1_{(-)} : G_0 \to G_1$, \end{itemize} such that for every $x \in G_0$,  Leonard Guetta committed Jun 25, 2020 12  $ Leonard Guetta committed Jul 02, 2020 13  \src(1_{x}) = \trgt (1_{x}) = x.  Leonard Guetta committed Jun 25, 2020 14 $  Leonard Guetta committed Jul 08, 2020 15  The vocabulary of categories is used : elements of $G_0$ are \emph{objects} or \emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells}, arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A \emph{morphism of reflexive graphs} $f : G \to G'$ consists of maps $f_0 : G_0 \to G'_0$ and $f_1 : G_1 \to G'_1$ that commute with sources, targets and units in an obvious sense. This defines the category $\Rgrph$ of reflexive graphs. There is a underlying reflexive graph'' functor  Leonard Guetta committed Jun 25, 2020 16 17 18 19 20 21 22 23 24  $U : \Cat \to \Rgrph,$ which has a left adjoint $L : \Rgrph \to \Cat.$ For a reflexive graph $G$, the objects of $L(G)$ are exactly the objects of $G$ and an arrow $f$ of of $L(G)$ is a chain $ Leonard Guetta committed Jul 02, 2020 25 26 27  \begin{tikzcd} X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} \end{tikzcd}  Leonard Guetta committed Jun 25, 2020 28 $  Leonard Guetta committed Jun 26, 2020 29  of arrows of $G$, such that \emph{none} of the $f_k$ are units. The integer $n$ is referred to as the \emph{length} of $f$. Composition is given by concatenation of chains.  Leonard Guetta committed Jun 25, 2020 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45  \end{paragr} \begin{lemma} A category $C$ is free in the sense of \todo{ref} if and only if there exists a reflexive graph $G$ such that $C \simeq L(G).$ \end{lemma} \begin{proof} If $C$ is free, consider the reflexive graph $G$ such that $G_0 = C_0$ and $G_1$ is the subset of $C_1$ whose elements are either generating $1$-cells of $C$ or units. It is straightforward to check that $C\simeq L(G)$. Conversely, if $C \simeq L(G)$ for some reflexive graph $G$, then the description of the arrows of $L(G)$ given in the previous paragraph shows that $C$ is free and that its set of generating $1$-cells is (isomorphic to) the non unital $1$-cells of $G$. \end{proof} \begin{remark} Note that for a morphism of reflexive graphs $f : G \to G'$, the functor $L(f)$ is not necessarily rigid in the sense of \todo{ref} because generating $1$-cells may be sent to units. \end{remark} \begin{paragr}  Leonard Guetta committed Jul 02, 2020 46  There is another important description of the category $\Rgrph$. Consider $\Delta_{\leq 1}$ the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$. Then, the category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq 1}}$, the category of pre-sheaves on $\Delta_{\leq 1}$. In particular, the canonical inclusion $i : \Delta_{\leq 1} \rightarrow \Delta$ induces by pre-composition a functor  Leonard Guetta committed Jun 26, 2020 47 48 49 50 51 52 53  $i^* : \Psh{\Delta} \to \Rgrph,$ which, by the usual technique of Kan extensions, has a left adjoint $i_! : \Rgrph \to \Psh{\Delta}.$  Leonard Guetta committed Jul 02, 2020 54  For a graph $G$, the simplicial set $i_!(G)$ has $G_0$ as its set of $0$-simplices, $G_1$ as its set of $1$-simplices and all $k$-simplices are degenerated for $k>2$. For future reference, we put here the following lemma.  Leonard Guetta committed Jun 26, 2020 55 \end{paragr}  Leonard Guetta committed Jun 26, 2020 56 \begin{lemma}\label{lemma:monopreserved}  Leonard Guetta committed Jun 26, 2020 57 58 59 60 61 62 63 64  The functor $i_! : \Rgrph \to \Psh{\Delta}$ preserves monomorphism. \end{lemma} \begin{proof} \end{proof} \begin{paragr} Let us denote by $N : \Psh{\Delta} \to \Cat$ (instead of $N_1$ as in Paragraph \todo{ref}) the usual nerve of categories and by $c : \Cat \to \Psh{\Delta}$ its left adjoint. Recall that for a (small) category $C$, an $n$-simplex of $N(C)$ is a chain $ Leonard Guetta committed Jul 02, 2020 65 66 67  \begin{tikzcd} X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} \end{tikzcd}  Leonard Guetta committed Jun 26, 2020 68 69 70 71 72 73 74 75 76 $ of arrows of $C$. Such an $n$-simplex is degenerated if and only if at least one of the $f_k$ is a unit. It is straightforward to check that the composite of $\Cat \overset{N}{\rightarrow} \Psh{\Delta} \overset{i^*}{\rightarrow} \Rgrph$ is nothing but the forgetful functor $U : \Cat \to \Rgrph$. Thus, the functor $L : \Rgrph \to \Cat$ is (isomorphic to) the composite of $\Rgrph \overset{i_!}{\rightarrow} \Psh{\Delta} \overset{c}{\rightarrow} \Cat.$  Leonard Guetta committed Jun 26, 2020 77 78 79  We now review a construction of Dwyer and Kan from \cite{dwyer1980simplicial}.  Leonard Guetta committed Jul 08, 2020 80  Let $G$ be a reflexive graph. For every $k\geq 1$, we define the simplicial set $N^k(G)$ as the sub-simplicial set of $N(L(G))$ whose $n$-simplices are chains  Leonard Guetta committed Jul 02, 2020 81 82 83 84  $\begin{tikzcd} X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} \end{tikzcd}  Leonard Guetta committed Jun 26, 2020 85 86 87 88 89 90 91 $ of arrows of $L(G)$ such that the length of each $f_i$ is \emph{at most} $k$. In particular, we have $N^1(G)=i_!(G)$ and the transfinite composition of $ Leonard Guetta committed Jul 02, 2020 92  i_!(G) = N^1(G) \hookrightarrow N^2(G) \hookrightarrow \cdots \hookrightarrow N^{k}(G) \hookrightarrow N^{k+1}(G) \hookrightarrow \cdots  Leonard Guetta committed Jun 26, 2020 93 94 95 96 97 98 $ is easily seen to be the map $\eta_{i_!(G)} : i_!(G) \to Nci_!(G),$ where $\eta$ is the unit of the adjunction $c \dashv N$.  Leonard Guetta committed Jun 26, 2020 99 \end{paragr}  Leonard Guetta committed Jun 26, 2020 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 \begin{lemma}[Dwyer-Kan]\label{lemma:dwyerkan} For every $k\leq 1$, the canonical inclusion map $N^{k}(G) \to N^{k+1}(G)$ is a weak equivalence of simplicial sets. \end{lemma} \begin{proof} \end{proof} From this lemma, we deduce the following propositon. \begin{proposition} Let $G$ be a reflexive graph. The map $\eta_{i_!(G)} : i_!(G) \to Nci_!(G),$ where $\eta$ is the unit of the adjunction $c \dashv N$, is a weak equivalence of simplicial sets.  Leonard Guetta committed Jul 02, 2020 117 \end{proposition}  Leonard Guetta committed Jun 26, 2020 118 \begin{proof}  Leonard Guetta committed Jul 02, 2020 119  It is an immediate consequence of Lemma \ref{lemma:dwyerkan} and the fact that filtered colimits are homotopic in a model category whose objects are all cofibrants \todo{ref}.  Leonard Guetta committed Jun 26, 2020 120 121 122 123 124 125 \end{proof} From the previous proposition, we deduce the following very useful corollary. \begin{corollary}\label{cor:hmtpysquaregraph} Let $\begin{tikzcd}  Leonard Guetta committed Jul 02, 2020 126 127  A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\ C \ar[r,"\gamma"]& D  Leonard Guetta committed Jun 26, 2020 128 129  \end{tikzcd}$  Leonard Guetta committed Jul 02, 2020 130  be a cocartesian square in $\Rgrph$. If either $\alpha$ or $\beta$ is a monomorphism, then the induced square  Leonard Guetta committed Jun 26, 2020 131 132  $\begin{tikzcd}  Leonard Guetta committed Jul 02, 2020 133 134  L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"]& L(B) \ar[d,"L(\delta)"] \\ L(C) \ar[r,"L(\gamma)"]& L(D)  Leonard Guetta committed Jun 26, 2020 135 136  \end{tikzcd}$  Leonard Guetta committed Jul 08, 2020 137  is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with the Thomason weak equivalences.  Leonard Guetta committed Jun 26, 2020 138 139 140 141 142 143 144 145 146 \end{corollary} \begin{proof} Since the nerve $N$ induces an equivalence of op-prederivators $\Ho(\Cat^{\Th}) \to \Ho(\Psh{\Delta}),$ it suffices to prove that the induced square of simplicial sets $\begin{tikzcd}  Leonard Guetta committed Jul 02, 2020 147 148  NL(A) \ar[d,"NL(\alpha)"] \ar[r,"NL(\beta)"]& NL(B) \ar[d,"NL(\delta)"] \\ NL(C) \ar[r,"NL(\gamma)"]& NL(D)  Leonard Guetta committed Jun 26, 2020 149 150 151 152 153  \end{tikzcd}$ is homotopy cocartesian. But, since $L \simeq c\circ i_!$, it follows from Lemma \ref{cor:hmtpysquaregraph} that this last square is weakly equivalent to the square of simplicial sets $\begin{tikzcd}  Leonard Guetta committed Jul 02, 2020 154 155  i_!(A) \ar[d,"i_!(\alpha)"] \ar[r,"i_!(\beta)"] &i_!(B) \ar[d,"i_!(\delta)"] \\ i_!(C) \ar[r,"i_!(\gamma)"]& i_!(D).  Leonard Guetta committed Jun 26, 2020 156 157  \end{tikzcd}$  Leonard Guetta committed Jul 02, 2020 158  This square is cocartesian because $i_!$ is a left adjoint. Since $i_!$ preserves monomorphisms (Lemma \ref{lemma:monopreserved}), the result follows from the fact that monomorphisms are cofibrations of simplicial sets for the standard Quillen model structure and \todo{ref}.  Leonard Guetta committed Jun 26, 2020 159 \end{proof}  Leonard Guetta committed Jun 28, 2020 160 161 162 163 164 Actually, by working a little more, we obtain the slightly more general result below. \begin{proposition} Let $\begin{tikzcd}  Leonard Guetta committed Jul 02, 2020 165 166  A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\ C \ar[r,"\gamma"]& D  Leonard Guetta committed Jun 28, 2020 167 168  \end{tikzcd}$  Leonard Guetta committed Jul 02, 2020 169  be a cocartesian square in $\Rgrph$. Suppose that the following two conditions are satisfied  Leonard Guetta committed Jul 02, 2020 170  \begin{enumerate}[label=\alph*)]  Leonard Guetta committed Jun 28, 2020 171  \item Either $\alpha$ or $\beta$ is injective on objects.  Leonard Guetta committed Jul 02, 2020 172  \item Either $\alpha$ or $\beta$ is injective on arrows.  Leonard Guetta committed Jul 02, 2020 173  \end{enumerate}  Leonard Guetta committed Jun 28, 2020 174 175 176  Then, the square $\begin{tikzcd}  Leonard Guetta committed Jul 02, 2020 177 178  L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"] &L(B) \ar[d,"L(\delta)"] \\ L(C) \ar[r,"L(\gamma)"] &L(D)  Leonard Guetta committed Jun 28, 2020 179 180  \end{tikzcd}$  Leonard Guetta committed Jul 08, 2020 181  is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with the Thomason weak equivalences.  Leonard Guetta committed Jun 28, 2020 182 183 \end{proposition} \begin{proof}  Leonard Guetta committed Jul 02, 2020 184 185 186 187 188  The case where $\alpha$ or $\beta$ is injective both on objects and arrows is Corollary \ref{cor:hmtpysquaregraph}. Hence, we only have to treat the case when $\alpha$ is injective on objects and $\beta$ is injective on arrows. The other case being symmetric. Let use denote by $E$ the set of objects of $B$ that lies in the image of $\beta$. For each element $x$ of $E$, we denote by $F_x$ the fiber'' of $x$, that is the set of objects of $A$ that $\beta$ sends to $x$. We consider the set $E$ and each $F_x$ as discrete reflexive graphs, i.e. reflexive graphs with no non-unital arrow. Now, let $G$ be the reflexive graph defined with the following cocartesian square $\begin{tikzcd}  Leonard Guetta committed Jul 08, 2020 189  \displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[d]\\  Leonard Guetta committed Jul 02, 2020 190  A \ar[r] & G,  Leonard Guetta committed Jul 08, 2020 191  \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]  Leonard Guetta committed Jul 02, 2020 192 193  \end{tikzcd}$  Leonard Guetta committed Jul 08, 2020 194 195  where the morphism $\coprod_{x \in E}F_x \to A$ is induced by the inclusion of each $F_x$ in $A$, and the morphism $\coprod_{x \in E}F_x \to E$ is the only one that sends an element $a \in F_x$ to $x$. In other words, $G$ is obtained from $A$ by collapsing the objects that are identified through $\beta$. Notice importantly that the morphism $\coprod_{x \in E}F_x \to A$ is a monomorphism, i.e. injective on objects and arrows, and the morphism $A \to G$ is injective on arrows.  Leonard Guetta committed Jun 28, 2020 196 \end{proof}  Leonard Guetta committed Jun 26, 2020 197 198 199 200 201 202 203 204 \begin{example}[Adding a generator] Let $C$ be a free category, $A$ and $B$ (possibly equal) two objects of $C$ and let $C'$ be the category obtained from $C$ by adding a generator $A \to B$, i.e. defined with the following cocartesian square: $\begin{tikzcd} \partial\sD_1 \ar[d,"i_1"] \ar[r,"{\langle A, B \rangle}"] & C \ar[d] \\ \sD_1 \ar[r] & C'. \end{tikzcd}$  Leonard Guetta committed Jul 02, 2020 205  Then, this square is homotopy cocartesian in $\Cat$ (when equipped with the Thomason equivalences). Indeed, it obviously is the image of a square of $\Rgrph$ by the functor $L$ and the morphism $i_1 : \partial\sD_1 \to \sD_1$ comes from a monomorphism of $\Rgrph$.  Leonard Guetta committed Jun 26, 2020 206 207 208 209 \end{example} \begin{remark} Since every free category is obtained by recursively adding generators starting from a set of objects (seen as a $0$-category), the previous example yields another proof that free (1-)categories are \good{} (which we already knew since we have seen that all (1-)categories are \good{}). \end{remark}  Leonard Guetta committed Jun 27, 2020 210 211 212 213 \begin{example}[Identyfing two generators] Let $C$ be a free category and $f,g : A \to B$ parallel generating arrows of $C$ such that $f\neq g$. Now consider the category $C'$ obtained from $C$ by identifying'' $f$ and $g$, i.e. defined with the following cocartesian square $\begin{tikzcd}  Leonard Guetta committed Jul 02, 2020 214  \sS_1\ar[d] \ar[r,"{\langle f, g \rangle}"] &C \ar[d] \\  Leonard Guetta committed Jun 27, 2020 215 216 217 218 219 220 221 222 223 224  \sD_1 \ar[r] & C', \end{tikzcd}$ where the morphism $\sS_1 \to \sD_1$ is the one that sends the two generating arrows of $\sS_1$ to the unique generating arrow of $\sD_1$. Let us prove that the previous square is homotopy cocartesian in $\Cat$ (when equipped with Thomason weak equivalences). Notice that this square is the image of a cocartesian square of $\Rgrph$ by the functor $L$ as in the previous example. We now distinguish two cases. First, if $A \neq B$, then the map $\sS_1 \to C$ comes from a monomorphism of reflexive graphs and we conclude as in the previous example. Now if $A=B$, notice that the map $\sS_1 \to C$ factorizes as $\sS_1 \to F_2 \to C$ where $F_2$ is the free monoid with two generators, seen as a category. In particular, it is free and notice that the map on the left comes from a monomorphism of reflexive graphs. Now, this factorization yields a factorization of our cocartesian square into two cocartesian squares $\begin{tikzcd}  Leonard Guetta committed Jul 02, 2020 225  a  Leonard Guetta committed Jun 27, 2020 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241  \end{tikzcd}$ \end{example} \begin{example}[Killing a generator] Let $C$ be a free category and let $f : A \to B$ one of its generating arrow such that $A \neq B$. Now consider the category $C'$ obtained from $C$ by killing'' $f$, i.e. defined with the following cocartesian square: $\begin{tikzcd} \sD_1 \ar[d] \ar[r,"\langle f \rangle"] & C \ar[d] \\ \sD_0 \ar[r] & C'. \end{tikzcd}$ Then, this above square is homotopy cocartesion in $\Cat$ (equipped with the Thomason weak equivalences). Indeed, it obviously is the image of a square in $\Rgrph$ by the functor $L$ and since the source and target of $f$ are different, the top map comes from a monomorphism of $\Rgrph$. \end{example} \begin{remark} Note that in the previous example, we see that it was useful to consider the category of reflexive graphs and not only the category of graphs because the map $\sD_1 \to \sD_0$ does not come from a morphism in the category of graphs. \todo{À mieux dire ?} \end{remark}  Leonard Guetta committed Jun 26, 2020 242