contractible.tex 33.3 KB
 Leonard Guetta committed Jan 21, 2021 1 2 3 \chapter{Homology of contractible \texorpdfstring{$\oo$}{ω}-categories and its consequences} \chaptermark{Contractible $\omega$-categories and consequences}  Leonard Guetta committed Oct 22, 2020 4 \section{Contractible \texorpdfstring{$\oo$}{ω}-categories}  5 Recall that for every $\oo$\nbd{}category $C$, we write $p_C : C \to \sD_0$ for the canonical morphism to the terminal object of $\sD_0$.  Leonard Guetta committed Sep 17, 2020 6 \begin{definition}\label{def:contractible}  Leonard Guetta committed Oct 01, 2020 7  An $\oo$\nbd{}category $C$ is \emph{oplax contractible} when the canonical morphism $p_C : C \to \sD_0$ is an oplax homotopy equivalence (Definition \ref{def:oplaxhmtpyequiv}).  Leonard Guetta committed Sep 17, 2020 8 9 10 11 12 13 14 15 16  %% there exists an object $x_0$ of $X$ and an oplax transformation %% $%% \begin{tikzcd} %% X \ar[r,"p_X"] \ar[rd,"1_X"',""{name=A,above}]& \sD_0 \ar[d,"\langle x_0 \rangle"]\\ %% &X. %% \ar[from=A,to=1-2,"\alpha",Rightarrow] %% \end{tikzcd} %%$  Leonard Guetta committed Jun 01, 2020 17 \end{definition}  Leonard Guetta committed Sep 17, 2020 18 19 20  %% \begin{paragr}  Leonard Guetta committed Oct 02, 2020 21 %% In other words, an $\oo$\nbd{}category $X$ is contractible when there exists an object $x_0$ of $X$ such that $\langle x_0 \rangle : \sD_0 \to X$ is a deformation retract (Paragraph \ref{paragr:defrtract}). It follows from Lemma \ref{lemma:oplaxloc} that $p_X : X \to \sD_0$ is a Thomason equivalence. In particular, we have the following lemma.  Leonard Guetta committed Sep 17, 2020 22 23 24 %% \end{paragr} %% \begin{lemma}\label{lemma:hmlgycontractible}  Leonard Guetta committed Oct 01, 2020 25 %% Let $X$ be a contractible $\oo$\nbd{}category. The morphism of $\ho(\Ch)$  Leonard Guetta committed Sep 17, 2020 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 %% $%% \sH(X) \to \sH(\sD_0) %%$ %% induced by the canonical morphism $p_X : X \to \sD_0$ is an isomorphism. %% \end{lemma} %% \begin{paragr} %% In addition to the previous result, it is immediate to check that $\sH(\sD_0)$ is nothing but $\mathbb{Z}$ seen as an object of $\ho(\Ch)$ concentrated in degree $0$. %% \end{paragr} %% \begin{corollary} %% Let $u : X \to Y$ be a morphism of $\oo\Cat$. If $u$ is a homotopy equivalence (Paragraph \ref{paragr:hmtpyequiv}), then %% $%% \sH^{\pol}(\gamma_{\folk}(u)) %%$ %% is an isomorphism. %% \end{corollary} %% \todo{Expliquer p-e pourquoi le corollaire précédent est important.} %% \begin{corollary}\label{cor:hmlgypolcontractible} %% Let $X$ be an $\oo$-category. If $X$ is contractible, then the morphism %% $%% \sH^{\pol}(\gamma_{\folk}(p_X)) : \sH^{\pol}(X) \to \sH^{\pol}(\sD_0) %%$ %% is an isomorphism of $\ho(\Ch)$. %% \end{corollary} %% We can now prove the main result of this section. \begin{proposition}\label{prop:contractibleisgood}  Leonard Guetta committed Oct 01, 2020 52  Every oplax contractible $\oo$\nbd{}category $C$ is \good{} and we have  Leonard Guetta committed Jun 03, 2020 53  $ Leonard Guetta committed Sep 17, 2020 54  \sH^{\pol}(C)\simeq \sH^{\sing}(C)\simeq \mathbb{Z}  Leonard Guetta committed Jun 03, 2020 55 $  Leonard Guetta committed Sep 17, 2020 56  where $\mathbb{Z}$ is seen as an object of $\ho(\Ch)$ concentrated in degree $0$.  Leonard Guetta committed Jun 03, 2020 57 58 \end{proposition} \begin{proof}  Leonard Guetta committed Sep 17, 2020 59 Consider the commutative square  Leonard Guetta committed Jun 03, 2020 60 61  $\begin{tikzcd}  Leonard Guetta committed Oct 25, 2020 62 63  \sH^{\pol}(C) \ar[d,"\sH^{\pol}(p_C)"] \ar[r,"\pi_C"] & \sH^{\sing}(C) \ar[d,"\sH^{\sing}(p_C)"] \\ \sH^{\pol}(\sD_0) \ar[r,"\pi_{\sS_0}"] & \sH^{\sing}(\sD_0).  Leonard Guetta committed Jun 03, 2020 64 65  \end{tikzcd}$  Leonard Guetta committed Dec 27, 2020 66 67 68 69 70 71 72  It follows respectively from Proposition \ref{prop:oplaxhmtpyisthom} and Proposition \ref{prop:oplaxhmtpypolhmlgy} that the right and left vertical morphisms of the above square are isomorphisms. Then, an immediate computation left to the reader shows that $\sD_0$ is \good{} and that $\sH^{\pol}(\sD_0)\simeq \sH^{\sing}(\sD_0)\simeq \mathbb{Z}$. By a 2-out-of-3 property, we deduce that $\pi_C : \sH^{\sing}(C)\to \sH^{\pol}(C)$ is an isomorphism and $\sH^{\pol}(C)\simeq \sH^{\sing}(C)\simeq \mathbb{Z}$.  Leonard Guetta committed Jun 03, 2020 73 \end{proof}  Leonard Guetta committed Sep 17, 2020 74 \begin{remark}  Leonard Guetta committed Oct 01, 2020 75  Definition \ref{def:contractible} admits an obvious lax'' variation and Proposition \ref{prop:contractibleisgood} is also true for lax contractible $\oo$\nbd{}categories.  Leonard Guetta committed Sep 17, 2020 76  \end{remark}  Leonard Guetta committed Dec 27, 2020 77 We end this section with an important result on slice $\oo$\nbd{}categories (Paragraph \ref{paragr:slices}).  Leonard Guetta committed Jun 09, 2020 78 \begin{proposition}\label{prop:slicecontractible}  Leonard Guetta committed Oct 01, 2020 79  Let $A$ be an $\oo$\nbd{}category and $a_0$ an object of $A$. The $\oo$\nbd{}category $A/a_0$ is oplax contractible.  Leonard Guetta committed Jun 03, 2020 80 81  \end{proposition} \begin{proof}  Leonard Guetta committed Sep 28, 2020 82  This follows from the dual of \cite[Proposition 5.22]{ara2020theoreme}.  Leonard Guetta committed Jun 03, 2020 83 84  \end{proof} \section{Homology of globes and spheres}  85 \begin{lemma}\label{lemma:globescontractible}  Leonard Guetta committed Oct 02, 2020 86  For every $n \in \mathbb{N}$, the $\oo$\nbd{}category $\sD_n$ is oplax contractible.  Leonard Guetta committed Jun 03, 2020 87 88 \end{lemma} \begin{proof}  Leonard Guetta committed Oct 03, 2020 89  Recall that we write $e_n$ for the unique non-trivial $n$\nbd{}cell of $\sD_n$ and that by definition $\sD_n$ has exactly two non-trivial $k$\nbd{}cells for every $k$ such that $0\leq kn$:] Since $\sS_n$ is an $n$\nbd{}category, the image of $\alpha_k$ is necessarily of the form $\varphi(\alpha_k)=\1^k_{x}$ with $x$ a cell of $\sS_n$ of dimension non-greater than $n$. If $x$ is a unit on a cell whose dimension is strictly lower than $n$, then everything works like in the case $k0. \] \emph{From now on, let us use the convention that$\trgt_0(x)=x$when$x$is a$0$\nbd{}cell of$X$}. When$n>0$, the source and target of an$n$\nbd{}cell$(x,p)$of$X/a$are given by  Leonard Guetta committed Jun 09, 2020 326  $ 327  \src((x,p))=(\src(x),p) \text{ and } \trgt((x,p))=(\trgt(x),p).  Leonard Guetta committed Jan 07, 2021 328 329 330 $ %% LA DESCRIPTION DU BUT AU DESSUS N'EST PAS BONNE POUR LA DIMENSION 1 %% A CORRIGER !!  331  Moreover, the$\oo$\nbd{}functor$f/a$is described as  Leonard Guetta committed Jun 09, 2020 332 333 334  $(x,p) \mapsto (f(x),p),$  Leonard Guetta committed Dec 27, 2020 335  and the canonical$\oo$\nbd{}functor$X/a \to X$as  Leonard Guetta committed Jun 09, 2020 336 337 338  $(x,p) \mapsto x.$  Leonard Guetta committed Jun 01, 2020 339 \end{paragr}  Leonard Guetta committed Jun 09, 2020 340 \begin{paragr}\label{paragr:unfolding}  341  Let$f : X \to A$be an$\oo$\nbd{}functor with$A$a$1$\nbd{}category. Every arrow$\beta : a \to a'$of$A$induces an$\oo\nbd{}functor  342  \begin{align*}  343  X/\beta : X/a &\to X/{a'} \\  Leonard Guetta committed Jun 09, 2020 344  (x,p) & \mapsto (x,\beta \circ p),  345  \end{align*}  Leonard Guetta committed Dec 27, 2020 346  which takes part in a commutative triangle  Leonard Guetta committed Jun 09, 2020 347 348  $\begin{tikzcd}[column sep=tiny]  349  X/{a} \ar[rr,"X/{\beta}"] \ar[dr] && X/{a'} \ar[dl] \\  Leonard Guetta committed Oct 14, 2020 350 351  &X&. \end{tikzcd}  Leonard Guetta committed Jun 09, 2020 352 353 354 $ This defines a functor \begin{align*}  355  X/{-} : A &\to \oo\Cat\\  356  a &\mapsto X/a  Leonard Guetta committed Jun 09, 2020 357  \end{align*}  Leonard Guetta committed Oct 25, 2020 358  and a canonical\oo$\nbd{}functor  Leonard Guetta committed Jun 09, 2020 359  $ 360  \colim_{a \in A} (X/{a}) \to X.  Leonard Guetta committed Jun 09, 2020 361 $  Leonard Guetta committed Nov 04, 2020 362  Let$f' : X' \to A$be another$\oo$\nbd{}functor and let  Leonard Guetta committed Jun 09, 2020 363 364  $\begin{tikzcd}  365  X \ar[rr,"g"] \ar[dr,"f"'] && X' \ar[dl,"f'"] \\  Leonard Guetta committed Jun 09, 2020 366 367 368  &A& \end{tikzcd}$  369  be a commutative triangle in$\oo\Cat$. Recall from \ref{paragr:slices} that for every object$a$of$A$, there is an$\oo$\nbd{}functor$g/adefined as  Leonard Guetta committed Jun 09, 2020 370  \begin{align*}  371 372  g/a : X/a &\to X'/a \\ (x,p) &\mapsto (g(x),p).  Leonard Guetta committed Jun 09, 2020 373  \end{align*}  374 375 376  This defines a natural transformation $g/- : X/- \Rightarrow X'/-,$ and thus induces an\oo$\nbd{}functor  Leonard Guetta committed Jun 09, 2020 377  $ 378  \colim_{a \in A}(X/a) \to \colim_{a \in A}(X'/a).  Leonard Guetta committed Jun 09, 2020 379 $  380  Furthermore, it is immediate to check that the square  Leonard Guetta committed Jun 09, 2020 381 382  $\begin{tikzcd}  383 384  \displaystyle \colim_{a \in A}(X/a) \ar[d] \ar[r] & X \ar[d,"g"] \\ \displaystyle\colim_{a \in A}(X'/a) \ar[r] & X',  Leonard Guetta committed Oct 14, 2020 385  \end{tikzcd}  Leonard Guetta committed Jun 09, 2020 386 387 388 $ is commutative. \end{paragr}  389 \begin{lemma}\label{lemma:colimslice}  390  Let$f : X \to A$be an$\oo$\nbd{}functor such that$A$is a$1$\nbd{}category. The canonical$\oo$\nbd{}functor  Leonard Guetta committed Jun 09, 2020 391  $ 392  \colim_{a \in A}(X/a) \to X  Leonard Guetta committed Jun 09, 2020 393 394 395 396 $ is an isomorphism. \end{lemma} \begin{proof}  397 398  We have to show that the cocone $ 399  (X/a \to X)_{a \in \Ob(A)}  400 401 402 403 $ is colimiting. Let $ 404  (\phi_{a} : X/a \to C)_{a \in \Ob(A)}  405 $  Leonard Guetta committed Oct 01, 2020 406  be another cocone and let$x$be a$n$\nbd{}arrow of$X$. Notice that the pair  407 408 409  $(x,1_{f(\trgt_0(x))})$  Leonard Guetta committed Oct 25, 2020 410  is a$n$\nbd{}arrow of$X/f(\trgt_0(x)). We leave it to the reader to check that the formula  411 412 \begin{align*} \phi : X &\to C \\  413  x &\mapsto \phi_{f(\trgt_0(x))}(x,1_{f(\trgt_0(x))}).  414  \end{align*}  415  defines an\oo$\nbd{}functor and it is straightforward to check that for every object$a$of$A$the triangle  416 417  $\begin{tikzcd}  418  X/a\ar[dr,"\phi_{a}"']\ar[r] & X \ar[d,"\phi"] \\  419 420 421  & C \end{tikzcd}$  422 423  is commutative. This proves the existence part of the universal property.  Leonard Guetta committed Nov 04, 2020 424  Now let$\phi' : X \to C$be another$\oo$\nbd{}functor that makes the previous triangles commute for every object$a$of$A$and let$x$be an$n$\nbd{}cell of$X$. Since the triangle  425 426  $\begin{tikzcd}  427  X/f(\trgt_0(x)) \ar[dr,"\phi_{f(\trgt_0(x))}"']\ar[r] & X \ar[d,"\phi'"] \\  428 429 430 431 432  & C \end{tikzcd}$ is commutative, we necessarily have $ 433  \phi'(x)=\phi_{f(\trgt_0(x))}(x,1_{f(\trgt_0(x))})  434 435 $ which proves that$\phi'=\phi$.  Leonard Guetta committed Jun 09, 2020 436 437 \end{proof} \begin{paragr}  438  In particular, when we apply the previous lemma to$\mathrm{id}_A : A \to A$,  439  we obtain that every$1$\nbd{}category$A$is (canonically isomorphic to) the colimit  Leonard Guetta committed Jun 09, 2020 440  $ 441  \colim_{a \in A} (A/a).  Leonard Guetta committed Jan 05, 2021 442 443 444 $ % In other words, this simply say that the colimit of the Yoneda embedding$A \to % \Psh{A}$is the terminal presheaves  445  We now proceed to prove that this colimit is homotopic with respect to  446  the folk weak equivalences.  Leonard Guetta committed Jun 09, 2020 447 \end{paragr}  Leonard Guetta committed Oct 01, 2020 448 Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an$\oo$\nbd{}functor$f : X \to A$with$A$a$1$\nbd{}category.  449 \begin{lemma}\label{lemma:sliceisfree}  450 451 452  If$X$is free, then for every object$a$of$A$, the$\oo$\nbd{}category$X/a$is free. More precisely, if$\Sigma^X_n$is the$n$\nbd{}basis of$X$, then the$n$\nbd{}basis of$X/a$is the set  Leonard Guetta committed Jun 09, 2020 453  $ 454  \Sigma^{X/a}_n := \{(x,p) \in (X/a)_n \vert x \in \Sigma^X_n \}.  Leonard Guetta committed Jun 09, 2020 455 456 457 $ \end{lemma} \begin{proof}  458  It is immediate to check that for every object$a$of$A$, the canonical  Leonard Guetta committed Dec 27, 2020 459  forgetful functor$\pi_{a} : A/a \to A$is a discrete Conduché functor (see Section  460  \ref{section:conduche}). Hence, from Lemma \ref{lemma:pullbackconduche} we  461  know that$X/a \to X$is a discrete Conduché$\oo$\nbd{}functor. The result follows  462  then from Theorem \ref{thm:conduche}.  Leonard Guetta committed Jun 09, 2020 463 464 \end{proof} \begin{paragr}  465  When$X$is free, every arrow$\beta : a \to a'$of$Ainduces a map  Leonard Guetta committed Jun 09, 2020 466  \begin{align*}  467  \Sigma^{X/a}_n &\to \Sigma^{X/a'}_n \\  Leonard Guetta committed Jun 09, 2020 468 469 470 471 472  (x,p) &\mapsto (x,\beta\circ p). \end{align*} This defines a functor \begin{align*} \Sigma^{X/{\shortminus}}_n : A &\to \Set \\  473  a &\mapsto \Sigma^{X/a}_n.  Leonard Guetta committed Jun 09, 2020 474 475  \end{align*} \end{paragr}  476 \begin{lemma}\label{lemma:basisofslice}  Leonard Guetta committed Sep 30, 2020 477  IfX$is free, then there is an isomorphism of functors  Leonard Guetta committed Jun 09, 2020 478  $ 479  \Sigma^{X/\shortminus}_n \simeq \coprod_{x \in \Sigma^X_n}\Hom_A\left(f(\trgt_0(x)),\shortminus\right).  Leonard Guetta committed Jun 09, 2020 480 481 482 $ \end{lemma} \begin{proof}  483  For every object$a$of$A$and every$x \in \Sigma_n^X, we have a canonical map  Leonard Guetta committed Jun 09, 2020 484  \begin{align*}  485  \Hom_A\left(f(\trgt_0(x)),a\right) &\to \Sigma^{X/a}_n \\  Leonard Guetta committed Jun 09, 2020 486 487 488 489  p &\mapsto (x,p). \end{align*} By universal property, this induces a map $ Leonard Guetta committed Dec 27, 2020 490  \coprod_{x \in \Sigma^X_n}\Hom_A\left(f(\trgt_0(x)),a\right) \to \Sigma^{X/a}_n,  Leonard Guetta committed Jun 09, 2020 491 $  492  which is natural ina$. A simple verification shows that it is a bijection.  Leonard Guetta committed Jun 09, 2020 493 \end{proof}  494 \begin{proposition}\label{prop:sliceiscofibrant}  Leonard Guetta committed Oct 25, 2020 495  Let$A$be a$1$\nbd{}category,$X$be a free$\oo$\nbd{}category and$f : X \to A$be an$\oo\nbd{}functor. The functor  Leonard Guetta committed Jun 13, 2020 496 497  \begin{align*} A &\to \oo\Cat \\  498  a &\mapsto X/a  Leonard Guetta committed Jun 13, 2020 499  \end{align*}  500 501  is a cofibrant object for the projective model structure on\oo\Cat(A)$induced by the folk model structure on$\oo\Cat$(\ref{paragr:projmod}).  Leonard Guetta committed Jun 09, 2020 502 \end{proposition}  Leonard Guetta committed Jun 13, 2020 503 \begin{proof}  504 505 506 507  Recall that the set $\{i_n: \sS_{n-1} \to \sD_n \vert n \in \mathbb{N} \}$  Leonard Guetta committed Dec 27, 2020 508  is a set of generating folk cofibrations.  509  From Lemmas \ref{lemma:sliceisfree} and \ref{lemma:basisofslice} we deduce  510  that for every object$a$of$A$and every$n \in \mathbb{N}$, the canonical square  511 512  $\begin{tikzcd}  513 514  \displaystyle\coprod_{x \in \Sigma^X_n}\coprod_{\Hom_A(f(\trgt_0(x)),a)}\sS_{n-1} \ar[r] \ar[d] & \sk_{n-1}(X/a) \ar[d]\\ \displaystyle\coprod_{x \in \Sigma^X_n}\coprod_{\Hom_A(f(\trgt_0(x)),a)}\sD_n \ar[r]& \sk_n{(X/a)}  515 516 517  \end{tikzcd}$ is cocartesian. It is straightforward to check that this square is natural in  518 $a$in an obvious sense, which means that we have a cocartesian square in$\oo\Cat(A)$:  Leonard Guetta committed Oct 12, 2020 519 520  $\begin{tikzcd}  Leonard Guetta committed Oct 25, 2020 521 522  \displaystyle\coprod_{x \in \Sigma^X_n}\sS_{n-1}\otimes f(\trgt_0(x)) \ar[r] \ar[d] & \sk_{n-1}(X/-) \ar[d]\\ \displaystyle\coprod_{x \in \Sigma^X_n}\sD_n\otimes f(\trgt_0(x)) \ar[r]& \sk_n{(X/-)}  Leonard Guetta committed Oct 13, 2020 523  \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]  Leonard Guetta committed Oct 12, 2020 524 525  \end{tikzcd}$  Leonard Guetta committed Oct 25, 2020 526  (see \ref{paragr:cofprojms} for notations). From the second part of Proposition \ref{prop:modprs}, we deduce that for every$n\geq 0$, $\sk_{n-1}(X/-) \to \sk_{n}(X/-)$ is a cofibration for the  Leonard Guetta committed Oct 12, 2020 527 528  projective model structure on$\oo\Cat(A)$. Thus, the transfinite composition $ Leonard Guetta committed Oct 13, 2020 529  \emptyset \to \sk_{0}(X/-) \to \sk_{1}(X/) \to \cdots \to \sk_{n}(X/-) \to \cdots,  Leonard Guetta committed Oct 12, 2020 530 $  Leonard Guetta committed Oct 13, 2020 531  which is canonically isomorphic to$\emptyset \to X/-$(see Lemma \ref{lemma:filtration}), is also a cofibration for the projective model structure.  Leonard Guetta committed Jun 13, 2020 532 \end{proof}  Leonard Guetta committed Oct 01, 2020 533 \begin{corollary}\label{cor:folkhmtpycol}  Leonard Guetta committed Oct 25, 2020 534  Let$A$be a$1$\nbd{}category and$f : X \to A$be an$\oo$\nbd{}functor. The canonical arrow of$\ho(\oo\Cat^{\folk})$ Leonard Guetta committed Jun 13, 2020 535  $ 536  \hocolim^{\folk}_{a \in A}(X/a) \to X,  Leonard Guetta committed Jun 13, 2020 537 $  538  induced by the co-cone$(X/a \to X)_{a \in \Ob(A)}$, is an isomorphism.  Leonard Guetta committed Jun 13, 2020 539 540 541 \end{corollary} Beware that in the previous corollary, we did \emph{not} suppose that$X$was free. \begin{proof}  542  Let$P$be a free$\omega$-category and$g : P \to X$a folk trivial fibration  Leonard Guetta committed Jan 05, 2021 543  and consider the following commutative diagram of$\ho(\oo\Cat^{\folk})$ 544 545  \label{comsquare} \begin{tikzcd}  Leonard Guetta committed Jan 05, 2021 546 547  \displaystyle\hocolim^{\folk}_{a \in A}(P/a) \ar[d] \ar[r] & \displaystyle\colim_{a \in A}(P/a) \ar[d] \ar[r] & P \ar[d]\\ \displaystyle\hocolim^{\folk}_{a \in A}(X/a) \ar[r] & \displaystyle\colim_{a \in A}(X/a) \ar[r] & X  548 549  \end{tikzcd}  Leonard Guetta committed Jan 05, 2021 550  where the middle and most left vertical arrows are induced by the arrows  551  $ Leonard Guetta committed Jan 05, 2021 552  g/a : P/a \to X/a,  553 $  Leonard Guetta committed Jan 05, 2021 554 555 556  and the most right vertical arrow is induced by$g$. Since trivial fibrations are stable by pullback,$g/a$is a trivial fibration. This proves that the most left vertical arrow of diagram \eqref{comsquare} is an isomorphism.  557   Leonard Guetta committed Jan 07, 2021 558  Now, from Proposition \ref{prop:sliceiscofibrant} and Corollary  Leonard Guetta committed Jan 05, 2021 559 560 561 562  \ref{cor:cofprojms}, we deduce that the arrow $\hocolim_{a \in A}^{\folk}(P/a)\to \colim_{a \in A}(P/a)$ is an isomorphism. Moreover, from Lemma \ref{lemma:colimslice}, we know that the arrows $\colim_{a \in A}(P/a)\to P$ and $\colim_{a \in A}(X/a)\to X$ are isomorphisms.  563   Leonard Guetta committed Jan 05, 2021 564 565 566 567 568  Finally, since$g$is a folk weak equivalence, the most right vertical arrow of diagram \eqref{comsquare} is an isomorphism and by an immediate 2-out-of-3 property this proves that all arrows of \eqref{comsquare} are isomorphisms. In particular, so is the composition of the two bottom horizontal arrows, which is what we desired to show.  Leonard Guetta committed Jun 13, 2020 569 \end{proof}  570 We now move on to the next step needed to prove that every$1$\nbd{}category is \good{}. For that purpose, let us recall a construction commonly referred to as the Grothendieck construction''.  Leonard Guetta committed Jun 13, 2020 571 \begin{paragr}  572  Let$A$be a$1$\nbd{}category and$F : A \to \Cat$a functor. We denote by$\int F$or$\int_{a \in A}F(a)$the category such that:  Leonard Guetta committed Oct 02, 2020 573  \begin{itemize}[label=-]  Leonard Guetta committed Jun 13, 2020 574 575 576 577 578 579 580 581 582 583  \item An object of$\int F$is a pair$(a,x)$where$a$is an object of$A$and$x$is an object of$F(a)$. \item An arrow$(a,x) \to (a',x')$of$\int F$is a pair$(f,k)$where $f : a \to a'$ is an arrow of$A$, and $k : F(f)(x) \to x'.$ \end{itemize}  Leonard Guetta committed Oct 25, 2020 584  The unit on$(a,x)$is the pair$(1_a,1_x)$and the composition of$(f,k) : (a,x) \to (a',x')$and$(f',k') : (a',x') \to (a'',x'')is given by:  Leonard Guetta committed Jun 13, 2020 585 586 587  $(f',k')\circ(f,k)=(f'\circ f,k'\circ F(f')(k)).$  Leonard Guetta committed Oct 02, 2020 588  Every natural transformation  Leonard Guetta committed Jun 13, 2020 589 590  $\begin{tikzcd}  Leonard Guetta committed Oct 02, 2020 591  A \ar[r,bend left,"F",""{name=A,below},pos=19/30]\ar[r,bend right,"G"',""{name=B, above},pos=11/20] & \Cat \ar[from=A,to=B,Rightarrow,"\alpha",pos=9/20]  Leonard Guetta committed Jun 13, 2020 592 593 594 595 596 597 598  \end{tikzcd}$ induces a functor \begin{align*} \int \alpha : \int F &\to \int G\\ (a,x) &\mapsto (a,\alpha_a(x)). \end{align*}  Leonard Guetta committed Oct 02, 2020 599  Altogether, this defines a functor  Leonard Guetta committed Jun 13, 2020 600 601 602 603 604 605  \begin{align*} \int : \Cat(A)&\to \Cat \\ F&\mapsto \int F, \end{align*} where\Cat(A)$is the category of functors from$A$to$\Cat$. \end{paragr}  606 We now recall an important theorem due to Thomason.  Leonard Guetta committed Jun 13, 2020 607 \begin{theorem}[Thomason]\label{thm:Thomason}  608  The functor$\int : \Cat(A) \to \Cat$sends the pointwise Thomason equivalences (\ref{paragr:homder}) to Thomason equivalences and the induced functor  Leonard Guetta committed Jun 13, 2020 609 610 611 612 613  $\overline{\int} : \ho(\Cat^{\Th}(A)) \to \ho(\Cat^{\Th})$ is canonically isomorphic to the homotopy colimit functor $ Leonard Guetta committed Nov 04, 2020 614  \hocolim^{\Th}_A : \ho(\Cat^{\Th}(A)) \to \ho(\Cat^{\Th}).  Leonard Guetta committed Jun 13, 2020 615 616 617 618 619 620 $ \end{theorem} \begin{proof} The original source for this Theorem is \cite{thomason1979homotopy}. However, the definition of homotopy colimit used by Thomason, albeit equivalent, is not the same as the one we used in this dissertation and is slightly outdated. A more modern proof of the theorem can be found in \cite[Proposition 2.3.1 and Théorème 1.3.7]{maltsiniotis2005theorie}. \end{proof} \begin{corollary}\label{cor:thomhmtpycol}  621  Let$A$be a$1$\nbd{}category. The canonical map  Leonard Guetta committed Jun 13, 2020 622  $ Leonard Guetta committed Nov 04, 2020 623  \hocolim^{\Th}_{a \in A}(A/a) \to A  Leonard Guetta committed Jun 13, 2020 624 625 626 627 628 629 630 631 $ induced by the co-cone$(A/a \to A)_{a \in \Ob(A)}$, is an isomorphism of$\ho(\Cat^{\Th})$. \end{corollary} \begin{proof} For every object$a$of$A$, the canonical map to the terminal category $A/a \to \sD_0$  Leonard Guetta committed Oct 02, 2020 632  is a Thomason equivalence. This comes from the fact that$A/a$is oplax contractible (Proposition \ref{prop:slicecontractible}), or from \cite[Section 1, Corollary 2]{quillen1973higher} and the fact that$A/a$has a terminal object.  Leonard Guetta committed Jun 13, 2020 633 634 635 636 637  In particular, the morphism of functors $A/(-) \Rightarrow k_{\sD_0},$  Leonard Guetta committed Oct 02, 2020 638  where$k_{\sD_0}$is the constant functor$A \to \Cat$with value the terminal category$\sD_0$, is a pointwise Thomason equivalence. It follows from the first part of Theorem \ref{thm:Thomason} that  Leonard Guetta committed Jun 13, 2020 639 640 641  $\int_{a \in A}A/a \to \int_{a \in A}k_{\sD_0}$  Leonard Guetta committed Nov 04, 2020 642  is a Thomason equivalence and an immediate computation shows that $\int_{a \in A}k_{\sD_0} \simeq A.$ From the second part of Theorem \ref{thm:Thomason}, we have that  Leonard Guetta committed Jun 13, 2020 643  $ Leonard Guetta committed Nov 04, 2020 644  \hocolim^{\Th}_{a \in A}(A/a) \simeq A.  Leonard Guetta committed Jun 13, 2020 645 $  646  A thorough analysis of all the isomorphisms involved shows that this last isomorphism is indeed induced by the co-cone$(A/a \to A)_{a \in \Ob(A)}$.  Leonard Guetta committed Jun 13, 2020 647 648 \end{proof} \begin{remark}  Leonard Guetta committed Nov 04, 2020 649  It is possible to extend the previous corollary to prove that for every functor$f : X \to A$($X$and$A$being$1$\nbd{}categories), we have $\hocolim^{\Th}_{a \in A} (X/a) \simeq X.$ However, to prove that it is also the case when$X$is an$\oo$\nbd{}category and$f$an$\oo$\nbd{}functor, as in Corollary \ref{cor:folkhmtpycol}, one would need to extend the Grothendieck construction to functors with value in$\oo\Cat$and to prove an$\oo$\nbd{}categorical analogue of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go beyond the scope of this dissertation.  Leonard Guetta committed Jun 13, 2020 650 651 \end{remark} Putting all the pieces together, we are now able to prove the awaited Theorem.  Leonard Guetta committed Oct 12, 2020 652 \begin{theorem}\label{thm:categoriesaregood}  653  Every$1$\nbd{}category is \good{}.  Leonard Guetta committed Jun 13, 2020 654 655 656 \end{theorem} \begin{proof} All the arguments of the proof have already been given and we sum them up here essentially for the sake of clarity.  657  Let$A$be a$1\nbd{}category. Consider the diagram  Leonard Guetta committed Jun 13, 2020 658 659 660 661 662 663 664 665 666 667 668  \begin{align*} A &\to \oo\Cat\\ a &\mapsto A/a \end{align*} and the co-cone $(A/a \to A)_{a \in \Ob(A)}.$ \begin{itemize}[label=-] \item The canonical map of\ho(\oo\Cat^{\folk})$$ Leonard Guetta committed Oct 04, 2020 669  \hocolim_{a \in A}^{\folk} (A/a) \to A  Leonard Guetta committed Jun 13, 2020 670 671 672 673 $ is an isomorphism thanks to Corollary \ref{cor:folkhmtpycol} applied to$\mathrm{id}_A : A \to A$. \item The canonical map of$\ho(\oo\Cat^{\Th})$$ Leonard Guetta committed Oct 04, 2020 674  \hocolim_{a \in A}^{\Th} (A/a) \to A  Leonard Guetta committed Jun 13, 2020 675 $  Leonard Guetta committed Jan 07, 2021 676 677 678  is an isomorphism thanks to Corollary \ref{cor:thomhmtpycol} and the fact that the canonical morphisms of op-prederivators$\Ho(\Cat^{\Th}) \to \Ho(\oo\Cat^{\Th})$is homotopy cocontinuous (see \ref{paragr:thomhmtpycol}).  679  \item Every$A/a$is \good{} thanks to Proposition \ref{prop:contractibleisgood} and Proposition \ref{prop:slicecontractible}.  Leonard Guetta committed Jun 13, 2020 680 681 682  \end{itemize} Thus, Proposition \ref{prop:criteriongoodcat} applies and this proves that$A\$ is \good{}. \end{proof}  683 684 685 686 687  %%% Local Variables: %%% mode: latex %%% TeX-master: "main" %%% End: