1.5 Examples of $\infty $-cosmoi

We briefly tour a few examples of $\infty $-cosmoi. The following theorem should be quite difficult to formalize:

Proposition 1.5.1 the $\infty $-cosmos of quasi-categories

The full subcategory ${\mathord {\mathcal{QCat}}}\subset {\mathord {\mathcal{sSet}}}$ of quasi-categories defines an $\infty $-cosmos in which the isofibrations, equivalences, and trivial fibrations coincide with the classes already bearing these names.

Proof ▶

The proof requires myriad combinatorial results about the class of isofibrations between quasi-categories. See [ RV22 , § D ] .

Two further examples fit into a common paradigm: both arise as full subcategories of the $\infty $-cosmos of quasi-categories and inherit their $\infty $-cosmos structures from this inclusion (see Lemma [ RV22 , 6.1.4 ] ), but it is also instructive, and ultimately takes less work, to describe the resulting $\infty $-cosmos structures directly.

Proposition 1.5.2 the $\infty $-cosmos of categories

The category ${\mathord {\mathcal{Cat}}}$ of 1-categories defines an $\infty $-cosmos whose isofibrations are the isofibrations: functors satisfying the displayed right lifting property:

$\begin{tikzcd} \catone \arrow[d, hook] \arrow[r] & A \arrow[d, "f", two heads] \\ \iso \arrow[r] \arrow[ur, dashed] & B \end{tikzcd}$

The equivalences are the equivalences of categories and the trivial fibrations are surjective equivalences: equivalences of categories that are also surjective on objects.

Proof ▶

It is well-known that the 2-category of categories is complete (and in fact also cocomplete) as a ${\mathord {\mathcal{Cat}}}$-enriched category (see [ Kel89 ] ). The categorically enriched category of categories becomes a quasi-categorically enriched category by applying the nerve functor to the hom-categories (see §1.6). Since the nerve functor is a right adjoint, it follows formally that these 2-categorical limits become simplicially enriched limits. In particular, as proscribed in Proposition 1.6.9, the cotensor of a category $A$ by a simplicial set $U$ is defined to be the functor category $A^{{\mathord {\mathsf{h}}}{U}}$. This completes the verification of axiom i.

Since the class of isofibrations is characterized by a right lifting property, the isofibrations are closed under all of the limit constructions of 1.4.1 ii except for the last two. For these, the Leibniz closure subsumes the closure under exponentiation.

To verify that isofibrations of categories $f \colon A \twoheadrightarrow B$ are stable under forming Leibniz cotensors with monomorphisms of simplicial sets $i \colon U \hookrightarrow V$, we must solve the lifting problem below-left

$\begin{tikzcd} \catone \arrow[r, "s"] \arrow[d, hook, "j"'] & A^{\ho{V}} \arrow[d, "{\ho{i}\leib\pwr f}"] \arrow[dr, phantom, "\leftrightsquigarrow"] & \ho{U} \times \iso \cup_{\ho{U}} \ho{V} \arrow[r, "{\langle \alpha, s\rangle}"] \arrow[d, hook, "\ho{i} \leib\times j"'] & A \arrow[d, two heads, "f"] \\ \iso \arrow[ur, dashed, "\gamma"] \arrow[r, "{\langle \beta, \alpha \rangle}"'] & B^{\ho V} \times_{B^{\ho U}} A^{\ho U} & \ho{V} \times \iso \arrow[r, "\beta"'] \arrow[ur, dashed, "\gamma"'] & B \end{tikzcd}$

which transposes to the lifting problem above-right, which we can solve by hand. Here the map $\beta $ defines a natural isomorphism between $fs \colon {\mathord {\mathsf{h}}}{V} \to B$ and a second functor. Our task is to lift this to a natural isomorphism $\gamma $ from $s$ to another functor that extends the natural isomorphism $\alpha $ along ${\mathord {\mathsf{h}}}{i} \colon {\mathord {\mathsf{h}}}{U} \to {\mathord {\mathsf{h}}}{V}$. Note this functor ${\mathord {\mathsf{h}}}{i}$ need not be an inclusion, but it is injective on objects, which is enough.

We define the components of $\gamma $ by cases. If an object $v \in {\mathord {\mathsf{h}}}{V}$ is equal to $i(u)$ for some $u \in {\mathord {\mathsf{h}}}{U}$ define $\gamma _{i(u)} :=\alpha _u$; otherwise, use the fact that $f$ is an isofibration to define $\gamma _v$ to be any lift of the isomorphism $\beta _v$ to an isomorphism in $A$ with domain $s(v)$. The data of the map $\gamma \colon {\mathord {\mathsf{h}}}{V} \times I\to A$ also entails the specification of the functor ${\mathord {\mathsf{h}}}{V} \to A$ that is the codomain of the natural isomorphism $\gamma $. On objects, this functor is given by $v \mapsto \textup{cod}(\gamma _v)$. On morphisms, this functor defined in the unique way that makes $\gamma $ into a natural transformation:

\[ (k \colon v \to v') \mapsto \gamma _{v'} \circ s(k) \circ \gamma _v^{-1}. \]

This completes the proof that ${\mathord {\mathcal{Cat}}}$ defines an $\infty $-cosmos. Since the nerve of a functor category, such as $A^I$, is isomorphic to the exponential between their nerves, the equivalences of categories coincide with the equivalences of Definition 1.2.35. It follows that the equivalences in the $\infty $-cosmos of categories coincide with equivalences of categories, and since the surjective equivalences are the intersection of the equivalences and the isofibrations, this completes the proof.

Similarly:

Proposition 1.5.3 the $\infty $-cosmos of Kan complexes

The category ${\mathord {\mathcal{Kan}}}$ of Kan complexes defines an $\infty $-cosmos whose isofibrations are the Kan fibrations: maps that lift against all horn inclusions $\Lambda ^k[n] \hookrightarrow \Delta [n]$ for $n \geq 1$ and $0 \leq k \leq n$.

One of the key advantages of the $\infty $-cosmological approach to abstract category theory is that there are a myriad varieties of “fibered” $\infty $-cosmoi that can be built from a given $\infty $-cosmos, which means that any theorem proven in this axiomatic framework specializes and generalizes to those contexts. The most basic of these derived $\infty $-cosmoi is the $\infty $-cosmos of isofibrations over a fixed base, which we introduce now.

Proposition 1.5.4 sliced $\infty $-cosmoi

For any $\infty $-cosmos ${\mathord {\mathcal{K}}}$ and any $\infty $-category $B \in {\mathord {\mathcal{K}}}$ there is an $\infty $-cosmos ${\mathord {\mathcal{K}}}_{/B}$ of isofibrations over $B$ whose

objects are isofibrations $p \colon E \twoheadrightarrow B$ with codomain $B$
functor spaces, say from $p \colon E \twoheadrightarrow B$ to $q \colon F \twoheadrightarrow B$, are defined by pullback

$\begin{tikzcd} [column sep=small] \Fun_B(p \colon E \fib B,q\colon F\fib B) \arrow[r] \arrow[d, two heads] \arrow[dr, phantom, "\lrcorner" very near start] & \Fun(E,F) \arrow[d, two heads, "{q_*}"] \\ \catone \arrow[r, "p"] & \Fun(E,B) \end{tikzcd}$

and abbreviated to ${\mathord {\mathsf{Fun}}}_B(E,F)$ when the specified isofibrations are clear from context
isofibrations are commutative triangles of isofibrations over $B$

$\begin{tikzcd} [row sep=small, column sep=small] E \arrow[rr,two heads, "r"] \arrow[dr, two heads, "p"'] & & F \arrow[dl, two heads, "q"] \\ & B \end{tikzcd}$
terminal object is $\textup{id}\colon B \twoheadrightarrow B$ and products are defined by the pullback along the diagonal

$\begin{tikzcd} \times^B_i E_i \arrow[r] \arrow[d, two heads] \arrow[dr, phantom, "\lrcorner" very near start] & \prod_i E_i \arrow[d, two heads, "\prod_i p_i"] \\ B \arrow[r, "\Delta"] & \prod_i B \end{tikzcd}$
pullbacks and limits of towers of isofibrations are created by the forgetful functor ${\mathord {\mathcal{K}}}_{/B} \to {\mathord {\mathcal{K}}}$
simplicial cotensor of $p \colon E \twoheadrightarrow B$ with $U \in {\mathord {\mathcal{sSet}}}$ is constructed by the pullback

$\begin{tikzcd} U \pwr_B p \arrow[d, two heads] \arrow[r] \arrow[dr, phantom, "\lrcorner" very near start] & E^U \arrow[d, two heads, "p^U"] \\ B \arrow[r, "\Delta"] & B^U \end{tikzcd}$
and in which a map over $B$

$\begin{tikzcd} [row sep=small, column sep=small] E \arrow[rr, "f"] \arrow[dr, two heads, "p"'] & & F \arrow[dl, two heads, "q"] \\ & B \end{tikzcd}$

is an equivalence in the $\infty $-cosmos ${\mathord {\mathcal{K}}}_{/B}$ if and only if $f$ is an equivalence in ${\mathord {\mathcal{K}}}$.

Proof ▶

The functor spaces are quasi-categories since axiom 1.4.1 ii asserts that for any isofibration $q \colon F \twoheadrightarrow B$ in ${\mathord {\mathcal{K}}}$ the map $q_* \colon {\mathord {\mathsf{Fun}}}(E,F) \twoheadrightarrow {\mathord {\mathsf{Fun}}}(E,B)$ is an isofibration of quasi-categories. Other parts of this axiom imply that each of the limit constructions — such as the products and cotensors constructed in iv and vi — define isofibrations over $B$. The closure properties of the isofibrations in ${\mathord {\mathcal{K}}}_{/B}$ follow from the corresponding ones in ${\mathord {\mathcal{K}}}$. The most complicated of these is the Leibniz cotensor stability of the isofibrations in ${\mathord {\mathcal{K}}}_{/B}$, which follows from the corresponding property in ${\mathord {\mathcal{K}}}$, since for a monomorphism of simplicial sets $i \colon X\hookrightarrow Y$ and an isofibration $r$ over $B$ as in iii above, the map $i \mathbin {\widehat{\pitchfork _B}} r$ is constructed by pulling back $i \mathbin {\widehat{\pitchfork }}r$ along $\Delta \colon B \to B^Y$.

The fact that the above constructions define simplicially enriched limits in a simplicially enriched slice category are standard from enriched category theory. It remains only to verify that the equivalences in the $\infty $-cosmos of isofibrations are created by the forgetful functor ${\mathord {\mathcal{K}}}_{/B} \to {\mathord {\mathcal{K}}}$. Suppose first that the map $f$ displayed in vii defines an equivalence in ${\mathord {\mathcal{K}}}$. Then for any isofibration $s \colon A \twoheadrightarrow B$ the induced map on functor spaces in ${\mathord {\mathcal{K}}}_{/B}$ is defined by the pullback:

$\begin{tikzcd} \Fun_B(A,E) \arrow[dr, "{f_*}", dashed, "\sim"'] \arrow[dd, two heads] \arrow[ddr, phantom, "\lrcorner" pos=.001] \arrow[rr] & & \Fun(A,E) \arrow[dd, two heads, "{p_*}"' near start] \arrow[dr, "{f_*}", "\sim"'] \\ & \Fun_B(A,F) \arrow[dl, two heads] \arrow[dr, phantom, "\lrcorner" very near start] \arrow[rr, crossing over] & & \Fun(A,F) \arrow[dl, two heads, "{q_*}"] \\ \catone \arrow[rr, "s"'] & ~& \Fun(A,B) \end{tikzcd}$

Since $f$ is an equivalence in ${\mathord {\mathcal{K}}}$, the map $f_* \colon {\mathord {\mathsf{Fun}}}(A,E) \to {\mathord {\mathsf{Fun}}}(A,F)$ is an equivalence, and so it follows that the induced map on fibers over $s$ is an equivalence as well. ¹

For the converse implication, we appeal to Lemma 1.4.9. If $f \colon E \to F$ is an equivalence in ${\mathord {\mathcal{K}}}_{/B}$ then it admits a homotopy inverse in ${\mathord {\mathcal{K}}}_{/B}$. The inverse equivalence $g \colon F \to E$ also defines an inverse equivalence in ${\mathord {\mathcal{K}}}$ and the required simplicial homotopies in ${\mathord {\mathcal{K}}}$

$\begin{tikzcd} E \arrow[r, "\alpha"] & \iso\pwr_Bp \arrow[r] & E^\iso & F \arrow[r, "\beta"] & \iso \pwr_Bq \to F^\iso \end{tikzcd}$

are defined by composing with the top horizontal leg of the pullback defining the cotensor in ${\mathord {\mathcal{K}}}_{/B}$.

Many, though not all, of the $\infty $-cosmoi we encounter “in the wild” satisfy an additional axiom: ²

Definition 1.5.5 cartesian closed $\infty $-cosmoi

An $\infty $-cosmos ${\mathord {\mathcal{K}}}$ is cartesian closed if the product bifunctor $- \times -\colon {\mathord {\mathcal{K}}}\times {\mathord {\mathcal{K}}}\to {\mathord {\mathcal{K}}}$ extends to a simplicially enriched two-variable adjunction

\[ {\mathord {\mathsf{Fun}}}(A \times B,C) \cong {\mathord {\mathsf{Fun}}}(A,C^B) \cong {\mathord {\mathsf{Fun}}}(B,C^A) \]

in which the right adjoints $(-)^A \colon {\mathord {\mathcal{K}}}\to {\mathord {\mathcal{K}}}$ preserve isofibrations for all $A \in {\mathord {\mathcal{K}}}$.

For instance, the $\infty $-cosmos of quasi-categories is cartesian closed, with the exponentials defined as (special cases of) simplicial cotensors. This is one of the reasons that we use the same notation for cotensor and for exponential. Note in this case the functor spaces and the exponentials coincide. The same is true for the cartesian closed $\infty $-cosmoi of categories and of Kan complexes. In general, the functor space from $A$ to $B$ is the “underlying quasi-category” of the exponential $B^A$ whenever it exists.

\(\infty \)-Cosmoi

1.5 Examples of \(\infty \)-cosmoi