1.4 $\infty $-Cosmoi

There are a variety of models of infinite-dimensional categories for which the category of “$\infty $-categories,” as we call them, and “$\infty $-functors” between them is enriched over quasi-categories and admits classes of isofibrations, equivalences, and trivial fibrations satisfying certain properties that are familiar from abstract homotopy theory. ¹ In particular, the use of isofibrations in diagrams guarantees that their strict limits are equivalence invariant, so we can take advantage of up-to-isomorphism universal properties and strict functoriality of these constructions while still working “homotopically.” This motivates the following axiomatization:

Definition 1.4.1 $\infty $-cosmos

An $\infty $-cosmos ${\mathord {\mathcal{K}}}$ is a category that is enriched over quasi-categories, ² meaning in particular that

its morphisms $f \colon A \to B$ define the vertices of a quasi-category denoted ${\mathord {\mathsf{Fun}}}(A,B)$ and referred to as a functor space,

that is also equipped with a specified collection of maps that we call isofibrations and denote by “$\twoheadrightarrow $” satisfying the following two axioms:

(completeness) The quasi-categorically enriched category ${\mathord {\mathcal{K}}}$ possesses a terminal object, small products, pullbacks of isofibrations, limits of countable towers of isofibrations, and cotensors with simplicial sets, each of these limit notions satisfying a universal property that is enriched over simplicial sets. ³
(isofibrations) The isofibrations contain all isomorphisms and any map whose codomain is the terminal object; are closed under composition, product, pullback, forming inverse limits of towers, and Leibniz cotensors with monomorphisms of simplicial sets; and have the property that if $f \colon A \twoheadrightarrow B$ is an isofibration and $X$ is any object then ${\mathord {\mathsf{Fun}}}(X,A) \twoheadrightarrow {\mathord {\mathsf{Fun}}}(X,B)$ is an isofibration of quasi-categories.

For ease of reference, we refer to the simplicially enriched limits of diagrams of isofibrations enumerated in i as the cosmological limit notions.

Definition 1.4.2

In an $\infty $-cosmos ${\mathord {\mathcal{K}}}$, a morphism $f\colon A \to B$ is an equivalence just when the induced map $f_* \colon {\mathord {\mathsf{Fun}}}(X,A) \rightsquigarrow {\mathord {\mathsf{Fun}}}(X,B)$ on functor spaces is an equivalence of quasi-categories for all $X \in {\mathord {\mathcal{K}}}$.

Definition 1.4.3

In an $\infty $-cosmos ${\mathord {\mathcal{K}}}$, a morphism $f\colon A \to B$ is a trivial fibration just when $f$ is both an isofibration and an equivalence.

These classes are denoted by “$\rightsquigarrow $” and “$\twoheadrightarrow $”, respectively. ⁴

Put more concisely, one might say that an $\infty $-cosmos is a “quasi-categorically enriched category of fibrant objects.”

Convention 1.4.4 $\infty $-category, as a technical term

Henceforth, we recast $\infty $-category as a technical term to refer to an object in an arbitrary ambient $\infty $-cosmos. Similarly, we use the term $\infty $-functor — or more commonly the elision “functor” — to refer to a morphism $f \colon A \to B$ in an $\infty $-cosmos. This explains why we refer to the quasi-category ${\mathord {\mathsf{Fun}}}(A,B)$ between two $\infty $-categories in an $\infty $-cosmos as a “functor space”: its vertices are the ($\infty $-)functors from $A$ to $B$.

Definition 1.4.5

The underlying category ${\mathord {\mathcal{K}}}_0$ of an $\infty $-cosmos ${\mathord {\mathcal{K}}}$ is the category whose objects are the $\infty $-categories in ${\mathord {\mathcal{K}}}$ and whose morphisms are the $0$-arrows, i.e., the vertices in the functor spaces.

In all of the examples to appear in what follows, this recovers the expected category of $\infty $-categories in a particular model and functors between them.

The following theorem should be quite difficult to formalize:

Proposition 1.4.6 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 1.4.7 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 §1.5). 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 1.5.9, 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 1.4.8 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 $0 \leq k \leq n$.

Several consequences of the $\infty $-cosmos axioms are mentioned in [ RV22 , § 1.2 ] . For now, we focus on just one.

By a Yoneda-style argument, the “homotopy equivalence” characterization of the equivalences in the $\infty $-cosmos of quasi-categories of Definition 1.2.35 extends to an analogous characterization of the equivalences in any $\infty $-cosmos:

Lemma 1.4.9 equivalences are homotopy equivalences

A map $f \colon A \to B$ between $\infty $-categories in an $\infty $-cosmos ${\mathord {\mathcal{K}}}$ is an equivalence if and only if it extends to the data of a “homotopy equivalence” with the free-living isomorphism $I$ serving as the interval: that is, if there exist maps $g \colon B \to A$

$\begin{tikzcd} & A & & & B \\ A \arrow[ur, equals] \arrow[dr, "gf"'] \arrow[r, "\alpha"] & A^\iso \arrow[u, two heads, "\sim", "\ev_0"'] \arrow[d, two heads, "\sim"', "\ev_1"] & \text{and} & B \arrow[dr, equals] \arrow[r, "\beta"] \arrow[ur, "fg"] & B^\iso \arrow[u, two heads, "\sim", "\ev_0"'] \arrow[d, two heads, "\sim"', "\ev_1"] \\ & A & & & B \end{tikzcd}$

in the $\infty $-cosmos.

Proof ▶

By hypothesis, if $f \colon A \to B$ defines an equivalence in the $\infty $-cosmos ${\mathord {\mathcal{K}}}$ then the induced map on post-composition $f_* \colon {\mathord {\mathsf{Fun}}}(B,A) \rightsquigarrow {\mathord {\mathsf{Fun}}}(B,B)$ is an equivalence of quasi-categories in the sense of Definition 1.2.35. Evaluating the inverse equivalence $\tilde{g} \colon {\mathord {\mathsf{Fun}}}(B,B) \rightsquigarrow {\mathord {\mathsf{Fun}}}(B,A)$ and homotopy $\tilde{\beta } \colon {\mathord {\mathsf{Fun}}}(B,B) \to {\mathord {\mathsf{Fun}}}(B,B)^I$ at the 0-arrow $\textup{id}_B \in {\mathord {\mathsf{Fun}}}(B,B)$, we obtain a 0-arrow $g \colon B \to A$ together with an isomorphism $\beta \colon I\to {\mathord {\mathsf{Fun}}}(B,B)$ from the composite $fg$ to $\textup{id}_B$. By the defining universal property of the cotensor 1.3.1, this isomorphism internalizes to define the map $\beta \colon B \to B^I$ in ${\mathord {\mathcal{K}}}$ displayed on the right of the displayed equation in the statement.

Now the hypothesis that $f$ is an equivalence also provides an equivalence of quasi-categories $f_* \colon {\mathord {\mathsf{Fun}}}(A,A) \rightsquigarrow {\mathord {\mathsf{Fun}}}(A,B)$, and the map $\beta f \colon A \to B^I$ represents an isomorphism in ${\mathord {\mathsf{Fun}}}(A,B)$ from $fgf$ to $f$. Since $f_*$ is an equivalence, we conclude from Lemma 1.2.36 that $\textup{id}_A$ and $gf$ are isomorphic in the quasi-category ${\mathord {\mathsf{Fun}}}(A,A)$: explicitly, such an isomorphism may be defined by applying the inverse equivalence $\tilde{h} \colon {\mathord {\mathsf{Fun}}}(A,B) \to {\mathord {\mathsf{Fun}}}(A,A)$ and composing with the components at $\textup{id}_A, gf \in {\mathord {\mathsf{Fun}}}(A,A)$ of the isomorphism $\tilde{\alpha } \colon {\mathord {\mathsf{Fun}}}(A,A) \to {\mathord {\mathsf{Fun}}}(A,A)^I$ from $\textup{id}_{{\mathord {\mathsf{Fun}}}(A,A)}$ to $\tilde{h}f_*$. Now by Proposition 1.2.32 this isomorphism is represented by a map $I\to {\mathord {\mathsf{Fun}}}(A,A)$ from $\textup{id}_A$ to $gf$, which internalizes to a map $\alpha \colon A \to A^I$ in ${\mathord {\mathcal{K}}}$ displayed on the left of the displayed equation in the statement.

The converse is easy: the simplicial cotensor construction commutes with ${\mathord {\mathsf{Fun}}}(X,-)$, so a homotopy equivalence induces a homotopy equivalence of quasi-categories as in Definition 1.2.35.

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

Definition 1.4.10 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

1.4 \(\infty \)-Cosmoi