7.7 Universality of Colimits
Recall that the set $\operatorname{\mathbf{Z}}$ of integers forms a commutative ring: in particular, addition and multiplication are related by the distributive law
7.77
\begin{eqnarray} \label{equation:distributive-law} (x \cdot y) + (x \cdot z) = x \cdot (y + z ). \end{eqnarray}
Now suppose that $x$, $y$, and $z$ are given as the cardinalities of finite sets $X$, $Y$, and $Z$. Then the sum $y + z$ is the cardinality of the disjoint union $Y \amalg Z$, so the right hand side of (7.77) is the cardinality of the cartesian product $X \times (Y \amalg Z)$. Equation (7.77) then follows from observation that $X \times (Y \amalg Z)$ factors as a disjoint union of subsets $X \times Y$ and $X \times Z$, having cardinalities $x \cdot y$ and $x \cdot z$. More precisely, for any sets $X$, $Y$, and $Z$, there is a canonical isomorphism
7.78
\begin{eqnarray} \label{equation:distributive-law-categorified} (X \times Y) \coprod (X \times Z) & \xrightarrow {\sim } & X \times (Y \coprod Z ), \end{eqnarray}
which categorifies the equation (7.77) in the case where $X$, $Y$, and $Z$ are finite.
The isomorphism (7.78) is a special case of a more general phenomenon. Let $\operatorname{\mathcal{K}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{K}}\rightarrow \operatorname{Set}$ be a $\operatorname{\mathcal{K}}$-indexed diagram of sets, and suppose we are given a function $\varinjlim (\mathscr {F} ) \rightarrow S$. Then, for any function $S' \rightarrow S$, the fiber product $S' \times _{ S } \varinjlim (\mathscr {F} )$ identifies with the colimit of another diagram $\mathscr {F}': \operatorname{\mathcal{K}}\rightarrow \operatorname{Set}$, given on objects by the formula $\mathscr {F}'(k) = S' \times _{S} \mathscr {F}(k)$. In this section, we study $\infty $-categories where an analogous phenomenon occurs.
Definition 7.7.0.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $\mathscr {F}: K \rightarrow \operatorname{\mathcal{C}}$. Suppose that $\mathscr {F}$ admits a colimit: that is, there exists an object $C \in \operatorname{\mathcal{C}}$ and a natural transformation $\beta : \mathscr {F} \rightarrow \underline{C}$ which exhibits $C$ as a colimit of $\mathscr {F}$. We say that the colimit of $\mathscr {F}$ is universal if the following condition is satisfied:
- $(\ast )$
For every morphism $f: C' \rightarrow C$ of $\operatorname{\mathcal{C}}$ and every levelwise pullback square
\[ \xymatrix { \mathscr {F}' \ar [r]^{ \beta ' } \ar [d] & \underline{C}' \ar [d]^{ \underline{f} } \\ \mathscr {F} \ar [r]^{\beta } & \underline{C} } \]
in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$, the natural transformation $\beta '$ exhibits $C'$ as a colimit of $\mathscr {F}'$.
We say that $K$-indexed colimits in $\operatorname{\mathcal{C}}$ are universal if, for every diagram $\mathscr {F}: K \rightarrow \operatorname{\mathcal{C}}$ which admits a colimit, the colimit of $\mathscr {F}$ is universal.
Example 7.7.0.3. Small colimits are universal in (the nerve of) the category of sets. In fact, this is true even for colimits which are not small, in cases where they exist (Corollary 7.7.3.21).
It will often be useful to work with a reformulation of Definition 7.7.0.1. Fix a simplicial set $K$ and an $\infty $-category $\operatorname{\mathcal{C}}$ which admits pullbacks and $K$-indexed colimits. Suppose we are given a diagram $\mathscr {F}: K \rightarrow \operatorname{\mathcal{C}}$ having a colimit $C = \varinjlim (\mathscr {F})$. For each vertex $x \in X$, the tautological map $\beta _{x}: \mathscr {F}(x) \rightarrow C$ determines a pullback map
\[ \beta _{x}^{\ast }: \operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{C}}_{ / \mathscr {F}(x) } \quad \quad C' \mapsto C' \times _{C} \mathscr {F}(x) \]
(see Proposition 7.6.2.24). Allowing $x$ to vary, these can pullback maps can be organized into a comparison functor
\[ T_{\mathscr {F}}: \operatorname{\mathcal{C}}_{/C} \rightarrow \varprojlim _{x \in K} \operatorname{\mathcal{C}}_{ / \mathscr {F}(x) }. \]
In §7.7.2, we show that the colimit of $\mathscr {F}$ is universal if and only if the functor $T_{\mathscr {F}}$ is fully faithful (Theorem 7.7.2.8). If the functor $T_{ \mathscr {F}}$ is an equivalence of $\infty $-categories, we will say that the colimit of $\mathscr {F}$ is strongly universal (see Definition 7.7.2.15 for a more precise formulation). Our primary objective in this section is to establish the following homotopy-theoretic counterpart of Example 7.7.0.3:
Theorem 7.7.0.4. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.5.1.1). Then the colimit of any small diagram $K \rightarrow \operatorname{\mathcal{S}}$ is strongly universal.
Warning 7.7.0.5. Theorem 7.7.0.4 is stronger than its classical counterpart: in the category of sets, small colimits are universal but are generally not strongly universal (see Example 7.7.2.22). In fact, Theorem 7.7.0.4 is in some sense the defining feature of unstable homotopy theory: we will return to this point in Chapter 
.
In this section, we will give two different proofs of Theorem 7.7.0.4, and one additional proof of the weaker assertion that colimits in $\operatorname{\mathcal{S}}$ are universal:
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits finite limits. We say that $\operatorname{\mathcal{C}}$ is locally cartesian closed if, for every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the pullback functor
\[ f^{\ast }: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}_{/X} \quad \quad C \mapsto C \times _{Y} X \]
has a right adjoint (see Definition 7.7.3.14 and Proposition 7.7.3.19). In §7.7.3, we show that the $\infty $-category $\operatorname{\mathcal{S}}$ is locally cartesian closed Proposition 7.7.3.17. Our proof is based the observation that, for every Kan complex $X$, the $\infty $-category $\operatorname{\mathcal{S}}_{/X}$ has an explicit model as the homotopy coherent nerve of a simplicially enriched category $\operatorname{KFib}(X)$ of Kan fibrations over $X$, where there is good supply of internal function objects. As a consequence, we deduce that $K$-indexed colimits in $\operatorname{\mathcal{S}}$ are universal for every simplicial set $K$ (even when $K$ is not small): see Corollary 7.7.3.23.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and small colimits. In §7.7.5, we explain that the (strong) universality of colimits in $\operatorname{\mathcal{C}}$ follows from the (strong) universality of coproducts and pushouts (Corollary 7.7.5.9). In the case $\operatorname{\mathcal{C}}= \operatorname{\mathcal{S}}$, the case of coproducts is easy to handle directly (Corollary 7.7.4.12). The strong universality of pushouts is a reformulation of the Mather cube theorems (Theorems 3.4.3.3 and Theorem 3.4.4.4), which we discussed in Chapter 3.
Recall that the $\infty $-category $\operatorname{\mathcal{S}}$ classifies left fibrations: that is, we can identify (isomorphism classes of) diagrams $\mathscr {F}: K \rightarrow \operatorname{\mathcal{C}}$ with (equivalence classes of) essentially small left fibrations $\widetilde{K} \rightarrow K$ (Corollary 5.6.0.6). In §7.7.6, we use this perspective to reformulate Theorem 7.7.0.4 as a statement about about left fibrations, which we prove directly (Proposition 7.7.6.2).
For later use, we record the following consequence of Theorem 7.7.0.4:
Corollary 7.7.0.6. Let $\mathscr {F}: K \rightarrow \operatorname{\mathcal{S}}$ be a small diagram. Suppose we are given a Kan complex $X \in \operatorname{\mathcal{S}}$ and a natural transformation $\beta : \mathscr {F} \rightarrow \underline{X}$. The following conditions are equivalent:
- $(1)$
The natural transformation $\beta $ exhibits $C$ as a colimit of the diagram $\mathscr {F}$.
- $(2)$
For every map of Kan complexes $f: X' \rightarrow X$ and every levelwise pullback square
\[ \xymatrix { \mathscr {F}' \ar [r]^{ \beta ' } \ar [d] & \underline{X}' \ar [d]^{ \underline{f} } \\ \mathscr {F} \ar [r]^{\beta } & \underline{X} } \]
the natural transformation $\beta '$ exhibits $X'$ as a colimit of $\mathscr {F}'$.
- $(3)$
For every vertex $x \in X$ and every levelwise pullback square
\[ \xymatrix { \mathscr {F}' \ar [r]^{ \beta ' } \ar [d] & \underline{ \{ x\} } \ar [d] \\ \mathscr {F} \ar [r]^{\beta } & \underline{X} } \]
the natural transformation $\beta '$ exhibits $\{ x\} $ as a colimit of $\mathscr {F}'$ (here the right vertical map is induced by the inclusion $\{ x\} \hookrightarrow X$).
- $(4)$
For every vertex $x \in X$, the diagram $\mathscr {F}' = \underline{ \{ x\} } \times _{ \underline{X} } \mathscr {F}$ has a contractible colimit in $\operatorname{\mathcal{S}}$.
Proof.
The implication $(1) \Rightarrow (2)$ follows from the universality of colimits in $\operatorname{\mathcal{S}}$ (Theorem 7.7.0.4) and the implications $(2) \Rightarrow (3) \Rightarrow (4)$ are immediate. We will complete the proof by showing that $(4)$ implies $(1)$. Choose a Kan complex $Y = \varinjlim (\mathscr {F} )$ and a natural transformation $\alpha : \mathscr {F} \rightarrow \underline{Y}$ which exhibits $Y$ as a colimit of $\mathscr {F}$. The natural transformation $\beta $ induces a morphism of Kan complexes $f: Y \rightarrow X$, which is characterized (up to homotopy) by the existence of a commutative diagram
\[ \xymatrix { & \underline{ Y } \ar [dr]^{ \underline{f} } & \\ \mathscr {F} \ar [ur]^{\alpha } \ar [rr]^{ \beta } & & \underline{X} } \]
in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{S}})$. To prove $(1)$, we must show that $f$ is a homotopy equivalence. By virtue of Remark 3.4.0.6, this is equivalent to the requirement that for each vertex $x \in X$, the homotopy fiber $Y_{x} = \{ x\} \times _{X}^{\mathrm{h}} Y$ is a contractible Kan complex. Using Example 7.6.3.12, we see that there is a pullback diagram $\tau :$
\[ \xymatrix { Y_{x} \ar [d] \ar [r] & \{ x\} \ar [d] \\ Y \ar [r]^{f} & X } \]
in the $\infty $-category $\operatorname{\mathcal{S}}$. As in Remark 7.7.0.2, we can choose a commutative diagram
\[ \xymatrix { \mathscr {F}' \ar [r]^{\alpha '} \ar [d] & \underline{ Y_ x } \ar [r] \ar [d] & \underline{ \{ x\} } \ar [d] \\ \mathscr {F} \ar [r]^{ \alpha } & \underline{Y} \ar [r]^{ \underline{f}} & \underline{X}, } \]
in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{S}})$, where the bottom row is given by the $2$-simplex $\sigma $, the right side of the diagram is the image of $\tau $ under the diagonal map $\operatorname{\mathcal{S}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{S}})$, and the left side of the diagram is a levelwise pullback square. It follows that the outer rectangle is also a levelwise pullback square (Proposition 7.6.2.28), so condition $(4)$ guarantees that the colimit of $\mathscr {F}'$ is a contractible Kan complex. Since $K$-indexed colimits in $\operatorname{\mathcal{S}}$ are universal (Theorem 7.7.0.4), the natural transformation $\alpha '$ exhibits $Y_{x}$ as a colimit of $\mathscr {F}'$, so that $Y_{x}$ is contractible as desired.
$\square$
Structure
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Subsection 7.7.1: Cartesian Natural Transformations
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Subsection 7.7.2: Descent Diagrams
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Subsection 7.7.3: Cartesian Closed $\infty $-Categories
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Subsection 7.7.4: Disjoint Coproducts
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Subsection 7.7.5: The Mather Cube Theorem Revisted
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Subsection 7.7.6: Descent in the $\infty $-Category of Spaces