7.1.2 Constant Diagrams
We now study limits and colimits which are indexed by constant diagrams of simplicial sets. These can be characterized by universal properties in the (enriched) homotopy category.
Definition 7.1.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of objects $X$ and $Y$, and let $e: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ be a morphism of simplicial sets. We will say that $e$ exhibits $X$ as a power of $Y$ by $K$ if, for every object $W \in \operatorname{\mathcal{C}}$, the composition law $\circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y)$ of Construction 4.6.9.9 induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X) \rightarrow \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y) )$.
We will say that $e$ exhibits $Y$ as a copower of $X$ by $K$ if, for every object $Z \in \operatorname{\mathcal{C}}$, the composition law $\circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \rightarrow \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) )$.
Warning 7.1.2.2. In the situation of Definition 7.1.2.1, the composition law
\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]
is only well-defined up to homotopy. However, the requirement that it induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \rightarrow \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) )$ depends only on its homotopy class.
Notation 7.1.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $Y$ be an object of $\operatorname{\mathcal{C}}$, and let $K$ be a simplicial set. Suppose that there exists an object $X \in \operatorname{\mathcal{C}}$ and a morphism $e: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ which exhibits $X$ as a power of $Y$ by $K$. In this case, the object $X$ is uniquely determined up to isomorphism. To emphasize this uniqueness, we will sometimes denote the object $X$ by $Y^{K}$.
Similarly, if there exists an object $Z \in \operatorname{\mathcal{C}}$ and a morphism $e: K \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z)$ which exhibits $Z$ as a copower of $Y$ by $K$, then $Z$ is uniquely determined up to isomorphism. We will sometimes emphasize this dependence by denoting the object $Z$ by $K \otimes Y$.
Example 7.1.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$ and $Y$. Then the unique morphism $e: \emptyset \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ exhibits $X$ as a power of $Y$ by the empty simplicial set $\emptyset $ if and only if $X$ is a final object of $\operatorname{\mathcal{C}}$. Similarly, $e$ exhibits $Y$ as a copower of $X$ by $\emptyset $ set if and only if $Y$ is an initial object of $\operatorname{\mathcal{C}}$.
Proposition 7.1.2.8. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, and let $e: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ be a morphism of simplicial sets. Let $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ denote the homotopy coherent nerve of $\operatorname{\mathcal{C}}$, and let $\theta _{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}( X, Y)$ denote the comparison map of Remark 4.6.8.6. Then:
- $(1)$
The morphism $\theta _{X,Y} \circ e$ exhibits $X$ as a power of $Y$ by $K$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ if and only if, for every object $W \in \operatorname{\mathcal{C}}$, composition with $e$ induces a homotopy equivalence of Kan complexes
\[ c_{W}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W, X)_{\bullet } \rightarrow \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y)_{\bullet } ). \]
- $(2)$
The morphism $\theta _{X,Y} \circ e$ exhibits $Y$ as a copower of $X$ by $K$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ if and only if, for every object $Z \in \operatorname{\mathcal{C}}$, precomposition with $e$ induces a homotopy equivalence of Kan complexes
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y, Z)_{\bullet } \rightarrow \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } ). \]
Proof.
We will prove $(1)$; the proof of $(2)$ is similar. Fix an object $W \in \operatorname{\mathcal{C}}$, so that the composition law
\[ \circ : \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X,Y) \times \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(W,X) \rightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(W,Y) \]
of Construction 4.6.9.9 determines a morphism of Kan complexes $c'_{W}: \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( W,X) \rightarrow \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(W,Y)_{\bullet } )$ (which is well-defined up to homotopy). To prove Proposition 7.1.2.8, it will suffice to show that $c'_{W}$ is a homotopy equivalence if and only if $c_{W}$ is a homotopy equivalence. Proposition 4.6.9.19 guarantees that the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X)_{\bullet } \ar [r]^-{ c_{W} } \ar [d]^-{ \theta _{W,X} } & \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W, Y)_{\bullet } ) \ar [d]^-{ \theta _{ W, Y} \circ } \\ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( W,X) \ar [r]^-{ c'_{W} } & \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(W,Y)_{\bullet }) } \]
commutes up to homotopy. We conclude by observing that the horizontal maps are homotopy equivalences, by virtue of Theorem 4.6.8.5 (and Remark 4.6.8.6).
$\square$
Example 7.1.2.9. Let $X$ and $Y$ be essentially small Kan complexes, let $e_0: K \rightarrow \operatorname{Fun}(X,Y)$ be a morphism of simplicial sets, and let $e: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}(X,Y)$ denote the composition of $e_0$ with the homotopy equivalence $\operatorname{Fun}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}(X,Y)$ of Remark 5.5.1.5. Then:
The morphism $e$ exhibits $X$ as a power of $Y$ by $K$ in the $\infty $-category $\operatorname{\mathcal{S}}$ and only the induced map $X \rightarrow \operatorname{Fun}(K,Y)$ is a homotopy equivalence of Kan complexes.
The morphism $e$ exhibits $Y$ as a copower of $X$ by $K$ in the $\infty $-category $\operatorname{\mathcal{S}}$ if and only if the induced map $K \times X \rightarrow Y$ is a weak homotopy equivalence of simplicial sets.
Example 7.1.2.10. Let $Y$ be an essentially small Kan complex. Suppose we are given a morphism of simplicial sets $e: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, Y)$. Then $e$ is a weak homotopy equivalence if and only if $\widetilde{f}$ exhibits $Y$ as a copower of $\Delta ^0$ by $K$ (in the $\infty $-category $\operatorname{\mathcal{S}}$). To prove this, we are free to modify the morphism $f$ by a homotopy (see Remark 7.1.2.3). We may therefore assume without loss of generality that $f$ factors through the homotopy equivalence $f: \operatorname{Fun}( \Delta ^0,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}(\Delta ^0,Y)$ of Remark 5.5.1.5, in which case the desired result follows from the criterion of Example 7.1.2.9 (applied in the case $X = \Delta ^0$). Taking $K = Y$ and $f = e$, we see that every essentially small Kan complex $Y$ can be viewed as a colimit of the constant diagram $Y \rightarrow \{ \Delta ^0 \} \hookrightarrow \operatorname{\mathcal{S}}$ (see Remark 7.1.2.6).
Variant 7.1.2.11. Let $\kappa $ be an uncountable cardinal and let $Y$ be a Kan complex which is essentially $\kappa $-small. Suppose we are given a morphism of simplicial sets $e: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, Y)$. If the simplicial set $K$ is essentially $\kappa $-small, then the following conditions are equivalent:
- $(1)$
The morphism $e$ is a weak homotopy equivalence.
- $(2)$
The morphism $e$ exhibits $Y$ as a copower of $\Delta ^0$ by $K$ in the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$.
Beware that, if $K$ is not assumed to be essentially $\kappa $-small, then it is possible for condition $(2)$ to be satisfied while condition $(1)$ is not (Exercise 7.1.2.12).
Exercise 7.1.2.12. Let $\kappa $ be an uncountable cardinal and let $K$ be a connected Kan complex satisfying the following conditions:
The fundamental group $\pi _{1}(K)$ is a simple group of cardinality $\geq \kappa $.
The Kan complex $K$ is acyclic: that is, the homology groups $\mathrm{H}_{n}(K; \operatorname{\mathbf{Z}})$ vanish for $n > 0$.
Show that the projection map $K \twoheadrightarrow \Delta ^0 \simeq \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, \Delta ^0)$ exhibits $\Delta ^0$ as a copower of itself by $K$ in the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$. That is, $\Delta ^0$ is a colimit of the constant diagram $\underline{\Delta ^0}_{K}$ in the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$, which is not preserved by the inclusion functor $\operatorname{\mathcal{S}}^{< \kappa } \hookrightarrow \operatorname{\mathcal{S}}^{< \lambda }$ for $\lambda \gg \kappa $.
Example 7.1.2.13. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be small $\infty $-categories, let $e_0: K \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$ be a morphism of small Kan complexes, and let $e$ denote the composition of $e_0$ with the homotopy equivalence $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{QC}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ of Remark 5.5.4.5. Combining Propositions 7.1.2.8 and 4.4.3.22, we obtain the following:
The morphism $e$ exhibits $\operatorname{\mathcal{C}}$ as a power of $\operatorname{\mathcal{D}}$ by $K$ in the $\infty $-category $\operatorname{\mathcal{QC}}$ if and only if the induced map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{D}})$ is an equivalence of $\infty $-categories.
The morphism $e$ exhibits $\operatorname{\mathcal{C}}$ as a copower of $\operatorname{\mathcal{D}}$ by $K$ in the $\infty $-category $\operatorname{\mathcal{QC}}$ if and only if the induced map $K \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories.
Warning 7.1.2.14. In the statement of Example 7.1.2.13, the assumption that $K$ is a Kan complex cannot be omitted.
Examples 7.1.2.9 and 7.1.2.13 show that the $\infty $-categories $\operatorname{\mathcal{S}}$ and $\operatorname{\mathcal{QC}}$ admit powers and copowers by any small simplicial set $K$. Beware that it is very rare for a small $\infty $-category to have the same property:
Proposition 7.1.2.15. Let $S$ be an infinite set of cardinality $\kappa $ and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\kappa ^{+}$-small. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to the nerve of a partially ordered set.
- $(2)$
For every nonempty simplicial set $K$ and every object $X \in \operatorname{\mathcal{C}}$, the constant map
\[ K \rightarrow \{ \operatorname{id}_ X \} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X) \]
exhibits $X$ as a power of itself by $K$.
- $(3)$
Every object $X \in \operatorname{\mathcal{C}}$ admits a power by $S$.
Proof.
The implications $(1) \Rightarrow (2) \Rightarrow (3)$ are immediate from the definitions. We will show that $(3)$ implies $(1)$. Assume that condition $(3)$ is satisfied and fix a pair of objects $X,Y \in \operatorname{\mathcal{C}}$; we wish to show that the morphism space $M = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)$ is either empty or contractible. Assume otherwise: then there exists a morphism $f: Y \rightarrow X$ in $\operatorname{\mathcal{C}}$ and an integer $n \geq 0$ such that the homotopy set $\pi _{n}(M,f)$ has at least two elements. Using assumption $(3)$, we can choose an object $X' \in \operatorname{\mathcal{C}}$ and a collection of morphisms $\{ g_ s: X' \rightarrow X \} $ which exhibit $X'$ as a power of $X$ by $S$. Choose a morphism $f': Y \rightarrow X$ such that $g_{s} \circ f'$ is homotopic to $f$ for each $s \in S$. Then $\pi _{n}(M', f')$ can be identified with the product ${\prod }_{s \in S} \pi _{n}(M,f)$. This set has cardinality larger than $\kappa $ (Proposition 4.7.2.8), contradicting our assumption that $\operatorname{\mathcal{C}}$ is locally $\kappa ^{+}$-small.
$\square$
We now consider a variant of Proposition 7.1.2.8. Suppose we are given a differential graded category $\operatorname{\mathcal{C}}$ containing objects $X$ and $Y$. Let
\[ \rho _{X,Y}: \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } ) \hookrightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X, Y) \]
denote the composition of the isomorphism $\mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } ) \simeq \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X, Y)$ of Example 4.6.5.15 with the pinch inclusion morphism $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X, Y) \hookrightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X, Y)$ of Construction 4.6.5.7.
Proposition 7.1.2.16. Let $\operatorname{\mathcal{C}}$ be a differential graded category, let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, and suppose we are given a morphism of simplicial sets $e_0: S \rightarrow \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } )$, which we identify with a morphism of chain complexes $f: \mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$. Let $e: S \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(X,Y)$ denote the composition of $e_0$ with the morphism $\rho _{X,Y}$. The following conditions are equivalent:
- $(1)$
The morphism $e$ exhibits $Y$ as a copower of $X$ by $S$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$.
- $(2)$
Let $Z$ be an object of $\operatorname{\mathcal{C}}$, so that $f$ induces a morphism of chain complexes
\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } \rightarrow \operatorname{Hom}_{\operatorname{Ch}(\operatorname{\mathbf{Z}})}( \mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}), \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast } )_{\ast }. \]
Then $\theta $ is an isomorphism on homology in degrees $\geq 0$.
Proof of Proposition 7.1.2.16.
Fix an object $Z \in \operatorname{\mathcal{C}}$. Using Proposition 4.6.9.21, we see that the diagram of Kan complexes
\[ \xymatrix@C =50pt@R=50pt{ \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } ) \ar [r]^-{ \mathrm{K}(\theta ) } \ar [dd]^{ \rho _{Y,Z} } & \mathrm{K}( \operatorname{Hom}_{\operatorname{Ch}(\operatorname{\mathbf{Z}})}( \mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}), \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast } )_{\ast } ) \ar [d]^-{\psi } \\ & \operatorname{Fun}(S, \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast } ) ) \ar [d]^-{ \rho _{X,Z} \circ } \\ \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(Y,Z) \ar [r] & \operatorname{Fun}(S, \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(X,Z))} \]
commutes up to homotopy, where $\psi $ is the homotopy equivalence of Example 3.1.6.11 and the bottom horizontal map is given by combining $e$ with the composition law on the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{\mathcal{C}})$. Note that condition $(1)$ is equivalent to the requirement that the bottom horizontal map is a homotopy equivalence (for each object $Z \in \operatorname{\mathcal{C}}$). Since the map $\rho _{Y,Z}$ and $\rho _{X,Z}$ are also homotopy equivalences (Proposition 4.6.5.10), this is equivalent to the requirement that $\mathrm{K}( \theta )$ is a homotopy equivalence (for each object $Z \in \operatorname{\mathcal{C}}$). The equivalence of $(1)$ and $(2)$ now follows from the criterion of Corollary 3.2.7.4.
$\square$
Example 7.1.2.17 (Homology as a Colimit). Let $\operatorname{\mathcal{C}}= \operatorname{Ch}(\operatorname{\mathbf{Z}})$ denote the category of chain complexes of abelian groups, which we regard as a differential graded category (see Example 2.5.2.5). Let $A$ be an abelian group, and let us abuse notation by identifying $A$ with its image in $\operatorname{\mathcal{C}}$ (by regarding it as a chain complex concentrated in degree zero). For every simplicial set $S$, let $\mathrm{N}_{\ast }(S; A)$ denote the normalized chain complex of $S$ with coefficients in $A$, given by the tensor product $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}) \boxtimes A$. Then the tautological map
\[ f: \mathrm{N}_{\ast }( S; \operatorname{\mathbf{Z}}) \rightarrow \operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathbf{Z}}) }( A, \mathrm{N}_{\ast }(S;A) )_{\ast } \]
satisfies condition $(2)$ of Proposition 7.1.2.16: in fact, for every object $M_{\ast } \in \operatorname{\mathcal{C}}$, precomposition with $f$ induces an isomorphism of chain complexes
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \mathrm{N}_{\ast }(S;A), M_{\ast } )_{\ast } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \mathrm{N}_{\ast }(S;\operatorname{\mathbf{Z}}), \operatorname{Hom}_{\operatorname{\mathcal{C}}}( A, M_{\ast } )_{\ast } )_{\ast }. \]
It follows that the induced map $S \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( A, \mathrm{N}_{\ast }(S; A) )$ exhibits $\mathrm{N}_{\ast }(S;A)$ as a copower of $A$ by $S$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$. In particular, the chain complex $\mathrm{N}_{\ast }(S;A)$ can be viewed as a colimit of the constant diagram $S \rightarrow \{ A \} \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{Ch}(\operatorname{\mathbf{Z}}) )$.
Variant 7.1.2.18 (Cohomology as a Limit). Let $A$ be an abelian group, let $S$ be a simplicial set, and let
\[ \mathrm{N}^{\ast }(S;A) = \operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathbf{Z}}) }( \mathrm{N}_{\ast }(S;\operatorname{\mathbf{Z}}), A) \]
denote the normalized cochain complex of $S$ with coefficients in $A$. Applying Proposition 7.1.2.16 to the differential graded category $\operatorname{Ch}(\operatorname{\mathbf{Z}})^{\operatorname{op}}$ (and using Remark 7.1.2.4), we see that the tautological chain map $\mathrm{N}_{\ast }( S; \operatorname{\mathbf{Z}}) \rightarrow \operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathbf{Z}})}( \mathrm{N}^{\ast }(S;A), A )_{\ast }$ induces a morphism of simplicial sets
\[ e: S \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{Ch}(\operatorname{\mathbf{Z}}) )}( \mathrm{N}^{\ast }(S;A), A) \]
which exhibits $\mathrm{N}^{\ast }(S;A)$ as a power of $A$ by $S$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{Ch}(\operatorname{\mathbf{Z}}) )$. In particular, $\mathrm{N}^{\ast }(S;A)$ can be viewed as a limit of the constant diagram $S \rightarrow \{ A\} \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{Ch}(\operatorname{\mathbf{Z}}) )$.