7.6.2 Powers and Tensors
We now study limits and colimits which are indexed by constant diagrams of simplicial sets. Like products and coproducts, these can be characterized by universal properties in the (enriched) homotopy category.
Definition 7.6.2.1. 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 tensor product 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 7.6.2.2. In the situation of Definition 7.6.2.1, 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 7.6.2.5. 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 tensor product 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 7.6.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$ and $Y$, and suppose we are given a collection of morphisms $\{ f_ j: X \rightarrow Y \} _{j \in J}$ indexed by a set $J$. If we abuse notation by identifying $J$ with the corresponding discrete simplicial set, then the collection $\{ f_ j \} _{j \in J}$ can be identified with a map $e: J \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. In this case:
The morphism $e$ exhibits $X$ as a power of $Y$ by $J$ (in the sense of Definition 7.6.2.1) if and only if the collection $\{ f_ j \} _{j \in J}$ exhibits $X$ as a product of the collection $\{ Y \} _{j \in J}$ (in the sense of Definition 7.6.1.3). Stated more informally, we have a canonical isomorphism $Y^{J} \simeq {\prod }_{j \in J} Y$ (provided that either side is defined).
The morphism $e$ exhibits $Y$ as a tensor product of $X$ by $J$ (in the sense of Definition 7.6.2.1) if and only if the collection $\{ f_ j \} _{j \in J}$ exhibits $Y$ as a coproduct of the collection $\{ X \} _{j \in J}$ (in the sense of Definition 7.6.1.3). Stated more informally, we have a canonical isomorphism $J \otimes X \simeq {\coprod }_{j \in J} X$ (provided that either side is defined).
Example 7.6.2.8. 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 if and only if $X$ is a final object of $\operatorname{\mathcal{C}}$. Similarly, $e$ exhibits $Y$ as a tensor product of $X$ by the empty simplicial set if and only if $Y$ is an initial object of $\operatorname{\mathcal{C}}$.
Notation 7.6.2.9 (Diagonal Morphisms). 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 which exhibits $X$ as a power of $Y$ by $K$. Then there exists a morphism $\delta : Y \rightarrow X$ which is characterized (up to homotopy) by the requirement that the diagram of simplicial sets
\[ \xymatrix { K \ar [r]^{e} \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [d]^{ \circ [\delta ] } \\ \{ \operatorname{id}_ Y \} \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Y) } \]
commutes up to homotopy. We will refer to $\delta $ as the diagonal morphism.
We will be particularly interested in the special case where $K = \operatorname{\partial \Delta }^1$, so that $X$ can be identified with the product $Y \times Y$ (Example 7.6.2.7). In this case, we will often denote $\delta $ by $\delta _{Y}: Y \rightarrow Y \times Y$ and refer to it as the diagonal of $Y$.
Proposition 7.6.2.10. 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 tensor product 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 7.6.2.10, 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 7.6.2.11. 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 tensor product 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 7.6.2.12. Let $Y$ be an essentially small Kan complex. Suppose we are given a morphism of simplicial sets $f: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, Y)$, which we identify with a morphism $\widetilde{f}: \underline{ \Delta ^0 }_{K} \rightarrow \underline{ Y }_{K}$ in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{S}})$. Then $f$ is a weak homotopy equivalence if and only if $\widetilde{f}$ exhibits $Y$ as a tensor product 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 7.6.2.3). We may therefore assume without loss of generality that $f$ factors through the homotopy equivalence $e: \operatorname{Fun}( \Delta ^0,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}(\Delta ^0,Y)$ of Remark 5.5.1.5, in which case the desired result follows from the criterion of Example 7.6.2.11 (applied in the case $X = \Delta ^0$). Taking $K = Y$ and $f = e$, we see that every Kan complex $Y$ can be viewed as a colimit of the constant diagram $Y \rightarrow \{ \Delta ^0 \} \hookrightarrow \operatorname{\mathcal{S}}$ (see Remark 7.6.2.6).
By virtue of Theorem 7.2.3.1 and Example 7.6.2.12, both conditions are equivalent to the requirement that, for every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $K_{/C} = K \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ is weakly contractible.
Example 7.6.2.14. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, let $e_0: K \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$ be a morphism of 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 7.6.2.10 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 tensor product 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 7.6.2.15. In the statement of Example 7.6.2.14, the assumption that $K$ is a Kan complex cannot be omitted.
Examples 7.6.2.11 and 7.6.2.14 show that the $\infty $-categories $\operatorname{\mathcal{S}}$ and $\operatorname{\mathcal{QC}}$ admit powers and tensor products by any small simplicial set $K$. Beware that it is very rare for a small $\infty $-category to have the same property:
Proposition 7.6.2.16. 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 can use Example 7.6.2.11 to give an alternative proof of the univerality of the left fibration $\operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{\mathcal{S}}$ (see Corollary 5.6.0.6).
Proposition 7.6.2.17 (Covariant Transport as a Kan Extension). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small left fibration of $\infty $-categories, let $\underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}}$ denote the constant functor $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}$ taking the value $\Delta ^0$, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be any functor. Suppose we are given a natural transformation $\beta : \underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F} \circ U$. The following conditions are equivalent:
- $(1)$
The natural transformation $\beta $ exhibits $\mathscr {F}$ as a left Kan extension of $\underline{ \Delta ^0}_{\operatorname{\mathcal{E}}}$ along $U$ (in the sense of Variant 7.3.1.5).
- $(2)$
The commutative diagram
7.56
\begin{equation} \begin{gathered}\label{equation:covariant-transport-as-Kan-extension} \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{\beta } & \{ \Delta ^0 \} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}\ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{ \mathscr {F} } & \operatorname{\mathcal{S}}} \end{gathered} \end{equation}
is a categorical pullback square.
Proof.
Fix an object $C \in \operatorname{\mathcal{C}}$ and let $\operatorname{\mathcal{E}}_{C}$ denote the fiber $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, so that the restriction of $\beta $ to $\operatorname{\mathcal{E}}_{C}$ can be identified with a morphism of Kan complexes $e_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, \mathscr {F}(C) )$. By virtue of Proposition 7.3.4.1 and Corollary 5.1.7.15, it will suffice to show that the following conditions are equivalent:
- $(1_ C)$
The morphism $e_ C$ exhibits $\mathscr {F}(C)$ as a tensor product of $\Delta ^0$ by $\mathscr {E}_{C}$ (as an object of the $\infty $-category $\operatorname{\mathcal{S}}$).
- $(2_ C)$
The morphism $e_ C$ is a homotopy equivalence.
This is a special case of Example 7.6.2.12.
$\square$
Corollary 7.6.2.18. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small left fibration of $\infty $-categories. Then a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is a covariant transport representation for $U$ (in the sense of Definition 5.6.5.1) if and only if it is a left Kan extension of the constant functor $\underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}}$ along $U$.
Proof.
Combine Proposition 7.6.2.17 with the equivalence $\operatorname{\mathcal{S}}_{\ast } \hookrightarrow \{ \Delta ^0 \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}$ of Theorem 4.6.4.17.
$\square$
Variant 7.6.2.19. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories, and suppose that the fibers of $U$ are essentially $\kappa $-small for some uncountable cardinal $\kappa $. Then, in the statements of Proposition 7.6.2.17 and Corollary 7.6.2.18, we can replace $\operatorname{\mathcal{S}}$ by the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$ of $\kappa $-small spaces (see Variant 5.5.4.12).
We now consider a variant of Proposition 7.6.2.10. 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 $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X, Y) \hookrightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X, Y)$ of Construction 4.6.5.7.
Proposition 7.6.2.20. 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 tensor product 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 7.6.2.20.
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 7.6.2.21 (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 7.6.2.20: 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 tensor product 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 7.6.2.22 (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 7.6.2.20 to the differential graded category $\operatorname{Ch}(\operatorname{\mathbf{Z}})^{\operatorname{op}}$ (and using Remark 7.6.2.4), 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 $S \rightarrow \{ A\} \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{Ch}(\operatorname{\mathbf{Z}}) )$.