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2.5.4 The Homotopy Category of a Differential Graded Category

Let $\operatorname{\mathcal{C}}$ be a differential graded category, and let $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ denote its differential graded nerve (Definition 2.5.3.7). Then $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.5.3.10). Moreover:

  • The objects of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ are the objects of $\operatorname{\mathcal{C}}$ (Example 2.5.3.2).

  • If $X$ and $Y$ are objects of $\operatorname{\mathcal{C}}$, then a morphism from $X$ to $Y$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ can be identified with a $0$-cycle in the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ (Example 2.5.3.3), or equivalently with a morphism from $X$ to $Y$ in the underlying category $\operatorname{\mathcal{C}}^{\circ }$ of Construction 2.5.2.4.

We now explain how to describe the homotopy category of $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ directly in terms of the differential graded category $\operatorname{\mathcal{C}}$ (Proposition 2.5.4.10).

Definition 2.5.4.1. Let $\operatorname{\mathcal{C}}$ be a differential graded category containing a pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, and let $f$ and $f'$ be $0$-cycles of the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$. A homotopy from $f$ to $f'$ is a $1$-chain $h \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{1}$ satisfying $\partial (h) = f'-f$. We will say that $f$ and $f'$ are homotopic if there exists a homotopy from $f$ to $f'$: that is, if we have an equality $[f] = [f']$ in the homology group $\mathrm{H}_{0}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) )$.

Example 2.5.4.2. Let $\operatorname{\mathcal{A}}$ be an additive category, let $C_{\ast }$ and $D_{\ast }$ be chain complexes with values in $\operatorname{\mathcal{A}}$, and let $f,f': C_{\ast } \rightarrow D_{\ast }$ be chain maps, which we regard as $0$-cycles in the mapping complex $\operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathcal{A}})}( C_{\ast }, D_{\ast } )_{\ast }$ in the differential graded category $\operatorname{Ch}(\operatorname{\mathcal{A}})$ of Example 2.5.2.5. Let $h = \{ h_ n: C_{n} \rightarrow D_{n+1} \} _{n \in \operatorname{\mathbf{Z}}}$ be a collection of morphisms, which we regard as a $1$-chain of $\operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathcal{A}})}( C_{\ast }, D_{\ast } )_{\ast }$. Then $h$ is a homotopy from $f$ to $f'$ (in the sense of Definition 2.5.4.1) if and only if it is a chain homotopy from $f$ to $f'$ (in the sense of Definition 2.5.0.5). In particular, $f$ and $f'$ are homotopic morphisms of the differential graded category $\operatorname{Ch}(\operatorname{\mathcal{A}})$ (in the sense of Definition 2.5.4.1) if and only if they are chain homotopic (in the sense of Definition 2.5.0.5).

Remark 2.5.4.3. Let $\operatorname{\mathcal{C}}$ be a differential graded category containing a pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, and let $f$ and $g$ be $0$-cycles of the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$. Then giving a homotopy from $f$ to $g$ in the sense of Definition 2.5.4.1 is equivalent to giving a homotopy from $g$ to $f$ as morphisms in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ (Definition 1.3.3.1): this follows from Example 2.5.3.4. In particular, $f$ and $g$ are homotopic in the sense of Definition 2.5.4.1 if and only if they are homotopic in the sense of Definition 1.3.3.1.

Remark 2.5.4.4. Let $\operatorname{\mathcal{C}}$ be a differential graded category containing objects $X$, $Y$, and $Z$, and suppose we are given $0$-cycles $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{0}$, $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{0}$, and $h \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Z)_{0}$. Then Example 2.5.3.4 supplies an equivalence between the following data:

  • The datum of a homotopy from $h$ to $g \circ f$, in the sense of Definition 2.5.4.1.

  • The datum of a $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{\mathcal{C}})$ witnessing $h$ as a composition of $f$ and $g$, in the sense of Definition 1.3.4.1.

In particular, $h$ is homotopic to the composition $g \circ f$ (in the differential graded category $\operatorname{\mathcal{C}}$) if and only if it is a composition of $g$ and $f$ (in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$).

Proposition 2.5.4.5. Let $\operatorname{\mathcal{C}}$ be a differential graded category containing a pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$. Let $f$ and $g$ be $0$-cycles of the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ which are homotopic. Then:

$(a)$

For any object $W \in \operatorname{Ob}(\operatorname{\mathcal{C}})$ and any $0$-cycle $u \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X)_{0}$, the composite cycles $f \circ u$ and $g \circ u$ are homotopic.

$(b)$

For any object $Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$ and any $0$-cycle $v \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y, Z)_{0}$, the composite cycles $v \circ f$ and $v \circ g$ are homotopic.

Proof. By virtue of Remarks 2.5.4.3 and 2.5.4.4, we can regard Proposition 2.5.4.5 as a special case of Proposition 1.3.4.7. However, it is easy to prove directly. If $h \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{1}$ is a homotopy from $f$ to $g$ and $u$ is a $0$-cycle in $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X)_{0}$, then the calculation

\begin{eqnarray*} \partial ( h \circ u ) & = & ((\partial h) \circ u) - (h \circ (\partial u)) \\ & = & (\partial h) \circ u \\ & = & (f - g) \circ u \\ & = & (f \circ u) - (g \circ u) \end{eqnarray*}

shows that $(h \circ u) \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y)_{1}$ is a homotopy from $f \circ u$ to $g \circ u$. This proves $(a)$, and $(b)$ follows from a similar argument. $\square$

Construction 2.5.4.6 (The Homotopy Category of a Differential Graded Category). Let $\operatorname{\mathcal{C}}$ be a differential graded category. We define a category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as follows:

  • The objects of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, we define

    \[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X, Y) = \mathrm{H}_0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ). \]

    If $f$ is a $0$-cycle of the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$, let $[f]$ denote its image in the homology group $\mathrm{H}_0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) = \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X, Y)$.

  • For each object $X \in \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the identity morphism from $X$ to itself in the category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is given by $[ \operatorname{id}_ X ]$, where $\operatorname{id}_ X$ is the identity morphism from $X$ to itself in $\operatorname{\mathcal{C}}$.

  • For every triple of objects $X,Y,Z \in \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the composition law

    \[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( Y,Z ) \times \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X, Y) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X, Z) \]

    is characterized by the formula $[g] \circ [f] = [g \circ f]$ for $f \in \mathrm{Z}_0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) )$ and $g \in \mathrm{Z}_0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) )$ (this composition law is well-defined by virtue of Proposition 2.5.4.5).

We will refer to $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as the homotopy category of the differential graded category $\operatorname{\mathcal{C}}$.

Remark 2.5.4.7. Passage from a differential graded category $\operatorname{\mathcal{C}}$ to its homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ can be regarded as a special case of Remark 2.1.7.4, applied to the lax monoidal functor

\[ \operatorname{Ch}(\operatorname{\mathbf{Z}}) \rightarrow \operatorname{Set}\quad \quad (C_{\ast }, d) \mapsto \mathrm{H}_0( C ) \]

with tensor constraints given by

\[ \mu _{C,D}: \mathrm{H}_0(C) \times \mathrm{H}_0(D) \rightarrow \mathrm{H}_0( C \boxtimes D) \quad \quad ( [x], [y] ) \mapsto [x \boxtimes y]. \]

Remark 2.5.4.8. Let $\operatorname{\mathcal{C}}$ be a differential graded category, with underlying category $\operatorname{\mathcal{C}}^{\circ }$ (Construction 2.5.2.4) and homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (Construction 2.5.4.6). There is an evident functor $\operatorname{\mathcal{C}}^{\circ } \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ which is the identity on objects, given on morphisms by the construction

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\circ }}(X,Y) = \mathrm{Z}_0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) \twoheadrightarrow \mathrm{H}_0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) = \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) \quad \quad f \mapsto [f]. \]

Example 2.5.4.9 (The Homotopy Category of Chain Complexes). Let $\operatorname{\mathcal{A}}$ be an additive category, and let $\operatorname{Ch}(\operatorname{\mathcal{A}})$ be the differential graded category of chain complexes with values in $\operatorname{\mathcal{A}}$ (Example 2.5.2.5). Then the homotopy category of $\operatorname{Ch}(\operatorname{\mathcal{A}})$ in the sense of Construction 2.5.4.6 agrees with the homotopy category $\operatorname{hCh}(\operatorname{\mathcal{A}})$ introduced in Construction 2.5.0.8.

Proposition 2.5.4.10. Let $\operatorname{\mathcal{C}}$ be a differential graded category and let $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ denote the differential graded nerve of $\operatorname{\mathcal{C}}$. Then the homotopy category $\mathrm{h} \mathit{\operatorname{N}}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ (Definition 1.3.5.3) is canonically isomorphic to the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (Construction 2.5.4.6).