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2.5.2 Differential Graded Categories

Let $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ denote the category of chain complexes of abelian groups, equipped with the monoidal structure described in Construction 2.5.1.17. A differential graded category is a category enriched over $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ (in the sense of Definition 2.1.7.1). For the convenient of the reader, we spell out this definition in detail.

Definition 2.5.2.1 (Differential Graded Categories). A differential graded category $\operatorname{\mathcal{C}}$ consists of the following data:

$(1)$

A collection $\operatorname{Ob}(\operatorname{\mathcal{C}})$, whose elements we refer to as objects of $\operatorname{\mathcal{C}}$. We will often abuse notation by writing $X \in \operatorname{\mathcal{C}}$ to indicate that $X$ is an element of $\operatorname{Ob}(\operatorname{\mathcal{C}})$.

$(2)$

For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a chain complex $(\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }, \partial )$. For each integer $n$, we refer to the elements of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{n}$ as morphisms of degree $n$ from $X$ to $Y$.

$(3)$

For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$ and every pair of integers $m,n \in \operatorname{\mathbf{Z}}$, a function

\[ c_{Z,Y,X}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{n} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{m} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Z)_{m+n}, \]

which we will refer to as the composition law. Given a pair of morphisms $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{m}$ and $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{n}$, we will often denote the image $c_{Z,Y,X}(g,f) \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{m+n}$ by $g \circ f$ or $gf$.

$(4)$

For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a morphism $\operatorname{id}_{X} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{0}$, which we will refer to as the identity morphism.

These data are required to satisfy the following conditions:

  • The composition law on $\operatorname{\mathcal{C}}$ is associative in the following sense: for every triple of elements $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X)_{\ell }$, $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{m}$, and $h \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{n}$, we have an equality $h \circ (g \circ f) = (h \circ g) \circ f$ (in the abelian group $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Z)_{\ell +m+n}$).

  • The composition law on $\operatorname{\mathcal{C}}$ is unital on both sides: for every element $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{n}$, we have $\operatorname{id}_{Y} \circ f = f = f \circ \operatorname{id}_{X}$.

  • For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the composition maps $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{n} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{m} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Z)_{m+n}$ are bilinear and satisfy the Leibniz rule of Exercise 2.5.1.15. In other words, we have

    \[ g \circ (f + f') = (g \circ f) + (g \circ f') \quad \quad (g + g') \circ f = (g \circ f) + (g' \circ f) \]
    \[ \partial (g \circ f) = (\partial g) \circ f + (-1)^{n} g \circ (\partial f). \]

Remark 2.5.2.2. Let $\operatorname{\mathcal{C}}$ be a differential graded category. For each object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the identity morphism $\operatorname{id}_{X}$ is a $0$-cycle of the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\ast }$: that is, it satisfies $\partial (\operatorname{id}_ X) = 0$. This follows from the calculation

\[ \partial ( \operatorname{id}_ X ) = \partial ( \operatorname{id}_ X \circ \operatorname{id}_ X) = \partial (\operatorname{id}_ X) \circ \operatorname{id}_ X + \operatorname{id}_ X \circ \partial (\operatorname{id}_ X) = \partial (\operatorname{id}_ X) + \partial (\operatorname{id}_ X). \]

Remark 2.5.2.3. Let $\operatorname{\mathcal{C}}$ be a differential graded category containing a pair of morphisms $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{m}$ and $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{n}$. It follows from the Leibniz rule

\[ \partial (g \circ f) = (\partial g) \circ f + (-1)^{n} g \circ (\partial f) \]

that if $f$ and $g$ are cycles (that is, if they satisfy $\partial f = 0$ and $\partial g =0$), then $g \circ f$ is also a cycle. In particular, we have a bilinear composition map

\[ \mathrm{Z}_{n}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) ) \times \mathrm{Z}_ m( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) \rightarrow \mathrm{Z}_{m+n}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) ). \]

Construction 2.5.2.4 (The Underlying Category of a Differential Graded Category). To every differential graded category $\operatorname{\mathcal{C}}$, we can associate an ordinary category $\operatorname{\mathcal{C}}^{\circ }$ as follows:

  • The objects of $\operatorname{\mathcal{C}}^{\circ }$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{Ob}( \operatorname{\mathcal{C}}^{\circ } ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, a morphism from $X$ to $Y$ in $\operatorname{\mathcal{C}}^{\circ }$ is a $0$-cycle of chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$.

  • For each object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}}^{\circ } ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the identity morphism from $X$ to itself in $\operatorname{\mathcal{C}}^{\circ }$ is the identity morphism $\operatorname{id}_{X} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{0}$ (which is a cycle by virtue of Remark 2.5.2.2).

  • Composition of morphisms in $\operatorname{\mathcal{C}}^{\circ }$ is given by the composition law on $\operatorname{\mathcal{C}}$ (which preserves $0$-cycles by virtue of Remark 2.5.2.3).

We will refer to $\operatorname{\mathcal{C}}^{\circ }$ as the underlying category of the differential graded category $\operatorname{\mathcal{C}}$ (note that $\operatorname{\mathcal{C}}^{\circ }$ can also be obtained by applying the general procedure described in Example 2.1.7.5).

Example 2.5.2.5 (Chain Complexes). Let $\operatorname{\mathcal{A}}$ be an additive category. We define a differential graded category $\operatorname{Ch}(\operatorname{\mathcal{A}})$ as follows:

  • The objects of $\operatorname{Ch}(\operatorname{\mathcal{A}})$ are chain complexes with values in $\operatorname{\mathcal{A}}$ (Definition 2.5.0.1).

  • If $C_{\ast }$ and $D_{\ast }$ are chain complexes with values in $\operatorname{\mathcal{A}}$, then $\operatorname{Hom}_{\operatorname{Ch}(\operatorname{\mathcal{A}})}( C_{\ast }, D_{\ast } )_{\ast }$ is the chain complex of abelian groups $[C,D]_{\ast }$ defined in Construction 2.5.0.9.

  • If $C_{\ast }$, $D_{\ast }$, and $E_{\ast }$ are chain complexes with values in $\operatorname{\mathcal{A}}$, then the composition law

    \[ \circ : [D,E]_{e} \times [C,D]_{d} \rightarrow [C,E]_{d+e} \]

    is given by the formula $\{ g_{n} \} _{n \in \operatorname{\mathbf{Z}}} \circ \{ f_ n \} _{n \in \operatorname{\mathbf{Z}}} = \{ g_{n+d} \circ f_{n} \} _{n \in \operatorname{\mathbf{Z}}}$.

Note that if $C_{\ast }$ and $D_{\ast }$ are chain complexes with values in $\operatorname{\mathcal{A}}$, then a collection of maps $f = \{ f_ n: C_ n \rightarrow D_ n \} _{n \in \operatorname{\mathbf{Z}}}$ is a $0$-cycle of the chain complex $[C,D]_{\ast }$ if and only if it is a chain map from $C_{\ast }$ to $D_{\ast }$. Consequently, applying Construction 2.5.2.4 to the differential graded category $\operatorname{Ch}(\operatorname{\mathcal{A}})$ yields the ordinary category of chain complexes and chain maps. In other words, this construction supplies a $\operatorname{Ch}(\operatorname{\mathbf{Z}})$-enrichment of the category $\operatorname{Ch}(\operatorname{\mathcal{A}})$ introduced in Definition 2.5.0.3.

Example 2.5.2.6 (Differential Graded Algebras). A differential graded algebra is a (not necessarily commutative) graded ring $A_{\ast } = \{ A_ n \} _{n \in \operatorname{\mathbf{Z}}}$ equipped with a differential $\partial : A_{\ast } \rightarrow A_{\ast -1}$ satisfying $\partial ^2 = 0$ and the Leibniz rule $\partial (x \cdot y) = (\partial x) \cdot y + (-1)^{m} x \cdot (\partial y)$ for $x \in A_{m}$ and $y = A_{n}$. If $\operatorname{\mathcal{C}}$ is a differential graded category containing an object $X$, then the composition law on $\operatorname{\mathcal{C}}$ endows the chain complex $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)_{\ast } = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\ast }$ with the structure of a differential graded algebra. Conversely, for every differential graded algebra $(A_{\ast }, \partial )$, there is a unique differential graded category $\operatorname{\mathcal{C}}$ with $\operatorname{Ob}(\operatorname{\mathcal{C}}) = \{ X \} $. In other words, the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{End}_{\operatorname{\mathcal{C}}}(X)_{\ast }$ induces a bijective correspondence

\[ \xymatrix { \{ \text{Differential graded categories $\operatorname{\mathcal{C}}$ with $\operatorname{Ob}(\operatorname{\mathcal{C}}) = \{ X\} $} \} \ar [d]^{\sim } \\ \{ \text{Differential graded algebras} \} . } \]

Example 2.5.2.7. Let $B_{\bullet }\operatorname{Ch}(\operatorname{\mathbf{Z}})$ denote the classifying simplicial set of the monoidal category of chain complexes. For each nonnegative integer $n \geq 0$, we can use the analysis of Remark 2.5.1.18 to identify $n$-simplices of $B_{\bullet }\operatorname{Ch}(\operatorname{\mathbf{Z}})$ with differential graded categories $\operatorname{\mathcal{C}}$ satisfying $\operatorname{Ob}(\operatorname{\mathcal{C}}) = \{ 0, 1, \cdots , n \} $ and

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(i, j)_{\ast } = \begin{cases} \operatorname{\mathbf{Z}}[0] & \text{ if } i = j \\ 0 & \text{ if } i > j. \end{cases} \]

Definition 2.5.2.8 (Differential Graded Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be differential graded categories. A differential graded functor $F$ from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ consists of the following data:

  • For each object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, an object $F(X) \in \operatorname{Ob}(\operatorname{\mathcal{D}})$.

  • For each pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a chain map $F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )_{\ast }$.

These data are required to satisfy the following conditions:

  • For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the chain map

    \[ F_{X,X}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\ast } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(X) )_{\ast } \]

    carries the identity morphism $\operatorname{id}_{X}$ to the identity morphism $\operatorname{id}_{ F(X)}$.

  • For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$ and pair of morphisms $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{m}$, $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{n}$, we have $F_{X,Z}( g \circ f) = F_{Y,Z}(g) \circ F_{X,Y}(f)$.

We let $\operatorname{Cat}^{\operatorname{dg}}$ denote the category whose objects are (small) differential graded categories and whose morphisms are differential graded functors.

Remark 2.5.2.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be differential graded categories. Then differential graded functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ (in the sense of Definition 2.5.2.8) can be identified with $\operatorname{Ch}(\operatorname{\mathbf{Z}})$-enriched functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ (in the sense of Definition 2.1.7.10).