# nLab internal hom of chain complexes

Contents

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## Idea

A natural internal hom of chain complexes that makes the category of chain complexes into a closed monoidal category.

## Definition

Let $R$ be a commutative ring and $\mathcal{A} = R$Mod the category of modules over $R$. Write $Ch_\bullet(\mathcal{A})$ for the category of chain complexes of $R$-modules.

###### Definition

For $X,Y \in Ch_\bullet(\mathcal{A})$ any two objects, define a chain complex $[X,Y] \in Ch_\bullet(\mathcal{A})$ to have components

$[X,Y]_n \,\coloneqq\, \prod_{i \in \mathbb{Z}} Hom_{R Mod}(X_i, Y_{i+n})$

(the collection of degree-$n$ maps between the underlying graded modules) and whose differential is defined on homogeneously graded elements $f \in [X,Y]_n$ by

$d f \,\coloneqq\, d_Y \circ f - (-1)^{n} f \circ d_X \,.$

This defines a functor

$[-,-] \;\colon\; Ch_\bullet(\mathcal{A})^{op} \times Ch_\bullet(\mathcal{A}) \longrightarrow Ch_\bullet(\mathcal{A}) \,.$

## Properties

###### Proposition

The internal hom $[-,-]$ (Def. ) together with the tensor product of chain complexes $(\text{-})\otimes (\text{-})$ endow $Ch_\bullet(\mathcal{A})$ with the structure of a closed monoidal category.

###### Proposition

The collection of cycles of the internal hom $[X,Y]$ in degree 0 coincides with the external hom functor

$Z_0\big([X,Y]\big) \,\simeq\, Hom_{Ch_\bullet}(X,Y) \,.$

The chain homology of the internal hom $[X,Y]$ in degree 0 coincides with the homotopy classes of chain maps.

###### Proof

By Definition the 0-cycles in $[X,Y]$ are collections of morphisms $\{f_k \colon X_k \to Y_k\}$ such that

$f_{k+1} \circ d_X \;=\; d_Y \circ f_k \,.$

This is precisely the condition for $f$ to be a chain map.

Similarly, the boundaries in degree 0 are precisely the collections of morphisms of the form

$\lambda_{k+1} \circ d_X + d_Y \circ \lambda_k$

for a collection of maps $\{\lambda_k : X_k \to Y_{k+1}\}$. This are precisely the null homotopies.

From the remark at tensor product of chain complexes we have that the canonical forgetful functor $U_{Ch} \coloneqq Ch_\bullet(\mathcal{A})(I_{Ch},-) \colon Ch_\bullet(\mathcal{A}) \to \mathcal{A}$ takes a chain complex to its 0-cycles.

Thus the description of the 0-cycles in the above proposition is equivalent to the statement $U_{Ch}([-,-]) \cong Ch_\bullet(\mathcal{A})(-,-)$, which is true in any closed category.

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