nLab internal hom of chain complexes

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.

Related concepts

• tensor product of chain complexes

References

• Samuel Eilenberg, G. Max Kelly, §IV.6 of: Closed Categories, in: Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 421-562 [doi:10.1007/978-3-642-99902-4]

Textbook accounts:

• Charles Weibel, An Introduction to Homological Algebra