(also nonabelian homological algebra)
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In higher category theory
A natural internal hom of chain complexes that makes the category of chain complexes into a closed monoidal category.
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.
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
(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
This defines a functor
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.
The collection of cycles of the internal hom $[X,Y]$ in degree 0 coincides with the external hom functor
The chain homology of the internal hom $[X,Y]$ in degree 0 coincides with the homotopy classes of chain maps.
By Definition the 0-cycles in $[X,Y]$ are collections of morphisms $\{f_k \colon X_k \to Y_k\}$ such that
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
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.
Textbook accounts:
Last revised on August 23, 2023 at 08:45:39. See the history of this page for a list of all contributions to it.