(also nonabelian homological algebra)
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category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A natural internal hom of chain complexes that makes the category of chain complexes into a closed monoidal category.
Let be a commutative ring and Mod the category of modules over . Write for the category of chain complexes of -modules.
For any two objects, define a chain complex to have components
(the collection of degree- maps between the underlying graded modules) and whose differential is defined on homogeneously graded elements by
This defines a functor
The collection of cycles of the internal hom in degree 0 coincides with the external hom functor
The chain homology of the internal hom in degree 0 coincides with the homotopy classes of chain maps.
By Definition the 0-cycles in are collections of morphisms such that
This is precisely the condition for 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 . This are precisely the null homotopies.
A standard textbook account is
Last revised on August 26, 2012 at 23:34:01. See the history of this page for a list of all contributions to it.