(also nonabelian homological algebra)
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 $R$ 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 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 1 the 0-cycles in $[X,Y]$ are collections of morphisms $\{f_k : X_k \to Y_k\}$ 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.
A standard textbook account is