An additive functor between abelian categories $F: \mathscr{C} \to \mathscr{D}$ induces a functor on categories of chain complexes $F: \mathscr{C}^\bullet \to \mathscr{D}^\bullet$. The internal hom functors $$\operatorname{Hom}_{\mathscr{C}}^\bullet: \mathscr{C}^{\bullet,\mathrm{op}} \times \mathscr{C}^\bullet \to \mathsf{Ab}^\bullet \qquad \text{ and } \qquad \operatorname{Hom}_{\mathscr{D}}^\bullet: \mathscr{D}^{\bullet,\mathrm{op}} \times \mathscr{D}^\bullet \to \mathsf{Ab}^\bullet$$ play nicely with $F$ in the following way: $F$ induces a map in $\mathsf{Ab}^\bullet$ $$\operatorname{Hom}_{\mathscr{C}}^\bullet(A^\bullet, B^\bullet) \to \operatorname{Hom}_{\mathscr{D}}^\bullet(F(A^\bullet), F(B^\bullet))$$ which is natural in $A^\bullet$ and $B^\bullet$.

What is the correct categorical way to state this interaction between these two functors? I could brute force it and just say that there's some natural transformation between properly defined bifunctors, but I'm wondering if this is a common situation with a name, or if there's a familiar phenomenon going on here that I'm missing.

One Possibility: The category $\operatorname{\mathsf{Gr}}\mathscr{C}$ of graded objects of $\mathscr{C}$ is enriched over $\operatorname{\mathsf{Gr}}\mathsf{Ab}$. The above property is mostly just saying that the functor $\operatorname{\mathsf{Gr}}\mathscr{C} \to \operatorname{\mathsf{Gr}}\mathscr{D}$ induced by $F$ is $\operatorname{\mathsf{Gr}}\mathsf{Ab}$ enriched, and translating that to the category $\mathscr{C}^\bullet$ where there is a bit more structure going on (and where $\operatorname{Hom}^\bullet_{\mathscr{C}^\bullet}$ becomes different than $\operatorname{Hom}_{\mathscr{C}^\bullet}$, unlike in $\operatorname{\mathsf{Gr}}\mathscr{C}$, where internal hom is just hom). But, there's one extra fact, which is that $$\operatorname{Hom}_{\mathscr{C}}^\bullet(A^\bullet, B^\bullet) \to \operatorname{Hom}_{\mathscr{D}}^\bullet(F(A^\bullet), F(B^\bullet))$$ is actually a chain map. I'm not sure how to fit that into the picture.

  • $\begingroup$ $\mathcal C^\bullet$ isn't just enriched over graded abelian groups, it's enriched over chain complexes! These are monoidal for a tensor product restricting to that in graded objects. $\endgroup$ – Kevin Carlson Jun 28 '16 at 18:07
  • $\begingroup$ @KevinCarlson What is the correct way to express the property that a cycle $f:B^\bullet \to C^\bullet$ induces a cycle $\operatorname{Hom}^\bullet(A^\bullet, B^\bullet) \to \operatorname{Hom}^\bullet(A^\bullet, C^\bullet)$? Also, when you say "these are monoidal" what are "these"? Not so familiar with monoidal categories yet. $\endgroup$ – Eric Auld Jun 29 '16 at 3:42

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