[Math] Is assigning the endomorphism object in some sense functorial

ct.category-theorymonoidal-categoriesreference-request

Let $\mathcal V$ be a monoidal category and let $\mathcal C$ be a $\mathcal V$-category. Let's denote the $\mathcal V$-valued hom-functor $[-,-]$. Now for every object $X\in\mathcal C$ we have it's endomorphism object $\mathcal End(X):=[X,X]$ – it is actually a monoid in $\mathcal V$. Can the assignment
$$X\mapsto \mathcal End(X)$$
be considered functorial in some way? Is there a language that captures the relations between $\mathcal End(X)$ when $X$ varies? What happens on the level of module categories $\mathcal V^{\mathcal End(X)}$?

I've allready thought about this a bit but i don't want to reinvent the wheel. [Edit] So here's what i've been thinking of:

For every object $X\in\mathcal C$ we get a functor $[X,-]:\mathcal C\to\mathcal End(X)-\operatorname{mod}$. I think these functors are connected in a vaguely functorial way – hopefully by adjunctions between the module categories (adjunction in the 2-category of categories under $\mathcal C$).

My idea:

  1. Let $f:X\to Y$ be a morphism in $\mathcal C_0$. On the one hand $[Y,X]$ is a bimodule from $\mathcal End(Y)$ to $\mathcal End(X)$. On the other hand $[Y,X]$ becomes a bimodule in the other direction – from $\mathcal End(X)$ $\mathcal End(Y)$ – by pre- and postcomposition with $f$ i.e. pulling back the module structure along $[f,f]:[Y,X]\to[X,Y]$. So assuming $\mathcal V$ is nice enough we have two functors $[Y,X]\otimes_{\mathcal End(X)}$ and $[Y,X]\otimes_{\mathcal End(Y)}$. However i can think of no canditate for a unit of a supposed adjunction between these two.
  2. As in (2) $[Y,X]$ also becomes a semigroup object in $\mathcal V$ that has a (left/right) unit – and thus is a monoid – precisely when $f$ has an (left/right) inverse.

My vague guess is that the framework where this question could be handled is that of extranatural transformations.

Best Answer

It's hard to answer this question on this abstract level in any other way that by saying "no, it's not a functor". Of course it is a bifunctor $\mathcal{C}^{op} \times \mathcal{C} \to \mathcal{V}$, and the language of ends and coends deals with such functors (Mac Lane's Categories for the working mathematician, or for the enriched version, Kelly's Basic concepts of enriched category theory).

You'll get functoriality if you restrict your category $\mathcal{C}$ to those morphisms $f\colon X \to Y$ such that $[Y,X] \to [X,X]$ is an iso (then the endomorphisms become a covariant functor) or those such that $[X,X] \to [X,Y]$ is an iso (then it's contravariant). The latter is related to the concept of centric maps in topology. It has been used to study realizations of diagrams in the homotopy category (Dwyer-Kan, Centric maps and realization of diagrams in the homotopy category, Proceedings of the AMS 114(2), 1992).

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