Is assigning the endomorphism object in some sense functorial?

Question

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.

Answer 1

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).

Answer 2

This reminds me of the discussion on pages 264-265 of Eilenberg and Mac Lane's General Theory of Natural Equivalences of the "degree of invariance" of various constructions on the category of groups; see in particular the table on the top of page 265.

For any $\mathcal{V}$-category $\mathcal{C}$, End defines a functor from the underlying groupoid iso$\mathcal{C}_0$ of $\mathcal{C}_0$ to the category of monoids in $\mathcal{V}$, where an isomorphism $f \in \mathcal{C}_0$ acts by conjugation $f \circ - \circ f^{-1}$.

One way to understand this construction is as a restriction of the hom bifunctor $\mathcal{C}_0^{\text{op}} \times \mathcal{C}_0 \to \mathcal{V}$ using the diagonal and the functor $(-)^{-1} \colon \text{iso}\mathcal{C}_0^{\text{op}} \to \mathcal{C}_0$. Depending on which $\mathcal{C}$ and which $\mathcal{V}$ you have in mind, there might be other (partially-defined) functors $\mathcal{C}^{\text{op}} \to \mathcal{C}$ (e.g., "transpose" or "adjoint") though they might give rise to morphisms in $\mathcal{V}$ that are not necessarily monoid homomorphisms.

Answer 3

One way to get a handle on the fact that $X\mapsto [X,X]$ is of 'mixed variance' was invented by Dana Scott around 1970. He showed how to solve the 'domain equation' $X\cong [X,X]$ to obtain a model of untyped lambda calculus.

The basic idea is to replace the category $\cal C$ by a category that has the same objects, but has as arrows 'embedding-projection pairs' $(e,p)$ which are specified in an order-enriched category as satisfying $pe=id$ and $ep\le id$. Now an arrow $X\to Y$ has a $X\to Y$ component and a $Y\to X$ component. In a nutshell, this is what allows us to deal with mixed variance.

In more detail.

1. General Method (covariant functors): Given an endofunctor $F$ on an category $\cal C$, one constructs solutions $X$ satisfying $X\cong FX$ via iterating $F$, either starting from the initial object 0 or from the final object 1. In a good situation, these iterations will give us the initial $F$-algebra as a colimit iterating from 0 and the final $F$-coalgebra as a limit iterating from 1. Both the initial algebra and the final coalgebra solve the equation $X\cong FX$ (this is known as Lambek's Lemma).

2. Mixed Variance: As Scott showed, in a suitable order-enriched category of embedding-projection pairs, on can define $X\mapsto[X,X]$ as a covariant functor. Moreover, the construction above leads to a coincidence of the initial algebra and the final coalgebra. This gives a solution for $X\cong [X,X]$.

3. Later Peter Freyd axiomatised this setting under the name of algebraically compact category.

A good first reference is available https://ncatlab.org/nlab/show/algebraically+compact+category but the current version ignores the earlier history and has no references to Scott's original papers and their importance for Domain Theory and the Semantics of Programming Languages.