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This is, in some sense, the homotopy version of this question.)

If $C$ is a category with $iC$ the subcategory of isomorphisms, there is a functor $X \mapsto End(X)$ from $iC$ to the category of monoids. Moreover, this extends to a result if $C$ has an enrichment in a monoidal category $\cal V$, and this changes the target to the category of $\cal V$-monoids.

My question is about the homotopical analogue of this. For example, if $C$ is a simplicial model category, Dwyer and Kan's sequence of papers on the simplicial localization show that there is a functor which captures the homotopy type of the monoid $End(X)$ properly and which is functorial in weak equivalences. (For a weak equivalence $X \to Y$, there are natural equivalences $End(X) \to Map(X,Y) \leftarrow End(Y)$, but these do not induce a map in the homotopy category of monoids.)

It is natural to ask this question in other circumstances. For instance, I was always taught that one of the operating principles of the theory of operads is that that algebra structures can be made invariant under weak equivalence. However, actually showing this seems to be surprisingly annoying, and requires one to make use of auxiliary constructions like the endomorphism operad of a map $f:X \to Y$. If this map is an acyclic cofibration or an acyclic fibration in an appropriate model category, this map has good homotopical properties. In general it seems to require constructing a factorization, and proving that we have a well-defined functor - from isomorphisms in the homotopy category of $C$ to the homotopy category of operads, or more generally some homotopy category of $\cal V$-monoids - seems to be ugly.

Is there a more direct proof that homotopy endomorphism monoids are functorial?

There are definitely obstructions to any direct constructions. For example, there is a quasi-isomorphism of chain complexes $$ (\mathbb Z \hookrightarrow \mathbb Z \oplus \mathbb Z) \to (\mathbb Z^2 \hookrightarrow \mathbb Z^2 \oplus \mathbb Z) $$ such that there is simply no map of differential graded algebras in either direction between the associated endomorphism objects.

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I'd say the standard proof, via endomorphisms of a map, is both cheap and direct – Fernando Muro May 19 '12 at 13:57
@Fernando: To me, this requires using $End(f)$ and the endomorphism operad of a composable pair $g \circ f$ of maps. Perhaps I am missing something, but the "usual proof" shows that the maps from $End(f)$ to $End(x)$ and $End(y)$ are weak equivalences, and this seems to require additional assumptions on the map $f$. If I am missing something, perhaps you can sketch it out? – Tyler Lawson May 19 '12 at 14:20
You have to use $End(f)$ but not $End(g\circ f)$, if I remember well. About the additional conditions, they are necessary to start the proof, but the one uses factorizations, any weak equivalence factors through a trivial cofibration and a trivial fibration. – Fernando Muro May 19 '12 at 14:31
Sure, but that requires making a choice of factorization, and so one first has to prove that the associated map in the homotopy category of operads is independent of the choice. – Tyler Lawson May 19 '12 at 14:52
I'm afraid that I don't follow. – Tyler Lawson May 19 '12 at 16:00

What about simple category theory?

Let $C_{X,Y}$ denotes the full simplicial subcategory of $C$ whose objects are $X$ and $Y$. If you see $End(X)$ as a simplicial category with one object, then you have a full embedding $End(X)\to C_{X,Y}$. The latter is a Dwyer-Kan equivalence if there exists a weak equivalence $X\to Y$, in which case the other full embedding $End(Y)\to C_{X,Y}$ is a Dwyer-Kan equivalence as well. In other words, $End(X)$ and $End(Y)$ are canonically isomorphic in the homotopy category of simplicial categories. Note that cofibrant/fibrant replacement functors of the Bergner-Dwyer-Kan model structure can be chosen so that they don't affect the objects, so that, at the end, we can promote this into an isomorphism in the homotopy category of simplicial monoids.

The same proof applies to a dg category $C$ (or to any $V$-enriched category $C$, where $V$ is a symmetric monoidal model category such that the category of $V$-enriched categories admits a Dwyer-Kan style model category structure).

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So Dundas has a paper where he works out the V-enriched localization, and it seems to rely on V being a model category where the monoidal structure commutes with colimits in each variable. However, it's not clear whether this is compatible with a V-enriched model category structure (in the Dwyer-Kan story they use the hammock localization to prove this). Do you happen to know what kinds of enriching categories might suffice? – Tyler Lawson May 20 '12 at 1:21
In any case, as I said, simplicial sets, complexes of modules over a commutative ring, and even categories enriched in spectra are eligible. For categories enriched in complexes or in spectra, there are these papers of Tabuada: "A Quillen model structure on the category of dg categories" C. R. Math. Acad. Sci. Paris, 340 (2005) (arXiv:math/0407338) and "Homotopy theory of spectral categories", Adv. Math., 221 (2009) (arXiv:0801.4524). For general sufficient conditions, there is Theorem A.3.2.24 in Lurie's book, as well as a result in a recent paper by Berger and Moerdijk (arXiv:1201.2134). – Denis-Charles Cisinski May 20 '12 at 2:14

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