In Jacob Lurie's book Higher Algebra, for an object $M$ of a monoidal $\infty$-category $\mathcal{C}$, he constructs a category $\mathcal{C}[M]$ which can be thought of as "maps in $\mathcal{C}$ of the form $A\otimes M\to M$ with associated coherence data". $\mathcal{C}[M]$ is shown to be monoidal, and algebra objects of this category are shown to be precisely the algebras which coherently act on $M$ (though one algebra that acts on $M$ in two different ways will be thought of as two different objects of $\mathcal{C}[M]$). Moreover, he shows that if $\mathcal{C}[M]$ has a final object, which we'll denote $End(M)$, this object is an algebra object of $\mathcal{C}$ and $M$ is an object of $LMod_{End(M)}$. Thus any algebra that acts on $M$ admits a morphism to $End(M)$.

I'm in particular thinking about the quasicategory of $\infty$-groupoids, $\mathcal{C}=Top$, so $M$ and $End(M)$ and everything else will just be spaces for the rest of this. We have another more down-to-earth notion of "endomorphisms of $M$." That is, the space of morphisms $Map_{Top}(M,M)$. This is a monoid object of $Top$ whose monoid structure is given by composition. Moreover there is an evaluation map $ev:Map_{Top}(M,M)\times M\to M$ so there is a corresponding object of $Top[M]$. Is this object clearly final in $Top[M]$?

In other words, if $M$ is an object of the quasicategory $LMod_A$ for $A$ some loop space in $Top$, is there induced a map $A\to Map_{Top}(M,M)$? Certainly given a map of topological spaces $A\times M\to M$ one can produce, by adjunction, a map $A\to Map_{Top}(M,M)$, but it's not clear to me that this is necessarily a map of algebras or that it carries all the necessary structure (though I suspect it does). In other words this would require a more complex kind of adjunction than just the one for tensor/hom.

qua algebrasrequires more than just the action $A \times M \to M$, but also the associativity data of that action. Does that work here? $\endgroup$ – Theo Johnson-Freyd Jun 21 '15 at 23:55