Question about topological monoid maps

Question

Let Mon be the category of topological monoids. I am happy to work with the model structure mentioned here:

Model Structure/Homotopy Pushouts in topological monoids?.

I'm looking for a reference for the following statement, which I believe to be true. Suppose $X$ and $Y$ are topological monoids, with $X$ cofibrant, and $Y$ group-like.

Assertion: The map of spaces

$$ \text{monoid-maps}(X,Y) \to \text{maps}_\ast(BX,BY) $$

is a weak homotopy equivalence.

My guess is that one could prove this by induction, using the fact that $X$ is a retract of an object given by attaching free things. For example, here is a verification of the statement when $X = FU$, the free monoid on the points of a based space $U$. In this case,

$$ \text{monoid-maps}(FU,Y) = \text{maps}_\ast(U,Y) , $$

whereas $$ \text{maps}_*(BFU,BY) \simeq \text{maps}_\ast(\Sigma U,BY) $$ using, say James theorem $FU \simeq \Omega\Sigma U$. Since $Y$ is group-like, we have $Y \simeq \Omega BY$ and we get

$$ \text{maps}_*(\Sigma U,BY) \simeq \text{maps}_\ast(U,Y) $$ verifying the assertion in this special case.

More generally, it seems to me that if $X = \text{colim}(X_0 \leftarrow FA \to FB)$ where $(B,A)$ is a cofibration pair (with the colimit taken in topological monoids), and if the assertion is true for $X_0$ then it is also true for $X$ using the above and by noting that (i) function spaces convert pushouts in the domain to pullbacks and (ii) the classifying space functor preserves homotopy pushouts. This would then give the inductive step.

Added Later: I'd like to reiterate that I'm really looking for a decent reference.

Answer 1

EDIT 2: Sorry about the confusion, I will try to be careful now, and I'll put some comments at the end to clear up the business about adjoints etc. Here's what we're going to do.

We'd like to show that $$ \text{Map}_{\text{Mon}}(X, Y) \rightarrow \text{Map}_{Spaces_*}(BX, BY) $$ is an equivalence when $X$ is cofibrant and $Y$ is grouplike. The strategy will be to consider the string $$ \text{Map}_{\text{Mon}}(X,Y) \rightarrow \text{Map}_{\text{Spaces}_*}(BX,BY) \rightarrow \text{Map}_{\text{Mon}}(\Omega'BX, \Omega'BY) \rightarrow $$

$$ \text{Map}_{\text{Mon}}(X, \Omega'BY) \cong \text{Map}_{\text{Mon}}(X,Y) $$ where $\Omega'$ denotes the Moore loop space (strictly associative multiplication), and show that the composite is homotopic to the identity. This clearly reduces down to showing that the third map is a weak equivalence, since I may as well have started the string with the inverse of the last equivalence ($\Omega' BY$ and $Y$ are interchangeable).

So we want to show that $$ \text{Map}_{\text{Mon}}(\Omega'BX, \Omega'BY) \rightarrow \text{Map}_{\text{Mon}}(X, \Omega'BY) $$ is a weak equivalence when $X$ is cofibrant ($Y$ can be arbitrary).

We will prove this by showing that these two spaces represent the same functor on the homotopy category of spaces, the fundamental fact being that the natural map $\Omega'B Y \rightarrow \Omega'B\Omega'B Y$ is a weak equivalence.

Given a map $K \rightarrow \text{Map}_{\text{Mon}}(X, \Omega'BY)$ we get a map $K \rightarrow \text{Map}_{\text{Mon}}(\Omega'BX, \Omega'B\Omega'BY)$, and composing with the natural weak equivalence gives us a map $K \rightarrow \text{Map}_{\text{Mon}}(\Omega'BX, \Omega'BY)$. One can check that composition gives back the original map, up to homotopy. This proves surjectivity of

$$ \text{Map}_{h\text{Spaces}}(K, \text{Map}_{\text{Mon}}(\Omega'BX, \Omega'BY)) \rightarrow \text{Map}_{h\text{Spaces}}(K, \text{Map}_{\text{Mon}}(X, \Omega'BY)) $$ to get injectivity just note that we can recover the original value by applying $\Omega B'$ and using the weak equivalence again.

This completes the proof.

---

In my "second edition" I erroneously stated that there was a left adjoint to the inclusion of group-like topological monoids into monoids. Ricardo gracefully explains why this can't be true. However, it is a consequence of the above argument that there is a homotopical left adjoint, namely $\Omega'B$. (We've shown that the image of $\Omega'B$ is a homotopically reflective subcategory, and so all that's left is to note that the essential image is all group-like topological monoids.)

I've also erased the original answer, since it seems silly now that we have this one.

A reference for the argument I gave that basically showed that $\Omega'B$ was a localization functor can be found in Higher Topos Theory Proposition 5.2.7.4. Lots more love can be found throughout Higher Algebra; one can, for example, deduce a similar statement about $\mathbb{E}_1$-spaces in general, which actually implies this result since the inclusion of monoids into $\mathbb{E}_1$-spaces is an equivalence of $\infty$-categories.

Hopefully this is all correct now! Let me know if there's still errors.