Brown representability in slice category

Question

Brown's representability theorem gives us a very nice set of conditions to check that a (contravariant) functor $Hot^{op}\rightarrow Set$ is representable. Choose an object $X$ in $Hot$. Then it is seems natural to ask whether or not an analogue of the Brown representability theorem is true for the slice category, i.e. if there exists a nice set of conditions to check whether or not a contravariant functor $$F:(Hot/_X)^{op} \rightarrow Set$$ is representable. Is there such an analogue? I've searched online, but couldn't find an comments on the subject. My hope would be that this functor is representable if $F$ respects coproducts and sends homotopy pushouts to weak pullbacks. One way I would hope to recover such a criterion is to consider the functor $$Hot\rightarrow Hot/_X, Y\mapsto (Y\rightarrow \{*\})$$ which by composition gives us a functor $Hot^{op}\rightarrow Set$, for which we know we can apply Brown's Theorem. I don't know however if this is enough to test representability of the functor $F$.

Answer 1

Yes, sliced homotopy categories of pointed connected spaces satisfy Brown representability. (We had better be using $Hot$ to denote the homotopy category of pointed connected spaces, as Brown representability is false for the homotopy category of unbased or non-connected spaces.)

The abstract version of Brown’s representability theorem has the following requirements on a category $C$: it must have coproducts, weak pushouts and a “compact generating set”. This is a set of objects, maps out of which jointly detect isomorphisms and commute with some choice of sequential weak colimits. Note this is more general than Neeman’s version for compactly generated triangulated categories, as $C$ need not be triangulated and indeed $Hot$ is not. That said, Neeman has proved vastly more general theorems in the well-generated and perfectly generated triangulated case which cannot be extended unstably. (At least not yet!)

As is common in these situations, the coproducts and weak pushouts in the slice category are no problem, while weak sequential colimits can be chosen, as in $Hot$, as the homotopy colimit/Milnor’s infinite mapping telescope.

As to the compact generators, we may take the set of all spheres over $X$. This is analogous to results on, say, local presentability of slice categories that may be familiar, but let’s go ahead and see the proof. Compactness is immediate from the case in $Hot$. For generation, if $f:A\to B$ is a map over $X$, then a map $S^n \to A$ is killed by $f$ equivalently whether we’re over $X$ or not, while if $S^n\to B$ fails to factor through $A$ in $Hot$, then it certainly fails to do factor over $X$. Thus the set of spheres over $X$ see $f$ as an isomorphism over $X$ if and only if the set of spheres sees $f$ as an isomorphism in $Hot$, if and only if $f$ is really as isomorphism.

Answer 2

Edgar Brown wrote two papers on this topic: one in the Annals in 1962 that focused on the category of topological spaces, and a second one "Abstract homotopy theory" Trans. Amer. Math. Soc. 119 (1965).

His second paper is very axiomatic, and I believe your situation is easily checked to satisfy his properties. (Check this!)

This paper obviously predated Quillen's work on model categories (it likely partially inspired Quillen), so of course he doesn't use that language. I confess that I have always been a bit disturbed by the number of papers on Brown Representability written by authors who show no indication that they have ever looked at the original papers. Yes, people have written about more general versions (and less general, when they assume a triangulated category!) but I encourage folks to use their library resources to look at Brown's own work.