(Pro-)representable functors and full subcategories in homotopy theory

Question

$\DeclareMathOperator\Ab{Ab}\DeclareMathOperator\Ho{Ho}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Hotc{Hotc}\DeclareMathOperator\Sm{Sm}$Let $\mathcal{C}\overset{\iota}{\longrightarrow} \mathcal{D}$ be the inclusion of a full subcategory. Consider a functor $$F:\mathcal{C}^{op}\rightarrow \Ab.$$ I've often seen examples where this functor might not be representable, but represented by a functor in the larger category, i.e. that there exists a $D\in \mathcal{D}$ such that $$F\cong \Hom_{\mathcal{D}}(-,D).$$ This is of course the idea behind say Schlessinger's criterion, which tells us when a deformation functor is pro-representable, or it is also the philosophy of stacks (extend the category of schemes with stacks to get representable functors).

My question is if there exists a nice set of criteria to study the representabilitly of $F$ in $\mathcal{D}$? A naïve approach I had was to Kan extend the functor to $\mathcal{D}$ and then to study the representability of the Kan extension. However this seems to be a bad idea in general.

A concrete situation I've encountered this is the following: consider the full subcategory $\Ho(\Sm)\subset \Hotc$ of smooth manifolds in the homotopy category of pointed CW-complexes. In the larger category, checking whether a functor is representable is "easy" by Brown representability. Consider for instance $$F:\Ho(\Sm)^{\text{op}}\rightarrow \Ab,\quad X\mapsto H^n(X,F)$$ the singular cohomology of degree $n$. In $\Hotc$, it is represented by the $n$-th Eilenberg-Maclane space, which is a pointed CW-complex but not a smooth manifold. Without using the fact that the singular cohomology on $\Ho(\Sm)$ is the restriction functor of singular cohomology on topological spaces, can we show that it is representable?

This illustrates also why it is not a good idea to study this via Kan extension, as the Kan extension of singular cohomology does not agree with singular cohomology.

Answer 1

This is a partial answer. Broadly speaking, representability theorems break down into two types. In both cases, the functor $F$ has to satisfy some exactness condition. For Freyd type theorems, $F$ must satisfy some set-theoretic condition such as accessibility or a solution set condition. For Brown type theorems, the domain category must satisfy some set-theoretic conditions, such as local presentability. This is nicely explained in a recent paper by Blanc and Chorny. I think you'd be interested in this paper. Like much of Chorny's work, it makes use of small functors, meaning functors that are the left Kan extension of some functor whose domain is a small category. Equivalently, a small functor is a small colimit of representable functors.

Brown type theorems further break down into two types. Cohomological Brown representability says, essentially, "any contravariant cohomological functor $F:\mathcal{T} \to Ab$ that takes coproducts to products is representable as $Hom(-,c)$. Homological Brown representability is about covariant functors $F:\mathcal{T}\to Ab$ being representable as $Hom(c,-)$, but more conditions are required, very much related to your question. For example, in Brown's original theorem, homological functors from the category of finite spectra to $Ab$ are represented by objects $c$ that are (not necessarily finite) spectra. This result was generalized in 1992 by Neeman.

You can also study representability for cohomological functors $F: \mathcal{T}^{op}_0 \to Ab$ defined on the full subcategory of small objects in $\mathcal{T}$. An excellent reference is Neeman's book, or this 2005 paper by Rosicky, which carries out a nice generalization that gets away from the need for triangulated categories and into the land of combinatorial model categories. Again, small functors play a crucial role. So this essentially answers your questions for situations where $\mathcal{C}$ is the full subcategory of small objects in $\mathcal{D}$, or where $\mathcal{D}$ is generated from $\mathcal{C}$ under filtered colimits.

Lastly, there are plenty of counterexamples where Brown representability fails, meaning the hypotheses are truly needed. Three such examples are mentioned in the Blanc and Chorny paper above, others are mentioned in Neeman's book, and another is in the paper Brown representability does not come for free by Casacuberta and Neeman.