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Let $J$ be a collection of small categories (to be thought of as diagrams in a category). Let $C$ be a small category with all $J$-limits (i.e. for every $J_0 \in J$ and every functor $F:J_0\rightarrow C$, $\lim F$ exists in $C$). Let $D$ be any category with all $J$-limits (such as Set).

Consider the category $[C,D]_J$ of functors from $C$ to $D$ which preserve all $J$-limits, with natural transformations as morphisms.

Question: What conditions on $J$ must hold in order for $[C,D]_J$ to be a reflective subcategory of $[C,D]$ (the category of all functors from $C$ to $D$)?

I ran into this question when investigating the case of two collections of diagrams $J \subseteq J'$, so that $[C,D]_{J'} \subseteq [C,D]_J$ (assuming $C$ and $D$ have all $J'$-limits). Since all these categories are full subcategories, it turns out that $[C,D]_{J'}$ is a reflective subcategory of $[C,D]_{J}$ if $[C,D]_{J'}$ is a reflective subcategory of $[C,D]$. (if $U_2$ is fully faithful, and $U_3 = U_2 \circ U_1$ has a left adjoint $F_3$, then $U_1$ has a left adjoint: $F_3 \circ U_2$.

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When $D$ has copowers, the reflectivity of continuous functors can be seen as a case of the orthogonal subcategory problem. This problem asks whether the full subcategory $\mathcal{X}_\Gamma$ of a category $\mathcal{X}$ on the objects orthogonal to a set of morphisms $\Gamma$ is reflective. There are many hypotheses that ensure a positive answer, a famous one being the condition that $\mathcal{X}$ is a locally presentable category and $\Gamma$ is a small set.

The problem so reduces because we can show that a functor $G : C \to D$ preserves the limit of $F : J_0 \to C$ if and only if it is orthogonal in the functor category $[C,D]$ to the morphisms $$\text{colim}_j C(Fj,-)\cdot d \longrightarrow C(\text{lim}_j Fj,-)\cdot d$$ for each $d\in D$. Letting $\Gamma$ be the set of these morphisms for all $F \in J$, we have therefore that the full subcategory $[C,D]_J$ of the $J$-continuous functors is the orthogonal subcategory $[C,D]_\Gamma$.

Hence (keeping your assumptions of smallness) the full subcategory $[C,D]_J$ is reflective in $[C,D]$ if $D$ is a locally presentable category and $J$ is a small set.

The classic paper on this subject is

Freyd, P. J.; Kelly, G. M. Categories of continuous functors. I. J. Pure Appl. Algebra 2 (1972), 169--191.

where more general hypotheses are considered, and see

Kelly, G. M. A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. Bull. Austral. Math. Soc. 22 (1980), no. 1, 1--83.

(in particular section 12) for a more complete and cleaner treatment.

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  • $\begingroup$ N.B. I have taken $J$ to be a set of functors $F : J_0 \to C$ rather than a set of categories $J_0$ as in your question, as it is the size of the former that needs to be bounded when constructing the reflection. $\endgroup$ – Alexander Campbell Feb 21 '16 at 13:31
  • $\begingroup$ what can be said about subcategories of cocontinuous functors? $\endgroup$ – Ivan Di Liberti Mar 12 '18 at 17:31
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This is heuristics only, not a rigorous argument, but I would strongly expect that if $D$ is sufficiently co-complete, then very little restriction on $J$ is needed.

The “moral adjoint functor theorem” says that if a functor preserves all limits, then it is probably a right adjoint. The inclusion functor $[C,D]_J \to [C,D]$ creates (hence reflects) all or most limits that exist in $[C,D]$, since they’re usually pointwise — always pointwise, if $D$ has a 1 and sufficient coproducts, since then we can run a version of the Yoneda lemma — and limits commute with limits.

So assuming some co-completeness for $D$ and smallness for either $J$ or $C$ (I guess something like “$J$ or $C$ is bounded by some $\kappa$, such that $D$ has all colimits of size $< \kappa$” should suffice), then one of the genuine adjoint functor theorems should imply that the inclusion of $[C,D]_J$ has a left adjoint.

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