MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose we have a span of categories $C \overset{F}{\hookleftarrow} D \overset{G}{\rightarrow} E$, where $F$ is a subcategory embedding. We can lift these normal functors to profunctors $\hat F$ and $\hat G$ and compose the "formal" adjoint $\hat F^\dagger$ with $\hat G$ to obtain a profunctor $\hat G \hat F^\dagger : C \nrightarrow E$. To what extent could one think of this as a partial functor, and what nice behaviours could it inherit from $F$ and $G$? For instance, if $F$ reflects products and $G$ preserves them, does $\hat G \hat F^\dagger$ preserve them where ever it is defined?

share|cite|improve this question

One thing along these lines that you can say is that if D has, and G preserves, finite limits (or more generally is flat), then so does $\hat{G}$ considered as a cocontinuous functor $[D^{op},Set] \to [E^{op},Set]$. Since $\hat{F}^\dagger : [C^{op},Set] \to [D^{op},Set]$ is just precomposition with F, it preserves all limits and colimits; thus $\hat{G} \hat{F}^\dagger$ preserves finite limits as soon as G does, without any hypothesis on F. I don't know whether this can be extended to other kinds of limits.

Regarding the more general question of whether $\hat{G} \hat{F}^\dagger$ could be considered a "partial functor," another way to describe it is as the left Kan extension of the composite $D \overset{G}{\to} E \hookrightarrow [E^{op},Set]$ along F. If F is fully faithful, then such an extension is an honest extension, i.e. it restricts back along F to the original functor. So one could think of it as obtained by extending G to objects not in D in the most universal way possible: it maps an object $c\in C$ to the formal colimit (viewing $[E^{op},Set]$ as the free cocompletion of E) over all approximations to c by objects of D.

On the other hand, every profunctor can be obtained as a composite $\hat{G} \hat{F}^\dagger$ for some functors G and F (not necessarily an embedding): let the intermediate category D be the two-sided discrete fibration corresponding to that profunctor. An arbitrary profunctor can be thought of as a "generalized functor," but usually not specifically a "partial functor." However, perhaps faithfulness, or full-and-faithfulness, of F implies some properties of the resulting profunctor which makes it seem more like a "partial functor."

share|cite|improve this answer
I guess this analogy goes through by replacing the notion of "subset where the image of a function is defined" with "subcategory where the image of a profunctor is representable." Interesting that a yes or no question is replaced by a question of approximations. Are there interesting ways one can gauge how good such approximations are? The examples I'm thinking of are categories of matrices over various rigs, especially Mat(C) <--< Mat(R+) -->> Rel. – Aleks Kissinger Aug 19 '10 at 18:15
Well, there are various sorts of "almost-representability" that one can consider, often phrased in terms of a presheaf being a certain sort of limit or colimit of representables. Replacing of yes/no questions by more refined information is quite common in categorification. – Mike Shulman Aug 20 '10 at 3:13

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.