Compatibility of functors with limits
I have the sensation that I'm about to ask a question to which half the readers of this list will be able to see an answer immediately. Unfortunately, I'm one of the other half. What should it mean for a functor to "respect limits"? Consider the following informal definition: a functor respects limits if given any diagram in the domain category, the limit of the image of the diagram is no bigger than it needs to be. Formally, let F: A ---> B be a functor, where B is a category with (for sake of argument) all small limits and colimits. Let I be a small category and D: I ---> A a diagram in A; write Cone(D) for the category of cones on D in A, write Cone(FD) for the category of cones on FD in B, and write F_*: Cone(D) ---> Cone(FD) for the induced functor. Then F can be said to "respect limits for D" if the colimit of F_* is the terminal object of Cone(FD) (that is, the limit cone on FD). * Example: if D has a limit in A then the limit is a terminal object of Cone(D), so F respects limits for D if and only if it preserves the limit in the usual sense. * Example: let B = Set and let A be the category consisting of a pair of parallel arrows; a functor F: A ---> B consists of sets and functions sigma, tau: F_0 ---> F_1. The condition that F respects pullbacks says that sigma and tau are monic and that the images of sigma and tau are disjoint. The thought behind "no bigger than it needs to be" (a very approximate description, I know) is that if we have a cone on D with vertex v then there's an induced map from F(v) to lim(FD), which in some sense places a "lower bound" on lim(FD): e.g. if B = Set and F(v) is nonempty then lim(FD) is nonempty. For F to respect limits for D means that lim(FD) is built up freely from these F(v)s. So the question is: is this notion of "respecting limits" well-known or well-understood? Is there, for instance, some way of rephrasing it that brings it into more familiar territory? Thanks very much, Tom
A week ago I asked this list a question, thinking that lots of people would know the answer. Either I was wrong or those who know are keeping it to themselves, as I didn't get any answers at all. But I now understand the issue better than before, so I'd like to try re-asking my question in a different way and see if that elicits a response. I'm trying to understand a certain notion of compatibility of a functor with limits. There are of course several well-known such notions: preserves, reflects, creates. I called mine (provisionally) "respecting" limits. It is close to preservation, but not quite the same; it seems more constructive and perhaps (dare I say it?) more natural. Preservation is all very well when the domain category has all limits, or all limits of whatever type we're concerned with. But otherwise, it seems a bit suspect. For let F: A ---> B be a functor; preservation says that given a diagram D: I ---> A in A, - if D *does* admit a limit cone, then the image of that cone under F is also a limit cone, - if D *doesn't* admit a limit cone, then... well, nothing. "Respect", on the other hand, says the same as preservation in the first case, but also says something in the second case. (So respect is in general stronger than preservation.) Last time I gave a definition of respect in terms of categories of cones; that definition is appended to this mail. Here's a different way to put it: F "respects limits for D" if the canonical map \int^a (\int_i A(a, Di)) \times Fa ----> \int_i FDi is an isomorphism. Here \int^a denotes coend over a in A, \int_i is limit over i in I, and I'm working under the assumption that these (co)limits exist in the codomain category B. Now, if D does have a limit in A then the left-hand side is \int^a A(a, \int_i Di) \times Fa which by density is just F \int_i Di; so, as claimed, respect is the same as preservation in the case where the limit exists. My question was whether anyone understood "respect of limits" well, or could shed any light on it. It seems to me that, as well as being just the right thing in certain examples I've been considering, it's a very natural concept. Tom The definition of respect from last time:
Let F: A ---> B be a functor, where B is a category with (for sake of argument) all small limits and colimits. Let I be a small category and D: I ---> A a diagram in A; write Cone(D) for the category of cones on/into D in A, write Cone(FD) for the category of cones on FD in B, and write
F_*: Cone(D) ---> Cone(FD)
for the induced functor. Then F can be said to "respect limits for D" if the colimit of F_* is the terminal object of Cone(FD) (that is, the limit cone on FD).
It just occurred to me that there is something closely related in lattice theory; unfortunately I cannot give a reference, but I remember that one calls a subposet P' of a poset P relatively (co)complete if whenever a subset of P' has an upper bound in P, it has a least upper bound in P'. A related question: does anybody know any analogs of the Freyd's Adjoint Functor Theorems for functors between in(co)complete categories? Mamuka
participants (3)
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leinster＠ihes.fr -
Mamuka Jibladze -
Tom LEINSTER