nLab
flat functor

Contents

Idea

If C is a finitely complete category (a category with all finite limits), then it's interesting to consider a left exact functor on C (a functor that preserves all finite limits). Even if C lacks some finite limits, then this concept still makes sense, but it may not be the correct one. Instead we use the stronger concept of a flat functor, which may be thought of as a functor that preserves all finite limits —even the ones that don't exist yet!

Definition

A functor F:CD is flat if for each object dD, the comma category (d/F) is cofiltered.

Remarks

Properties

If F is flat, then it preserves any finite limits that exist in C. A partial converse holds: if C has enough finite limits and F preserves these, then F is flat.

If is a cocomplete category and F:CD is flat, then the functor (F):[C op,Set] (the Yoneda extension of F, i.e. the left Kan extension of F along the Yoneda embedding) preserves all finite limits (and now all finite limits exist).

A similar statement holds when C is a site and we extend a cocontinuous functor F to Sh(C). But a cocontinuous functor from Sh(C) preserving finite limits is (almost) the same thing as (the inverse image part of) a geometric morphism. So cocontinuous flat functors out of a site C characterise (almost) geometric morphisms into Sh(C). This can be very useful when a Grothendieck topos has a presentation by a particularly simple site.