nLab
flat functor

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 $F:C\to D$ is flat if for each object $d\in D$, the comma category $(d/F)$ is cofiltered.

Remarks

• Some authors call this precisely a left exact functor, see the definition there.

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 $ℰ$ is a cocomplete category and $F:C\to D$ is flat, then the functor $(-\otimes F):[C^{\mathrm{op}},\mathrm{Set}]\to ℰ$ (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 $\mathrm{Sh}(C)$. But a cocontinuous functor from $\mathrm{Sh}(C)$ 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 $\mathrm{Sh}(C)$. This can be very useful when a Grothendieck topos has a presentation by a particularly simple site.