If is a finitely complete category (a category with all finite limits), then it is interesting to consider a left exact functor on (a functor that preserves all finite limits). Even if 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!
It turns out that the most appropriate generality in which to speak of a flat functor is when is a site. We build up to this definition in stages through several more classical notions, remarking at each stage on some basic properties and equivalences. Proofs will be given in the following section.
The most classical notion is the following.
For disambiguation with the later notions, we may refer to such a functor as being -valued flat.
Spelled out explicitly, this means that is flat precisely if the following three conditions hold.
If is flat, then it preserves all finite limits that exist in . Conversely, if has finite limits and preserves them, then it is flat.
Since is equivalent to the category of elements of the composite , this is equivalent to saying that is Set-valued flat for every . Hence, this notion of flatness may be called representably flat. Spelled out explicitly as we did above for flat set-valued functors, this means that for every , we have:
There is an object and a morphism .
For any and morphisms and , there exists an object , morphisms , in , and a morphism such that and .
For two parallel morphisms in , and a morphism such that , there exists a morphism in and a morphism such that and .
A functor between small categories is representably flat if and only if the operation of left Kan extension preserves finite limits.
A proof of this is given below as prop. 6.
If is representably flat, then it preserves all finite limits that exist in . Conversely, if has finite limits and preserves them, then it is representably flat.
If has finite limits, then a functor is representably flat if and only if it is Set-valued flat, if and only if it preserves finite limits.
However, if lacks finite limits, then representable flatness of can be stronger than Set-valued flatness.
For disambiguation, this notion of flatness may be called internally flat since it refers to the internal logic of . Internally flat functors have multiple other names:
Since the internal logic of is just ordinary logic, a functor is internally flat just when it is -valued flat, def. 1.
If is internally flat, then it preserves all finite limits that exist in . Conversely, if has finite limits and preserves them, then it is internally flat.
Finally, we can give the most general definition, due to Karazeris
is a covering sieve of in .
For disambiguation, we may refer to this notion as being covering-flat. This subsumes the other three definitions as follows:
If is a small category and is a small-generated site, then a functor is covering-flat if and only if its extension preserves finite limits.
If is covering-flat, where has finite limits and all covering families in are extremal-epic, then preserves all finite limits that exist in . Conversely, if has finite limits and preserves them, then it is covering-flat.
We now prove the asserted propositions about the equivalence of flatness with finite-limit-preserving extensions to presheaf categories.
This is prop. 6.1.3 in (Borceux).
where is the corresponding comma category and is the canonical projection.
Now, by definition being representably-flat means that is a filtered category. So this is a filtered colimit. By the discussion there, it is precisely the filtered colimits that commute with finite limits.
This is VII.9.1 in Mac Lane-Moerdijk.
If is a site, is a sheaf topos, and is internally flat, then the restriction of to still preserves finite limits, and it is cocontinuous just when preserves covering families. Since cocontinuous left-exact functors between sheaf toposes are precisely the inverse image parts of geometric morphisms, we conclude that cover-preserving internally-flat functors out of a site characterise geometric morphisms into . In other words, is the classifying topos for such functors. This can be very useful when a Grothendieck topos has a presentation by a particularly simple site.
of the category of presheaves on (which is the free cocompletion of ) on the flat functors is the free cocompletion under filtered colimits. When regarded in this way, flat functors are also known as ind-objects.
This is due to (KarazerisVelebil).
The following statement is known as Diaconescu's theorem, see there for more details. It says that the internally flat functors, def. 3 are precisely the inverse images of geometric morphisms from into the presheaf topos over .
There is an equivalence of categories
This equivalence takes to the composite
One says that is the classifying topos for internally flat functors out of .
Morphisms of sites are flat functors which additionally preserve covering families.
internally flat functors (“torsors”) are discussed around B3.2, and representably flat functors around C2.3.7.
-valued flat functors are discussed in VII.6, and internally flat functors in VII.8 (both called “filtering functors”).
In section 2 of
internally flat functors with values in a topos with enough points are discussed.
For the relationship between the various notions of flatness, and the notion of covering-flatness, see
Limits in the category of flat functors are discussed in