nLab
flat functor

Idea

If $C$ is a finitely complete category (a category with all finite limits), then it is 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

There are two different definitions that both go by the name flat functor in the literature. For emphasis, we speak here of representably and internally flat functors, following (Shulman).

Definition

A functor $F:C\to D$ is

• representably flat if for each object $d\in D$, the opposite comma category $(d/F{)}^{\mathrm{op}}$ is a filtered category

• internally flat if …

For instance Johnstone, C2.3.7 and B3.2.3, where “internally flat functors” are called “torsors”.

Remark

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

Properties

General

Proposition

If $F:C\to D$ 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.

Proposition

If $F:C\to D$ is a flat functor and

$$[C^{\mathrm{op}},\mathrm{Set}]\leftarrow [D^{\mathrm{op}},\mathrm{Set}]:F^{*}$$[C^{op}, Set] \leftarrow [D^{op},Set] : F^*

is its action on presheaves by precomposition, then the corresponding left adjoint $F_{!}$

$$(F_{!}⊣F^{*}):[D^{\mathrm{op}},\mathrm{Set}]\stackrel{\stackrel{F_{!}}{\to }}{\underset{F^{*}}{\leftarrow }}[C^{\mathrm{op}},\mathrm{Set}]$$(F_! \dashv F^*) : [D^{op}, Set] \stackrel{\overset{F_!}{\to}}{\underset{F^*}{\leftarrow}} [C^{op},Set]

preserves finite limits.

Proof

The left adjoint is the left Kan extension along $F$

$$F_{!}\simeq {\mathrm{Lan}}_F.$$F_! \simeq Lan_F \,.

Since presheaf toposes have all colimits, this is computed on any object $c\in C$ (as discussed at Kan extension) by the colimit

$$(F_{!}X(c)=\underrightarrow{\lim }((d/F{)}^{\mathrm{op}}\to C^{\mathrm{op}}\stackrel{F}{\to }\mathrm{Set}),$$(F_! X(c) = \lim_{\to} \left( (d/F)^{op} \to C^{op} \stackrel{F}{\to} Set \right) \,,

where $(d/F)$ is the corresponding comma category and $(d/F{)}^{\mathrm{op}}\to C^{\mathrm{op}}$ is the canonical projection.

Now, by definition $F$ being flat is equivalent to $(d/F{)}^{\mathrm{op}}$ being a filtered category. So this is a filtered colimit. By the discussion there, it is precisely the filtered colimits that commute with finite limits.

Proposition

If $ℰ$ is a topos and $F:C\to ℰ$ 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).

For $ℰ=$ Set this is prop. 6.1.3 in (Borceux).

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.

Category of flat functors

For $A$ a category the full subcategory

$$\mathrm{FlatFunc}(A^{\mathrm{op}},\mathrm{Set})\subset \mathrm{Func}(A^{\mathrm{op}},\mathrm{Set})$$FlatFunc(A^{op}, Set) \subset Func(A^{op}, Set)

of the category of presheaves on $A$ (which is the free cocompletion of $A$) on the flat functors is the free cocompletion under filtered colimits.

Proposition

$\mathrm{FlatFunc}(A^{\mathrm{op}},\mathrm{Set})$ has finite limits precisely if for every finite diagram $D$ in $A$, the category of cones on $D$ is filtered.

This is due to (KarazerisVelebil).

Examples

• Morphisms of sites are given by flat functors.

References

Representbaly flat functors are the topic of chapter 6 in

• Francis Borceux, Handbook of categorical algebra , volume I, Basic category theory

Internally flat functors (“torsors”) are discussed around B3.2 and representably flat functors around … in C2.3.7 of

• Peter Johnstone, Sketches of an Elephant .

The difference between internally and representably flat functors is discussed in

• Mike Shulman, Flat Functors and Morphisms of Sites

Limits in the category of flat functors are discussed in

• Panagis Karazeris, Jiří Velebil, Representability relative to a doctrine , Cahiers de Topologie et Géometrie Différentielle Catégoriques 50 (2009), 3–22.