nLab
flat functor

Contents

Idea

If CC is a finitely complete category (a category with all finite limits), then it is interesting to consider a left exact functor on CC (a functor that preserves all finite limits). Even if CC 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!

Definitions

It turns out that the most appropriate generality in which to speak of a flat functor CDC \to D is when DD 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.

SetSet-valued functors

The most classical notion is the following.

Definition

A functor CSetC\to Set is flat if the opposite of its category of elements, el(C) opel(C)^{op}, is a filtered category.

For disambiguation with the later notions, we may refer to such a functor as being SetSet-valued flat.

Remark

Spelled out explicitly, this means that E:CSetE : C \to Set is flat precisely if the following three conditions hold.

  1. (non-emptiness) There is at least one object cCc \in C such that E(c)E(c) is an inhabited set.

  2. (transitivity) For objects c,dCc,d \in C and elements yE(c)y \in E(c), zE(d)z \in E(d), there exists an object bCb \in C, morphisms α:bc\alpha : b \to c, β:bd\beta : b \to d and an element wE(b)w \in E(b) such that E(α):wyE(\alpha) : w \mapsto y and E(β):wzE(\beta) :w \mapsto z .

  3. (freeness) For two parallel morphisms α,β:cd\alpha, \beta : c \to d and yE(c)y \in E(c) such that E(α)(y)=E(β)(y)E(\alpha)(y) = E(\beta)(y), there exists a morphism γ:bc\gamma : b \to c and an element zE(b)z \in E(b) such that αγ=βγ\alpha \circ \gamma = \beta \circ \gamma and E(γ):zyE(\gamma) : z \mapsto y.

Proposition

When CC is small, a functor F:CSetF\colon C\to Set is SetSet-valued flat if and only if its Yoneda extension [C op,Set]Set[C^{op},Set] \to Set preserves finite limits.

This partially explains the terminology “flat”, since the Yoneda extension is a sort of tensoring with FF, and a flat module is one such that tensoring with it preserves finite limits.

Corollary

If F:CSetF\colon C\to Set is flat, then it preserves all finite limits that exist in CC. Conversely, if CC has finite limits and FF preserves them, then it is flat.

Representable flatness

Definition

A functor F:CEF\colon C \rightarrow E is flat if for each object eEe \in E, the opposite comma category (e/F) op(e / F)^{op} is a filtered category.

Since (e/F)(e/F) is equivalent to the category of elements of the composite CFEE(e,)SetC \xrightarrow{F} E \xrightarrow{E(e,-)} Set, this is equivalent to saying that E(e,F):CSetE(e,F-)\colon C\to Set is Set-valued flat for every eEe\in E. 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 eEe\in E, we have:

  1. There is an object cCc\in C and a morphism eF(c)e\to F(c).

  2. For any c,dCc,d\in C and morphisms y:eF(c)y:e\to F(c) and z:eF(d)z:e\to F(d), there exists an object bCb\in C, morphisms α:bc\alpha : b \to c, β:bd\beta : b \to d in CC, and a morphism w:eF(b)w: e\to F(b) such that F(α)w=yF(\alpha)\circ w = y and F(β)w=zF(\beta)\circ w = z.

  3. For two parallel morphisms α,β:cd\alpha, \beta : c \to d in CC, and a morphism y:eF(c)y : e \to F(c) such that F(α)y=F(β)yF(\alpha)\circ y = F(\beta)\circ y, there exists a morphism γ:bc\gamma : b \to c in CC and a morphism z:eF(b)z : e \to F(b) such that αγ=βγ\alpha \circ \gamma = \beta \circ \gamma and F(γ)z=yF(\gamma) \circ z = y.

Representably flat functors are sometimes referred to simply as “left exact functors”. On the nnLab we try to generally reserve the latter terminology for the case when CC has finite limits.

Proposition

A functor F:CEF \colon C \to E between small categories is representably flat if and only if the operation Lan F:[C op,Set][E op,Set]Lan_F\colon [C^{op}, Set] \to [E^{op},Set] of left Kan extension preserves finite limits.

A proof of this is given below as prop. 6.

Corollary

If F:CEF\colon C\to E is representably flat, then it preserves all finite limits that exist in CC. Conversely, if CC has finite limits and FF preserves them, then it is representably flat.

Corollary

If CC has finite limits, then a functor CSetC\to Set is representably flat if and only if it is Set-valued flat, if and only if it preserves finite limits.

However, if CC lacks finite limits, then representable flatness of CSetC\to Set can be stronger than Set-valued flatness.

Topos-valued functors

Definition

Let EE be a cocomplete topos (for instance a Grothendieck topos). A functor F:CEF\colon C\to E is flat if the statement “FF is SetSet-valued flat, def. 1.” is true in the internal logic of EE.

Explicitly, this means that for any finite diagram D:ICD\colon I\to C, the family of factorizations through lim(FD)\lim (F\circ D) of the FF-images of all cones over DD in CC is epimorphic in EE.

For disambiguation, this notion of flatness may be called internally flat since it refers to the internal logic of EE. Internally flat functors have multiple other names:

Remark

Since the internal logic of SetSet is just ordinary logic, a functor CSetC\to Set is internally flat just when it is SetSet-valued flat, def. 1.

More generally:

Example

If EE has enough points, then FF is internally flat precisely if for all stalks x *:ESetx^* : E \to Set the composite x *Fx^* \circ F is SetSet-valued flat.

Proof

In a topos EE with enough points, a morphism f:XYf : X \to Y is an epimorphism precisely if x *fx^* f is an epimorphism in SetSet. By definition, the stalks x *:ESetx^* : E \to Set commute with finite limits.

Proposition

When CC is small, a functor F:CEF\colon C\to E is internally flat if and only if its Yoneda extension [C op,Set]E[C^{op},Set] \to E preserves finite limits.

Corollary

If F:CEF\colon C\to E is internally flat, then it preserves all finite limits that exist in CC. Conversely, if CC has finite limits and FF preserves them, then it is internally flat.

Site-valued functors

Finally, we can give the most general definition, due to Karazeris

Definition

Let EE be any site. A functor F:CEF\colon C\to E is flat if for any finite diagram D:ICD\colon I\to C and any cone TT over FDF\circ D in EE with vertex uu, the sieve

{h:vu|Th factors through the F-image of some cone over D} \{ h\colon v\to u | T h \;\text{ factors through the }\; F\text{-image of some cone over }\; D \}

is a covering sieve of uu in EE.

For disambiguation, we may refer to this notion as being covering-flat. This subsumes the other three definitions as follows:

  • If E=SetE=Set with its canonical topology, then covering-flatness reduces to Set-valued flatness.
  • More generally, if EE is a cocomplete topos with its canonical topology, then covering-flatness reduces to internal flatness.
  • On the other hand, if EE has a trivial topology?, then covering-flatness reduces to representable flatness.
Proposition

If CC is a small category and EE is a small-generated site, then a functor F:CEF \colon C \to E is covering-flat if and only if its extension [C op,Set]Sh(E)[C^{op}, Set] \to Sh(E) preserves finite limits.

Corollary

If F:CEF\colon C\to E is covering-flat, where EE has finite limits and all covering families in EE are extremal-epic, then FF preserves all finite limits that exist in CC. Conversely, if CC has finite limits and FF preserves them, then it is covering-flat.

Properties

Yoneda extensions

We now prove the asserted propositions about the equivalence of flatness with finite-limit-preserving extensions to presheaf categories.

Proposition

When CC is small, a functor F:CSetF\colon C\to Set is Set-valued flat if and only if its Yoneda extension [C op,Set]Set[C^{op},Set] \to Set preserves finite limits.

Proof

This is prop. 6.1.3 in (Borceux).

Proposition

When CC and EE are small, a functor F:CEF \colon C \to E is representably flat if and only if its Yoneda extension Lan F:[C op,Set][E op,Set]Lan_F\colon [C^{op}, Set] \to [E^{op},Set] preserves finite limits.

Proof

Since presheaf toposes have all colimits, F !=Lan FF_! = Lan_F is computed on any object eEe \in E (as discussed at Kan extension) by the colimit

(F !X(e)=lim ((e/F) opC opFSet) (F_! X(e) = \lim_{\to} \left( (e/F)^{op} \to C^{op} \stackrel{F}{\to} Set \right)

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

Now, by definition FF being representably-flat means that (e/F) op(e/F)^{op} 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.

Proposition

When CC is small and EE is a cocomplete topos, a functor F:CEF\colon C\to E is internally flat if and only if its Yoneda extension [C op,Set]E[C^{op},Set] \to E preserves finite limits.

Proof

This is VII.9.1 in Mac Lane-Moerdijk.

If CC is a site, EE is a sheaf topos, and F:CEF\colon C\to E is internally flat, then the restriction of [C op,Set]E[C^{op},Set] \to E to Sh(C)Sh(C) still preserves finite limits, and it is cocontinuous just when FF 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 CC characterise geometric morphisms into Sh(C)Sh(C). In other words, Sh(C)Sh(C) is the classifying topos for such functors. This can be very useful when a Grothendieck topos has a presentation by a particularly simple site.

Category of flat functors

For AA a category the full subcategory

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

of the category of presheaves on AA (which is the free cocompletion of AA) on the flat functors is the free cocompletion under filtered colimits. When regarded in this way, flat functors are also known as ind-objects.

Proposition

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

This is due to (KarazerisVelebil).

Classifying toposes and Diaconescu’s theorem

The following statement is known as Diaconescu's theorem, see there for more details. It says that the internally flat functors, def. 3 F:CF \colon C \to \mathcal{E} are precisely the inverse images of geometric morphisms from EE into the presheaf topos over CC.

Theorem

(Diaconescu’s theorem)

There is an equivalence of categories

Topos(,PSh(C))FlatFunc(C,) Topos(\mathcal{E}, PSh(C)) \simeq FlatFunc(C, \mathcal{E})

between the category of geometric morphisms f:PSh(C)f : \mathcal{E} \to PSh(C) and the category of internally flat functors CC \to \mathcal{E}.

This equivalence takes ff to the composite

CjPSh(C)f *, C \stackrel{j}{\to} PSh(C) \stackrel{f^*}{\to} \mathcal{E} \,,

where jj is the Yoneda embedding and f *f^* is the inverse image of ff.

One says that PSh(C)PSh(C) is the classifying topos for internally flat functors out of CC.

Examples

References

  • Francis Borceux, Handbook of categorical algebra , volume I, Basic category theory. Representable flatness is discussed in chapter 6.

In

internally flat functors (“torsors”) are discussed around B3.2, and representably flat functors around C2.3.7.

In

SetSet-valued flat functors are discussed in VII.6, and internally flat functors in VII.8 (both called “filtering functors”).

In section 2 of

  • Ieke Moerdijk, Classifying spaces and classifying topoi, Lecture Notes in Mathematics 1616, Springer 1995. vi+94 pp. ISBN: 3-540-60319-0

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

  • Panagis Karazeris, Notions of flatness relative to a Grothendieck topology, Theory and Applications of Categories, 12 (2004), 225-236 (TAC)

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.

Discussion of left exact functors or flat functors in the context of (∞,1)-category theory is in

Revised on December 2, 2014 17:34:52 by Alex? (128.232.100.139)