Contents

category theory

# Contents

## 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!

## Definitions

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

### $Set$-valued functors

The most classical notion is the following.

###### Definition

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

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

###### Remark

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

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

2. (transitivity) For objects $c,d \in C$ and elements $y \in E(c)$, $z \in E(d)$, there exists an object $b \in C$, morphisms $\alpha : b \to c$, $\beta : b \to d$ and an element $w \in E(b)$ such that $E(\alpha) : w \mapsto y$ and $E(\beta) :w \mapsto z$.

3. (freeness) For two parallel morphisms $\alpha, \beta : c \to d$ and $y \in E(c)$ such that $E(\alpha)(y) = E(\beta)(y)$, there exists a morphism $\gamma : b \to c$ and an element $z \in E(b)$ such that $\alpha \circ \gamma = \beta \circ \gamma$ and $E(\gamma) : z \mapsto y$.

###### Proposition

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

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

###### Corollary

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

### Representable flatness

###### Definition

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

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

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

2. For any $c,d\in C$ and morphisms $y:e\to F(c)$ and $z:e\to F(d)$, there exists an object $b\in C$, morphisms $\alpha : b \to c$, $\beta : b \to d$ in $C$, and a morphism $w: e\to F(b)$ such that $F(\alpha)\circ w = y$ and $F(\beta)\circ w = z$.

3. For two parallel morphisms $\alpha, \beta : c \to d$ in $C$, and a morphism $y : e \to F(c)$ such that $F(\alpha)\circ y = F(\beta)\circ y$, there exists a morphism $\gamma : b \to c$ in $C$ and a morphism $z : e \to F(b)$ such that $\alpha \circ \gamma = \beta \circ \gamma$ and $F(\gamma) \circ z = y$.

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

###### Proposition

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

A proof of this is given below as prop. .

###### Corollary

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

###### Corollary

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

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

### Topos-valued functors

###### Definition

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

Explicitly, this means that for any finite diagram $D\colon I\to C$, the family of factorizations through $\lim (F\circ D)$ of the $F$-images of all cones over $D$ in $C$ is epimorphic in $E$.

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

###### Remark

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

More generally:

###### Example

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

###### Proof

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

###### Proposition

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

###### Corollary

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

### Site-valued functors

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

###### Definition

Let $E$ be any site. A functor $F\colon C\to E$ is flat if for any finite diagram $D\colon I\to C$ and any cone $T$ over $F\circ D$ in $E$ with vertex $u$, the sieve

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

is a covering sieve of $u$ in $E$.

For disambiguation, we may refer to this notion as being covering-flat.

This subsumes the other three definitions as follows:

• If $E=Set$ with its canonical topology, then covering-flatness reduces to Set-valued flatness, def. .

• More generally, if $E$ is a cocomplete topos with its canonical topology, then covering-flatness reduces to internal flatness, def. .

• On the other hand, if $E$ has a trivial topology, then covering-flatness reduces to representable flatness, def. .

###### Proposition

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

###### Corollary

If $F\colon C\to E$ is covering-flat, where $E$ has finite limits and all covering families in $E$ are extremal-epic, then $F$ preserves all finite limits that exist in $C$. Conversely, if $C$ has finite limits and $F$ 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 $C$ is small, a functor $F\colon C\to Set$ is Set-valued flat if and only if its Yoneda extension $[C^{op},Set] \to Set$ preserves finite limits.

###### Proof

This is prop. 6.3.8 in (Borceux).

###### Proposition

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

###### Proof

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

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

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

Now, by definition $F$ being representably-flat means that $(e/F)^{op}$ 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 $C$ is small and $E$ is a cocomplete topos, a functor $F\colon C\to E$ is internally flat if and only if its Yoneda extension $[C^{op},Set] \to E$ preserves finite limits.

###### Proof

This is VII.9.1 in Mac Lane-Moerdijk.

If $C$ is a site, $E$ is a sheaf topos, and $F\colon C\to E$ is internally flat, then the restriction of $[C^{op},Set] \to E$ to $Sh(C)$ still preserves finite limits, and it is cocontinuous just when $F$ 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 $C$ characterise geometric morphisms into $Sh(C)$. In other words, $Sh(C)$ 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 $A$ a category the full subcategory

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

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. When regarded in this way, flat functors are also known as ind-objects.

###### Proposition

$FlatFunc(A^{op},Set)$ 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).

### 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. $F \colon C \to \mathcal{E}$ are precisely the inverse images of geometric morphisms from $E$ into the presheaf topos over $C$.

###### Theorem

(Diaconescu’s theorem)

There is an equivalence of categories

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

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

This equivalence takes $f$ to the composite

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

where $j$ is the Yoneda embedding and $f^*$ is the inverse image of $f$.

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

## Examples

• 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

$Set$-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

A notion of “flat 2-functor” is discussed, with an eye towards applications with 2-toposes, in the article

• M.E. Descotte, E.J. Dubuc, M. Szyld, On the notion of flat 2-functors, arXiv:1610.09429