flag variety

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## Idea

A __[[flag]]__ in a [[vector space]] or a [[projective space]] is a nested system of linear/projective subspaces, one of each [[dimension]]
from $0$ to $n-1$.

Given a [[field]] $k$, the space of all flags in an $n$-dimensional $k$-vector space has the structure of a [[projective variety]] over $k$, this is the __flag variety__.

If $k$ is the field of real or complex numbers, then it has a structure of a smooth [[manifold]]. It can be considered as the [[homogeneous space]] $SL(n,k)/B$ where $B$ is the subgroup of lower (or upper if you like) triangular matrices. By the [Gram--Schmidt orthogonalization procedure](http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process), one shows the isomorphism as real manifolds $SL(n,\mathbb{C})/B\cong SU(n)/T$ where $T$ is the subgroup of the diagonal $n\times n$-matrices.

More generally, the __generalized flag variety__ is the complex projective variety obtained as the [[coset space]] $G/T\cong G^{\mathbb{C}}/B$ where $G$ is a [[compact Lie group]], $T$ its [[maximal torus]], $G^{\mathbb{C}}$ the [[complexification]] of $G$, which is a complex semisimple group, and $B\subset G^{\mathbb{C}}$ is the [[Borel subgroup]]. It has a structure of a compact [[Kähler manifold]]. It is a special case of the larger family of coset spaces of semisimple groups modulo [[parabolic subgroup|parabolics]] which includes, for example, [[Grassmannians]]. There are quantum, noncommutative and infinite-dimensional generalizations. Flag varieties have rich combinatorial and geometric structure and play an important role in [[representation theory]] and [[integrable systems]].

## Related concepts

* [[Grassmannian]]

* [[coset space]]

* [[coadjoint orbit]]

* [[building]]

* [[Schubert variety]]

* [[Schubert calculus]]

* [[character sheaf]], [[horocycle correspondence]], [[Harish Chandra transform]]

[[!include local and global geometry - table]]

## References

* D. Monk, _The geometry of flag manifolds_, Proceedings of the London Mathematical Society 1959 s3-9(2):253--286; [doi:10.1112/plms/s3-9.2.253](http://dx.doi.org/10.1112/plms/s3-9.2.253)

* Wikipedia, _[Generalized flag variety](http://en.wikipedia.org/wiki/Generalized_flag_variety)_

* M. Brion, _Lectures on the geometry of flag varieties_, [pdf](http://www-fourier.ujf-grenoble.fr/~mbrion/lecturesrev.pdf)

* [[Armand Borel]], _Linear algebraic groups_, Springer

On flag varieties of [[loop groups]]:

* {#Kumar02} [[Shrawan Kumar]], _Kac-Moody Groups, their Flag Varieties and Representation Theory_, Birkhäuser 2002

On [[flag manifold sigma-models]]:

* [[Ian Affleck]], [[Dmitri Bykov]], [[Kyle Wamer]], *Flag manifold sigma models: spin chains and integrable theories*,  Phys. Rept. **953** (2022) 1-93 [[arXiv:2101.11638](https://arxiv.org/abs/2101.11638), [doi:10.1016/j.physrep.2021.09.004](https://doi.org/10.1016/j.physrep.2021.09.004)]

[[!redirects flag varieties]]
[[!redirects flag manifold]]
[[!redirects flag manifolds]]

[[!redirects partial flag variety]]
[[!redirects partial flag varieties]]

[[!redirects flag space]]
[[!redirects flag spaces]]


flagged category

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## Definition

A flagged category consists of a [[category]] $C$, a [[groupoid]] $X$, and a surjection $p \colon X\to C$ of groupoids from $X$ to the underlying groupoid of objects of $C$ (hence an [[essentially surjective functor]]).

One may also call this a *[[category with an atlas]]*, where the atlas is allowed to be groupoidal itself.

## In type theory

A flagged category is a [[1-truncated]] [[precategory]].

## See also

* [[category with an atlas]]

* [[univalent category]]

* [[gaunt category]]

## References

The terminology "flagged" for this notion appears (in generalization to [[higher category theory]]) in:

* {#AyalaFrancis18} [[David Ayala]], [[John Francis]], Def. 0.12 of: *Flagged higher categories*, in: *Topology and Quantum Theory in Interaction*,  Contemporary Mathematics **718**, AMS (2018) [[arXiv:1801.08973](https://arxiv.org/abs/1801.08973), [ISBN:978-1-4704-4941-4](https://bookstore.ams.org/conm-718)]

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flat (infinity,1)-functor

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## Idea

The generalization of the notion of [[flat functor]] from [[category theory]] to [[(∞,1)-category theory]].

## Definition

As for 1-categorical flat functors, there is a general definition of flat functors that restricts, in the case when [[finite limits]] exist, to the condition that these are preserved.

+-- {: .un_defn}
###### Definition

For $\kappa$ a [[regular cardinal]], an [[(∞,1)-functor]] $F : C \to D$ is **$\kappa$-flat**, if, when modeled as a morphism of [[quasicategories]], for any [[left Kan fibration]] $D' \to D$ with $D'$ a $\kappa$-[[cofiltered (∞,1)-category]], the [[pullback]] $C' := C \times_D D'$ (in [[sSet]]) is also $\kappa$-cofiltered.

If $\kappa = \omega$ then we just say $F$ is **flat**.

=--

The dual of this is [[Higher Topos Theory|HTT, def. 5.3.2.1]], under the name "$\kappa$-right exact".  But in 1-category theory, the terminology "left/right exact" is almost universally reserved for the case when finite limits/colimits *do* exist, so we continue that tradition in the $\infty$-case.  We do have:

+-- {: .un_prop}
###### Proposition

If $C$ has $\kappa$-small [[limits]], then $F$ is $\kappa$-flat precisely if it preserves these $\kappa$-small limits.

In particular, if $C$ has all [[finite (infinity,1)-limit|finite limits]], then $F$ is flat precisely if it preserves these.

=--

The dual of this is [[Higher Topos Theory|HTT, prop. 5.3.2.9]].

## Properties

+-- {: .un_prop}
###### Proposition

1. $\kappa$-flat $(\infty,1)$-functors are closed under composition.

1. Every [[(∞,1)-equivalence]] is $\kappa$-flat.

1. An $(\infty,1)$-functor equivalent (in the [[(∞,1)-category of (∞,1)-functors]]) to a $\kappa$-flat one is itself $\kappa$-flat.

=--

This is [[Higher Topos Theory|HTT, prop. 5.3.2.4]].

## Internal flatness

For 1-categories, there are two notions of [[flat functor]]: the above corresponds to the "representable" one, while (for functors valued in a [[topos]]) there is also a notion of "internal flatness" (and a notion of "covering flatness" that generalizes them both.  I do not know whether internally-flat or covering-flat $(\infty,1)$-functors have been defined, but the following shows that left exact $(\infty,1)$-functors valued in an $(\infty,1)$-topos, at least, satisfy a condition that ought to characterize internally-flat ones.

+-- {: .un_prop}
###### Proposition
If $C$ is an $(\infty,1)$-category with finite limits, $D$ is an $(\infty,1)$-topos, and $F:C\to D$ preserves finite limits, then its Yoneda extension $\mathcal{P}(C) \to D$ also preserves finite limits.
=--

This is [[Higher Topos Theory|HTT, prop. 6.1.5.2]].

## Related concepts

* [[flat functor]]
* [[excisive (∞,1)-functor]]

## References

Section 5.3.2 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

[[!redirects flat (infinity,1)-functor]]
[[!redirects flat (∞,1)-functor]]

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flat bundle

A [[bundle]] equipped with a [[flat connection]].

[[!redirects flat bundles]]


flat connection

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## Definition

A [[connection on a bundle|connection]] on a [[fibre bundle]] is **flat** if its [[curvature]] is zero.

The same definition of flatness holds for connections in various algebraic setups and for connections on [[quasicoherent sheaves]].

The condition of flatness is usually expressed via the [[Maurer-Cartan equation]]. Flat connections on bundles are also referred to as [[local systems]] (by the _Riemann-Hilbert correspondence_ discussed [below](#RiemannHilbertCorrespondence)).

## Properties

### Riemann-Hilbert correspondence
 {#RiemannHilbertCorrespondence}

That flat connections are equivalently [[representations]] of the [[fundamental groupoid]] of the base space/[[local systems]] is known as the _[[Riemann-Hilbert correspondence]]_.

### Flatness as integrability

In geometry one says instead of flat connection, **integrable connection**. The reason is roughly the following: in the theory of systems of differential equations the flatness of the corresponding connection is the condition of the integrability of the system.

(...elaborate on this with equations)

The condition of flatness is usually expressed via the [[Maurer-Cartan equation]], which is in integrable systems theory often called *zero curvature equation*. For example, the Lax equations can always be written in the form of the zero curvature equation.

### Over a complex manifold

Over a [[complex manifold]]/[[complex variety]] the [[Koszul-Malgrange theorem]] identifies holomorphic flat connections on [[complex vector bundles]] with [[holomorphic vector bundles]]. See there for more

### Over a Riemann surface

#### Existence

In ([Milnor](#Milnor)) it is shown that a [[vector bundle]] over a [[surface]] of [[genus]] $g$ admits flat connections iff its [[Euler class]] is less than $g$ by an [[absolute value]] (see also Wood, Bundles with totally disconnected structure group). [Sullivan](#Sullivan) gives a refinement.

#### Narasimhan–Seshadri theorem

The _[[Narasimhan-Seshadri theorem]]_ identifies [[moduli spaces of flat connections]] over a [[Riemann surface]] with that of certain [[stable vector bundles]].

### Isometric embeddings

Maurer-Cartan equation is called also structure equation when used to treat the conditions for isometric embeddings of Riemannian submanifolds in an Euclidean space.

...

## Related concepts

* [[local system]], [[inner local system]]

* [[projectively flat connection]]

* [[moduli space of flat connections]], [[Narasimhan-Seshadri theorem]]

* [[flat ∞-connection]]

* [[flat section]]

## References

On the equivalence between flat connections ([[local systems]]) with [[group homomorphisms]] out of the [[fundamental group]] (over each [[connected component]]):

* [[Pierre Deligne]], §I.1 of: *Equations différentielles à points singuliers réguliers*, Lecture Notes  Math. **163**, Springer (1970) $[$[publications.ias:355](https://publications.ias.edu/node/355)$]$

and generally with [[functors]] out of the [[fundamental groupoid]]:

* [[Alexandru Dimca]], Prop. 2.5.1 of: *Sheaves in Topology*, Universitext, Springer (2004) [[doi:10.1007/978-3-642-18868-8](https://doi.org/10.1007/978-3-642-18868-8)]

* [[Urs Schreiber]], [[Konrad Waldorf]], p. 194 of: *Parallel Transport and Functors*, J. Homotopy Relat. Struct. **4** (2009) 187-244 [[arXiv:0705.0452](https://arxiv.org/abs/0705.0452), [pdf](https://tcms.org.ge/Journals/JHRS/xvolumes/2009/n1a10/v4n1a10hl.pdf)]

* [[Florin Dumitrescu]], Cor. 3 of: *Connections and Parallel Transport*, Journal of Homotopy and Related Structures **5** 1 (2010) 171–175 [[arXiv:0903.0121](https://arxiv.org/abs/0903.0121), [pdf](https://tcms.org.ge/Journals/JHRS/xvolumes/2010/n1a10/v5n1a10hl.pdf)]

The latter articles also discuss the generalization of this statement to non-flat connections ,for more see the references at *[[parallel transport]]* [here](parallel+transport#ReferencesInTermsOfParallelTransport).

See also

* {#Milnor} [[John Milnor]], *On the existence of a connection with curvature zero*, Comm. Math. Helv. **32** (1957/58) 215-223 [[dml:139154](http://eudml.org/doc/139154)]

* {#Sullivan} [[Dennis Sullivan]], _A generalization of Milnor's inequality_ Comm. Math. Helv. v. 51

* [[Hélène Esnault]], _Algebraic Differential Characters of Flat Connections with Nilpotent Residues_, in [[Nils Baas]] et al., The Abel Symposium 2007 ([pdf](bib.tiera.ru/b/91797))

[[!redirects integrable connection]]

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[[!redirects projectively flat connection]]
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flat functor

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## 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 limit]]s).  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.

+-- {: .num_defn #SetValuedFlat}
###### 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**.

+-- {: .num_remark }
###### 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]].

1. (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 $.

1. (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 $E(\gamma) : z \mapsto y$.

=--

+-- {: .num_prop}
###### Proposition
When $C$ is [[small category|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 [[tensor product of functors|tensoring with]] $F$, and a [[flat module]] is one such that tensoring with it preserves finite limits.

+-- {: .num_cor}
###### 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

+-- {: .num_defn #RepresentablyFlat}
###### Definition
A functor $F\colon C \rightarrow E$ is **flat** if for each [[object]] $e \in E$, the [[opposite category|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 $e\to F(c)$.
1. 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 $F(\beta)\circ w = z$.

1. 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 "[[exact functor|left exact functors]]".  On the $n$Lab we try to generally reserve the latter terminology for the case when $C$ has [[finite limits]].

+-- {: .num_prop}
###### 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. \ref{FlatnessIfYonedaExtensionPreservesFiniteLimits}.
+-- {: .num_cor}
###### 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.
=--

+-- {: .num_cor}
###### 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
 {#ToposValuedFlatFunctors}

+-- {: .num_defn #InternallyFlat}
###### Definition

Let $E$ be a [[cocomplete category|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. \ref{SetValuedFlat}." 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 [[epimorphism|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:

* In [Johnstone, C2.3.7 and B3.2.3](#Johnstone) they are called "torsors".
* In [Mac Lane-Moerdijk, VII.8](#MLM) they are called "filtering functors".
* In [Moerdijk, II.2](#Moerdijk) they are called "principal functors".

+-- {: .num_remark }
###### 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. \ref{SetValuedFlat}.

=--

More generally:

+-- {: .num_example }
###### Example

If $E$ has [[point of a topos|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}
###### 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]].

=--

+-- {: .num_prop}
###### Proposition
When $C$ is [[small category|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.
=--

+-- {: .num_cor}
###### 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
 {#SiteValuedFunctors}

Finally, we can give the most general definition, due to [Karazeris](#Karazeris)

+-- {: .num_defn #CoveringFlat}
###### 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. \ref{SetValuedFlat}.

* More generally, if $E$ is a cocomplete [[topos]] with its [[canonical topology]], then covering-flatness reduces to internal flatness, def. \ref{InternallyFlat}.

* On the other hand, if $E$ has a [[trivial topology]], then covering-flatness reduces to representable flatness, def. \ref{RepresentablyFlat}.

+-- {: .num_prop}
###### 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.
=--

+-- {: .num_cor}
###### Corollary
If $F\colon C\to E$ is covering-flat, where $E$ has finite limits and all covering families in $E$ are [[extremal epimorphism|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.

+-- {: .num_prop}
###### Proposition
When $C$ is [[small category|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}
###### Proof
This is prop. 6.3.8 in ([Borceux](#Borceux)).
=--

+-- {: .num_prop #FlatnessIfYonedaExtensionPreservesFiniteLimits}
###### Proposition

When $C$ and $E$ are [[small category|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}
###### Proof
Since [[presheaf topos]]es have all [[colimit]]s, $F_! = Lan_F$ is computed on any object $e \in E$ (as discussed at [[Kan extension]]) by the [[colimit]]
$$ (F_! X(e) = \lim_{\to} \left( (e/F)^{op} \to C^{op} \stackrel{X}{\to} Set \right) $$
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.
=--

+-- {: .num_prop}
###### Proposition
When $C$ is [[small category|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}
###### Proof
This is VII.9.1 in [Mac Lane-Moerdijk](#MLM).
=--

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
 {#CategoryOfFlatFunctors}

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]].

+-- {: .num_prop}
###### 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 category|filtered]].

=--

This is due to ([KarazerisVelebil](#KarazerisVelebil)).

### Classifying toposes and Diaconescu's theorem
  {#DiaconescuTheorem}

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

+-- {: .num_theorem}
###### 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

* [[morphism of sites|Morphisms of sites]] are flat functors which additionally preserve covering families.

* [[tensor product of modules]] with a [[flat module]]

## Related concepts

* [[theory of flat functors]]

* [[exact (∞,1)-functor]]

* [[ind-object]]

## References

* {#Borceux} [[Francis Borceux]], Section 6.3  in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory* [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

In

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_ .
 {#Johnstone}

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

In

* [[Saunders Mac Lane]] and [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ .
 {#MLM}

$Set$-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
 {#Moerdijk}

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](http://www.tac.mta.ca/tac/volumes/12/5/12-05abs.html))
 {#Karazeris}

* [[Mike Shulman]], _[Flat Functors and Morphisms of Sites](http://golem.ph.utexas.edu/category/2011/06/flat_functors_and_morphisms_of.html)_
 {#Shulman}

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.
 {#KarazerisVelebil}

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

* [[Jacob Lurie]], def. 5.3.2.1 in _[[Higher Topos Theory]]_

A notion of "flat 2-functor" is discussed, with an eye towards applications with [[2-topos]]es, in the article

* M.E. Descotte, [[Eduardo J. Dubuc]], M. Szyld, _Sigma limits in 2-categories and flat pseudofunctors_, (v1: On the notion of flat 2-functors) [arXiv:1610.09429](https://arxiv.org/abs/1610.09429) Adv. Math. __333__ (2018) 266--313

Enriched flat functors are studied and characterized in

* [[Stephen Lack]], [[Giacomo Tendas]], _Flat vs. filtered colimits in the enriched context_, Advances in Mathematics, 404(A):108381, 2022. [arXiv:2107.08612](https://arxiv.org/abs/2107.08612), [doi](https://doi.org/10.1016/j.aim.2022.108381).

[[!redirects flat functor]]
[[!redirects flat functors]]

[[!redirects representably flat functor]]
[[!redirects representably flat functors]]

[[!redirects internally flat functor]]
[[!redirects internally flat functors]]

[[!redirects covering-flat functor]]
[[!redirects covering-flat functors]]


flat infinity-connection

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the notion of [[flat connection]] from [[differential geometry]] to [[higher differential geometry]] and generally to [[higher geometry]].

## Definition

Given a [[cohesive (∞,1)-topos]] $(\esh \dashv \flat \dashv \sharp)$ with [[shape modality]] $\esh$ and [[flat modality]] $\flat$, a **flat $\infty$-connection** an an [[object]] $X$ with [[coefficients]] in an object $A$ is a morphism

$$
  \nabla \;\colon\; X \to \flat A
$$

or equivalently a morphism

$$
  \nabla \;\colon\; \esh(X) \to A
  \,.
$$

This is also sometimes called a _[[local system]]_ on $X$ with [[coefficients]] in $A$, or a _[[cocycle]]_ in [[nonabelian cohomology]] of $X$ with _constant_ [[coefficients]] $A$.

For more see at _[structures in a cohesive (∞,1)-topos -- flat ∞-connections](cohesive+%28infinity,1%29-topos+--+structures#FlatDifferentialCohomology)_.

## Examples

### Flat principal $\infty$-bundles

For $A = \mathbf{B}G$ the [[delooping]] of an [[∞-group]], flat $\infty$-connections with coefficients in $A$ are a special case of $G$-[[principal ∞-connections]].

### Flat $(\infty,1)$-vector bundles ($\infty$-local systems)

For $A = Core(Ch_k)$ the [[core]] of an [[(infinity,1)-category of chain complexes]], functors $\esh X \longrightarrow A$ are [[(infinity,1)-vector bundles|$(\infty,1)$-vector bundles]] with [[flat infinity-connections|flat $\infty$-connections]].

In parts of the literature this case is understood by default when speaking of "$\infty$-local systems". Other parts refer to this as "[[representations up to homotopy]]" (really: up to [[coherence|coherent]] [[higher homotopy]]).

## Related concepts

* [[flat connection]]

* [[Galois theory]]

* [[local system]]

* [[constant infinity-stack]]

* [[Riemann-Hilbert correspondence]]

## References

### General

On higher version of [[Galois theory]] via [[automorphisms]] of [[locally constant infinity-stacks|locally constant $\infy$-stacks]]:

* [[Bertrand Toën]], *Vers une interprétation galoisienne de la théorie de l'homotopie*, Cahiers de Topologie et Géométrie Différentielle Catégoriques **43** 4 (2002) 257-312 [[numdam:CTGDC_2002__43_4_257_0](http://www.numdam.org/item/?id=CTGDC_2002__43_4_257_0)]

* {#PoleselloWaschkies05} [[Pietro Polesello]], [[Ingo Waschkies]], *Higher monodromy*, Homology, Homotopy and Applications **7** 1 (2005) 109-150 [[arXiv:0407507](http://arxiv.org/abs/math/0407507), [eudml:51918](https://eudml.org/doc/51918)]

In view of [[cohesive homotopy theory]]:

* [[schreiber:differential cohomology in a cohesive topos|dcct §3.8.5]]

### Flat $(\infty,1)$-vector bundles ($\infty$-local systems)
  {#ReferencesFlatInfinityVectorBundles}

On [[infinity-local systems|$\infty$-local systems]] in the sense of [[(infinity,1)-vector bundles|$(\infty,1)$-vector bundles]] with [[flat infinity-connections|flat $\infty$-connections]]:

Component-definitions are due to:

* [[Camilo Arias Abad]], [[Florian Schätz]]: *The $A_\infty$ de Rham theorem and integration of representations up to homotopy*, International Mathematics Research Notices, **2013** 16 (2013) 3790–3855 [[arXiv:1011.4693](https://arxiv.org/abs/1011.4693), [doi:10.1093/imrn/rns166](https://doi.org/10.1093/imrn/rns166)]

* {#BlockSmith14} [[Jonathan Block]], [[Aaron M. Smith]], *The higher Riemann--Hilbert correspondence*, Advances in Mathematics **252** (2014) 382-405 [[arXiv:0908.2843](https://arxiv.org/abs/0908.2843), [doi:10.1016/j.aim.2013.11.001](https://doi.org/10.1016/j.aim.2013.11.001)]

Identification with [[(infinity,1)-functors|$(\infty,1)$-functors]] is made explicit in:

* [Block & Smith (2014), below Def. 2.3 (§2.2 in the preprint)](#BlockSmith14)

* [[Manuel Rivera]], [[Mahmoud Zeinalian]], §5 of: *The colimit of an $\infty$-local system as a twisted tensor product*, Higher Structures **4** 1 (2020) 33-56  [[arXiv:1805.01264](https://arxiv.org/abs/1805.01264), [higher-structures:Vol4Iss1](https://higher-structures.math.cas.cz/api/files/issues/Vol4Iss1/RiveraZeinalian)]

Enhancement of the [[Chern-Weil homomorphism]] from [[ordinary cohomology]]-groups to [[dg-categories]] of [[infinity-local systems|$\infty$-local systems]]:

* [[Camilo Arias Abad]], Santiago Pineda Montoya, Alexander Quintero Velez, *Chern-Weil theory for $\infty$-local systems* $[$[arXiv:2105.00461](https://arxiv.org/abs/2105.00461)$]$

[[!redirects flat infinity-connections]]

[[!redirects flat ∞-connection]]
[[!redirects flat ∞-connections]]

[[!redirects flat infinity-vector bundle]]
[[!redirects flat infinity-vector bundles]]
[[!redirects flat ∞-vector bundle]]
[[!redirects flat ∞-vector bundles]]

[[!redirects infinity-local system]]
[[!redirects infinity-local systems]]

[[!redirects ∞-local system]]
[[!redirects ∞-local systems]]

[[!redirects flat (∞,1)-bundle]]
[[!redirects flat (∞,1)-bundles]]

[[!redirects flat (infinity,1)-bundle]]
[[!redirects flat (infinity,1)-bundles]]

[[!redirects flat principal ∞-connection]]
[[!redirects flat principal ∞-connections]]
[[!redirects flat principal infinity-connection]]
[[!redirects flat principal infinity-connections]]

[[!redirects flat ∞-bundle]]
[[!redirects flat ∞-bundles]]

[[!redirects flat infinity-bundle]]
[[!redirects flat infinity-bundles]]


flat K-theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[topological K-theory]] (at least), *flat K-theory* (first considered in [Karoubi 1987, §7.21](#Karoubi87), [1990, Ex. 3](#Karoubi90)) is the [[Whitehead-generalized cohomology theory]] which is ([[Brown representability theorem|represented by]]) the [[homotopy fiber]] of the [[Chern character]]; equivalently the restriction (namely the [[homotopy pullback]]) of [[differential K-theory]] (any version of it) to vanishing generalized-[[curvature]].

Since the [[coefficients]] of flat [[KU]] form the [[circle group]] $U(1) \simeq \mathbb{R}/\mathbb{Z}$, flat K-theory is also known as *$\mathbb{R}/\mathbb{Z}$-K-theory* or as *K-theory with $\mathbb{R}/\mathbb{Z}$*-coefficients ([Lott 1994](#Lott94)), or similar.

In the [[differential cohomology hexagon]] for [[topological K-theory]], flat K-theory appears in outer [[long exact sequence]] ([Simons & Sullivan 2008, p. 2](#SimonsSullivan08)):

> (graphics from [[schreiber:Anyonic defect branes in TED-K-theory|SS22]])

## Related concepts

* [[differential K-theory]]

* [[flat connection]]

* [[flat vector bundle]]

* [[secondary characteristic class]]

* [[differential cohomology hexagon]]

## References

Original accounts:

* {#Karoubi87} [[Max Karoubi]], §7.21 in: *Homologie cyclique et K-théorie*, Astérisque, **149** (1987) 156 $[$[numdam:AST_1987__149__1_0](http://www.numdam.org/item/AST_1987__149__1_0)$]$

* {#Karoubi90} [[Max Karoubi]], Ex. 3 in: *Théorie générale des classes caractéristiques secondaires*, K-Theory **4** 1 (1990) 55-87 $[$[doi:10.1007/BF00534193](http://dx.doi.org/10.1007/BF00534193)$]$

* {#Lott94} [[John Lott]], (16) in: *$\mathbf{R}/\mathbf{Z}$ index theory*, Communications in Analysis and Geometry **2** 2 (1994) 279-311 $[$[doi:10.4310/CAG.1994.v2.n2.a6](https://dx.doi.org/10.4310/CAG.1994.v2.n2.a6)$]$

Discussion as a [[vertex]] in the [[differential cohomology hexagon]]:

* {#SimonsSullivan08} [[James Simons]], [[Dennis Sullivan]], p. 2 of: *Structured vector bundles define differential K-theory*, Astérisque, **321** (2008)   $[$[arXiv:0810.4935](https://arxiv.org/abs/0810.4935), [numdam:AST_2008__321__1_0](http://www.numdam.org/item/?id=AST_2008__321__1_0)$]$

[[!redirects R/Z K-theory]]
[[!redirects R/Z-K-theory]]


flat modality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

On a [[local topos]]/[[local (∞,1)-topos]] $\mathbf{H}$, hence with extra [[full and faithful (infinity,1)-functor|fully faithful]] [[right adjoint]] $coDisc$ to the [[global section geometric morphism]] $(Disc \dashv \Gamma)$, is canonically induced the [[idempotent monad|idempotent]] [[comonad]] $\flat \coloneqq Disc\circ \Gamma$. This [[modality]] sends for instance pointed [[connected]] objects $\mathbf{B}G$ to [[coefficients]] $\flat \mathbf{B}G$ for [[flat principal ∞-connections]], and may therefore be referred to as the _flat modality_. It is itself the [[left adjoint]] in an [[adjoint modality]] with the [[sharp modality]] $\sharp \coloneqq coDisc \circ \Gamma$. If $\mathbf{H}$ is in addition a [[cohesive (∞,1)-topos]] then it is also the [[right adjoint]] in an [[adjoint modality]] with the [[shape modality]] $\int$.

### In type theory

There are multiple ways to define the flat modality in type theory, by using [[axioms]] and unjustified inference rules, and by using [[natural deduction]] [[inference rules]]. There are advantages and disadvantages to both:

* The [[natural deduction]] [[inference rules]] for flat modalities are simple; they involve the usual 4 rules for positive types: [[formation rules]], [[introduction rules]], [[elimination rules]], and [[computation rules]], and they could be constructed for all types. However, the syntax of the type theory itself becomes significantly more complex compared to vanilla [[dependent type theory]]. One needs, in addition to element judgments $x:A$, crisp element judgments $x::A$, as well as some way of distinguishing between element judgments and crisp element judgments in the [[context]] such as [[split contexts]].

* When using [[axioms]] or unjustified inference rules, the flat modality could be defined in vanilla [[dependent type theory]]. However, one needs the [[sharp modality]] and a [[type universe]] $U$ already defined in the dependent type theory, and the flat modality is only defined for types in the universe $\sharp U$, rather than all possible types.

#### Natural deduction inference rules

We assume a [[dependent type theory]] with crisp term judgments $a::A$ in addition to the usual (cohesive) type and term judgments $A \; \mathrm{type}$ and $a:A$, as well as context judgments $\Xi \vert \Gamma \; \mathrm{ctx}$ where $\Xi$ is a list of crisp term judgments, and $\Gamma$ is a list of cohesive term judgments. A crisp type is a type in the context $\Xi \vert ()$, where $()$ is the empty list of cohesive term judgments. In addition, we also assume the dependent type theory has [[typal equality]] and [[judgmental equality]].

From here, there are two different notions of the flat modality which could be defined in the type theory, the **strict flat modality**, which uses [[judgmental equality]] in the [[computation rule]] and [[uniqueness rule]], and the **weak flat modality**, which uses [[typal equality]] in the [[computation rule]] and [[uniqueness rule]]. When applied to a type they result in **strict flat types** and **weak flat types** respectively. The [[natural deduction]] rules for strict and weak flat types are provided as follows:

* Formation rule for weak and strict flat types:

$$\frac{\Xi, \Gamma \vert () \vdash A \; \mathrm{type}}{\Xi \vert \Gamma \vdash \flat A \; \mathrm{type}}\flat-\mathrm{form}$$

* Introduction rule for weak and strict flat types:

$$\frac{\Xi, \Gamma \vert () \vdash a:A}{\Xi \vert \Gamma \vdash a^\flat:\flat A}\flat-\mathrm{intro}$$

* Elimination rule for weak and strict flat types:

$$\frac{\Xi \vert \Gamma, x:\flat A \vdash C \; \mathrm{type} \quad \Xi \vert \Gamma \vdash M:\flat A \quad \Xi, u::A \vert \Gamma \vdash N : C[u^\flat/x]}{\Xi \vert \Gamma \vdash (\mathrm{let}\; u^\flat \coloneqq M \;\mathrm{in}\; n):C[M/x]}\flat-\mathrm{elim}$$

* Computation rule for weak and strict flat types respectively:

$$\frac{\Xi \vert \Gamma, x:\flat A \vdash C \; \mathrm{type} \quad \Xi \vert () \vdash M:\flat A \quad \Xi, u::A \vert \Gamma \vdash N : C[u^\flat/x]}{\Xi \vert \Gamma \vdash (\mathrm{let}\; u^\flat \coloneqq M \;\mathrm{in}\; N) \equiv N[M/u]:C[M^\flat/x]}\flat-\mathrm{comp}\mathrm{strict}$$

$$\frac{\Xi \vert \Gamma, x:\flat A \vdash C \; \mathrm{type} \quad \Xi \vert () \vdash M:\flat A \quad \Xi, u::A \vert \Gamma \vdash N : C[u^\flat/x]}{\Xi \vert \Gamma \vdash \beta_{\flat}(M, N):(\mathrm{let}\; u^\flat \coloneqq M \;\mathrm{in}\; N) =_{C[M^\flat/x]} N[M/u]}\flat-\mathrm{comp}\mathrm{weak}$$

Weak flat modalities are primarily used in cohesive [[weak type theories]], while strict flat modalities are typically used in cohesive type theories which are not weak, such as cohesive [[Martin-Löf type theory]] ([[cohesive homotopy type theory]] or cohesive [[higher observational type theory]].

#### Axioms for the flat modality

See [Shulman & Schreiber 2014](#ShulmanSchreiber14) for the time being.

...

We assume that the dependent type theory has the [[sharp modality]] $\sharp$ defined by natural deduction inference rules and a [[Tarski universe]] $(U, T)$. The codiscrete universe is given by the type $\sharp U$ with type families $A:\sharp U \vdash \sharp(T)(A)$ The axioms are given by

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, A:\sharp U \vdash \mathrm{escape}(A):U} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, A:\sharp U \vdash T(\mathrm{escape}(A)) \equiv \sum_{B:\sharp \sum_{X:U} T(X)} \sharp(\pi_1)(B) =_{\sharp U} A}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, A:\sharp U \vdash \mathrm{eIsDisc}(A) \; \mathrm{type}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, A:\sharp U \vdash \mathrm{eIsDiscIsProp}(A):\mathrm{isProp}(\mathrm{eIsDisc}(A))}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, A:\sharp U \vdash \flat A:\sharp U}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, A:\sharp U \vdash \mathrm{flatIsDisc}(A):\mathrm{eIsDisc}(\sharp(T)(\flat A))}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, A:\sharp U \vdash \epsilon(A):T(\mathrm{escape}(\sharp(\flat A \to A)))}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, A:\sharp U, B:\sharp U \vdash \mathrm{flr}(A, B):\mathrm{eIsDisc}(A) \to \mathrm{isEquiv}(\lambda f:T(\mathrm{escape}(\sharp(A \to \flat B))).\epsilon(B) \circ f)}$$

...

## Properties

### In type theory

In [[type theory]], there are commuting conversion rules for the flat modality, which are derivable from the inference rules for the flat modality.

\begin{theorem}
The $\mathrm{let}$ in the elimination rules for the flat modality commutes with itself:

$$\frac{
\begin{array}{c}
\Xi \vert () \vdash A \; \mathrm{type} \quad \Xi \vert () \vdash B \; \mathrm{type} \quad \Xi \vert \Gamma, y:\flat B \vdash C \; \mathrm{type} \\
\Xi \vert \Gamma \vdash M:\flat A \quad \Xi, u::A \vert \Gamma \vdash N:\flat B \quad \Xi, v::B \vert \Gamma \vdash P:C[v^\flat/y]
\end{array}
}{\Xi \vert \Gamma \vdash (\mathrm{let}\; v^\flat \coloneqq :(\mathrm{let}\; u^\flat \coloneqq M \;\mathrm{in}\; N) \;\mathrm{in}\; P) =_{C[v^\flat/y]} (\mathrm{let}\; u^\flat \coloneqq M \;\mathrm{in}\; (\mathrm{let}\; v^\flat \coloneqq N \;\mathrm{in}\; P))}$$
\end{theorem}

\begin{proof}
to be done...
\end{proof}

### Relation to discrete and codiscrete objects

* The [[codiscrete objects]] in a local topos are the [[modal types]] for the [[sharp modality]].

* The [[discrete objects]] in a local topos are the [[modal types]] for the flat modality.

## Related concepts

[[!include cohesion - table]]

## References

The terminology of the flat-modality in the above sense was introduced -- in the language of [[(infinity,1)-toposes|$(\infty,1)$-toposes]] and as part of the axioms on "[[cohesive (infinity,1)-toposes|cohesive $(\infty,1)$-toposes]]" -- in:

* [[Urs Schreiber]], [[Zoran Škoda]], [§7.4.2](https://arxiv.org/pdf/1004.2472.pdf#page=31) of: *Categorified symmetries* [[arXiv:1004.2472](https://arxiv.org/abs/1004.2472)]

* [[Urs Schreiber]], *[[schreiber:differential cohomology in a cohesive topos|Differential cohomology in a cohesive $\infty$-topos]]* (2013)

See also the references at _[[local topos]]_.

Early discussion in view of [[homotopy type theory|homotopy]] [[type theory]] and as part of a set of  axioms for [[cohesive homotopy type theory]] is in

* {#ShulmanSchreiber14} [[Mike Shulman]], [[Urs Schreiber]], *[[schreiber:Quantum gauge field theory in Cohesive homotopy type theory]]* (2014)

The dedicated type theory formulation with "crisp" types, as part of the formulation of [[real cohesive homotopy type theory]], is due to:

* [[Mike Shulman]], *Brouwer's fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

[[!redirects flat modality]]
[[!redirects flat type]]
[[!redirects flat types]]

[[!redirects strict flat modality]]
[[!redirects strict flat type]]
[[!redirects strict flat types]]

[[!redirects weak flat modality]]
[[!redirects weak flat type]]
[[!redirects weak flat types]]


flat module

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In [[algebra]], the notion of *flat* [[modules]] (due to [GAGA](#GAGA)) is a generalisation of the class of [[torsion-free modules]] over [[rings]] $R$ that are not [[PID]].

An $R$-module $M$ is _flat_ (from the French word “*plat*”) if it has “no torsion” in the sense that the [[Tor]]-functor vanishes for every $R$-module $X$:
$$
  M\;\text{is flat}
  \;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;\;
  \underset{X \in R Mod}{\forall}
  \;\;\;
  \mathrm{Tor}_1^R(M, X) \,=\, 0
  \,.
$$

\begin{remark}\label{TerminologyAndRelationToFiberBundles}
**(terminology and relation to fiber bundles)**
\linebreak
[[finitely presented module|Finitely presented]] flat modules are locally free \[\ref{FinitelyPresentedFlatIsProjective}\], hence
under the [[duality between algebra and geometry|dual geometric interpretation]] of [modules as generalized vector bundles](module#RelationToVectorBundlesInIntroduction) over the space on which $R$ is the [[ring of functions]], flatness of a finitely presented module is essentially the _[[local triviality]]_ of this bundle, hence in particular the fact that the [[fibers]] of the bundle do not change, up to isomorphism (cf. *[[fiber bundle]]*).

On the other hand there is **no** relation to "flat" as in [[flat connection]] on such a bundle.
\end{remark}

## Definition
 {#Definition}

We first state the definition and its equivalent reformulations over a [[commutative ring]] abstractly in _[In terms of exact functors and Tor-functors](#InTermsOfExactFunctorsAndTorFunctors)_. Then we give an explicit element-wise characterization in _[Explicitly in terms of identities](#InTermsOfIdentities)_.

Much of this discussion also works in the more general case where the ring is not-necessarily taken to be commutative and not necessarily required to be equipped with a unit, this we indicate in _[For more general rings](#spring)_.

### In terms of exact functors and Tor-functors
 {#InTermsOfExactFunctorsAndTorFunctors}

Let $R$ be a [[commutative ring]].

+-- {: .num_defn #FlatModule}
###### Definition

An $R$-[[module]] $N$ is __flat__ if [[tensor product of modules|tensoring]] with $N$ over $R$ as a [[functor]] from $R$[[Mod]] to itself
$$
  (-)\otimes_R N : R Mod \to R Mod
$$

is an [[exact functor]] (sends [[short exact sequences]] to short exact sequences).

=--

+-- {: .num_defn #FaithfullyFlatModule}
###### Definition

A module as above is _faithfully flat_ if it is flat and tensoring in addition _reflects exactness_, hence if the tensored sequence is exact if and only if the original sequence was.

=--

+-- {: .num_remark #ImmediateReformulationOfFlatness}
###### Remark

The condition in def. \ref{FlatModule} has the following immediate equivalent reformulations:

1. $N$ is flat precisely if $(-)\otimes_R N$ is a [[left exact functor]],

   because tensoring with any module is generally already a [[right exact functor]];

1. $N$ is flat precisely if $(-)\otimes_R N$ sends [[monomorphisms]] ([[injections]]) to monomorphisms,

   because for a right exact functor to also be left exact the only remaining condition is that it preserves the monomorphisms on the left of a [[short exact sequence]];

1. $N$ is flat precisely if $(-)\otimes_R N$ is a [[flat functor]],

   because [[Mod]] is [[finitely complete category|finitely complete]];

1. $N$ is flat precisely if the degree-1 [[Tor]]-[[functor]] $Tor_1^{R Mod}(-,N)$ is zero,

   because by the general properties of [[derived functors in homological algebra]], $L_1 F$ is the obstruction to a [[right exact functor]] $F$ being left exact;

1. $N$ is flat precisely if all higher [[Tor]] functors $Tor_{\geq 1}(R Mod)(-,N)$ are zero,

    because the higher [[derived functor in homological algebra|derived functors]] of an [[exact functor]] vanish;

1. $N$ is flat precisely if $N$ is an [[acyclic object]] with respect to the tensor product functor;

   because the [[Tor]] functor is symmetric in both arguments and an object is called tensor-[[acyclic object]] if all its positive-degree $Tor$-groups vanish.

=--

The condition in def. \ref{FlatModule} also has a number of not so immediate equivalent reformulations. These we discuss in detail below in _[Equivalent characterizations](#EquivalentCharacterizations)_. One of them gives an explicit characterization of flat modules in terms of relations beween their elements.  An exposition of this we give now in _[In terms of identities](#InTermsOfIdentities)_.

### Explicitly in terms of identities
 {#InTermsOfIdentities}

There is a characterisation of flatness that says that a left $A$-module $M$ is flat if and only if "everything (that happens in $M$) happens for a reason (in $A$)". We indicate now what this means. Below in prop. \ref{RelationsFromCharacterizationonIdealInclusion} it is shown how this is equivalent to def. \ref{FlatModule} above.

The meaning of this is akin to the existence of [[basis of a vector space|bases in vector spaces]].  In a [[vector space]], say $V$, if we have an identity of the form $\sum_i \alpha_i v_i = 0$ then we cannot necessarily assume that the $\alpha_i$ are all zero.  However, if we choose a basis then we can write each $v_i$ in terms of the basis elements, say $v_i = \sum_j \beta_{i j} u_j$, and substitute in to get $\sum_{i j} \alpha_i \beta_{i j} u_j = 0$.  Now as $\{u_j\}$ forms a basis, we can deduce from this that for each $j$, $\sum_i \alpha_i \beta_{i j} = 0$.  These last identities happen in the coefficient field, which is standing in place of $A$ in the analogy.

When translating this into the language of modules we cannot use bases so we have to be a little more relaxed.  The following statement is the right one.

Suppose there is some identity in $M$ of the form $\sum_i a_i m_i = 0$ with $m_i \in M$ and $a_i \in A$.  Then there is a family $\{n_j\}$ in $M$ such that every $m_i$ can be written in the form $m_i = \sum_j b_{i j} n_j$ and the coefficients $b_{i j}$ have the property that $\sum_i a_i b_{i j} = 0$.

The module $M$ being flat is equivalent to being able always to do this.

There is an alternative way to phrase this which is less element-centric.  The elements $m_i$ correspond to a [[morphism]] into $M$ from a [[free module]], say $m \colon F \to M$.  The $a_i$ correspond to a morphism $a \colon F \to F$, multiplying the $i$th term by $a_i$.  That we have the identity $\sum_i a_i m_i = 0$ says that the composition $m a$ is zero, or that $m \colon F \to M$ factors through the [[coequaliser]] of $a$ and $0$.

Now we consider the elements $n_j$.  These define another morphism from a free module, say $n \colon E \to M$.  That the $m_i$ can be expressed in terms of the $n_j$ says that the morphism $m$ factors through $n$.  That is, there is a morphism $b \colon F \to M$ such that $m = n b$.  We therefore have two factorisations of $m$: one through $n$ and one through the [[cokernel]] $\coker a$.  The question is as to whether these have any relation to each other.  In particular, does $\coker a \to M$ factor through $n$?  We can represent all of this in the following diagram.

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=--

Saying that $M$ is flat says that this lift always occurs.

+-- {: .num_prop #RelationsFromCharacterizationonIdealInclusion}
###### Proposition

A module $N$ is flat precisely if for every finite linear combination of zero, $\sum_i r_i n_i  = 0\in N$ with $\{r_i \in R\}_i$,  $\{n_i \in N\}$ there are elements $\{\tilde n_j \in N\}_j$ and linear combinations

$$
  n_i = \sum_j b_{i j} \tilde n_j \;\;\in  N
$$

with $\{b_{i j} \in R\}_{i,j}$ such that for all $j$ we have

$$
  \sum_i r_i b_{i j}  = 0\;\;\; \in R
  \,.
$$

=--

+-- {: .proof}
###### Proof

A finite set $\{r_i \in R\}_i$ corresponds to the inclusion of a finitely generated ideal $I \hookrightarrow R$.

By theorem \ref{CharacterizationOnIdealInclusions} $N$ is flat precisely if $I \otimes_R N \to N$ is an injection. This in turn is the case precisely if the only element of the tensor product $I \otimes_R R$ that is 0 in $R \otimes_R N = N$ is already 0 on $I \otimes_R N$.

Now by definition of [[tensor product of modules]] an element of $I \otimes_R N$ is of the form $\sum_i (r_i ,n_i)$ for some $\{n_i \in N\}$. Under the inclusion $I \otimes_R N \to N$ this maps to the actual linear combination $\sum_i r_i n_i$. This map is injective if whenever this linear combination is 0, already $\sum_i (r_i, n_i)$ is 0.

But the latter is the case precisely if this is equal to a combination $\sum_j (\tilde r_j  , \tilde n_j)$ where all the $\tilde r_j$ are 0. This implies the claim.

=--

Taking this a step further, we consider the [[filtered category|filtered family]] of all finite [[subsets]] of $M$.  This generates a filtered family of [[finitely generated module|finitely generated]] [[free modules]] with compatible morphisms to $M$.  So there is a morphism from the [[colimit]] of this family to $M$.  This morphism is [[surjection|surjective]] by construction.  To show that it is [[injection|injective]], we need to show that any element in one of the terms in the family that dies by the time it reaches $M$ has actually died on the way.  This is precisely what the above characterisation of flatness is saying: the element corresponding to $\sum_i a_i m_i$ that dies in $M$ is already dead by the time it reaches $E$.

We have thus arrived at the following result:

+-- {: .num_theorem #filtfree}
###### Theorem
A module is flat if and only if it is a [[filtered colimit]] of finitely generated [[free modules]].
=--

This observation (Wraith, Blass) can be put into the more general context of modelling [[geometric theory|geometric theories]] by [[geometric morphism|geometric morphisms]] from their [[classifying topos|classifying toposes]], or equivalently, certain [[flat functor|flat functors]] from [[site|sites]] for such topoi.

### For more general rings

Even if the [[ring]] $R$ is not necessarily commutative and not necessarily unital, we can say:

A left $R$-[[module]] is flat precisely if the [[tensor product of modules|tensoring]] functor

$$
  (-)\otimes_R \colon Mod_R \to Ab
$$

from _right_ $R$ modules to [[abelian groups]] is an [[exact functor]].

##  General properties

### Properties of the category of flat modules

\begin{proposition}
The full subcategory of flat modules over a ring $R$ is stable under
the following categorical constructions :

1. (infinite) sums ;
1. filtered colimits ;
1. retracts;
1. tensor product.

\end{proposition}

+-- {: .num_prop #ProjectiveModulesAreFlat}
###### Corollary

Every [[projective module]] is flat.

=--

+-- {: .proof}
###### Proof

Every projective module is a retract of a free module, which in turn is a sum of copies of the base ring $R$, which is flat over itself.

=--

\begin{corollary}
Let $R$ be a commutative ring and $S \subset R$ be a multiplicative subset.
Then the localisation $R[S^{-1}]$ is a flat $R$-module.
\end{corollary}

\begin{proof}
Regard the set $S$ as a poset with de divisibility relation. The fact that $S$ be multiplicative means that the poset is directed.
One thus have that $R[S^{-1}] = \underset{\longrightarrow}{\lim}_s \;\frac{1}{s} \; R$ is a filtered colimit of free modules, hence it is flat.
\end{proof}

### Flatness is a local property

\begin{proposition}(Stability under base change)
Let $R \to S$ be a morphism of commutative rings and let $M$ be a flat $R$-module, then the $S$-module $M \otimes_R S$ is also flat.
\end{proposition}

\begin{proof}
Let $X \to Y$ be an injective morphism of $S$-modules, then
the map $(M \otimes_R S) \otimes_S X \to (M \otimes_R S) \otimes_S Y$ is
isomorphic to the map $M \otimes_R X \to M \otimes_R X$ which is injective because $M$ is flat.
\end{proof}

\begin{proposition}(Flatness is a local property)
Let $R$ be a commutative ring and let $M$ be an $R$-module. Then the following assertions are equivalent :

1. $M$ is flat;
1. $M_\mathfrak{p}$ is a flat $R_\mathfrak{p}$-module, for every prime ideal $\mathfrak{p} \subset R$;
1. $M_\mathfrak{m}$ is a flat $R_\mathfrak{m}$-module, for every maximal ideal $\mathfrak{m} \subset R$.

\end{proposition}

\begin{proof}

$1 \Rightarrow 2$. By base change.

$2 \Rightarrow 3$. Obvious.

$3 \Rightarrow 1$. Let $X \to Y$ be an injective morphism of $R$-modules and let $K$ denote the kernel of the induced map $M \otimes_R X \to M \otimes_R Y$. Since $R_\mathfrak{m}$ is a flat $R$-module it follows that
$X_\mathfrak{m} \to Y_\mathfrak{m}$ is also injective ; and $K_\mathfrak{m}$ is the kernel of $M_\mathfrak{m} \otimes_{R_\mathfrak{m}} X_\mathfrak{m}\to M_\mathfrak{m} \otimes_{R_\mathfrak{m}} Y_\mathfrak{m}$.
By the flatness assumption, we deduce that $K_\mathfrak{m} = 0$ for every maximal ideal $\mathfrak{m} \subset R$, so $K = 0$ and $M$ is flat.
\end{proof}

### Flat modules are torsion-free

\begin{proposition}
For any module $M$ over a ring $R$, one has
$$
  M\;\text{is flat}
  \;\;\;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;\;\;
  M\;\text{is torsion-free}
  \,.
$$
\end{proposition}

\begin{proof}
For every regular element $p \in R$, the $p$-torsion submodule of $M$ is isomorphic to $\mathrm{Tor}^R_1(M, R/(p))$.
\end{proof}

The converse holds for semi-hereditary rings.

## Reduction to cyclic modules

\begin{theorem}(Reduction to cyclic modules)
Let $R$ be a commutative ring and let $M$ be an $R$-module.
If
$$
  Tor_1^R(M, R/I) = 0
$$
for every finitely generated ideal $I \subset R$, then $M$ is a flat $R$-module.

Equivalently, if
$$
  I \otimes_R M \to IM
$$
is an isomorphism for every finitely generated ideal $I \subset R$, then
$M$ is flat.
\end{theorem}

\begin{proof}
The proof goes in 3 steps.

1. $Tor_1^R(M, R/I) = 0$ for every ideal $I \subset R$. Indeed, write $I$ as the union of the finitely generated ideals $I_\mathrm{fin} \subset I$, then $R/I = \underset{\longrightarrow}{\lim}_{I_\mathrm{fin} \subset I} R/I_\mathrm{fin}$. Then
because $Tor_1$ commutes with filtered colimits
$$
  Tor_1(M, R/I) = Tor_1(M, \underset{\longrightarrow}{\lim}_{I_fin \subset I} R/I_fin) = \underset{\longrightarrow}{\lim}_{I_fin \subset I} Tor_1(M, R/I_fin) = 0
$$
1. $Tor_1^R(M, X) = 0$ for every finitely generated module $X$. We can show this by induction on the number of generators of the module. The case of $n = 1$ generator is covered by the previous step. Assume that $Tor_1(M, X) = 0$ for every module generated by $n$ elements. Let $X$ be generated by $n+1$ elements $x_0, \dots, x_n$. Let us call $Y$ the submodule of $X$ generated by $x_1, \dots, x_n$, then one has a short exact sequence $0 \to Y \to X \to X/Y \to 0$ leading to the sequence
$$
  0 = Tor_1(M, Y) \to Tor_1(M, X) \to Tor_1(M, X/Y) = 0
$$
which is exact in the middle, so $Tor_1(M, X) =  0$.
1. For the last step, we use the fact that every module is the union of its finitely generated submodules and that $Tor_1(M,-)$ commutes with filtered colimits.
$$
  Tor_1(M, X) = Tor_1\left(M, \bigcup_{X_fin \subset X} X_fin\right) = \underset{\longrightarrow}{\lim}_{X_fin \subset X} Tor_1(M, X_fin) = 0
$$
for every module $X$.

The equivalence with the second characterisation comes from the fact that
using the exact sequence $0 \to I \to R \to R/I \to 0$ and tensoring it with $M$, one gets the long exact sequence
$$
  0 \to Tor_1(M, R/I) \to I \otimes R \to R \to R/I \to 0
$$
identifying $Tor_1(M, R/I)$ as the kernel of the always surjective map
$I \otimes M \to IM \subset R$.
\end{proof}

## Finitely generated flat modules

Thanks to the general structure theorem of flat modules, one gets

\begin{corollary}
Given a [[finitely presented]] module $M$, one has
$$
  M\;\text{is flat}
  \;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;\;
  M\;\text{is projective}
  \,.
$$
\label{FinitelyPresentedFlatIsProjective}
\end{corollary}

\begin{proof}
If $M$ is at the same time finitely presented and a filtered colimit of finitely generated free modules, the identity morphism $M \to M$ must factor as $M \to F \to M$ where $F$ is a finitely generated free module. This implies that $M$ is projective.
\end{proof}

In many cases, “finite presentation” can be replaced with “finite generation”.

\begin{proposition}(When $R$ is local)

Let $M$ be a [[finitely generated module]] over a [[local ring]] $R$.
Then one has
$$
  M\;\text{is flat}
  \;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;\;
  M\;\text{is free}
  \,.
$$

\end{proposition}

This is [Matsumara, Theorem 7.10](#Matsumara).

Thus we see that finitely generated flat modules are locally free in the weak sense. Let us recall the definition of locally free modules and their relation with projective modules.

+-- {: .num_defn #LocallyFreeModule}
###### Definition

An $R$-module $M$ is called a **locally free module in the weak sense** if for every [[prime ideal]] $\mathfrak{p} \hookrightarrow R$ the localised module $M_\mathfrak{p}$ is a [[free module]] over the [[localization of a ring|localization]] $R_\mathfrak{p}$.

=--

\begin{proposition}
Let $M$ be a [[finitely generated module]], then the following are equivalent :

1. $M$ is projective ;
1. $M$ is locally free in the weak sense and the rank function $\mathfrak{p} \mapsto \mathrm{r}(\mathfrak{p})$ is locally constant.

\end{proposition}

Then one has the following characterisation of finitely generated flat modules.

\begin{proposition}
Let $M$ be a [[finitely generated module]], then
$$
  M\;\text{is flat}
  \;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;\;
  M\;\text{is locally free in the weak sense}
  \,.
$$
\end{proposition}

This characterisation can be strengthened in two very large cases

\begin{proposition}(When $R$ is Noetherian)
If $R$ is a [[Noetherian ring]] and $M$ is a [[finitely generated module]], then
$$
  M\;\text{is flat}
  \;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;\;
  M\;\text{is projective}
  \,.
$$
\end{proposition}

\begin{proof}
Over a Noetherian ring all finitely generated modules are finitely presented.
So this is a simple consequence of Corollary (\ref{FinitelyPresentedFlatIsProjective})
\end{proof}

\begin{proposition}(When $R$ is an integral domain)
If $R$ is an [[integral domain]] and $M$ is a [[finitely generated module]], then
$$
  M\;\text{is flat}
  \;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;\;
  M\;\text{is projective}
  \,.
$$
\end{proposition}

\begin{proof}
Given two prime ideals $\mathfrak{p} \subset \mathfrak{q}$, one gets
$\mathrm{r}(\mathfrak{p}) = \mathrm{r}(\mathfrak{q})$.
Since $R$ is an integral domain $(0)$ is a prime ideal and $(0) \subset \mathfrak{p}$ for every prime ideal $\mathfrak{p}$.
The function $\mathfrak{p} \mapsto \mathrm{r}(\mathfrak{p})$ is thus constant.
\end{proof}

\begin{remark}
The proof of the above proposition shows that the same result is true for every ring $R$ with a unique minimal ideal.
\end{remark}

In the general case, it is not true that all finitely generated flat modules are projective.

\begin{example}([[counter-example]])
Let $R = \prod_{i = 0}^\infty k$ be an infinite product of fields.
Let $I = \oplus_{i = 0}^\infty k \subset R$ its ideal.

Then $R/I$ is a finitely generated $R$-module, flat (all $R$-modules are flat), but not projective.
\end{example}

The class of commutative rings $R$ over which all finitely generated flat modules are projective has been determined in the 1970s.

\begin{theorem}
Let $R$ be a commutative ring. The following two are equivalent :

1. Every finitely generated flat $R$-module is projective ;
1. Every sequence $(a_0, a_1, \dots)$ of elements of $R$ such that
$a_{n+1} a_n = a_n$ for every $0 \leq n$, is eventually constant.

\end{theorem}

One can see Gena Puninski; Philipp Rothmaler (2004). _When every finitely generated flat module is projective_. Journal of Algebra, 277(2), 542–558.         doi:10.1016/j.jalgebra.2003.10.027

## Countably presented flat modules

As we have seen finitely presented flat module are always projective ; their projective dimension is equal to zero.
For countably presented flat modules, the dimension cannot exceed one.

\begin{theorem}
A countably presented flat module has [[projective dimension]] inferior or equal to $1$.
\end{theorem}

\begin{proof}
Let $M$ be a countably generated flat module. Being flat, it can be written as an inductive limit of finitely generated free modules $M = \underset{\longrightarrow}{\lim}_A M_\alpha$. Since $M$ is also countably presented, there exists a countable cofinal filtered subset $H \subset A$, so one can reduce to the case where $A = \mathbb{N}$.

Denoting by $g_n : M_n \to M_{n+1}$ the transition maps, one can build the short exact sequence
$$
  0 \to \oplus M_n \overset{G}{\to} \oplus M_n \to M \to 0
$$
where the map $G$ restricted to $M_n$ is given by $1 - g_n$.
\end{proof}

## Classes of rings, related to flatness

As we have seen, there is not much difference between flat modules and projective modules in the finitely generated case.
The notion of flatness starts becoming really distinct in the case of infinitely generated modules. For example the $\mathbb{Z}$-module $\mathbb{Q}$ is flat (since it is torsion-free) but far from being free.

On the other side, for infinitely generated module, the notion of projectivity is less useful since for example the module $\mathbb{Z}[[X]]$ which is the infinite product of copies of $\mathbb{Z}$ is _not free_ but still flat.
It is common to encounter infinitely generated flat modules and uncommon to encounter infinitely generated projective modules.

To better understand the general notion of flatness from a geometric perspective, very early after the definition of the notion, two questions have been asked

1. A product of torsion-free modules is always torsion-free, is it true for flat modules ?
1. What type of rings $R$ have all their modules flat?
1. For what type of rings $R$ is it true that all torsion-free modules are flat?
1. For what type of rings can flat modules be the filtered union of their finitely generated projective submodules?

### Coherent rings

\begin{proposition}
For a commutative ring $R$, the following propositions are equivalent :

1. $R$ is a [[coherent ring]] ;
1. the product of a family of flat $R$-modules is flat.

\end{proposition}

\begin{proof}

$1 \Rightarrow 2$. Let $\{Q_\alpha\}_{\alpha \in A}$ be a set of flat $R$-modules. By cyclic reduction, we only need to show that the map $I \otimes \prod_\alpha Q_\alpha \to \prod_\alpha Q_\alpha$ is injective for every finitely generated ideal $I \subset R$. Since $R$ is coherent every such ideal $I$ is a [[finitely presented module]] and thus the map $I \otimes \prod_\alpha Q_\alpha \to \prod_\alpha Q_\alpha$ is the product of the maps $I \otimes Q_\alpha \to Q_\alpha$. It is injective since each individual map is injective by flatness of each $Q_\alpha$.

$2 \Rightarrow 1$. Let $I \subset R$ be a finitely generated ideal.
Using one of the equivalent characterisations of [[finitely presented modules]], we shall show
that the canonical map
$I \otimes R^A \to I^A$ is an isomorphism, for every set $A$.
By assumption, since $R^A$ is a flat module, the map $I \otimes R^A \to R^A$ is injective and since $I$ is finitely generated, its image is $I^A$.
\end{proof}

### Absolutely flat rings

\begin{definition}
We say that a ring $R$ is absolutely flat if every $R$-module is flat.
\end{definition}

\begin{theorem}
The following propositions are equivalent for a commutative ring $R$:

1. $R$ is absolutely flat;
1. every ideal $I \subset R$ is idempotent, i.e. $I^2 = I$;
1. for every $x \in R$ there exists an element $y \in R$ such that $x = x^2 y$ ;
1. $R_\mathfrak{p}$ is a field for every prime ideal $\mathfrak{p} \subset R$;
1. $R_\mathfrak{m}$ is a field for every maximal ideal $\mathfrak{m} \subset R$.

\end{theorem}

\begin{proof}
1 $\Rightarrow$ 2. Let $I \subset R$ be any ideal of $R$, then since $R$ is absolutely flat, the $R$-module $R/I$ must be flat.
One has
$$
  Tor_1(R/I, R/I) \simeq I/{I^2}
$$
the ideal $I$ must thus be idempotent;

2 $\Rightarrow$ 3. For every $x \in R$, the ideal $(x)$ must be idempotent, that is $(x)(x) = (x)$ which means that there must exist $y$ such that $x = x^2y$;

3 $\Rightarrow$ 4. Let $r \in \mathfrak{p}$. Let $u \in R$ such that $r u r = r$. Then $r ( 1 - u r ) = 0$. Since $u r \in \mathfrak{p}$ it follows that the image of $1- u r$ is invertible in $R_\mathfrak{p}$, so the image of $r$ is null in $R_\mathfrak{p}$. Thus the maximal ideal $R_\mathfrak{p}$ is trivial.

4 $\Rightarrow$ 5. Obvious.

5 $\Rightarrow$ 1. Because flatness is a local property.

\end{proof}

### Semi-hereditary rings

\begin{definition}
Recall that a commutative ring $R$ is  [[semi-hereditary]] if every finitely generated ideal is projective.
\end{definition}

\begin{theorem}
For a commutative ring $R$, the following are equivalent :

1. $R$ is [[semi-hereditary]] ;
1. The total ring of fractions $\mathrm{Q}(R)$ of $R$ is absolutely flat and for every maximal ideal $\mathfrak{m}$ of $R$, the local ring $R_\mathfrak{m}$ is a [[valuation ring]] ;
1. Every torsion-free $R$-module is flat.

\end{theorem}

\begin{proof}
First, if $R$ is a local ring, this follows from the equivalent characterisations of [[valuation rings]]. So we only need to take care of the global part of the theorem.

$1 \Rightarrow 2$. Every $R_\mathfrak{m}$ is a localisation of $R$, so it is semi-hereditary and hence a valuation ring.
To show that $\mathrm{Q}(R)$ is absolutely flat, it shall be enough to show that for every $x \in R$, there exists a regular $r$ such that $x^2 = r x$. Let $x \in R$, then $(x)$ is projective. As a consequence the annihilator $Ann(x) \subset R$ is a direct summand, so there exists an idempotent $\pi \in R$ such that $Ann(x) = (\pi)$. Then $r \coloneqq x + \pi$ is a regular element such that $x^2 = r x$.

$2 \Rightarrow 3$. When $\mathrm{Q}(R)$ is absolutely flat, being torsion-free becomes a local property, as it is for flatness.

$3 \Rightarrow 1$. Torsion-free modules are stable under product, so it follows that flat modules are also stable under product which means that $R$ is coherent. Then every finitely generated ideal $I \subset R$ is finitely presented. Since they are also torsion-free, they are flat and thus projective.
\end{proof}
## Examples
 {#Examles}

\begin{example}
Since $\mathbb{Z}$ is a [[PID]], an [[abelian group]] is flat (regarded as a $\mathbb{Z}$-module) precisely if it is [[torsion subgroup|torsion-free]].

In particular, the modules $\mathbb{Q}$ and $\mathbb{Z}[[X]]$ are flat.
\end{example}

\begin{example}
The ring of power series $R[[X]]$ (which as an $R$-module is simply an infinite product of the base ring) is an interesting example.

1. It is always torsion-free;
1. it is a flat $R$-module if and only if $R$ is a [[coherent ring]];
1. it is almost never free. In particular $\mathbb{Z}[[X]]$ is not free.

\end{example}

For the flatness of $R[[X]]$ see [_Direct products of modules_, Theorem 2.1](#Chase).

\begin{example}
Let $R$ be an integral domain, with field of fractions $K$. Then $K$ is a flat $R$-module, as it can be written as the filtered colimit
$$
  K = \underset{\longrightarrow}{\lim}_r \;\frac{1}{r}\;R
$$
of free $R$-modules.
\end{example}

\begin{example}
If $\pi$ is an idempotent element of a commutative ring $R$, then
the cyclic module $R/(\pi)$ is projective, hence flat.
\end{example}

## Related concepts

* [[flat morphism of schemes]]

* [[flat resolution lemma]]

* [[projective object]], [[projective presentation]], [[projective cover]], [[projective resolution]]

* [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

  * [[injective module]]

* [[free object]], [[free resolution]]

* flat object, [[flat resolution]]

* [[free module]] $\Rightarrow$ [[projective module]] $\Rightarrow$ **flat module** $\Rightarrow$ [[torsion-free module]]

## References

Flat modules (“modules plats” in French) where first defined in [[GAGA]]

* Jean-Pierre Serre, _Géométrie Algébrique et Géométrie Analytique_ ([pdf](http://www.numdam.org/article/AIF_1956__6__1_0.pdf))
{#GAGA}

Original articles include

* Shizuo Endo, _On flat modules over commutative rings_,  J. Math. Soc. Japan Volume 14, Number 3 (1962), 284-291. ([EUCLID](http://projecteuclid.org/euclid.jmsj/1261060583))

* Shizuo Endo, _On semi-hereditary rings_,  J. Math. Soc. Japan Vol. 13, No. 2, 1961

* Michel Raynaud, Laurent Gruson, _Critères de platitude et de projectivité_, Techniques de "platification" d'un module. Invent. Math. 13 (1971), 1–89.
 {#RaynaudGruson}

* S. Jondrup, _Flat and projective modules_, Math, Scand. 43 (1978) ([pdf](http://www.mscand.dk/article.php?id=2456))

The characterization of flat modules as filtered colimits of [[projective modules]] is due to

* [[Daniel Lazard]], _Sur les modules plats_ C. R. Acad. Sci. Paris 258, 6313–6316 (1964)
 {#Lazard}

For a general account see for instance section 3.2 of

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_

For more details see

* Matsumura's CRT book
 {#Matsumara}

Lecture notes include

* Arthur Ogus, _Flatness -- a brief overview_ ([pdf](http://math.berkeley.edu/~ogus/Math%20_256B--09/Supplements/flat.pdf))

Further resources include

* MO discussion _[flatness and local freeness](http://mathoverflow.net/questions/33522/flatness-and-local-freeness)_

* Stephen Chase, _[Direct product of modules](https://www.ams.org/journals/tran/1960-097-03/S0002-9947-1960-0120260-3/S0002-9947-1960-0120260-3.pdf)
{#Chase}

[[!redirects flat modules]]

[[!redirects faithfully flat module]]
[[!redirects faithfully flat modules]]


flat morphism

#Contents#
* table of contents
{:toc}

## Definition

A [[homomorphism]] of (not necessarily commutative nor unital) [[rings]] $u:R\to S$ is __flat__ if the [[extension of scalars]] (= [[inverse image]]) [[functor]] $u^*:{}_R Mod\to {}_S Mod$, $M\mapsto S\otimes_R M$ is an [[exact functor|exact]] [[additive functor]].

Recall that every [[extension of scalars]] functor for [[modules]] is cocontinuous and a fortiori admits a [[right adjoint]]. Hence this definition is in an agreement with the more general notion of a [[flat functor]] in category and topos theory.

If one pictures a morphism as a family over its [[codomain]], then for many purposes good families are flat (e.g. in [[deformation theory]]).

**Flat modules:** A right $R$-module $M_R$ is **flat** if the functor ${}_R N\mapsto M \otimes_R N$ from the category of left $R$-modules to the category [[Ab]] of abelian groups is exact.

A morphism $f:X\to Y$ of [[schemes]] is **flat** if the induced map on [[stalks]] is a flat morphism of (commutative unital) rings.

A morphism of schemes is **faithfully flat** if it is flat and [[epimorphism|epi]].

A morphism of [[affine schemes]] $Spec(B) \to Spec(A)$, hence coming from a [[ring]] [[homomorphism]] $f \colon A \to B$

* flat precisely if $f$ exhibits $B$ as a [[flat module]] over $A$;

* faithfully flat if $f$ exhibits $B$ as a [[faithfully flat module]] over $A$.

## Properties

+-- {: .num_prop }
###### Proposition

A flat morphism $Spec(B) \to Spec(A)$ is faithfully flat if it is an [[epimorphism]].

=--

(e.g [Milne, footnote 18](#Milne))

## Flat topologies

Given a base scheme $S$, the [[slice category]] $Sch/S$ of [[relative schemes]] over $S$ is equipped with several _flat [[Grothendieck topologies]]_.

These are the [[Grothendieck topologies]] in which covers consist of flat morphisms which set-theoretically cover the target scheme with some additional finiteness conditions. The usual choices are the [[fppf topology|fppf]] (fr. _fidèlement plat de presentation finie_, 'faithfully flat and of finite presentation') and [[fpqc topology|fpqc]] (fr. _fidèlement plat et quasicompact_, 'faithfully flat and quasicompact') topologies.  Cf. [wikipedia:flat topology](http://en.wikipedia.org/wiki/Flat_topology)

## Related concepts

* [[étale morphism of schemes]], [[formally étale morphism of schemes]]

## References

The standard reference is [[EGA IV]]. See also [[flat morphism in derived geometry]].

See also

* [[James Milne]], _[[Lectures on Étale Cohomology]]_
 {#Milne}

[[!redirects flat morphisms]]
[[!redirects flat topology]]
[[!redirects flat morphism of schemes]]

[[!redirects faithfully flat]]

[[!redirects faithfully flat morphism]]
[[!redirects faithfully flat morphisms]]

[[!redirects faithfully flat morphism of schemes]]
[[!redirects faithfully flat morphisms of schemes]]


flat morphism in derived geometry

Let $A$ be an $E_\infty$-ring and let $M$ be an $A$-module.  $M$ is **flat** if

1. $\pi_0 M$ is a [[flat morphism|flat module]] over $\pi_0 A$ in the classical sense (i.e. $\otimes_{\pi_0 A} M$ is an exact functor);

1. For each $n$, the induced map
$$
\pi_n A \otimes_{\pi_0 A} \pi_0 M \to \pi_n M
$$
is an isomorphism.

A map $A \to B$ of $E_\infty$-rings is **flat** if $B$ is flat when regarded as an $A$-module.

The same definitions work for some other contexts of derived local algebra, e.g. dg-algebras.

A morphism $X \to Y$ of [[derived scheme]]s is **flat** if for all affine subsets $U \subset X$ and $ f(U) = V \subset Y$ the induced map on global sections $\Gamma (V) \to \Gamma (U)$ is flat as a map of $E_\infty$-rings.


flat resolution

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* table of contents
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## Idea

In [[homological algebra]] a _flat resolution_ is a [[resolution]] of a [[chain complex]] by a complex of [[flat modules]].

## Related concepts

* [[flat resolution lemma]]

* [[projective object]], [[projective presentation]], [[projective cover]], [[projective resolution]]

  * [[projective module]]

* [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

  * [[injective module]]

* [[free object]], [[free resolution]]

  * [[free module]]

* flat object, **flat resolution**

  * [[flat module]]

[[!redirects flat resolutions]]


flat resolution lemma

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#Contents#
* table of contents
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## Idea

The _flat resolution lemma_ in [[homological algebra]] say that the [[Tor]]-[[derived functor]], which a-priori is defined via [[projective resolutions]], may also be computed via [[resolution]] by [[flat modules]].

## References

This appears for instance as lemma 3.2.8 in

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_

and in lemma 19.17.9 and following in

* [[Aise Johan de Jong]], _[[The Stacks Project]]_


flat section

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#### Differential cohomology
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* table of contents
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## Idea

A [[section]] of a [[bundle]] [[connection on a bundle|with connection]] is called _flat_ or _covariantly constant_ if its [[covariant derivative]] vanishes.

## Related concepts

* [[local section]], [[global section]]

* [[flat connection]]

* [[Killing vector]], [[Killing spinor]]

[[!redirects flat sections]]

[[!redirects covariantly constant section]]
[[!redirects covariantly constant sections]]


flat space holography

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* table of contents
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## Idea

What is called *flat space holography* is a -- somewhat more hypothetical -- generalization of the [[holographic principle]] in the form of the [[AdS-CFT correspondence]], which should be applicable not to the usual [[bulk]] [[spacetimes]] that are [[asymptotic boundary|asymptotically]] [[anti de Sitter spacetime|anti de Sitter]], but to [[bulks]] that are [[asymptotic boundary|asymptotically]] [[Minkowski spacetime|Minkowskian]].

Notice here that the  *[[far horizon geometry]]* of all [[BPS state|BPS]] branes is a [[flat orbifold]] spacetime (see [here](near-horizon+geometry#FarHorizonGeometry)) and that "far" is [[Planck length|Planck scale]] for single branes ("[[cone branes]]"), hence in the [[large 1/N limit]] (opposite to the [[large N limit]] in which the [[AdS-CFT correspondence]] applies).

In one more well-developed version of flat space holography, [[scattering amplitudes]] in [[asymptotic boundary|asymptotically]] flat [[spacetimes]] are related to [[conformal field theories]] on [[celestial spheres]] in two dimensions lower -- "celestial amplitudes". This goes back to an observation in [de Boer & Solodukhin 2003](#deBoerSolodukhin03).

## Related concepts

* [[cone brane]]

* [[holographic light front QCD]]

* [[Penrose limit]]

## References

### General

Early comments:

* [[Edward Witten]], *Baryons and Branes in Anti de Sitter Space* talk at *[String98](https://online.kitp.ucsb.edu/online/strings98/)* (1998) [[web](https://online.kitp.ucsb.edu/online/strings98/witten/)]

  > (by the title, apparently originally intended to touch on [[AdS-QCD]], but de facto ending with focus on the problem of flat space holography)

Dedicayed introduction and survey:

* [[Daniel Grumiller]], *Flat Space Holography*, 2015 ([pdf](http://quark.itp.tuwien.ac.at/~grumil/pdf/schwarzschild2015.pdf), [[Grumiller_FlatSpaceHolography.pdf:file]])

Activities:

* Janell Rodgers, _[Flat Holography: April 4 – 8, 2016](http://scgp.stonybrook.edu/archives/14035)

Original articles:

* [[Leonard Susskind]], *Holography in the flat space limit*, AIP Conf.Proc. 493 (1999) 1, 98-112,  ([spire](https://inspirehep.net/literature/482388), [arXiv:hep-th/9901079](https://arxiv.org/abs/hep-th/9901079), [doi:10.1063/1.1301570](https://doi.org/10.1063/1.1301570))

  > (in relation to the [[BFSS matrix model]])

* {#deBoerSolodukhin03} [[Jan de Boer]], Sergey N. Solodukhin, *A holographic reduction of Minkowski space-time*, Nucl. Phys. B665 (2003) 545-593 ([arXiv:hep-th/0303006](https://arxiv.org/abs/hep-th/0303006))

* [[David Berenstein]], *A toy model for the AdS/CFT correspondence*,  	JHEP 0407 (2004) 018 ([arXiv:hep-th/0403110](https://arxiv.org/abs/hep-th/0403110))

* R. B. Mann, *Flat space holography*, Can. J. Phys. 86 (2008) 563-570 ([spire:1119742](https://inspirehep.net/literature/1119742), [doi:10.1139/p07-188](https://doi.org/10.1139/p07-188))

* Arjun Bagchi, [[Daniel Grumiller]], *Holograms of Flat Space*, International Journal of Modern Physics DVol. 22, No. 12, 1342003 (2013) ([doi:10.1142/S0218271813420030](https://doi.org/10.1142/S0218271813420030))

* Arjun Bagchi, Rudranil Basu, Ashish Kakkar, Aditya Mehra, _Flat Holography: Aspects of the dual field theory_, JHEP 12(2016)147 ([arXiv:1609.06203](https://arxiv.org/abs/1609.06203))

* Reza Fareghbal, Isa Mohammadi, *Flat-space holography and correlators of Robinson–Trautman stress tensor*, Annals of Physics Volume 411, December 2019, 167960 ([doi:10.1016/j.aop.2019.167960](https://doi.org/10.1016/j.aop.2019.167960))

For [[JT-gravity]]/[[SYK model]]-duality (i.e. analogous to [[AdS2/CFT1 duality]]):

* Hamid Afshar, Hernan Gonzalez, [[Daniel Grumiller]], Dmitri Vassilevich, *Flat space holography and complex SYK*, Phys. Rev. D 101, 086024 ([arXiv:1911.05739](https://arxiv.org/abs/1911.05739), [doi:10.1103/PhysRevD.101.086024](http://dx.doi.org/10.1103/PhysRevD.101.086024))

### Celestial amplitudes and celestial holography

Introduction and survey:

* [[Sabrina Pasterski]], *Lectures on Celestial Amplitudes*, The European Physical Journal C **81** 1062 (2021) [[arXiv:2108.04801](https://arxiv.org/abs/2108.04801), [doi:10.1140/epjc/s10052-021-09846-7](https://doi.org/10.1140/epjc/s10052-021-09846-7)]

* [[Sabrina Pasterski]], *Celestial Amplitudes*, talk at [Strings 2021](https://www.ictp-saifr.org/strings2021) [slides: [pdf](https://www.ictp-saifr.org/wp-content/uploads/2021/07/Sabrina-G.-Pasterski.pdf), video: [YT](https://youtu.be/cXSv_SWQsTs)]

* [[Ana-Maria Raclariu]], *Lectures on Celestial Holography* [[arXiv:2107.02075](https://arxiv.org/abs/2107.02075)]

Original articles:

* [[Nima Arkani-Hamed]], Monica Pate, [[Ana-Maria Raclariu]], [[Andrew Strominger]], *Celestial Amplitudes from UV to IR*, JHEP 08 (2021) 062 ([arXiv:2012.04208](https://arxiv.org/abs/2012.04208))

(...)

* Charlotte Sleight, Massimo Taronna, *Celestial Holography Revisited* ([arXiv:2301.01810](https://arxiv.org/abs/2301.01810))

Description in terms of [[topological strings]] on [[twistor space]]:

* [[Kevin Costello]], [[Natalie M. Paquette]], [[Atul Sharma]], *Top-down holography in an asymptotically flat spacetime* [[arXiv:2208.14233](https://arxiv.org/abs/2208.14233)]

* [[Kevin Costello]], [[Natalie M. Paquette]], [[Atul Sharma]], *Burns space and holography* [[arXiv:2306.00940](https://arxiv.org/abs/2306.00940)]

See also:

* Niklas Garner, [[Natalie M. Paquette]], *Twistorial monopoles & chiral algebras* [[arXiv:2305.00049](https://arxiv.org/abs/2305.00049)]

Review:

* [[Sabrina Pasterski]], *A Chapter on Celestial Holography*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, [[arXiv:2310.04932](https://arxiv.org/abs/2310.04932)]

* [[Laura Donnay]], *Celestial holography: An asymptotic symmetry perspective* [[arXiv:2310.12922](https://arxiv.org/abs/2310.12922)]

[[!redirects celestial holography]]

[[!redirects celestial amplitude]]
[[!redirects celestial amplitudes]]


flat vector bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A *flat vector bundle* is a [[vector bundle]] with [[connection on a vector bundle]] which is [[flat connection|flat]].
Specifically a *flat line bundle* is a [[line bundle]] with [[flat connection]].

Under the identification of [[local systems]] --  in the sense of [[locally constant sheaf|locally constant]] and locally [[free module|free]] [[abelian sheaves]] -- with flat vector bundles, the [[flat connection]] induced on these [[vector bundles]] is called the *[[Gauss-Manin connection]]* (e.g. [Voisin 2002, Def. 9.13](#Voisin02)).

## Related entries

* [[local system]]

* [[twisted de Rham cohomology]]

* [[Knizhnik-Zamolodchikov equation]]

## References

Textbook accounts:

* {#Voisin02} [[Claire Voisin]] (translated by [[Leila Schneps]]), Section I 9.2.1 of: _[[Hodge theory and Complex algebraic geometry]] I,_, Cambridge Stud. in Adv. Math. __76, 77__, 2002/3 ([doi:10.1017/CBO9780511615344](https://doi.org/10.1017/CBO9780511615344))

[[!redirects flat vector bundles]]

[[!redirects flat line bundle]]
[[!redirects flat line bundles]]


Flavio da Silveira

Flavio Ernesto Azeredo da Silveira

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=64334)

## writings

* _Rational homotopy theory of fibrations_, Pacific Journal of Mathematics, Vol. 113, No. 1 (1984) ([pdf](http://msp.org/pjm/1984/113-1/pjm-v113-n1-p01-s.pdf))

## related $n$Lab entries

* [[rational homotopy theory]]

* [[KS-model]]

[[!redirects Flavio Ernesto Azeredo da Silveira]]

category: people


flavor on heterotic M5-branes -- references

### $SU(2)$-flavor symmetry on heterotic M5-branes
 {#FlavorOnSingleHeteroticM5BranesReferences}

Emergence of [[SU(2)]]  [[flavor (particle physics)|flavor]]-[[chiral symmetry|symmetry]] on single [[M5-branes]] in [[heterotic M-theory on ADE-orbifolds]] (in the [[D=6 N=(1,0) SCFT]] on [[small instantons]] in [[heterotic string theory]]):

* Abhijit Gadde, Babak Haghighat, [[Joonho Kim]], [[Seok Kim]], Guglielmo Lockhart, [[Cumrun Vafa]], Section 4.2 of: _6d String Chains_,   J. High Energ. Phys. 2018, 143 (2018) ([arXiv:1504.04614](https://arxiv.org/abs/1504.04614), doi:10.1007/JHEP02(2018)143)

* [[Kantaro Ohmori]], Section 2.3.1 of: _Six-Dimensional Superconformal Field Theories and Their Torus Compactifications_, Springer Theses 2018 ([springer:book/9789811330919](https://www.springer.com/gp/book/9789811330919))

Argument for this by translation under [[duality between M-theory and type IIA string theory]] to [[half NS5-brane]]/[[D6/D8-brane bound state]] systems in [[type I' string theory]]:

Reviewed in:

* Santiago Cabrera, [[Amihay Hanany]], Marcus Sperling, Section 2.3 of: _Magnetic Quivers, Higgs Branches, and 6d $\mathcal{N}=(1,0)$ Theories_, JHEP06(2019)071, JHEP07(2019)137 ([arXiv:1904.12293](https://arxiv.org/abs/1904.12293))

The emergence of flavor in these
[[half NS5-brane]]/[[D6/D8-brane bound state]] systems, due to the semi-infinite extension of the [[D6-branes]] making them act as [[flavor branes]]:

* {#HananyZaffaroni98} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Branes and Six Dimensional Supersymmetric Theories_, Nucl.Phys. B529 (1998) 180-206 ([arXiv:hep-th/9712145](https://arxiv.org/abs/hep-th/9712145))

* {#BrunnerKarch97} [[Ilka Brunner]], [[Andreas Karch]], _Branes at Orbifolds versus Hanany Witten in Six Dimensions_, JHEP 9803:003, 1998 ([arXiv:hep-th/9712143](https://arxiv.org/abs/hep-th/9712143))

Reviewed in:

* {#ApruzziFazzi17} Fabio Apruzzi, [[Marco Fazzi]], Section 2.1 of: _$AdS_7/CFT_6$ with orientifolds_, J. High Energ. Phys. (2018) 2018: 124 ([arXiv:1712.03235](https://arxiv.org/abs/1712.03235))

See also:

* [[Amihay Hanany]], Noppadol Mekareeya, _The Small $E_8$ Instanton and the Kraft Procesi Transition_, JHEP07 (2018) 098 ([arXiv:1801.01129](https://arxiv.org/abs/1801.01129))