coherence theorem for monoidal bicategories

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## Idea

The _coherence theorem for [[monoidal bicategories]]_, like many [[coherence theorems]], has several forms (or, alternatively, refers to several different theorems):

1. Every diagram of constraint 2-cells in a free [[monoidal bicategory]] commutes; in other words, any two parallel composites of constraint 2-cells are equal.  Moreover, any two parallel composites of constraint 1-cells are uniquely isomorphic.

1. Every monoidal bicategory is equivalent to a [[Gray-monoid]].

The second version is a direct corollary of the [[coherence theorem for tricategories]].  The first can then be deduced from it (not entirely trivially).

## References

* Gordon, Power, Street, *Coherence for tricategories*, Mem. Amer. Math Soc. 117 (1995) no 558

* [[Nick Gurski]], *Coherence in three-dimensional category theory*


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## Idea

The _coherence theorem for [[monoidal categories]]_, like many [[coherence theorems]], has several forms (or, alternatively, refers to several different theorems):

1. Every "formal" diagram in a [[monoidal category]] made up of [[associators]] and [[unitors]] commutes.

1. Every diagram in a *free* monoidal category made up of associators and unitors commutes.

1. The free monoidal category on some given data is equivalent to the free [[strict monoidal category]] on the same data.

1. Every monoidal category is monoidally equivalent to a [[strict monoidal category]].

1. Every monoidal category is equivalent to an [[unbiased monoidal category]].

1. The forgetful 2-functor $StrMonCat \to MonCat$ has a strict [[left adjoint]] and the components of the unit are equivalences in $MonCat$.

## Proofs

* [[Mac Lane's proof of the coherence theorem for monoidal categories]]

## Discussion ##

(I plan to get back to this and make emendations, but for now I've just pasted in some discussion from the nForum -- Todd.)

Most people, when they first hear about strictifications, imagine that strictifying a monoidal category is like taking a quotient, in order to identify associativities with identities. That's actually the wrong picture, and it leads to a lot of confusion. The right picture is that a monoidal category $M$ can be monoidally *embedded* in a strict monoidal category $M^{st}$, in which associativity and unit isomorphisms of $M$ are assembled into what Joyal calls "contractible networks", or [[clique|cliques]] (aka anaobjects). The cliques that so arise are then the objects of the strictification.

So, for example, a 4-fold tensor product like $(a^b \otimes (b^c \otimes c^d)) \otimes d$ is a vertex in a clique which consists of all five ways of bracketing $a^b \otimes b^c \otimes c^d \otimes d$ and the groupoid of associativities between them. Or more precisely, it consists of the infinitely many such bracketings with units inserted (e.g., $((a^b \otimes I) \otimes (b^c \otimes c^d)) \otimes (I \otimes d)$), as objects of the groupoid generated by associativity and unit isomorphisms between them. By Mac Lane's coherence theorem ("all diagrams commute"), there is exactly one morphism between any two vertices in the clique, meaning that the groupoid is equivalent to a terminal groupoid, and therefore contractible.

Formally, a *clique* in a category $C$ consists of a contractible groupoid $G$ and a functor $F: G \to C$. A *morphism* between two cliques $(G, F: G \to C)$, $(G', F': G' \to C)$ is a collection of morphisms $F(g) \to F'(g')$, where $(g, g')$ ranges over $Ob(G) \times Ob(G')$, making all triangles in sight commute. (In fact, by contractibility, any one of the $F(g) \to F'(g')$ uniquely determines all the rest.) To form the monoidal strictification $M^{st}$, we take as objects those cliques in the monoidal category $M$ which arise by application of the "all diagrams commute" formulation of Mac Lane's coherence theorem (which specifies the structure of the free monoidal category on one generator as a disjoint sum of contractible groupoids), and the morphisms in $M^{st}$ are defined to be morphisms of cliques. The precise description is laid out here, where it is indicated that the evident embedding $i: M \to M^{st}$ is an equivalence in $MonCat$.

In any case, because $M$ is *embedded* in $M^{st}$, any diagram we are trying to prove commutative in $M$ is *physically there* in $M^{st}$, but any associativities and units in the diagram get absorbed into cliques, or rather: any such associativity is one of the components $F(g) \to F'(g')$ of, and uniquely determines, an identity morphism of cliques. Therefore any such associativity in $M$ is reinterpreted as an identity in $M^{st}$, and the diagram we are trying to prove commutative in $M$ uniquely generates a corresponding "larger" diagram of cliques in the strict monoidal category $M^{st}$. So it is enough to prove the diagram commutes when interpreted in a strict monoidal category: we can then interpret the result in $M^{st}$, and the original diagram in $M$, which is embedded in the clique diagram, is then automatically commutative as well.

One may have to practice visualizing this before it all sinks in, but it's a tremendously potent principle.

## References

* {#MacLane63} [[Saunders Mac Lane]]. "Natural associativity and commutativity". Rice University Studies 49, 28-46 (1963). ([Rice Digital Scholarship Archive](https://scholarship.rice.edu/handle/1911/62865))

* [[Saunders Mac Lane]]. [[Categories for the Working Mathematician]] (Chapter 7).

* {#JoyalStreet} [[André Joyal]], [[Ross Street]], _Braided tensor categories_,  Adv. Math. 1993 ([pdf](http://web.science.mq.edu.au/~street/JS1.pdf))

See also section 5 of

* [[Saunders Mac Lane]], Topology and Logic as a Source of Algebra (Retiring Presidential Address), _Bulletin of the AMS_ 82:1, January 1976. ([euclid](https://projecteuclid.org/euclid.bams/1183537593))

* {#Schauenburg01} [[Peter Schauenburg]], *Turning Monoidal Categories into Strict Ones*, New York Journal of Mathematics **7** (2001) 257-265 [[nyjm:j/2001/7-16](http://nyjm.albany.edu/j/2001/7-16.html), [eudml:121925](https://eudml.org/doc/121925)]

## See also

- [[coherence theorem for symmetric monoidal categories]]


coherence theorem for symmetric monoidal bicategories

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## Idea

The _coherence theorem for [[symmetric monoidal bicategories]]_, like many [[coherence theorems]], has several forms (or, alternatively, refers to several different theorems):

1. Every diagram of constraint 2-cells in a free [[symmetric monoidal bicategory]] commutes; in other words, any two parallel composites of constraint 2-cells are equal. Moreover, two parallel composites of constraint 1-cells are isomorphic if and only if they have the same underlying [[permutation]], in which case they are uniquely isomorphic.

1. Every symmetric monoidal bicategory is equivalent to a strict symmetric monoidal bicategory.

## References

* [[Nick Gurski]] and [[Angelica Osorno]], "Infinite loop spaces, and
coherence for symmetric monoidal bicategories", [arXiv](http://arxiv.org/abs/1210.1174)

* [[Bruce Bartlett]], "Quasistrict symmetric monoidal 2-categories via wire diagrams", [arXiv](http://arxiv.org/abs/1409.2148)

* [[Bruce Bartlett]], "Quasistrict symmetric monoidal 2-categories via wire diagrams", [n-Cafe](https://golem.ph.utexas.edu/category/2014/09/quasistrict_symmetric_monoidal.html)

Despite the terminology, the result by Chris Schommer-Pries explained by Bruce Bartlett goes further than the earlier result of Gurski and Osorno.  The result in the n-Cafe post seems to go even further.


coherence theorem for symmetric monoidal categories

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#Contents#
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## Idea

The _coherence theorem for [[symmetric monoidal categories]]_, like many [[coherence theorems]], has several forms (or, alternatively, refers to several different theorems):

1. Every diagram in a free [[symmetric monoidal category]] made up of [[associators]] and [[unitors]] and symmetries ([[braidings]]), and in which both sides have the same underlying [[permutation]], commutes.

1. The free symmetric monoidal category on some given data is equivalent to the free [[symmetric strict monoidal category]] on the same data.

1. Every symmetric monoidal category is symmetric-monoidally equivalent to a symmetric strict monoidal category.

1. Every symmetric monoidal category is equivalent to an [[unbiased symmetric monoidal category]].

1. The forgetful 2-functor $SymStrMonCat \to SymMonCat$ has a strict [[left adjoint]] and the components of the unit are equivalences in $SymMonCat$.

Note that in a symmetric strict monoidal category, the associators and unitors are identities, but the symmetry is not in general.

## References

* [[Saunders Mac Lane]]. "Natural associativity and commutativity". Rice University Studies 49, 28-46 (1963). ([Rice Digital Scholarship Archive](https://scholarship.rice.edu/handle/1911/62865))

* [[Saunders Mac Lane]]. [[Categories for the Working Mathematician]] (Chapter 11).

* [[Andre Joyal]] and [[Ross Street]].  "Braided tensor categories".  *Adv. Math* 1993

See also section 5 of

* [[Saunders Mac Lane]], Topology and Logic as a Source of Algebra (Retiring Presidential Address), _Bulletin of the AMS_ 82:1, January 1976. ([euclid](https://projecteuclid.org/euclid.bams/1183537593))


coherent (infinity,1)-topos

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## Idea

The concept of _coherent $(\infty,1)$-topos_ is a notion of _[[compact topos]]_ in the context of [[(∞,1)-topos theory]] ([[Spectral Schemes|Lurie VII, def. 3.1]]).

## Definitions

+-- {: .num_defn }
###### Definition
An (∞,1)-topos $\mathbf{H}$ is called **[[quasicompact morphism|quasi-compact]]** if, for every [[effective epimorphism]]

$$ \coprod_{i\in I} U_i\to * $$

there exists a [[finite set|finite]] [[subset]] $J\subset I$ such that $\coprod_{i\in J} U_i\to *$ is an effective epimorphism. An object $X\in\mathbf{H}$ is called quasi-compact if the [[slice (∞,1)-topos]] $\mathbf{H}_{/X}$ is quasi-compact.
=--

We then define $n$-coherence by [[induction]] on $n$.

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

Let $\mathbf{H}$ be an [[(∞,1)-topos]]. We say that $\mathbf{H}$ is 0-coherent if it is quasi-compact. If $n\geq 1$, we say that $\mathbf{H}$ is **n-coherent** if

1. it is **locally $(n-1)$-coherent**, i.e., for every $X\in\mathbf{H}$ there exists an effective epimorphism $\coprod_{i\in I} U_i\to X$ such that each $U_i$ is (n-1)-coherent;
2. the [[sub-(∞,1)-category]] of (n-1)-coherent objects in $\mathbf{H}$ is closed under [[finite products]].

We say that $\mathbf{H}$ is **coherent** if it is $n$-coherent for every $n\geq 0$, and **locally coherent** if for every $X\in\mathbf{H}$ there exists an effective epimorphism $\coprod_{i\in I} U_i\to X$ such that each $U_i$ is coherent.

=--

([[Spectral Schemes|Lurie SpecSch, def. 3.1, def. 3.12]])

+-- {: .num_remark}
###### Remark
This terminology differs from the one in [[SGA]]4: a [[topos]] is a [[coherent topos]] in the sense of [[SGA]]4 if and only if it is 2-coherent according to the above definition.
=--

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

An [[object]] $X \in \mathcal{X}$ in an [[(∞,1)-topos]] is a _[[n-coherent object]]_ if the [[slice (∞,1)-topos]] $\mathcal{X}_{/X}$ is $n$-coherent according to def. \ref{nCoherentTopos}.

=--

## Properties

### Commutativity with filtered colimits

Notice that a [[compact object in an (∞,1)-category]] is one that distributes over [[filtered (∞,1)-colimits]].

In an $n$-coherent $\infty$-topos the [[global section geometric morphism]] (given by homming out of the [[terminal object]]) preserves [[filtered (∞,1)-colimits]] of [[truncated object in an (infinity,1)-topos|(n-1)-truncated objects]].

### In terms of sites

An [[(∞,1)-site]] is *finitary* if every [[covering]] [[sieve]] is generated by a finite family of morphisms. If $C$ is a finitary (∞,1)-site with [[finite (∞,1)-limits]], then the [[(∞,1)-topos of (∞,1)-sheaves]] on $C$ is coherent and locally coherent.

### Deligne-Lurie completeness theorem

The following generalizes the [[Deligne completeness theorem]] from [[topos theory]] to [[(∞,1)-topos theory]].

+-- {: .num_theorem }
###### Theorem
**[[Deligne-Lurie completeness theorem]]**

An [[hypercomplete (∞,1)-topos]] which is locally coherent has [[enough points]].

=--

([[Spectral Schemes|Lurie SpecSchm, theorem 4.1]]).

## Examples
 {#Examples}

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

[[∞Grpd]] is coherent and locally coherent. An [[object]] $X$, hence an [[∞-groupoid]], is an [[n-coherent object]] if all its [[homotopy groups]] in degree $k \leq n$ are [[finite set|finite]]. Hence the fully coherent objects here are the [[homotopy types with finite homotopy groups]].

=--

([[Spectral Schemes|Lurie SpecSchm, example 3.13]])

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

Let $X$ be a [[scheme]] and let $Sh_\infty(X_{Zar})$ be the [[(∞,1)-topos of (∞,1)-sheaves]] on the [[small site|small]] [[Zariski site]] of $X$. Then the following assertions are equivalent:

1. $Sh_\infty(X_{Zar})$ is coherent;
2. $Sh_\infty(X_{Zar})$ is 1-coherent;
3. $X$ is quasi-compact and quasi-separated.

=--

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

A [[spectral scheme]] or [[spectral Deligne-Mumford stack]], regarded as a [[structured (∞,1)-topos]] is locally coherent.

=--

([[Quasi-Coherent Sheaves and Tannaka Duality Theorems|Lurie QCoh, cor. 1.4.3]])

## References

* [[Jacob Lurie]], section 3 of _[[Spectral Schemes]]_

* [[Jacob Lurie]], section 2.3 of _[[Rational and p-adic Homotopy Theory]]_

* [[Jacob Lurie]], _[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]_

[[!redirects coherent (infinity,1)-toposes]]

[[!redirects coherent (∞,1)-topos]]
[[!redirects coherent (∞,1)-toposes]]
[[!redirects n-coherent (∞,1)-topos]]
[[!redirects n-coherent (infinity,1)-topos]]
[[!redirects n-coherent (∞,1)-toposes]]
[[!redirects n-coherent (infinity,1)-toposes]]

[[!redirects locally coherent (∞,1)-topos]]
[[!redirects locally coherent (∞,1)-toposes]]
[[!redirects locally n-coherent (∞,1)-topos]]
[[!redirects locally n-coherent (infinity,1)-topos]]
[[!redirects locally n-coherent (∞,1)-toposes]]
[[!redirects locally n-coherent (infinity,1)-toposes]]


coherent 2-category

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## Idea

The generalization of the notion of _[[coherent category]]_ from [[category theory]] to [[2-category theory]].

## Definition

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###### Definition

A [[2-category]] is called **coherent** if
1. it has finite [[2-limits]],
1. finite jointly-[[eso]] families are stable under [[2-pullback]], and
1. every finitary [[2-polycongruence]] which is a kernel can be completed to an exact [[2-polyfork]].

=--

Here a family $\{f_i:A_i \to B\}$ is said to be **jointly-eso** if whenever $m:C\to B$ is ff and every $f_i$ factors through $m$ (up to isomorphism), then $m$ is an equivalence.

Likewise, we have **infinitary coherent** 2-categories in which "finite" in the second two conditions is replaced by "small."

## Examples

* [[Cat]] is coherent.

* A [[1-category]] is coherent as a 2-category iff it is [[coherent category|coherent]] as a 1-category.

* A [[(0,1)-category]] (= [[poset]]) is coherent iff it is a [[distributive lattice|distributive lattice]], and infinitary-coherent iff it is a [[frame]].

## Properties

### Factorizations

The following are proven just like their unary analogues in a [[regular 2-category]].

+--{: .num_lemma}
###### Lemma
**(Street's Lemma)** In a finitely complete 2-category where finite jointly-eso families are stable under pullback, if $\{e_i:A_i \to B\}$ is finite and jointly-eso and $n:B\to C$ is such that the induced functor $\ker(e_i) \to ker(n e_i)$ is an equivalence, then $n$ is ff.
=--

+--{: .num_theorem}
###### Theorem
A 2-category is coherent if and only if
1. it has finite limits,
1. finite jointly-eso families are stable under pullback,
1. every finite family $\{f_i\}$ factors as $f_i = m e_i$ where $m$ is ff and $\{e_i\}$ is jointly-eso, and
1. every jointly-eso family is the quotient of its kernel.
=--

Of course, there are infinitary versions.  In particular, we conclude that in a coherent (resp. infinitary-coherent) 2-category, the posets $Sub(X)$ have finite (resp. small) unions that are preserved by pullback.

### Colimits

+--{: .num_lemma}
###### Lemma
A coherent 2-category has a strict [[initial object]]; that is an initial object $0$ such that any morphism $X\to 0$ is an equivalence.
=--
+--{: .proof}
###### Proof
The empty 2-congruence is the kernel of the empty family (over any object), so it must have a quotient $0$, which is clearly an initial object.  The empty family over $0$ is jointly-eso, so for any $X\to 0$ the empty family over $X$ is also jointly-eso; but this clearly makes $X$ initial as well.
=--

Two [[nLab:ff morphism|ffs]] $m:A\to X$ and $n:B\to X$ are said to be **disjoint** if the _comma objects_ $(m/n)$ and $(n/m)$ are initial objects.  If initial objects are strict, then this implies that the pullback $A\times_X B$ is also initial, but it is strictly stronger already in $Pos$.

+--{: .num_lemma #DisjointUnion}
###### Lemma
In a coherent 2-category, if $A\to X$ and $B\to X$ are disjoint subobjects, then their union $A\cup B$ in $Sub(X)$ is also their coproduct $A+B$.
=--
+--{: .proof}
###### Proof
If $A$ and $B$ are disjoint subobjects of $X$, then the kernel of $\{A\to X, B\to X\}$ is the disjoint union of $ker(A)$ and $ker(B)$.  Therefore, a quotient of it (which is a union of $A$ and $B$ in $Sub(X)$) will be  a coproduct of $A$ and $B$.
=--

A coproduct $A+B$ in a 2-category is **disjoint** if $A$ and $B$ are disjoint subobjects of $A+B$.  We say a coherent 2-category is **positive** if any two objects have a disjoint coproduct.  By Lemma \ref{DisjointUnion}, this is equivalent to saying that any two objects can be embedded as disjoint subobjects of some other object.  Disjoint coproducts in a coherent 2-category are automatically stable under pullback, so a positive coherent 2-category is necessarily [[extensive 2-category|extensive]].  Conversely, we have:

+--{: .num_lemma}
###### Lemma
A regular and extensive 2-category is coherent (and positive).
=--
+--{: .proof}

In the presence of finite coproducts, a family $\{e_i:A_i\to B\}$ is jointly-eso iff $\coprod_i A_i \to B$ is eso; thus regularity and universal coproducts imply that finite jointly-eso families are stable under pullback.  And assuming extensivity, any 2-polycongruence $\{C_{i j}\} \rightrightarrows \{C_i\}$ gives rise to an ordinary 2-congruence $\coprod_{i j} C_{i j} \rightrightarrows \coprod_i C_i$.  Likewise, 2-polyforks $\{C_{i j}\} \rightrightarrows \{C_i\} \to X$ correspond to 2-forks $\coprod_{i j} C_{i j} \rightrightarrows \coprod_i C_i \to X$, and the property of being a kernel or a quotient is preserved; thus regularity implies coherency.

=--

### Preservation

If $K$ is coherent, then easily so are $K^{co}$, $disc(K)$, $gpd(K)$, $pos(K)$, and $Sub(1)$.  Moreover, we have:

+--{: .num_theorem}
###### Theorem
If $K$ is a coherent 2-category, so are the [[fibrational slice]]s $Opf(X)$ and $Fib(X)$ for any $X\in K$.
=--

## References

This is due to

* [[Michael Shulman]], _[[michaelshulman:coherent 2-category]]_

based on

* [[Ross Street]], _[[StreetCBS]]_

[[!redirects coherent 2-categories]]


coherent category

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### Context
#### Regular and Exact categories
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#### Category Theory
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# Contents
* table of contents
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## Definition

A **coherent category** (also called a **pre-[[logos]]**) is a [[regular category]] in which the [[subobject posets]] $Sub(X)$ all have finite [[unions]] which are preserved by the [[base change]] functors $f^*:Sub(Y)\to Sub(X)$.

A [[coherent functor]] serves as a morphism between coherent categories.

The [[internal logic]] of a coherent category is [[coherent logic]].

An **infinitary coherent category** is a [[regular category]] in which the subobject posets $Sub(X)$ have all _[[small set|small]]_ unions which are [[pullback stability|stable under pullback]].  Infinitary coherent categories are also called __[[geometric category|geometric categories]]__.  (These are sometimes also required to be [[well-powered category|well-powered]]; see their page for details.)

See [[familial regularity and exactness]] for a general description of the spectrum from regular categories through finitary and infinitary coherent categories.

## Examples

* [[Set]] is a coherent category.

* More generally, any [[topos]] is coherent and any [[pretopos]] is coherent.

* Even more generally, any [[locally cartesian closed category]] with finite [[colimits]] is coherent, and thus any [[quasitopos]] is coherent.

* Every [[Heyting category]] is a [[coherent category]].

* If $C$ is coherent, so is the [[functor category]] $C^D$ for any $D$.

* A [[slice category]] of a coherent category is coherent.

* Any [[distributive lattice]], regarded as a [[thin category]], is coherent.

* The [[category of PERs]] over a [[PCA]] is coherent.

## Properties

### Extensivity

Any coherent category automatically has a strict [[initial object]].  Moreover, if an object $X$ is the [[union]] of two subobjects $A\hookrightarrow X$ and $B\hookrightarrow X$ such that $A\cap B = 0$, then $X = A\amalg B$ is their [[coproduct]].  Thus, if every pair of objects can be embedded [[disjoint coproduct|disjointly]] in some third object, then a coherent category has disjoint finite coproducts and is [[extensive category|extensive]] (or "positive").

Extensivity is the analogue for coherent categories of [[exact category|exactness]] for regular categories.  A coherent category which is both extensive and exact is called a [[pretopos]].

### The coherent topology

Any [[coherent category]] $C$ admits a [[subcanonical site|subcanonical]] [[Grothendieck topology]] in which the covering families are generated by finite, jointly [[regular epimorphism|regular-epimorphic]] families: the [[coherent coverage]].  Equivalently, they are generated by single [[regular epimorphisms]] and by finite unions of [[subobject]]s.  If $C$ is [[extensive category|extensive]], then its coherent topology is generated by the regular topology together with the extensive topology.  (In fact, the coherent topology is [[superextensive site|superextensive]].)

If $C$ is a pretopos, then its self-indexing is a [[stack]] for its coherent topology.  Exactness and extensivity are stronger than necessary, however; a pair of necessary and sufficient conditions for this to hold are that

1. If $R\;\rightrightarrows\; A$ is a kernel pair, then for any $f\colon B\to A$, the pullback $f^*R \;\rightrightarrows\; B$ is also a kernel pair (this is equivalent to the [[codomain fibration]] being a stack for the regular topology).

1. If $A,B \rightarrowtail C$ are a pair of subobjects, then for any $f\colon X\to A$ and $g\colon Y\to B$ and any isomorphism $f^*(A\cap B) \cong g^*(A\cap B)$, the pushout
  $$\array{f^*(A\cap B) & \overset{}{\to} & X\\
  \downarrow && \downarrow\\
  Y& \underset{}{\to} & Z}$$
  exists and is also a pullback.

Topoi of sheaves for coherent topologies on coherent categories are called [[coherent topos|coherent topoi]].  (The terminology is slightly confusing, though, because every topos is a coherent category.)

### Making categories coherent

Just like the [[reg/lex completion]], there is a "coh/lex completion" which makes an arbitrary [[finitely complete category]] into a coherent one in a universal way.

Similarly, there are "[[pretopos completion]]s" analogous to the [[ex/reg completion]] and the [[ex/lex completion]].

### Subobjects, slices and internal logic
 {#SlicesAndInternalLogic}

For any object $X$ in a coherent category $\mathcal{C}$, the [[poset of subobjects]] $Sub_{\mathcal{C}}(X)$ is a [[distributive lattice]]. For $f : X \to Y$ any [[morphism]], the [[base change]] $f^* : Sub(Y) \to Sub(X)$ has a [[left adjoint]] $\exists_f : Sub(X) \to Sub(Y)$: the [[dependent sum]]/[[existential quantifier]] along $f$.

The corresponding [[functor]]/[[indexed category]]

$$
  Sub : \mathcal{C}^{op} \to DistLat
$$

to the category of [[distributive lattices]] is the _[[coherent hyperdoctrine]]_ of the coherent category $\mathcal{C}$.

### Topos of types

See _[[topos of types]]_.

## Related concepts

* [[2-category]] version: [[coherent 2-category]]

[[!include categories and logic - table]]

## References

Section A1.4 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

[[!redirects coherent category]]
[[!redirects coherent categories]]

[[!redirects pre-logos]]
[[!redirects pre-logoses]]
[[!redirects pre-logoi]]

[[!redirects prelogos]]
[[!redirects prelogoses]]
[[!redirects prelogoi]]


coherent cohomology

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### Context
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## Idea

Given a (Noetherian) [[ringed topos]] $(\mathcal{X}, \mathcal{O}_X)$, then a [[chain complex]] $V_\bullet$ of [[modules]] over the [[structure sheaf]] is said to have _(quasi-)coherent cohomology_ if all its [[chain homology]] [[groups]]  are ([[quasi-coherent sheaf|quasi-]])[[coherent sheaves]] ([[coherent objects]]).

## Properties
 {#Properties}

Over a (finite-dimensional) [[Noetherian scheme]] $X$ the [[derived category]] of [[quasi-coherent sheaves]] is canonically [[equivalence of categories|equivalent]] to that of sheaves with quasicoherent cohomology.

The coherent version of the statement is ([[SGA|SGA 6, Exp. II, Corollaire 2.2.2.1]]) while the quasi-coherent version is ([[SGA|SGA 6, Exp. II, Proposition 3.7, b)]]). A review appears also as ([Orlov 03, prop. 1.3.2](#Orlov03)).

See also the discussion at _[[triangulated categories of sheaves]]_.

## References

* {#Orlov03} [[Dmitri Orlov]], _Derived categories of coherent sheaves and equivalences between them_, Russian Math. Surveys, 58 (2003), 3, 89-172 ([English transl. pdf](http://www.mi.ras.ru/~orlov/papers/Uspekhi2003.pdf), [Russian orig. pdf](http://www.mi.ras.ru/~orlov/papers/uspehwin.pdf))

[[!redirects quasi-coherent cohomology]]


coherent coverage

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## Definition

For $C$ a [[coherent category]] $C$ the **coherent coverage** on $C$ is the [[coverage]] in which the [[covering]] families are generated by finite, jointly [[regular epimorphism|regular-epimorphic]] families.  Similarly, on an [[infinitary-coherent category]] (a.k.a. a "geometric category"), the **infinitary-coherent coverage** (a.k.a. *geometric coverage*) is generated by all *small* jointly regular-epimorphic families.

The corresponding [[Grothendieck topology]] is called the **coherent topology**, making $C$ into a __coherent site__.

Equivalently, this coverage is generated by single [[regular epimorphisms]] and by finite [[union]]s of [[subobject]]s (resp. small unions in the infinitary case).

Topoi of sheaves for (finitary) coherent topologies on coherent categories are called [[coherent topos|coherent toposes]].  (The terminology is slightly confusing, though, because every topos is a coherent category.)  Note that every topos is equivalent to a topos of sheaves for the *infinitary* coherent topology on an infinitary-coherent site, namely itself.

## Properties

The coherent coverage is [[subcanonical site|subcanonical]].

* If $C$ is [[extensive category|extensive]], then its coherent topology is generated by the [[regular coverage|regular topology]] together with the [[extensive topology]].  (In fact, the coherent topology is [[superextensive site|superextensive]].)

* If $C$ is a [[pretopos]], then its self-indexing is a [[stack]] for its coherent topology.  Exactness and extensivity are stronger than necessary, however; a pair of necessary and sufficient conditions for this to hold are that

  1. If $R\;\rightrightarrows\; A$ is a kernel pair, then for any $f\colon B\to A$, the pullback $f^*R \;\rightrightarrows\; B$ is also a kernel pair (this is equivalent to the [[codomain fibration]] being a stack for the regular topology).
  2. If $A,B \rightarrowtail C$ are a pair of subobjects, then for any $f\colon X\to A$ and $g\colon Y\to B$ and any isomorphism $f^*(A\cap B) \cong g^*(A\cap B)$, the pushout
    $$\array{f^*(A\cap B) & \overset{}{\to} & X\\
    \downarrow && \downarrow\\
    Y& \underset{}{\to} & Z}$$
    exists and is also a pullback.

## Related concepts

[[!include categories and logic - table]]

## References

Section A1.4 and example C2.1.12 in

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

[[!redirects coherent coverage]]
[[!redirects coherent coverages]]
[[!redirects coherent topology]]
[[!redirects coherent topologies]]
[[!redirects coherent Grothendieck topology]]
[[!redirects coherent Grothendieck topologies]]
[[!redirects coherent site]]
[[!redirects coherent sites]]

[[!redirects geometric coverage]]
[[!redirects geometric coverages]]
[[!redirects geometric topology]]
[[!redirects geometric topologies]]
[[!redirects geometric Grothendieck topology]]
[[!redirects geometric Grothendieck topologies]]
[[!redirects geometric site]]
[[!redirects geometric sites]]


coherent D-module

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## Idea

[[coherent module|coherent]] [[D-module]] (...)

## Related concepts

* [[holonomic D-module]]

* [[arithmetic D-module]]

## References

* B. Kaup, _Coherent D-modules_, chapter II of [[Armand Borel]] et al., _Algebraic D-modules_, Perspectives in Mathematics, Academic Press, 1987 ([djvu](http://www.math.columbia.edu/~scautis/dmodules/boreletal.djvu))

[[!redirects coherent D-modules]]


coherent dagger 2-poset

[[!redirects coherent dagger 2-posets]]

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## Idea ##

A coherent dagger 2-poset is a dagger 2-poset whose [[category of maps]] is a [[coherent category]].

## Definition ##

A **coherent dagger 2-poset** is a [[dagger 2-poset]] $C$ such that

* There is an object $0 \in Ob(C)$ such that for each object $A \in Ob(C)$, there is a [[monic map in a dagger 2-poset|monic map]] $i_{0,A} \in Hom(0,A)$ such that for each object $B \in Ob(C)$ with a monic map $i_{B,A} \in Hom(B,A)$, there is a monic map $i_{0,B} \in Hom(0,B)$ such that $i_{B,A} \circ i_{0,B} = i_{0,A}$.

* For each object $A \in Ob(C)$, $B \in Ob(C)$, $E \in Ob(C)$ with monic maps $i_{B,A} \in Hom(B,A)$, $i_{E,A} \in Hom(E,A)$, there is an object $B \cup E \in Ob(C)$ with monic maps $i_{B \cup E,A} \in Hom(B \cup E,A)$, $i_{B,B \cup E} \in Hom(B,B \cup E)$, $i_{E,B \cup E} \in Hom(E,B \cup E)$, such that $i_{B \cap E,A} \circ i_{B,B \cap E} = i_{B,A}$ and $i_{B \cap E,A} \circ i_{E,B \cap E} = i_{E,A}$,and for every object $D \in Ob(C)$ with monic maps $i_{D,A} \in Hom(D,A)$ $i_{B,D} \in Hom(B,D)$, $i_{E,D} \in Hom(E,D)$ such that $i_{D,A} \circ i_{B,D} = i_{B,A}$ and $i_{D,A} \circ i_{E,D} = i_{E,A}$, there is a monic map $i_{B \cup E,D} \in Hom(B \cup E,D)$ such that $i_{D,A} \circ i_{B \cup E,D} = i_{B \cup E,A}$

* For each object $A \in Ob(C)$, $B \in Ob(C)$, $E \in Ob(C)$ with monic maps $i_{B,A} \in Hom(B,A)$, $i_{E,A} \in Hom(E,A)$, there is an object $B \cap E \in Ob(C)$ with monic maps $i_{B \cap E,A} \in Hom(B \cap E,A)$, $i_{B \cap E,B} \in Hom(B \cap E,B)$, $i_{B \cap E,E} \in Hom(B \cap E,E)$, such that $i_{B,A} \circ i_{B \cap E,B} = i_{B \cap E,A}$ and $i_{E,A} \circ i_{B \cap E,E} = i_{B \cap E,A}$, and for every object $D \in Ob(C)$ with monic maps $i_{D,A} \in Hom(D,A)$ $i_{D,B} \in Hom(D,B)$, $i_{D,E} \in Hom(D,E)$ such that $i_{B,A} \circ i_{D,B} = i_{D,A}$ and $i_{E,A} \circ i_{D,E} = i_{D,A}$, there is a monic map $i_{D,B \cap E} \in Hom(D,B \cap E)$ such that $i_{B \cap E,A} \circ i_{D,B \cap E} = i_{D,A}$.

* For each object $A \in Ob(C)$, $B \in Ob(C)$, $D \in Ob(C)$, $E \in Ob(C)$ with monic maps $i_{B,A} \in Hom(B,A)$, $i_{D,A} \in Hom(D,A)$, $i_{E,A} \in Hom(E,A)$, there is a [[unitary isomorphism]]
$$j_{B,C,E} \in E \cap (B \cup C) \cong^\dagger (E \cap B) \cup (E \cap C)$$

## Properties ##

* For each object $A \in Ob(C)$, the identity function $1_A \in Hom(A,A)$ is a monic map, and for each object $B \in Ob(C)$ with a monic map $i_{B,A} \in Hom(B,A)$, $1_A \circ i_{B,A} = i_{B,A}$.

* The [[isomorphism classes]] of monic maps is a [[distributive lattice]]. Since every monic map is a map, the [[category of maps]] is a [[coherent category]].

## Examples ##

The dagger 2-poset [[Rel]] of [[sets]] and [[relations]] is a coherent dagger 2-poset.

## See also ##

* [[dagger 2-poset]]
* [[coherent dagger 2-poset]]
* [[geometric dagger 2-poset]]
* [[Heyting dagger 2-poset]]
* [[Boolean dagger 2-poset]]
* [[elementarily topical dagger 2-poset]]


coherent formula > history

< [[coherent formula]]


coherent functor

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## Definition

A [[functor]] $F : C \to D$ between [[coherent categories]] is called **coherent** if it is a [[regular functor]] and in addition preserves finite [[union]]s.

## Properties

* For $\mathcal{T}$ a [[coherent theory]] and $\mathcal{C}_{\mathbb{T}}$ its [[syntactic category]], coherent functors $\mathcal{C}_{\mathbb{T}} \to \mathcal{E}$ into some [[topos]] $\mathcal{E}$ are precisely [[model]]s of the theory in $\mathcal{E}$.

  For $\mathcal{C}_{\mathbb{T}}$ equipped with the structure of the [[syntactic site]] (the [[coherent coverage]]), this is in turn equivalent to [[geometric morphism]]s $\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})$ into the [[sheaf topos]] over $\mathcal{C}_{\mathbb{T}}$ (the [[classifying topos]] for the theory).

## Related concepts

[[!include categories and logic - table]]

## References

Page 34 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_


coherent hyperdoctrine

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## Definition

Let $C$ be a [[category]] with [[finite limits]]. A **[[coherent logic|coherent]] [[hyperdoctrine]]** over $C$ is a [[functor]]

$$
  P : C^{op} \to DistLattice
$$

from the [[opposite category]] of $C$ to the category of [[distributive lattices]], such that for every [[morphism]] $f : A \to B$ in $C$, the functor $P(f) : P(B) \to P(A)$ has a [[left adjoint]] $\exists_f$ satisfying

1. [[Frobenius reciprocity]];

1. [[Beck-Chevalley condition]].

## Properties
 {#Properties}

The assignment ([here](#OverACoherentCategory)) of a coherent hyperdoctrine $S(C)$ to a [[coherent category]] $C$ extends to a [[2-adjunction]]

$$
  (A \dashv S)
  :
  CoherentCat
   \stackrel{\overset{A}{\leftarrow}}{\underset{S}{\hookrightarrow}}
  CoherentHyperdoctrine
$$

with the [[right adjoint]] being a [[full and faithful 2-functor]], hence exhibiting $CoherentCat$ as a [[reflective sub-2-category]] of $CoherentHyperdoctrine$.

(Here $CoherentCat$ has as [[2-morphisms]] those [[natural transformations]] that preserve finite products.)

This appears as ([Coumans, prop. 8](#Coumans)).

Coherent hyperdoctrines are closed under [[canonical extension]] $(-)^\delta : DistLattice \to DistLattice$, in that for $P : C^{op} \to DistLattic$ a coherent hyperdoctrine, so is $(-)^\delta \circ P$.

This appears as ([Coumans, prop. 9](#Coumans)).

## Examples

### Powersets

The [[powerset]] functor

$$
  P := \{0,1\}^{(-)} : Set^{op} \to DistLattice
$$

(sending a [[set]] to its [[power set]] and a [[function]] to the [[preimage]]-assignment) is a coherent hyperdoctrine.

### Over a coherent category
 {#OverACoherentCategory}

Let $C$ be a [[coherent category]]. For every [[object]] $A \in C$ the [[poset of subobjects]] $Sub_C(A)$ is a [[distributive lattice]].

The corresponding [[functor]]

$$
  C^{op} \to DistLattice
$$

from the [[opposite category]] of $C$ to the category of distributive lattices is called the **coherent hyperdoctrine** of $C$.

### For a coherent theory

Accordingly, there is a coherent hyperdoctrine associated with any [[coherent theory]], where the objects of $C$ are lists of [[free variables]] in the theory, and the lattice assigned to them is that of [[propositions]] of the theory in this [[context]].

## Related concepts

[[!include categories and logic - table]]

## References

* Dion Coumans, _Generalizing canonical extensions to the categorical setting_ ([Arxiv](http://arxiv.org/abs/1204.3745))
 {#Coumans}

* Graham Manuell, *Quantalic spectra of semirings*, ([arXiv:2201.06408](https://arxiv.org/abs/2201.06408))

[[!redirects coherent hyperdoctrines]]


coherent logic

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## Definitions

**Coherent logic** is a fragment of (finitary) [[first-order logic]] which allows only the connectives and quantifiers

* $\wedge$ ([[and]]),

* $\vee$ ([[or]]),

* $\top$ ([[true]]),

* $\bot$ ([[false]]),

* $\exists$ ([[existential quantifier]]).

A **coherent formula** is a [[formula]] in coherent logic.

A **coherent sequent** is a [[sequent]] of the form $\varphi \vdash \psi$, where $\varphi$ and $\psi$ are coherent formulas, possibly with [[free variables]] $x_1,\dots,x_n$.

In full first-order logic, such a sequent is equivalent to the single formula

$$
  \forall x_1, \dots, \forall x_n (\varphi \Rightarrow \psi)
$$

(in the empty [[context]]).  Of course, this latter formula is not coherent, but this shows that when we deal with coherent *sequents* rather than merely formulas, it can also be thought of as allowing one instance of $\Rightarrow$ and a string of $\forall$s at the very outer level of a formula.

Coherent logic (including sequents, as above) is the [[internal logic]] of a [[coherent category]].  The [[classifying topos]] of a coherent theory is a [[coherent topos]].

## Examples

* Any (finitary) [[algebraic theory]] is a coherent theory, as is any theory in [[regular logic]].  Syntactically this is obvious, as algebraic theories and regular theories use only a fragment of the logical connectives available in coherent logic.  However, it should be noted that the [[syntactic category]] of an algebraic theory is not the same as its syntactic category when regarded as a coherent theory: the latter is the "free coherent category" generated by the former (which is a category with finite products).

* A good example of a coherent theory that is not [[algebraic theory|algebraic]] (in any of the usual senses, although it comes from algebra) is the theory of a [[local ring]], defined by axioms $0 = 1 \vdash \bot$ and
$$x + y = 1 \vdash (\exists z.x \cdot z = 1) \vee (\exists z.y \cdot z = 1)$$

* The theory of a [[totally ordered set]] is another example of a coherent theory that is not algebraic. One adds to the algebraic theory of a [[idempotent]] [[commutative semigroup]] the totality axiom $\vdash_{x,y} (x \sqcup y = x) \vee (x \sqcup y = y)$.

* Similar examples are the theories of a [[discrete field]].

* The theory of an [[apartness relation]] is coherent. The irreflexivity axiom is expressed as $(x \# x) \vdash \bot$, the symmetry axiom is expressed as $(x \# y) \vdash (y \# x)$, and the comparison axiom is expressed as $(x \# z) \vdash \exists y.(x \# y) \vee (y \# z)$. However, the "tightness" axiom $\neg(x\# y) \vdash (x=y)$ in [[tight apartness relations]] is not coherent since it uses negation.

* The theory of a [[total order]] is coherent, though also not algebraic. The theory of a [[partial order]] is [[essentially algebraic theory|essentially algebraic]], but the totality axiom $\vdash_{x,y} (x\le y) \vee (y\le x)$ is coherent but not essentially algebraic.

* The theory of a [[strict total order]] is not coherent if we use the "connectedness" axiom $(x\nless y), (y\nless x) \vdash (x=y)$, which is not coherent since negation is not allowed in coherent formulas. We can express one outer negation, however, as in the irreflexivity axiom $(x\lt x)\vdash \bot$. In [[classical mathematics]], there is another solution, to use the "[[trichotomy]]" axiom $\top \vdash (x=y) \vee (x\lt y) \vee (y\lt x)$ instead, in order to get an axiomatisation of "coherent" strict total orders. However, in [[constructive mathematics]], strict total orders are in general not trichotomous, so this solution cannot be used.

* The theory of a [[strict weak order]] is coherent, since it doesn't have the connectedness axiom that [[strict total orders]] do. The irreflexivity axiom is expressed as $(x\lt x)\vdash \bot$, and the asymmetry axiom is expressed as $(x \lt y) \wedge (y \lt x) \vdash \bot$.

* Since the theory of a [[local ring]] and a [[strict weak order]] is coherent, this means that the theory of an [[ordered local ring]] is also coherent, where we have
$$(x \lt y) \vee (y \lt x) \vdash \exists z.y \cdot z = 1 + x \cdot z$$
and
$$\exists z.x \cdot z = 1 \vdash (0 \lt x) \vee (x \lt 0)$$

* The theory of a [[reduced ring]] is coherent, with the [[trivial nilradical]] axiom being expressed as $x \cdot x = 0 \vdash x = 0$.

* Similarly, the theory of an [[integral domain]] is coherent, with the [[zero-product property]] axiom being expressed as $(x \cdot y = 0) \vdash (x = 0) \vee (y = 0)$

* Since [[GCD rings]] and [[Bézout rings]] are [[algebraic theories]] and [[integral domains]] are coherent, [[GCD domains]] and [[Bézout domains]] are coherent as well.

* The theory of an [[elementary topos]] is coherent. However, the well-pointedness condition for a [[well-pointed topos]] is not coherent, which means that the theory [[ETCS]] is not coherent. See [[fully formal ETCS]] for more details.

## Properties

Coherent logic has many pleasing properties.

* Every finitary [[first-order logic|first-order theory]] is equivalent, over [[classical logic]], to a coherent theory.  This theory is called its *Morleyization* and can be obtained by adding new relations representing each first-order formula and its negation, with axioms that guarantee (over classical logic) these relations are interpreted correctly (using the facts that $(P\Rightarrow Q) \dashv\vdash (\neg P \vee Q)$ and $(\forall x, P) \dashv\vdash (\neg \exists x, \neg P)$ in classical logic).  See D1.5.13 in [[Sketches of an Elephant]], or Prop. 3.2.8 in Makkai-Paré.

* By (one of the theorems called) [[Deligne completeness theorem|Deligne's theorem]], every [[coherent topos]] has [[point of a topos|enough points]].  In particular, this applies to the classifying toposes of coherent theories.  It follows that models in [[Set]] are sufficient to detect provability in coherent logic.  By Morleyization, we can obtain from this the classical [[completeness theorem for first-order logic]].  See for instance 6.2.2 in Makkai-Reyes.

* Coherent logic also satisfies a *definability theorem*: if a relation can be constructed in every [[Set]]-model of a coherent theory $T$, in a [[natural transformation|natural]] way, then that relation is named by some coherent formula in $T$.  See chapter 7 of Makkai-Reyes or D3.5.1 in [[Sketches of an Elephant]].

* It follows that if a morphism of coherent theories (i.e. an interpretation of one coherent theory in another) induces an equivalence of their categories of models in $Set$, then it is a [[Morita equivalence]] of theories (i.e. induces an equivalence of classifying toposes, hence an equivalence of categories of models in all [[Grothendieck toposes]]).  This is called [[conceptual completeness]]; see 7.1.8 in Makkai-Reyes or D3.5.9 in [[Sketches of an Elephant]].  (Note, though, that two coherent theories can have equivalent categories of models in $Set$ without being Morita equivalent, if the former equivalence is not induced by a morphism of theories; see for instance D3.5.4 in the Elephant.)

However, here is a property which one might expect coherent theories to have, but which they do not.

* The category of models of a coherent theory (in [[Set]]) is always an [[accessible category]] (this is true more generally for models of any logical theory).  However, it need not be *finitely* accessible (i.e. $\omega$-accessible).  An example is given in Adamek-Rosicky 5.36: the theory of sets equipped with a binary relation $\prec$ such that for all $x$ there exists a $y$ with $x\prec y$.  Then the model $(\mathbb{N},\lt)$ of this theory does not admit any morphism from an ($\omega$-)compact object.  (However, many coherent theories do have a finitely accessible category of $Set$-models.)

## Variations

* Sometimes coherent logic is called *geometric logic*, but that term is now more commonly used for the analogous fragment of infinitary logic which allows disjunctions over arbitrary sets (though still only finitary conjunctions).  See [[geometric logic]].

* Conversely, geometric logic is sometimes called _coherent_ , e.g. in Reyes ([1977](#Reyes)), so that coherent logic in the nLab terminology corresponds to the _finitary_ fragment only.

* Occasionally the [[existential quantifiers]] in coherent logic are further restricted to range only over *finitely presented types*.

## Related concepts

* [[finitely complete category]], [[cartesian functor]], [[cartesian logic]], [[cartesian theory]]

* [[regular category]], [[regular functor]], [[regular logic]], [[regular theory]], [[regular coverage]], [[regular topos]]

* [[coherent category]], [[coherent functor]], **coherent logic**, [[coherent theory]], [[coherent coverage]], [[coherent topos]]

* [[geometric category]], [[geometric functor]], [[geometric logic]], [[geometric theory]]

* [[coherent mathematics]]

## References

An early reference is

* M.-F. Coste, [[Michel Coste|M. Coste]], _Théories cohérentes et topos cohérents_ , Séminaire de théorie des catégories dirigé par Jean Bénabou Mai 1975. ([pdf](https://perso.univ-rennes1.fr/michel.coste/publis/coherent.pdf))

A survey of results on geometric and coherent logic is in

* {#Reyes77}[[Gonzalo Reyes]], _Sheaves and concepts: A model-theoretic interpretation of Grothendieck topoi_ , Cah. Top. Diff. Géo. Cat. **XVIII** no.2 (1977) pp.405-437. ([numdam](http://www.numdam.org/item?id=CTGDC_1977__18_2_105_0))

A standard textbook account of coherent logic (called 'geometric logic' there) can be found in

* {#MM94} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994.

Properties of the generic model of a coherent theory are investigated in

* Marc Bezem, [[Ulrik Buchholtz]], [[Thierry Coquand]], _Syntactic Forcing Models for Coherent Logic_ , arXiv:72.07743 (2017). ([abstract](https://arxiv.org/abs/1712.07743))

Else consider the monographs

* [[Michael Makkai]] and [[Gonzalo Reyes]], *First Order Categorical Logic:  Model-theoretical methods in the theory of topoi and related categories*, Springer-Verlag, 1977.

* [[Michael Makkai]] and [[Robert Paré]], *Accessible categories*

* [[Peter Johnstone]], [[Sketches of an Elephant]] vol 2. Part D

* [[Jiri Adamek]] and [[Jiri Rosicky]], *Locally presentable and accessible categories*

[[!redirects coherent logic]]
[[!redirects coherent logics]]
[[!redirects coherent theory]]
[[!redirects coherent theories]]
[[!redirects coherent formula]]
[[!redirects coherent formulas]]
[[!redirects coherent sequent]]
[[!redirects coherent sequents]]
[[!redirects Morleyization]]


coherent mathematics

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## Idea

Coherent mathematics is mathematics done using only [[coherent logic]].

Examples of coherent mathematics include mathematics done in an [[arithmetic pretopos]], which can be defined using only [[coherent logic]], as well as mathematics done in any [[dependent type theory]] which does not have [[function types]] nor [[dependent function types]], such as the internal type theory of an [[arithmetic pretopos]] as a fragment of [[geometric type theory]].

Since coherent logic does not have a [[negation]] operator, it is manifestly [[constructive]], since one cannot even express the law of [[excluded middle]] or [[double negation]]. Similarly, coherent mathematics is manifestly [[strongly predicative]], since having [[function sets]] in the theory implies having [[implication]] and $\Delta_0$-[[universal quantifiers]], as every cartesian closed coherent category is a [[Heyting category]].

## Equality and apartness

Since coherent mathematics does not have [[negation]], one cannot use negation to define [[denial inequality]] as the [[negation]] of [[equality]]. In particular, what fails is that the definition of denial inequality involves a double implication
$$
  \big(
    (a = b)
      \implies
    \bot
  \big)
    \implies
  (a \neq b)
$$
which is not allowed in coherent mathematics. On the level of coherent [[sequents]], single implications $\phi \implies \psi$ are expressed as [[entailment]], the sequent $\phi \vdash \psi$. But double implications $(\phi \implies \psi) \implies \varphi$ become the sequent $(\phi \implies \psi) \vdash \varphi$, which cannot be simplified down further to get rid of the remaining implication. Instead, one has to use the positive notion of [[apartness relation]] $a \# b$ to express the notion of "not being equal".

Similarly to the fact that one cannot define denial inequality in coherent mathematics, one cannot define a [[tight apartness relation]], since the definition involves negation of an apartness relation implying equality, which is a double implication
$$((a \# b) \implies \bot) \implies (a = b)$$
As a result, concepts such as [[Heyting fields]] which are important in [[constructive mathematics]] cannot be defined, and one has to use alternative concepts such as [[local rings]] instead.

Other things have to be expressed positively as well - for example, in the definition of a [[local ring]], one cannot say that the non-invertible elements form an [[ideal]] $I \subseteq A$, since that would involve an implication
$$((\exists y \in R.x \cdot y = 1) \implies \bot) \vdash x \in I$$
or in the structural case with injection $i:I \hookrightarrow R$,
$$((\exists y \in R.x \cdot y = 1) \implies \bot) \vdash \exists z \in I.i(z) = x$$
Instead, one has to say that the invertible elements form an [[anti-ideal]] $A \subseteq R$.
$$\exists y \in R.x \cdot y = 1 \vdash x \in A \quad \mathrm{material}$$
$$\exists y \in R.x \cdot y = 1 \vdash \exists z \in A.i(z) = x \quad \mathrm{structural}$$
Furthermore, in the definition of an anti-ideal itself, when one says that $0$ is not in the anti-ideal, one has to express it as a sequent $0 \in A \vdash \bot$, or in the structural case with injection $i:A \hookrightarrow R$, that $\exists x \in A.i(x) = 0 \vdash \bot$. Alternatively, if the set has an apartness relation one could just say that every element in the ideal is apart from zero, $0 \# x$.

The [[degree of a polynomial]] is only well-defined in any ring with an apartness relation. Anything that depends on well-defined degrees of a polynomial, such as [[algebraic closure]], also is only well-defined for a ring with an apartness relation.

### Decidable equality

Sets with [[decidable equality]] are however still definable in coherent mathematics; they are sets $S$ equipped with an [[apartness relation]] such that for all elements $a \in S$ and $b \in S$, $a = b$ or $a \# b$. This is expressed by the sequent
$$a \in S, b \in S \vdash (a = b) \vee (a \# b)$$
This means that one could work in predicative [[classical mathematics]] (i.e. a [[Boolean category]]) by stipulating that every set has decidable equality.

Nonetheless, even without assuming the above, a lot of [[algebraic number theory]] is coherent because the rings and [[number fields]] studied have decidable equality.

## Predicates, subsets, and restricted separation

The [[axiom schema]] of [[restricted separation]] in the set theory axioms states that for each set $S$ and each predicate $x \in S \vdash P(x)$, one could construct

* a set $\{x \in S \vert P(x)\}$
* a function $i:\{x \in S \vert P(x)\} \to S$
* a predicate $y \in \{x \in S \vert P(x)\}, z \in \{x \in S \vert P(x)\}, i(y) = i(z) \vdash y = z$
* a predicate $x \in R, \exists y \in \{x \in S \vert P(x)\}.x = i(y) \vdash P(x)$,
* a predicate $x \in R, P(x) \vdash \exists y \in \{x \in S \vert P(x)\}.x = i(y)$

This implies that $\{x \in S \vert P(x)\}$ is a [[subset]] of $S$ in the structural sense.

Assuming that the ambient logic is [[coherent logic]], restricted separation automatically implies that the sets and functions form a [[coherent category]], since the [[poset]] of [[subsets]] of each set being a [[distributive lattice]] follows from the rules for [[coherent logic]] and restricted separation.

## See also

* [[coherent logic]]
* [[foundations of mathematics]]
* [[constructive mathematics]]
* [[predicative mathematics]]
* [[classical mathematics]]
* [[geometric mathematics]]


coherent module

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## Definition

Consider a (not necessarily [[commutative ring|commutative]]) [[unital ring]] $R$. Recall that a left $R$-[[module]] $M$ is _[[finitely generated module|finitely generated]]_ if there is an [[exact sequence]] $R^n\to M\to 0$ of left $R$-modules where $n$ is a [[natural number]], and that it is _[[finitely presented module|finitely presented]]_ (or of finite presentation) if there exists an [[exact sequence]] $R^q\to R^p\to M\to 0$ where $p,q$ are [[natural numbers]].

A left _coherent module_ is a left $R$-module which is finitely generated and such that every finitely generated $R$-submodule $N\subset M$ is finitely presented (equivalently: such that the kernel of any morphism $R^n \to M$ is finitely generated).

Coherent modules behave well over [[noetherian rings]] and to some extent more generally over [[coherent rings]].

A geometric globalization of a notion of coherent module is a notion of a [[coherent sheaf]] of $\mathcal{O}$-modules for a ringed space $(X,\mathcal{O})$.

## Related concepts

* [[quasicoherent module]]

* [[dualizable module]]

* [[coherent D-module]]

## References

* [[James Milne]], section 6 of _[[Lectures on Étale Cohomology]]_

* B. Kaup, _Coherent D-modules_, chapter II of [[Armand Borel]] et al., _Algebraic D-modules_, Perspectives in Mathematics, Academic Press, 1987 ([djvu](http://www.math.columbia.edu/~scautis/dmodules/boreletal.djvu))

[[!redirects sheaf of coherent modules]]
[[!redirects sheafs of coherent modules]]


coherent object

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## Definition

### In Abelian categories

An object $X$ in an [[AB5-category]] $C$ is __of [[finite type]]__ if one of the following equivalent conditions hold:

(i) any complete directed set $\{X_i\}_{i\in I}$ of [[subobject]]s of $X$ is stationary

(ii) for any complete directed set $\{Y_i\}_{i\in I}$ of subobjects of an object $Y$ the natural morphism $\operatorname{colim}_i C(X,Y_i) \to C(X,Y)$ is an isomorphism.

An object $X$ is __finitely presented__ if it is of finite type and if for any epimorphism $p : Y \to X$ where $Y$ is of finite type, it follows that $\operatorname{ker} p$ is also of finite type. An object $X$ in an AB5 category is __coherent__ if it is of finite type and for any morphism $f : Y \to X$ where $Y$ is of finite type $\operatorname{ker} f$ is of finite type.

For an exact sequence $0 \to X' \to X \to X'' \to 0$ in an AB5 category the following hold:

1. if $X'$ and $X''$ are finitely presented, then $X$ is finitely presented;
2. if $X$ is finitely presented and $X'$ of finite type, then $X''$ is finitely presented;
3. if $X$ is coherent and $X'$ of finite type then $X''$ is also coherent.

For a module $M$ over a ring $R$ this is equivalent to $M$ being finitely generated $R$-module. It is finitely presented if it is finitely presented in the usual sense of existence of short exact sequence of the form $R^I \to R^J \to M \to 0$ where $I$ and $J$ are finite.

An AB5-category is __locally coherent__ if it has a [[generator|generating set]] of coherent objects. If it is such, than every finitely presented object is coherent, and the full subcategory of finitely presented objects is therefore abelian.

### In 1-topos theory

Let $C$ be a [[topos]].

+-- {: .num_defn}
###### Definition
An object $X$ of $C$ is called **compact** if the [[top]] element of the [[poset of subobjects]] $Sub(X)$ is a [[compact element]].
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###### Definition
An [[object]] $X$ of $C$ is called **stable** if for all morphisms $Y \to X$ from a compact object $Y$, the [[domain]] of the [[kernel pair]] $R \rightrightarrows Y$ of $f$ is also a compact object.
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###### Definition
An object $X$ of $C$ is called **coherent** if it is compact and stable.
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###### Theorem
Let $(C, \tau)$ be a [[small site|small]] [[cartesian site|cartesian]] [[site]], and suppose that $\tau$ is generated by finite [[covering families]].  For an object $X$ of $C$, let $l(X)$ denote the [[sheaf]] [[sheafification|associated]] to the [[presheaf]] [[representable functor|represented]] by $X$.  Then

* $l(X)$ is a coherent object of the [[topos]] $Sh(C, \tau)$, for all objects $X$ in $C$,
* if $(C, \tau)$ is further a [[pretopos]] with its [[coherent coverage]], then every [[coherent object]] of $Sh(C, \tau)$ is isomorphic to $l(X)$ for some $X$.
=--

This is ([Johnstone, Theorem D3.3.7](#JohnstoneSketches)).

### In $(\infty,1)$-topos theory
 {#InInfinityToposTheory}

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###### Definition

An [[object]] $X$ in an [[(∞,1)-topos]] $\mathbf{H}$ is an _$n$-coherent object_ if the [[slice (∞,1)-topos]] $\mathbf{H}_{/X}$ is an [[n-coherent (∞,1)-topos]]

=--

([Lurie, def. 3.1](#Lurie)).

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###### Remark

A coherent object which is also [[n-truncated object of an (∞,1)-category|n-truncated]] for some $n$ is called a _[[finitely constructible object]]_.

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([[Rational and p-adic Homotopy Theory|Lurie pAdic, def. 2.3.1]])

## Examples

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###### Example

[[∞Grpd]] is a [[coherent (∞,1)-topos]] and a [[locally coherent (∞,1)-topos]]. An [[object]] $X$, hence an [[∞-groupoid]], is an $n$-coherent object if all its [[homotopy groups]] in degree $k \leq n$ are [[finite set|finite]]. Hence the fully coherent objects here are the [[homotopy types with finite homotopy groups]].

=--

([[Spectral Schemes|Lurie SpecSchm, example 3.13]])

## See also

* [[compact object]]

* [[coherent topos]]

* [[coherent cohomology]]

* [[(∞,1)-pretopos]]

## References

* {#PopescuAbelian} N. Popescu, _Abelian categories with applications to rings and modules_, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. [MR0340375](http://www.ams.org/mathscinet-getitem?mr=0340375)

* {#JohnstoneSketches} [[Peter Johnstone]], [[Sketches of an elephant]], D3.3.

* {#Lurie} [[Jacob Lurie]], section 3 of _[[Spectral Schemes]]_

* Ivo Herzog, _Contravariant functors on the category of finitely presented modules_,  Israel J. Math. [pdf](http://lima.osu.edu/people/iherzog/contra.pdf):
_The Ziegler spectrum of a locally coherent Grothendieck category_, Proc. London Math. Soc. 74(3) (1997), 503-558 [pdf](http://lima.osu.edu/people/iherzog/Ziegler%20spec.pdf)

[[!redirects coherent objects]]
[[!redirects locally coherent category]]
[[!redirects n-coherent object]]
[[!redirects n-coherent objects]]


coherent product

## Definition

A **coherent product** on an [[inhabited set]] $Z$ is a [[binary function]] $k:Z \times Z \to \mathbb{C}$ of [[function of positive type|positive type]].

An inhabited set equipped with the structure of a coherent product is a **coherent space**

## Examples

* The inner product of every [[Euclidean space]] is a coherent product.

## See also

* [[inner product space]]

## References

* [[Arnold Neumaier]], *Introduction to coherent spaces*, ([arXiv:1804.01402](https://arxiv.org/abs/1804.01402))

* [[Arnold Neumaier]], *Coherent Quantum Physics: A Reinterpretation of the Tradition*, 1st edition, De Gruyter, 2019. ISBN:9783110667295


coherent ring

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## Definition

A [[ring]] is (left) **coherent** if every [[finitely generated object|finitely generated]] (left) [[ideal]] of this ring is [[finitely presented object|finitely presented]].   Equivalently, a ring is coherent if and only if it is a (left) [[coherent module]] over itself.

Every (left) [[noetherian ring]] is a coherent  ring.

[Lombardi & Quitté 2015](#LombardiQuitte15) argue that coherent rings are more suited to [[constructive mathematics]] than noetherian rings.  This is probably connected to the importance of [[coherent sheaves]].

## Related concepts

* [[coherent module]], [[coherent D-module]]

## References

* V. E. Govorov in Springer [[eom]]: *[Coherent ring](https://www.encyclopediaofmath.org//index.php?title=Coherent_ring)*

* {#LombardiQuitte15} Henri Lombardi and Claude Quitté, *Commutative Algebra: Constructive Methods: Finite Projective Modules*, Springer, 2015 ([arXiv:1605.04832](https://arxiv.org/abs/1605.04832), [doi:10.1007%2F978-94-017-9944-7](https://link.springer.com/book/10.1007%2F978-94-017-9944-7))

[[!redirects coherent rings]]


coherent sheaf

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## Idea

A _coherent sheaf of modules_ is a geometric globalization of the notion of [[coherent module]].

## Definition

### Over a ringed topos

Let $(X,\mathcal{O})$ be a [[ringed space]] or, more generally, a [[ringed site]].

A [[sheaf]] $\mathcal{E}$ on $X$ of $\mathcal{O}$-[[module]]s is

*  __finitely generated__, or __[[of finite type]]__ , if every point $x \in X$ has an open neighbourhood such that there is a surjective morphism

   $$\mathcal{O}^n|_U \to \mathcal{E}|_U$$

   from a [[free module]] to $\mathcal{E}|_{U}$,  where $n$ is finite.

* __coherent__ if it is

  1. finitely generated

  1. for every open $U$ in the base space (resp. every object $U$ in the base site), every finite $p \in \mathbb{N}$ and every morphism

     $$\mathcal{O}^p|_U\to \mathcal{E}|_U$$

     of $\mathcal{O}|_U$-modules has a finitely generated [[kernel]].

* __finitely presented__ if there is an exact sequence of the form

  $$\mathcal{O}^p\to\mathcal{O}^n\to\mathcal{E}\to 0$$

  with $p$ and $n$ finite.

  Every finitely presented $\mathcal{O}$-module is finitely generated.

* __[[quasicoherent sheaf|quasi coherent]]__ if it is _locally_   -- on a cover $\{U_i\}$ -- _presentable_, i.e. for each $i$ there is an [[exact sequence]]s

  $$
    \mathcal{O}^{I_i}|_{U_i} \to \mathcal{O}^{J_i}|_{U_i}
    \to \mathcal{E}|_{U_i}
    \to 0\,,
  $$

  where $I_i$ and $J_i$ may be infinite, i.e. $\mathcal{E}$ is locally the [[cokernel]] of [[free module]]s. For more see [[quasicoherent sheaf]].

### Over a structured $(\infty,1)$-topos

Over a [[spectral Deligne-Mumford stack]]:

([[Quasi-Coherent Sheaves and Tannaka Duality Theorems|Lurie QCoh, def. 2.6.20]])

## Properties

For a coherent sheaf $\mathcal{E}$ over a [[ringed space]], for every  point $y$ in the base space $X$ there is a neighborhood $V$ such that the $\mathcal{O}_X(V)$-module $\mathcal{E}(V)$ of sections of $\mathcal{E}$ over $V$ is finitely presented.
On a [[noetherian scheme]] the notions of finitely presented and coherent sheaves of $\mathcal{O}$-modules agree, but this is not true on a general scheme or general analytic space; sometimes even the structure sheaf $\mathcal{O}$ itself is a counterexample (not coherent while finitely presented).

The notion of coherent sheaf behaves well on the category of noetherian schemes. On a general topological space, by a basic result of Serre, if two of the sheaves of $\mathcal{O}$-modules in a [[short exact sequence]]
$$
0\to \mathcal{E}\to\mathcal{E}'\to \mathcal{E}''\to 0
$$
are coherent then so is the third. All this holds even if $\mathcal{O}$ is a sheaf of noncommutative [[rings]]. For commutative $\mathcal{O}$, the inner hom $Hom_{\mathcal{O}}(\mathcal{E},\mathcal{F})$ in the category of sheaves of $\mathcal{O}$-modules is coherent if $\mathcal{E},\mathcal{F}$ are coherent.

A theorem of Serre says that the category of coherent sheaves over a projective variety of the form $Proj R$ where $R$ is a graded commutative Noetherian ring is equivalent to the localization of the category of finitely generated graded $R$-modules modulo its ("torsion") subcategory of (finitely generated graded) $R$-modules of finite length.

## Historical note and definition variants

First works on coherent sheaves in complex analytic geometry. Serre adapted their work to algebraic framework in his famous article [[FAC]]. Hartshorne's definitions are changed/adapted to the special setup of Noetherian schemes with the excuse that the coherence does not behave that well otherwise; thus they differ from the definitions in [[EGA]] and [[FAC]].

[[A. Vistoli]] commented at MathOverflow [here](http://mathoverflow.net/questions/68150) that for some categorical purposes

> one should interpret "coherent" as meaning "quasi-coherent of finite presentation". The notion of coherent sheaf, as defined in EGA, is not functorial, that is, pullbacks of coherent sheaves are not necessarily coherent. Hartshorne's book defines "coherent" as "quasi-coherent and finitely generated", but this is a useless notion when working with non-noetherian schemes.

## Related concepts

* [[degree of a coherent sheaf]], [[rank of a coherent sheaf]] [[slope of a coherent sheaf]],

* [[stable coherent sheaf]]

* [[quasicoherent sheaf]]
* [[triangulated categories of sheaves]]
* [[Bondal-Orlov reconstruction theorem]]

* [[coherent cohomology]]

##References

* [[J-P. Serre]], _[[Faisceaux algébriques cohérents]]_, Ann. of Math. (2) __61__,  (1955) 197--278, [doi](http://dx.doi.org/10.2307/1969915).

* H. Grauert, R. Remmert, _Coherent analytic sheaves_, Grundlehren der Math. Wissenschaften __265__, Springer 1984. xviii+249 pp.

* M. M. [[Kapranov]], _On the derived categories of coherent sheaves on some homogeneous spaces_,  Invent. Math.  92  (1988),  no. 3, 479--508, [doi](http://dx.doi.org/10.1007/BF01393744).

* [[D. O. Orlov]], _Производные категории когерентных пучков и эквивалентности между ними_ ([pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=629&what=fullt&option_lang=rus), Russian)  Uspekhi Mat. Nauk __58__  (2003),  no. 3(351), 89--172;  Engl. transl. _Derived categories of coherent sheaves and equivalences between them_, Russian Math. Surveys __58__ (2003),  no. 3, 511--591.

* V. D. Golovin, _Homology of analytic sheaves and duality theorems_, Contemporary Soviet Mathematics (1989) viii+210 pp. transl. from Russian original Гомологии аналитических пучков и теоремы двойственности, Moskva, Nauka 1986. (192 pp.)

* [[EGA]] 0.5.3.1
* Qing Liu, _Algebraic geometry and arithmetic curves_, 5.1.3

Categories of ind-coherent sheaves on schemes and stacks are studied  in [[Dennis Gaitsgory]], _Notes on Geometric Langlands: ind-coherent sheaves_, [arxiv/1105.4857](http://arxiv.org/abs/1105.4857)

Discussion in [[(∞,1)-topos theory]]

* [[Jacob Lurie]], _[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]_

category: sheaf theory, algebraic geometry

[[!redirects coherent sheaves]]
[[!redirects coherent modules]]


coherent space

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> Not to be confused with the notion of *[[coherence space]]* in [[models]] of [[linear logic]].

***

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## Idea

A _coherent space_ (alias _spectral topological space_) is a [[topological space]] which is [[homeomorphism|homeomorphic]] to the [[spectrum of a commutative ring]], hence to the topological space underlying an [[affine scheme]].

Equivalently, it is a [[compact]] [[sober]] [[topological space]]
whose collection of compact open subsets is closed under finite intersections and forms a topological base.

Morphisms of coherent spaces are continuous maps
such that preimages of compact open subsets are again compact.

## Stone duality for coherent spaces

Passing from a coherent space to its lattice of compact open subsets
establishes a contravariant equivalence
from the category of coherent spaces
to the category of (bounded) [[distributive lattices]].

A coherent space is Hausdorff if and only if it is a [[Stone space]].
Under Stone duality for coherent spaces,
this corresponds to the fact that in a [[distributive lattice]] $L$
every element has a complement if and only if $L$ is a [[Boolean algebra]].

In particular, restricting the Stone duality equivalence between
coherent spaces and distributive lattices
to Stone spaces and Boolean algebras
recovers the classical Stone duality.

## Coherent locales

A [[locale]] is coherent if its compact elements
are closed under finite meets
and any element is a join of compact elements.

A morphism of coherent locales is a morphism $f$ of [[locales]]
such that $f^*$ preserves compact elements.

Assuming the [[axiom of choice]], coherent locales are
automatically spatial.

The category of coherent locales is contravariantly equivalent
to the category of [[distributive lattices]].
This statement does not depend on the [[axiom of choice]],
unlike the point-set version above.

## Related notions

* [[coherent topos]]

## References

* [[Peter Johnstone]], _[[Stone Spaces]]_.

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Countable meets in coherent spaces with applications to the cyclic spectrum*, Theory Appl. Categories **25** 19 (2011), 508–232 [[tac:25-19](http://www.tac.mta.ca/tac/volumes/25/19/25-19abs.html)]

* Max Dickmann, Niels Schwartz, [[Marcus Tressl]], _Spectral Spaces_. New Mathematical Monographs 35 (2019).  Cambridge: Cambridge University Press. ISBN 9781107146723.

See also:

* Wikipedia, _[Spectral space](http://en.wikipedia.org/wiki/Spectral_space)_

[[!redirects spectral topological space]]
[[!redirects spectral topological spaces]]

[[!redirects spectral space]]
[[!redirects spectral spaces]]

[[!redirects coherent topological space]]
[[!redirects coherent topological spaces]]

[[!redirects coherent spaces]]
[[!redirects coherent locale]]
[[!redirects coherent locales]]


coherent state

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## Idea

_Coherent states_ are members of special overcomplete families of elements (usually in the projectivization) of [[Hilbert space]] of states of certain systems in [[quantum mechanics]]. This entry is mainly about Perelomov style CS, see a separate page for [[Hall coherent state]]s. Coherence in the sense of quantum optics, and in the sense of statistical density matrix is one of the features of these very specifical objects in mathematical physics (thus coherent quantum states would be states of high degree of features of coherence in those senses, rather than refering to the specific families in mathematical physics).

## Examples

### Of the harmonic oscillator

The most classical coherent states of E. Schrödinger (1926, _coherent_ refers to its meaning in optics) are the elements in $L^2(\mathbb{R}^n)$ which are the eigenfunctions $|z\rangle = |z_1,\ldots,z_n\rangle$ of the annihilation operators $a_i$, $i=1,\ldots,n$; the eigenvalues $z_i$ are complex numbers and all complex numbers appear in that way; more precisely the coherent states are parametrized by numbers $z=(z_1,\ldots,z_n)$ in $\mathbb{C}^n$, and there is a measure $d\mu$ on $\mathbb{C}^n$ providing the resolution of unity formula
$$
\int_{\mathbb{C}^n} |z\rangle \langle z| d\mu = 1
$$

Coherent states however are not mutually orthogonal; they form an overcomplete family of states: not only that every vector in $L^2$ can be expressed in terms of them but rather in more than one way and we can throw certain coherent states out and not to spoil the property (how many and which is a more subtle question). Thus expressing states in terms of coherent states is sort of Fourier analysis; in fact this approach is taken in a variant called wavelet analysis. Coherent states are minimizing the uncertainty relations, closest to classical states in their behaviour; their probability density is Gaussian. Vectors in the Hilbert space can be represented in the coherent state representation: $|f\rangle = \int |z\rangle\langle z|f\rangle d\mu$; if $f$ is in $L^2$ then $\langle z|f\rangle$ is a holomorphic function and this passage is called the Bargmann-Segal transform (referring to [Irving Segal](http://en.wikipedia.org/wiki/Irving_Segal)); this way certain Hilbert space of holomorphic function appears, the Bargmann-Fock space. The evolution of $\langle z|f\rangle$ formally satisfies the classical Newton equations of motion.

In general, projection operators $|z\rangle \langle z|$ and the scalar measure $d\mu$ do not separate, in fact instead of $|z\rangle \langle z| d\mu$ one considers more general [[projection measure]] (descriptively a "measure" with values in self-adjoint projection operators on a separable Hilbert space) on a more general geometric space $X$ replacing $\mathbb{C}^n$ which is normalized to 1 on the whole space. Symbollically we can still pretend that the projective measure is of the form $\mu(A) = \int_A |z\rangle \langle z| d\mu$ where $d\mu$ is an ordinary (scalar-valued) measure.

### Over a flag manifold

The basic example of the space $X$ is the homogeneous space $G^{\mathbb {C}}/B = G/K$ where $G$ is a compact Lie group, $G^{\mathbb{C}}$ its complexification, $K\subset G$ maximal compact subgroup and $B\subset G^{\mathbb{C}}$ the Borel subgroup. Generalized [[flag manifold]] $G/K$ is naturally a compact complex manifold, in fact Kähler. Fix a unitary character $\chi:G\to S^1$ which uniquely extends to a character $\chi:G^{\mathbb{C}}\to \mathbb{C}^*$ of the complexification; let $\mathbf{C}_\chi$ be the corresponding 1-dimensional representation. Then let $L_\chi = G^{\mathbb{C}}\times_{G^{\mathbb{C}}} \mathbb{C}_\chi$ be the associated line bundle. It is equipped with a natural holomorphic structure and hermitean scalar product; the space of holomorphic (or antiholomorphic depending on conventions in the construction) sections of that bundle is finite-dimensional and irreducible by [[Borel-Weil theorem]]. __Perelomov coherent states__ on that bundle are the elements of the orbit of $G$ of the heighest (equivalently lowest) weight vector (or equivalently of $G^{\mathbb{C}}$: the real and complex orbits are equal). The covariant generalized uncertainty relations (ref. by Onofri, below) correspond to the expression for a quantity proportional to negative of the square of the moment map (see ref. by Spera, below), which is naturally extremal on symplectic orbits. These coherent states can also be realized as the duals (according to Riesz theorem, that is dual vectors in a Hilbert space) to the evaluation functionals on the space of sections: take a point $q$ in the space of a line bundle and then divide it by the value of the section on the projection of $q$ to the homogeneous space. Up to a scalar, coherent states do not depend on the choice of the point on the line, hence this yields a holomorphic embedding of $G^{\mathbb{C}}/B$ to the projectivization of the representation space, so called __coherent states embedding__. The rays are the coherent states and the choices with scalar multiple accounted for are the coherent vectors. These coherent vectors are normalized by the measure which is the push down of the measure form $G$ to $G/K$. The classical coherent states appear in noncompact case where the group in question is the Heisenberg group, and the covariant uncertainty relations are the usual ones.

## Related concepts

* [[quasi-free quantum state]]

* [[coherent state (in geometric quantization)]],

* [[Hall coherent state]]

* [[Berezin quantization]]

* [[wavelet]], [[curvelet]]

## References

Textbook account:

* [[Ingemar Bengtsson]], [[Karol Życzkowski]], Chapter 6 of: *Geometry of Quantum States --- An Introduction to Quantum Entanglement*, Cambridge University Press (2006) [[doi:10.1017/CBO9780511535048](https://doi.org/10.1017/CBO9780511535048)]

See also:

* A. M. Perelomov, [Coherent states for arbitrary Lie groups](http://projecteuclid.org/euclid.cmp/1103858078),
Comm. Math. Phys., 26 (1972), pp. 222--236; archived as [arXiv:math-ph/0203002](http://arxiv.org/abs/math-ph/0203002).

* A. Perelomov, _Generalized coherent states and their applications_, Springer 1986

* J. H. Rawnsley, _Coherent states and Kähler manifolds_,
Quart. J. Math. Oxford (2), 28 (1977), pp. 403--415

* V. Bargmann, _On a Hilbert space of analytic functions and an associated integral transform_, Communications on Pure and Applied Mathematics __14__ (1961) 187-214 [MR0157250](https://www.ams.org/mathscinet-getitem?mr=157250) [doi](ttp://dx.doi.org/10.1002/cpa.3160140303)

* F. A. Berezin, _Quantization_, Math. USSR Izv., 8 (1974), 1109-1163. MR 0395610 (52:16404)

* [[Mauro Spera]], _On a generalized uncertainty principle, coherent states and the moment map_, J. of Geometry and Physics 12 (1993) 165-182.

* Enrico Onofri, _A note on coherent state representations of Lie groups_, J. Math. Phys. __16__:5, 1087-1089, may 1975.

* for physical aspects [wikipedia:Coherent_state](https://en.wikipedia.org/wiki/Coherent_state).

* V. V. Kisil, _Integral representations and coherent states_, Bulletin of the Belgian Mathematical Society, v. 2 (1995), No 5, 529--540.

* Thomas Appl,  Diethard H Schiller, _Generalized hypergeometric coherent states_, J. Phys A37:7 (2004) 2731 [doi](https://doi.org/10.1088/0305-4470/37/7/015)
* [[Mauro Spera]], _On Kahlerian coherent states_, in Proc. Int. Conf. on Geometry, Integrability and Quantization 241-256, Institute of Biophysics and Biomedical Engineering, Bulg. Acad. Sci. 2000.
* Rukmini Dey, Kohinoor Ghosh, Pullback coherent and squeezed states and quantization, [arXiv:2108.08082](https://arxiv.org/abs/2108.08082)
* JP Antoine, F. Bagarello, JP Gazeau (eds) Coherent States and Their Applications. Springer Proceedings in Physics 205

Projection valued measures are central objects in the theory of (general) [[coherent state]]s.

* Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau, _Coherent states, wavelets and their generalizations_, Graduate Texts in Contemporary Physics. Springer 2000

Coherent states can be generalized to [[noncommutative geometry]], most notably for [[quantum groups]]:

* H. Sazdjian, Y.S. Stanev, I.T. Todorov, _SU(3)-coherent state
operators and invariant correlation functions and their quantum group
counterparts_, J. Math. Phys. __36__ (1995), pp. 2030--2052.

* [[Branislav Jurčo|B. Jurčo]], P. Štoviček, [Coherent states for quantum compact groups](http://projecteuclid.org/euclid.cmp/1104288026), Comm. Math. Phys. __182__ (1996), pp. 221--251; [hep-th/9403114](https://arXiv.org/abs/hep-th/9403114)

* [[Zoran Škoda|Z. Škoda]], [Coherent states for Hopf algebras](https://doi.org/10.1007/s11005-007-0166-y), Letters in Mathematical Physics __81__, N.1, pp. 1-17, July 2007; distinct earlier version, arXiv: [math.QA/0303357](http://arXiv.org/abs/math/0303357)

* N. Aizawa, R. Chakrabarti, _Coherent state on $SU_q(2)$ homogeneous space_, J. Phys. A __42__:29 [arXiv:0905.0194](http://arxiv.org/abs/0905.0194).

[[!redirects coherent states]]

[[!redirects coherent quantum state]]
[[!redirects coherent quantum states]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Geometric quantization
+-- {: .hide}
[[!include geometric quantization - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of _[[coherent state]]_ in [[quantum mechanics]]/[[quantum physics]] may be formalized in the context of _[[geometric quantization]]_, The original definition is due to ([Rawnsley 77](#Rawnsley77)), further developments include ([Kirwin 07](#Kirwin07)).

## Related concepts

* [[quantization of Chern-Simons theory]]

## References

The original reference that interpreted coherent states in [[geometric quantization]] over [[Kähler manifold]]s is

* {#Rawnsley77} J. H. Rawnsley, _Coherent states and Kähler manifolds_, Quart. J. Math. Oxford (2), 28 (1977), pp. 403–415

A review and further developments of these considerations are in

* [[Mauro Spera]], _On Kählerian coherent states_ ([djvu](http://www.bio21.bas.bg/proceedings/Proceedings_files/vol1/GEOM19.DJVU))

Generalization to non-Kähler symplectic manifolds is in

* {#Kirwin07} [[William Kirwin]], _Coherent states in geometric quantization_,  J. Geom. Phys.  57  (2007),  no. 2, 531–548. [arXiv:0502026](http://arxiv.org/abs/math/0502026) [MR2008b:53114](http://www.ams.org/mathscinet-getitem?mr=2271203) [doi](http://dx.doi.org/10.1016/j.geomphys.2006.04.007)

See also

* Anatol Odzijewicz, _Coherent state method in geometric quantization_, in Twenty Years of Bialowiez: a mathematical anthology, Aspects of Differential Geometric Methods in Physics_ (pp 47-78) , World Sci. Monogr. Ser. Math. 8 (2005); _Coherent states and geometric quantization_,  Comm. Math. Phys.  __150__  (1992),  no. 2, 385–413 [doi](http://ams.mpim-bonn.mpg.de/leavingmsn?url=http://projecteuclid.org/getRecord?id=euclid.cmp/1104251870) [94c:58077](http://www.ams.org/mathscinet-getitem?mr=1194023)

* Brian Charles Hall, _Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type_, Comm. Math. Phys., __226__:233–268, 2002. [doi](http://dx.doi.org/10.1007/s002200200607)

* D. J. Rowe, _Coherent states, induced representations, geometric quantization, and their vector coherent state extensions_, Symmetry in physics, 177–188, CRM Proc. Lecture Notes __34__, Amer. Math. Soc. 2004.
[MR2005a:81100](http://www.ams.org/mathscinet-getitem?mr=2056987)

* C. Florentino, P. Matias, J. Mourão, J. P. Nunes, _Geometric quantization, complex structures and the coherent state transform_,
J. Funct. Anal. __221__(2):303–322, 2005 [MR2005m:53179](http://www.ams.org/mathscinet-getitem?mr=2124866) [doi](http://dx.doi.org/10.1016/j.jfa.2004.10.021)

[[!redirects coherent state (in geometric quantization)]]
[[!redirects coherent states (in geometric quantization)]]
[[!redirects coherent state (in geometric quantisation)]]
[[!redirects coherent states (in geometric quantisation)]]
[[!redirects coherent state in geometric quantization]]
[[!redirects coherent states in geometric quantization]]
[[!redirects coherent state in geometric quantisation]]
[[!redirects coherent states in geometric quantisation]]


coherent states > history

see at _[[coherent state]]_


coherent topological space > history

< [[coherent space]]


coherent topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A **coherent topos** is a [[sheaf topos]] with a [[site]] of definition that is a [[coherent category]] with [[coherent coverage]].

Notice that the terminology is slightly confusing: every topos is a [[coherent category]].

## Properties

+-- {: .num_theorem }
###### Theorem
**[[Deligne completeness theorem]]**

Every coherent topos has [[point of a topos|enough points]].

=--

## Examples

* The [[classifying topos]] of a [[coherent theory]] is a coherent topos.

## Related concepts

* [[coherent (∞,1)-topos]]

[[!include categories and logic - table]]

## References

Section D3.3 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

[[!redirects coherent toposes]]
[[!redirects coherent topoi]]


cohesion - table

**[[cohesion]]**

* ([[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]])

   $(\esh \dashv \flat \dashv \sharp )$

  * [[discrete object]],  [[codiscrete object]], [[concrete object]]

  * [[points-to-pieces transform]]

  * [[cohesive (infinity,1)-topos -- structures|structures in cohesion]]

* [[dR-shape modality]] $\dashv$ [[dR-flat modality]]

  $\esh_{dR} \dashv \flat_{dR}$

**[[infinitesimal cohesion]]**

* [[classical modality]]

**[[tangent cohesive (∞,1)-topos|tangent cohesion]]**

* [[differential cohomology diagram]]

**[[differential cohesion]]**

* ([[reduction modality]] $\dashv$ [[infinitesimal shape modality]] $\dashv$ [[infinitesimal flat modality]])

  $(\Re \dashv \Im \dashv \&)$

  * [[reduced object]], [[coreduced object]], [[formally smooth object]]

  * [[formally étale map]]

  * [structures in differential cohesion](cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion)

**[[super smooth infinity-groupoid|graded differential cohesion]]**

* [[fermionic modality]] $\dashv$ [[bosonic modality]] $\dashv$ [[rheonomy modality]]

  $(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

**[[orbifold cohomology|singular cohesion]]

$$
  \array{
    && id &\dashv& id
    \\
    && \vee && \vee
    \\
    &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{}
    \\
    && \bot && \bot
    \\
    &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{}
    \\
    && \vee && \vee
    \\
    &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{}
    \\
    && \bot && \bot
    \\
    &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{}
    \\
    && \vee && \vee
    \\
    &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{}
    \\
    && \bot && \bot
    \\
    &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{}
    \\
    && \vee && \vee
    \\
    && \emptyset &\dashv& \ast
  }
$$
{#Diagram}