discrete fibration

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## Definition

A [[functor]] $F: C \to B$ is a **discrete fibration** if for every object $c$ in $C$, and every morphism of the form $g : b\to F(c)$ in $B$ there is a unique morphism $h : d\to c$ in $C$ such that $F(h) = g$.

A functor $F: C \to B$ is a **discrete opfibration** if $F^{op}:C^{op}\to B^{op}$ is a discrete fibration.

A discrete fibration is a special case of a [[Grothendieck fibration]], namely one where each [[fiber]] is a [[set]].

Given a cartesian category $E$, [[internal categories]] $C,B$ in $E$, an internal functor $F: C \to B$ is a **discrete fibration of internal categories** if the square
$$\begin{matrix}
C_1 &\stackrel{F_1}\to& B_1\\
d_0\downarrow && \downarrow d_0\\
C_0 &\stackrel{F_0}\to& B_0
\end{matrix}$$
is cartesian.

## Discussion via category of elements

Given a discrete fibration $F: C \to B$, define a functor $F^*: B^{op} \to Set$ as follows:

*  For $x$ an object of $B$, let $F^*(x)$ be the set of objects $y$ of $C$ such that $F(y) = x$.
*  For $g: x \to y$ a morphism of $B$, let $F^*(g): F^*(y) \to F^*(x)$ be the function that maps each element of $F^*(y)$ to the unique $d$ determined by the definition of discrete fibration above.

There is a size issue here, is $F^*(x)$ in fact [[small category|small]]?  We say that the fibration has **small fibres** if so; else we must pass to a larger [[universe]] when we define [[Set]].

Conversely, give a functor $F^*: B^{op} \to Set$, define a category $C$ and a discrete fibration $F: C \to B$ as follows:

*  Let $C$ be the [[category of elements]] of the functor $F^*$; that is:
   *  an object of $C$ is a pair consisting of an object $x$ of $B$ and an element of $F^*(x)$,
   *  a morphism from $(x,a)$ to $(y,b)$ in $C$ is a morphism $g: x \to y$ in $B$ such that $F^*(g)$ maps $b$ to $a$.
*  The functor from $C$ to $B$ is the obvious [[forgetful functor]].

If you start from $F^*$, construct $C$ and $F$, and then construct a new $F^*$, it will be equal to the original $F^*$.  Conversely, if you start with $C$ and $F$, construct $F^*$, and then construct a new $C'$ and $F'$, then there will be an [[isomorphism of categories]] between $C$ and $C'$, relative to which $F$ and $F'$ are equal.

Under this equivalence, the [[representable presheaf]] on an object $X$ corresponds to the canonical functor $B/X \to B$ from the [[slice category]] over $X$.

## Invariance under equivalence

Note that the definition of fibration refers to [[equality]] of morphisms without previously assuming that the sources match, while the construction of $F^*$ from $F$ refers to equality of objects.  This is also why we get [[equality]] of [[functors]] and [[isomorphism]] of [[categories]] in the immediately preceding paragraph.  So the only thing on this page which respects the [[principle of equivalence]] is the idea of a functor to [[Set]].  That is the fundamental invariant notion; a discrete fibration is just a convenient way of talking about it.

## Further properties

* Just as general [[Grothendieck fibration|fibrations]], discrete fibrations are stable under [[pullbacks]] in [[Cat]].

* Discrete (op)fibrations are the right class of the [[comprehensive factorisation system]].

## Generalization for spans internal to a category

Let $E$ be a cartesian category.
A span of [[internal categories]] $A\stackrel{p}\leftarrow C\stackrel{q}\to B$ in $Cat(E)$ is called a **discrete fibration from $A$ to $B$** if in the diagram

$$\begin{matrix}
A_0 & \leftarrow & C_l && \\
\downarrow &&\downarrow i_l &&\\
A &\stackrel{p}\leftarrow & C &\stackrel{i_r}\leftarrow & C_r\\
&&q \downarrow && \downarrow \\
&& B &\leftarrow & B_0
\end{matrix}$$

in which the two squares are the cartesian satisfies the following 3 properties:

* $p\circ i_r : C_1\to A$ is a discrete fibration

* $q\circ i_l: C_l\to B$ is a discrete opfibration

* Let $X$ be defined as the pullback
$$\begin{matrix}
X & \to & (C_r)_1 \\
\downarrow &&\downarrow \\
(C_l)_1 &\to & C_0
\end{matrix}$$
and $j:X\hookrightarrow C_1\times_{C_0} C_1$ the canonical inclusion. Then
the morphism $c\circ j : X\to C_1$, where $c: C_1\times_{C_0} C_1\to C_1$ is the composition morphism of internal category $C$, is invertible.

**Example.** Given internal functors $a : A\to D$ and $b : B\to D$ in $E$, the obvious span $A\leftarrow a\downarrow b\rightarrow B$ is a discrete fibration from $A$ to $B$.

## Related concepts

* [[fibration fibered in groupoids]]

* [[Grothendieck fibration]], [[Grothendieck construction]]

## References

Review:

* [[Angelo Vistoli]], §3.4 in: *Grothendieck topologies, fibered categories and descent theory*, in: *[[Fundamental algebraic geometry -- Grothendieck's FGA explained]]*, Mathematical Surveys and Monographs **123**, Amer. Math. Soc. (2005) 1-104 [[ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), [math.AG/0412512](http://arxiv.org/abs/math/0412512)]

See also:

* [[Emily Riehl]] and [[Fosco Loregian]], _Categorical notions of fibration_, Expositiones Mathematicae **38** 4 (2020) 496-514 [[arXiv:1806.06129](https://arxiv.org/abs/1806.06129), [doi:10.1016/j.exmath.2019.02.004](https://doi.org/10.1016/j.exmath.2019.02.004)]

A joint generalisation of the notions of discrete fibrations and [[discrete opfibrations]] are studied in the following paper under the name *unique factorization liftings*, i.e. [[discrete Conduché functors]]:

* [[Marta Bunge]], [[Susan Niefield]], _Exponentiability and single universes_, JPAA **148** (2000) 217--250

[[!redirects discrete fibration]]
[[!redirects discrete fibrations]]
[[!redirects discrete opfibration]]
[[!redirects discrete opfibrations]]


discrete Fourier transform

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## Idea

The special case of the [[Fourier transform]] for a [[cyclic group]] (a discretized [[circle]])

## References

* Wikipedia, _[Discrete Fourier transform](https://en.wikipedia.org/wiki/Discrete_Fourier_transform)_

* Audrey Terras, _Fourier Analysis on Finite Groups and Applications_


discrete group

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#### Cohesion
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## Idea

A __discrete group__ is a [[group object]] in [[discrete spaces]].

Equivalently this is just an ordinary [[group]] -- a [[set]] with a group structure. One says _discrete group_ for emphasis in a context where by default one is speaking about [[topological group]]s or [[Lie group]]s or groups equipped with other [[stuff, structure, property|structure]]. See [[discrete ∞-groupoid]] for more discussion.

## Related concepts

* [[discrete space]]

* **discrete group**

  * [[acyclic group]]

* [[discrete groupoid]]

* [[discrete ∞-groupoid]]

* [[good orbifold]]
[[!redirects discrete groups]]


discrete mathematics

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## Idea
 {#Idea}

Taken at its verbatim face value, the term *discrete mathematics* refers to [[mathematics]] concerned with [[mathematical structures]] which are *discrete* in the sense of *[[discrete topological spaces]]*, hence which do not involve [[topology]] and in particular do *not* involve [[analysis]] ("[[calculus]]").

With the hindsight of the [[nPOV]] one could usefully say that discrete mathematics, in this sense, is the topic of ([[categorical semantics|models]] of) bare [[homotopy type theory]], in contrast to its [[cohesion|cohesive]] refinement to [[cohesive homotopy type theory]].

However, in common parlance the term *discrete mathematics* is used much more restrictively:

{#AsUndergraduateCourses} On the one hand, it often appears as the title of those introductory courses in the undergraduate [[mathematics]] and [[computer science]] curricula which are expressly independent of [[differential calculus]] and instead concerned with discussing prerequisites such as naïve [[set theory]], basic [[predicate logic]], an introduction to [[proofs]], and maybe elementary [[number theory]], and possibly aimed at providing an overview of [[combinatorics]], [[graph theory]] and associated [[algorithms]] (e.g. [Molluzzo & Buckley 1986](#MolluzzoBuckley86)).

On the other hand, in research mathematics the term *discrete mathematics* is mainly used (see the references [below](#ResearchLevelReferences)) as a synonym for *[[combinatorics]]* and closely related fields (or subfields): finite geometry and [[incidence geometry]], [[coding theory]] and [[Boolean functions]], combinatorics of [[posets]], [[matroid]] theory, convex geometry and [[polytopes]], [[packings]], [[tilings]], etc.

## Related concepts

* [[combinatorics]]

* [[graph theory]]

## References

See also:

* Wikipedia, *[Discrete mathematics](https://en.m.wikipedia.org/wiki/Discrete_mathematics)

### Undergraduate level

Textbook accounts for undergraduates:

* {#MolluzzoBuckley86} John C. Molluzzo, Fred Buckley, *A First Course in Discrete Mathematics*, Waveland Press (1986) $[$[ISBN 13:978-0-88133-940-6](https://www.waveland.com/browse.php?t=482&r=d|274)$]$

* Gary Chartrand, Ping Zhang: *Discrete Mathematics* Waveland Press (2011) $[$[ISBN:978-1577667308](https://www.waveland.com/browse.php?t=479&r=d|274)$]$

* Kenneth Rosen, *Discrete Mathematics and its Applications*, McGraw Hill (2019) $[$ISBN:978-1-259-67651-2, [pdf](https://lms.su.edu.pk/download?filename=1588493195-discretemathematics-text-book.pdf&lesson=536)$]$

### Research level
 {#ResearchLevelReferences}

* ScienceDirect: *[Discrete Mathematics](https://www.sciencedirect.com/journal/discrete-mathematics)*


discrete morphism

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# Discrete morphisms
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## Definition

A [[morphism]] $f\colon A\to B$ in a [[2-category]] $K$ is called **discrete** if it is representably [[faithful morphism|faithful]] and [[conservative morphism|conservative]], i.e. if for any [[object]] $X$ the [[hom functor|induced]] [[functor]]

$$
  K(X,A) \to K(X,B)
$$

is [[faithful functor|faithful]] and [[conservative functor|conservative]].

An object $A$ is called a **[[discrete object]]** if for any $X$, the category $K(X,A)$ is ([[equivalence of categories|equivalent to]]) a [[discrete category]], i.e. a [[set]].  If $K$ has a ([[2-limit|2-]])[[terminal object]] $1$, this is equivalent to saying that the unique map $A\to 1$ is a discrete morphism.

### Caveat

**NB:** it is more common to define these concepts in the other order: to first define an object to be discrete, as we have done, and then say that $f\colon A\to B$ is a discrete morphism if is a discrete object in the [[slice 2-category]] $K/B$.  In general this does *not* result in the same notion of "discrete morphism" as the definition we have given.  For instance, if $B$ is the [[interval category]] $(0\to 1)$ and $A$ is the free [[parallel pair]] $(0 \rightrightarrows 1)$, then the obvious functor $A\to B$ is a discrete object of $Cat/B$, but is not faithful.

However, the two definitions do coincide for [[fibration in a 2-category|fibrations]], opfibrations, and [[two-sided fibration]]s.  That is, if $f\colon A\to B$ is a fibration or an opfibration in $B$, then it is faithful and conservative if and only if it is a discrete object of $K/B$, and similarly if $A\leftarrow E \to B$ is a two-sided fibration, then $E\to A\times B$ is faithful and conservative if and only if it is a discrete object of $K/(A\times B)$.  Since this is usually the case of most interest (giving rise to [[discrete fibrations]] and, dually, [[codiscrete cofibrations]]), the difference between the two definitions is usually unimportant.

## Examples

### Discrete categories

A discrete object in the [[2-category]] [[Cat]] is, of course, a [[discrete category]]. A functor is a discrete morphism if and only if the functor itself is [[faithful functor|faithful]] and [[conservative functor|conservative]].

### Discrete groupoids

A discrete object in the [[(2,1)-category]] [[Grpd]] of [[groupoids]] is also called a [[0-truncated]] object or [[0-groupoid]] or [[homotopy n-type|homotopy 0-type]] or just [[h-level|0-type]] or [[h-set]].

See also the discussion at [[discrete space]] and [[discrete groupoid]].

## Factorization systems and discrete reflections

Discrete morphisms are often the right class of a [[factorization system in a 2-category|factorization system]].  This factorization system, or one related to it, plays a role in the construction of a [[proarrow equipment]] from [[codiscrete cofibrations]].

[[!redirects discrete morphism]]
[[!redirects discrete morphisms]]

[[!redirects discrete object in a 2-category]]


discrete Morse theory

## Discrete Morse Theory ##

Discrete Morse theory aims to do for [[simplicial complexes]] and more generally, finite [[CW complexes]] what standard [[Morse theory]] does for manifolds.

### Partial Matchings on CW Complexes ###

Let $X$ be a finite CW complex. The central idea is to make extensive use of [[simple homotopy theory]]. Although the initial formulation of discrete Morse theory involved certain families of locally constant functions $f:X \to \mathbb{R}$ playing the role of Morse functions, a cleaner treatment may be given as follows. An **acyclic matching** on $X$ consists of the following data: a partition of the cells into three disjoint classes $A, B$ and $C$ along with a distinguished set-bijection $w: A \to B$. This data is required to obey the following axioms:

* for each cell $a \in A$, the matched cell $w(a)$ is a regular co-face of $a$, meaning that $\dim (w(a)) = \dim (a) + 1$ and the degree of the attaching map from $w(a)$ to $a$ is $\pm 1$,
* the relation $\prec$ on $A$ generated by $a \prec a'$ whenever $a$ is a (not necessarily regular) face of $w(a')$ defines a partial order.

### Homotopy ###

A central result of discrete Morse theory is as follows: if $X$ is any finite CW complex which admits an acyclic matching with $n$ critical cells, then $X$  is homotopy equivalent to a CW complex with $n$ cells. The proof is similar in spirit to the traditional Morse lemmas: instead of deforming along flow lines between adjacent critical points, one uses simple homotopy collapses to obtain the desired equivalence.

### Gradient Paths and Morse Homology ###

The unmatched cells $C$ play the role of critical points, while the matched cells in $A$ and $B$ generate gradient-like paths between critical cells. In fact, if we denote the face relation in $X$ by $\leq$, then a gradient path from critical cell $c$ to critical cell $c'$ consists precisely of an alternating sequence of distinct matched cells which fit together like this:

$$c \geq a_0 \leq w(a_0) \geq a_1 \leq w(a_1) \geq \ldots \geq a_n \leq w(a_n) \geq c'$$

Each such path can be assigned an **multiplicity** given by a product of attaching degrees: reading from left to right, each time one sees $x \geq y$, one multiplies by the degree of the attaching map from $x$ to $y$, and each time one sees $y \leq x$ one divides by this degree. The first axiom of the partial matching guarantees that one only divides by $\pm 1$, while the second axiom guarantees that no such path can go "backwards" from $c'$ up to $c$.

The **Morse Complex** $(C,d)$ is the free chain complex built on the critical cells in $C$ where the component of the boundary operator $d$ from $c$ to $c'$ is the sum of multiplicity over all gradient paths from $c$ to $c'$. The main *Theorem* is that the Morse complex has the same integer homology groups as $X$.

### Applications ###

Discrete Morse theory has been used in topological combinatorics to show that certain simplicial complexes are homotopy-equivalent to wedges of spheres. It has also been used in commutative algebra to construct minimal resolutions of ideals in polynomial rings. Discrete Morse theory is also used extensively in [[homology]] and [[persistent homology]] computations as in the [CHomP](http://chomp.rutgers.edu) and [Perseus](http://www.sas.upenn.edu/~vnanda/perseus) software packages.


discrete object

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# Discrete objects
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## Idea

A __discrete space__ is, in general, an object of a [[concrete category]] $Sp$ of spaces that is [[free functor|free]] on its own underlying set.  More generally, the notion can be applied relative to any [[forgetful functor]].

**Note:** *This page is about the "[[cohesive]]" or "[[topology|topological]]" notion of discreteness.  In [[2-category theory]] the term "discrete object" is also often used for [[n-truncated object|0-truncated objects]].  For this usage, see _[[discrete morphism]]_ instead.*

## Definition

A discrete space must, in particular, be a [[free object]] for the forgetful functor $U\colon Sp\to Set$, i.e. in the image of its [[adjoint functor|left adjoint]] $F: Set \to Sp$.  However, this is not sufficient for it to be free on its *own* underlying set; we must also require that the counit $F U X\to X$ be an [[isomorphism]].

Thus, we say that $U\colon Sp \to Set$ (or more generally, any functor) **has discrete spaces** or **discrete objects** if it has a [[fully faithful functor|fully faithful]] left adjoint.  This ensures that the functor
$$ Set \stackrel{F}\to Sp \stackrel{U}\to Set $$
is ([[natural isomorphism|naturally isomorphic]] to) the [[identity functor]] on [[Set]].  This is true, for example, if $Sp$ is [[Top]], [[Diff]], [[Loc]], etc.

Assuming that $U$ is [[faithful functor|faithful]] (as it is when $Sp$ is a concrete category), we can characterise a discrete space $X$ as one such that every function from $X$ to $Y$ (for $Y$ any space) is a morphism of spaces.  (More precisely, this means that every function from $U(X)$ to $U(Y)$ is the image under $U$ of a morphism from $X$ to $Y$.)

The dual notion is a [[codiscrete object]].

## Examples

### Discrete geometric spaces

The best known example is a discrete [[topological space]], that is one, $X$, in which all subsets of $X$ are [[open subset|open]] in the topology.  This is the [[discrete topology]] on $X$.  If $X$ is discrete in this sense, then its [[diagonal map]] $\Delta: X \to X \times X$ is [[open map|open]]. The converse also holds: if the diagonal $\Delta(X)$ is open, then so is $i_x^{-1}(\Delta(X)) = \{x\}$ for any $x \in X$, where $i_x(y) \coloneqq (x, y)$.

This same space serves as a discrete object in many subcategories and supercategories of $Top$, from [[convergence space]]s (where the only proper filter that converges to a point is the free ultrafilter at that point) to (say) paracompact Hausdorff spaces or [[Diff|manifolds]] (because a discrete topological space has those properties).

It is also [[sober space|sober]] and thus serves as a discrete [[locale]], whose corresponding [[frame]] is the [[power set]] of $X$; see [[CABA]].  (Note that [[Loc]] is not concrete over [[Set]].).  A locale is discrete if and only if $X \to X \times X$ is open and $X \to 1$ is also open.  A locale that satisfies the latter condition is called [[overt locale|overt]]; note that every locale is $T_0$ while every topological space is overt.  Moreover, in [[classical mathematics]], every locale is overt, but the notion is important when internalizing in [[toposes]].

A discrete [[uniform space]] $X$ has all [[reflexive relations]] as [[entourages]], or equivalently all [[covers]] as [[uniform cover]]s.  It is the only uniformity (on a given set) whose underlying topology is discrete.

Strictly speaking, there is no discrete [[metric space]] on any set with more than one element, because the forgetful functor has no left adjoint.  However, there is a discrete *extended* metric space, given by $d(x,y) = \infty$ whenever $x \ne y$.  More usually, the term 'discrete metric' is used when $d(x,y) = 1$ for $x \ne y$, which is discrete in the category of metric spaces of diameter at most $1$.  (Comparing the [[adjoint functor theorem]], the problem with $Met$ is that it generally lacks infinitary [[product]]s; in contrast, $Ext Met$ and $Met_1$ are [[complete category|complete]].)

In [[Abstract Stone Duality]], a space is called __discrete__ if the diagonal map $\delta: X \to X \times X$ is open, which corresponds to the existence of an [[equality]] relation on $X$; discrete spaces as described above correspond to discrete *overt* spaces in ASD.

### Local toposes

Any [[local topos]] has discrete and [[codiscrete object|codiscrete]] objects.  By definition, a local topos $\mathbf{H}$ comes with an [[adjoint triple]] of [[functors]]

$$
  \mathbf{H}
   \stackrel{\overset{Disc}{\hookleftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\hookleftarrow}}}
  \mathbf{B}
$$

to a [[base topos]] $\mathbf{B}$ (for instance [[Set]]), for which both $Disc$ and $Codisc$ are fully faithful.  Thus, a _discrete object_ is one in the [[essential image]] of the functor $Disc$.  Note that $\Gamma$ is not generally faithful in this case.

Even more generally, $\mathbf{H}$ may be a [[local (∞,1)-topos]]. For more on the discrete objects in such a context see _[[discrete ∞-groupoid]]_ .

Equivalently, this [[adjoint triple]] induces an [[adjoint pair]] of [[modalities]]

$$
  (\flat \dashv \sharp)
  \coloneqq
  ( Disc \Gamma \dashv coDisc \Gamma)
  \,,
$$

the _[[flat modality]]_ and the _[[sharp modality]]_. The discrete objects are precisely the [[modal types]] for the [[flat modality]]. The [[codiscrete objects]] are the modal types for the [[sharp modality]].

### Topological categories, fibrations, and final lifts {#Fibrations}

Every [[topological concrete category]] has discrete (and also codiscrete) spaces

More generally, if $U$ is an [[opfibration]] and $Sp$ has an initial object preserved by $U$, then $Sp$ has discrete objects: the discrete object on $X$ can be obtained as $i_!(0)$ where $0$ is the initial object of $Sp$ and $i\colon \emptyset \to X$ is the unique map from the initial object in $Set$ (or whatever underlying category).  (Conversely, if $Sp$ has discrete objects and pushouts preserved by $U$, then $U$ is an opfibration.)

Discrete objects can also be characterized as [[final lifts]] for empty [[sinks]].

### In simplicial sets

The category [[sSet]] of [[simplicial sets]] is a [[local topos]] (in fact a [[cohesive topos]]).

* A _discrete object_ in $sSet$ is precisely the [[nerve]] of a [[discrete groupoid]].

* A _[[codiscrete object]]_ in $sSet$ is precisely the [[nerve]] of a [[codiscrete groupoid]].

### Discrete cellular/categorical structures
 {#DiscreteCellularStructures]

Often one calls a cellular structure, such as those appearing in [[higher category theory]], _discrete_ if it is in the [[essential image]] of the inclusion of [[Set]].

For instance, one may speak of a _[[discrete category]]_ as a category that is equivalent (or, in some cases, isomorphic) to one which has only [[identity]] morphisms.  This concept has a generalization to a notion of [[discrete object in a 2-category]].

An alternative terminology for this use of "discrete" is _[[truncated object|0-truncated]]_, or more precisely (0,0)-truncated.  A discrete groupoid in this sense is a _[[homotopy n-type|homotopy 0-type]]_, or simply a [[h-level|0-type]].  This terminology may be preferable to "discrete" in this context, notably when one is dealing with higher categorical structures that are in addition equipped with geometric structure. For instance, when dealing with a [[topological category]] there is otherwise ambiguity in what it means to say that it is "discrete": it could either mean that its underlying topological spaces (of objects and of morphisms) are discrete spaces,  or it could mean that it has no nontrivial morphisms, but possibly a non-discrete topological space of objects.

+--{: .num_remark}
###### Remark
In some cases, the cellular notion of "discreteness" for higher categories can be seen as a special case of the *spatial* notion of discreteness --- often the 1-category of shapes will have a functor to sets for which the cellularly discrete objects are the discrete objects in the sense considered on this page.  For instance, this is the case for [[simplicial sets]], which form a [[local topos]] over [[Set]].  The discrete objects relative to this notion of cohesion are precisely the simplicial sets that are constant on a given ordinary set, hence those that are "discrete" in the cellular sense.
=--

### In $\infty$-toposes
 {#ExamplesInInfinityToposes}

The definition of discrete objects has the evident generalization from [[category theory]] to [[(∞,1)-category theory]]/[[homotopy theory]]. One noteworthy aspect of discrete objects in the context of homotopy theory is that there they are intimately related to the notion of _[[cohomology]]_.

For $\mathcal{X}$ an [[(∞,1)-sheaf (∞,1)-topos]] with [[global section]] [[geometric morphism]]

$$
  (\Delta \dashv \Gamma)
  \colon
  \mathcal{X}
  \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}
 \infty Grpd
$$

then for $X \in \mathcal{X}$ any object and $A \in $ [[∞Grpd]] any object, one says that

$$
  H(X,A)
  \coloneqq
  \pi_0 \mathcal{X}(X,\Delta(A))
  \in
  Set
$$

is the [[cohomology]] of $X$ [[cohomology with constant coefficients|with (locally) constant coefficients]] in $A$. (Here on the right $\mathcal{X}(-,-)$ denotes the [[(∞,1)-categorical hom-space]].)

Now if $\mathcal{X}$ _has discrete objects_ in the sense that $\Delta \colon \infty Grpd \to \mathcal{X}$ is a [[full and faithful (∞,1)-functor]], then it follows immediately from the definitions that the cohomology of discrete objects with constant coefficients in $\mathcal{X}$ equals the cohomology in [[∞Grpd]], which is standard "[[nonabelian cohomology]]":

$$
  (\mathcal{X} \, has\, discrete\, objects)
  \Leftrightarrow
  ( \forall_{S,A \in \infty Grpd} \mathcal{X}(\Delta S, \Delta A) \simeq \infty Grpd(S,A) )
  \,.
$$

Conversely: _the failure of the cohomology with constant coefficients of objects in the image of $\Delta$ to coincide with standard cohomology is a measure for $\Delta$ not respecting discrete objects.

For example the [[natural numbers object]] $\mathbf{N} \simeq \Delta \mathbb{N}$ in the [[(∞,1)-sheaf (∞,1)-topos]] over some [[topological spaces]] fails to be a discrete object. Accordingly in this case the natural numbers object can have nontrivial higher cohomology with constant coefficients, see for instance ([Blass 83](#Blass83), [Shulman 13](#Shulman13)).

## Related concepts

* **discrete space**

* [[discrete group]]

* [[discrete groupoid]]

* [[discrete ∞-groupoid]]

* [[codiscrete space]], [[Freyd cover]]

* [[adjoint cylinder]]

* [[category of being]]

[[!include cohesion - table]]

## References

* *Discreteness, concreteness, fibrations, and scones*: [blog post](http://golem.ph.utexas.edu/category/2011/11/discreteness_concreteness_fibr.html)

* [[Andreas Blass]], _Cohomology detects failures of the axiom of choice_, Trans. Amer. Math. Soc. 279 (1983), 257-269 ([web](http://www.ams.org/journals/tran/1983-279-01/S0002-9947-1983-0704615-7/))
 {#Blass83}

* [[Mike Shulman]], _Cohomology_ on the [[homotopy type theory]] blog [here](http://homotopytypetheory.org/2013/07/24/cohomology/)
 {#Shulman13}

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[[!redirects discrete spaces]]

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[[!redirects discrete uniform spaces]]

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[[!redirects discrete extended metric spaces]]

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discrete object classifier

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#### Universes
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## Idea

Just as a [[subobject classifier]] in a [[(1,1)-category]] classifies the [[monomorphisms]] or [[(-1)-truncated]] morphisms (and thus the [[subobjects]]) of the category, a _discrete object classifier_ in a [[(2,1)-category]] should classify the [[faithful functor|faithful]] or [[0-truncated]] morphisms (and thus the [[discrete morphism|discrete objects]]) in the (2,1)-category, see at _[n-truncated morphisms -- between groupoids](n-truncated+object+of+an+infinity-category#TruncatedMorphismsBetweenGroupoids)_.

## Definition

In a (2,1)-category $C$ with [[terminal object]] $*$, [[interval object]] $I$, and finite [[2-pullback|(2,1)-pullbacks]], a __discrete object classifier__ is a morphism $inhabited: [I, Set] \rightarrow Set$ whose target is a nonterminal object $Set$ and whose source is the [[internal hom|internal]] [[hom-object]] $[I, Set]$ such that for every [[faithful morphism]] $B \colon U \rightarrow G$ in $C$, there is a unique morphism $F:G \rightarrow Set$ such that there is a (2,1)-pullback [[diagram]] of the form

$$\array{U & \to & [I,Set] \\
  ^{B}\downarrow & \cong & \downarrow^{inhabited}\\
  G & \underset{\chi_U}{\to} & Set}
$$

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

If $Set$ exists, it is typically called the **[[universe]] of [[set]]s** or **[[groupoid]] of sets**. A global element $F:G \rightarrow Set$ is typically called an **[[indexed family]]**, an element $A:* \rightarrow Set$ is typically called a **set**, and a set $A:* \rightarrow Set$ is **[[inhabited object|inhabited]]** if there exists a morphism $B:* \rightarrow [I, Set]$ such that the $A$ [[factor]]s into $inhabited \circ B$. All these terms refer to the [[internal set theory]] of the (2,1)-category $C$.

The morphism $\chi_U$ is also called the **[[classifying morphism]]** of the discrete object $U$ and morphism $B:U \rightarrow G$.

=--

## Examples

### In $Grpd$

In the (2,1)-category [[Grpd]] of groupoids and functors between groupoids, the discrete object classifier is the groupoids of sets and bijections (i.e. the [[groupoid core]] of [[Sets]]).

## See also

* [[subobject classifier]]

* [[(sub)object classifier in an (infinity,1)-topos]]

* [[classifying morphism]]

* [[type of types]], [[universe in a topos]]

* [[class object]]

## References

* David Corfield: _101 things to do with a 2-classifier_ ([blog](http://golem.ph.utexas.edu/category/2008/01/101_things_to_do_with_a_2class.html))


discrete Segal type

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\tableofcontents

## Definition

In [[simplicial type theory]], a [[Segal type]] $A$ is a **discrete Segal type** if for all elements $x:A$ and $y:A$ there is an equivalence between the [[identity type]] $x =_A y$ and the [[hom type]] $\mathrm{hom}_A(x, y)$:

$$x:A, y:A \vdash \mathrm{disc}(x, y):\mathrm{isEquiv}(\mathrm{idToHom}(x, y))$$

where

$$
  \mathrm{idToHom}(x, y)
  \;\colon\;
  (x =_A y)
  \;
  \to
  \;
  \mathrm{hom}_A(x, y)
$$

$$
  \mathrm{idToHom}(x, y)
  \big(
    \mathrm{refl}_A(x)
  \big)
    \;\coloneqq\;
  \mathrm{id}_A(x)
$$

## Related concepts

* [[hom type]]
* [[Segal type]]
* [[Rezk type]]
* [[univalent groupoid]]

## References

* {#RiehlShulman17} [[Emily Riehl]], [[Michael Shulman]], *A type theory for synthetic $\infty$-categories* $[$[arXiv:1705.07442](https://arxiv.org/abs/1705.07442)$]$

[[!redirects discrete Segal type]]
[[!redirects discrete Segal types]]


discrete torsion

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#### Differential cohomology
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#Contents#
* table of contents
{:toc}

## Idea

What is known in the literature as _discrete torsion_ ([Vafa 86](#Vafa86)) are phenomena of [[equivariant ordinary differential cohomology]], specifically of [[circle n-bundle with connection|circle 2-bundles with connection]] (e.g. represented as [[bundle gerbes]]) modelling the [[B-field]] in [[string theory]] over [[orbifold]] [[spacetimes]] ([Sharpe 99](#Sharpe99)) and of [[circle n-bundle with connection|circle 3-bundles with connection]] (e.g. represented as [[bundle 2-gerbes]]) modelling the [[supergravity C-field]] on [[orbifolds]] ([Sharpe 00](#Sharpe00)), as in [[M-theory on G2-manifolds]] with [[ADE-singularities]].

## Classification

Discrete torsion first arose in the context of 2d orbifold partition functions in [Vafa 86](#Vafa86), where it was noted that the twisted sectors could be multiplied by nontrivial $U(1)$ phases. By imposing requirements on these phases, such as respecting modular invariance, and multi-loop factorization (that the genus $2$ phase factor is equal to the product of two genus $1$ phase factors whenever the generators $a_{1(2)}$ of the fundamental group commute with the generators $b_{1(2)}$, respectively), it was observed that consistent phases are described by group 2-cocycles

$$
\omega\in Z^2(G,U(1))
$$

for $G$ the orbifold group with trivial action on $U(1)$, and that 2-coboundaries yield trivial phases. This gives a classification of discrete torsion in terms of the second group cohomology group $H^2(G,U(1))$. Concretely, the orbifold partition function of a genus $1$ surface with discrete torsion $\omega\in Z^2(G,U(1))$ is

$$
Z = \sum_{g,h\in G} \frac{\omega(g,h)}{\omega(h,g)} Z_{g,h}
$$

for $g,h$ commuting pairs. The phase factor, even though it is formulated in terms of cocycles, only depends on the cohomology class of discrete torsion, since it is clearly invariant under shifts by coboundaries.

This classification was observed again in [Sharpe 99](#Sharpe99), where such 2-cocycles were interpreted as actions of the orbifold group $G$ on a U(1) 2-bundle, and extended to the three-dimensional case in [Sharpe 00](#Sharpe00), with the actions now being classified by the *third* cohomology group $H^3(G,U(1))$ (under simplifying assumptions).

## Related concepts

* [[equivariant ordinary differential cohomology]]

## References

### For the B-field

Early discussion of classification in [[2d CFTs]] includes

* [[Maximilian Kreuzer]], [[A. N. Schellekens]], _Simple currents versus orbifolds with discrete torsion -- a complete classification_,  	Nucl. Phys. B411 (1994) 97-121 ([arXiv:hep-th/9306145](https://arxiv.org/abs/hep-th/9306145))

and more specifically for [[orbifolds]] in [[string theory]] in

* {#Vafa86} [[Cumrun Vafa]], _Modular Invariance and Discrete Torsion on Orbifolds_, Nucl. Phys. B273 (1986) 592-606 ([spire:227126](http://inspirehep.net/record/227126/))

* [[Cumrun Vafa]], [[Edward Witten]], _On Orbifolds with Discrete Torsion_, J. Geom. Phys. 15:189-214, 1995 ([arXiv:hep-th/9409188](https://arxiv.org/abs/hep-th/9409188))

The identification of discrete torsion in [[type II string theory]] as a choice of [[orbifold]] [[equivariant cohomology|equivariance]] on a [[principal 2-bundle]]/[[bundle gerbe]] is due to

* [[Eric Sharpe]], _Discrete torsion_, Phys. Rev. D68 (2003) 126003 ([arXiv:hep-th/0008154](https://arxiv.org/abs/hep-th/0008154), [doi:10.1103/PhysRevD.68.126003](https://doi.org/10.1103/PhysRevD.68.126003))

based on

* {#Sharpe99} [[Eric Sharpe]],

  _Discrete Torsion and Gerbes I_ ([arXiv:hep-th/9909108](http://arxiv.org/abs/hep-th/9909108))

  _Discrete Torsion and Gerbes II_ ([arXiv:hep-th/9909120](http://arxiv.org/abs/hep-th/9909120))

  _Discrete Torsion, Quotient Stacks, and String Orbifolds_, in _[[Orbifolds in Mathematics and Physics]]_ ([arXiv:math/0110156](http://arxiv.org/abs/math/0110156))

In relation to twisted [[Chen-Ruan cohomology|Chen-ruan]] [[orbifold cohomology]]:

* [[Yongbin Ruan]], *Discrete torsion and twisted orbifold cohomology*, J. Sympl. Geom. 2 (2003), 1–24 ([arXiv:math/0005299](https://arxiv.org/abs/math/0005299))

See also

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], Nikolaos Prezas, _Discrete Torsion, Non-Abelian Orbifolds and the Schur Multiplier_, JHEP 0101:033, 2001 ([arXiv:hep-th/0010023](https://arxiv.org/abs/hep-th/0010023))

* [[Kiyonori Gomi]], [[Yuji Terashima]], _Discrete Torsion Phases as Topological Actions_, Commun. Math. Phys. (2009) 287: 889 ([doi:10.1007/s00220-009-0736-1](https://doi.org/10.1007/s00220-009-0736-1))

The case of [[heterotic string theory]] is discussed in

* [[Eric Sharpe]], _Discrete Torsion in Perturbative Heterotic String Theory_, Phys. Rev. D68 (2003) 126005 ([arXiv:hep-th/0008184](https://arxiv.org/abs/hep-th/0008184))

### For the C-field
 {#ReferencesForTheCField}

The higher version of discrete torsion for [[circle n-bundle|circle 3-bundles]] describing the [[supergravity C-field]] is discussed in

* {#Sharpe00} [[Eric Sharpe]], _Analogues of Discrete Torsion for the M-Theory Three-Form_, Phys. Rev. D68 (2003) 126004 ([arXiv:hep-th/0008170](https://arxiv.org/abs/hep-th/0008170))

* Shigenori Seki, _Discrete Torsion and Branes in M-theory from Mathematical Viewpoint_, Nucl. Phys. B606 (2001) 689-698 ([arXiv:hep-th/0103117](https://arxiv.org/abs/hep-th/0103117))

* {#BDHKMMS01} [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], section 4.6.2 of _Triples, Fluxes, and Strings_, Adv. Theor. Math. Phys. 4 (2002) 995-1186 ([arXiv:hep-th/0103170](https://arxiv.org/abs/hep-th/0103170))

and applied to discussion of [[black brane|black]] [[M2-brane]] [[worldvolume]] [[field theory]] ([[BLG model]]/[[ABJM model]], see at _[[fractional M2-brane]]_) in

* {#Sethi98} [[Savdeep Sethi]], _A Relation Between $N=8$ Gauge Theories in Three Dimensions_, JHEP 9811:003,1998 ([arXiv:hep-th/9809162](https://arxiv.org/abs/hep-th/9809162))

* [[Neil Lambert]], [[David Tong]], _Membranes on an Orbifold_, Phys. Rev. Lett. 101:041602, 2008 ([arXiv:0804.1114](https://arxiv.org/abs/0804.1114))

* {#AharonyBergmanJafferis08} [[Ofer Aharony]], [[Oren Bergman]], Daniel Louis Jafferis, _Fractional M2-branes_, JHEP 0811:043,2008 ([arXiv:0807.4924](https://arxiv.org/abs/0807.4924))

* [[Mauricio Romo]], _Aspects of ABJM orbifolds with discrete torsion_, J. High Energ. Phys. (2011) 2011 ([arXiv:1011.4733](https://arxiv.org/abs/1011.4733))

See also at _[[finite subgroup of SU(2)]]_ the section [on group cohomology](finite+rotation+group#GroupCohomology).


discrete valuation

#Contents#
* table of contents
{:toc}

## Definition

There are two different notions of a discrete valuation.

A __discrete valuation__ on a field $K$ is a function $v:K\to \mathbf{Z}\cup \{\infty\}$ such that

* $v$ defines a surjective homomorphism of groups by $v|_{K^*} : K^*\to \mathbf{Z}$ where $K^*$ is the multiplicative group of $K$

* $v(0) = \infty$

* $v(x+y) \geq inf\{ v(x),v(y)\}$

with usual conventions for $\infty$.

The set $R_v = \{x\in K\,|\, v(x)\geq 0\}$ is called the [[valuation ring]] of the valuation $v$, and it is an integral domain with quotient field $K$; its part $\mathfrak{p}_v = \{x\in K\,|\, v(x)\gt 0\}$ is a maximal ideal of $R_v$ and is called the __valuation ideal__ of $v$. This can be generalized to other [[valuation]]s.

Let $0\lt \rho\lt 1$ and define $|x|_v = \rho^{v(x)}$. Then $||$ will be a nonarchimedean multiplicative discrete valuation in the sense of the following definition. If we change a $\rho$ then we get an *equivalent* multiplicative valuation.

A valuation on a field $K$ is a function $| |:K\to \mathbf{R}_{\geq 0}$ such that

* $|x| = 0$ iff $x=0$

* $|x y| = |x| |y|$

* there is a constant $C$ such that $|1+x|\leq C$ if $|x|\leq 1$

Two valuations on the same field are equivalent if one is the positive power of another i.e. $\exists c\gt 0$ such that $|x|_1 = |x|_2^c$ for all $x\in K$.

A valuation is __non-archimedean__ if $C$ above can be taken $1$ and archimedean otherwise. A (multiplicative) valuation is __discrete__ if there is a neighborhood $U\ni 1\in \mathbf{R}_+$ such that the only $x\in K$ such that $|x|\in U$ is $1_K\in K$.

## Properties

By a theorem of Gel'fand and Tornheim the only archimedean valuation fields are the subfields of $\mathbf{C}$ with a valuation which is equivalent to the valuation obtained by resriction from the standard absolute value on $\mathbf{C}$.

## Literature

* {#FroehlichCassels67} [[Albrecht Fröhlich]], [[J. W. S. Cassels]] (eds.), _Algebraic number theory_, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965 (ISBN:9780950273426, [pdf](https://www.math.arizona.edu/~cais/scans/Cassels-Frohlich-Algebraic_Number_Theory.pdf), [errata pdf](https://www.ma.imperial.ac.uk/~buzzard/errata.pdf) by [[Kevin Buzzard]])

* [[Serge Lang]], _Algebraic number theory_. GTM 110, Springer 1970, 2000

[[!redirects discrete valuations]]


discrete valuation ring

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## Idea

A *discrete valuation ring* is a [[principal ideal domain]] which has exactly one non-[[zero object|zero]] [[maximal ideal]]. This means that it is a [[local ring|local]] [[principal ideal domain]], or equivalently, a [[local integral domain]] with a [[Dedekind-Hasse norm]].

## Properties

Every [[discrete valuation ring]] is a [[local integral domain]].

The ring of [[formal power series]] of a [[field]] is a discrete valuation ring.

## See also

* [[local ring]]
* [[local integral domain]]
* [[Dedekind-Hasse norm]]
* [[principal ideal domain]]

## References

See also:

* Wikipedia, *[Discrete valuation ring](https://en.m.wikipedia.org/wiki/Discrete_valuation_ring)*

[[!redirects discrete valuation rings]]

[[!redirects discrete-valuation ring]]
[[!redirects discrete-valuation rings]]


discriminant

## Idea

### General

(...)

### For cubic curves and elliptic curves

In the context of [[cubic curves]] and [[elliptic curves]] "the discriminant" refers to that of the [[Weierstrass equation]], see at _[elliptic curve -- Discriminant](elliptic+curve#WeierstrassCubicDiscriminantAndJInvariant)_

[[!redirects discriminants]]


discussion on terminology -- omega-category

The original entry [[omega-category]] suggested that '$\omega$-category' is supposed to implicitly mean ‹strict $\omega$-category›.

After the following discussion, we renamed that entry to [[strict omega-category]].

***

[[Toby Bartels]]: Who actually uses the language this way? If anything, it should be the reverse, as 'omega' is Greek (like 'bi', 'tri', and 'tetra') and 'infinity' is not.

[[John Baez]]: For a long time, for most category theorists, the default meaning of '$n$-category' has been 'strict globular $n$-category' — especially since weak $n$-categories had not yet been defined (except for very low $n$).   So when Street invented '$\omega$-categories' in his paper on orientals, he meant 'globular strict $\infty$-categories'.  Now there are many definitions of 'weak $n$-category', terminology is in flux, and people like [[Tom Leinster]] and myself advocate a new system in which '$n$-category' means 'some sort of possibly weak $n$-category' — at least until one has defined it more precisely in a specific context.  It is still true that most papers referring to $\omega$-categories mean 'globular strict $\infty$-categories'.

We may want to discuss whether the $n$Lab adopts traditional terminology, or takes this historic opportunity to play Bourbaki and choose consistent terminology for the future (while still explaining the traditional terms, or course).  In any event, globular strict $\omega$-categories are worth talking about, and that's what this page is about!

[[Toby Bartels]]: I agree with you and Tom about '$n$-category', and I agree that the strict globular case is worth talking about. I only question making a terminological distinction between '$\omega$-category' for the strict globular concept and '$\infty$-category' for the general weak case; (this seems to go against both the Greek-prefixes-for-the-weak-concept convention *and* the usual-name-for-the-weak-concept convention!). And I'm wondering if this distinction is established somewhere.

[[Urs Schreiber]]: I am not dogmatic about this, but I did not make this up, I think. I thought I followed the standard convention.

I jumped on the $\omega$-category train after

1. I learned about [Sjoerd Crans' work](http://crans.fol.nl/papers/comb.html) on $\omega$-categories from Todd. Sjoerd Crans uses the term "$\omega$-cvategory" for strict $\infty$-categories throughout;

and

2. after Mike Shulman pointed out that there is now an extension of the [[folk model structure]] to it, described by Lafont, Métayer and Worytkiewicz [here](http://arxiv.org/abs/0712.0617). They use the term "$\omega$-category" in this article and in all their other previous articles for strict $\infty$-catgeories.

Together with Ross Street's original work on $\omega$-categories this seems to say that _all_ the substantial work on the theory of strict $\infty$-categories uses the term "$\omega$-category" for them.

I have added more references below, all of which use the term $\omega$-category exclusively for strict $\infty$-categories. Please have a look.

Finally, I thought that the terminology was actually a good idea: I was thinking of the "$\omega$" as indicating the (important) fact that $\omega$-categories are the directed limit of the iterated enrichment process

$$
  Cat \hookrightarrow
  2Cat = Cat-Cat
  \hookrightarrow
  2Cat = (Cat-Cat)-Cat
  \hookrightarrow
  3Cat = ((Cat-Cat)-Cat)-Cat
  \hookrightarrow
  \cdots
  \,.
$$

[[Toby Bartels]]: Hey, you don\'t have to give me references that use '$\omega$-category' for the strict notion; I know all about that. What I want are references that use '$\infty$-category'! Keep in mind what John said that he and Tom propose above: that the usual term '$n$-category', although historically most used for the strict concept, should really be applied to the weak concept; you add the word 'strict' (or 'globular strict' to be more clear) if that\'s what you want. Maybe you\'re not folowing this … but then shouldn't you use the adjective 'weak' instead?

In other words: Who makes this *distinction* between '$\omega$-category' and '$\infty$-category'? (And secondarily, of course, is this a good way to do things?)

[[Urs Schreiber]]: All right, now I am confused about what I am being asked! :-)
I read you very first remark above -- "Who actually uses the language this way?" -- following the statement that "$\omega$-Categories are precisely those [[globular set|globular]] [[higher category theory|infinity-categories]] which are _strict_.
" as asking "Who says '$\omega$-category' for 'strict $\infty$-category'. "

But you maybe meant: "Who uses '$\infty$'-category to mean 'weak $\infty$-category'?"? The answer to that seems to be: pretty much everybody nowadays. I happen to have the following example in fron of me as we speak: Toën, [Higher and derived stacks: a global overview](http://arxiv.org/abs/math.AG/0604504). See in particular on [p. 4](http://arxiv.org/PS_cache/math/pdf/0604/0604504v3.pdf#page=4) the first lines of section 2.2 and the footnote on that page.

This follows of course what seems to have become the "Lurie school". Jacob Lurie has introduced his own, slightly weird, convention, to say "$\infty$-category" for "weak $(\infty,1)$-category".

It seems to me the upshot is:

* at least in more recent texts "$\infty$-category" is always meant to include the weak version

* no reference I know of uses "$\omega$-category" for anything but strict $\infty$-categories.

(If and when we have sorted this out, the essence of our discussion should be turned into content on the entry [[higher category theory]])

[[Toby Bartels]]: Who uses '$\omega$-category' for the weak notion? I know one reference: John Baez! (from whom I\'ve learned most of my terminology). The first use of '$\omega$-category' for the weak notion that I can find in TWF is [weak 100](http://math.ucr.edu/home/baez/week100.html); search for 'ω' (second hit). Maybe it\'s just him (and presumably Tom, and of course some of his students like me). If the distinction between '$\omega$-category' and '$\infty$-category' is established, then I guess that we have to live with it. But I don\'t have to like it! ^_^

[[Urs Schreiber]]: You don't have to like it. And I don't really care as long as we can reduce the general confusion of terms. But one question: why would one say _$\omega$-category_ and not _$\infty$-category_ if one wants to refer to _weak_ $\infty$-categories. I mean, if the "$\omega$" does not indicate the degree of weakening, what then is it supposed to indicate in this context?

[[Toby Bartels]]: In general, '$\omega$' and '$\infty$' mean the same thing: infinity; that\'s what it means in this context too. It\'s kind of arbitrary that one would use one symbol instead of the other; I certainly don\'t think that Street (or whoever originally picked the name '$\omega$-category') meant anything special in picking the symbol '$\omega$', since after all the term '$n$-category' defaulted to the strict concept for them too (as it did for everybody at the time, I believe). In other words, the strictness was in the term '$n$-category', not in the symbol '$\omega$'. Somehow that symbol has acquired extra connotations, and accordingly the new term '$\infty$-category' has been invented, I\'m not sure how or why.

Also note that your directed limit above applies equally well to the weak case as long $2Cat$ etc are interpreted in a weak sense. So again, $\omega$ means infinity, which is independent of strictness.

[[Urs Schreiber]] says: Hm, in TWF 100 I see Batanin's title with "weak $\omega$-categories". Is there anyone saying just "$\omega$-category" to mean "weak $\infty$-category"??

[[Toby Bartels]]: Yes, John says that, just above that title. He states (earlier in TWF 100) his general philosophy (also given earlier in this query box) that one should use '$n$-category' for the general weak concept, then applies this later to casually refer to weak $\omega$-categories simply as '$\omega$-categories', then points to a reference on that subject, the paper by Batanin (which does *not* follow John\'s convention and so includes the adjective 'weak', thereby confirming that John meant the weak concept). And I like John\'s convention, so I\'ve used it too (not that I\'ve published anything), which is why it\'s now disconcerting to be told that now '$\omega$' *implies* strictness even when (following John\'s convention) '$n$-category' does not, so now I have to relearn all my language.

Which, in light of the references that use '$\infty$-category' for the weak concept, is apparently what I really will have to do. \(;_;\) \([crying emoticon](https://secure.wikimedia.org/wikipedia/en/wiki/List_of_common_emoticons#Simple_examples)\)

[[Urs Schreiber]] say: Okay, I have modified the first sentence in the entry a bit.

_[[Mike Shulman|Mike]]_: I don't want to reopen this can of worms, although I am just as sad as Toby.  Can I at least request that we at the nLab try to say "strict $\omega$-category" when that is what we mean, rather than just saying "$\omega$-category" and assuming that the reader knows that we mean the strict version?  In particular, since we have a page called [[strict 2-category]] rather than "2-category," it would seem only fair for this page to be called "strict omega-category," with the page "omega-category" containing links and pointing out the variations in terminology.

_Urs_: I really don't want to come across as dogmatic here, but I am confused: if "$\omega$-category" is supposed to have precisely the same (vague) meaning as "$\infty$-category", why should we have the two different terms then in the first place?

_Mike_: No particular reason, except that we're already stuck with both of them.  I would be just as happy to use one exclusively, with adjectives.  (In particular, I wouldn't mind being able to use "(weak) $\omega$-category" for the weak version, to avoid confusion with things like $\infty$-pretoposes and Lurie's use of "$\infty$-category".)  I just get confused by trying to force them to have different, specific meanings, with no mnemonic to help me remember which is supposed to imply "weak" and which "strict" (except the "anti-mnemonic" that $\omega$, being Greek, implies strictness, while the prefixes "bi-" and "tri-", also Greek, imply weakness).  Especially since they have each already been used by different people with different meanings.  And especially if we are trying to make "$n$-category" not imply strictness (in the face of tradition), it seems inconsistent to insist that "$\omega$-category" does imply strictness.  I feel sure that there is far more literature using "2-category" to mean "strict 2-category" than there is using "$\omega$-category" to mean "strict $\omega$-category," so why should we feel bound by the latter but not the former?

_Toby_: We could have the general concept at [[infinity-category]], the strict concept at [[strict omega-category]], and the discussion of terminology at [[omega-category]]. (It sounds like a joke, but it really may be the best use for each term!)

_[[Urs Schreiber|Urs]]_: Okay. In that case let me suggest the following: instead of "strict omega-category" let's have an entry "strict globular infinity-category" in which we give the above material and remark that "strict globular infinity-categories" are sometimes just called "omega-categories". Then we need another entry "strict cubical infinity-category" or "infinity-fold category" and mention the relation to "strict globular infinity-category".

_[[Todd Trimble|Todd]]_:
I'm pretty sure the literature is mixed on this point. I don't recall seeing infinity-category being used much at all in the days when the subject was first born (mid to late nineties); people said either "strict" or "weak" omega-category. Tom Leinster in his
book for example has an entry in the index of his book:
"infinity-category: see omega-category" and never uses the term infinity-category anywhere else, as far as I know. I don't think I ever heard Ross Street say infinity-category -- I really think this
usage must have caught on more recently, with a different crowd.

I don't really mind infinity-category; I'm just not used to saying it myself, and until now was confused by what people like you meant when you used that or omega-category, except in specific contexts, as for example discussing the work of Crans. So now I know!

[I should add that the above was given in am email to Urs, which he reproduced here; hence the phrase "people like you". I added at the end that I was not pushing an agenda.]

_Mike_: I was going to say that I liked Toby's suggestion, but it seems to have already been implemented, so I am satisfied.  Let me just say, in response to Urs' suggestion, that since [[2-category|2-categories]] don't need the word "globular" to distinguish them from [[double category|double categories]] (aka 2-fold categories), I would rather not have to say "globular $\infty$-category (or $\omega$-category)" to distinguish it from cubical or $\infty$-fold ones.  Of course, it's fine to add "globular" for emphasis occasionally.


disjoint coproduct

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# Contents
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## Idea

The notion of _disjoint coproduct_ is a generalization to arbitrary [[categories]] of that of _[[disjoint union]]_ of sets.

One says that a [[coproduct]] $X + Y$ of two [[objects]] $X, Y$ in a [[category]] $\mathcal{C}$ is _disjoint_ if the [[intersection]] of $X$ with $Y$ in $X + Y$ is [[initial object|empty]]. In this case one often writes $X \coprod Y \coloneqq X + Y$ for the coproduct, particularly if the coproduct is [[pullback-stable colimit|stable under pullbacks]], and there one speaks of the _[[disjoint union]]_ of $X$ with $Y$.

## Definition

### In a category

A binary [[coproduct]] $a+b$ in a [[category]] is **disjoint** if

1. the coprojections $a\to a+b$ and $b\to a+b$ are [[monomorphism|monic]], and

2. their [[intersection]] is an [[initial object]].

Equivalently, this means we have [[pullback]] squares

$$
\array{ a & \to & a &&&
b & \to & b &&&
0 & \to & b\\
\downarrow && \downarrow &&&
\downarrow && \downarrow &&&
\downarrow && \downarrow \\
a & \to & a+b &&&
b & \to & a+b &&&
a & \to & a+b}
$$

An arbitrary coproduct $\sum_i a_i$ is disjoint if each coprojection $a_i\to \sum_k a_k$ is monic and the intersection of any two distinct ones is initial.  Note that every 0-ary coproduct (that, is initial object) is disjoint.

A more [[constructive mathematics|constructive]] way to phrase disjointness of an arbitrary coproduct is that the pullback of any two [[coprojections]] $a_i\to \sum_k a_k$ and $a_j\to \sum_k a_k$ is the coproduct $\sum_{i=j} a_i$, where $i=j$ denotes the [[subsingleton]] corresponding to the [[proposition]] $i=j$, a.k.a. $\{ \ast \mid i=j \}$.  (Since $a_i=a_j$ as soon as this indexing set is [[inhabited set|inhabited]], this coproduct could equally be written $\sum_{i=j} a_j$.)

## Examples

* In the category $Set$ of sets and functions, the coproduct is given by [[disjoint union]] and is, unsurprisingly, disjoint. In the category $Pfn$ of sets and [[partial function|partial functions]] the coproduct is equally  given by disjoint union and total injections and is disjoint as well.

* Since having all finitary disjoint coproducts is half of the condition for a category to be [[extensive category|extensive]], extensive categories provide [[extensive category#examples|examples for categories with disjoint finite coproducts]]. In the preceding discussion $Set$ instantiates this case whereas $Pfn$ does not: since the initial object $\emptyset$ in $Pfn$ is not [[strict initial object|strict]], the latter category is not extensive.

* In the category $Vect$ of (real) [[vector space|vector spaces]] coproducts are given by [[direct sum]] and are disjoint but not [[pullback-stable colimit|stable under pullback]]: pulling back the colimit diagram $\mathbb{R}\to\mathbb{R}\oplus\mathbb{R}\leftarrow\mathbb{R}$ along the diagonal morphism $\Delta:\mathbb{R}\to\mathbb{R}\oplus\mathbb{R}\; ,\; x\mapsto x\oplus x$ yields $0\to\mathbb{R}\leftarrow 0$ which is not a colimit diagram. Whence $Vect$ is not [[extensive category|extensive]].

* Non-example: the [[interval category]]  $\left\{
    0 \to 1
  \right\}$ has coproducts but they are not all disjoint: $1+1=1$.
There are plenty more examples of [[posets]] that have non-disjoint coproducts besides this one. In a [[Boolean algebra]], two elements $a$ and $b$ are disjoint in the sense that $a\wedge b=0$ if and only if $a\vee b$ is their disjoint coproduct.

## Properties

### Characterization of sheaf toposes

Having all small disjoint coproducts is one of the conditions in [[Giraud's theorem]] characterizing [[sheaf toposes]].

### In coherent categories

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

Let $\mathcal{C}$ be a [[coherent category]]. If $X, Y \hookrightarrow Z$ are two [[subobjects]] of some [[object]] $Z \in \mathcal{C}$ and are disjoint, in that their [[intersection]] in $Z$ is [[initial object|empty]], $X \cap Y \simeq\emptyset$, then their [[union]] $X \cup Y$ is their (disjoint) [[coproduct]].

=--

This apears as ([Johnstone, cor. A1.4.4](#Johnstone)).

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

A coherent category in which all coproducts are disjoint is also called a **[[positive category|positive coherent category]]**.

=--

([Johnstone, p. 34](#Johnstone))

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

Every [[extensive category]] is in particular positive, by definition.

=--

In a positive coherent category, every morphism into a coproduct factors through the coproduct coprojections:

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

Let $\mathcal{C}$ be a postive coherent category, def. \ref{PositiveCategory}, and let $f \colon A \to X \coprod Y$ be a [[morphism]]. Then the two [[subobjects]] $f^*(X) \hookrightarrow A$ and $f^*(Y) \hookrightarrow Y$ of $A$, being the [[pullbacks]] in

$$
  \array{
    f^* (X) &\to& X
    \\
    \downarrow && \downarrow^{\mathrlap{i_X}}
    \\
    A &\stackrel{f}{\to}& X \coprod Y
  }
  \;\;\;\;
  \array{
    f^* (Y) &\to& Y
    \\
    \downarrow && \downarrow^{\mathrlap{i_Y}}
    \\
    A &\stackrel{f}{\to}& X \coprod Y
  }
$$

are disjoint in $A$ and $A$ is their disjoint coproduct

$$
  A \simeq f^*(X) \coprod f^*(Y)
  \,.
$$

=--

This appears in ([Johnstone, p. 34](#Johnstone)).

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

This means that if $A \in \mathcal{C}$ itself is indecomposable in that it is not a coproduct of two objects in a non-trivial way, for instance if $\mathcal{C}$ is an [[extensive category]] and $A \in \mathcal{C}$ is a [[connected object]], then every morphism $A \to X \coprod Y$ into a disjoint coproduct factors through one of the two canonical inclusions.

=--

## Related concepts

* [[disjoint subset]]

* [[disjunctive logic]]

## References

* {#CarboniLackWalters93} [[Aurelio Carboni]], [[Stephen Lack]], [[Bob Walters|R. F. C. Walters]], around Def. 2.5 in: _Introduction to extensive and distributive categories_, JPAA **84** (1993) pp. 145-158 (doi:10.1016/0022-4049(93)90035-R)

* [[Peter Johnstone]], Sec. A1.4.4, p. 34 in: _[[Sketches of an Elephant]]_

[[!redirects disjoint coproduct]]
[[!redirects disjoint coproducts]]


disjoint subsets

# Disjoint subsets
* table of contents
{: toc}

## Idea

Two [[concrete sets]] are disjoint if they have no common [[elements]].

## Definitions

Given an [[abstract set]] $V$, two [[subsets]] $A$ and $B$ of $V$ __meet__ if their [[intersection]] is [[inhabited subset|inhabited]]; $A$ and $B$ are __disjoint__ if they do not meet, in other words if their intersection is [[empty subset|empty]].

## Foundational issues

In [[material set theory]], we may speak of take $A$ and $B$ to be simply [[sets]], rather than subsets of some [[ambient set]] $V$.  Equivalently, one may take $V$ to be the [[von Neumann universe|class of all sets]] by default.  (In this context, it's important that whether $A$ and $B$ meet or are disjoint is independent of the ambient set or class.)

In [[constructive mathematics]], the default meaning of 'disjoint' is as above, but sometimes one wants a definition relative to some [[inequality relation]] $\ne$ on $V$.  Then $A$ and $B$ are __$\ne$-disjoint__ if, whenever $x \in A$ and $y \in B$, $x \ne y$.  (Ordinary disjointness is relative to the [[denial inequality]].)

## Properties

### Relation to disjoint unions

The concrete sets $A$ and $B$ are disjoint iff they have an internal [[disjoint union]], in other words if their [[inclusions]] into their [[union]] $A \cup B$ form a [[coproduct]] diagram in [[the category of sets]].  (Etymologically, of course, this is backwards.)

Many authors are unfamiliar with disjoint unions.  When the disjoint union oid two [[abstract sets]] $A$ and $B$ is needed, they will typically lapse into [[material set theory]] (even when the work is otherwise perfectly [[structural set theory|structural]]), and make some comment such as 'without loss of generality, assume that $A$ and $B$ are disjoint' or (especially when $A = B$) 'take two [[isomorphic]] copies of $A$ and $B$', then call the disjoint union simply a 'union'.  (This works by the previous paragraph.)

### Internalization

In any [[category]] with an [[object]] $V$, two [[subobjects]] $A$ and $B$ are __disjoint__ if their [[pullback]] is [[initial object|initial]] in $C$.  Then disjoint subsets are precisely disjoint subobjects in [[Set]].

To internalize the characterization in terms of internal disjoint unions is harder.  If $A$ and $B$ have a [[join]] in the [[poset of subobjects]] $Sub(V)$, then we may ask whether this forms a [[coproduct]] diagram in $C$.  This should be equivalent if $C$ has [[disjoint coproducts]].

## Related concepts

* [[disjoint coproduct]]

[[!redirects disjoint set]]
[[!redirects disjoint sets]]
[[!redirects disjoint subset]]
[[!redirects disjoint subsets]]

[[!redirects disjoint]]


disjoint union

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# Disjoint unions
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## Idea

The _disjoint union_ is a [[coproduct]] in [[Set]], the category of [[sets]].

In a general [[category]] [[coproducts]] need not have the expected disjointness property of those in [[Set]]. If they do they are called _[[disjoint coproducts]]_.

## Definition

Given any family $(A_i)_{i:I}$ of sets, the (external) __disjoint union__ $\biguplus_i A_i$ (also written $\sum_i A_i$, $\coprod_i A_i$, etc) of the family is the set of all (ordered) pairs $(i,a)$ with $i$ in the index set $I$ and $a$ in $A_i$.

As stated, the type of the second element of such a pair depends on the first element, which is natural in [[dependent type theory]] (see at _[[dependent sum type]]_) and no problem for [[material set theory]], but it may be ill formed in a [[structural set theory]] or in some forms of [[type theory]], especially those based on the [[internal language]] of [[topos theory]].  Alternatively, one may define $\biguplus_i A_i$ to be the set of those elements $x$ of the [[cartesian product]] $\prod_i \mathcal{P}A_i$ of the [[power sets]] such that there is exactly one index $j$ such that $x_j$ is [[inhabited set|inhabited]] and that $x_j$ is a [[singleton]].  If you\'re trying to be [[predicative mathematics|predicative]] too, then you may need to simply adopt the existence of disjoint unions as an axiom (the __axiom of disjoint unions__) in your [[foundations]], stating the following facts about it.

There is a natural [[injection]] $A_j \to \biguplus_i A_i$ (mapping $a$ to $(j,a)$, or mapping $a$ to $(i \mapsto \{a \;|\; i = j\})$) for each index $j$.  Conversely, for each element $x$ of $\biguplus_i A_i$, there is a unique index $j$ and such that $x$ is in the [[image]] of the injection from $A_j$.  It is common to treat $A_j$ as a [[subset]] of $\biguplus_i A_i$; so if no confusion can result (in particular, when the notation for an element of $A_j$ always makes the ambient set clear), one often suppresses the index in the notation for an element of the disjoint union.

## Special cases

Given sets $A$ and $B$, the disjoint union of the binary family $(A,B)$ is written $A \uplus B$ (also $A + B$, $A \amalg B$, etc); its elements may be written (if care is needed) as $(0,a)$ and $(1,b)$, $(1,a)$ and $(2,b)$, $\iota{a}$ and $\kappa{b}$, and in many other styles.

Given sets $A_1$ through $A_n$, the disjoint union of the $n$-ary family $(A_1,\ldots,A_n)$ is written $\biguplus_{i=1}^n A_i$ (or similarly); its elements may be written (if care is needed) as $(i,a)$ for $1 \leq i \leq n$ and $a \in A_i$.

Given sets $A_1$, $A_2$, etc, the disjoint union of the countably infinitary family $(A_1,A_2,\ldots)$ is written $\biguplus_{i=1}^\infty A_i$ (or similarly); its elements may be written (if care is needed) as $(i,a)$ for $i$ a natural number and $a \in A_i$.

Given a set $A$, the disjoint union of the unary family $(A)$ may be identified with $A$ itself; that is, we identify $(i,a)$ for the unique index $i$ with $a$.

The disjoint union of the empty family $()$ is [[empty set|empty]]; it has no elements.

## Internal version

(This is internal in the sense of 'internal direct sum', not [[internalization]].  For that, just see [[coproduct]].)

If a family $(A_i)_{i: I}$ of [[subsets]] of a given set $X$ are all pairwise disjoint (that is, $A_i \cap A_j$ has an element only if $i = j$, for any indices $i$ and $j$), then the [[union]] $\bigcup_i A_i$ is [[natural isomorphism|naturally bijective]] with the (external) disjoint union defined above.  Conversely, given an external disjoint union $\biguplus_i A_i$, each $A_j$ may be identified with a subset of $\biguplus_i A_i$ (as explained above); these subsets are all pairwise disjoint, and their union is the entire disjoint union.

Accordingly, a union of pairwise disjoint subsets may be called an __internal disjoint union__.  (Compare the internal vs external notions of [[direct sum]].)

## Examples

* [[countable unions of countable sets are countable]]

## Related concepts

* [[union]]

* [[dependent sum]], [[dependent sum type]]

* [[Cartesian product]]

* [[disjoint union topological space]]

category: foundational axiom

[[!redirects disjoint union]]
[[!redirects disjoint unions]]

[[!redirects external disjoint union]]
[[!redirects external disjoint unions]]

[[!redirects internal disjoint union]]
[[!redirects internal disjoint unions]]

[[!redirects axiom of disjoint unions]]


disjoint union topological space

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#Contents#
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## Definition

Given a [[set]] $X_i$, $i \in I$ of [[topological spaces]], then their _disjoint union space_ $\underset{i \in I}{\sqcup} X_i$ is the topological space whose underlying set is the [[disjoint union]] of the underlying sets of the $X_i$, and whose [[open subsets]] are precisely the disjoint unions of the open subsets of the $X_i$.

More abstractly, this is the [[coproduct]] in the [[category]] [[Top]] of topological spaces.

## Related concepts

* disjoint union of spaces is used to characterize [[connected topological spaces]]

[[!include universal constructions of topological spaces -- table]]

[[!redirects disjoint union topological spaces]]

[[!redirects disjoint union space]]
[[!redirects disjoint union spaces]]


disjunction

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# Logical disjunction
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## Definitions

In [[logic]], logical disjunction is the [[join]] in the [[poset]] of [[truth values]].

Assuming that (as in [[classical logic]]) the only truth values are true ($T$) and false ($F$), then the disjunction $p \vee q$ of the truth values $p$ and $q$ may be defined by a truth table:

|$p$|$q$| |$p \vee q$|
|---|---|-|------------|
|$T$|$T$| |$T$|
|$T$|$F$| |$T$|
|$F$|$T$| |$T$|
|$F$|$F$| |$F$|

That is, $p \vee q$ is true if and only if at least one of $p$ and $q$ is true.  Disjunction also exists in nearly every non-classical logic.

More generally, if $p$ and $q$ are any two [[relations]] on the same domain, then we define their disjunction pointwise, thinking of a relation as a [[function]] to truth values.  If instead we think of a relation as a [[subset]] of its domain, then disjunction becomes [[union]].

In [[natural deduction]] the [[inference rules]] for disjunction are given as

$$\frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma \vdash P \vee Q \; \mathrm{prop}} \qquad \frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma, P \; \mathrm{true} \vdash P \vee Q \; \mathrm{true}} \qquad \frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma, Q \; \mathrm{true} \vdash P \vee Q \; \mathrm{true}}$$

$$\frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop} \quad \Gamma, P \vee Q \; \mathrm{true} \vdash R \; \mathrm{prop} \quad \Gamma, P \; \mathrm{true} \vdash R \; \mathrm{true} \quad \Gamma, Q \; \mathrm{true} \vdash R \; \mathrm{true}}{\Gamma, P \vee Q \; \mathrm{true} \vdash R \; \mathrm{true}}$$

## Remarks

Disjunction as defined above is sometimes called __inclusive disjunction__ to distinguish it from [[exclusive disjunction]], where *exactly* one of $p$ and $q$ must be true.

In the context of [[substructural logics]] such as [[linear logic]], we often have both _[[additive disjunction]]_ $\oplus$ and _[[multiplicative disjunction]]_ $\parr$; see the [Rules of Inference](#RoI) below for the distinction.  In [[linear logic]], additive disjunction is the [[join]] under the [[entailment]] relation, just like disjunction in [[classical logic]] (and [[intuitionistic logic]]), while multiplicative disjunction is something different.

Disjunction is [[de Morgan dual]] to [[logical conjunction|conjunction]].

Like any join, disjunction is an associative operation, so we can take the disjunction of any finite positive whole number of truth values; the disjunction is true if and only if at least one of the various truth values is true.  Disjunction also has an [[identity element]], which is the [[falsity|false]] truth value.  Some logics allow a notion of infinitary disjunction.  Indexed disjunction is [[existential quantification]].

## In dependent type theory

In [[dependent type theory]], the disjunction of two [[mere propositions]], $P$ and $Q$, is the [[bracket type]] of their [[sum type]], $\| P + Q \|$. Disjunction types in general could also be regarded as a particular sort of [[higher inductive type]]. In [[Coq]] syntax:

    Inductive disjunction (P Q:Type) : Type :=
    | inl : P -> disjunction P Q
    | inr : Q -> disjunction P Q
    | contr0 : forall (p q : disjunction P Q) p == q

If the [[dependent type theory]] has a [[type of propositions]] $\mathrm{Prop}$, such as the one derived from a type universe $U$ - $\sum_{A:U} \mathrm{isProp}(A)$, then the disjunction of two types $A$ and $B$ is defined as the [[dependent function type]]

$$A \vee B \equiv \prod_{P:\mathrm{Prop}} ((A \to P) \times (B \to P)) \to P$$

By [[weak function extensionality]], the disjunction of two types is a proposition.

\begin{theorem}
The two definitions above are equivalent.
\end{theorem}

\begin{proof}
The [[propositional truncation]] of a type $A$ is equivalent to the following dependent function type

$$\| A \| \simeq \prod_{P:\mathrm{Prop}} (A \to P) \to P$$

Substituting the sum type $A + B$ for $A$, we have

$$\| A + B \| \simeq \prod_{P:\mathrm{Prop}} ((A + B) \to P) \to P$$

Given any type $C$, there is an equivalence

$$((A + B) \to C) \simeq (A \to C) \times (B \to C)$$
and if $A \simeq B$, then $(A \to C) \simeq (B \to C)$. In addition, for all type families $x:A \vdash B(x)$, and $x:A \vdash C(x)$, if there is a family of equivalences $e:\prod_{x:A} B(x) \simeq C(x)$, then there is an equivalence $\left(\prod_{x:A} B(x)\right) \simeq \left(\prod_{x:A} C(x)\right)$. All this taken together means that there are equivalences

$$\| A + B \| \simeq \left(\prod_{P:\mathrm{Prop}} ((A + B) \to P) \to P\right) \simeq \left(\prod_{P:\mathrm{Prop}} ((A \to P) \times (B \to P)) \to P\right)$$
\end{proof}

If one has the [[boolean domain]] and the [[existential quantifier]], then the disjunction of two types $A$ and $B$ is given by the following type:

$$A \vee B \coloneqq \exists b:\mathrm{bool}.((b = \mathrm{true}) \to A) \times ((b = \mathrm{false}) \to B)$$

## Classical vs constructive

There are a variety of connectives that are distinct in [[intuitionistic logic]] but are all equivalent to disjunction in [[classical logic]].  Here is a Hasse diagram of some of them, with the strongest statement at the bottom and the weakest at the top (so that each statement entails those above it):
$$ \array {           &          & \neg(\neg{P} \wedge \neg{Q}) \\
                      & ⇗ &          & ⇖ \\
\neg{P} \rightarrow Q &          &          &          & P \leftarrow \neg{Q} \\
                      & ⇖ &          & ⇗ \\
                      &          & (\neg{P} \rightarrow Q) \wedge (P \leftarrow \neg{Q}) \\
                      &          & \Uparrow \\
                      &          & P \vee Q } $$
(A single arrow is implication in the object language; a double arrow is entailment in the metalanguage.)  Note that $\neg{P} \wedge \neg{Q}$ is the [[negation]] of every item in this diagram.

In the [[double-negation interpretation]] of classical logic in intuitionistic logic, $\neg(\neg{P} \wedge \neg{Q})$ is the interpretation in intuitionistic logic of disjunction in classical logic.  For this reason, $\neg(\neg{P} \wedge \neg{Q})$ is sometimes called _classical disjunction_.  But this doesn\'t mean that it should always be used when turning classical mathematics into [[constructive mathematics]].  Indeed, a stronger statement is almost always preferable, if one is valid; $\neg(\neg{P} \wedge \neg{Q})$ is merely the fallback position when nothing better can be found.  (And as can be seen in the example in the paragraph after next, sometimes even this is not valid.)

In the [[antithesis interpretation]] of [[affine logic]] in intuitionistic logic, $(\neg{P} \rightarrow Q) \wedge (P \leftarrow \neg{Q})$ is the interpretation of the [[multiplicative disjunction]] $P \parr Q$ for affirmative propositions.  More generally, a statement $P$ in affine logic is interpreted as a pair $(P^+,P^-)$ of mutually contradictory statements in intuitionistic logic; $P^-$ is simply the negation of $P^+$ for affirmative propositions, but in general, $P^-$ only entails $\neg{P^+}$.  Then $P \parr Q$ is interpreted as $\big((P^- \rightarrow Q^+) \wedge (P^+ \leftarrow Q^-), P^- \wedge Q^-\big)$; that is, $(P \parr Q)^+$ is $(P^- \rightarrow Q^+) \wedge (P^+ \leftarrow Q^-)$, and $(P \parr Q)^-$ is $P^- \wedge Q^-$.  (In contrast, the additive disjunction $P \oplus Q$ is interpreted as $(P^+ \vee Q^+, P^- \wedge Q^-)$.  Note that $P \oplus Q$ entails $P \parr Q$ in affine logic, even though they are independent in linear logic.)

For a non-affirmative example, in the arithmetic of (located) [[real numbers]], it is not constructively valid to derive $(a = 0) \vee (b = 0)$ from $a b = 0$, and it\'s not even valid to derive $\neg\big(\neg(a = 0) \wedge \neg(b = 0)\big)$ without [[Markov's principle]] (or at least some weak version of it), but it *is* valid to derive $(a \# 0) \rightarrow (b = 0)$ (and conversely), where $\#$ is the usual [[apartness relation]] between real numbers.  (Here, $P^+$ is $a = 0$ and $P^-$ is $a \# 0$, and similarly for $Q$ and $b$.)  Of course, it\'s also valid to derive $\neg\big((a \# 0) \wedge (b \# 0)\big)$ (which is actually equivalent).

## Rules of inference {#RoI}

The [[rule of inference|rules of inference]] for disjunction in [[sequent calculus]] are dual to those for [[logical conjunction|conjunction]]:

$$ \begin {gathered}
   \frac { \Gamma , p , \Delta \vdash \Sigma ; \; \Gamma , q , \Delta \vdash \Sigma } { \Gamma , p \vee q , \Delta \vdash \Sigma } \; \text {left additive} \\
   \frac { \Gamma \vdash \Delta , p , \Sigma } { \Gamma \vdash \Delta , p \vee q , \Sigma } \; \text {right additive 0} \\
   \frac { \Gamma \vdash \Delta , q , \Sigma } { \Gamma \vdash \Delta , p \vee q , \Sigma } \; \text {right additive 1} \\
\end {gathered} $$

Equivalently, we can use the following rules with weakened contexts:
$$ \begin {gathered}
   \frac { \Gamma , p \vdash \Delta ; \; q , \Sigma \vdash \Pi } { \Gamma , p \vee q , \Sigma \vdash \Delta , \Pi } \; \text {left multiplicative} \\
   \frac { \Gamma \vdash \Delta , p , q , \Sigma } { \Gamma \vdash \Delta , p \vee q , \Sigma } \text {right multiplicative} \\
\end {gathered} $$

The rules above are written so as to remain valid in logics without the [[exchange rule]].  In [[linear logic]], the first batch of sequent rules apply to additive disjunction (interpret $p \vee q$ in these rules as $p \oplus q$), while the second batch of rules apply to multiplicative disjunction (interpret $p \vee q$ in those rules as $p \parr q$).

The [[natural deduction]] rules for disjunction are a little more complicated than those for conjunction:
$$ \begin {gathered}
   \frac { \Gamma , p \vdash r ; \; \Gamma , q \vdash r } { \Gamma , p \vee q \vdash r } \; \text {elimination} \\
   \frac { \Gamma \vdash p } { \Gamma \vdash p \vee q } \; \text {introduction 0} \\
   \frac { \Gamma \vdash q } { \Gamma \vdash p \vee q } \; \text {introduction 1} \\
\end {gathered} $$

## Related concepts

[[!include logic symbols -- table]]

## References

* Landon D. C. Elkind, Richard Zach, _The Genealogy of $\vee$_ ([arXiv:2012.06072](https://arxiv.org/abs/2012.06072))

The definition of the disjunction of two types in dependent type theory as the propositional truncation of the sum type is found in:

* {#UFP13} [[Univalent Foundations Project]], §3.7 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

And the disjunction of two types defined from the type of propositions and dependent product types can be found in:

* Madeleine Birchfield, *Constructing coproduct types and boolean types from universes*, MathOverflow ([web](https://mathoverflow.net/questions/457904/constructing-coproduct-types-and-boolean-types-from-universes))

[[!redirects disjunction]]
[[!redirects disjunctions]]
[[!redirects logical disjunction]]
[[!redirects inclusive disjunction]]
[[!redirects or]]
[[!redirects inclusive or]]


disjunctive (∞,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Regular and Exact categories
+-- {: .hide}
[[!include regular and exact categories - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[(∞,1)-category]] satisfying the analogs of the properties of a [[disjunctive category]].

## Related concepts

* [[Burnside category]]

## References

* {#Barwick14} [[Clark Barwick]], section 4 of _Spectral Mackey functors and equivariant algebraic K-theory (I)_, Adv. Math., 304:646–727 ([arXiv:1404.0108](http://arxiv.org/abs/1404.0108))

[[!redirects disjunctive (∞,1)-categories]]

[[!redirects extensive (∞,1)-category]]
[[!redirects extensive (∞,1)-categories]]

[[!redirects lextensive (∞,1)-category]]
[[!redirects lextensive (∞,1)-categories]]

[[!redirects infinitary extensive (∞,1)-category]]
[[!redirects infinitary extensive (∞,1)-categories]]

[[!redirects positive (∞,1)-category]]
[[!redirects positive (∞,1)-categories]]
[[!redirects positive coherent (∞,1)-category]]
[[!redirects positive coherent (∞,1)-categories]]


disjunctive logic

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

\tableofcontents

## Idea

**Disjunctive logic** is the [[internal logic]] of [[pre-lextensive categories]].  Roughly speaking this means it is [[cartesian logic]], with conjunction and "provably-unique existential quantification", together with "provably-disjoint disjunction".

## Definition

...

## Example

The (coherent) theory of (geometric) [[field|fields]] is obtained from the axioms for the algebraic theory of commutative unital rings by adding the sequents

$$(0=1)\vdash \bot\,\text{ (nontriviality) and }\,\top \vdash_x ((x=0) \vee
((\exists y)(xy=1)))\quad.$$

Since inverse elements in a commutative ring are unique when they exist the second sequent involves a legitimate existential quantification plus a legitimate disjunction (due to the nontriviality) whence the resulting theory is (finitary) disjunctive.

## Properties

...

## Related entries

* [[extensive category]]

* [[disjoint union]]

* [[connected limit]]

[[!include categories and logic - table]]

## Link

* [category theory mailing list: discussion on extensive categories](http://www.mta.ca/~cat-dist/catlist/1999/extensive)

## References

The main reference on disjunctive logic is Johnstone ([1979](#Johnstone79)) which was inspired by a non-syntactic concept of [[Yves Diers]]. The [elephant](#Johnstone02) has some additional cursory remarks. Freyd ([2002](#Freyd02)) gives disjunctive logic a short treatment under the name 'alternating logic' and discusses the [[real closed field|theory of real closed fields]] as an example. Barr and Wells ([1985](#BarrWells85)) discuss the corresponding class of sketches without mentioning the syntactic side.

* {#AdamekRosicky94}[[Jiří Adámek]], [[Jiří Rosický]], _Locally presentable and accessible categories_,  Cambridge UP 1994. (exercise 5f., p.238)

* {#BarrWells85}[[Michael Barr]], [[Charles Wells]], _Toposes, Triples and Theories_ , Springer Heidelberg 1985. (Reprinted as [TAC reprint no.12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html) (2005); section 8.2, pp.292f)

* {#Freyd02}[[Peter Freyd]], _Cartesian Logic_ , Theor. Comp. Sci. **278** (2002) pp.3-21.

* {#Johnstone79}[[Peter Johnstone]], _A Syntactic Approach to Diers' Localizable Categories_ , pp.466-478 in Springer LNM **753** Heidelberg 1979.

* {#Johnstone02}[[Peter Johnstone]], _Sketches of an [[Elephant]] II_, Oxford UP 2002. (D1.3.6 p.834, D2.1.2(e) p.863, D2.2.6 p.872, D2.4.8 p.886)

[[!redirects Disjunctive logic]]
[[!redirects Disjunctive Logic]]
[[!redirects disjunctive theory]]


dispersion relation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Harmonic analysis
+-- {: .hide}
[[!include harmonic analysis - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[harmonic analysis]]a _dispersion relation_ is a relation between the [[frequency]] and the [[wavelength]] of [[plane waves]]. Typically this relation expresses the frequency $\nu(\lambda)$ as a [[function]] of the [[wavelength]].

In [[special relativity]] the [[frequency]] of a [[plane wave]] is proportional to its [[energy]], and its [[wave vector]] is proportional to its [[momentum]], so that now a dispersion relation becomes an _energy-momentum relation_.

[[!include plane waves -- table]]

## Related conceptss

* [[energy band]]

* [[energy-momentum tensor]]

## References

* Wikipedia, _[Dispersion relation](https://en.wikipedia.org/wiki/Dispersion_relation)_

* Wikipedia, _[Energy-momentum relation](https://en.wikipedia.org/wiki/Energy%E2%80%93momentum_relation)_

[[!redirects dispersion relations]]

[[!redirects energy-momentum relation]]
[[!redirects energy-momentum relations]]


display locale

# Display locales
* table of contents
{: toc}

## Idea

The category of [[cosheaves]] of sets on a [[locale]] $X$
is equivalent to the category of [[complete spreads]] over $X$.
The equivalence functor sends a [[cosheaf]] $D$ of sets over $X$
to its display locale over $X$.

Thus, the functor described here
plays the same role in the equivalence between
[[cosheaves]] of sets on $X$ and [[complete spreads]] over $X$
as the [[etale locale]] construction
plays in the equivalence between
[[sheaves]] of sets on $X$ and [[etale locales]] over $X$.

## Definition

The __display locale__ of a [[precosheaf]] $D$ on a [[locale]] $X$
is a locale over $X$ (i.e., an object in the [[slice category]] $Loc/X$)
given by the [[base change]]
of the morphism
$$Alex(elem(D))\to Alex(elem(1_X))$$
(induced by the unique morphism $D\to 1_X$)
along the morphism $X\to Alex(elem(1_X))$.

Here $1_X$ denotes the [[terminal]] [[precosheaf]] on $X$.

The functor $elem\colon Precosheaf(X) \to Poset$
computes the [[category of elements]] of a [[precosheaf]] on $X$.

The functor $Alex\colon Poset \to Locale$
computes the [[Alexandroff locale]] of a poset $P$
by sending it the [[locale]] whose [[frame]]
consists of downward closed subsets of $P$.

The morphism $X\to Alex(elem(1_X))$ has as its underlying
[[morphism of frames]] the map that sends a downward closed subset of $X$
to its [[supremum]].

## Adjunction

The display locale functor
$$Cosheaf(X) \to LocConLoc/X$$
is [[right adjoint]]
to the [[cosheaf of connected components]] functor
$$LocConLoc/X \to Cosheaf(X).$$

(The above adjunction also makes holds for [[precosheaves]]
if we generalize the [[cosheaf of connected components]] construction
accordingly, see Section 2 in [Funk](#Funk).)

This adjunction exhibits the category of [[cosheaves]] of sets on $X$
as a (full) [[reflective subcategory]] of [[locally connected locales]] over $X$.
The [[essential image]] of the inclusion is known as the category
of [[complete spreads]] over $X$.

## References

* [[Jonathon Funk]], _The display locale of a cosheaf_.
{#Funk}

[[!redirects display locale]]
[[!redirects display locales]]


display logic

# Display logic

* table of contents
{:toc}

## Idea

Display logic is a general kind of [[sequent calculus]] containing enough [[structural connectives]] and [[structural rules]] so that every [[logical formula]] appearing in a sequent can be "displayed" all by itself on the left or the right in an equivalent sequent.  This enables a simple formulation of the [[cut rule]].

## Informal Description

In the simplest sort of display logic, the [[antecedent]] and [[consequent]] are lists of formulas, some of which are marked with a star $*$, and there are [[structural rules]] allowing formulas to be moved back and forth between the antecedent and consequent by adding or removing a star:

$$ \frac{\Gamma,A \vdash \Delta}{\Gamma \vdash A*,\Delta} \qquad
 \frac{\Gamma,A* \vdash \Delta}{\Gamma \vdash A,\Delta} $$
$$ \frac{\Gamma \vdash A*,\Delta}{\Gamma,A \vdash \Delta} \qquad
 \frac{\Gamma \vdash A,\Delta}{\Gamma,A* \vdash \Delta} $$

By repeated application, this means that if $A$ appears anywhere in the antecedent --- or in the consequent with a star --- then we can rearrange the sequent to look like $A \vdash \Delta$.   Dually, if $A$ appears anywhere in the consequent, or anywhere in the antecedent with a star, we can rearrange the sequent to look like $\Gamma \vdash A$.  This enables a simple formulation (and proof) of the cut rule:

$$ \frac{\Gamma \vdash A \quad A \vdash \Delta}{\Gamma\vdash \Delta} $$

The general form of display logic allows [[bunches]] as antecedents and consequents.  There are some number of "families" of connectives, each of which comes with a [[structural connective]] for joining formulas in a context, like comma or semicolon, and also a structural connective for "negation" like star, with structural rules like those above formulated for each family.  (Each "comma" also has a corresponding nullary version, i.e. there is one "empty context" for each family.)  The structural comma is internalized on the left by a [[logical conjunction|conjunction]] and on the right by a [[logical disjunction|disjunction]], while the structural negation is internalized on both sides by a [[negation]].

Each family can also have other logical connectives like [[implication]] and [[modal operators]].  The families are distinguished from each other by the *other* structural rules that their commas satisfy; for instance, a "boolean family" satisfies contraction and weakening, while a "[[relevance logic|relevance]]" family satisfies only contraction.

## Properties

Any display logic satisfying a list of eight natural conditions automatically satisfies [[cut admissibility]].

## References

* Nuel D. Belnap, Jr.  *Display logic*, Journal of Philosophical Logic 11 (1982) 375-417.  [PDF](http://www.pitt.edu/~belnap/87displaylogic.pdf)

[[!redirects display logics]]


display map

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In [[type theory]], a **display map** [[Taylor 1983-7, §4.3.2](#Taylor87)] is a [[morphism]] $p\colon B\to A$ in a [[category]] which represents a [[dependent type]] under [[categorical semantics of dependent type theory]].

That is, $B$ represents (interprets) a [[type]] [[dependent type|dependent]] on a [[variable]] of type $A$, usually written syntactically as $x\colon A \vdash B(x)\colon Type$.  The intended intuition is that $B(x)$ is the [[fiber]] of the map $p$ over $x\colon A$.

More specifically, the [[syntactic category]] of a type theory is naturally equipped with a class of maps called "display maps" which come from its type dependencies, while [[models]] for such a [[theory]] can be studied in any other category equipped with a suitable class of maps called "display maps."  By the [usual](type%20theory#CategoricalSemantics) [[adjunction]], such models are then equivalent to functors out of the syntactic category which preserve the relevant structure, including the display maps.

Since the most common dependent type theories include [[dependent product]] types, and any display map in a category that represents semantics for such a theory must be [[exponentiable morphism|exponentiable]], the term **display map** is sometimes used to mean any exponentiable morphism.

Sometimes all maps are display maps.  This happens particularly in
[[extensional type theory]], such as versions of the [[internal logic]] of categories that include dependent types.

In the context of _[[homotopy type theory]]_ display maps are [[fibrations]], for instance in a [[type-theoretic model category]].

## Definition

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

For $\mathcal{C}$ a [[category]], a [[class]] $D \subset Mor(\mathcal{C})$ of [[morphisms]] of $\mathcal{C}$ is called a **class of displays** if

* All [[pullbacks]] of elements of $D$ exist and belong to $D$.
=--

It follows automatically that the [[full subcategory]] on $D$ of the [[arrow category]] $\Arr(\mathcal{C})$ is [[replete subcategory|replete]]; in other words, the [[property]] of $\mathcal{C}$-morphisms given by membership in $D$ respects the [[principle of equivalence]].

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

The category with displays is called **well rooted** if it has a [[terminal object]] $1$ and all the morphisms to $1$ are display maps.
=--

It follows that binary [[products]] exist and their [[projections]] are display maps.

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

Often (which simplifies matters) $D$ is __closed under composition__:

1. Every [[identity morphism]] (and hence every [[isomorphism]]) belongs to $D$.

2. The [[composite]] of a [[composable pair]] of elements of $D$ belongs to $D$.
=--

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

An __interpretation__ or __[[model]]__ of a category with class of displays $(\mathcal{C}, D)$ in another category with displays $(\mathcal{C}', D')$ is a  [[functor]]
$$
  S\colon \mathcal{C} \to \mathcal{C}'
$$
which preserves display maps and their pullbacks.
=--

This appears as ([Taylor 1999, def. 8.3.2](#Taylor99)).

## Examples

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

Examples of categories with displays, def. \ref{CategoryWithDisplays}, include

1. One extreme: any [[locally cartesian category]] $\mathcal{C}$ with the class of all morphisms. (This is well rooted ---if $\mathcal{C}$ has a terminal object--- and closed under composition.)

2. Another extreme: any category with the empty class of no morphisms.

3. Modifying (2) for an extreme case closed under composition: any category with the class of [[isomorphisms]].

4. Modifying (2) for an extreme well-rooted case: A category $\mathcal{C}$ with finite [[products]] and $D$ the class of product [[projections]] $X \times A \to X$. (This is also closed under composition.)

5. A [[locally cartesian category]] $\mathcal{C}$ (such as a [[topos]]) equipped with its class of [[monomorphisms]]. (This is closed under composition.)

6. A category equipped with a [[singleton pretopology]]. (This is closed under composition.)

7. (...)
=--

See also [Taylor 1999, Exp. 8.3.6](#Taylor99).

## Related pages

* [[clan]]

* [[bundle]]

* [[categorical semantics of dependent type theory]]

## References

The notion is due to

* {#Taylor87} [[Paul Taylor]], §4.3.2 in: *Recursive Domains, Indexed Category Theory and Polymorphism*, Cambridge (1983-7) [[[Taylor-IndexedCategoryTheory.pdf:file]]]

* {#Taylor99} [[Paul Taylor]], Section 8.3 in: _[[Practical Foundations of Mathematics]]_, Cambridge University Press (1999) [[webpage](http://www.paultaylor.eu/~pt/prafm/index.html)]

[[!redirects display map]]
[[!redirects display maps]]
[[!redirects display morphism]]
[[!redirects display morphisms]]