directed homotopy type theory

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### Context
#### Directed Type Theory
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[[!include directed homotopy type theory - contents]]
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#### Higher category theory
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[[!include higher category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

What [[homotopy type theory]] is for [[homotopy theory]]/[[(∞,1)-category theory]], _directed homotopy type theory_ is (or should be) for [[directed homotopy theory]]/[[(∞,n)-category]] [[higher category theory|theory]].

More in detail: Where

* the [[types]] of [[homotopy type theory]] have [[relation between category theory and type theory|interpretation]] as [[∞-groupoids]] aka [[(∞,0)-categories]]  (and more generally: as [[∞-stacks]]), so that [[homotopy type theory]] itself is an [[internal language]] for the [[(∞,1)-categories]] (specifically, for [[univalence axiom|univalent HoTT]]: [[(∞,1)-toposes]]) that these [[(∞,0)-categories]] form

so

* the [[types]] of directed homotopy type theory ought to have [[relation between type theory and category theory|interpretation]] as [[(∞,n)-categories]] for some $n \in \{1, 2, \cdots, \infty\}$ (and more generally: as [[(∞,n)-sheaves|(∞,n+1)-sheaves]]) so that directed homotopy type theory itself should be an [[internal language]] for the [[(∞,n)-category|(∞,n+1)-categories]] (and in the suitably directed [[univalence axiom|univalent]] case: [[(∞,n)-toposes|(∞,n+1)-toposes]]) that these (geometric) [[(∞,n)-categories]] form.

A proposal for the case $n = \infty$ (potentially describing [[(∞,∞)-categories]] aka [[omega-categories]]) is *[[opetopic type theory]]*, going back to [Finster (2012)](#opetopic+type+theory#Finster12).

A proposal for the case $n = 1$ with the directed analog of the [[univalence axiom]] included --- hence for a type theory whose types may be interpreted as [[(∞,1)-categories]]/[[(∞,2)-sheaves]] and which itself would be the [[internal language]] of the [[(∞,2)-category|$(\infty,2)$-categories]]/[[(∞,2)-topos|$(\infty,2)$-toposes]] that these form --- is announced in [Cisinski et al. (2023)](#CisinskiEtAl23).

## Appeal and practicality

One obvious reason for studying directed homotopy type theory (especially when $n = \infty$ in the above) is generality:  $(\infty,\infty)$-categories are among the most general higher categorical structures. Besides this reason, access to higher directed structure may offer, in particular, the following conceptual advantages.

* _Universe types can retain their higher directed structure._ [[type universe|Universe types]] are (small) 'internal' reflections of the category of all types of our type theory, but without sufficient higher structure in the theory, this reflection process must forget structure present in the external category (for instance, the [[category of sets]] can only contain a 'set of sets' as an object which does not remember functions; similarly, the universe type in [[homotopy type theory]] must forget about maps between types that are not equivalences).

* _Dependent function types become 'just' function types._ In the presence of higher directed structure, equipped with a universe type $\mathcal{U}$ and unit type $\mathbf{1}$, dependent function types $\Pi_{x : A} F(x)$, for $F$ a type family $F : A \to \mathcal{U}$, can be understood in terms of 'just' function types, namely, as $\mathsf{Fun}_{\mathsf{Fun}(A,\mathcal{U})}(\mathrm{const}_{\mathbf{1}}, F)$ (where $\mathrm{const}_{\mathbf{1}} : A \to \mathcal{U}$ is the constant functor with image $\mathbf{1}$).

* _Inductive types become substantially more expressive._ Inductive types with dependent constructors can also be expressed in terms of non-dependent constructors: for instance, for $F$ a type family $F : A \to \mathcal{U}$, the dependent pair type $\Sigma_{x : A} F(x)$ can be introduced with a single constructor $\mathrm{in} : F \to \mathrm{const}_{\Sigma_{x : A} F(x)}$ (which is a map in the function type $\mathsf{Fun}(A, \mathcal{U})$ and should thus be thought of as a natural transformation between functors). Moreover, there is the topic of [[higher inductive types]]; a useful example of a 'true' directed higher inductive type is the [[poset]] type $(\mathbb{N},\leq)$ with constructors
$$ 0 : \mathbf{1} \to \mathbb{N} $$
$$ \mathsf{succ} : \mathbb{N} \to \mathbb{N} $$
$$ \mathsf{less} : \id_{\mathbb{N}} \to \mathsf{succ} $$

* _Formalization of (higher) categorical semantics._ On a more practical side, access to higher directed structure could allow us to formalize the categorical semantics of other types theories, such as [[homotopy type theory]], as discussed in [Cisinki et al 2023](https://ncatlab.org/nlab/show/directed+homotopy+type+theory#CisinskiEtAl23).

On the other hand, working directed higher type theory may not be very practical.

* While it is clear that one eventually wants to speak about higher categorical concepts with type theory, it is not a priori clear that this motivates the dedicated formulation of new rules for directed higher type theory: it might still be more convenient to instead work internal to ordinary [[homotopy type theory]].

* More generally, it has been argued that directed higher type theories may not aid the practical usability of proof assistants due to the potential complexity of the rules involved.

## Related concepts

* [[directed univalence axiom]]

* [[directed type theory]], [[2-type theory]]

* [[opetopic type theory]]

* [[semi-simplicial type]]

* [[geometric homotopy type theory]]

* [[modal homotopy type theory]]

* [[fibered category theory]]

## References

### General

* [[Robert Harper]], [[Dan Licata]], _Canonicity for 2-Dimensional Type Theory_ ([pdf](http://www.cs.cmu.edu/~drl/pubs/lh112tt/lh112tt.pdf))

* [[Robert Harper]], [[Dan Licata]], _2-Dimensional directed dependent type theory_ ([pdf](http://www.cs.cmu.edu/~drl/pubs/lh102dtt/lh102dtt.pdf) [slides](http://www.cs.cmu.edu/~drl/pubs/lh102dtt/lh102dtt-slides.pdf))

* [[Michael Warren]], _Directed Type Theory_ ([video lecture](https://video.ias.edu/univalent/1213/0410-MichaelWarren))

* [[Dan Licata]], Chapters 7 and 8 of _Dependently Typed Programming with Domain-Specific Logics_ PhD thesis, ([pdf](http://www.cs.cmu.edu/~drl/pubs/thesis/thesis.pdf))

* [[Emily Riehl]], [[Mike Shulman]], *A type theory for synthetic ∞-categories*, Higher Structures **1** 1 (2017) [[arxiv:1705.07442](https://arxiv.org/abs/1705.07442), [published article](https://higher-structures.math.cas.cz/api/files/issues/Vol1Iss1/RiehlShulman)]

* [[Paige Randall North]], _Towards a directed homotopy type theory_ [[arXiv:1807.10566](https://arxiv.org/abs/1807.10566)]

* [[Alex Kavvos]], *A quantum of direction* (2019) [[pdf](https://seis.bristol.ac.uk/~tz20861/papers/meio.pdf)]

* {#Nuyts23} [[Andreas Nuyts]], _Higher Pro-arrows: Towards a Model for Naturality Pretype Theory_, HoTT/UF 2023, [PDF](https://hott-uf.github.io/2023/HoTTUF_2023_paper_1410.pdf)

* [[Andreas Nuyts]], _Towards a Directed Homotopy Type Theory based on 4 Kinds of Variance_, 2015, [PDF](https://anuyts.github.io/files/mathesis.pdf)

### Cisinski et al.
 {#ReferencesCisinskiEtAl}

* {#CisinskiEtAl23} [[Denis-Charles Cisinski]], [[Hoang Kim Nguyen]], Tashi Walde: *Univalent Directed Type Theory*, lecture series in the *[CMU Homotopy Type Theory Seminar](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html)* (13, 20, 27 Mar 2023) [[web](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html#230313), video 1:[YT](https://www.youtube.com/watch?v=5YOltuTcBK8), 2:[YT](https://www.youtube.com/watch?v=xWmELBvHMPo), 3:[YT](https://www.youtube.com/watch?v=P0Cfb4eUJo4); slides 0:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-intro_talk1.pdf), 1:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk1.pdf), 2:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk2.pdf), 3:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk3.pdf)]

  > **Abstract:** We will introduce a version of [[dependent type theory]] that is suitable to develop a [[synthetic mathematics|synthetic]] theory of [[1-categories]]. The [[axioms]] are both a [[fragment]] and an extension of ordinary [[dependent type theory]]. The [[axioms]] are chosen so that [[(∞,1)-category theory]] (in the form of [[quasi-categories]] or [[complete Segal spaces]]) gives a [[categorical semantics|semantic interpretation]], in a way which extends [[Vladimir Voevodsky|Voevodsky]]'s interpretation of [[univalence axiom|univalent]] [[dependent type theory]] in the [[homotopy theory]] of [[Kan complexes]]. More generally, using a slight generalization of [Shulman's methods](relation+between+type+theory+and+category+theory#Shulman19), we should be able to see that the theory of [[internal (∞,1)-category|(∞,1)‑categories internally]] in any [[(∞,1)-topos|∞‑topos]] (as [developed by Martini and Wolf](internal+infinity1-category#ReferencesInternalToInfinityTopos)) is a [[categorical semantics|semantic interpretation]] as well (hence so is [[Parametrized Higher Category Theory and Higher Algebra|parametrized higher category theory]] introduced by Barwick, Dotto, Glasman, Nardin and Shah). There are of course strong links with [[∞-cosmoi]] of Riehl and Verity as well as with [[cubical type theory|cubical HoTT]] (as strongly suggested by the work of Licata and Weaver), or [[simplicial type theory|simplicial HoTT]] (as in the work of Buchholtz and Weinberger). We will explain the axioms in detail and have a glimpse at basic theorems and constructions in this context ([[(∞,1)-Yoneda lemma|Yoneda Lemma]], [[(∞,1)-Kan extension|Kan extensions]], [[localization of an (∞,1)-category|Localizations]]). We will also discuss the perspective of reflexivity: since the theory speaks of itself (through directed [[univalence axiom|univalence]]), we can use it to justify new deduction rules that express the idea of working up to [[equivalence]] natively (e.g. we can produce a logic by rectifying the idea of having a locally cartesian type). In particular, this logic can be used to produce and study [[categorical semantics|semantic interpretations]] of [[HoTT]].

This is based on the discussion of [[straightening and unstraightening]] entirely within the context of [[quasi-categories]] from

* [[Denis-Charles Cisinski]], [[Hoang Kim Nguyen]], *The universal coCartesian fibration* [[arXiv:2210.08945](https://arxiv.org/abs/2210.08945)]

which (along the lines of the discussion of the universal left fibration from [Cisinski 2019](universal+left+fibration#Cisinski19)) allows to understand the [[universal coCartesian fibration]] as [[categorical semantics]] for the [[univalence axiom|univalent]] [[type universe]] in [[directed homotopy type theory]] (see [video 3 at 1:16:58](https://youtu.be/P0Cfb4eUJo4?t=4618) and [slide 3.33](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk3.pdf#page=33)).

{#CisinskiOnSyntax} But the actual type-theoretic syntax ([[inference rules]]) for this intended semantics remains to be given:

> [[[Denis-Charles Cisinski|Cisinski]] in [video 3 at 1:27:43](https://youtu.be/P0Cfb4eUJo4?t=5263)]: I won't provide the full [[syntax]] yet and actually I would be very happy to discuss that, because we don't know yet and I have questions myself, actually.

> [[[Steve Awodey|Awodey]] in [video 3 at 1:46:23](https://youtu.be/P0Cfb4eUJo4?t=6383)]: Maybe I'll suggest something, you tell me if you agree: What we have is a kind of axiomatization of the semantics of a system for type theory, so that we know what exactly we want formalize in the type theory, and what depends on what, and it articulates and structures the intended interpretation of the type theory in a very useful way. Maybe in the way that the axiomatic description of a cartesian closed category was very good to have for formulating the [[lambda-calculus]]. But I think that what we have is more on the side of the axiomatic description of the semantics, like the cartesian closed category, that it is on the side of the lambda-calculus itself. So, maybe I would suggest the term "abstract type theory" to describe this system as an intermediate in between an actual formally implemented system of type theory and the big unclear world of possible semantics and all the different structures that one could try to capture with a type theory, in between is this abstract type theory which specifies a particular structure that we want to capture in our type theory, which is a very very useful methodological step. [...] I am trying to maybe reconcile:

> Some people would prefer to call a type theory only something which can immediately be implemented in a computer. So that's different than an abstract description of a structure that we would want to describe in such a type theory.

> [[[Denis-Charles Cisinski|Cisinski]] in [video 3 at 1:49:28](https://youtu.be/P0Cfb4eUJo4?t=6568)]: I agree with what you say but I still have the hope to be able to produce an actual syntax [...] that's really the goal.

More on the [[directed univalence axiom]] in this context:

* [[Hoang Kim Nguyen]], *Directed univalence in simplicial sets*, talk in *[Homotopy Type Theory Electronic Seminar Talks](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html)* (March 2023) [[video](https://www.youtube.com/watch?v=GgcJqzGvq80), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Nguyen-2023-03-09-HoTTEST.pdf)]

[[!redirects directed homotopy type theories]]

[[!redirects directed homotopy type]]
[[!redirects directed homotopy types]]


directed homotopy type theory - contents

**[[directed homotopy type theory]]**

  ([[directed type theory|directed]] enhancement of [[homotopy type theory]] with [[types]] behaving like [[(infinity,n)-category|$(\infty,n)$-categories]])

* [[directed univalence]]

* [[simplicial homotopy type theory]]

  * [[hom type]]

  * [[Segal type]]

  * [[isomorphism in a Segal type]]

  * [[Rezk type]]

  * [[discrete Segal type]]

  * [[covariant type family]]

  * [[contravariant type family]]

* [[opetopic type theory]]


directed join

# Directed joins
* table of contents
{: toc}

## Idea

A __directed join__ is simply a [[join]] of a [[directed set]].

## Definitions

More precisely, if $P$ is a [[poset]] and $D$ is a [[subset]] of $P$, then we can consider the join $\bigvee D$ (if it exists) of $D$ in $P$.  Since $D$ is a poset in its own right, we can also consider whether $D$ is directed set.  If so, then $\bigvee D$ (if it exists) is a __directed join__ in $P$.  Sometimes one denotes that $\bigvee D$ is a directed join by making a little arrow out of the upper-right flank of the symbol (so it\'s a mix of '$\bigvee$' and '$\nearrow$').  Unfortunately, I haven\'t found that symbol in LaTeX or Unicode. Possible workaround is $\bigvee{}^{\uparrow} D$.

A __codirected meet__ in $P$ is a directed join in $P^\op$, but people don\'t talk about those so much.

By default, we mean *finitely* directed sets, that is $\aleph_0$-directed.  If instead we take the join of a $\kappa$-directed set (for some [[regular cardinal]] $\kappa$), then we have a __$\kappa$-directed join__.

## Properties

If a join-[[semilattice]] (a poset with all finitary joins) has all directed joins, then it has all joins (and so is a [[suplattice]], equivalently a [[complete lattice]]).  More generally, if a poset has all joins of fewer than $\kappa$ elements and all $\kappa$-directed joins, then it is a suplattice.

A [[topological space]] (or [[locale]]) $X$ is [[compact space|compact]] if and only if $X$ may be expressed as a directed join of open subsets only trivially.  That is, whenever $D$ is a directed collection of opens, if $X = \bigcup D$, then $X \in D$.

[[Directed colimits]] and [[filtered colimits]] are two slightly different [[categorifications]] of directed joins.

## DCPOs

A poset which has all directed joins is called a __directed-complete partial order__, or __[[dcpo]]__.  The [[homomorphisms]] of DCPOs are those [[functions]] that preserve directed joins; these are also called __Scott-continuous__ because they are precisely the [[continuous maps]] relative to the [[Scott topology]] on the DCPOs.

A __pointed dcpo__ is a DCPO with a [[bottom element]] (which is rather more specific than a [[pointed object]] in the category of DCPOs).

DCPOs are studied widely in [[domain theory]].

[[!redirects directed supremum]]
[[!redirects directed suprema]]
[[!redirects directed join]]
[[!redirects directed joins]]

[[!redirects directed meet]]
[[!redirects directed meets]]
[[!redirects codirected meet]]
[[!redirects codirected meets]]


directed limit

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### Context
#### Limits and colimits
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[[!include infinity-limits - contents]]
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# Directed limits
* table of contents
{: toc}

## Abstract definition

A __directed limit__ (or __codirected limit__) is a [[limit]] $\underset{\leftarrow}\lim F$ of a functor $F\colon J \to C$ whose [[source]] [[category]] $J$ is a downward-[[direction|directed set]].

More generally, for $\kappa$ a [[cardinal number|regular cardinal]] say that a **$\kappa$-directed set** $J$ is a [[poset]] in which every subset of cardinality $\lt \kappa$ has an upper bound. Then a limit over a functor $J \to C$ is called **$\kappa$-directed limit**.

If the directed set is an [[ordinal]], one speaks of a [[sequential limit]].

The [[duality|dual]] notion is that of [[directed colimit]], a [[colimit]] of a functor whose source is a upward-directed set.

## Terminology

Note that the terminology varies.  Especially in algebra, a directed limit may be called a '[[projective limit]]' or '[[inverse limit]]'; it\'s also possible to distinguish these so that an inverse limit may have an arbitrary (possibly undirected) [[partial order|poset]] as its source.  On the other hand, both terms are often used for arbitrary [[limits]] as an alternative to the 'co-' method of distinction.  (The corresponding dual terms are '[[inductive limit]]' and '[[direct limit]]', with no 'co-' even though these are [[colimits]].)

Directed (co)limits were studied in algebra (as projective and inductive limits) before the general notion of limit in category theory.  The elementary definition still seen there follows.

## Explicit definition

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

A __projective system__ in $C$ consists of a [[direction|directed set]] $I$ (which we will write directed-upward as usual), a family $(A_i)_{i: I}$ of objects of $C$, and a family $(f_{ij}: A_j \to A_i)_{i \leq j: I}$ of morphisms, such that:
* $f_{ii}: A_i \to A_i$ is the [[identity morphism]] on $A_i$;
* $f_{ik}: A_k \to A_i$ is the [[composition|composite]] $f_{ij} \circ f_{jk}$.

Then a __projective cone__ of this projective system is an object $X$ and a family of __projections__ $\pi_i: X \to A_i$ such that
$$ \pi_i = f_{ij} \circ \pi_j .$$

Finally, a __projective limit__ of the projective system is a projective cone $\underset{\leftarrow}\lim_i A_i$ (where both $f$ and $\pi$ are suppressed in the notation, each in its own way) which is [[universal property|universal]] in that, given any projective cone $X$, there exists a unique morphism $u\colon X \to \underset{\leftarrow}\lim_i A_i$ such that
$$ \pi_i = \pi_i \circ u $$
(where the left-hand $\pi$ is from the cone $X$ and the right-hand $\pi$ is from the limit).

Notice that a projective system in $C$ consists precisely of a directed set $I$ and a [[contravariant functor]] from $I$ (thought of as a category) to $C$, while a projective cone or limit of such a projective system is precisely a cone or limit of the corresponding functor.  So this is a special case of [[limit]].

As with other limits, a projective limit, if any exists at all, is unique up to a given isomorphism, so we speak of [[generalized the|the]] projective limit of a given projective system.

## In algebra

A projective limit in algebra is usually defined as a [[subobject|subalgebra]] of a [[cartesian product]].  To be precise, $\underset{\leftarrow}\lim_i A_i$ consists of those elements $(x_i)_{i: I}$ of $\prod_{i: I} A_i$ such that:
$$ x_i = f_ij(x_j) .$$
This can be seen as a special case of the construction of an arbitrary limit out of [[product]]s and [[equalizer]]s.

## Examples

Directed limits over the codirected set $(\mathbb{N},\geq)$ of [[natural numbers]], the [[tower]]-diagram,
$$
  \array{
     && && \lim_{\leftarrow_n} X(n) &&
     \\
     && &\swarrow& \downarrow & \searrow&
     \\
     \cdots & \to & X(2) & \to  & X(1) & \to & X(0)
  }
$$
are extremely common. Classical examples occur in the theory of [[Postnikov towers]] and also in the definition of the [[solenoids]].

A [[ring]] $ K [ [ x ] ] $ of formal [[power series]] (for $K$ a [[field]]) is a projective limit of the rings $K[x]/x^n$ (for $n$ a [[natural number]]).  Here, $C$ is the category of rings, $I$ is the directed set of natural numbers, $A_i = K[x]/x^i$, and $f_{ij}: A_j \to A_i$ is induced by the quotient map $K[x] \to K[x]/x^i$ (which must be proved well defined on $K[x]/x^j$ for $i \leq j$).

Similarly, a ring $\mathbf{Z}_p$ of $p$-[[adic integer]]s (for $p$ a [[prime number]]) is a projective limit of the rings $\mathbf{Z}/p^n$.

A set of infinite [[sequences]] is a projective limit of sets of [[list|finite sequences]] (which, at the level of [[sets]], includes the above examples).

[[!redirects directed limit]]
[[!redirects directed limits]]
[[!redirects codirected limit]]
[[!redirects codirected limits]]


directed n-graph > history

< [[directed n-graph]]


directed object

#Contents#
* table of contents
{:toc}

## Idea

The idea of the following text is to begin with a category of objects presumed [[undirected object|undirected]] and construct from that a supercategory of directed objects, analogous to how [[Marco Grandis]] developed [[directed space|directed topological spaces]] out of the usual undirected ones.

(Rather different approaches to a notion of "directed object" will exist. See also at _[[directed homotopy theory]]_ and _[[directed homotopy type theory]]_.)

## Definition (tentative)

Let $C$ be a [[Trimble n-category|Trimble omega-category]] with [[interval object]] $pt\stackrel{\sigma}{\to}I\stackrel{\tau}{\leftarrow}pt$, and suppose that every object $X$ of $C$ is $I$-[[undirected object|undirected]] (i.e. $[pt,X]\simeq [I,X]$).

Let $d_I \subset {}_{pt}hom(I,I)_{pt}$ be a subset of the set of [[co-span]]-endomorphisms of  $pt\stackrel{\sigma}{\to}I\stackrel{\tau}{\leftarrow}pt$. Let $dX\subset [I,X]$ be a subset of the [[hom-set]] $[I , X]$.

Then we call the pair $(X, dX)$ an *object with directed path space $dX$* (or *directed object*) if the following conditions (attributed to Marco Grandis) are satisfied:

1. (Constant paths) Every map $I \to \pt \to X$ is directed;

2. (Reparametrisation) If $\gamma\in d_I$, $\phi \in dX$, then $\gamma \circ \phi\in dX$. If e.g $d_I=[I,I]$, then this condition means that $d X$ is a sieve in $I/C$.

3. (Concatenation) Let $a,b:I\to X$ be *consecutive wrt. $I$* (i.e. $\pt \to^{\tau} I \to^{a} X$ equals $\pt \to^{\sigma} I \to^{b} X$), let $I^{v2}$ denote the pushout of $\sigma$ and $\tau$, then by the universal property of the pushout there is a map $\phi:I^{v2}\to X$. By definition of the [[interval object]] (described there in the section "Intervals for Trimble $\omega$-categories") there is a unique morphism $\psi:I\to I^{v2}$. Then the *composition of $a$ and $b$* is defined by $a\bullet b:=\phi\circ \psi$. Then $dX$ shall be closed under composition of consecutive paths.

We define a *morphism of objects with directed path space* to be a morphism of their underlying objects that preserves directed paths (i.e. if $p\in dX$, $\phi:X\to Y$, then $\phi\circ p\in d_Y$). Objects with directed path space and morphisms thereof define a category denoted by $d_I{C}$.

$C$ is a subcategory of $d_I{C}$.

## Examples

* The category of [[directed space|directed topological spaces]] according to Grandis is of the above form $d_I{C}$ for $C = $ [[Top]], $I = [0,1]$ and $d_I = \{monotonic maps I \to I\}$.

## References

The definition and study of directed topological spaces was undertaken in

* [[Marco Grandis]], _Directed homotopy theory, I. The fundamental category_ ([arXiv](http://arxiv.org/abs/math.AT/0111048))

Applications of categories regarded as models for directed spaces are discussed in

* [[Tim Porter]]: _Enriched categories and models for spaces of evolving states_, Theoretical Computer Science, 405, (2008), pp. 88 - 100.

* [[Tim Porter]], _Enriched categories and models for spaces of
dipaths. A discussion document and overview of some techniques_ ([pdf](http://drops.dagstuhl.de/opus/volltexte/2007/898/pdf/06341.PorterTimothy.Paper.898.pdf))


directed set > history

< [[directed set]]
[[!redirects directed set -- history]]


directed topological space

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###Context###
#### Topology
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[[!include topology - contents]]
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# Contents
* automatic table of contents goes here
{:toc}

## Idea

A _directed topological space_ is a [[topological space]] $X$ in which there is some 'sense of direction'. This can happen in various different ways and the level of the 'directedness' can be different in different situations, so naturally there are several 'competing' ideas, but the beginning of a consensus on what the overarching idea is.

If one bases homotopy theory on the idea of a [[singular simplex]] or more generally a singular cell of any shape, then there is no way in which a 'sense of direction' can be encode.  If we have a path in a space we can go along it (traverse it) in either direction, from 0 to 1 or from 1 to 0.  From this perspective a directed space is one in which not every singular cell $\Delta^n \to X$ (for $\Delta^n$ the standard topological [[simplex]]) is supposed to be _traversable_ in all directions, in some sense: instead these $k$-dimensional _paths_ may have a _direction_ .

As an example one can base the 'sense of direction' on a closed [[preorder]] or [[partial order]], (that is a [[pospace]]),so that the paths from the directed interval $[0,1]$ with the usual order to the space $X$, can only be 'traversed' in one direction. Another example which does not fit into this first type would be the directed circle.

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In other words, a circle with direction determined by the anticlockwise sense. Again it is easy to see that there are certain paths that respect the direction, 'directed paths' whilst others do not.

So far there exists a well-developed theory for a notion of _directed spaces_ $X$ where 1-dimensional paths given by maps $[0,1] \to X$ from the [[interval]] into the space are equipped with a direction. See in particular the book by  [[Marco Grandis]] on [[Directed Algebraic Topology]] listed below. Another suggested notion for modelling directed spaces is that of _[framed spaces](#framed_spaces)_, which is tailored towards certain higher categorical applications.

Note that a directed space is like a generalised space; not every directed space need be a space in the traditional sense, in accordance with the [[red herring principle]]. As an instance of this, note that [[Marco Grandis]] in his book [[Directed Algebraic Topology]] handles the directed homotopy of small categories, and of cubical complexes, since this is useful for comparison an interpretation of directed homotopy 'invariants'.

Directed spaces are studied in [[directed homotopy theory]], a relatively young topic. In generalization of how a [[topological space]] has a [[fundamental groupoid]], a [[directed space]] has a [[fundamental category]].

### Homotopy-theoretic perspective

From a [[homotopy theory|homotopy theoretic]] perspective one would wish that notions of directed spaces might serve to generalize the [[homotopy hypothesis]] -- which identifies ordinary (undirected) [[topological space]]s with [[∞-groupoid]]s, i.e., with [[(∞,0)-category|(∞,0)-categories]] -- to a more general context where [[(∞,0)-category|(∞,0)-categories]] are generalized to [[(∞,n)-category|(∞,r)-categories]] with $r \gt 0$:

An [[(∞,r)-category]] in this context might correspond to a **$r$-directed topological space** , one that comes equipped with a notion of orientation of its $k$-cells for $0 \leq k \leq r$, but was impartial on direction above that dimension.

If such a definition exists, it may need to use [[filtered topological space]]s instead of bare topological spaces.

Even in the absence of a homotopy-theoretic definition of $r$-directed space in this sense, from the perspective of [[homotopy theory]] one might take the standpoint of the [[homotopy hypothesis]] and _define_ a (nice) $r$-directed space to be an [[(∞,n)-category|(∞,r)-category]], just as it makes good sense and is nowadays common practice in [[algebraic topology]] to _define_ a nice [[topological space]] to be an [[∞-groupoid]].

See [[(n,r)-category]] for more on that.

## Variants

### $d$-Spaces

A **directed topological space** or **d-space** is pair $(X, d X)$ consisting of a [[topological space]] $X$ and a subset $d X \subset C(I,X)$ of continuous maps from the interval $I = [0,1]$ into $X$ -- called _directed paths_ or d-paths -- satisfying the following conditions:

1. (constant paths) every constant map $I\to X$ is directed,

2. (reparametrisation) $dX$ is closed under composition with increasing maps $I\to I$,

3. (concatenation) $dX$ is closed under path-concatenation: if the d-paths $a, b$ are consecutive in $X$ $(a(1) = b(0))$, then their ordinary concatenation $a+b$ is also a d-path

$$(a+b)(t) = a(2t),\,\text{if}\, 0\le t\le \frac{1}{2},$$

$$(a+b)(t) = b(2t-1),\,\text{if}\, \frac{1}{2}\le t\le 1.$$

A **morphism of directed topological spaces**  $f : (X, d X)\to (Y , d Y)$ is a morphism of topological spaces $f: X \to Y$ which preserves directed paths in that for every $\gamma: I \to X$ in $d X$ the path $f_* \gamma : I \stackrel{\gamma}{\to} X \stackrel{f}{\to} Y$ is in $d Y$.

\begin{example} Basic examples of $d$-spaces include:

* The _standard directed interval_ is $I_d = ([0,1], d I)$ with $d I$ the set of all _monotonic_ continuous maps $[0,1] \to [0,1]$ is a d-space

* Any [[pospace]] $X$ gives rise to a d-space by taking the directed paths to be, well, directed paths, i.e. continuous order-preserving maps from $I_d$ to $X$.

Many other example can be found in the references.
\end{example}

### Streams

A different definition comes from [[Sanjeevi Krishnan]], _A Convenient Category of Locally Preordered Spaces_, Applied Categorical Structures, 2009, vol. 17, no 5, p. 445-466 ([arxiv](https://arxiv.org/abs/0709.3646)):

__Definition__ A **stream** is a tuple $X, \leq_{-}$, where $\leq_{-}$ assigns to each open subset $U \subset X$ a preorder $\leq_U$, such that:
$$\leq_{\bigcup_i U_i} = \bigvee_i \leq_{U_i}$$
Here, $U_i, i \in I$ is a collection of open sets, and $\bigvee$ is pointwise or of relations.

_Remarks_

* Morally, a point in a stream is less or equal to another point if it is less or equal in any open set. The relation $\leq_X$ for the whole space does not hold much information.
* In the typical example of the clockwise oriented circle, each point is less or equal to every other one in the relation $\leq_{S^1}$; but for a contractible subset $U \subset S^1$, a point $x$ is less or equal than $y$ if $y$ can be reached from $x$ in a monotonous clockwise path.
* The defining equation of a stream can be thought of as a (co)sheaf condition, and there is indeed a cosheafification result.
* Every d-space gives rise to a stream.
* The category of streams has good properties. In particular, there is an further notion of compactly flowing streams_ extending the notion of compactly generated Hausdorff spaces, and indeed the forgetful functor creates limits and colimits.
* The category of compactly flowing streams is Cartesian closed.

### Framed spaces {#framed_spaces}

#### Motivation

Another way to endow spaces with directions is via [[frame of a vector space|framings]], i.e. choices of a "basis of the vector space of tangential directions at each point". Somewhat abstracting this idea, frames may also be thought about in terms of [[projection#in_linear_algebra|projections]], as the following remark motivates.

\begin{rmk} Let $V$ be an $n$-dimensional [[vector space]] with an [[inner product]] $g$. The following structures on $V$ are equivalent.

* An orthonormal frame of $V$, i.e. an ordered sequence of vectors $v_i$, $1 \leq i \leq n$, such that  $g(v_i,v_j)=\delta_{ij}$.
* A sequence of surjections $V_i \to V_{i-1}$, $1 \leq i \leq n$, where $V_i$ is an oriented $i$-dimensional vector space (and $V_n = V$).

The two structures are related by setting $v_i$ to be the unit vector spanning $\ker(V_i \to V_{i-1})$ such that $V_{i-1} \oplus v_i$ recovers the orientation of $V_i$ (note that all $V_i$ canonically embed in $V$ as the orthogonal complement of the kernel of the composite map $V \to V_i$).

In the absence of inner products, one cannot speak of orthonormal frames any longer. However, sequences of projections can still be defined, and may be regarded as playing the role of "metric-free orthonormal" frames. (A vaguely analogous line of thinking is that a [[Morse function]] $M \to \mathbb{R}$ provides useful "direction" information on $M$, e.g. for the construction of handlebodies, that is ultimately independent from any chosen metric on $M$.) Moreover, this approach offers room for generalization by varying the length of the projection sequence and the vector space dimensions.
\end{rmk}

\begin{example} The standard orthonormal frame of $n$-dimensional euclidean space $\mathbb{R}^n$ consists for the ordered sequence of vectors $e_1 = (1,0,...,0)$, $e_2 = (0,1,0,...,0)$, ..., $e_n = (0,...,0,1)$. By the previous remark, this orthonormal frame is equivalently described by the sequence of projections $\pi_i : \mathbb{R}^i = \mathbb{R}^{i-1} \times \mathbb{R} \to \mathbb{R}^{i-1}$ (each $\mathbb{R}^i$ being endowed with standard orientation).
\end{example}

When forgetting basepoints, then the previous remark and example equally apply to [[affine space|affine spaces]], now endowing each point in the space with a basis of frames. Using affine standard framed $\mathbb{R}^n$ (or rather [[compact]] [[contractible]] patches of it, that interact nicely with the projections, see below) as our "local models" for framed spaces one may define global framed spaces and their maps.

#### Definition

\begin{terminology} Inductively in $n \in \mathbb{N}$, an *$n$-framed patch* $U \subset \mathbb{R}^n$ is a non-empty subspace of $\mathbb{R}^n$ with the property that its projection $\pi_n(U)$ is an $(n-1)$-framed patch, such that $\pi_n : U \to \pi_n(U)$ has fibers $\pi^{-1}_n(x)$ of the form $[\gamma_-(u),\gamma_+(u)]$ for two continuous sections $\gamma_\pm : \pi_n(U) \to \pi_n(U) \times \mathbb{R}$. Given two $n$-framed patches $U$ and $V$, a *(partial) $n$-framed patch map* $F : U \to V$ is a (partial) continuous map which descends along $\pi_n$ to a (partial) $(n-1)$-framed patch map $F_{n-1} : \pi_n (U) \to \pi_n(V)$ such that mappings of fibers $F : \pi^{-1}_n(x) \to \pi^{-1}_n(F_{n-1}(x))$ are monotone.
\end{terminology}

Note that $n$-framed patches are [[compact]] and [[contractible]] spaces.

\begin{example} The standard example of an $n$-framed patch is the closed $n$-cube $\mathbf{I}^n = [-1,1]^n \subset \mathbb{R}^n$. However, in general $n$-framed patches need not be "$n$-dimensional": for instance, the $0$th slice $\{0\} \times \mathbf{I}^{n-1}$ of the $n$-cube is itself an $n$-framed patch, and so are the "$k$-directed intervals" $\mathbf{I}_k := \{0\}^{k-1} \times \mathbf{I} \times \{0\}^{n-k}$.
\end{example}

\begin{defn} Let $X$ be a topological space. Fix $n \in \mathbb{N}$.

* An *$n$-framed chart* $(U,\gamma)$ in $X$ is an embedding $\gamma : U \hookrightarrow \mathbb{R}^n$ of a subspace $U \subset X$ whose image $\im(\gamma)$ is an $n$-framed patch.

* Two $n$-framed charts $(U,\gamma)$, $(V,\rho)$ in $X$ are *compatible* if  $\rho \circ \gamma^{-1}$ is a partial map of $n$-framed patches.

An **$n$-framed space** is a space $X$ endowed with an $n$-framing structure $\mathcal{A}$, which is an "atlas" of compatible $n$-framed charts $\{(U_i,\gamma_i)\}$ such that $U_i$ are a [[locally finite]] cover of $X$.
\end{defn}

\begin{rmk} The condition for covers to be locally finite is convenient as it describes the situation of locally finite [[cell complex|cell complexes]]. However, the condition could be replaced, and the definition generalized, in several ways. The above version of the definition should be considered a first approximation to a potentially more general notion of framed spaces.
\end{rmk}

Maps of framed spaces could be defined along the following lines.

\begin{defn} Given spaces with $n$-framing structure $(X,\mathcal{A}) \to (Y,\mathcal{B})$ then a **framed map** $F : X \to Y$ is a map such that for any charts $(U,\gamma) \in \mathcal{A}$ and $(V,\rho) \in \mathcal{B}$, $F$ yields a partial map $\rho \circ F \circ \gamma^{-1} : \mathrm{im}(\gamma) \to \mathrm{im}(\rho)$ of $n$-framed patches.
\end{defn}

Framed spaces and framed maps are, in a sense, "very rigid" variants of directed spaces. Nonetheless, they are interesting to study as they turn out to have a rich [[framed combinatorial topology|combinatorial theory associated to them]]. This combinatorial counterpart is particular useful when translating between the topology of directed spaces and the combinatorics of higher categories: an example of this is the definition of [[manifold diagram|manifold diagrams]] in the language of framed spaces.

\begin{example} Realizations of [[geometric computad|geometric computads]] (resp. their functors) have the structure of framed spaces (resp. of framed maps), see [Dorn23](#Dorn23).
\end{example}

#### Comparison to $d$-spaces

\begin{rmk} One may want to define directed paths in an $n$-framed space as maps from "the framed interval" into that space: but, in fact, there are now $n$ different framed intervals, corresponding to the $n$ possible directions of $n$-framings. These are precisely modelled by the $n$-patches $\mathbf{I}_k$ mentioned in a previous example. Note, there is a framed bijection $\mathbf{I}_k \to \mathbf{I}_j$ if and only if $k \leq j$ (in particular, directions aren't interchangeable at all: analogously, note that in an [[n-category]] a $(n-k)$-morphisms can be (possibly degenerate) $(n-j)$-morphisms only if $k \leq j$). Given a framed space $(X,\mathcal{A})$, and defining $d_k X = \mathrm{Map}_{\mathrm{fr}}(\mathbf{I}_k, (X,\mathcal{A}))$, we thus obtain a filtration of directed paths:
\[
  d_n X \subset d_{n-1} X \subset ... \subset d_1 X \subset \Map(\mathbf{I},X)
\]
instead of a single path subset $dX \subset \Map(\mathbf{I},X)$, as required in the definition of $d$-spaces.
\end{rmk}

## Remarks

* If we can equip directed spaces with an internal hom, then a directed space with at least one directed path should be a strictly directed object in the category of directed spaces, with respect to the standard directed interval as the [[interval object]], while an ordinary topological space regarded as a directed space should be an [[undirected object]].

## Related concepts

* [[directed object]]

* [[directed homotopy theory]]

* [[directed homotopy type theory]]

## References

The above definition of $d$-spaces is from

* [[Marco Grandis]], _Directed homotopy theory, I. The fundamental category_ ([arXiv](http://arxiv.org/abs/math.AT/0111048))

This has now developed into a book

* [[Marco Grandis]], _[[Directed Algebraic Topology]], Models of non-reversible worlds_ , Cambridge University Press, 2009.

A discussion of reparameterization of directed paths in directed topological spaces is in

* Ulrich Fahrenberg and Martin Raussen, _Reparametrizations of Continuous Paths_ ([arXiv](http://arxiv.org/abs/0706.3560), ([blog](http://golem.ph.utexas.edu/category/2006/09/fahrenberg_and_raussen_on_cont.html)))

Further references are given in [[directed homotopy theory]].

* Emmanuel Haucourt, [_Streams, d-Spaces and Their Fundamental Categories_, Electronic Notes in Theoretical Computer Science, Volume 283, 15 June 2012, Pages 111-151](https://doi.org/10.1016/j.entcs.2012.05.008)

Some 'local combinatorial' aspects of framed spaces are discussed in:

* {#DornDouglas21} [[Christoph Dorn]] and [[Christopher Douglas]], _Framed combinatorial topology_, 2021 ([arXiv](https://arxiv.org/abs/2112.14700))

A global (higher-categorical) perspective on directed spaces is taken in:

* {#Dorn23} [[Christoph Dorn]], _Nine short stories about geometric higher categories_, 2023 ([pdf](https://cxdorn.github.io/assets/pdfs/nine-stories.pdf))

[[!redirects directed space]]
[[!redirects directed spaces]]

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[[!redirects d-spaces]]


directed type theory

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#### Directed Type Theory
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#### 2-category theory
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Any kind of [[type theory]] with [[categorical semantics]] in [[2-categories]] or [[(n,r)-category|(n,2)-categories]], such as [[(infinity,2)-categories|$(\infty,2)$-categories]]. See:

* [[2-type theory]]

* [[directed homotopy type theory]]

category: disambiguation

[[!redirects directed type theories]]


directed univalence axiom

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#### Directed type theory
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#### Universes
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\tableofcontents

## Idea

In [[dependent type theory]] with a notion of [[hom type]] (i.e. modelling [[(infinity,1)-category|$\infty$-categories]] as envisioned in [[directed homotopy type theory]]) and a notion of [[type universe]], the *directed univalence axiom* for a type universe $U$ states that given two $U$-small types $A \colon U$ and $B \colon U$ there is an [[equivalence of types]] between the hom-type $\mathrm{hom}_U(A, B)$ and the [[function type]] $A \to B$:

$$
  \mathrm{dua}_U(A, B)
  \;\colon\;
  \mathrm{hom}_U(A, B)
    \,\simeq\,
  (A \to B)
  \,.
$$

The definition also makes sense in analytic models of [[(infinity,1)-categories]] such as those constructed from [[simplicial sets]].

## Definition

### In simplicial homotopy type theory

In [[simplicial homotopy type theory]], let $A$ be a [[Segal type]] and let $x:A \vdash B(x)$ be a [[covariant type family]]. The covariant type family $x:A \vdash B(x)$ satisfies the **directed univalence axiom** if given two elements $x \colon A$ and $y \colon A$ there is an [[equivalence of types]] between the [[hom type]] $\mathrm{hom}_A(x, y)$ and the [[function type]] $B(x) \to B(y)$:

$$
  \mathrm{dua}_A(x, y)
  \;\colon\;
  \mathrm{hom}_A(x, y)
    \,\simeq\,
  (B(x) \to B(y))
  \,.
$$

## See also

* [[univalence axiom]]

* [[directed homotopy type theory]]

## References

* [[Hoang Kim Nguyen]], *Directed univalence in simplicial sets*, talk in *[Homotopy Type Theory Electronic Seminar Talks](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html)* (March 2023) [[video](https://www.youtube.com/watch?v=GgcJqzGvq80), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Nguyen-2023-03-09-HoTTEST.pdf)]

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

* César Bardomiano Martínez, *Limits and colimits of synthetic $\infty$-categories* $[$[arXiv:2202.12386](https://arxiv.org/abs/2202.12386)$]$

[[!redirects directed univalence]]


directed-complete poset > history

> see at *[[DCPO]]*


direction

> This entry is about the concept in [[order theory]]. For the concept in [[analytic geometry]] see at [[direction of a vector]].

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#### $(0,1)$-Category theory
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# Contents
* table of contents
{: toc}

## Idea and terminology

A **direction** on a set $S$ is a [[preorder]] on $S$ in which any finite subset has an [[upper bound]].  A **directed set** is a [[set]] equipped with a direction.  A **directed poset** is a directed set whose preorder is a [[partial order]] (so _directed proset_ might be a better term than _directed set_). Note that many authors by directed set mean directed poset.

Directedness is an asymmetric condition.  Sometimes a direction as defined here is called **upward-directed**; a preorder whose [[opposite category|opposite]] is upward-directed is called **downward-directed** or **codirected**.

A [[subset]] of a poset (or proset) which is a directed set (when regarded as a poset in its own right) is a **directed subset**, and dually.  More generally, a [[diagram]] in a [[category]] whose domain is a directed set (when regarded as a [[thin category]]) is called a **directed diagram**, and dually.

## Definitions

### Finitely directed set

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

A **finitely upward-directed set** (which is the default notion) is a [[set]] with a [[preorder]] $\leq$ such that:

1. there exists an element (so the set is [[inhabited set|inhabited]]); and

2. given elements $x, y$, there exists an element $z$ such that $x \leq z$ and $y \leq z$.

=--

It follows that, given any [[finite set]] $x_1, \dots, x_n$ of elements, there exists an element $z$ such that $x_i \leq z$ for all $i$.

(For [[constructive mathematics|constructive]] purposes, one should interpret 'finite set' above as a [[finite set|finitely indexed set]], as shown.)

Equivalently, this says:

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

A directed set is a [[proset]] which is a [[filtered category]]: a **filtered [[(0,1)-category]].

=--

In analogy with the definition there, we can provide an [[biased definition|unbiased definition]] that is also commonly used in the literature:

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

A directed set is a [[proset]] in which every finite subset has an upper bound.

=--

### Higher cardinality

More generally, if $\kappa$ is a [[cardinal number]], then a **$\kappa$-directed set** is equipped with a preorder $\leq$ such that, given any index set $A$ with $|A| \lt \kappa$ and function $i \mapsto x_i$ from $A$, there exists an element $z$ such that $x_i \leq z$ for all $i$. Then a finitely directed set is the same as an $\aleph_0$-directed set.  An **infinitely directed set** allows any index set $A$ whatsoever, but this reduces to the statement that the proset has a [[top]] element.

## Remarks

Directions on the [[real line]] are quite interesting; there\'s a textbook (probably _LIMITS: A New Approach to Real Analysis_) that does [[infinitesimal calculus|ordinary calculus]] rigorously from scratch using directions, and there\'s a paper (which I can\'t find now) generalising interval arithmetic to arithmetic on directions.

As a partially ordered set is a special kind of [[category]], so a (finitely) directed set is such a category in which all (finite) diagrams admit a [[cocone]].  If the category actually has finite coproducts (equivalently, all finite colimits), then it has all [[joins]] and so is a join-[[semilattice]].  (In particular, every join-semilattice is a directed set.)

Directed sets are heavily used in point-set [[topology]] and [[analysis]], where they serve as index sets for [[nets]] (aka Moore--Smith sequences).  In this application, it is important that a direction need not be a partial order, since a net need not preserve the preorder in any way but by default still preserves [[equality]].  (But in principle, one could force a directed set to be a poset by allowing a net to be a [[multi-valued function]]; this has practical consequences for the meaning of [[sequence]] in the absence of [[countable choice]].)

[[join|Joins]] over directed index sets are [[directed joins]]; [[colimits]] over directed index sets are [[directed colimits]].  These play an important role in the theory of [[locally presentable category|locally presentable]] and [[accessible category|accessible]] categories; see also [[filtered category]].

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direction of a vector

> This entry is about the concepts of direction of a vector and of a line in [[linear algebra]]/[[analytic geometry]]. For the concept in [[order theory]] see at _[[direction]]_.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[vector space|vector]]/[[affine space|affine]] [[geometry]] the [[parallel]] [[vectors]] with the same [[orientation]] form an [[equivalence class]].  The same holds for oriented lines and even higher dimensional oriented subspaces.  The equivalence class of a vector (or other object) is called its *direction*.  (In the case of higher dimension we may say "$n$-direction"; see also [[bivector]].)  If the vector space is [[normed space|normed]], then each such equivalence class has a canonical [[unit vector]] representative, called a *direction vector*.

If we neglect the orientation we get a bigger equivalence class, called the *unoriented direction*.  Unoriented directions have a representing unit vector only up to sign.

## Definition

### Oriented direction

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

For $V$ a [[real vector space]], or more generally a vector space over an [[ordered field]], the _direction_ of a non-zero vector $v \in V$ is its [[equivalence class]] under the multiplication with [[positive number|positive]] scalars.

If $V$ is a [[normed vector space]] (such as an [[inner product space]] with the induced [[norm]] $v\mapsto \sqrt{\langle v, v\rangle}$), the _direction_ of $v$ is canonically represented by the unit norm vector obtained by multiplying $v$ with the inverse of its [[norm]]

$$
  \tfrac{1}{\vert v\vert} v \;\in \; S(V) \subset V
  \,,
$$

which may be regarded as an element of the [[unit sphere]] $S(V)$ of $V$.

=--

(e.g. [MathWorld](#MathWorld))

Equivalently:

Recall that a nonzero vector $v$ is _parallel_ to a [[linear subspace]] $W$ of a vector space $V$ if $v$ has a representative of the form $\vec{A B}$ (i.e. $v = B - A$) with $A,B\in W$.  When $W$ is a [[line]], we say that $v$ furthermore has the same [[orientation]] as $W$ if $\vec{A B}$ points (in the sense of def. \ref{DirectionOfAVector}) in the positive direction along $W$.

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

The _direction_ of a non-zero vector $v \in V$ is the [[equivalence class]] of oriented lines parallel to $v$ and having the same orientation.

=--

### Unoriented direction

Similarly, the _unoriented direction_ of a non-zero vector in any vector space is its [[equivalence class]] under multiplication by non-zero elements of the [[ground field]], hence the element it represents in the corresponding [[projective space]] $P V$.  Unlike oriented direction, this makes sense over an arbitrary field.

The unit direction vector can be attached not only to a (direction of) a vector but also to a direction of lines per se (see Wikipedia, _[Direction vector](https://en.wikipedia.org/wiki/Direction_vector)_).

J. Lelong-Ferrand defines the direction of an affine subspace of an affine space as its orbit under the action of the translation group. This agrees with the parallelness in Choquet where two affine subspaces of the same dimension are parallel if their translation group of vectors (as a subgroup of the group of vectors of translations of the entire space) is the same.

## Examples
 {#Examples}

+-- {: .num_example #WaveFronts}
###### Example
**([[wave front set|wave fronts]])

In [[microlocal analysis]] the _[[wave front set]]_ of a [[distribution]] records the direction (def. \ref{DirectionOfAVector}) of all those [[wave vectors]] along which, locally, the [[Fourier transform of distributions|Fourier transform of the distribution]] is not a [[rapidly decaying function]].

=--

## Related concepts

* [[slope of a line]], [[tangent]]

* [[norm]], [[length]], [[absolute value]]

* [[wave vector]], [[orientation]], [[bivector]]

## References

* {#MathWorld} MathWorld, _[Unit vector](http://mathworld.wolfram.com/UnitVector.html)_

* Gustave Choquet, _Geometry in a modern setting_, Hermann 1969

* K. Horvatić, _Linear algebra_ (in Croatian), 2004

* Jacquelline Lelong-Ferrand, _Les fondements de la géométrie_, Presses Universitaires de France 1985.

[[!redirects directions of vectors]]
[[!redirects direction vector]]
[[!redirects direction of a line]]


directional derivative

# Directional derivatives
* table of contents
{: toc}

## Idea

A directional derivative, or Gâteaux derivative, is a [[partial derivative]] of a [[function]] on a [[manifold]] along the direction given by a [[tangent vector]].

## Definitions

Let $F$ and $G$ be [[locally convex topological vector space]]s, $U \subseteq F$ an [[open subspace]] and $P\colon U \to G$ a [[continuous map]]. The __derivative__ of $P$ at the point $f \in U$ in the __direction__ $h \in F$ is the [[convergence|limit]]
$$
D P_f h = \lim_{t \to 0} \frac{1}{t} (P(f + t h) - P(f))
.$$

If the limit exists for every $f \in U$ and every $h \in F$ then one can define a map
$$
D P\colon U \times F \to G
.$$
If the limit exists and $D P$ is [[continuous map|continuous]] (jointly in both variables), we say that $P$ is __[[continuously differentiable map|continuously differentiable]]__ or $C^1$.

A simple but nontrivial example is the operator
$$
P\colon C^{\infty}[a, b] \to C^{\infty}[a, b]
$$
given by
$$
P(f) \coloneqq f f'
$$
with the derivative
$$
D P(f) h = f' h + f h'
.$$

In the context of a [[Fréchet space]], it may be that the directional derivative in every direction exists but the [[Fréchet derivative]] does not; however the existence of Fréchet derivative implies the existence of directional derivatives in all directions.

The notion of directional derivatives extends to [[smooth manifolds]] (including infinite-dimensional ones based on [[Fréchet spaces]]) using local coordinates; the differentiability does not depend on the choice of a local chart. In this case we have (if everything is defined)
$$
D P\colon T(U) \to G
,$$
where $T(U)$ is the [[tangent space]] of $U$ (an open subspace of $T(F)$.

## References

* Wikipedia (English): [Gâteaux derivative](http://en.wikipedia.org/wiki/G%C3%A2teaux_derivative)

* [[eom]]: [Gâteaux derivative](http://eom.springer.de/g/g043360.htm), [Gâteaux variation](http://eom.springer.de/G/g043390.htm), [René Gâteaux](http://en.wikipedia.org/wiki/Ren%C3%A9_G%C3%A2teaux)

* R. Gâteaux, _Sur les fonctionnelles continues et les fonctionnelles analytiques_, C.R. Acad. Sci. Paris Sér. I Math. __157__ (1913)  pp. 325–327; _Fonctions d'une infinités des variables indépendantes_,  Bull. Soc. Math. France __47__  (1919) 70–96, [numdam](http://www.numdam.org/item?id=BSMF_1919__47__70_1); _Sur diverses questions du calcul fonctionnel_, Bulletin de la Société Mathématique de France tome __50__ (1922) 1–37, [numdam](http://www.numdam.org/item?id=BSMF_1922__50__1_0)

An analogue of the directional derivative and Faa di Bruno formula in the Goodwillie calculus are in

* [[Kristine Bauer]], Brenda Johnson, Christina Osborne, [[Emily Riehl]], Amelia Tebbe, _Directional derivatives and higher order chain rules for abelian functor calculus_ ([arxiv/1610.01930](https://arxiv.org/abs/1610.01930))

[[!redirects directional derivative]]
[[!redirects directional derivatives]]
[[!redirects Gâteaux derivative]]
[[!redirects Gâteaux derivatives]]
[[!redirects Gateaux derivative]]
[[!redirects Gateaux derivatives]]
[[!redirects Gâteaux differentiability]]


Dirichlet character

## Definition

e.g. [Goldfeld-Hundley 11 (2.1.5) (2.1.6)](#GoldfeldHundley11)

## Properties

### Relation to automorphic forms on the idele group

Dirichlet characters are essentially [[automorphic forms]] for the [[group of ideles]] in disguise. ([Goldfeld-Hundley 11, theorem 2.1.9](#GoldfeldHundley11))

## Related concepts

* [[Dirichlet theta function]]

* [[Dirichlet L-function]]

## References

* {#GoldfeldHundley11} [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of _Automorphic representations and L-functions for the general linear group_, Cambridge Studies in Advanced Mathematics 129, 2011 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf))

[[!redirects Dirichlet characters]]


Dirichlet L-function

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Theta functions
+--{: .hide}
[[!include theta functions - contents]]
=--
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Dirichlet L-functions_ $L_\chi$ are a kind of [[L-function]] induced by [[Dirichlet characters]] $\chi$

$$
  L_\chi(s) \coloneqq \underoverset{n=1}{\infty}{\sum} \chi(n)n^{-s}
  \,.
$$

(e.g. [Goldfeld-Hundley 11 (2.2.1)](#GoldfeldHundley11), [Continuations](#Continuations)).

The completion of this by a [[Gamma function]] factor (and a power of $\pi$ and of the [[conductor]]) is the [[Mellin transform]] of the [[Dirichlet theta function]] (e.g. [Continuations, p. 8](#Continuations)).

By [[Artin reciprocity]] Dirichlet L-function are equal to suitable [[Artin L-functions]] induced by 1-dimensional [[Galois representations]].

In terms of the [[adelic integral]] representation of L-functions via [[Iwasawa-Tate theory]], given a [[Dirichlet character]] $\chi$ then the corresponding Dirichlet L-functions are simply the [[adelic integrals]] (e.g. [Garrett 11, section 2.2](#Garrett11))

$$
  s \mapsto \int_{\mathbb{I}} \chi(x) \;f(x)\; {\vert x\vert}^s
$$

for suitable Schwartz functions $f$ on the [[idele group]].

Hence Dirichlet L-functions are [[Mellin transforms]] of suitable [[theta functions]] ("theta kernels"), see. e.g. ([Stopple, p. 3](#Stopple)).

Where [[Dirichlet characters]] (see there) are essentially [[automorphic forms]] for the [[idele group]] (hence for $n = 1$) the generalization to automorphic forms for $n \geq 1$ is the concept of _[[automorphic L-function]]_.

The differential geometric analog of a Dirichlet L-function is the _[[eta function]]_ (see there for more) of a differential operator.

## Related concepts

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

## References

* E. Kowalski, section 1.3 of  _Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures_ ([pdf](http://www.math.ethz.ch/~kowalski/lectures.pdf))

* {#Continuations} section 3 of _Continuations and functional equations_ ([pdf](http://people.reed.edu/~jerry/361/lectures/feqn.pdf))

* Wikipedia, _[Dirichlet L-function](http://en.wikipedia.org/wiki/Dirichlet_L-function)_

* {#GoldfeldHundley11} [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of _Automorphic representations and L-functions for the general linear group_, Cambridge Studies in Advanced Mathematics 129, 2011 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf))

* {#Garrett11} [[Paul Garrett]], section 1.6 _Iwasawa-Tate on ζ-functions and L-functions_
([pdf](http://www-users.math.umn.edu/~garrett/m/mfms/notes_c/Iwasawa-Tate.pdf))

* {#Stopple95} [[Jeffrey Stopple]], _Theta and $L$-function splittings_, Acta Arithmetica LXXII.2 (1995) ([pdf](http://matwbn.icm.edu.pl/ksiazki/aa/aa72/aa7221.pdf))

Analogs of Dirichlet L-functions in [[chromatic homotopy theory]] are constructed in

* Ningchuan Zhang, _Analogs of Dirichlet L-functions in chromatic homotopy theory_, ([arXiv:1910.14582](https://arxiv.org/abs/1910.14582))

[[!redirects Dirichlet L-functions]]

[[!redirects Dirichlet L-series]]


Dirichlet theta function

#Contents#
* table of contents
{:toc}

## Idea

Given a [[Dirichlet character]]

$$
  \chi \;\colon\; (\mathbb{Z}/N\mathbb{Z})^\times \longrightarrow \mathbb{C}^\times
$$

it is called _even_ if $\chi(-1) = 1$ and _odd_ if $\chi(-1) = -1$.

The _[[theta function]]_ associated with $\chi$ is the function on $\mathbb{R}_+$ defined by

$$
  \theta_\chi(\tau)
  \coloneqq
  \left\{
    \array{
      \underset{n \in \mathbb{Z}}{\sum} \chi(n) \exp(-\pi n^2 t/N) & if\; \chi\; even
      \\
      \underset{n \in \mathbb{Z}}{\sum} n \chi(n) \exp(-\pi n^2 t/N) & if\; \chi\; odd
    }
  \right.
$$

where $N$ is the [[conductor]] of $\chi$.

## Properties

### Relation to Jacobi theta function

For trivial $\chi$ the Dirichlet theta reproduces the [[Jacobi theta function]].

### Mellin transform and relation to Dirichlet L-function

The [[Mellin transform]] of this is the completed [[Dirichlet L-function]] associated with $\chi$ (e.g. [Continuations, p. 8](#Continuations))

## Related concepts

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

## References

* section 3 of _Continuations and functional equations_ ([pdf](http://people.reed.edu/~jerry/361/lectures/feqn.pdf))


Dirichlet's unit theorem

[[!redirects Dirichlet unit theorem]]

## Related concepts

* [[regulator of a number field]]

## References

* Wikipedia, _[Dirichlet's unit theorem](http://en.wikipedia.org/wiki/Dirichlet's_unit_theorem)_


Dirk Bouwmeester

* [institute page](https://www.universiteitleiden.nl/en/staffmembers/dirk-bouwmeester)

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Dirk_Bouwmeester)

## Selected writings

On [[quantum teleportation]] realized in [[experiment]]:

* [[Dirk Bouwmeester]], Jian-Wei Pan, Klaus Mattle, Manfred Eibl, [[Harald Weinfurter]], [[Anton Zeilinger]], *Experimental quantum teleportation*, Nature **390** 575–579 (1997) [[doi:10.1038/37539](https://doi.org/10.1038/37539)]

On the [[quantum physics]] of [[quantum information]]:

* [[Dirk Bouwmeester]], [[Artur Ekert]], [[Anton Zeilinger]] (eds.), *The Physics of Quantum Information -- Quantum Cryptography, Quantum Teleportation, Quantum Computation*, Springer (2020) [[doi:10.1007/978-3-662-04209-0](https://doi.org/10.1007/978-3-662-04209-0)]

category: people


Dirk Ferus

* [website](http://www.math.tu-berlin.de/~ferus/)

## Selected writings

On [[analysis]]/[[differential geometry]]:

* {#Ferus07} [[Dirk Ferus]], _Analysis III_ (2007) ([pdf](http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf), [[Ferus_AnalysisIII.pdf:file]])

## Related $n$Lab entries

* [[open ball]]

* [[good open cover]]

category: people


Dirk Hofmann

Dirk Hofmann is Associate Professor at the Department of Mathematics of the University of Aveiro.

* [webpage](http://sweet.ua.pt/dirk/)

## Thesis

* D. Hofmann, _Natürliche Dualitäten und das verallgemeinerte Stone-Weierstraß-Theorem_, Ph.D. Thesis, University of Bremen, 1999.

##Selected papers

* Dirk Hofmann and Pedro Nora,
_Enriched Stone-type dualities_, Advances in Mathematics 330 (2018) 307–360. ([ArXiv version](https://arxiv.org/abs/1605.00081)).

* Dirk Hofmann, Renato Neves and Pedro Nora,
_Generating the algebraic theory of C(X): the case of partially ordered compact spaces_,
[Theory and Applications of Categories, 33 (12) (2018), 276–295](http://www.tac.mta.ca/tac/volumes/33/12/33-12abs.html).

category:people


Dirk Kreimer

* [website](http://math.bu.edu/people/dkreimer/)

## related $n$Lab entries

* [[renormalization]]

category: people


Dirk Pattinson

Dirk Pattinson is a mathematician _cum_ computer scientist at the A.N.U.

* [webpage](http://users.cecs.anu.edu.au/~dpattinson/)

Pattison's research interests include:

* [[modal logics]], in partiular coalgebraic semantics, automated reasoning

* [[domain theory]], in particular computation over continuous data

* [[formal logic]] and [[category theory]] in general

## Selected writings

On the theory of [[coalgebras over an endofunctor|coalgebras]]:

* {#Pattinson03} [[Dirk Pattinson]], *An Introduction to the Theory of Coalgebras* (2003) [[pdf](https://nasslli.sitehost.iu.edu/2003/datas/DirkPattinson.pdf), [[Pattinson-Coalgebras.pdf:file]]]

## Related entries

* [[Mealy machine]]

category:people


Dirk Rischke

* [research group page](https://www.goethe-university-frankfurt.de/66535811/Workgroup_Dirk_Rischke)

* [GoogleScholar page](https://scholar.google.com/citations?user=ON462YgAAAAJ&hl=en)

## Selected writings

On [[path integrals]] in [[quantum mechanics]]:

* {#Rischke21} [[Dirk Rischke]], *Path Integrals in Quantum Mechanics*, lecture notes (2021) [[pdf](https://itp.uni-frankfurt.de/~drischke/Script_Path_Integrals_GU2021.pdf)]

category: people


Dirk van Dalen

* [Wikipedia entry](https://en.wikipedia.org/wiki/Dirk_van_Dalen)

* [institute page](https://www.uu.nl/staff/DvanDalen)

## Selected writings

On [[constructive mathematics]] (including [[intuitionistic logic]] and the [[BHK interpretation]], etc.):

* [[Anne Sjerp Troelstra]], [[Dirk van Dalen]], *Constructivism in Mathematics -- An introduction*, Vol 1, Studies in Logic and the Foundations of Mathematics **121**, North Holland (1988) [[ISBN:9780444702661](https://www.elsevier.com/books/constructivism-in-mathematics-vol-1/troelstra/978-0-444-70266-1)]

category: people


DisCoPy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###

#### Proof assistants and formalization projects
+--{: .hide}
[[!include proof assistants and formalization projects -- list]]
=--

#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--

=--
=--

# Contents
* table of contents
{:toc}

## Overview

_DisCoPy_ is a toolbox for computing with [[string diagram|string diagrams]], [[monoidal categories]] and [[monoidal functor|strong monoidal functors]] in [[Python]]. It is available on [GitHub](#GitHubDisCoPy).

The name stands for _Distributional Compositional Python_, indeed the package was first intended as an implementation of the [[DisCoCat]] models of [[natural language]].

## String diagrams

The core data structure is [`Diagram`](https://discopy.readthedocs.io/en/main/_autosummary/discopy.monoidal.Diagram.html), an implementation of [[string diagram|string diagrams]] in the [[free construction|free]] [[premonoidal category]].

Fix a monoidal [[signature (in logic) |signature]] $\Sigma = \Sigma_1 \xrightarrow{\text{dom}, \text{cod}} \Sigma_0^\star$ for a set of _boxes_ $\Sigma_1$, a set of objects $\Sigma_0$ and its [[free monoid]] $\Sigma_0^\star$, we write $f \colon s \to t$ for $f \in \Sigma_1$ whenever $\text{dom}(f) = s$ and $\text{cod}(f) = t$.
Let $L(\Sigma) = \Sigma_0^\star \times \Sigma_1 \times \Sigma_0^\star$, a _layer_ is a triple $l = (u, f \colon s \to t, v) \in L(\Sigma)$, where we define $\text{dom}(l) = u s v$ and $\text{cod}(l) = u t v$.
It is depicted, in [[string diagram]]-notation, as:

\begin{tikzpicture}
\node () at (0.25, 1) {$u$};
\node () at (1.25, 1) {$s$};
\node () at (2.25, 1) {$v$};
\node () at (1.25, 0.0) {$t$};
\draw [out=-90, in=90] (0, 1) to (0, 0);
\draw [out=-90, in=90] (1, 1) to (1, 0.75);
\draw [out=-90, in=90] (2, 1) to (2, 0);
\draw [out=-90, in=90] (1.0, 0.25) to (1.0, 0);
\draw (0.75, 0.25) -- (1.25, 0.25) -- (1.25, 0.75) -- (0.75, 0.75) -- (0.75, 0.25);
\node () at (1.0, 0.5) {$f$};
\end{tikzpicture}

The set of _diagrams_ $D(\Sigma)$ is the [[hom-set|set of arrows]] in the [[free category]] [[generators and relations|generated]] by the layers $L(\Sigma)$, considered as a [[directed graph]]. Concretely, $d$ is given by a length $\text{len}(d) = n \in \mathbb{N}$, a domain $\text{dom}(d) \in \Sigma_0^\star$ and a list of layers $\layers(d) = (d_1, \dots, d_n) \in L(\Sigma)^n$ such that $\text{dom}(d) = \text{dom}(d_0)$ and $\text{cod}(d_i) = \text{dom}(d_{i + 1})$ for each $i \leq n$.
The codomain of a diagram is the codomain of its last layer $\cod(d) = \cod(d_n)$.

Note that in order to define a diagram uniquely, its layers have more data than necessary. It is enough to specify a list $\boxes(d) \in \Sigma_1^n$ of generators together with a list $\offsets(d) \in \N^n$: the number of wires on the left of each box.
This is the definition used in [Delpeuch and Vicary (2018)](#DelpeuchVicary18), [[Globular]], [[homotopy.io]] as well as DisCoPy.

We can define identity, composition and tensor of diagrams in terms of this combinatorial encoding:

* The identity diagram $\text{id}(x)$ for $x \in \Sigma_0^\star$ is defined by $\text{len}(\text{id}(x)) = 0$ and $\text{dom}(\text{id}(x)) = x$. It is depicted as vertical wires labeled with $x$.
* Given two diagrams $d_0, d_1 \in D(\Sigma)$ with $\text{cod}(d_0) = \text{dom}(d_1)$, their composition $d_1 \circ d_0 \in D(\Sigma)$ is defined by vertical concatenation, i.e. $\text{layers}(d_1 \circ d_0) = \text{layers}(d_0) \text{layers}(d_1)$. In Python code, this is written `d0 >> d1` or `d1 << d0`.
* Given a diagram $d \in D(\Sigma)$ and a type $x \in \Sigma_0^\star$, the [[whiskering|whiskerings]] $x \otimes d$ and $d \otimes x$ are defined by concatenating $d$ with $x$-labeled wires on its left and right. First, define whiskering for layers $l = (u, f, v) \in L(\Sigma)$ as $x \otimes l = (x u, f, v)$ and $l \otimes x = (u, f, v x)$. Then, define whiskering for diagrams as $\text{layers}(x \otimes d) = (x \otimes d_1, \dots, x \otimes d_n)$ and $\text{layers}(d \otimes x) = (d_1 \otimes x, \dots, d_n \otimes x)$.
* Given two diagrams $d_0, d_1 \in D(\Sigma)$, their tensor is defined as $d_0 \otimes d_1 = \text{cod}(d_0) \otimes d_1 \circ d_0 \otimes \text{dom}(d_1)$. In Python, this is written `d0 @ d1 == d0 @ Id(d1.dom) >> Id(d0.cod) @ d1`.

Diagrams in the free [[monoidal category]] can be seen as premonoidal diagrams quotiented by the _[[interchanger]]_, i.e. the naturality equation for the [[tensor product|monoidal product]]: $f \otimes 1_c \circ 1_b \otimes g = 1_a \otimes g \circ f \otimes 1_d$ for all $f : a \to b$ and $g : c \to d$. Diagrams in the free [[rigid monoidal category]] can be seen as monoidal diagrams with generators for cups and caps quotiented by the [[triangle identities|snake equations]]. In practice, these quotients are computed by calling `Diagram.normal_form`.

Diagrams are equipped with an involution `Diagram.dagger` that implements [[dagger categories]]. Furthermore, formal sums of diagrams can also be composed and tensored, making all the diagams [[enriched category|enriched]] in [[monoid|monoids]].
Although their normal form is not yet implemented, DisCoPy also implements the syntax for diagrams in:

* [[braided monoidal categories]]
* [[balanced monoidal categories]]
* [[symmetric monoidal categories]]
* [[traced monoidal categories]]
* [[closed monoidal categories]]
* [[pivotal categories]]
* [[ribbon categories]]
* [[compact categories]]

DisCoPy also implements diagrams in [[hypergraph categories]] in two different ways:

* as diagrams with [[Frobenius algebras]] on every object,
* as cospans of labeled hypergraphs, following [Bonchi et al. (2022)](#BonchiEtAl22).

## Monoidal functors

Diagrams can be evaluated in concrete monoidal categories by applying [`Functor`](https://discopy.readthedocs.io/en/main/_autosummary/discopy.monoidal.Functor.html) instances to them.

Let $\mathcal{F} : \text{MonSig} \to \text{MonCat}$ be the functor that sends a monoidal signature to its free monoidal category, and its right adjoint $\mathcal{U} : \text{MonCat} \to \text{MonSig}$ the [[forgetful functor]]. From the [[free-forgetful adjunction]], we get that monoidal functor $F : \mathcal{F}(\Sigma) \to S$ from a free monoidal category generated by a signature $\Sigma$ into some monoidal category $S$ is uniquely defined by a morphism of signature $F : \Sigma \to \mathcal{U}(S)$. That is, in order to define a functor $F : \mathcal{F}(\Sigma) \to S$, it's enough to define two mappings $F_0 : \Sigma_0 \to \text{Ob}(S)$ and $F_1 : \Sigma_1 \to \text{Ar}(S)$ which commute with $\text{dom}$ and $\text{cod}$.

Concretely, a DisCoPy functor is defined by `F = Functor(ob, ar)` where `ob` and `ar` are mappings from generating object to types and from box to diagram respectively. These mappings can be either given as Python dictionaries if they have finite domains or as Python functions otherwise. Once its image for each generating object and each box is given, `F` can be applied to arbitrary diagrams. An optional argument `cod` can be used to define functors with arbitrary categories as codomain. The domain of the functor is given implicitly by the domain of the `ob` and `ar` maps.
By default DisCoPy functors are endofunctors from the free category to itself, i.e. they send diagrams to diagrams.

Currently, DisCoPy implements [[strong monoidal functors]] into:

* the category of [[Python]] functions,

* the category of [[matrices]] in arbitrary [[rig|semirings]], including [[finite dimensional vector spaces]] and finite [[relation|relations]], with either the [[direct sum]] or the [[tensor product]] as monoidal product,

* the category of [[quantum channels]], i.e. finite dimensional Hilbert spaces and completely positive trace preserving (CPTP) maps,

* the category of [[quantum circuits]], i.e. with generating objects for classical and quantum digits and generating arrows for quantum gates and measurements.

Thus, DisCoPy can be used as an implementation of [[categorical quantum mechanics]]: it can encode any quantum process as a diagram and evaluate it either on quantum hardware or as a classical simulation using monoidal functors.

## Natural language processing

Diagrams can be used to encode derivations in a [[formal grammar]]. Indeed, the first use case of DisCoPy was to apply monoidal diagrams to these grammar diagrams in order to compute their [[semantics]]. Currently, DisCoPy can handle the following frameworks:

* [[formal grammar]] where derivations are diagrams in a [[monoidal category]] and in particular [[context-free grammar]] where derivations are syntax trees,
* [[categorial grammar]] where derivations are diagrams in a [[closed monoidal category]],
* [[pregroup grammar]] and [[dependency grammar]] where derivations are diagrams in a [[rigid monoidal category]].

## References

* Giovanni de Felice, [[Alexis Toumi]], [[Bob Coecke]]. _DisCoPy: Monoidal Categories in Python_. Applied Category Theory, 2020. ([arXiv:2005.02975](https://arxiv.org/abs/2005.02975 ))

* Konstantinos Meichanetzidis, Stefano Gogioso, Giovanni De Felice, Nicolò Chiappori, [[Alexis Toumi]], [[Bob Coecke]]. _Quantum Natural Language Processing on Near-Term Quantum Computers_. Quantum physics & logic, 2020. ([arXiv:2005.04147](https://arxiv.org/abs/2005.04147 ))

* {#DelpeuchVicary18} Delpeuch, Antonin, and [[Jamie Vicary]], _Normalization for planar string diagrams and a quadratic equivalence algorithm._ arXiv preprint [arXiv:1804.07832](https://arxiv.org/abs/1804.07832) (2018).

* {#BonchiEtAl22} Filippo Bonchi, Fabio Gadducci, Aleks Kissinger, Pawel Sobocinski, Fabio Zanasi, _String Diagram Rewrite Theory I: Rewriting with Frobenius Structure._ [DOI:10.1145/3502719](https://doi.org/10.1145/3502719) (2022).

* {#DisCoPyDoc} _DisCoPy's documentation._ [discopy.readthedocs.io](https://docs.discopy.org/)

* {#GitHubDisCoPy} _DisCoPy's repository._ [github.com/discopy/discopy](https://github.com/discopy/discopy)


discovery > history


discrete and concrete objects - contents

**[[discrete objects]]**

* [[discrete space]], [[discrete ∞-groupoid]]

**[[codiscrete objects]]** ([[chaos]])

* [[codiscrete space]], [[codiscrete groupoid]]

**[[concrete objects]]**

* [[concrete sheaf]], [[concrete (∞,1)-sheaf]]

**[[co-concrete object]]**

* [[ionad]]

**[[reduced object]]**

**[[coreduced object]]**

**typical contexts**

* [[local topos]], [[connected topos]], [[cohesive topos]],

* [[local (∞,1)-topos]], [[∞-connected (∞,1)-topos]], [[cohesive (∞,1)-topos]]

[Edit this sidebar](/nlab/edit/discrete+and+concrete+objects+-+contents)


discrete and indiscrete topology

[[!redirects discrete and codiscrete topology]]

+-- {: .rightHandSide}
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###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The [[forgetful functor]] $\Gamma : Top \to Set$ from [[Top]] to [[Set]] that sends any [[topological space]] to its underlying [[set]] has a [[left adjoint]] $Disc : Set \to Top$ and a [[right adjoint]] $Codisc : Set \to Top$.
$$
  (Disc \dashv \Gamma \dashv Codisc) :
  Top
    \stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{Codisc}{\leftarrow}}}
  Set
  \,.
$$

For $S \in Set$

* $Disc(S)$ is the [[topological space]] on $S$ in which every subset is an [[open set]],

  this is called the **discrete topology** on $S$, it is the [[finer topology|finest topology]] on $S$; $Disc(S)$ is called a [[discrete space]];

* $Codisc(S)$ is the topological space on $S$ whose only open sets are the [[empty set]] and $S$ itself, which is called the **indiscrete topology** on $S$ (rarely also __antidiscrete topology__ or __codiscrete topology__ or __trivial topology__ or __chaotic topology__ ([[SGA4|SGA4-1, 1.1.4]])), it is the [[coarser topology|coarsest topology]] on $S$; $Codisc(S)$ is called a __indiscrete space__ (rarely also __antidiscrete space__, even more rarely __codiscrete space__).

For an axiomatization of this situation see _[[codiscrete object]]_.

## Properties

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

Let $S$ be a [[set]] and let $(X,\tau)$ be a [[topological space]]. Then

1. every [[continuous function]] $(X,\tau) \longrightarrow Disc(S)$ is [[locally constant function|locally constant]];

1. every [[function]] (of sets) $X \longrightarrow CoDisc(S)$ is [[continuous function|continuous]].

=--

### The left adjoint of the discrete space functor

The functor $Disc$ does not [[preserved limit|preserve]] [[infinite products]]
because the infinite [[product topological space]] of discrete spaces may be nondiscrete.
Thus, $Disc$ does not have a [[left adjoint functor]].

However, if we restrict the [[codomain]] of $Disc$
to [[locally connected spaces]], then the [[left adjoint functor]]
of $Disc$ does exist and it computes the set of [[connected components]] of a given [[locally connected space]], i.e., is the $\pi_0$ functor.

This is discussed at _[locally connected spaces -- cohesion over sets](locally+connected+topological+space#CohesionOverSets)_
and [[cosheaf of connected components]].

## References
 {#References}

For [[Grothendieck topologies]], the terminology "chaotic" is due to

* [[SGA4|SGA4-1, 1.1.4]] (1972)

reviewed, e.g., in:

* [[The Stacks Project]], _[Example 7.6.6](https://stacks.math.columbia.edu/tag/07GE)_

Conceptualization of the terminology via [[right adjoints]] to [[forgetful functors]] (see also at _[[chaos]]_) is due to

* {#Lawvere84} [[William Lawvere]], _Functorial remarks on the general concept of chaos_ IMA preprint #87, 1984 ([pdf](http://www.ima.umn.edu/sites/default/files/87s.pdf))

and via footnote 1 (page 3) in:

* [[William Lawvere]],  _Categories of spaces may not be generalized spaces, as exemplified by directed graphs_, preprint, State University of New York at Buffalo, (1986) Reprints in Theory and Applications of Categories, No. 9, 2005, pp. 1–7 ([tac:tr9](http://www.tac.mta.ca/tac/reprints/articles/9/tr9abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/9/tr9.pdf)).

[[!redirects discrete topology]]

[[!redirects discrete topologies]]
[[!redirects discrete topological structure]]
[[!redirects discrete topological structures]]
[[!redirects discrete topological space]]
[[!redirects discrete topological spaces]]

[[!redirects codiscrete topology]]
[[!redirects codiscrete topologies]]
[[!redirects codiscrete topological structure]]
[[!redirects codiscrete topological structures]]

[[!redirects codiscrete topological space]]
[[!redirects codiscrete topological spaces]]

[[!redirects co-discrete topological space]]
[[!redirects co-discrete topological spaces]]

[[!redirects chaotic topology]]
[[!redirects chaotic topologies]]
[[!redirects chaotic topological space]]
[[!redirects chaotic topological spaces]]

[[!redirects indiscrete topology]]
[[!redirects indiscrete topologies]]
[[!redirects antidiscrete topology]]
[[!redirects antidiscrete topologies]]
[[!redirects indiscrete space]]
[[!redirects indiscrete spaces]]
[[!redirects antidiscrete space]]
[[!redirects antidiscrete spaces]]


discrete category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

The term _discrete category_ (or _discrete groupoid_) is used in two different ways:

1. A category is _discrete_ if it is [[equivalence of categories|equivalent]] to (or, sometimes, isomorphic to) a category whose only morphisms are identities; i.e., if [[small]], to a [[set#AsCategory|set]].  In other words, this is a [[truncated object|0-truncated]] object of [[Cat]]; see also [[discrete object in a 2-category]].  This meaning is discussed below in [Categorical meaning](#CategoricalMeaning).

1. A category with additional "topological" or "cohesive" structure is _discrete_ if that topological structure consists only of [[discrete spaces]].  This meaning is discussed below in [Topological meaning](#TopologicalMeaning).

See also the related discussion at [[discrete space]].

## Categorical meaning
{#CategoricalMeaning}

A [[category]] is **discrete** if it is both a [[groupoid]] and a [[preorder]].  That is, every morphism should be invertible, any two [[parallel morphisms]] should be equal.  The idea is that in a discrete category, no two distinct (nonisomorphic) objects are connectable by any path (morphism), and an object connects to itself only through its [[identity morphism]].

Often one also assumes that a discrete category is [[skeletal category|skeletal]]; a category is both discrete and skeletal if and only if it contains only identity morphisms.  However, this definition is [[evil]], because it states that objects (the source and target of the identity morphism in question) are equal; it is cleaner to separate the discreteness from the skeletality.

A ([[small category|small]]) discrete category may be identified with its [[set]] of isomorphism classes.  Conversely, given a [[collection]] $S$ of objects, the __discrete category over $S$__ is the category with $S$ as its collection of objects and only identity morphisms.

## Topological meaning
{#TopologicalMeaning}

If $C$ is a category [[enriched category|enriched]] or [[internal category|internal]] to [[topological space|topological spaces]], then there is another completely different meaning of **discrete**: that the _topology_ on the arrows (and the objects, in the internal case) is the
[[discrete topology]]. In this sense a _discrete category_ is an [[internal category]] in [[discrete spaces]] sitting inside a more general kind of spaces.  This usage can be applied more generally for internal or enriched categories in any context with a notion of [[discrete space]].

This is especially confusing if one extends the use of "discrete category over $S$" to the case of internal categories, when $S$ is an object of some ambient category.  With this usage, if $S$ is a topological space, then the "discrete internal category over $S$" in [[Top]] will not be discrete in the topological sense: it still remembers the topology on that space.

Because of potential confusion, in cases when topology or cohesion is present it is perhaps better to use *0-truncated* for categorically-discrete categories and groupoids, reserving *discrete* for those which are topologically-discrete.

## Adjoints

The discrete category construction extends to a [[fully faithful functor]] $Set \to Cat$. It has a [[left adjoint]] given by taking connected components and a [[right adjoint]] given by the underlying set of objects functor (which itself has a right adjoint given by the [[indiscrete category]] construction).

## Related concepts

* [[discrete space]]

* [[discrete group]]

* **discrete groupoid**, [[discrete object in a 2-category]]

* [[discrete ∞-groupoid]]

[[!redirects discrete category]]
[[!redirects discrete categories]]

[[!redirects discrete groupoid]]
[[!redirects discrete groupoids]]


Discrete causal spaces

* tic
{:toc}

This is an experiment in collaboration. I want to write an article about discrete causal spaces. Please help. Coauthors welcome! - [[Eric Forgy|Eric]]

# Goals #

1. Construct a generalization of [[ericforgy:Discrete differential geometry on causal graphs|Discrete differential geometry on causal graphs]] to $n$-dimensional discrete causal spaces that are _locally_ like $n$-diamonds (see the paper for a description of $n$-diamonds until a page is created).

1. The continuum limit of a discrete causal space should be a [[directed space]], or more specifically, a [[smooth Lorentzian space]].

1. Etc

# Abstract #

...

# Introduction #

We need a good category-friendly definition of an $n$-diamond. Here's a first stab that is incomplete, but hopefully gets the ball rolling.

**(Tentative/Incomplete) Definition:** An $n$-diamond is a minimal [[causet]].

**(Tentative/Incomplete) Definition:** An $n$-diamond complex is a [[directed n-graph]] in which each node has exactly $n$ edges directed into it and exactly $n$ nodes directed away from it _and each ??? has fillers_.

**Example:** $\mathbb{Z}^n$ with the its obvious $r$-diamond faces, $0\le r\le n$ is an $n$-diamond complex.

# References #

* Marco Grandis, _Directed homotopy theory, I. The fundamental category_ ([arXiv](http://arxiv.org/abs/math.AT/0111048))
* Tim Porter: _Enriched categories and models for spaces of evolving states_, Theoretical Computer Science, 405, (2008), pp. 88--100.
* Tim Porter, _Enriched categories and models for spaces of
dipaths. A discussion document and overview of some techniques_ ([pdf](http://drops.dagstuhl.de/opus/volltexte/2007/898/pdf/06341.PorterTimothy.Paper.898.pdf))
* Fajstrup, Rosicky, _A convenient category for directed homotopy_ ([arXiv](http://arxiv.org/abs/0708.3937))
* Dan Christensen, and Louis Crane, _Causal sites as quantum geometry_ ([arXiv](http://arxiv.org/abs/gr-qc/0410104))
+--{.query}
[[Eric]]: They use the word "diamond". I hope that catches on.
=--
* Louis Crane, _Model categories and quantum gravity_ ([arXiv](http://arxiv.org/abs/0810.4492))
* Bombelli, Henson, Sorkin, _Discreteness without symmetry breaking: a theorem_ ([arXiv](http://arxiv.org/abs/gr-qc/0605006))
* Dagstuhl Seminar Proceedings 04351, _Spatial Representation: Discrete vs. Continuous Computational Models_, ([web](http://drops.dagstuhl.de/portals/index.php?semnr=04351))

See also:

* [[directed homotopy theory]].

***

# Discussion #

_Note.  Topics will be separated by lines and each topic is presented in reverse chronological order._

[[JA]]: Just starting reading this and don't know much about diamonds yet, but I have been working on logic-based approaches to discrete dynamics — that I sometimes think of as differential geometry over GF(2) — for quite a while now.  You might take a gander at my [Magnum Opiate](http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0) and we could see if it fits in here somewhere.

***

[[Eric]]: Motivated by some emails from [[Tim Porter|Tim]], I think we could use pages on [[causal site]] and [[dg-quiver]]. I am particularly interested in seeing a definition of [[dg-quiver]].

***

[[Eric]]: I wonder if $n$-diamonds should be more closely related to [[n-fold categories]] rather than [[posets]]?

***

(From nCafe: Authorship)

_[[Urs Schreiber|Urs]] says_: I think Eric wants a [[partial order|poset]] of sorts.

It seems that a good formalization of "[[smooth Lorentzian space|smooth Lorentzian space(time)]]" is something like: a [[partial order|poset]] [[internal category|internal]] to a category of measure spaces.

There are some immediate possibilities about graph versions of this statement that come to mind. For one, a discrete "Lorentzian spacetime" should be a poset such that all causal subsets are finite set.

A [[causet|causal subset]] in a poset $X$ is what John in his latest entry calls an "interval", namely for two objects $x,y$ in the poset the [[under category|under-over category]]

$$x\darr X\darr y$$

of all objects in the past of $x$ and in the future of $y$.

_[[Tim Porter|Tim]]_: In a poset one can define a diamond as being an order convex subset possibly with extra properties: U is order convex if $x$, $z$ are in $U$ with $x \lt z$, and $y$ lies between $x$ and $z$ ($x\lt y \lt z$).

One possible interpretation of what Eric wants is that one specifies a collection of order convex subsets with inclusion on them making a directed set or possibly a lattice. (sort of needing to capture that diamonds should intersect in diamonds  (or not at all). Something like this is used by John Roberts in some of his work I seem to remember.

 Reminder ('cos I thought this had been mentioned in the café) it may also pay to look at gr-qc/0410104 for some ideas but I never convinced myself that that was really what was needed.

category: drafts

[[!redirects Discrete Causal Spaces]]


discrete event system

#Contents#
* table of contents
{:toc}

## Idea

A __discrete event system__ is a mathematical model of a system (such as a computational device) that communicates with its environment by atomic actions (called _events_). For example, a user of the system pressing a button could send a signal to a controller. These events are assumed to be discrete in the sense that they occur instantaneously (as opposed to over a period of time).

## Related concepts

* [[tropical semiring]]

## Literature

See also:

* Wikipedia, *[Discrete event dynamic system](https://en.wikipedia.org/wiki/Discrete_event_dynamic_system)*

[[!redirects discrete event systems]]