John von Neumann

John von Neumann (German: Johann von Neumann; Hungarian: Neumann János Lajos) was a Hungarian mathematician.

He defended his PhD thesis in 1925 advised by [[Lipót Fejér]],
with the title _Az általános halmazelmélet axiomatikus felépítése_
(Axiomatic construction of general set theory),
which introduced the [[NBG set theory]],
as well as [[classes]] and von Neumann [[ordinals]].

* [Wikipedia entry](http://en.wikipedia.org/wiki/John_von_Neumann)

## Selected writings

On [[set theory]]:

* [[J. v. Neumann]], _Eine Axiomatisierung der Mengenlehre_, Journal für die reine und angewandte Mathematik 154 (1925), 219–240.  [doi](http://dx.doi.org/10.1515/crll.1925.154.219).

PhD thesis (journal version):

* [[J. v. Neumann]], _Die Axiomatisierung der Mengenlehre_, Mathematische Zeitschrift 27 (1928), 669–752.  [doi](https://doi.org/10.1007/BF01171122).

On [[Hermitian operators]] and introducing the formal definition of [[Hilbert spaces]]:

* {#vonNeumann30} [[John von Neumann]], §I in: *Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren*, Math. Ann. **102** (1930) 49–131 [[doi:10.1007/BF01782338](https://doi.org/10.1007/BF01782338)]

On the mathematical foundations of [[quantum mechanics]] based on [[Hilbert spaces]] [[space of quantum states|of quantum states]]:

* {#HilbertvonNeumannNordheim} [[David Hilbert]], [[John von Neumann]], [[Lothar W. Nordheim]], *Über die Grundlagen der Quantenmechanik*,  Math. Ann. **98** (1928) 1–30 [[doi:10.1007/BF01451579](https://doi.org/10.1007/BF01451579)]

Proving the [[Stone-von Neumann theorem]]:

* [[John von Neumann]], *Die Eindeutigkeit der Schrödingerschen Operatoren*, Mathematische Annalen **104**  (1931) 570–578 [[doi:10.1007/BF01457956](https://doi.org/10.1007/BF01457956)]

* [[John von Neumann]], _Über Einen Satz Von Herrn M. H. Stone_, Annals of Mathematics, Second Series **33** 3 (1932) 567-573 [[doi:10.2307/1968535](https://doi.org/10.2307/1968535), [jstor:1968535](https://www.jstor.org/stable/1968535)]

* [[John von Neumann]]:

  *Mathematische Grundlagen der Quantenmechanik* (German) (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]

  *Mathematical Foundations of Quantum Mechanics* Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), [Wikipedia entry](https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics)]

Co-introducing [[Jordan algebras]] as [[algebras of quantum observables]]:

* {#JordanvNeumannWigner34} [[Pascual Jordan]], [[John von Neumann]], [[Eugene Wigner]], *On an algebraic generalization of the quantum mechanical formalism*, Ann. Math. **35** (1934) 29-64 [[jstor:1968117](https://www.jstor.org/stable/1968117), [doi:10.1007/978-3-662-02781-3_21](https://doi.org/10.1007/978-3-662-02781-3_21)]

On [[quantum logic]]:

* {#BirkhoffvonNeumann36} [[Garrett Birkhoff]], [[John von Neumann]]: *The logic of quantum mechanics*, Annals of Mathematics **37** 823-843 (1936) [[doi:10.2307/1968621](https://doi.org/10.2307/1968621), [pdf](http://www.fulviofrisone.com/attachments/article/451/the%20logic%20of%20quantum%20mechanics%201936.pdf)]

Introducing the theory of what came to be known as *[[von Neumann algebra factors]]*:

The classification of factors into types I, II, III and the construction of examples not of type I:

* [[Francis J. Murray]], [[John von Neumann]], *On Rings of Operators*, Annals of Mathematics, Second Series, **37** 1 (1936) 116-229 [[doi:10.2307/1968693](https://doi.org/10.2307/1968693), [jstor:1968693](https://www.jstor.org/stable/1968693)]

Discussion of [[traces]] on these factors:

* [[Francis J. Murray]], [[John von Neumann]], *On rings of operators. II*, Trans. Amer. Math. Soc. **41** (1937) 208-248  [[doi:10.2307/1989620](https://doi.org/10.2307/1989620), [jstor:1989620](https://www.jstor.org/stable/1989620)]

On [[isomorphism]] of factors and proof of a single isomorphism class of approximately finite type $II_1$ factors:

* [[Francis J. Murray]], [[John von Neumann]], *On rings of operators. IV*, Annals of Mathematics, Second Series, **44** 4 (1943) 716-808 [[doi:10.2307/1969107](https://doi.org/10.2307/1969107), [jstor:1969107](https://www.jstor.org/stable/1969107)]

On decomposing [[von Neumann algebras]] as a [[direct integral]] of factors:

* {#vNeumann49} [[John von Neumann]], _On rings of operators, reduction theory_, Annals of Mathematics Second Series, Vol. 50, No. 2 (1949) [[jstor:1969463]( http://www.jstor.org/stable/1969463)]

Recollection of the history which made von Neumann turn to discussion of these "factors", motivated from considerations in the foundations of [[quantum mechanics]] and [[quantum logic]]:

* {#Rédei96} [[Miklos Rédei]], *Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead)*, Studies in History and Philosophy of Modern Physics **27** 4 (1996) 493-510 [doi:10.1016/S1355-2198(96)00017-2]

category: people

[[!redirects von Neumann]]
[[!redirects J. v. Neumann]]


John W. Gray

[[!redirects John Gray]]

John Walker Gray ([1932? -- 2017](https://www.news-gazette.com/obituaries/john-gray/article_fb2520a5-d6c4-5087-9ac1-ed6613823b9a.html)) was a category theorist known for work in, among other areas, [[2-category theory]].

* [website](http://www.math.uiuc.edu/~gray/)

## Selected writings

On (co-)[[fibered categories]]:

* [[John W. Gray]], *Fibred and Cofibred Categories*, in: [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

On [[category theory]] formulated "[[formal category theory|formally]]" in the [[2-category]] [[Cat]] of all [[categories]]:

* [[John Gray]], _[[Adjointness for 2-Categories|Formal category theory: adjointness for 2-categories]]_, Lecture Notes in Mathematics **391**  Springer (1974) [[doi:10.1007/BFb0061280](https://doi.org/10.1007/BFb0061280)]

Early discussion of *[[adjunctions of two variables]]* under the name "[[THC-situation]]":

* [[John Gray]], *Closed categories, lax limits and homotopy limits*. J. Pure Appl. Algebra **19** (1980) 127 - 158 [doi:10.1016/0022-4049(80)90098-5]

## Related entries

* [[Gray tensor product]]

* [[Gray category]]

category:people


John W. Negele

* [institute page](https://physics.mit.edu/faculty/john-negele/)

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

## Selected writings

On [[solid state physics]] (with an emphasis on [[path integral]] methods):

* [[John W. Negele]], [[Henri Orland]], *Quantum Many-Particle Systems*, Westview Press (1988, 1998) [[doi:10.1201/9780429497926](https://doi.org/10.1201/9780429497926)]

category: people


John Ward

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

## Selected writings

Co-introducing the theory of the [[electroweak field]]:

* [[Abdus Salam]], [[John Clive Ward]], *Weak and electromagnetic interactions*, Nuovo Cimento. **11** 4 (1959) 568–577 [[doi:10.1007/BF02726525](https://doi.org/10.1007/BF02726525)]

## Related $n$Lab entries

* [[electroweak field]]

* [[Ward identity]]

category: people

[[!redirects John Clive Ward]]


John Watrous

* [personal page](https://cs.uwaterloo.ca/~watrous/)

## Selected writings

On [[quantum information theory]]:

* [[John Watrous]], *The Theory of Quantum Information*, Cambridge University Press (2018) [[doi:10.1017/9781316848142](https://doi.org/10.1017/9781316848142), [webpage](https://cs.uwaterloo.ca/~watrous/TQI/), [pdf](https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf)]

category: people


John Wheater

* [institute page](https://www2.physics.ox.ac.uk/contacts/people/wheater)

## Selected writings

On [[quantum field theory]] and [[string theory]]:

* [[Mikhail Shifman]], [[Arkady Vainshtein]], [[John Wheater]] (eds.),

  _[[From Fields to Strings -- Circumnavigating Theoretical Physics]]_

  Ian Kogan Memorial Collection (In 3 Volumes)

  World Scientific 2005

  [doi:10.1142/5621](https://doi.org/10.1142/5621)

category: people


John Wheeler

* [Wikipedia entry](http://en.wikipedia.org/wiki/John_Archibald_Wheeler)

* [[Kip Thorne]], _John Archibald Wheeler: A Biographical Memoir_ ([arXiv:1901.06623](https://arxiv.org/abs/1901.06623))

* list of video recordings of Wheeler: [YT](https://www.youtube.com/watch?v=8dQUaO_mwmI&list=PLVV0r6CmEsFzVlqiUh95Q881umWUPjQbB)

## Selected writings

On [[geometrodynamics]]:

* [[Charles W. Misner]], [[John A. Wheeler]], *Classical physics as geometry*, Annals of Physics **2** 6 (1957) 525-603 \[doi:10.1016/0003-4916(57)90049-0\]

On the [[classical field theory]] of [[Einstein gravity]] ("[[general relativity]]"):

* [[Charles Misner]], [[Kip Thorne]], [[John Wheeler]]: *[[Gravitation]]* W. H. Freeman, San Francisco (1973) [ISBN:9780716703440, 9780691177793, [inspire:95654](https://inspirehep.net/literature/95654), Wikipedia entry]

## Related entries

* [[gravity]]

* [[geometrodynamics]]

* [[Wheeler superspace]]

## Quotes
 {#Quotes}

### On commanding equations to fly
 {#CommandingEquationsToFly}

[[Christopher Fuchs]] recounts (in an email from Dec. 1997, reproduced on [p. 292](https://arxiv.org/pdf/1405.2390.pdf#page=350) of "*[My Struggles with the Block Universe](Christopher+Fuchs#FuchsSchlosshauerStacey14)"*, 2014):

> a little anecdote about John Wheeler that I heard from [[John Preskill]] a few days ago. In 1972 he had Wheeler for his freshman classical mechanics course at Princeton.

> One day Wheeler had each student write all the equations of physics s/he knew on a single sheet of paper. He gathered the papers up and placed them all side-by-side on the stage at the front of the classroom. Finally, he looked out at the students and said,

>> “These pages likely contain all the fundamental equations we know of physics. They encapsulate all that’s known of the world.”

> Then he looked at the papers and said,

>> “Now fly!”

> Nothing happened. He looked out at the audience, then at the papers, raised his hands high, and commanded,

>> “Fly!”

> Everyone was silent, thinking this guy had gone off his rocker. Wheeler said,

>> “You see, these equations can’t fly. But our universe flies. We’re still missing the single, simple ingredient that makes it all fly.”

On this point, in the video documentary

* Timothy Ferris, *Creation of the Universe* (1985) [[imdb:tt0204224](https://www.imdb.com/title/tt0204224)]

Wheeler appears (cf. [[Blake Stacey]], [Jan 2016](https://www.sunclipse.org/?m=201601)) saying:

> There’s nothing deader than an equation. You write that down in a square on a tile floor. And on another tile on the floor you write down another equation, which you think might be a better description of the Universe. And you keep on writing down equations hoping to get a better and better equation for what the Universe is and does.

> And then, when you’ve worked your way out to the end of the room and have to step out, you wave your wand and tell the equations to fly.

> And not one of them will put on wings and fly.

> Yet the Universe flies!

> It has a life to it that no equation has, and that life to it is a life with which we are also tied up.

[[John Preskill]] re-enacting the episode at the conference banquet of *[It from QBit 2023](https://events.perimeterinstitute.ca/event/43/)*:

\linebreak

[[!redirects John Archibald Wheeler]]
[[!redirects John A. Wheeler]]

category: people


John Whitehead > history

> see _[[John Henry Constantine Whitehead]]_


Johnson-Wilson spectrum

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#Contents#
* table of contents
{:toc}

## Idea

A [[localization of spectra|localization]] of the truncated [[Brown-Peterson spectrum]]. A form of [[Morava E-theory]].

## Properties

### Orientation

The [[orientation in generalized cohomology|orientation]] of [[Morava E-theory]] is discussed in ([Sati-Kriz 04](#SatiKriz04), [Buhné 11](#Buhne11)). It is essentially given by the vanishing of the seventh [[integral Stiefel-Whitney class]] $W_7$.

Notice that this is in analogy to how [[orientation in generalized cohomology|orientation]] in [[complex K-theory]] is given by the vanishing third [[integral Stiefel-Whitney class]] $W_3$ ([[spin^c-structure]]).

Precisely:

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

Let $X$ be a [[connected topological space|connected]] [[closed manifold]] of [[dimension]] 10 with [[spin structure]]. This is [[orientation in generalized cohomology|generalized oriented]] in second [[integral Morava K-theory]] $\tilde K(2)$ (for $p = 2$) precisely if its seventh [[integral Stiefel-Whitney class]] vanishes, $W_7(X) = 0$.

=--

This is ([Buhné 11, prop. 8.1.13](#Buhne11)), following ([Sati-Kriz 04](#SatiKriz04)).

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

Let $X$ be a [[connected topological space|connected]] [[closed manifold]] of [[dimension]] 10 with [[spin structure]]. This is [[orientation in generalized cohomology|generalized oriented]] in second Johnson-Wilons cohomology theory ([[Morava E-theory]]) $E(2)$ (for $p = 2$) precisely if its seventh [[integral Stiefel-Whitney class]] vanishes, $W_7(X) = 0$.

=--

This is ([Buhné 11, cor. 8.1.14](#Buhne11)), following ([Sati-Kriz 04](#SatiKriz04)).

## References

The original article:

* [[David Copeland Johnson]], [[W. Stephen Wilson]], _BP operations and Morava's extraordinary K-theories_, Math. Z. 144 (1): 55−75 (1975) ([pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/jw-morava.pdf))

Discussion of [[orientation in generalized cohomology|orientation]] in Johnson-Wilson theory and relation to the [[Diaconescu-Moore-Witten anomaly]] is in

* {#SatiKriz04} [[Igor Kriz]], [[Hisham Sati]], _M-theory, type IIA superstrings, and elliptic cohomology_, Adv. Theor.
Math. Phys. 8 (2004), no. 2, 345–394 ([arXiv:hep-th/0404013](http://arxiv.org/abs/hep-th/0404013))

* {#Buhne11} [[Lukas Buhné]], _[[Properties of Integral Morava K-Theory and the Asserted Application to the Diaconescu-Moore-Witten Anomaly]]_, Diploma thesis, Hamburg (2011)

[[!redirects Johnson-Wilson spectra]]

[[!redirects Johnson-Wilson cohomology theory]]
[[!redirects Johnson-Wilson cohomology theories]]

[[!redirects Johnson-Wilson theory]]
[[!redirects Johnson-Wilson theories]]


Johnstone's topological topos

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## Idea

The category $Top$ of topological spaces notoriously lacks some desirable features. There are therefore quite a few attempts to define "big" categories that contain (all) interesting topological spaces but are better behaved categorically. The **topological topos** was proposed in this vein by [[Peter Johnstone]] in the seventies. As it is conceptually based on sequential convergence it plays an important role in constructive mathematics.

## Definition

Let [[Top]] be the [[category]] of [[topological space]]s, and let $\Sigma$ be the [[full subcategory]] whose only two [[object]]s are a one-point space and $\mathbb{N}^+$, the [[one-point compactification]] of the [[discrete space]] of [[natural number]]s. Let $J$ be the [[canonical Grothendieck topology]] on $\Sigma$.

Johnstone's **topological topos** (specifically, the one described in the [eponymous paper](#Johnstone79)) is the [[topos]] of canonical [[sheaves]] $Sh_J(\Sigma)$ on $\Sigma$. The functor

$$Top \to Set^{\Sigma^{op}}: X \mapsto Top(-, X)$$

is [[faithful functor|faithful]] and factors through $Sh_J(\Sigma)$, and its restriction to the category of [[sequential space|sequential spaces]] is full.

The category of [[subsequential spaces]] can also be found as a full subcategory of this topos (in fact, it consists of the separated objects for the [[double negation|double negation topology]]  $\neg\neg$).  A general object of the topos can be thought of as being like a subsequential space, but such that a given sequence can converge to a given point in "more than one way."

Importantly, following an idea by Joyal, the topological topos $\mathcal{E} \coloneqq Sh_J(\Sigma)$ allows one to represent the [[geometric realization]] functor $sSet \to \mathcal{E}$ as the inverse image of a [[geometric morphism]] from $\mathcal{E}$ to the topos of [[simplicial set|simplicial sets]]. As Johnstone points out, this approach fails for the [[big topos]] on $Top$ and also for Lawvere's topos for [[continuum mechanics]].

## Internal logic

In the [[internal logic]] of the topological topos, the [[limited principle of omniscience]] (LPO) for the natural numbers is false (and thus [[excluded middle]] also fails).  Specifically, the space $\mathbb{N}^+$ can be constructed internally as the subset of $2^{\mathbb{N}}$ consisting of non-decreasing sequences, and the discrete space $\mathbb{N}+1$ includes into it as the set of sequences that are 1 at exactly one place or 0 everywhere.  Thus, the non-isomorphy of this map exhibits the failure of LPO.

Since the topological topos is generated by $\mathbb{N}^+$, it is in some sense a "canonical" topos where LPO fails.  Moreover, if we [[localization|localize]] at the inclusion $\mathbb{N}+1 \to \mathbb{N}^+$, we get the subtopos of [[codiscrete objects]], which is equivalent to $Set$; so "forcing LPO to hold gives [[classical logic]].

On the other hand, the *lesser* limited principle of omniscience (LLPO) *does* hold in the topological topos.  This is easiest to see from its "analytic" version, which is equivalent to saying that every real number $x$ is either $\ge 0$ or $\le 0$.  Equivalently, this means that $(-\infty,0] + [0,\infty) \to \mathbb{R}$ is surjective.   This follows from Johnstone's Proposition 6.2 on the preservation of closed unions, which implies that $\mathbb{R}$ is the pushout of $(-\infty,0]$ and $[0,\infty)$ over $\{0\}$.

In particular, as Johnstone remarks, this implies that the quotient $[0,1]/(0\sim 1)$ of the unit [[interval]] identifying its endpoints coincides with a "correct" constructive definition of the circle $S^1 = \mathbb{R}/\mathbb{Z}$.  It also implies that "every real number has a decimal expansion" (though there is no *function* assigning to every real number a decimal expansion, since no such function can be continuous).

## The real numbers

One of the important facts about the topological topos is that its [[real numbers object]] is the set of classical [[real numbers]] equipped with its usual [[topology]] --- or more precisely, with the structure of [[subsequential space]] given by the usual notion of convergence for sequences of real numbers (arising from the usual Euclidean topology).

In fact, there are two commonly used definitions of "real number" in a topos: the [[Dedekind real numbers]] and the [[Cauchy real numbers]].  In a general topos the two may not coincide, although they do coincide if the topos is [[Boolean topos|Boolean]] or satisfies the [[axiom of countable choice]].  The topological topos is not Boolean and might not satisfy countable choice (this is unclear at the moment); but in any case the two real numbers objects nevertheless do coincide therein, and both are the set of classical real numbers with its usual topology.

For the Dedekind real numbers, a proof of this can be found in Johnstone's paper.  Here we sketch a proof for the Cauchy reals.

+-- {: .un_theorem}
###### Theorem
The Cauchy real numbers object in the topological topos $\mathcal{E}$ is the classical set of real numbers with its usual notion of sequential convergence.
=--
+-- {: .proof}
###### Proof
For simplicity, we use Cauchy sequences with a fixed modulus: $x:\mathbb{N} \to \mathbb{Q}$ is Cauchy if ${|x_m - x_n|} \lt \frac{1}{m+1} + \frac{1}{n+1}$ for all $m,n$.  Note that if $y$ is the limit of such a Cauchy sequence, then ${|y - x_n|} \le \frac{1}{n+1}$ for all $n$, so convergence also has a fixed modulus.

Since the [[inverse image functor]] $\Delta : Set \to \mathcal{E}$ preserves [[natural numbers objects]], the NNO $\mathbb{N}$ of $\mathcal{E}$ has the discrete structure (only eventually-constant sequences converge).  And since $\mathbb{Q} \cong \mathbb{N}$ constructively, the "rational numbers object" $\mathbb{Q}$ of $\mathcal{E}$ is also discrete.  It follows, by definition of exponentials in $\mathcal{E}$, that the object $\mathbb{Q}^{\mathbb{N}}$ of sequences of rational numbers has the topology of "pointwise-eventually-constant convergence": a sequence of sequences $((x_{n,k})_k)_n$ converges to a sequence $(y_k)_k$ if for each $k$ the sequence $(x_{n,k})_n$ is eventually constant at $y_k$.

Now the object of Cauchy sequences is a subobject of $\mathbb{Q}^{\mathbb{N}}$ defined by
$$ \left\{ (x_n) \mid \forall n,m, {|x_m - x_n|} \lt \frac{1}{m+1} + \frac{1}{n+1} \right\}. $$
The predicate ${|x_m - x_n|} \lt \frac{1}{m+1} + \frac{1}{n+1}$ defines a subobject of $\mathbb{Q}^{\mathbb{N}} \times \mathbb{N}\times \mathbb{N}$, which is the pullback of the predicate $\lt$ on $\mathbb{Q}\times\mathbb{Q}$ along the map $\mathbb{Q}^{\mathbb{N}} \times \mathbb{N}\times \mathbb{N} \to \mathbb{Q}\times\mathbb{Q}$ defined by $(x,m,n) \mapsto ({|x_m-x_n|},\frac{1}{m+1}+\frac{1}{n+1})$.  Now $\mathbb{Q}\times \mathbb{Q}$ is discrete, and every subobject of a discrete object in $\mathcal{E}$ is (discrete and) relatively codiscrete, i.e. it has the induced topology.  Thus, the predicate ${|x_m - x_n|} \lt \frac{1}{m+1} + \frac{1}{n+1}$ also has the induced topology.  And since relatively codiscrete maps are the modal maps for a [[modality]], they are closed under arbitrary universal quantification.  So the object of Cauchy sequences has the topology induced from $\mathbb{Q}^{\mathbb{N}}$ of pointwise eventually constant convergence.

In any topos, there is a monomorphism from the Cauchy real numbers to the Dedekind real numbers, and as remarked above, Johnstone proved that the latter have the usual topology.  Moreover, it is easy to see that the *points* of the Cauchy real numbers object are precisely the classical real numbers: any classical Cauchy sequence certainly induces a Cauchy sequence internal to $\mathcal{E}$.  Thus, it will suffice to show that the Cauchy subobject of the Dedekind reals also contains all convergent sequences.

To this end, suppose we have a sequence of Cauchy sequences $((x_{n,k})_k)_n$ and a Cauchy sequence $(y_k)_k$ (all in the above fixed-modulus sense) such that the sequence of real numbers $([x_n])_n$ converges to the real number $[y]$ in the usual sense.  We want to show that this convergence also happens in the Cauchy real numbers object.  To this end, define a new sequence of Cauchy sequences $z_{n,k}$ by setting $z_{n,k} = y_{2k+1}$ if ${|[x_n] - y_{2k+1}|} \lt \frac{1}{k+1}$ and $z_{n,k} = x_{n,k}$ otherwise.  To show that $z_n$ is Cauchy, the only nontrivial case is to show that ${|y_{2k+1} - x_{n,\ell}|} \lt \frac{1}{k+1} + \frac{1}{\ell+1}$ if ${|[x_n] - y_{2k+1}|} \lt \frac{1}{k+1}$; but since $x_{n,\ell}$ converges to $[x_n]$ with fixed modulus, we have ${|[x_n] - x_{n,\ell}|} \le \frac{1}{\ell+1}$ and the triangle inequality is our friend.

It is also evident that $z_n$ converges to $[x_n]$, i.e. $[z_n] = [x_n]$.  Thus it will suffice to show that $([z_n])_n$ converges to $[y]$ in the Cauchy real numbers object.  Let $w_k = y_{2k+1}$; then $[w]=[y]$ and $w$ converges twice as fast, i.e. ${|[w] - w_k|} \le \frac{1}{2(k+1)}$.  We claim that $((z_{n,k})_k)_n$ is pointwise-eventually-constant convergent to $w_k$.  To see this, by the ordinary convergence of $[x_n]$ to $[y]$, for each $k$ there exists an $N$ such that $n\gt N$ implies ${|[x_n] - [w]|}\lt \frac{1}{2(k+1)}$, and hence by the triangle inequality ${|[x_n] - y_{2k+1}|} = {|[x_n] - w_{k}|} \lt \frac{1}{k+1}$.  Thus, by definition of $z_n$, we have $z_{n,k} = w_k$ for all $n\gt N$, which is to say that $(z_{n,k})_n$ is eventually constant at $w_k$.

Finally, since $((z_{n,k})_k)_n$ converges to $w_k$ in the object of Cauchy *sequences*, the image sequence $([z_n])_n$ in the object of Cauchy *real numbers* (the quotient by equivalence of Cauchy sequences) must also converge to $[w]$ there.  Since $[z_n]=[x_n]$ and $[w]=[y]$, this is what we wanted.
=--

## Related entries

* [[gros topos]]

* [[convenient category of topological spaces]]

* [[bornological topos]]

* [[topos of recursive sets|recursive topos]]

* [[effective topos]]

* [[generic interval]]

* [[topological site]]

## Links

* n-cafe [blog on Johnstone's paper](https://golem.ph.utexas.edu/category/2014/04/on_a_topological_topos.html)

## References

The topological topos was introduced in

* [[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977. (Dover reprint 2014; exercise 0.10, p.21)

* {#Johnstone79}[[Peter Johnstone]], _On a topological topos_ , Proc. London Math. Soc. (3) **38** (1979) pp.237–271. ([pdf](http://plms.oxfordjournals.org/content/s3-38/2/237.full.pdf))

Some information with a somewhat changed terminology can be found in

* [[Peter Johnstone]], _Sketches of an [[Elephant]] vols.I&II_, Oxford UP 2002. (A2.1.11(j), A2.6.4(c), C2.2.14(b); pp.81,123,537f)

The topological topos is discussed in the context of [[convenient category of topological spaces|convenient categories of spaces]] in

* H. A. Harasani, _Topos theoretic methods in general topology_ , PhD Thesis University of Wales Bangor 1988.

Recently, the topos has received attention in the context of [[Homotopy type theory]]. Shulman has shown that there is a [[higher topos]] analogue which models [[homotopy type theory]].

* [[Michael Shulman]], _Brouwer's fixed-point theorem in real-cohesive homotopy type theory_, 2015 [arXiv](https://arxiv.org/abs/1509.07584)

* [[Martín Escardó]], [[Thomas Streicher]], _The intrinsic topology of Martin-Lof universes_, APAL (2016). ([pdf](http://www.cs.bham.ac.uk/~mhe/papers/universe-indiscrete.pdf))

* [[Bas Spitters]], _Cubical sets and the topological topos_ , arXiv:1610.05270 (2016).  ([abstract](https://arxiv.org/abs/1610.05270))

Generalizations of its construction occur in the following two papers:

* R. Montañez, _Topoi generated by topological spaces_ , Talk CT15 Aveiro 2015. ([pdf-slides](http://sweet.ua.pt/dirk/ct2015/slides/Montanez.pdf))

* C. Ruiz, R. Montañez, _Elevadores de Estructura_ , Boletín de Matemáticas **XIII** no.2 (2006) pp.111–135. ([link](http://www.revistas.unal.edu.co/index.php/bolma/article/view/40451))

[[!redirects topological topos]]


join

> this entry is about the notion of [[colimits]] in [[posets]]. For the concepts of _[[join of topological spaces]]_, _[[join of simplicial sets]]_, _[[join of categories]]_ and _[[join of quasi-categories]]_ see there.

***

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

If $x$ and $y$ are elements of a [[partial order|poset]] $P$, then their __join__ (or __supremum__, abbreviated _sup_, or __least upper bound__, abbreviated _lub_), is an element $x \vee y$ of the poset such that:

* $x \leq x \vee y$ and $y \leq x \vee y$;
* if $x \leq a$ and $y \leq a$, then $x \vee y \leq a$.

(These may be combined as: for all $a$, $x \vee y \leq a$ iff $x \leq a$ and $y \leq a$.)  Such a join may not exist; if it does, then it is unique.

If $P$ is a [[preorder|proset]], then join may be defined similarly, but it need not be unique.  (However, it is still unique up to the natural [[equivalence]] in $P$.)

The above definition is for the join of two elements of $P$, but it can easily be generalised to any number of elements.  It may be more common to use 'join' for a join of finitely many elements and 'supremum' for a join of (possibly) infinitely many elements, but they are the same concept.  The join may also be called the **maximum** if it equals one of the original elements.

A poset that has all finite joins is a **join-[[semilattice]]**.  A poset that has all suprema is a **[[suplattice]]**.

A join of [[subset]]s or [[subobject]]s is called a [[union]].

## Special cases

A join of zero elements is a [[bottom]] element.  Any element $a$ is a join of that one element.

## Properties

As a poset is a special kind of [[category]], so a join is simply a [[coproduct]] in that category.

## In constructive mathematics {#constructive}

In [[constructive analysis]], we sometimes want a stronger notion of supremum.  (Dual remarks apply to [[infima]].)

Let $S$ be a set of [[real numbers]], and let $M$ be a real number.  We say (as above) that $M$ is a __least upper bound__ (lub) of $S$ if for each real number $a$, $M \leq a$ iff for each member $x$ of $S$, $x \leq a$.  But we say that $M$ is a __supremum__ of $S$ if for each real number $a$, $M \gt a$ iff for some member $x$ of $S$, $x \gt a$.  In [[constructive mathematics]], we can prove that lubs and suprema are both unique when they exist and that every supremum is an lub, but we cannot prove that every lub is a supremum.  (We can prove that, if $M$ is an lub of $S$ and $M \gt a$, then there is [[double negation|not not]] some member $x$ of $S$ such that $x \gt a$, but not that there is such an member $x$.  For a specific [[weak counterexample]], let $p$ be any [[truth value]], and let $S$ be the [[subsingleton]] $\{0 \;|\; p\}$.  Then $0$ is a supremum of $S$ iff $p$ is true, while $0$ is an lub of $S$ iff $p$ is not not true.)

This generalizes to any set $P$ equipped with a relation $\gt$ (better written $\nleq$ in the general case) that is an [[irreflexive relation|irreflexive]] [[connected relation|connected]] [[comparison]] (properties [[de Morgan duality|dual]] to the properties that define a [[partial order]]) if $\leq$ is defined as the [[negation]] of $\nleq$ (which forces $\leq$ to be a partial order).  It\'s not even necessary for $\nleq$ to be a comparison, as long as its negation is a partial order (which still forces $\nleq$ to be irreflexive and connected).

Still more generally, let $P$ be a set equipped with the [[antithesis interpretation]] of a partial order.  This consists of two [[binary relations]] $\leq$ and $\nleq$ such that $\leq$ is a partial order, $\nleq$ is irreflexive, and $\leq$ and $\nleq$ are compatible:

* $y \geq x \nleq z$ implies $y \nleq z$,
* $x \nleq z \geq y$ implies $x \nleq y$.

Then we have two versions of a join $M$:

* for each element $a$ of $P$, $M \leq a$ iff for each member $x$ of $S$, $x \leq a$;
* for each element $a$ of $P$, $M \nleq a$ iff for some member $x$ of $S$, $x \nleq a$.

Then neither of these implies the other, and we probably really want to demand *both* at once.  The extended [[MacNeille real numbers]] provide a good example here.

## Related concepts

* **join**

* [[meet]]

[[!redirects join]]
[[!redirects joins]]
[[!redirects supremum]]
[[!redirects supremums]]
[[!redirects suprema]]
[[!redirects sup]]
[[!redirects sups]]
[[!redirects least upper bound]]
[[!redirects least upper bounds]]
[[!redirects lub]]
[[!redirects lubs]]
[[!redirects (0,1)-colimit]]
[[!redirects (0,1)-colimits]]


join of categories

+-- {: .rightHandSide}
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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents #
* table of contents
{: toc}

## Idea

The _join_ of two categories $C$ and $C'$ is obtained from the [[disjoint union]] of $C$ with $C'$ by throwing in a unique morphism from every object of $C$ to every object of $C'$.

## Definition

The **join** of [[category|categories]] $C$ and $C'$ is the category with

* objects $Obj(C \star C') := Obj(C) \amalg Obj(C')$;

* morphisms given by
$$
  Mor_{C \star C'}(a,b) :=
  \left\lbrace
    \array{
      Mor_C(a,b) & \text{ if  }\;\; a,b \in C
      \\
      Mor_{C'}(a,b) & \text{ if }\;\; a,b \in C'\\
      \emptyset & \text{ if  }\;\;  a \in C', b \in C;
      \\
      \mathrm{pt} & \text{ if  } \;\; a \in C, b \in C';
    }
  \right.
$$

### In terms of profunctors

The join of categories $C,C'$ can also be described to be the [[cograph of a profunctor|cograph]] of the unique profunctor $W\colon C ⇸ \; C'$ sending all objects $(c,c')$ to the terminal set (the definition speaks for itself).

### In terms of adjoint functors

Consider the inclusion of the [[boundary of a simplex|boundary]] of the standard 1-simplex, $i\colon \{0,1\}\to [1]$ as a functor between the [[discrete category]] with two elements and the [[interval category|walking arrow]] $I=\{0 \leq 1\}$. It induces a functor
$$
i^*\colon \Cat / I
\to
\Cat \times \Cat
$$
which admits a right adjoint. This right adjoint is precisely the [[bifunctor]] $\star\colon \Cat \times \Cat \to \Cat / I$, once we noticed that the category $C\star C'$ comes naturally equipped with an arrow $C\star C'\to I=1\star 1$ induced by (bi)functoriality of $\star$, starting from the canonical arrows $C\to 1, C'\to 1$ to the [[terminal category]].

+-- {: .proof}
###### Proof
It is quite clear that $i^*$ is defined by sending $C\to I$ to the pair of categories $i^\leftarrow(0)=C_0, C_1=i^\leftarrow(1)$. The bijection
$$
\Cat^{\mathbf{2}}\Big(i^*\big( C\to I\big), (A,B)\Big)\cong
\Cat / I \Big( C, A\star B \Big)
$$
is now rather obvious, since any functor
$i^*
\Big(
\array{ C\\ \downarrow \\ I }
\Big)
\to (A,B)$
determines a functor $C\to A\star B$ and viceversa.
=--

## Properties

* If the small categories $C,C'$ are two posets, their join consists of their [[ordinal sum]];
* The monoidal structure induced on $\Cat$ by $\star$ is not symmetric (if it was, then the right cone of $C$ would be equivalent to the left cone, which is blatantly false);
* The monoidal structure induced on $\Cat$ by $\star$ is not closed, since the functor $A\star -$ does not preserve colimits.
* The functor $-\star C'\colon \Cat \to C'\!/\!\Cat\colon A\mapsto (A\to A\star C')$ is a left adjoint, and similarly is the functor $C\star-$; see [Joyal, Ch. 3.1.1-2](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf#page=96) for a detailed description.

## Examples

* The **cone** below a category $C$ is the join $C \star \mathrm{pt}$. The cone above $C$ is the join $\mathrm{pt} \star C$.

## Related concepts

* [[join of topological spaces]]

* [[join of quasi-categories]];

* [[join of simplicial sets]].

* [[ordinal sum]]

## References

See [p. 42](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=42) of

* [[Jacob Lurie]], _Higher topos theory_ ([arXiv](http://arxiv.org/abs/math.CT/0608040))

See also [Ch. 3](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf#page=95) of

* [[André Joyal]], _The theory of quasicategories and its applications_ lectures at [Simplicial Methods in Higher Categories](http://www.crm.es/HigherCategories/), ([pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf))


join of maps

+-- {: .rightHandSide}
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###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

##Idea

Given two maps between homotopy types with a common codomain, $f: A \to X$ and $g: B \to X$, we denote by $A \ast_X B$ the [[pushout]] of their [[pullback]], $A \times_X B$. Then the **join** of $f$ and $g$, written $f \ast g$, is the induced map from $A \ast_X B$ to $X$.

When $X \equiv \mathbf{1}$, the unit type, we write $A \ast_{\mathbf{1}} B$ as $A \ast B$, the join of the two types.

The join of two maps is the fiberwise join of their respective fibres.

## Examples

The [[sequential colimit]] $f \ast^{\infty}$ of the join
powers $f \ast^{n}$ is equivalent to the [[image]] inclusion of $f$.

When $X \equiv \mathbf{1}$, the infinite join power $A \ast^{\infty}$ is the [[propositional truncation]] of $A$.

When $pt: \mathbf{1} \to \mathbf{B} G$, the point in a delooped group, $pt \ast^{\infty}$ is the [[Milnor construction]].

The join of two [[embeddings]] is an embedding, and an embedding is [[idempotent]] for the join construction.

The unique map of type $\mathbf{0} \to X$ is a unit for the join operation.

The join of two [[mere propositions]], $P \ast Q$, is equal to the [[propositional truncation]] of their sum, $\vert \vert P + Q \vert \vert$.

##Related concepts

* [[join of simplicial sets]]

##References

* {#Rijke17} [[Egbert Rijke]], _[[The join construction]]_ ([arXiv:1701.07538](https://arxiv.org/abs/1701.07538))


join of quasi-categories

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

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

The _join of two [[quasi-category|quasi-categories]]_ is the generalization of the [[join of categories]] from ordinary [[category|categories]] to [[quasi-category|quasi-categories]].

The  _join_ of quasi-categories $C$ and $C'$ is a [[quasi-category]] $C \star C'$ which looks roughly like the disjoint union of $C$ with $C'$ with one morphisms from every object of $C$ to every object of $C'$ thrown in.

## Definition

Two different definitions are used in the literature, which are not isomorphic, but are weakly equivalent with respect to the [[model structure on quasi-categories]].

1. The **join** $C \star C'$ of two [[quasi-category|quasi-categories]] $C$ and $C'$ is the [[join of simplicial sets]] of their underlying [[simplicial set]]s.

1. The alternate join $C \diamondsuit D$ of two quasi-categories should be thought of as something like the [[2-limit|pseudopushout]]

   $$
     \array{
       C \times D &\stackrel{p_2}{\to}& D
       \\
       {}^{\mathllap{p_1}} &\swArrow& \downarrow
       \\
       C &\to& C \diamondsuit D
     }
   $$

   Explicitly (compare [[mapping cone]]) it is the ordinary [[colimit]]

   $$
     \array{
       && C \times D \times {1} &\to&  D
       \\
       &&\downarrow && \downarrow
       \\
       C \times D \times {0}
       &\to& C \times D \times \Delta^1
       \\
       \downarrow &&&& \downarrow
       \\
       C &\to& &\to& C \diamondsuit D
     }
   $$

   in [[sSet]].

The join of two [[simplicial set]]s that happen to be [[quasi-category|quasi-categories]] is itself a quasi-category.

## Properties

For $C$ and $D$ [[simplicial set]]s, the canonical morphism

$$
  C \diamondsuit D \to C \star D
$$

is a weak equivalence in the [[model structure on quasi-categories]].

## Examples {#Examples}

### Joins with the point

Let $* = \Delta[0]$ be the [[terminal object|terminal]] quasi-category. Then for $X$ any quasi-category,

* the join $X^{\triangleleft} := (*)\star X$ is the quasi-category obtained from $X$ by freely adjoining a new [[terminal object in a quasi-category|initial object]];

* the join $X^{\triangleright} := X \star (*)$ is the quasi-category obtained from $X$ by freely adjoining a new [[terminal object in a quasi-category|terminal object]].

For instance for $X = \Delta[1] = \{ 0 \to 1 \}$ be have

$$
  X^{\triangleright} =
  \left\{
    \array{
      0 &&\to&& 1
      \\
      & \searrow &\swArrow& \swarrow
      \\
      && \bottom
    }
  \right\}
  \,.
$$

## Related concepts

* [[join of topological spaces]]

* [[join of categories]];

* [[join of simplicial sets]].

## References

The operation $\star_s$
is discussed around [proposition 1.2.8.3, p. 43](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=43) of

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

The operation $\diamondsuit$ is discussed in
[chapter 3](http://www.crm.cat/HigherCategories/hc2.pdf#page=95) of

* [[André Joyal]], _The theory of quasicategories and its applications_ lectures for the advanced course, at the conference on _Simplicial Methods in Higher Categories_, ([pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf))

and in [[Higher Topos Theory|section 4.2.1]] of

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

where also the relation between the two constructions is established.


join of simplicial sets

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

#Contents#
* table of contents
{:toc}

## Idea

The **join** $S \star T$ of two [[simplicial set]]s $S$ and $T$ is a new simplicial set that may geometrically be thought of as a [[cone]] over $T$ with tip of shape $S$. Topologically, it can also be thought of as the union of line segments connecting $S$ to $T$ if both are placed in general position.

If the simplicial sets in question are [[quasi-categories]] the notion of join on them produces the notion of [[join of quasi-categories]] that underlies many constructions in [[higher category theory]] such as the definition of [[limit in a quasi-category]].

The join of [[simplicial set]]s extends that historically given for [[simplicial complex|simplicial complexes]], cf. for instance the description and discussion in [[Spanier|Spanier's classical text]] (page 109 and then pages 114 -116).

The adaptation of this to simplicial sets reveals a neat link with some categorical structure in the category, $\Delta_a$, of finite ordinals (including the empty one).

## Motivating examples

When $S = \Delta^0$ is the [[point]], then the join $S \star T$ is a genuine [[cone]] over $T$. Or if $S = 2$ is the discrete two-point space, the join is the [[suspension]] of $T$.

For example, consider the two cones over $[2]$, the standard 2-simplex.  The first picture represents $[0]\star [2]$, while the second represents $[2]\star [0]$.

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[0]\star [2] & [2]\star [0]
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$$

If you take two non-coplanar line segments in $\mathbb{R}^3$ (such as $A B$ and $C D$ in the picture below), then join every point in one to every point in the other, you get a 3-simplex (the tetrahedron in the picture). You can think of this as being the union of all the cones on the first segment, with cone points on the second one. We have that the join $\Delta[1]\star \Delta[1]$ is $\Delta[3]$.

$$
\begin{gathered}
\begin{matrix}
0\overset{f}{\longrightarrow}1\\
\\
0'\underset{g}{\longrightarrow}1'
\end{matrix}
\overset{\quad f\star g\quad}{\longrightarrow}
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\cong
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f\cong g \cong [1] \Rightarrow f\star g \cong \Delta[3]
\end{gathered}
$$

## Definition

We first define the **join of simplicial sets** as the restriction to simplicial sets of the extension of the [[ordinal sum]] operation on the augmented [[simplex category]] $\Delta_a$ to augmented simplicial sets.

Then we give the more explicit definition in terms of concrete formulas. We first refer to the description of [[ordinal sum]], and then how it induces structure on the category of augmented simplicial sets.

### By Day convolution {#DayConvolution}

Via the general process of [[Day convolution]], the [[ordinal sum]] [[monoidal structure]] on $\Delta_a$ is lifted to a [[monoidal structure]] on [[presheaf|presheaves]] on $\Delta_a$, i.e. to the the category [[asSet]] or $sSet_+$ of [[augmented simplicial set]]s. This is given by a [[end|coend]] formula:

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

The join of simplicial set is equivalently expressed as

$$
  \star : sSet_+ \times sSet_+ \to sSet_+
$$
$$
  (S \star S')(-)
  :=
  \int^{[i],[j] \in \Delta_a}
    (S_i \times S'_j) \times Hom_{\Delta_a}(-,[i] \boxplus [j])
  \,.
$$

=--

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

This is an abuse of notation because $Hom_{\Delta_a}(-,[i] \boxplus [j])$ is a functor, while $(S_i \times S'_j)$ is a set.  To be precise, the second $\times$ should be replaced with $\cdot$, which denotes the [[copower|indexed copower]].

=--

Note that the join of simplicial sets $S \star T$ is [[cocontinuous functor|cocontinuous]] in each of its separate arguments $S$, $T$ (this is true generally of Day convolution products).

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

This join tensor product forms part of a [[closed monoidal category|closed monoidal structure]] on the category of
augmented simplicial sets, [[asSet]] $ := Sets^{\Delta_a^{op}}$. The [[internal hom]] is given by
$$[X, Y ]_n =asS(X; Dec^{n+1}Y )\,,$$
where $Dec$ is the [[total décalage]] functor (see also at _[[décalage]]_).

=--

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

For $S$ a simplicial set, let $\hat S$ denote the augmented simplicial set which equals $S$ in all degrees except in degree -1, where it is the point, $({\hat S})_{-1} = pt$. This is the **trivial augmentation** of $S$.

=--

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

The **join** of two ordinary [[simplicial set]]s $S_1$ and $S_2$ is the join of their _trivial augmentation_ :

$$
  S_1 \star S_2 := {\hat S_1} \star {\hat S_2}
  \,.
$$

=--

+-- {: .un_remark}
###### Remark
Note that the join operation is not commutative.  This comes from the fact that the [[monoidal structure]] given by [[ordinal sum]] on the [[augmented simplex category]] is not [[symmetric monoidal category|symmetric monoidal]].  See the discussion at [[augmented simplex category]] for more details.

=--

### Concrete formulas

The join of two non-augmented simplicial sets is given by the formula

$$
  (S \star S')_n := S_n \cup S'_n \cup
   (\cup_{i+j = n-1} S_i \times S'_j)
  \,.
$$

The $i$-th boundary map

$$
  d_i : (S \star T)_n \to (S \star T)_{n-1}
$$

is defined on $S_n$ and $T_n$ using the $i$th boundary map on $S$ and $T$.

Given $\sigma \in S_j$ and $\tau \in T_k$ , we have:

$$
  d_i (\sigma, \tau) =
  \left\{
    \array{
      (d_i \sigma, \tau) & if\; i \leq j , j \neq 0
      \\
      (\sigma, d_{i-j-1} \tau) & if\; i \gt j, k \neq 0
    }
  \right.
$$

If $j = 0$ then

$$
  d_0(\sigma, \tau) = \tau \in T_{n-1} \subset (S \star T)_{n-1}
  \,.
$$

If $k = 0$ then

$$
  d_n(\sigma, \tau) = \sigma \in S_{n-1} \subset (S \star T)_{n-1}
  \,.
$$

### Join of quasi-categories

If the simplicial sets in question are [[quasi-categories]], their join computes the corresponding [[join of quasi-categories]], effectively an [[over quasi-category]] construction.

In this sense the join can then also be computed -- up to equivalence of quasi-categories -- as the [[homotopy colimit|homotopy pushout]] of the two projections out of $S \times S'$.

In this form, the join is used
in [definition 1.2.8.1, p. 42](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=42) of [[Higher Topos Theory|HTT]]

## Examples {#Examples}

Recall that the join of augmented simplicial sets $S \star T$ is a [[cocontinuous functor]] in each of its separate arguments $S$, $T$ (this is true generally of [[Day convolution]] products).

This observation can help simplify calculations. For example, simplicial joins preserve unions in the first argument $S$, and inasmuch as [[horn]]s are unions of face simplices, this allows one to compute joins of horns with simplices.

As an operation on simplicial sets, $S \star T$ merely commutes with connected colimits (since these preserve the property of having trivial augmentation). However, for each $S$, the natural inclusion $S \cong S \star \varnothing \to S \star T$ induces a functor $S \star (-) : sSet \to S/sSet$ that does preserve colimits. Similarly, $(-) \star T : sSet \to T/sSet$ preserves colimits.

### Joins with the point: cones

For $\{v\} = \Delta[0]$ the [[point]], a join with the point is called a **cone** with cone vertex $v$: for $S \in sSet$ we say

* $S^\triangleleft := \{v\} \star S$ is the **cone** over $S$;

* $S^\triangleright :=  S \star \{v\}$ is the **co-cone** under $S$;

Universal images of cones and cocones over a fixed base $S$ in a [[quasi-category]] $C$ are [[limit in a quasi-category|limits and colimits in that quasi-category]].

For instance the cone over the interval $\Delta[1]$ is the 2-simplex

$$
  \{v\} \star \Delta[1]
  =
  \left(
    \array{
      && v
      \\
      & \swarrow &\swArrow& \searrow
      \\
      0 &&\to&& 1
    }
  \right)
  \simeq
  \Delta[2]
  \,.
$$

More generally, the cone over the $n$-[[simplex]] is the $(n+1)$-simplex

$$
  \Delta[n]^{\triangleleft} \simeq \Delta[n+1]
  \,.
$$

Cones of 2-[[horn]]s are simplicial 2-squares $\simeq \Delta[1] \times \Delta[1]$:

$$
  \Delta[1] \times \Delta[1]
  \simeq
  \{v\} \star \Lambda_2[2]
  =
  \left(
  \array{
    v &\to& 1
    \\
    \downarrow &{}_{\swArrow}\searrow^{\swArrow}& \downarrow
    \\
    0 &\to& 2
  }
  \right)
$$

and

$$
  \Delta[1] \times \Delta[1]
  \simeq
  \Lambda_0[2] \star \{v\}
  =
  \left(
  \array{
    0&\to& 1
    \\
    \downarrow &{}_{\swArrow}\searrow^{\swArrow}& \downarrow
    \\
    2 &\to& v
  }
  \right)
  \,.
$$

### Joins of simplices

Effectively by the definition from ordinal sum, we have that the join of two [[simplex|simplices]] is another simplex:

$$
  \Delta[k] \star \Delta[l] = \Delta[k + l + 1]
  \,.
$$

In particular the cone over the $n$-simplex is the $(n+1)$-simplex

$$
  \Delta[0] \star \Delta[n] = \Delta[n+1]
$$

and hence

$$
  \Delta[n] = \Delta[0] \star \cdots \star \Delta[0]
  \,.
$$

Notice that while thus $\Delta[n+1] \simeq  \Delta[0]\star\Delta[n] \simeq \Delta[n] \star \Delta[0]$ the identifications of the cone point of course differ in both cases. The asymmetry is seen for instance by restricting attention to the cone over the boundary of the $n$-simplex, where we have

$$
  \partial \Delta[n] \star \Delta[0] = \Lambda_{n+1}[n+1]
$$

and

$$
  \Delta[0] \star \partial \Delta[n] = \Lambda_0[n+1]
  \,.
$$

### Simplicial $n$-sphere

Let $\partial \Delta[1] = \Delta[0] \sqcup \Delta[0]$ the **simplicial 0-sphere**: just the disjoint union of the point. Then the $n$-fold join of $\partial \Delta[1]$ with itself is a simplicial model for the $n$-sphere
$$\mathbf{S}^0 := \partial \Delta[1]$$
$$\mathbf{S}^n := \mathbf{S}^0 \star \mathbf{S}^{n-1}$$
for $n \in \mathbb{N}$, $n \gt 0$. The [[geometric realization]] of $\mathbf{S}^n$ is equivalent to the topological $n$-sphere.

See [Ehlers/Porter p. 8](http://arxiv.org/PS_cache/math/pdf/9904/9904039v1.pdf#page=8).

## Properties

### Compatibility with quasi-categories

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

If $S, S' \in $ [[sSet]] are both [[quasi-categories]], then so is their join $S \star S'$.

=--

This is due to [[Andre Joyal]]. A proof appears as [[Higher Topos Theory|HTT, prop. 1.2.8.3]].

### Compatibility with homotopy coherent nerve

There is also a join operations on [[categories]] and [[sSet-categories]]:

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

Let $C,D \in sSet Cat$. Then define $C \star D$ to be the $sSet$-category given by

$$
  Obj(C \star D) = Obj(C) \coprod Obj(D)
$$

$$
  C \star D(x,y) =
  \left\{
    \array{
       C(x,y) & for\; x,y \in C
       \\
       D(x,y) & for\; x,y \in D
       \\
       \emptyset & for \;x \in D, y \in C
       \\
       * & for\; x \in C , y \in D
   }
  \right.
$$

with the obvious composition operations.

=--

Write

$$
  \tau_{hc} : sSet \to sSet Cat
$$

for the [[left adjoint]] of the [[homotopy coherent nerve]] functor (denoted $\mathfrak{C}$ in [[Higher Topos Theory|HTT]]. )

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

For $S, S'$ two [[simplicial set]]s we have that

* the two inclusions $\tau_{hc}(S), \tau_{hc}(S') \to \tau_{hc}(S\star S')$ are [[full and faithful functor|full and faithful]].

* $\tau_{hc}(S \star S')$ is in general not [[isomorphic]] to $\tau_{hc}(S) \star \tau_{hc}(S')$;

* the canonical morphism

  $$
    \tau_{hc}(S \star S') \to \tau_{hc}(S) \star \tau_{hc}(S')
  $$

  is an equivalence in the [[model structure on sSet-categories]].

=--

This is [[Higher Topos Theory|HTT, corollary 4.2.1.4]].

## Related concepts

* [[join of topological spaces]]

* [[join of categories]], [[join of quasi-categories]]

* [[join of maps]]

## References

The join operation was studied by P. J. Ehlers, in his thesis

* _Algebraic Homotopy in Simplicially Enriched Groupoids_, 1993,
University of Wales Bangor, available [[Ehlers-thesis.pdf|here:file]], (see also the reference below and the Menagerie notes available from [[Tim Porter]]'s homepage.),

but was there ascribed to [[Jack Duskin]] and [[Don van Osdol]] in some unpublished notes.  The main ideas were derived there from earlier work of [[Bill Lawvere]].

A useful published reference is

* [[Phil Ehlers|P. J. Ehlers]] and [[Tim Porter]], _Joins for (Augmented) Simplicial Sets_,  Jour. Pure Applied Algebra, 145 (2000) 37-44, ([arXiv](http://arxiv.org/abs/math.CT/9904039)) .

A useful discussion emphasizing the Day convolution operation is also in section 3.1 of

* [[Dominic Verity]], _Weak complicial sets I_ ([arXiv](http://arxiv.org/abs/math/0604414))

Discussion in [[homotopy type theory]] (with application to [[n-image]] factorization) is in

* {#Rijke17} [[Egbert Rijke]], _The join construction_ ([arXiv:1701.07538](https://arxiv.org/abs/1701.07538))


join of topological spaces

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

#Contents#
* table of contents
{:toc}

## Idea

The *join* $X \star Y$ of [[topological spaces]] $X$ and $Y$ consists of $X$, $Y$, and [[convex set|convex combinations]] of a point of $X$ and a point of $Y$ (with some suitable topology).

## Definition

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

Write $I \coloneqq [0,1]$ for the unit [[interval]], regarded as a [[topological space]].

For $X,Y $ two [[topological spaces]], the [[join]] $X \star Y$ is the [[colimit]] in [[Top]] of the [[diagram]]

$$
  \array{
   & & X \times Y & & & & X \times Y & &
   \\
   & \mathllap{\pi_1} \swarrow & & \searrow \mathrlap{i_0} & & \mathllap{i_1} \swarrow & & \searrow \mathrlap{\pi_2} &
   \\
   X & & & & X \times I \times Y & & & & Y
}
$$

where $i_0$ is the inclusion $(x, y) \mapsto (x, 0, y)$, and $i_1$ is the inclusion $(x, y) \mapsto (x, 1, y)$.

=--

(This in turn, by the discussion at _[[mapping cone]]_, is a model for the [[homotopy type]] of the [[homotopy pushout]] of the two projections $X \leftarrow X \times Y \to Y$.)

Intuitively, $X \star Y$ is the union of all line segments connecting $X$ to $Y$ when these are placed in general position in an ambient [[Euclidean space]].

The join is [[monoidal category|associative]]; intuitively, a join of three spaces $X, Y, Z$ is the union of 2-[[simplices]] whose vertices lie in $X, Y, Z$ respectively when these are placed in general position. Thus the join endows $Top$ with a [[monoidal category]] structure whose unit is the empty space.

## Milnor’s join construction

There is another topology on the join of spaces, due to [[John Milnor]], which appears in the *[[Milnor construction]]* of [[universal principal bundles]] ([Milnor 1956, section 2](#Milnor1956)):

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

Milnor’s join $X \overline{\star} Y$ has the same underlying set as $X \star Y$ and the [[coarsest topology]] which makes $t \colon X \overline{\star} Y \to [0,1]$ (given on $X \times I \times Y$ by $(x,t,y)\mapsto t$, constantly $0$ on $X$, and constantly $1$ on $Y$) and the projections $\pi_1 \colon t^{-1}([0,1)) \to X$ and $\pi_2 \colon t^{-1}((0,1]) \to Y$ continuous.

If $p\colon X \to S$ and $p\colon Y \to S$ are spaces over $S$, then their _fibrewise join_ or _Whitney sum_ $X \overline{\star}_S Y$ is the space $\{ x \in X \overline{\star} Y : t(x) \in (0,1) \implies p(\pi_1(x))=q(\pi_2(y)) \}$.

=--

The inclusions $X \hookrightarrow X \overline{\star} Y$ and $Y \hookrightarrow X \overline{\star} Y$ are [[Hurewicz cofibration|closed Hurewicz cofibrations]]: the open set $t^{-1}[[0,1)]$ deforms onto $X$ and $t^{-1}[(0,1]]$ deforms onto $Y$.

\begin{proposition} The identity map $X \star Y \to X \overline{\star} Y$ is a homotopy equivalence. If $X$ and $Y$ are compact, then it is a homeomorphism.
\end{proposition}
(See [Fritsch–Golasiński 2004, section 1](#FG2004))

\begin{proposition}\label{FibrewiseJoinOfFibrationsIsFibration}
If $p \colon X \to S$ and $q \colon Y \to S$ are [[Hurewicz fibration|Hurewicz fibrations]], then so is $X \overline{\star}_S Y \to S$
\end{proposition}
(See [Hall 1965](#Hall1965), [Lemmens 1970, theorem I](#Lemmens1970))

## Examples

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

The join of any topological space $X$ with the [[point]] is the [[cone]] on $X$:

$$
  X \star 1 =  C X
  \,.
$$

=--

+-- {: .num_example #JoinWith0Sphere}
###### Example

The join of any topological space $X$ with the [[0-sphere]] $S^0$
is the [[suspension|(unreduced) suspension]] of $X$:

$$
  X \star S^0 \cong S X
  \,.
$$

=--

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

The join of [[n-spheres]] with each other is

$$
  S^m \star S^n \cong S^{m+n+1}
$$

=--

+-- {: .num_example}
###### Example
For $X = S^1, Y = S^1$, we may consider $X$ to consist of [[quaternions]] $a + b i$ and $Y$ to consist of quaternions $c j + d k$ such that $a^2 + b^2 = 1 = c^2 + d^2$. Then $X$ and $Y$ are in general position in $\mathbb{H} \cong \mathbb{R}^4$ with respect to each other, and the quotient map

$$X \times I \times Y \to S^3$$

$$\,$$

$$(a + b i, t, c + d i) \mapsto t^{1/2}(a + b i) + (1 - t)^{1/2}(c + d i)$$

is an explicit realization of the [[unit sphere]] $S^3$ as $X \star Y$.
=--

+-- {: .num_example}
###### Example
For a topological group $G$, the [[Milnor construction]] of the total space $E G$ of the [[classifying space|classifying bundle]] is an iterated join, i.e., the colimit of a diagram of inclusions

$$G \to G \overline{\star} G \to G \overline{\star} G \overline{\star} G \to \ldots$$

where the identity element of $G$ is used to embed each $G^{\bar \star n}$ into its successor $G^{\bar \star (n+1)}$. The idea is that passing to higher joins kills off more and more lower-dimensional [[homotopy groups]], until one reaches the colimit which is then weakly contractible.

The same idea applies to a general space $X$; an $H$-space structure and higher homotopy associativities (collectively embodied in a structure of algebra over the [[associahedron|Stasheff operad]]) may be used to build a classifying bundle in iterative fashion, with the [[Hopf construction]] giving the first stage of the iteration.
=--

+-- {: .num_example}
###### Example
If $p \colon X \to Y$ is a fibration, then $p$ factors as a [[Hurewicz cofibration]] $X \hookrightarrow X \overline{\star}_Y Y$ followed by an [[acyclic fibration|acyclic]] [[Hurewicz fibration]] $X \overline{\star}_Y Y \to Y$ by proposition \ref{FibrewiseJoinOfFibrationsIsFibration}: the embedding $Y \hookrightarrow X \overline{\star}_Y Y$ and retraction $X \overline{\star}_Y Y \to Y$ exhibit $Y$ as a strong deformation retract of $X \overline{\star}_Y Y$.

(This fact is used in Strøm’s construction of his [[Strøm model structure|model structure on topological spaces]].)

=--

## Related concepts

* [[suspension]]

* [[Milnor construction]]

* [[Hopf construction]]

* [[join type]]

* [[join of simplicial sets]],

* [[join of categories]], [[join of quasi-categories]]

## References

* {#Hall1965} I.M. Hall, _The generalized Whitney sum_, The Quarterly Journal of Mathematics 16(4): 360–384, December 1965. [10.1093/qmath/16.4.360](https://doi.org/10.1093/qmath/16.4.360)

* {#Lemmens1970} P.W.H. Lemmens, _A note on the join of fibrations_, Indagationes Mathematicae 73: 53–56 (1970) [doi:10.1016/S1385-7258(70)80008-7](https://doi.org/10.1016/S1385-7258%2870%2980008-7)

* {#Milnor1956} [[John Milnor]], _Construction of Universal Bundles_, _II_, Ann. of Math. __63__ 3 (1956) 430-436 [[doi:10.2307/1970012](https://doi.org/10.2307/1970012)]

Discussion of relation to [[homotopy limits]] etc.

* Jean-Paul Doeraene, _Homotopy pull backs, homotopy push outs and joins_, Bull. Belg. Math. Soc. Simon Stevin,    Volume 5, Number 1 (1998), 15-37. [pdf on Project Euclid](https://projecteuclid.org/euclid.bbms/1103408963)

Comparison with other forms of join:

* {#FG2004}  [[Rudolf Fritsch]],  [[Marek Golasiński]], _Topological, simplicial and categorical joins_, Archiv der Mathematik **82** 5 (2004) 468-480

Discussion in [[homotopy type theory]] (applied to [[n-image]] factorization) is in

* {#Rijke17} [[Egbert Rijke]], _The join construction_ ([arXiv:1701.07538](https://arxiv.org/abs/1701.07538))

[[!redirects joins of topological spaces]]


join type

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

#Contents#
* table of contents
{:toc}

## Idea

The _join type_ is an [[axiom|axiomatization]] of the [[join of topological spaces]] in [[algebraic topology]] in the context of [[homotopy type theory]].

## Definition

The join type of a [[type]] $A$ and $B$, $A * B$, is the [[pushout type]] of the following span

$$A \xleftarrow{\text{fst}} A \times B \xrightarrow{\text{snd}} B$$

## See also

* [[higher inductive type]]

* [[join of topological spaces]]

* [[bracket type]]

## References

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

* [[Egbert Rijke]], *The join construction*, 26 Jan 2017, ([arXiv:1701.07538](https://arxiv.org/abs/1701.07538))

* [[Egbert Rijke]], *Homotopy pushouts*, Lecture 13 in: *[[Introduction to Homotopy Type Theory]]*, lecture notes, CMU (2018) [[pdf](http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf), [[Rijke-IntroductionHoTT-2018.pdf:file]], [webpage](https://www.andrew.cmu.edu/user/erijke/hott/)]

* {#RijkeDraft22} [[Egbert Rijke]] (2022), *[[Introduction to Homotopy Type Theory]]*, draft. ([pdf](https://raw.githubusercontent.com/martinescardo/HoTTEST-Summer-School/main/HoTT/hott-intro.pdf))


join-semilattice object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The notion of a *join-semilattice object* is the generalization of that of *[[join-semilattice]]* as one passes from the ambient [[category of sets]] [[internalization|into]] more general ambient  [[categories]] with suitable properties.

## Definition

In a [[finitely complete category]] $C$, a **join-semilattice object** is a [[cocartesian monoidal preordered object]] that is also a [[partially ordered object]].

## See also

* [[cocartesian monoidal preordered object]]

* [[partially ordered object]]

* [[meet-semilattice object]]

* [[lattice object]]

* [[semilattice object]]

* [[join-semilattice]]

[[!redirects join-semilattice objects]]


joint and marginal probability

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[probability theory]], it is a well known fact that [[events]] are not always [[independent events|independent]], i.e. that in general
$$
P(A,B) \ne P(A)\,P(B).
$$
One says that *the probability of a product is not the product of the probabilities*.
The term on the left is called the *joint probability* (the probability that the events $A$ and $B$ happen *jointly*) and the terms on the right are called *marginal probabilities* (the probabilities that $A$ and $B$ happen *separately*.)
In general, the joint probability has more information than the marginal probabilities alone, because of the presence of [[correlation]] or other [[stochastic interactions]].

From the [[category-theoretic approaches to probability theory|category-theoretic perspective]], one can encode this idea using the following (related) concepts:

* A [[monoidal category|monoidal structure]] on a [[category of kernels]] or [[Markov category]] which is [[semicartesian monoidal category|semicartesian]], but not [[cartesian monoidal category|cartesian]], i.e. for which the tensor product is not a [[product|categorical product]];

or

* A [[probability monad]] which is [[affine monad|affine]] but not [[strong monoidal functor|strong monoidal]], i.e. which does not preserve [[products]].

When one takes as [[Markov category]] the [[Kleisli category]] of a [[probability monad]], the two concepts coincide.

## In measure-theoretic probability

Let $(\Omega,\mathcal{F},p)$ be a [[probability space]], and let $A,B\subseteq\mathcal{F}$ be [[events]], i.e. [[measurable subsets]] of $\Omega$.
We call the **joint probability** of $A$ and $B$ the probability of their [[intersection]], $p(A\cap B)$.

More generally, let $f:(\Omega,\mathcal{F})\to(X,\mathcal{A})$ and $g:(\Omega,\mathcal{F})\to(Y,\mathcal{B})$ be [[random variables]] (or [[random elements]]) on $\Omega$.
The **joint random variable** of $f$ and $g$ is the random variable $(f,g):(\Omega,\mathcal{F})\to(X\times Y,\mathcal{A}\otimes\mathcal{B})$ given by the [[universal property]] of the [[product]]:
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large,%
]
& \Omega \ar{dl}[swap]{f} \ar{dr}{g} \ar[dashed]{d}[near end]{(f,g)} \\
X & X\times Y \ar{l}{\pi_1} \ar{r}[swap]{\pi_2} & Y
\end{tikzcd}

We call

* The **joint distribution** of $f$ and $g$ (or of $X$ and $Y$, when the maps are clear from the context) the [[pushforward measure]] $p_{X,Y}=(f,g)_*p$ on $X\times Y$;
* The **marginal distributions** of $f$ and $g$ (or of $X$ and $Y$) the [[pushforward measures]] $p_X=f_*p$ and $p_Y=g_*p$ on $X$ and $Y$, respectively.

Note that $(\pi_1)_*p_{X,Y} = p_X$ and $(\pi_2)_*p_{X,Y} = p_Y$, so that the marginalization maps can be seen as the functorial images of the [[product]] projections under a [[probability monad]].

In absence of the space $\Omega$ one can call "joint distribution" any probability distribution on a product space $X\times Y$, and define its marginal distributions by pushing forward along the product projections. This is equivalent to taking $X\times Y$ as $\Omega$. The same is true for products of more spaces, including [[infinite tensor product|infinite products]].

Given two (marginal) distributions $p$ and $q$ on spaces $(X,\mathcal{A})$ and $(Y,\mathcal{B})$, in principle there could be many joint distributions on $X\times Y$ admitting $p$ and $q$ as their marginals, corresponding to different possible [[stochastic dependence and independence|stochastic interactions]].
A canonical choice of joint distribution is given by the **product distribution** $p\otimes q$, defined on product sets in the form $A\times B\subseteq X\times Y$ by
$$
(p\otimes q)\,(A\times B) = p(A)\,q(B) ,
$$
which makes $X$ and $Y$ [[stochastic dependence and independence|independent]].

### In terms of probability monads

Recall that the [[Giry monad]], as well as other [[probability monads]], encodes the idea of forming, from a space $X$, a space $P X$ of [[probability measures]] over it.
A joint distribution is then encoded by an element of the space $P(X\times Y)$, and the marginals are elements of $P X$ and $P Y$ obtained via applying $P$ to the product projections.
Additionally, by the [[universal property]] of [[products]], there exists a unique map $\Delta:P(X\times Y)\to P X\times P Y$ making the following diagram commute.
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large,%
]
& P(X\times Y) \ar{dl}[swap]{P(\pi_1)} \ar{dr}{P(\pi_2)} \ar[dashed]{d}{\Delta} \\
PX & PX\times PY \ar{l}{\pi_1} \ar{r}[swap]{\pi_2} & PY
\end{tikzcd}
This can be interpreted as forming, from a joint distribution, the pair of its marginals, which in general encode less information. The same can be done for joint distributions over a larger number of factors, including [[infinite tensor product|infinite products]].

In order to form product distributions, one needs additional compatibility conditions between the [[monad]] structure (giving probability measures) and the [[products]] of the category.
These conditions are exactly captured by the idea of a [[monoidal monad]], or equivalently, a [[commutative monad]].
This amounts in particular to a [[natural transformation|natural map]] $\nabla:P X\times P Y\to P(X\times Y)$ satisfying particular compatibility conditions. For the [[Giry monad]], it is the following map.
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
]
PX\times PY \ar{r}{\nabla} & P(X\times Y) \\
(p,q) \ar[mapsto]{r} & p\otimes q
\end{tikzcd}

Consider now the following two diagrams.
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large%
]
PX \times PY \ar{dr}[swap]{\mathrm{id}} \ar{r}{\nabla} & P(X\times Y) \ar{d}{\Delta} & P(X\times Y) \ar{dr}[swap]{\mathrm{id}} \ar{r}{\Delta} & PX\times PX \ar{d}{\nabla} \\
& PX\times PY & & P(X\times Y)
\end{tikzcd}

We have that

* The commutativity of the diagram on the left says that if one forms a product measure and then takes its marginals, one recovers the two factors. In other words, *every product probability measure is necessarily the product of its marginals*.
One can show that the diagram on the left commutes if and only if $P$ preserves the terminal object, i.e. $P(1)\cong 1$ (see at [[affine monad]]). This is the case for the [[Giry monad]] and for most [[probability monads]], since the one-point space admits a unique probability measure. (It can be considered a "normalization" condition for probability measures: if one drops normalization, the one-point space admits several measures.)

* In general, the diagram on the right does not commute: if one starts with a joint distribution and then forms its marginals, the original distribution may be different from the product of the marginals. Therefore marginalization is a destructive operation, forgetting possible [[stochastic interaction]].
Indeed, if the diagram on the left commutes, then the diagram on the right commutes if and only if the monad $P$ preserves [[products]], i.e. it is [[strong monoidal functor|strong monoidal]]. The "random" behavior of the [[Giry monad]] and of other [[probability monads]] can be explained, to a large degree, by their failure to preserve products. (More on this below, and at [[Markov category#eqdet|Markov category]]).

### In the category of Markov kernels

As we have seen above, a joint distribution, in general, encodes more information than the pair of its marginals, and cannot always be recovered by forming the product distribution.

We can express this equivalently by looking at the category [[Stoch]] of [[Markov kernels]], as well as at other [[Kleisli categories]] of similar [[probability monads]].

First of all, recall that the category [[Stoch]] inherits the [[monoidal category|monoidal structure]] from the [[product]] in [[Meas]], the [[cartesian product]] of [[measurable spaces]].

This product in [[Stoch]], however, is not a categorical product: the [[universal property]] of a [[product]] requires that given [[objects]] $\Omega$, $X$ and $Y$ in a [[category]], for every pair of morphisms $f:\Omega\to X$ and $g:\Omega\to Y$ there exists a unique morphism $\Omega\to X\otimes Y$ making the following diagram commute.
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large,%
]
& \Omega \ar{dl}[swap]{f} \ar{dr}{g} \ar[dashed]{d}[near end]{(f,g)} \\
X & X\otimes Y \ar{l}{\pi_1} \ar{r}[swap]{\pi_2} & Y
\end{tikzcd}
For Markov kernels, this property does not hold: setting $\Omega=1$, the one-point space, the property would require in particular that there is a unique joint distribution on $X\times Y$ for every pair of marginals $X$ and $Y$. As we have seen above, this is not the case.
What fails is not the *existence* part of the universal property: a joint distribution always exists, the product distribution. What fails is the *uniqueness* part. In other words, the monoidal structure in [[Stoch]] is a [[weak limit|weak product]] but not a categorical [[product]].

More generally, given a [[monoidal monad]] on a [[cartesian monoidal category]], the induced monoidal structure on the [[Kleisli category]] is monoidal only when the monad is [[strong monoidal functor|strong monoidal]], and we saw that this is not the case for most probability monads.
(As mentioned at [[Markov category#eqdet|Markov category]], a cartesian monoidal Kleisli category can be seen as having "no randomness".)

## In Markov categories

In [[Markov categories]], one can define joint and marginal states similarly to what happens in the category [[Stoch]]. Recall that a state on $X$

\begin{tikzpicture}[%
thick,%
scale=0.8,%
every node/.style={scale=1.25},%
none/.style={fill=none,draw=none},%
morphism/.style={fill=white, draw=black, shape=rectangle},%
bn/.style={fill=black, draw=black, shape=circle, inner sep=1.8pt},%
over arrow/.style={-, black, preaction={draw=white, double}},%
]

		\node [style=none] (0) at (0, -0.5) {};
		\node [style=none] (1) at (0, 1) {};
                \node [style=none] (2) at (0, 1.5) {$X$};

		\draw (0.center) to (1.center);

        \node [style=morphism] (4) at (0, -0.5) {$p$};
\end{tikzpicture}

has the interpretation of a [[probability measure]] on $X$, and in [[Stoch]] it is exactly a [[Markov kernel]] $1\to X$, i.e. a probability measure.

A *joint state* is a state in the form $I\to X\otimes Y$:

\begin{tikzpicture}[%
thick,%
scale=0.8,%
every node/.style={scale=1.25},%
none/.style={fill=none,draw=none},%
morphism/.style={fill=white, draw=black, shape=rectangle},%
bn/.style={fill=black, draw=black, shape=circle, inner sep=1.8pt},%
over arrow/.style={-, black, preaction={draw=white, line width=2mm}},%
]

		\node [style=none] (10) at (-3, 0) {};
		\node [style=none] (11) at (-2, 0) {};
		\node [style=none] (12) at (-2, 1.25) {};
		\node [style=none] (13) at (-3, 1.25) {};
		\node [style=none] (14) at (-3, 1.75) {$X$};
		\node [style=none] (15) at (-2, 1.75) {$Y$};
		\node [style=none] (20) at (-2.5, 0) {};

		\draw (10.center) to (13.center);
		\draw (11.center) to (12.center);

                \node [style=morphism] (27) at (-2.5, 0) {$\;\, p \;\,$};
\end{tikzpicture}

In *Stoch*, this is exactly a probability measure on $X\times Y$.

Given a joint state $p$ on $X\otimes Y$, its *marginals* are the states $p_X$ and $p_Y$ on $X$ and $Y$ respectively given by discarding the unobserved variable:

\begin{tikzpicture}[%
thick,%
scale=0.8,%
every node/.style={scale=1.25},%
none/.style={fill=none,draw=none},%
morphism/.style={fill=white, draw=black, shape=rectangle},%
bn/.style={fill=black, draw=black, shape=circle, inner sep=1.8pt},%
over arrow/.style={-, black, preaction={draw=white, line width=2mm}},%
]

		\node [style=none] (0) at (-3.5, -0.75) {};
		\node [style=none] (2) at (-4, 1) {};
		\node [style=none] (3) at (-4, 1.5) {$X$};
		\node [style=none] (4) at (-4, -0.5) {};
		\node [style=none] (5) at (-3, -0.5) {};
		\node [style=bn] (6) at (-3, 0.5) {};
		\node [style=none] (9) at (-8.25, 1) {};
		\node [style=none] (10) at (-8.25, 1.5) {$X$};
		\node [style=none] (11) at (-8.25, -0.5) {};
		\node [style=none] (12) at (-6, 0) {$=$};
		\node [style=none] (13) at (7.75, -0.75) {};
		\node [style=none] (15) at (8.25, 1) {};
		\node [style=none] (16) at (8.25, 1.5) {$Y$};
		\node [style=none] (17) at (8.25, -0.5) {};
		\node [style=none] (18) at (7.25, -0.5) {};
		\node [style=bn] (19) at (7.25, 0.5) {};
		\node [style=none] (21) at (3, 1) {};
		\node [style=none] (22) at (3, 1.5) {$Y$};
		\node [style=none] (23) at (3, -0.5) {};
		\node [style=none] (24) at (5.25, 0) {$=$};

		\draw (2.center) to (4.center);
		\draw (5.center) to (6);
		\draw (9.center) to (11.center);
		\draw (15.center) to (17.center);
		\draw (18.center) to (19);
		\draw (21.center) to (23.center);

		\node [style=morphism] (1) at (-3.5, -0.75) {$\;\, p \;\,$};
		\node [style=morphism] (8) at (-8.25, -0.75) {$p_X$};
		\node [style=morphism] (14) at (7.75, -0.75) {$\;\, p \;\,$};
		\node [style=morphism] (20) at (3, -0.75) {$p_Y$};
\end{tikzpicture}

In *Stoch*, these correspond exactly to marginal distributions.

More on this (and examples in other Markov categories) can be found at [[Markov category#states_channels_joints_marginals|Markov category - States, channels, joints, marginals]].

## See also

* [[stochastic dependence and independence]]
* [[affine monad]]
* [[probability theory]], [[categorical probability]]
* [[probability monad]], [[Giry monad]]
* [[Markov category]], [[Markov kernel]]
* [[infinitary tensor product]], [[Kolmogorov extension theorem]]

## References

* {#bimonoidal_monads} [[Tobias Fritz]], [[Paolo Perrone]], _Bimonoidal Structure of Probability Monads_, Proceedings of MFPS, 2018, ([arXiv:1804.03527](https://arxiv.org/abs/1804.03527))

* {#Perrone_thesis} [[Paolo Perrone]], _Categorical Probability and Stochastic Dominance in Metric Spaces_, ([thesis](http://www.paoloperrone.org/phdthesis.pdf))

* {#cd_categories} Kenta Cho, [[Bart Jacobs]], _Disintegration and Bayesian Inversion via String Diagrams_, Mathematical Structures of Computer Science 29, 2019. ([arXiv:1709.00322](https://arxiv.org/abs/1709.00322))

* {#fritzmarkov} [[Tobias Fritz]], _A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics_, Advances of Mathematics 370, 2020. ([arXiv:1908.07021](http://arxiv.org/abs/1908.07021))

* [[Anders Kock]], _Bilinearity and cartesian closed monads_, Mathematica Scandinavica 29(2), 1971.

category: probability

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jointly epimorphic family

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## Definition

Given an [[object]] $X$ in some [[category]], a family $(f_i\colon U_i \to X)_i$ of [[morphisms]] to $X$ is an __epic [[sink]]__, or a __jointly epic family__ if, given any two morphisms $g, h\colon X \to Y$ such that $ g \circ f_i= h\circ f_i$ for all $i$, it follows that $g = h$.

Dually, a family $(f_i\colon X \to U_i)_i$ of morphisms from $X$ is a __monic source__, or a __jointly monic family__ if, given any two morphisms $g, h\colon Y \to X$ such that $f_i \circ g = f_i \circ h$ for all $i$, it follows that $g = h$.

Sometimes we are interested only in [[small families]] of morphisms, but if so then it is best to say so explicitly.

A single morphism $U \to X$ is an [[epimorphism]] if and only it forms an epic sink by itself; conversely, a sink $(f_i\colon U_i \to X)_i$ is epic iff the induced map $\coprod_i U_i \to X$ is an epimorphism, assuming that the [[coproduct]] $\coprod_i U_i$ exists.  (Note, though, that for a large family of morphisms, this coproduct might not exist even if the category has all small coproducts.)  Dual results hold for [[monomorphisms]] and [[products]].

Finally, the [[empty family]] of morphisms with domain $X$ is a monic source iff $X$ is a [[subterminal object]] (and dually).

## Examples

If a functor $F \colon J \to C$ has a colimit $\mathrm{colim}F$, with maps $\iota_i \colon F i \to \mathrm{colim}F$ for $ i \in J$, then the family $(\iota_i)_{i \in J}$ is jointly epic. Similarly, the maps $(\pi_i \colon \mathrm{lim}F \to F i)_{i\in J}$ are jointly monic, when the limit exists.

## Related concepts

* [[cancellative category]]

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jointly monic morphisms

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#### Category theory
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#Contents#
* table of contents
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## Definition

Given a [[category]] $C$ and objects $a, b \in Ob(C)$, a pair of [[morphisms]] $f, g \in C(a, b)$ is **jointly monic** if for every object $c: Ob(C)$ and pair of morphisms $h, k \in C(c, a)$, $f \circ h = f \circ k$ and $g \circ h = g \circ k$ imply that $h = k$.

In a [[well-pointed category]] $C$, given objects $a, b \in Ob(C)$, a pair of [[morphisms]] $f, g \in C(a, b)$ is **jointly injective** if for every [[global element]] $h, k \in C(1, a)$, $f \circ h = f \circ k$ and $g \circ h = g \circ k$ imply that $h = k$.

More generally, we can consider a **jointly monic family of morphisms**, where we indexed over a set $I$. When $I$ is a [[singleton set]], this reduces to a [[monomorphism]], and when $I$ is a two-element set, this reduces to a jointly monic pair. (And similarly for joint injectivity.)

## In tablular allegories

In every [[tabular allegory]], a relation $R$ could be factored into jointly monic [[map in a dagger 2-poset|maps]] $f$ and $g$ such that $f^\dagger \circ g = R$.

## Examples

* Taking $C = Set$, and ${ F_i : X \to Y }_{i \in I}$ to be a family of [[functors]], joint monicity of $(F_i)_{x, x'} : X(x, x') \to Y(F_i x, F_i x')$ is the notion of **joint faithfulness** (which specialises to the notion of [[faithful functor]] when $I$ is a singleton).

## See also

* [[monomorphism]]

* [[faithful functor]]

* [[internal relation]]

* [[regular category]]

* [[tabular allegory]]

* [[bicategory of relations]]

* [[Rel]]

* [[SEAR]]

* [[ETCR]]

* [[digraph]]

* [[loop digraph object]]

* [[toddtrimble:Theory of units and tabulations in allegories]]

## References

* [[Peter Freyd]], [[Andre Scedrov]]:

  **Categories, Allegories**

  Mathematical Library Vol 39, North-Holland (1990).

  [ISBN 978-0-444-70368-2](https://www.elsevier.com/books/categories-allegories/freyd/978-0-444-70368-2)

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