pure subobject

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Compact objects
+-- {: .hide}
[[!include compact object - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

A pure subobject is a [[monomorphism]] $A \rightarrowtail B$ -- hence a [[subobject]] $A$ of some [[object]] $B$ in some [[category]] -- which is a _[[pure morphism]]_: such that any sufficiently small system of [[equations]] involving constants in $A$ that admits a solution in $B$ also admits a solution in $A$. This generalises the classical notions of 'pure group' and 'pure submodule'.

## Definition

Let $\kappa$ be a regular [[cardinal]]. A **$\kappa$-[[pure morphism]]** in a [[category]] $\mathcal{C}$ is a morphism $f : A \to B$ with the following extension property:

* Given any morphisms $f' : A' \to B'$, $a : A' \to A$,  $b : B' \to B$ in $\mathcal{C}$, if both $A'$ and $B'$ are $\kappa$-[[compact object|compact]] and $f \circ a = b \circ f'$,

  $$
    \array{
      A' &\stackrel{a}{\to}& A
      \\
      \downarrow^{\mathrlap{f'}} && \downarrow^{\mathrlap{f}}
      \\
      B' &\stackrel{b}{\to}& B
    }
    \,,
  $$

  then there exists a (not necessarily unique) morphism $\bar{a} : B' \to A$ in $\mathcal{C}$ such that $a = \bar{a} \circ f'$. (We do not assert any compatibility with $b$, however.)

A **$\kappa$-pure subobject** is a $\kappa$-pure monomorphism.

## Examples

* A [[retract]] is a $\kappa$-pure subobject in any category, for any $\kappa$.

* Conversely, any $\kappa$-pure subobject in [[Set]] is a retract.

* If $A$ is an [[injective]] [[module]] and $B$ is any module containing $A$ as a submodule, then the inclusion $A \hookrightarrow B$ is $\kappa$-pure. (This can be checked directly without recourse to the fact that any injective submodule is a retract!)

* The torsion subgroup of any [[abelian group]] is a $\kappa$-pure subgroup, since it is a filtered colimit of [[direct sum|direct summands]]. (See below.)

## Properties
 {#Properties}

+-- {: .num_lemma}
###### Lemma

In any category:

* The class of $\kappa$-pure morphisms is closed under composition.

* If $g \circ f$ is a $\kappa$-pure morphism, then so is $f$.

* If $\kappa' \le \kappa$, then any $\kappa$-pure morphism is also $\kappa'$-pure.

=--

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

In a $\kappa$-[[accessible category]], any $\kappa$-pure morphism is necessarily monic.

=--

This is [LPAC, Prop. 2.29](#AdamekRosicky).

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

If $\mathcal{C}$ is a $\kappa$-accessible category, then $\kappa$-pure subobjects in $\mathcal{C}$ are closed under $\kappa$-[[filtered limit|filtered colimits]] in the [[arrow category]] $Arr (\mathcal{C})$.

If $\mathcal{C}$ is a $\kappa$-accessible category with [[pushout|pushouts]], then any $\kappa$-pure subobject in $\mathcal{C}$ is a $\kappa$-filtered colimit in $Arr (\mathcal{C})$ of retracts in $\mathcal{C}$.

=--

This is [LPAC, Prop. 2.30](#AdamekRosicky).

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

In a locally $\kappa$-[[locally presentable category|presentable category]], every $\kappa$-pure morphism is a [[regular monomorphism]].

=--

This is [LPAC, Prop. 2.31](#AdamekRosicky).

## Applications

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

Let $\mathcal{C}$ be a $\kappa$-accessible category, and let $\mathcal{D}$ be a full subcategory of $\mathcal{C}$ that is closed under $\kappa$-filtered colimits for some regular cardinal $\kappa$. Then, $\mathcal{D}$ is a $\mu$-accessible category for some regular cardinal $\mu$ sharply larger than $\kappa$ if and only if $\mathcal{D}$ is closed under $\kappa$-pure subobjects in $\mathcal{C}$.

In particular, a category $\mathcal{D}$ is accessible if and only if there is a fully faithful functor $R : \mathcal{D} \to Set^{\mathcal{A}}$ where $\mathcal{A}$ is small, $R$ [[created limit|creates]] colimits for all $\kappa$-filtered diagrams, and $\mathcal{D}$ is closed under $\kappa$-pure subobjects in $Set^{\mathcal{A}}$.

=--

This is [LPAC, Cor. 2.36](#AdamekRosicky).

## References

* [LPAC] [[Jiri Adamek]], [[Jiri Rosicky]], _[[Locally presentable and accessible categories]]_
 {#AdamekRosicky}

[[!redirects pure morphism]]
[[!redirects pure monomorphism]]

[[!redirects pure subobjects]]
[[!redirects pure morphisms]]
[[!redirects pure monomorphisms]]


pure tone

\tableofcontents

\section{Idea}

In [[acoustics]], [[signal processing]], and [[music]], a **pure tone** is a [[sound]] or [[signal]] with a [[sine function|sinusoidal]] [[waveform]].

\section{See also}

* [[overtone series]]

* [[sine function]]

* [[complex tone]]

\section{References}

See also:

* Wikipedia, _[Pure tone](https://en.wikipedia.org/wiki/Pure_tone)_


pure type system

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

In [[type theory]], under a *pure type system* one understand and explicitly typed [[lambda calculus]]
using [[dependent product]] as the type of lambda expressions:
the basic idea is that
$$\Gamma, x:A \vdash b:B$$
implies
$$\Gamma \vdash (\lambda x:A . b) : (\prod x:A . B).$$

In other words a _pure type system_ is

1. a system of [[natural deduction]];

1. for [[dependent types]];

1. and with (only) the [[dependent product type]] [[type formation|formation]] rule specified.

## Definition ##

A _pure type system_ is defined by

* a set $S$ of _sorts_, all of which are constants,
* a set $A$ of _axioms_ of the form $c : s$ with $c$ a constant and $s$ a sort,
* a set $R \hookrightarrow S \times S \times S$ of _rules_: triples $(s_{1}, s_{2}, s_{3}) $ of sorts.

  Rules $(s_{1}, s_{2}, s_{2})$ are abbreviated as $(s_{1}, s_{2})$.

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

These _rules_ will appear in the [[type formation]] rule for [[dependent product types]] below. They will say that for a type of sort $s_2$ [[dependent type|depending]] on a type of sort $s_1$ its dependent product is a type of sort $s_3$.

=--

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

In fact all rules $(s_{1}, s_{2}, s_{2})$ appearing in the following have $s_2 = s_3$. So we can just as well regard $R$ as a binary relation $R \hookrightarrow S\times S$ (rather than a ternary one).

=--

From the above input data we derive the following

1. The **terms** of a pure type system are the following (recursive definition):
   * variables and constants
   * $(\lambda x : A . B)$ (abstraction)
   * $(\prod x : A . B)$ (product)
   * $A B$ (application)

   Here $x$ is a variable and $A$, $B$ are terms. The operators $\lambda$ and $\prod$ bind the variable $x$.

1. The **typing of terms** is inductively defined by the following rules.

   A typing is of the form

   $$
     x_{1} : A_{1}, \dots, x_{n} : A_{n} \vdash A : B
   $$

   meaning that the types of the variables declared on the left induces the term $A$ has type $B$. Note that types are also terms.

   The order of variable  declarations is significant:
the declaration $x_{i} : A_{i}$ may depend on declarations to its left.

The [[natural deduction]] rules are defined to be the following, for all $s \in S$ and where $x$ ranges over the set of variables.

name          | rule                                                                                | condition                         |
--------------|---------------------------------------------------------------------------------------------------------------------------|-----------------------------------|
(axioms)      | $\vdash c : s$                                                                                | if $(c : s)$ is an axiom;         |
(start)       | $\frac{\Gamma \vdash A:s}{\Gamma, x : A \vdash x : A}$                                                             | if $x \notin \Gamma$              |
(weakening)   | $\frac{\Gamma \vdash A:B; \quad \Gamma \vdash C:s}{\Gamma, x : C \vdash A : B}$                                           | if $x \notin \Gamma$              |
(product)     | $\frac{\Gamma \vdash A : s_{1}; \quad \Gamma, x:A \vdash B : s_{2}}{\Gamma \vdash (\prod x:A . B) : s_{3}}$                | if $(s_{1}, s_{2}, s_{3}) \in R$; |
(application) | $\frac{\Gamma \vdash F : (\prod x:A . B); \quad \Gamma \vdash a : A}{\Gamma \vdash Fa : B [x := a]}$                      |                                   |
(abstraction) | $\frac{\Gamma, x:A \vdash b : B; \quad \Gamma \vdash (\prod x:A . B) : s}{\Gamma \vdash (\lambda x:A.b) : (\prod x:A.B)}$ |                                   |
(conversion)  | $\frac{\Gamma \vdash A : B; \quad \Gamma \vdash B' : s; \quad B =_{\beta} B'}{\Gamma \vdash A : B'}$                                   |                                   |

## Examples ##

### Strongly normalizing systems ###

#### Intuitionistic (higher-order) logic ####

#### Lambda cube ####

The _lambda cube_ ([Barendregt 91](#Barendregt91)) consists of eight systems arranged in a cube. The most expressive is given by the following choice of sorts, axioms and rules:

| symbol | actual value
|--------|-------------
| $S = $ | $\{\ast, \square\}$
| $A =$  | $\{(\ast : \square)\}$
| $R =$  | $\{(\ast, \ast), (\ast, \square), (\square, \ast), (\square, \square)\}$

(Here $\{\ast, \Box\}$ denotes the 2-element set, see [Barendregt 91, 2.1](#Barendregt91))

The other systems omit some of the last three rules.
Some specific systems are the following:

name | $(\ast, \ast)$ | $(\ast, \square)$ | $(\square, \ast)$ | $(\square, \square)$ |
-----|----------------|-------------------|-------------------|-----------------------|
$\lambda\rightarrow$ [[simply typed lambda calculus]] | yes |   |   |   |
${\lambda}P$ [[logical framework]] | yes | yes |   |   |
$\lambda2$ [[System F]] | yes |   | yes |   |
$\lambda\underline{\omega}$ | yes |   |   | yes |
${\lambda}C$ [[calculus of constructions]] | yes | yes | yes | yes |

#### Calculus of constructions {#CalculusOfConstructions}

For instance for the [[calculus of constructions]] we have

* $* =$ [[Prop]], the _[[type of propositions]]_

* $\Box = $ [[Type]], the _[[type of types]]_.

The single axiom $* \colon \Box$ hence says that $Prop \colon Type$, hence that [[Prop]] is a [[type]].

The rules express the usual rule for [[dependent product type]]:

* a dependent product over a generic [[dependent type]] is itself some type;

* but the dependent product of a dependent [[proposition]] is itself a proposition, namely the [[universal quantifier|universal quantification]].

### Inconsistent systems ###

The most permissive pure type system:

| symbol | actual value
|--------|-------------
| $S = $ | $\{\ast\}$
| $A = $ | $\{(\ast : \ast)\}$
| $R = $ | $\{(\ast, \ast)\}$

But there is an example with even non-circular system of axioms (System $\mathsf{U}^-$):

| symbol | actual value
|--------|-------------
| $S =$  | $\{ \ast, \square, \triangle\}$
| $A =$  | $\{(\ast : \square), (\square : \triangle)\}$
| $R = $ | $\{(\ast, \ast), (\square, \ast), (\square, \square), (\triangle, \square)\}$

## Related concepts

* [[simple type theory]]

* Pure type systems can be augmented with a *cumulativity* relation between sorts, so that if $s_1 \preceq s_2$, then any type in $s_1$ is also in $s_2$; see [Barras-Gregoire](#BG05).

* The _[[calculus of inductive constructions]]_ can be formulated as a particular pure type system (with a hierarchy of [[type of types]]) augmented by rules for introducing [[inductive types]].

## References ##

* {#Barendregt91} Henk Barendregt, _Introduction to generalized type systems_, J. Funct. Program. 1(2), 1991 ([pdf](http://patryshev.com/books/barendregt.pdf))

* Henk Barendregt (Catholic University Nijmegen), _[[Lambda calculi with types]]_, To appear in [[Samson Abramsky]], D.M. Gabbay and T.S.E. Maibaum
(eds.) _Handbook of Logic in Computer Science_, Volume II,
Oxford University Press. ([preprint pdf](http://ttic.uchicago.edu/~dreyer/course/papers/barendregt.pdf))

Survey includes

* Frade, _Calculus of inductive constructions_ ([pdf](http://www4.di.uminho.pt/~mjf/pub/SFV-CIC-2up.pdf))

* {#Cody15} [[Cody Roux]], _Pure type systems: Dependents when you need them_, talk at Boston Haskell Meetup 2015 ([slides](http://de.slideshare.net/imalsogreg/cody-roux-pure-type-systems-boston-haskell-meetup),[video](https://www.youtube.com/watch?v=ZGqKsalJi4s))

A generalization to cumulativity can be found in

* Bruno Barras and Benjamin Gregoire, *On the role of type decorations in the Calculus of Inductive Constructions*, Lecture Notes in Computer Science Volume 3634, 2005, pp 151-166, [PDF](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.61.2666&rep=rep1&type=pdf)
 {#BG05}

[[!redirects pure type systems]]


Pursuing Stacks

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

This page collects material related to the text:

* {#Reference} [[Alexander Grothendieck]],

  _À la poursuite des champs_

  (1983)

  [English pdf](https://thescrivener.github.io/PursuingStacks/ps-online.pdf)

  \linebreak

  [[Georges Maltsiniotis]] (ed.)

  Documents mathématiques **20**

  Société mathématique de France (2022)

  [ISBN:978-2-85629-958-6](https://smf.emath.fr/publications/la-poursuite-des-champs-volume-i)

on [[homotopy theory]] via [[category theory]] ([[test categories]]) and (rather informally) on [[stacks]] and [[higher toposes]].

\linebreak

_Pursuing Stacks_ is an influential manuscript written by [[Alexander Grothendieck]] in 1983, and of which copies were sent to [[Ronnie Brown]] and [[Larry Breen]]. It is written in English "in response to a correspondence in English". He intended that later volumes would be in French, entitled  _À la poursuite des [[stack|champs]]_ (literally _In Pursuit of Stacks_).  Grothendieck gave permission to Ronnie Brown to copy  the correspondence, and so this volume gradually circulated.

Further ideas in a later manuscript, *[[Les Derivateurs]]*, were made available by [[Georges Maltsiniotis]]. A main theme is the [[homotopy coherent diagram|homotopy theory of diagrams]].

Preliminary to the circulated Pursuing Stacks is a 12 page letter to [[Daniel Quillen]], dated 19-02-1983, who did not respond. The work then proceeds as a sort of research diary of about 600 pages, including many back-trackings and corrections.  The ideas presented have proved hugely influential, even to this day, in [[homotopy theory]], [[higher category theory]] and [[higher geometry]]. Indeed, many of them have now been thoroughly worked over, 'pursued' and published (see below).  We  note however that this does not mean "Pursuing stacks" did not contain many rigorously worked notions and results itself. On the other hand one of the values of the document is to show how Grothendieck went about developing his ideas, and he was insistent that the document should be published, if at all,  "as is", so that young people could see that even well known people could make errors.

#Contents#
* table of contents
{:toc}

## Download

For an account of the origins of the manuscript, see [Ronnie Brown's account](https://groupoids.org.uk/pstacks.html), where a large downloadable .ps file may be found; see the note for February 7, 2013 on that webpage (but the links are, at the moment, dead, so use the one in the next line here.

Scanned copies of the original typescript are available [here](http://modular.fas.harvard.edu/sga/djvu/Pursuing%20Stacks.tif.djvu) (in djvu, warning: 23MB), [here](http://www.math.umn.edu/~messing/docs/) (djvu and pdf - 252MB), or [here](http://preschema.com/pursuing-stacks/) (png images) (some of these links seem no longer to work!). [Page](http://webusers.imj-prg.fr/~georges.maltsiniotis/ps.html) by Maltsiniotis.

There is now a set of [LaTeX files on GitHub](https://github.com/thescrivener/PursuingStacks) with corresponding [PDFs on GitHub Pages](https://thescrivener.github.io/PursuingStacks/). The original source tarball is still here, but an administrator should delete it: [[PursuingStacks.tar.gz:file]].

## Content and influence on current research

Grothendieck considered among other things, the notion of $n$-[[n-groupoid|groupoids]] and $\infty$-[[omega-groupoid|groupoids]], [[homotopy types]] and how to model them, [[homology]] and [[cohomology]] theories defined on categories of models and schematisation of homotopy types. This last is an attempt to define homotopy theory relative to a base ring, say, such that over $\mathbb{Z}$ ordinary homotopy theory is recovered.

The following summary is due to [[David Roberts]] (from this [discussion on MO](http://mathoverflow.net/questions/115549/conjectures-in-grothendiecks-pursuing-stacks)):

Pursuing Stacks is composed of (if memory serves correctly) three themes. The first was [[homotopy types]] as [[infinity-groupoid|higher (non-strict) groupoids]]. This part was first considered in Grothendieck's letters to [[Larry Breen]] from 1975, and is mostly contained in the letter to [[Daniel Quillen]] which makes up the first part of PS (about 12 pages or so). [[Georges Maltsiniotis]] has extracted [[Grothendieck-Maltsiniotis omega-groupoid|Grothendieck's proposed definition for a weak ∞-groupoid]], and there is by [[Dimitri Ara]] ([Ara 10](Grothendieck-Maltsiniotis+infinity-category#Ara10)) towards showing that this definition satisfies the [[homotopy hypothesis]].

The other parts (not entirely inseparable) are the first thoughts on [[derivators]], which were later taken up in great detail in Grothendieck's 1990-91 [notes](http://www.math.jussieu.fr/~maltsin/groth/Derivateursengl.html) (see there for extensive literature relating to derivators, the first 15 of 19 chapters of [[Les Dérivateurs]] are themselves available), and the 'schematisation of homotopy types', which is covered by work of [[Bertrand Toën]], [[Gabriele Vezzosi]] and others on [[homotopical algebraic geometry]] (e.g. [HAG I](http://www.dma.unifi.it/~vezzosi/papers/hagI.pdf), [HAG II](http://www.dma.unifi.it/~vezzosi/papers/hagIIfin.pdf)) using [[simplicial sheaves]] on schemes. This has taken off with work of [[Jacob Lurie]], [[Charles Rezk]] and others dealing with [[(infinity,1)-topos theory]]  and [[derived algebraic geometry]], which is going far ahead of what Grothendieck probably envisaged.

{#JoyalLetter}  In conversation with Grothendieck on these matters, [André Joyal, 1984](model+structure+on+simplicial+sheaves#Joyal1984) described the *[[model structure on simplicial sheaves]]* which, in hindsight, is an early model (along with that by [[BrownAHT|Brown 1973]]) for [[infinity-stack|$\infty$-stacks]] as now in use in modern [[(infinity,1)-topos theory|higher topos theory]].

The theory of [[localizers]] and [[modelizers]], Grothendieck's conception of homotopy theory, is covered in the work of [[Denis-Charles Cisinski]].

## History
 {#History}

In his _Obituary to Aurelio Carboni_ ([pdf](https://www.acsu.buffalo.edu/~wlawvere/FarewellAurelio.htm)) [[William Lawvere]] writes the following:

In 1989 Aurelio proposed to accompany me with his Alfa Romeo to visit Alexander Grothendieck in his stone hut in the middle of a lavender field near Mormoiron in the South of France. I had already visited Grothendieck in 1981 at that stone
hut, but this
time there
was a specific
item
that
needed
to
be
negotiated.
Grothendieck
had
stated
that
I
might
be
the
appropriate
person
to
edit
and
publish
his
great
work
'Pursuing
Stacks'.
But
I
had
very
serious
questions
about
how
that
editing
should
be
done
in
particular.
Aurelio
was
attentively
studying
the
situation.
The
discussion
was
complicated
by
the
fact
that
Grothendieck
was
bound
by
a
religious
vow
of
silence,
so
that
he
could
only
communicate
by
writing
(though
he
had
given
me
a
very
friendly
one-word
greeting:
'Bill!')
He
wrote
immediately
that
he
was
under
a
vow
not
to
discuss
mathematics.
But
his
mathematical
soul
soon
triumphed
and
he
was
writing
mathematical
statements
and
questions.
(Aurelio
observed
all
this
with
valiant
equanimity.)
The
1983
work
of
Grothendieck
under
discussion
concerned
among
many
other
questions
the
homotopy
theory
of
presheaf
toposes.
It
is
a
long
manuscript,
produced
by
the
basic
method
of
typing
all
night
for
several
weeks.
Naturally,
mistakes
occurred
due
to
fatigue,
but
the
fever
to
press
on
had
meant
that
each
morning
when
Alexander
noted
the
previous
evening's
errors,
he
did
not
delete
them,
but
simply
kept
them,
and
added
the
corrected
version
before
proceeding.
I
said
that
as
a
conscientious
editor
I
would
have
to
delete
the
erroneous
passages
and
give
substantial
explanations
of
my
own.
Grothendieck
insisted
that
the
errors
be
kept,
so
that
students
would
have
the
opportunity
to
learn
that
even
famous
mathematicians
make
mistakes.
I
objected
to
this,
pointing
out
that
students
have
ample
opportunity
to
see THAT and
that learning
the actual
scientific material
is difficult enough without the extra punishment that he was in effect advocating. But our parting was very amicable with the agreement to further consider the matter. Aurelio agreed with my point. In the end the publication did not take place. Only several years later did Maltsiniotis and Cisinski work out the mathematics and bring this magnificent homotopy program to an initial fruition.

category: reference

[[!redirects Pursuing stacks]]
[[!redirects À la poursuite des champs]]


Pursuing stacks > history

< [[Pursuing stacks]]
[[!redirects Pursuing stacks -- history]]


pushforward

Pushforward tends to refer to [[covariant]] operations,
typically defined in a geometric setting.
It can refer to

* [[pushforward measure]]
* [[fiberwise integral]]
* [[direct image]] of a [[sheaf]]


pushforward measure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

In [[measure theory]], the _pushforward_ $f_\ast \mu$ of a [[measure]] $\mu$ on a [[measurable space]] $X$ along a [[measurable function]] $f \colon X \to Y$ to another [[measure space]] $Y$ assigns to a subset the original measure of the [[preimage]] under $f$ of that subset:

$$
  (f_\ast\mu)( - ) \coloneqq \mu(f^{-1}(-))
  \,.
$$

## Properties

### General

* The pushforward of a [[τ-additive measure]] along a [[continuous map]] is again τ-additive.

* If a [[continuous valuation]] $\nu$ on a topological space $X$ is [[valuation (measure theory)#extending_valuations_to_measures|extendable to a measure]] $\nu$, its [[valuation (measure theory)#pushforward|pushforward (as a valuation)]] $f_*\mu$ along a [[continuous map]] $f:X\to Y$ is again extendable, and it extends to $f_*\nu$.

* The pushforward measure along a [[product]] projection is called a [[marginal measure]].

### Relation to entropy

\begin{prop}\label{EntropyDoesNotIncreaseUnderPushforward}
  **([[entropy]] does not increase under [[pushforward measure|pushforward]])**
  Let $f \colon X \longrightarrow Y$ be a [[measurable function]] between [[measure spaces]], and let $\mu$ be a [[probability distribution]] on $X$. Then the [[entropy]] of $\mu$ is larger or equal to that of its [[pushforward measure|pushforward]] distribution $f_\ast \mu$:

$$
  S(\mu)
  \;\geq\;
  S
  \big(
    f_\ast(\mu)
  \big)
  \,.
$$

\end{prop}

(e.g. [Austin, Prop. 2.7](#Austin))

### Relation to the Giry monad

The construction of pushforward measures is how one can make the [[Giry monad]], or other [[measure monads]], [[functor|functorial]]. Given a measurable (or continuous, etc.) map $f \colon X\to Y$, the pushforward gives a well-defined, measurable map $P X\to P Y$ (where $P$ denotes the [[Giry monad]]), making $P$ into a [[functor]].

## Related concepts

* [[Borel measure]], [[Radon measure]], [[τ-additive measure]]

* [[monads of probability, measures, and valuations]]

* [[Giry monad]]

* [[conditional expectation]]

* [[pullback of a function]]

* [[pullback in cohomology]]

## References

See also

* Wikipedia, _[Pushforward measure](https://en.wikipedia.org/wiki/Pushforward_measure)_

In relation to [[entropy]]:

* {#Austin} Tim D. Austin, *Entropy and Sinai’s Theorem* ([pdf](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.135.8492&rep=rep1&type=pdf), [[AustinEntropyAndSinaiTheorem.pdf:file]])

[[!redirects pushforward measures]]
[[!redirects push-forward measure]]
[[!redirects push-forward measures]]
[[!redirects pushforward of a measure]]
[[!redirects pushforward of measures]]
[[!redirects push-forward of a measure]]
[[!redirects push-forward of measures]]
[[!redirects image measure]]
[[!redirects image measures]]


pushout

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

# Contents
* table of contents
{: toc}

## Idea

A **pushout** is an ubiquitous construction in [[category theory]] providing a [[colimit]] for the diagram $\bullet\leftarrow\bullet\rightarrow\bullet$. It is dual to the notion of a [[pullback]].

## Pushouts in $Set$

In the category [[Set]] a 'pushout' is a quotient of the disjoint union of two sets.  Given a diagram of sets and functions like this:

\begin{center}
  \begin{tikzcd}
      & C \ar[ld, "f"'] \ar[rd, "g"] \\
    A &   & B
  \end{tikzcd}
\end{center}

the 'pushout' of this diagram is the set $X$ obtained by taking the disjoint union $A + B$ and identifying $a \in A$ with $b \in B$ if there exists $x \in C$ such that $f(x) = a$ and $g(x) = b$ (and all identifications that follow to keep [[equality]] an [[equivalence relation]]).

This construction comes up, for example, when $C$ is the intersection of the sets $A$ and $B$, and $f$ and $g$ are the obvious inclusions.  Then the pushout is just the union of $A$ and $B$.

Note that there are maps $i_A : A \to X$, $i_B : B \to X$ such that $i_A(a) = [a]$ and $i_B(b) = [b]$ respectively.  These maps make this square commute:

\begin{center}
  \begin{tikzcd}
                      & C \ar[ld, "f"'] \ar[rd, "g"] \\
    A \ar[rd, "i_A"'] &   & B \ar[ld, "i_B"] \\
                      & X
  \end{tikzcd}
\end{center}

In fact, the pushout is the [[universal property|universal]] solution to finding a [[commutative square]] like this.  In other words, given _any_ commutative square

\begin{center}
    \begin{tikzcd}
                             & C \arrow[ld, "f"'] \arrow[rd, "g"] &                     \\
        A \arrow[rd, "j_A"'] &                                    & B \arrow[ld, "j_B"] \\
                             & Y                                  &
    \end{tikzcd}
\end{center}

there is a unique function $h: X \to Y$ such that
$$ h i_A = j_A $$
and
$$ h i_B = j_B .$$
Since this universal property expresses the concept of pushout purely arrow-theoretically, we can formulate it in any category.  It is, in fact, a simple special case of a [[colimit]].

## Definition

A **pushout** is a [[colimit]] of a [[diagram]] like this:

\begin{center}
    \begin{tikzcd}
          & c \arrow[ld, "f"'] \arrow[rd, "g"] &   \\
        a &                                    & b
    \end{tikzcd}
\end{center}

Such a diagram is called a [[span]].  If the colimit exists, we obtain a [[commutative square]]

\begin{center}
    \begin{tikzcd}
                             & c \arrow[ld, "f"'] \arrow[rd, "g"] &                     \\
        a \arrow[rd, "i_a"'] &                                    & b \arrow[ld, "i_b"] \\
                             & \chi                               &
    \end{tikzcd}
\end{center}

and the object $x$ is also called the **pushout**.  It has the universal property already described above in the special case of the category $Set$.

Other terms: $x$ is a __cofibred coproduct__ of $a$ and $b$, or (especially in [[algebraic categories]] when $f$ and $g$ are [[monomorphisms]]) a free product of $a$ and $b$ with $c$ __[[amalgamation|amalgamated]] sum__ ([Gabriel & Zisman (1967), p. 1](#GabrielZisman67)) or more simply an __[[amalgamation]]__ (or __amalgam__) of $a$ and $b$.

The concept of pushout is a special case of the notion of **[[cofiber coproduct|wide pushout]]** (compare [[wide pullback]]), where one takes the colimit of a diagram which consists of a set of arrows $\{f_i: c \to a_i\}_{i \in I}$. Thus an ordinary pushout is the case where $I$ has cardinality $2$.

Note that the concept of pushout is dual to the concept of [[pullback]]: that is, a pushout in $C$ is the same as a pullback in $C^{op}$.

See [[pullback]] for more details.

## Properties

### In any category

+-- {: .num_prop #PushoutsAsCoequalizers}
###### Proposition
**(pushouts as coequalizers)

If [[coproducts]] exist in some [[category]], then the pushout

\begin{center}
    \begin{tikzcd}
        a \ar[r, "f"] \ar[d, "g"'] & b \ar[d] \\
        c \ar[r]                   & b \underset{a}{\sqcup} c
    \end{tikzcd}
\end{center}

is equivalently the [[coequalizer]]

\begin{center}
  \begin{tikzcd}
     a
     \ar[r, "i_1 \circ f", shift left]
     \ar[r, "i_2 \circ g"', shift right]
     &
     b \sqcup  c
     \arrow[r]
     &[-35pt]
     b
       \underset{a}{\sqcup}
     c
  \end{tikzcd}
\end{center}

of the two morphisms induced by $f$ and $g$ into the [[coproduct]] of $b$ with $c$.

=--

+-- {: .num_prop #PushoutPreservesEpimorphisms}
###### Proposition
**(pushouts preserves epimorphisms and isomorphisms)

Pushouts preserve [[epimorphisms]] and [[isomorphisms]]:

If

\begin{center}
    \begin{tikzcd}
        a \arrow[r, "f"] \arrow[d, "g"'] & b \arrow[d, "f_\ast g"] \\
        c \arrow[r]                      & d
    \end{tikzcd}
\end{center}

is a pushout square in some category then:

1. if $g$ is a [[epimorphism]] then $f_\ast g$ is an epimorphism;

1. if $g$ is an [[isomorphism]] then $f_\ast g$ is an isomorphism.

=--

+-- {: .num_prop}
###### Proposition
**([[pasting law for pushouts]])**

Consider a [[commuting diagram]] of the following shape in any category:

\begin{center}
    \begin{tikzcd}
        x \arrow[r] \arrow[d] & y \arrow[r] \arrow[d] & z \arrow[d] \\
        u \arrow[r]           & v \arrow[r]           & w
    \end{tikzcd}
\end{center}

If the left square is a [[pushout]], then the total rectangle is a pushout if and only if the right square is a pushout.

=--

+-- {: .proof}
###### Proof
See the proof of the dual property for [[pullbacks]].
=--

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

The converse implication does not hold: it may happen that the outer and the right square are pushouts, but not the left square.

=--

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

See the proof of the dual proposition for [[pullbacks]].

=--

### In a quasitopos

+-- {: .num_prop #PushoutOfStrongMonomorphismInQuasitopos}
###### Proposition
**pushout of [[strong monomorphism]] in [[quasitopos]]**

Suppose that $(\mathrm{T},\mathcal{C})$ is either

* ([[monomorphism]],[[topos]]), or
* ([[strong monomorphism]],[[quasitopos]])

Suppose that

\begin{center}
    \begin{tikzcd}
        {O_{0,1}} \arrow[r] \arrow[d, "m"'] & {O_{1,1}} \arrow[d, "h"] \\
        {O_{0,0}} \arrow[r]                 & {O_{1,0}}
    \end{tikzcd}
\end{center}

is a [[commutative diagram]] in $\mathcal{C}$ such that

* $m$ is $\mathrm{T}$ in $\mathcal{C}$
* the diagram is a pushout in $\mathcal{C}$

Then

* $h$ is $\mathrm{T}$ in $\mathcal{C}$
* the diagram is a pullback in $\mathcal{C}$

=--

See at _[[quasitopos]]_ [this lemma](quasitopos#PushoutOfStrongMonos). Note that the result for quasitoposes immediately implies the result for toposes, since all monomorphisms $i: A \to B$ in a topos are [[regular monomorphism|regular]] ($i$ being the equalizer of the arrows $\chi_i, t \circ !: B \to \Omega$ in

\begin{center}
    \begin{tikzcd}
                                               & 1 \arrow[d, "t"] \\
        B \arrow[ru, "!"] \arrow[r, "\chi_i"'] & \Omega
    \end{tikzcd}
\end{center}

where $\chi_i$ is the classifying map of $i$) and therefore strong.

## Examples
 {#Examples}

\begin{example}
  A [[pushout]] of [[injections]] of [[Sets]] is called the [[union]] of the sets.
\end{example}

\begin{example}
  A pushout of [[groups]] in [[Grps]] is called their *[[amalgamated free product]]*
\end{example}

\begin{example}
  In [[topology]], [[space attachments]] are pushouts in [[Top]].
\end{example}

## Related entries

* [[amalgamation property]]

* [[wide pushout]]

* [[coequalizer]]

* [[colimit]]

* [[pullback]]

* [[pushout-product]]

* [[homotopy pushout]]

## References

Early use of the terminology "pushout":

* {#Mitchell65} [[Barry Mitchell]], Section I.7 of: *Theory of categories*, Pure and Applied Mathematics **17**, Academic Press (1965) [[ISBN:978-0-12-499250-4](https://www.elsevier.com/books/theory-of-categories/mitchell/978-0-12-499250-4)]

Early use of the terminology "amalgamated sums":

* {#GabrielZisman67} [[Pierre Gabriel]], [[Michel Zisman]], p 1 of: *[[Calculus of fractions and homotopy theory]]*, Ergebnisse der Mathematik und ihrer Grenzgebiete **35**, Springer (1967) [[doi:10.1007/978-3-642-85844-4](https://link.springer.com/book/10.1007/978-3-642-85844-4), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf)]

Textbook accounts:

* [[Saunders MacLane]], p. 65-66 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Francis Borceux]], Section 2.5 in Vol. 1: *Basic Category Theory* of: *[[Handbook of Categorical Algebra]]*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858))

[[!redirects pushout]]
[[!redirects pushouts]]
[[!redirects pushout diagram]]
[[!redirects pushout diagrams]]
[[!redirects cocartesian diagram]]
[[!redirects cocartesian diagrams]]
[[!redirects cocartesian square]]
[[!redirects cocartesian squares]]

[[!redirects cofibred coproduct]]
[[!redirects cofibred coproducts]]
[[!redirects cofibered coproduct]]
[[!redirects cofibered coproducts]]
[[!redirects cofiber coproduct]]
[[!redirects cofiber coproducts]]


pushout complement

# Pushout complements

* table of contents
{: toc}

## Idea

A *pushout complement* is a "completion of a pair of [[arrows]] to a [[pushout]]".  Intuitively, it is a way to "remove part of an [[object]] of a [[category]]" while retaining information about how that part is "connected" to other parts.  Pushout complements play an important role in some kinds of [[span rewriting]].

## Definition

Given [[morphisms]] $m:C\to A$ and $g:A\to D$ in a [[category]], a **pushout complement** is a pair of arrows $f:C\to B$ and $n:B\to D$ such that the square
$$ \array{ C & \to & A \\ \downarrow & & \downarrow \\ B & \to & D }$$
[[commutative square|commutes]] and is a [[pushout]].

In general, even in good categories, pushout complements may not exist.  However, if they do exist, then as long as the category is [[adhesive category|adhesive]] (such as a [[topos]]) and $m$ is a [[monomorphism]], they are unique up to unique isomorphism.  In the language of [[homotopy type theory]], the [[type]] of such pushout complements is an [[h-proposition]].

## Examples

* In the category [[Set]], if $m$ and $g$ are [[injections]] then a pushout complement always exists: take $B = D \setminus (A\setminus C)$.

* In a [[coherent category]], if $C=\emptyset$ and $g$ is a [[monomorphism]], then a pushout complement is the same as an ordinary [[complement]] for the [[subobject]] $A\to D$.

* In a category of [[graphs]], pushout complements play an important role in [[double pushout graph rewriting]].

* The dual of a pushout complement is a *pullback complement*.  Important pullback complements include [[final pullback complements]], which arise from [[exponential objects]] of [[monomorphisms]].

## References

* H. Ehrig, M. Pfender and H. J. Schneider. _Graph-grammars: An algebraic approach_, 14th Annual Symposium on Switching and Automata Theory, 1973, pp. 167-180. ([doi:10.1109/SWAT.1973.11](https://doi.org/10.1109/SWAT.1973.11))

* {#LS} [[Steve Lack]] and [[Pawel Sobocinski]], _Adhesive categories_, Foundations of Software Science and Computation Structures. FoSSaCS 2004. Lecture Notes in Computer Science, vol 2987. Springer, Berlin, Heidelberg. doi:[10.1007/978-3-540-24727-2_20](https://doi.org/10.1007/978-3-540-24727-2_20), [PDF](https://www.ioc.ee/~pawel/papers/adhesive.pdf)

[[!redirects pushout complements]]
[[!redirects pullback complement]]
[[!redirects pullback complements]]


pushout type

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

#Contents#
* table of contents
{:toc}

## Idea

In [[homotopy type theory]], a _homotopy pushout type_ is a kind of [[higher inductive type]] whose [[categorical semantics]] is that of [[homotopy pushouts]].

## Definition

### As a higher inductive type
 {#AsAHigherInductiveType}

The (homotopy) pushout of a span $A \xleftarrow{f} C \xrightarrow{g} B$ is the [[higher inductive type]] $A \sqcup^{C}_{f ; g} B$ (or $A \sqcup^C B$) generated by:

* a function $inl : A \to A \sqcup^{C} B$
* a function $inr : B \to A \sqcup^{C} B$
* for each $c:C$ a path $glue(c) : inl (f (c)) = inr (g( c))$

In [[Coq]] pseudocode:

    Inductive hpushout {A B C : Type} (f : A -> B) (g : A -> C) : Type :=
    | inl : B -> hpushout f g
    | inr : C -> hpushout f g
    | glue : forall (a : A), inl (f a) == inr (g a).

### Inference rules
{#InferenceRules}

Explicitly, this means that the [[inference rules]] for pushout types are as follows (where we now label terms as shown in the following diagram):

\begin{tikzcd}[sep=40pt]
  &
  Y
  \ar[
    dr,
    "{ \mathrm{cpr} }"{description, sloped}
  ]
  \ar[
    dd,
    Rightarrow,
    shorten=11pt,
    start anchor={[xshift=+6pt]},
    end anchor={[xshift=-6pt]},
    "{ \mathrm{hmt} }"{description}
  ]
  \ar[
    rrrd,
    bend left=30,
    "{ \mathrm{cpr}_P }",
    "{\ }"{name=s, swap, pos=.55}
  ]
  \\[-20pt]
  X
  \ar[
    ur,
    "{ f }"
  ]
  \ar[
    dr,
    "{ f' }"{swap}
  ]
  &&
  Y \underset{X}{\sqcup} Y'
  \ar[ddr, equals, gray]
  \ar[
    rr,
    dashed,
    "{ \mathrm{ind}_{(P,\, \cdots)} }"
  ]
  &&
  P
  \ar[ddl, gray, "{ \pi_P }"{description}]
  \\
  &
  Y'
  \ar[
    ur,
    shorten >=-6pt,
    "{ \mathrm{cpr}' }"{description, sloped}
  ]
  \ar[
    rrru,
    bend right=45,
    "{ \mathrm{cpr}_{P'} }"{swap, pos=.35},
    "{\ }"{name=t, pos=.4}
  ]
  \ar[
    from=s,
    to=t,
    shorten=8pt,
    Rightarrow,
    crossing over,
    "{\mathrm{hmt}_P}"{description, pos=.7}
  ]
  \\[-100pt]
  && &
  \color{gray}
  Y \underset{X}{\sqcup} Y'
\end{tikzcd}

**[[type formation rule]]:**

$$
  \frac{
    \begin{array}{l}
    X,\, Y,\, Y^' \,\colon\, Type
    \;;
    \;\;
    f \,\colon\, X \to Y
    \;,
    \;\;
    f^' \,\colon\, X \to Y^'
    \mathclap{\phantom{\vert_{\vert}}}
    \end{array}
  }
  {
    \mathclap{\phantom{\vert^{\vert}}}
    Y \underset{X}{\sqcup} Y^'
    \,\colon\,
    Type
  }
$$

\linebreak

**[[term introduction rules]]:**

$$
  \frac{
    y \,\colon\, Y
    \mathclap{\phantom{\vert_{\vert}}}
  }{
    \mathclap{\phantom{\vert^{\vert}}}
    cpr(y)
    \,\colon\,
    Y \underset{X}{\sqcup} Y^'
  }
  \;\;\;\;\;\;\;\;\;
  \frac{
    y^' \,\colon\, Y^'
    \mathclap{\phantom{\vert_{\vert}}}
  }{
    \mathclap{\phantom{\vert^{\vert}}}
    cpr^'(y^')
    \,\colon\,
    Y \underset{X}{\sqcup} Y^'
  }
  \;\;\;\;\;\;\;\;\;
  \frac{
    x \,\colon\, X
    \mathclap{\phantom{\vert}}
  }{
    \mathclap{\phantom{\vert^{\vert^{\vert}}}}
    hmt(x)
    \,\colon\,
    Id_{\big(Y \underset{X}{\sqcup} Y^'\big)}
    \Big(
      cpr\big(f(x)\big)
      ,\,
      cpr^'\big(f^'(x)\big)
    \Big)
  }
$$

\linebreak

**[[term elimination rule]]:**

$$
  \frac{
    \begin{array}{l}
    \hat{y}
      \,\colon\,
    Y \underset{X}{\sqcup} Y^'
    \;\vdash\;
    P(\hat{y})
    \,\colon\,
    Type
    \;;
    \\
    cpr_P
    \,\colon\,
    \underset{ y \colon Y }{\prod}
    P\big(
      cpr(y)
    \big)
    \;;
    \;\;
    cpr^'_P
    \,\colon\,
    \underset{ y^' \colon Y^' }{\prod}
    P\big(
      cpr^'(y^')
    \big)
    \;;
    \\
    hmt_P
    \,\colon\,
    \underset{x \colon X}{\prod}
    Id_{P\Big( cpr^'\big(f^'(x)\big) \Big)}
    \bigg(
      hmt(x)_\ast
      \Big(
        cpr_P\big(
          f(x)
        \big)
      \Big)
      ,\,
        cpr^'_P\big(
          f^'(x)
        \big)
    \bigg)
    \end{array}
  }{
    \mathclap{\phantom{\vert^{\vert}}}
    ind_{\big(P,\, cpr_P,\,cpr^'_P,\,hmt_P\big)}
    \,\colon\,
    \underset{
      \hat{y} \,\colon\, Y \underset{X}{\sqcup} Y^'
    }{\prod}
    \,
    P(\hat{y})
  }
$$

\linebreak

**[[computation rules]]:**

$$
  \frac{
    \begin{array}{l}
    \hat{y}
      \,\colon\,
    Y \underset{X}{\sqcup} Y^'
    \;\vdash\;
    P(\hat{y})
    \,\colon\,
    Type
    \;;
    \\
    cpr_P
    \,\colon\,
    \underset{ y \colon Y }{\prod}
    P\big(
      cpr^'(y)
    \big)
    \;;
    \;\;
    cpr^'_P
    \,\colon\,
    \underset{ y^' \colon Y^' }{\prod}
    P\big(
      cpr^'(y^')
    \big)
    \;;
    \\
    hmt_P
    \,\colon\,
    \underset{x \colon X}{\prod}
    Id_{P(x)}
    \bigg(
      hmt(x)_\ast
      \Big(
        cpr_P\big(
          f(x)
        \big)
      \Big)
      ,\,
        cpr^'_P\big(
          cpr^'(x)
        \big)
    \bigg)
    \end{array}
  }{
    \begin{array}{rlc}
    \mathclap{\phantom{\vert^{\vert}}}
    ind_{\big(P,\, cpr_P,\,cpr^'_P,\,hmt_P\big)}
      \,\circ\,
    cpr
    &=&
    cpr_P
    \;;
    \\
    \mathclap{\phantom{\vert^{\vert}}}
    ind_{\big(P,\, cpr_P,\,cpr^'_P,\,hmt_P\big)}
      \,\circ\,
    cpr^'
    &=&
    cpr^'_P
    \;;
    \\
    happly_{\big(
      ind_{\big(P,\, cpr_P,\,cpr^'_P,\,hmt_P\big)}
    \big)}
    \,\circ\,
    hmt
    &=&
    hmt_P
    \end{array}
  }
$$

\linebreak

Here:

* "$Id_X$" denotes the [[identity type]] of the type $X$;

* "=" denotes [[judgemental equality]];

*  "$happly$" denotes the [[dependent type|dependently typed]] version of the operation [here](function+extensionality#eq:happlyDefinition).

## Examples ##

* [[suspension type]]

* [[join type]]

* [[wedge sum type]]

* [[smash product type]]

* [[cofiber type]]

## See also

* [[higher inductive type]]

* [[dependent pushout type]]

* [[pushout]], [[homotopy pushout]]

## References ##

Early discussion of the idea of homotopy pushouts as [[higher inductive types]]:

* [[Mike Shulman]], _Homotopy Type Theory VI_ (2011) [[blog post](http://golem.ph.utexas.edu/category/2011/04/homotopy_type_theory_vi.html)]

Many references on [[higher inductive types]] mention pushout types and highlight their categorical semantics, but skip type-theoretic details, e.g.:

* [[Univalent Foundations Project]], §6.8 of: *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

The full [[inference rules]] are stated in:

* {#FFLL16} [[Kuen-Bang Hou]] ([[Favonia]]), [[Eric Finster]], [[Dan Licata]], [[Peter LeFanu Lumsdaine]], p. 4 of: *A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory*, LICS '16 (2016) 565–574  [[arXiv.1605.03227](https://arxiv.org/abs/1605.03227), [doi:10.1145/2933575.2934545](https://doi.org/10.1145/2933575.2934545)]

as well as in section 6 of:

* {#LumsdaineShulman17} [[Peter LeFanu Lumsdaine]], [[Mike Shulman]], *Semantics of higher inductive types*, Math. Proc. Camb. Phil. Soc. **169** (2020) 159-208 [[arXiv:1705.07088](https://arxiv.org/abs/1705.07088), talk slides [pdf](http://home.sandiego.edu/~shulman/papers/cellcxs.pdf), [doi:10.1017/S030500411900015X](https://doi.org/10.1017/S030500411900015X)]

See also:

* [[Egbert Rijke]], *Homotopy pushouts* [[pdf](https://www.andrew.cmu.edu/user/erijke/hott/pushout.pdf)], 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/)]

Formalization in the *[[Coq]]* [[proof assistant]]:

* {#Brunerie} [[Guillaume Brunerie]], _[HoTT/Coq/Limits/Pullbacks.v](https://github.com/guillaumebrunerie/HoTT/blob/master/Coq/Limits/Pullbacks.v)_

[[!redirects pushout types]]

[[!redirects homotopy pushout type]]
[[!redirects homotopy pushout types]]


pushout-product

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

#Contents#
* table of contents
{:toc}

## Idea

The concept of _pushout product_ is a natural kind of pairing operation on [[morphisms]] in [[categories]] equipped with a pairing operation on [[objects]] (e.g. a [[tensor product]]) and having [[pushouts]]. It sends two morphisms to the universal morphism out of the [[pushout]] of the [[span]]-diagram they form by pairing their [[domain]] objects. Regarding arrows in a category as diagrams with domain the [[interval category]], and giving the interval category the natural monoidal product, this is a kind of [[Day convolution]] [[tensor product]] with the natural [[copower|copowering]] over [[Set]] replacing a tensor in the [[end|coend]].

## Definition
 {#Definition}

Consider

* three [[categories]] $\mathcal{E}_i$, $i \in \{1,2,3\}$

  (often taken to be the same),

* a [[functor]]  $\otimes \colon \mathcal{E}_1 \times \mathcal{E}_2 \to \mathcal{E}_3$ out of the [[product category]]

  (often being the [[tensor product]] on a [[monoidal category]] or the [[tensoring]] of an [[enriched category]] over its [[cosmos]]).

Assume that $\mathcal{E}_3$ has all [[pushouts]].

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

Given a [[pair]] of [[morphisms]]

* $f \colon A \to B$ in $\mathcal{E}_1$

* $g \colon X \to Y$ in $\mathcal{E}_2$,

their **pushout product** is the [[morphism]] in $\mathcal{E}_3$

$$
  f \Box g
  \;\;\colon\;\;
  A \otimes Y
    \overset
      {A \otimes X}
      {\amalg}
  B \otimes X
  \longrightarrow
  B \otimes Y
$$

out of the [[pushout]] of the two morphisms, which is induced by the [[universal property]] of the pushout by the following [[commuting diagram]] (induced, in turn, from the [[functor|functoriality]] of the [[tensor product]] as a functor on a [[product category]]):

$$
  \array{
    A \otimes X
    &\overset{f \otimes id_X}{\longrightarrow}&
    B \otimes X
    \\
    \mathllap{{}^{id_A \otimes g}}\Big\downarrow
    &&
    \Big\downarrow\mathrlap{{}^{id_B \otimes g}}
    \\
    A \otimes Y
    &\underset{f \otimes id_Y}{\longrightarrow}&
    B \otimes Y
    \mathrlap{\,.}
  }
$$

\begin{tikzcd}[
 row sep=40pt,
 column sep=20pt
]
  A \otimes X
  \ar[rr, "{ f \otimes \mathrm{id}_X }"{description}]
  \ar[dd, "{ \mathrm{id}_A \otimes g }"{description}]
  \ar[dr, phantom, "{ \scalebox{.6}{(po)} }"{pos=.8}]
  &&
  B \otimes X
  \ar[dd, "{ \mathrm{id}_B \otimes g }"{description}]
  \ar[dl]
  \\
  &
  A \otimes Y
    \overset
      {A \otimes X}
      {\sqcup}
  B \otimes X
  \ar[dr, dashed, "{ f \,\Box\, g }"{sloped, description}]
  \\
  A \otimes Y
  \ar[rr, "{ f \otimes \mathrm{id}_Y }"{description}]
  \ar[ur]
  &&
  B \otimes Y
\end{tikzcd}

=--

## Properties
 {#Properties}

For $\mathcal{C}$ any [[category]] and $K\subset Mor(\mathcal{C})$ any [[class]] of its morphisms, write $K Inj$ for the $K$-[[injective morphisms]] and $K Cof \coloneqq  (K Inj)Proj$ for the $K Inj$-[[projective morphisms]].

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

Let $\mathcal{C}$ be a [[symmetric monoidal category|symmetric]] [[closed monoidal category]] with [[finite limits]] and [[finite colimits]], and let $I_1, I_2\subset Mor(\mathcal{C})$ be two [[classes]] of its morphisms.

Then under [[pushout product]] $\Box$:

$$
  (I_1 Cof) \Box (I_2 Cof)
  \subset
  (I_1 \Box I_2) Cof
  \,.
$$

=--

([Hovey-Shipley-Smith 00](#HoveyShipleySmith00))

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

By a little [[Joyal-Tierney calculus]].

=--

\begin{remark}\label{PushoutProductAxiomInCofibrantlyGeneratedModelCategories}
In the context of [[monoidal model category]] theory, prop. \ref{PushoutProductOfCofClassIsCofClassOfPushoutProduct} implies that for checking the [[pushout-product axiom]] in the case of [[cofibrantly generated model categories]] it is sufficient to check it on [[generating cofibrations]].
\end{remark}

## Examples
 {#Examples}

### In Sets

It is instructive to consider the elementary case of pushout-products in [[Sets]] with respect to the [[Cartesian product]]

$$
  \times
    \;\colon\;
  Set \times Set
    \longrightarrow
  Set
  \,.
$$

For a [[pair]] of [[maps]]/[[functions]] of sets

$$
  \array{
    f \; \colon & X &\longrightarrow& X'
    \\
    g \; \colon & Y &\longrightarrow& Y'
  }
$$

the relevant [[pushout]] diagram is

\begin{tikzcd}
  X \times Y
  \ar[d, "{ f \times \mathrm{id} }"{description}]
  \ar[r, "{ \mathrm{id} \times g }"{description}]
  \ar[dr, phantom, "{\scalebox{.55}{(po)}}"]
  &
  X \times Y'
  \ar[d, " q_r "{description}]
  \ar[ddr, bend left=15, "{ f \times \mathrm{id} }"{description}]
  &[-10pt]
  \\
  X' \times Y
  \ar[r, "{ q_l }"{description}]
  \ar[drr, bend right=15, "{ \mathrm{id} \times g }"{description}]
  &
  f \,\widehat{\times}\, g
  \ar[dr, dashed]
  \\[-10pt]
  &
  & X' \times Y'
\end{tikzcd}

where the dashed arrow denotes the pushout-product map.

\begin{example}
\label{PushoutProductOfMapsOfSets}
The pushout-product of sets may be described as the [[quotient set]]
$$
  f \,\widehat{\times}\, g
  \;\;
  \simeq
  \;\;
  \big\{
  (x,y'),
  \,
  (x',y)
  \big\}_{
    \bigg/
    \Big(
      \big(x,\,g(y)\big) \,\sim\, \big(f(x),\,y\big)
    \Big)
  }
$$
(where each variable ranges over the set denoted by the corresponding capital symbol).

Moreover, if we denote the [[equivalence class]] of $(x,y')$ by $[x,y']$, etc. then the [[coprojections]] into the pushout product are given by

$$
  \array{
    X' \times Y
    &\xrightarrow{\;\; q_l\;\;}&
    f \,\widehat{\times}\, g
    \\
    (x',\,y) &\mapsto& [x',\, y]
  }
  \;\;\;\;\;\;\;\;\;
  \text{and}
  \;\;\;\;\;\;\;\;\;
  \array{
    X \times Y'
    &\xrightarrow{\;\; q_r\;\;}&
    f \,\widehat{\times}\, g
    \\
    (x,\,y') &\mapsto& [x,\, y']
  }
$$
and the pushout-product map itself (the dashed arrow) is given as follows:
\[
  \label{ThePushoutProductMapForSets}
  \array{
    f \widehat{\times} g
    &\xrightarrow{\phantom{---}}&
    X' \times Y'
    \\
    [x',\, y] &\mapsto& \big(x',\, g(y)\big)
    \\
    [x,\, y'] &\mapsto& \big(f(x),\, y' \big)
    \mathrlap{\,.}
  }
\]
More informatively, the [[fibers]] of the pushout-product map over any $(x',y') \,\in\, X' \times Y'$ look as follows:
\[
  \label{FibersOfPushoutProductOfMapsOfSets}
  \big(
    f \widehat{\times} g
  \big)_{(x',y')}
  \;\;
  =
  \;\;
  \left\{
  \array{
    \ast &\vert& (x',y') \;\in\; im(f) \times im(g)
    \\
    f^{-1}\big(\{x'\}\big)  &\vert& y' \;\in\; Y' \setminus im(g)
    \\
    g^{-1}\big(\{y'\}\big)  &\vert& x' \;\in\; X' \setminus im(f)
  }
  \right.
\]
where

* "$\ast$" denotes the [[singleton set]]

* "$(-)^{-1}$" denotes forming [[preimages]]

* "$\setminus$" denotes forming [[complements]].

\end{example}
\begin{proof}
We discuss in more detail how to obtain the fibers (eq:FibersOfPushoutProductOfMapsOfSets).

To see the first case: By the assumption that $x' \,\in\, im(f)$ we find $x_1 \in f^{-1}\big(\{x'\}\big)$ and hence by (eq:ThePushoutProductMapForSets) we find $[x_1,y'] \,\in\, \big(f \widehat{\times} g\big)_{(x',y')}$.
But since also $y' \,\in\, im(g)$ there is also $y \in g^{-1}\big(\{y'\}\big)$ and hence
$$
  \big[x_1,\, y'\big]
    \,=\,
  \big[x_1,\, g(y)\big]
    \,=\,
  \big[f(x_1),\, y\big]
    \,=\,
  \big[x',\, y\big]
  \,.
$$
This shows that all elements of the form $[x,y']$ in the fiber are in fact equal, and also equal to $[x', y]$, which, by the symmetric argument, are in turn equal for all choices of $y$. Therefore there is one single element in the fiber of the pushout-product map, in this case.

To see the second case: Since $y'$ is not in the image of $g$, by (eq:ThePushoutProductMapForSets) the elements in the fiber can only be of the form $[x,y']$ with $x \in f^{-1}\big(\{x'\}\big)$. None of these is contained in the relation -- again by the assumption that $y'$ is not in the image of $g$ -- and hence they are all distinct.

The third case works the same way, under swapping $X \leftrightarrow Y$.
\end{proof}

\begin{example}\label{InjectionsOfSets}
**(Pushout-product of injections of sets)**
\linebreak
In the special case that both maps are [[injections]], the fibers appearing in (eq:FibersOfPushoutProductOfMapsOfSets) are all [[(-1)-truncated]] (either [[empty sets]] or [[singleton sets]]).

This shows that the pushout-product map of two [[injections]] of [[Sets]] is itself an [[injection]]. The following graphics illustrates this for [[interval]]-subsets of the [[plane]]:

<img src="/nlab/files/PushoutProductOfInjections-230425.jpg" width="300">

\end{example}

### In Topological Spaces

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

For $n \in \mathbb{N}$, let

$$
  i_n \;\colon\; S^{n-1}\hookrightarrow D^n
$$

be the canonical [[sphere]] inclusions in [[Top]] (the generating cofibrations of the [[classical model structure on topological spaces]]). Their pushout product (with respect to [[Cartesian product|cartesian]] [[topological product space|product of topological spaces]]) is given by addition of indices:

$$
  i_{n_1} \Box i_{n_2} \simeq i_{n_1 + n_2}
  \,.
$$

Let moreover

$$
  j_n \coloneqq (id,\delta_0) \;\colon\; D^n \hookrightarrow D^n \times I
  \,.
$$

Then

$$
  i_{n_1}\Box j_{n_2} \simeq j_{n_1 + n_2}
  \,.
$$

=--

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

To see this, it is profitable to model [[n-disks]] and [[n-spheres]], up to [[homeomorphism]], as n-cubes and their boundaries.

To see the idea of the proof, consider the situation in low dimensions, where one readily sees that

$$
  i_1 \Box i_1
  \;\colon\;
    (\; = \;\cup\; \vert\vert\;)
    \hookrightarrow
    \Box
$$

and

$$
  i_1 \Box j_0
  \;\colon\;
    (\; = \;\cup\; \vert \; )
    \hookrightarrow
    \Box
  \,.
$$

Generally, $D^n$ may be represented as the space of $n$-tuples of elements in $[0,1]$, and $S^n$ as the suspace of tuples for which at least one of the coordinates is equal to 0 or to 1.

Accordingly $S^{n_1} \times D^{n_2} $ is the spaces of $(n_1+n_2)$-tuples, such that one of the first $n_1$ coordinates is equal to 0 or 1, and hence

$$
  S^{n_1}
    \times
  D^{n_2}
    \;\cup\;
  D^{n_1}
   \times
  S^{n_2}
   \;\;\simeq\;\;
  S^{n_1 + n_2}
  \,.
$$

And of course it is clear that $D^{n_1} \times D^{n_2} \simeq D^{n_1 + n_2}$. This shows the first case.

For the second, use that $S^{n_1} \times D^{n_2} \times I$ is contractible to $S^{n_1} \times D^{n_2}$ in $D^{n_1} \times D^{n_2} \times I$, and that $S^{n_1} \times D^{n_2}$ is a subspace of $D^{n_1} \times D^{n_2}$.

=--

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

The relations in example \ref{PushoutProductOfSpheresInclusionsIntoDisks} are the key in proving that the [[classical model structure on topological spaces]] (on [[compactly generated topological spaces]]) is an [[enriched model category]] over itself. See there at _[topological enrichment](classical+model+structure+on+topological+spaces#TopologicalEnrichment)_ for more.

=--

### General

\begin{example}
\label{PushoutProductWithAnInitialMorphism}
**(pushout-product with initial morphism is tensor product)**
\linebreak
With respect to a [[bifunctor]]
$
  \otimes
  \;\colon\;
  \mathcal{C} \times \mathcal{C}
  \longrightarrow
  \mathcal{C}
$ which [[preserved colimit|preserves]] [[initial objects]] $\varnothing$ in, say, the first variable, the pushout-product with an [[initial object|initial morphisms]] $\varnothing \to X$ is given by $X \otimes (-) $:

$$
  \big(
    \varnothing \to X
  \big)
  \widehat{\otimes}
  g
  \;\simeq\;
  id_X
   \otimes
  g
  \,.
$$

(Of course, in applications $(-)\otimes(-)$ is typically a [[closed monoidal category|closed]] [[tensor product]] which hence even [[preserved colimit|preserves all colimits]] in each variable separately.)

This is because the defining pushout diagram now looks like this:

\begin{tikzcd}
  \varnothing
  \ar[r, "{ \mathrm{id} }"{description}]
  \ar[d]
  \ar[
    dr,
    phantom,
    "{ \scalebox{.6}{$(po)$} }"
  ]
  &
  \varnothing
  \ar[d]
  \ar[ddr, bend left=15]
  \\
  X \otimes Y
  \ar[
    drr,
    bend right=15,
    "{ \mathrm{id} \otimes g }"{description}
  ]
  \ar[r, "{ \mathrm{id} }"{description}]
  &
  X \otimes Y
  \ar[dr, dashed]
  \\
  &&
  X \otimes Y'
\end{tikzcd}

where the top row is the unique map on the initial object by assumption whence its pushout to the bottom horizontal morphism is also an identity, as shown.
\end{example}

\begin{example}
\label{PushoutProductWithAnIdentity}
**(pushout product with identity is identity morphism)**
\linebreak
With respect to any [[bifunctor]] $\otimes \,\colon\, \mathcal{C} \times \mathcal{C} \to \mathcal{C}$, the pushout-product with with an [[identity morphism]] is an identity morphism:
$$
  f \widehat{\otimes} id
  \;\simeq\;
  id
  \;\;\;\;\;\;\;\;\;
  \text{and}
  \;\;\;\;\;\;\;\;\;
  id \,\widehat{\otimes}\, g
  \;\simeq\;
  id
  \,.
$$
This is because the defining pushout-diagram, say in the first case, now  looks like this:
\begin{tikzcd}
  X \otimes Y
  \ar[r, "{ \mathrm{id} }"{description}]
  \ar[d, "{ f \otimes \mathrm{id} }"{description}]
  \ar[dr, phantom, "{\scalebox{.6}{(po)}}"]
  &
  X \otimes Y
  \ar[d, "{ f \otimes \mathrm{id} }"{description}]
  \ar[
    ddr,
    bend left=15,
    "{ f \otimes \mathrm{id} }"{description}
  ]
  \\
  X' \otimes Y
  \ar[r, "{ \mathrm{id} }"{description}]
  \ar[drr, bend right=15, "{ \mathrm{id} }"{description}]
  &
  X' \otimes Y
  \ar[dr, dashed, "{ \mathrm{id} }"{description}]
  \\
  & & X' \otimes Y
\end{tikzcd}
because the top morphism is an identity morphism by assumption, so that also its pushout is given by an identity, as shown.

The analogous argument applies in the other variable.
\end{example}

\begin{example}
\label{PushoutProductOfSplitEpimorphisms}
With respect to any [[bifunctor]]
$
  \otimes
  \;\colon\;
  \mathcal{C} \times \mathcal{C}
  \longrightarrow
  \mathcal{C}
$
the pushout-product of two [[split epimorphisms]]
$$
  \array{
    f \,\colon &  X
      &\overset{split}{\twoheadrightarrow}&
    X'
    \\
    g \,\colon &  Y
      &\overset{split}{\twoheadrightarrow}&
   Y'
  }
$$
is an [[isomorphism]], in that the following [[commuting square]] is already a [[pushout]]
$$
  \array{
    X \otimes Y
      &\overset{id \times g}{\longrightarrow}&
    X \otimes Y'
    \\
    \mathllap{{}^{f \otimes id}}
    \Big\downarrow
      &&
    \Big\downarrow
    \mathrlap{{}^{ f \otimes id}}
    \\
    X' \otimes Y
      &\underset{id \otimes g}{\longrightarrow}&
    X' \otimes Y'
    \mathrlap{\,.}
  }
$$
\end{example}
\begin{proof}
Given any [[cocone]] under the given [[span]], such as shown in the outer part of the following diagram
\begin{tikzcd}[sep=30pt]
  X \otimes Y
  \ar[dd, "{ f \otimes \mathrm{id} }"{description}]
  \ar[rr, "{ \mathrm{id} \times g }"{description}]
  &&
  X \otimes Y'
  \ar[dd, "{ f \otimes \mathrm{id} }"{description}]
  \ar[dddr, bend left=20, "{ r }"{description}]
  \\
  \\
  X' \otimes Y
  \ar[rr, "{ \mathrm{id} \otimes g }"{description}]
  \ar[drrr, bend right=20, "{ s }"{description}]
  &&
  X' \otimes Y'
  \ar[dr, dashed, "{ \phi }"{description}]
  \\
  && &
  Z
\end{tikzcd}
we need to see that there is a unique dashed morphism $\phi$ making the diagram commute, as shown.

Now, by assumption, we have [[sections]]
$$
  \array{
    \overline{f} \,\colon &
    X'
      &\overset{\phantom{--}}{\hookrightarrow}&
    X
    &
    \text{with}
    &
    f \circ \overline{f} \,=\, id
    \\
    \overline{g} \,\colon &
    Y'
      &\overset{\phantom{--}}{\hookrightarrow}&
    Y
    &
    \text{with}
    &
    g \circ \overline{g} \,=\, id
  }
$$
from which we obtain a candidate dashed morphism by setting:
$$
  \begin{array}{l}
    \mathllap{
      \phi \; \coloneqq \;
    }
    r
      \circ
    (\overline{f} \otimes id)
    \\
    \;=\;
    r
      \circ
    (\overline{f} \otimes id)
      \circ
    (id \otimes g)
      \circ
    (id \otimes \overline{g})
    \\
    \;=\;
    r
      \circ
    (id \otimes g)
      \circ
    (\overline{f} \otimes id)
      \circ
    (id \otimes \overline{g})
    \\
    \;=\;
    s
      \circ
    (f \otimes id)
      \circ
    (\overline{f} \otimes id)
      \circ
    (id \otimes \overline{g})
    \\
    \;=\;
    s
      \circ
    (id \otimes \overline{g})
    \,.
  \end{array}
$$
This does make the bottom triangle commute
$$
  \begin{array}{l}
    \phi
      \circ
    (id \otimes g)
    \\
    \;=\;
    r
      \circ
    (\overline{f} \otimes id)
      \circ
    (id \otimes g)
    \\
    \;=\;
    r
      \circ
    (id \otimes g)
      \circ
    (\overline{f} \otimes id)
    \\
    \;=\;
    s
      \circ
    (f \otimes id)
      \circ
    (\overline{f} \otimes id)
    \\
    \;=\;
    s
  \end{array}
$$
and analogously for the right triangle
$$
  \begin{array}{l}
    \phi
      \circ
    (f \otimes id)
    \\
    \;=\;
    s
      \circ
    (id \otimes \overline{g})
      \circ
    (f \otimes id)
    \\
    \;=\;
    s
      \circ
    (f \otimes id)
      \circ
    (id \otimes \overline{g})
    \\
    \;=\;
    r
      \circ
    (id \otimes g)
      \circ
    (id \otimes \overline{g})
    \\
    \;=\;
    r
    \,;
  \end{array}
$$
and given any $\phi$ making the diagram commute, we find that it equals the previous one:
$$
  \begin{array}{l}
    \phi
    \\
    \;=\;
    \phi
      \circ
    (f \otimes id)
      \circ
    (\overline{f} \otimes id)
    \\
    \;=\;
    r
      \circ
    (\overline{f} \otimes id)
    \,.
  \end{array}
$$
\end{proof}

## Related concepts

* [[pullback power]] -- the dual concept

* [[pushout-product axiom]]

* [[Joyal-Tierney calculus]]

## References

Pushout-products are prominently discussed in the context of [[monoidal model category]]-theory (where they appear in a *[[pushout-product axiom]]*), and here a key motivation are constructions of [[symmetric monoidal smash products of spectra]]. See for instance:

* {#HoveyShipleySmith00} [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], _Symmetric spectra_, J. Amer. Math. Soc. 13 (2000), 149-208 ([arXiv:math/9801077](http://arxiv.org/abs/math/9801077))

[[!redirects pushout product]]
[[!redirects pushout-products]]
[[!redirects pushout products]]


pushout-product axiom

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The *pushout-product axiom* for a [[two-variable adjunction]] between [[model categories]] is a condition on [[pushout-products]] of certain [[cofibrations]] which ensures (together with its equivalent dual *[[pullback-power axiom]]*) that the two-variable adjunction is [[homotopical category|homotopically]] meaningful in a way (see the remark [here](monoidal+model+category#MeaningOfPushoutProductAxiom)) analogous to how the axioms of a [[Quillen adjunction]] ensure that an ordinary [[adjoint functor|adjunction]] between model categories is. Therefore one refers to such two-variable adjunctions also as *[[Quillen bifunctors]]*.

In particular, in the definition of **[[enriched model categories]]** the pushout-product axiom ensures that the [[enriched category|enriched]] [[tensoring]]/[[cotensoring]] is homotopically meaningful.

Specialized to the case of [[simplicial model categories]] this is the origin of the notion of the pushout-product axiom, in its dual guise as the [[pullback-power axiom]] as "axiom SM7" in [Quillen (1967)](#Quillen67).

Specialized, alternatively, to the case of self-enrichment the pushout-product axiom for **[[monoidal model categories]]** ensures that the [[tensor product]] in [[two-variable adjunction]] with its [[internal hom]]-functor is homotopically well-behaved.

This situation of [[monoidal model categories]] has come to be the case where the pushout-product axiom is most prominently discussed in the literature, and where it has received its name, see the references [there](monoidal+model+category#OriginalReferences).

## Definition

Let $C$ be a [[closed monoidal category|closed]] [[monoidal category|symmetric monoidal category]] equipped with a [[model category]] structure.

Then $C$ satisfies the _pushout-product axiom_ if for any pair of [[cofibrations]] $f : X \to Y$ and $f' \colon X' \to Y'$ their [[pushout-product]], hence the induced morphism out of the [[coproduct]]

$$
  (X \otimes Y')
  \overset
    {X \otimes X'}
    {\amalg}
  (Y \otimes X')
  \longrightarrow
  Y \otimes Y'
  \,,
$$

is itself a cofibration, which, furthermore, is [[acyclic cofibration|acyclic]] if $f$ or $f'$ is.

This means that the [[tensor product]]

$$
  \otimes \colon C \times C \to C
$$

is a left [[Quillen bifunctor]].

If the monoidal category underlying $C$ is a [[closed monoidal category]], one can [[formal duality|dually]] define the _pullback-power axiom_ to mean that if $f \colon A \to B$ is a [[cofibration]] and $g \colon X \to Y$ is a [[fibration]], their [[pullback power]]

$$
  [B, X]
  \to
  [A, X] \times_{[A,Y]} [B, Y]
  \,,
$$

is a fibration, which, furthermore, is [[acyclic fibration|acyclic]] if $f$ or $g$ is.

By [[Joyal-Tierney calculus#IteratedLifting|Joyal-Tierney calculus]], the [[pullback-power axiom]] holds if and only if the [[pushout-product axiom]] holds.

\begin{remark}
The pushout-product axiom implies in particular that [[tensoring]] with [[cofibrant objects]] preserves [[cofibrations]] and [[acyclic cofibrations]].

However the plain tensor product of a pair of (acyclic) cofibrations is in general not an (acyclic) cofibration.
\end{remark}

\begin{remark}
\label{InCofibrantlyGeneratedModelCategories}
  In a [[cofibrantly generated model category]] the pushout product axiom holds as soon as it holds for (acyclic) [[generating cofibration]] (see [here](pushout-product#PushoutProductAxiomInCofibrantlyGeneratedModelCategories)).
\end{remark}

\begin{remark}
The pushout-product axiom makes sense more generally in the context of a [[two-variable adjunction]] between model categories.  This is important in [[enriched homotopy theory]].
\end{remark}

## Related concepts

* [[monoidal model category]]

* dually: [pullback-power axiom](enriched+model+category#PullbackPowerAxiom) in [[enriched model categories]]

* [[Joyal-Tierney calculus]]

## References

The pullback-power axiom in its role in [[enriched model categories]], specifically in [[simplicial model categories]] originates in:

* {#Quillen67} [[Daniel Quillen]], axiom "SM7" in §2.2 of: *[[Homotopical Algebra]]*, Lecture Notes in Mathematics **43**, Springer (1967) [[doi:10.1007/BFb0097438](https://link.springer.com/book/10.1007/BFb0097438)]

For more see the references at:

* *[[monoidal model category]]*

* *[[enriched model category]]*

* *[[pushout-product]]*.

Discussion for [[model (infinity,1)-categories|model $\infty$-categories]] (such as with [[homotopy Kan fibrations]]):

* {#MazelGee15} [[Aaron Mazel-Gee]], around Def. 4.3 in: *Model ∞-categories II: Quillen adjunctions*, New York Journal of Mathematics **27** (2021) 508-550.  [[arXiv:1510.04392](https://arxiv.org/abs/1510.04392), [nyjm:27-21](http://nyjm.albany.edu/j/2021/27-21.html)]

[[!redirects pushout product axiom]]

[[!redirects pullback-power axiom]]
[[!redirects pullback power axiom]]


pyknotic set

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

#Contents#
* table of contents
{:toc}

## Idea and definition

Pyknotic sets (from the ancient Greek πυκνός, meaning thick, dense, or compact) aim to provide a convenient setting
in the framework of [[homotopical algebra]] for working with algebraic objects that have some sort of [[topological space|topology]] on them.

\begin{definition}
A _pyknotic set_ is a [[sheaf]] of sets on the site of [[compact]] [[Hausdorff]]
[[topological spaces]] whose underlying set has [[rank]] smaller
than the smallest [[strongly inaccessible cardinal]].
The [[Grothendieck topology]] on the latter category
is generated by finite jointly effective epimorphic families of maps.
\end{definition}

Pyknotic sets form a [[coherent topos]], whereas the closely related category of [[condensed sets]] does not form a topos ([BarHai 19, Sec. 0.3](#BarHai19)), only an [[infinitary pretopos]].

This construction generalizes to pyknotic objects for any [[(∞,1)-category]].

Compactly generated topological spaces embed fully faithfully into pyknotic sets, in a manner that preserves limits.

> One key point, however, is that the relationship between compactly generated topological spaces and pyknotic sets is _dual_ to the relationship between compactly generated topological spaces and general topological spaces: in topological spaces, compactly generated topological spaces are stable under colimits but not limits; in pyknotic sets, compactly generated topological spaces are stable under limits but not colimits ([BarHai 19, 0.2.3](#BarHai19)).

The concept of pyknotic structure can be compared to that of [[motivation for cohesive toposes|cohesive structure]]:

> one of the main peculiarities of the theory of pyknotic structures ... is also one of its advantages: the forgetful functor is not faithful. ([BarHai 19, 0.2.4](#BarHai19))

The global sections functor is given by evaluation at the one-point compactum. This has a left adjoint sending a set, $X$, to the discrete pyknotic set attached to $X$, and a right adjoint to the *indiscrete* pyknotic set attached to $X$. The left adjoint, the discrete functor, does not preserve limits and so does not possess a further left adjoint. The topos of pyknotic sets is thus not [[cohesive]] ([BarHai 19, 2.2.14](#BarHai19)).

## Examples

For example [[locally compact topological space|locally compact]] [[abelian groups]], [[normed rings]], and [[complete metric space|complete]] [[locally convex topological vector spaces]] are pyknotic objects.

## Alternative definitions

Pyknotic sets can be described as sheaves on several different sites.

1. Take $\mathcal{C}$ to be the Kleisli category for the ultrafilter monad: objects are sets and morphisms are functions between the sets of ultrafilters borne by those sets. Pyknotic sets are functors $\mathcal{C}^{op}\to Set$ that carry finite coproducts in $\mathcal{C}$ to products in $Set$. So they can be understood as models for a large [[Lawvere theory]].

1. Take $\mathcal{C}$  to be the category of compact Hausdorff spaces (i.e., compacta) but where the sheaf condition is somewhat more elaborate, requiring in addition that $F$ carries the coequalizer displaying an epimorphism $K \to L$ as the cokernel of its kernel pair to monomorphism.

1. Take $\mathcal{C} \subset Top$ to be the subcategory of tiny compact hausdorff spaces that are extremally disconnected, called [[Stonean spaces]]. A pyknotic set carries finite coproducts to products in $Set$.

## Motivation

In the first of a series of talks ([BarHaiMSRI](#BarHaiMSRI)), Barwick motivates the concept of pyknotic set as follows:

The category of [[topological abelian groups]], $AbTop$, is not [[abelian category|abelian]]. This can be seen by taking an abelian group and imposing two topologies, one finer than the other. Both the [[kernel]] and [[cokernel]] of the continuous map which is the identity on elements are $0$. This is an indication that $AbTop$ does not have enough objects. To rectify this, we can modify the category to allow 'pyknotic' structures on $0$, which can act as a cokernel here.

In [[Pontrjagin duality]], the pairing and double dual maps are not necessarily continuous outside of locally compact abelian groups. This occurs because the category of topological spaces does not have an [[internal hom]]. This is an indication of too many objects, and restriction to [[compactly generated spaces]] may be imposed.

## Contrast with condensed mathematics

> ...the indiscrete topological space $\{0,1\}$, viewed as a sheaf on the site of compacta, is pyknotic but not [[condensed set|condensed]] (relative to any universe). By allowing the presence of such pathological objects into the category of pyknotic sets, we guarantee that it is a topos, which is not true for the category of condensed sets. ([BarHai 19, sec 0.3](#BarHai19))

> [Barwick and Haine] assume  the  existence  of  universes,  fixing  in  particular  a  "tiny" and a "small" universe, and look at sheaves on tiny profinite sets with values in small sets; they term these pyknotic sets. In our language, placing ourselves in the small universe, this would be $\kappa$-condensed sets for the first strongly inaccessible cardinal $\kappa$ they consider (the one giving rise to the tiny universe). ([Scholze 19, p. 7]())

## Related concepts

* [[condensed set]]

* [[pro-étale site]]

* [[quasi-topological space]] (roughly, the [[concrete sheaf|concrete]] pyknotic sets)

## References

* {#BarHai19} [[Clark Barwick]], [[Peter Haine]], _Pyknotic objects, I.  Basic notions_, ([arXiv:1904.09966](https://arxiv.org/abs/1904.09966))

* {#BarHaiMSRI} [[Clark Barwick]], [[Peter Haine]], _MSRI: Pyknotic/Condensed Seminar_,
  * 1: _Who cares about pyknosis?_, ([video](https://www.msri.org/seminars/24809));
  * 2: _Pyknotic sets_, ([video](https://www.msri.org/seminars/24915))
  * 3: _Pyknotic abelian groups_, ([video](https://www.msri.org/seminars/24919))
  * 4: _Animating pyknotic sets (AKA pyknotic spaces)_, ([video](https://www.msri.org/seminars/24750))

* {#Scholze19} [[Peter Scholze]], _Lectures on Condensed Mathematics_, ([pdf](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf))

* Sebastian Wolf, _The pro-étale topos as a category of pyknotic presheaves_, [arxiv/2012.10502](https://arxiv.org/abs/2012.10502)

[[!redirects pyknotic sets]]
[[!redirects pyknotic object]]
[[!redirects pyknotic objects]]


Pyotr Sergeyevich Novikov

Pjotr Sergeevič Novikov also spelled Pyotr Sergeyevich Novikov was a mathematicial logician from Soviet Union.
He has shown that the word problem in some abstract groups is undecidable. Topologist [[Sergei Novikov]], Fields medalist, is his son.

* [wikipedia](https://en.wikipedia.org/wiki/Pyotr_Novikov)

category: people

[[!redirects P. S. Novikov]]
[[!redirects Pyotr Novikov]]
[[!redirects Pjotr Sergeevič Novikov]]


Pythagoras

# Pythagoras
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__Pythagoras__ (*Πυθαγόρης* in ancient Ionian, _Πυθαγόρας_ in modern Greek) of Samos was an ancient Greek philosopher, the founder of a religious and philosophic tradition that lasted for centuries, the __Pythagoreans__ (*πυθαγόρειοι*).

## Legacy

Credited by his followers and later ancient historians with significant developments in mathematics, it seems likely that these were only done by his followers.  Most famously, the [[Pythagorean Theorem]] was known to be true long before Pythagoras, while the earliest proofs in the western tradition (although one appears earlier in China) came after Pythagoras, but possibly by the Pythagoreans.  More significantly, the idea that numerical and geometric theorems require [[proof]] at all (rather than merely empirical verification) is connected with the branch of Pythagoreans who called themselves _learners_ ('μαθηματικοί', 'mathematicians').

Even if he did no true mathematics, Pythagoras himself seems to have had a mystical conception of [[number]] (meaning, more or less, [[natural number]]) as part of the order underlying the universe.  This inspired the mathematical Pythagoreans to develop mathematical ideas to explain Pythagoras\'s moral precepts.  The other branch of Pythagoreans, calling themselves _hearers_ ('ακουσματικοί'), concentrated on these precepts ('ακούσματα') and considered the mathematicians to be heretical.  The mathematicians eventually joined the [[Plato|Platonists]], and the acousmaticians (if we may call them that) joined the Cynics.  There was a later revival of mathematical Pythagoreanism (the _Neopythagoreans_), influenced by Platonism and in turn influencing Neoplatonism, in the Hellenistic world.  There was also a more typically religious group of Pythagoreans in Italy who influenced the Orphics.

## Related entries

* the [[Pythagorean theorem]],
* [[Pythagorean triples]],
* the [[Pythagorean temperament]] (in music).

## References

* <https://plato.stanford.edu/entries/pythagoras/>

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

category: people

[[!redirects Pythagoras]]
[[!redirects Πυθαγόρης]]
[[!redirects Πυθαγορης]]
[[!redirects Πυθαγόρας]]
[[!redirects Πυθαγορας]]

[[!redirects Pythagorean]]
[[!redirects Pythagoreans]]
[[!redirects πυθαγόρειος]]
[[!redirects πυθαγορειος]]
[[!redirects πυθαγόρειοι]]
[[!redirects πυθαγορειοι]]
[[!redirects Πυθαγόρειος]]
[[!redirects Πυθαγορειος]]
[[!redirects Πυθαγόρειοι]]
[[!redirects Πυθαγορειοι]]


Pythagorean ring

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## Idea

Pythagorean rings are rings in which all elements satisfy the common Pythagorean identity, generalizing from the [[real numbers]].

## Definitions

Let $R$ be a (possibly [[nonassociative ring|nonassociative]] and/or possibly [[nonunital ring|nonunital]]) [[ring]]. Then $R$ is a __Pythagorean ring__ if it has a binary operation $p:R \times R \to R$ such that for every element $a$ and $b$ in $R$, $a^2 + b^2 = p(a,b)^2$. (Note all binary operations $\cdot$ have an associated cartesian square $(-)^2$ defined as $a^2 = a \cdot a$.)

There is also a $n$-[[arity|ary]] version of $p$, which is a finite sum

$$\sum_{i=0}^n a_i^2 = p_n(a_0,a_1,\ldots,a_n)^2$$

for a natural number $n:\mathbb{N}$.

## Properties

Due to the nature of addition in an [[abelian group]], the set $R$ with the binary operation $p$ is a [[commutative magma|commutative]] [[semigroup]].

Every [[finitely generated]] $R$-[[module]] $A$ for a Pythagorean ring $R$ by a set of finite cardinality $n$ has an absolute value $\vert (-) \vert: A \to R$ given by the $n$-ary version of $p$ in $R$ for the scalars $a_i$ of an element $a$ in $A$, and a [[quadratic form]] given by $\vert(-)\vert^2$. As a result, for every [[finitely generated]] $R$-[[module]] $A$ for a Pythagorean ring $R$ there is an associated [[Clifford algebra]] $Cl(R,\vert(-)\vert^2)$.

## Examples

* The real numbers are a Pythagorean ring.

* A Pythagorean ring that is a [[field]] is a __Pythagorean field__.

## Related concepts

* [[Euclidean geometry]]

* [[absolute value]]

* [[Clifford algebra]]

* [[exponential ring]]

## External links

* Wikipedia, *[Pythagorean field](https://en.wikipedia.org/wiki/Pythagorean_field)*


Pythagorean theorem

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> To this day, the theorem of Pythagoras remains the most important single theorem in the whole of mathematics. -- Jacob Bronowski, _The Ascent of Man_, page 160.

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## Introduction and History

In [[Euclidean geometry]], the _Pythagorean theorem_ expresses a necessary and sufficient condition on the [[lengths]] of the three sides of a [[triangle]] in order for it to be a right triangle. In the popular mind it is associated with the equation

$$a^2 + b^2 = c^2,$$

where $c$ denotes the length of the hypotenuse of a right triangle, and $a, b$ are the lengths of the other two sides.

It was apparently known (in some form or other) by a number of ancient civilizations (Mesopotamia, circa 1800 BCE[^1]; China, Zhou Dynasty, circa 1000 BCE[^2]), and appears in [[Euclid]]'s _[[Elements]]_ as a capstone result of Book 1 (as Proposition 47, and Proposition 48 for the converse). [Wikipedia](http://en.wikipedia.org/wiki/Pythagorean_theorem#Pythagorean_triples) mentions, but without a citation, that certain megalithic monuments in Northern Europe indicate that [[Pythagorean triples]] were known even before the invention of writing systems.[^2.5]

[^1]: See [Maor 07](#Maor07), chapter 1.
[^2]: See [Wikipedia](http://en.wikipedia.org/wiki/Zhou_Bi_Suan_Jing), which exhibits an apparent proof based on the famous 'pinwheel' configuration.
[^2.5]: On the other hand, the oft-repeated mention of the use of knotted ropes or the like in Ancient Egypt, to mark off 3-4-5 triangles and hence right angles, appears to have no scholarly backing whatsoever.

While it was traditional among ancient historians to credit [[Pythagoras]] with the first recorded [[proof]] (at least the first from Hellenic civilization), there is cause to believe that this (semi-legendary) individual had little to do personally with such a proof or even with mathematics generally within his school -- without however there being doubt that this result and other mathematical ideas were vigorously discussed and developed by the Pythagoreans.

## Proof

There are apparently hundreds of proofs. It should be noted that the Pythagorean theorem is a [[theorem]] of [[Euclidean geometry]] and depends on Euclid's [[parallel postulate]], particularly the consequence that the sum of the interior angles of a triangle is invariably $180^\circ$.

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###### Remark
As perhaps goes without saying, very few of these proofs could be considered even remotely serious as "proofs" by the strict standards of modern axiomatic mathematics developed in the last one hundred years or so. That remark applies in particular to the pretend "proof" that follows, where there is for example some unanalyzed notion of adding angles and so on. Of course it can be made rigorous, but just how one goes about it is a matter of interest. One could for example situate it within a grand edifice (based on a [[set theory|theory of sets]]) that allows one to construct the [[real number]] system which in turn is used to model [[Euclidean spaces]]. Or one could proceed from a much more local axiomatic system, such as the axiomatics in Hilbert's *Grundlagen der Geometrie* or Tarski's Elementary Euclidean Geometry. In any case, in what follows we'll brush all such considerations under the rug, and beg the reader's indulgence to take the word "proof" here with whatever size grain of salt he or she requires.
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One proof[^3] proceeds by constructing, given a right triangle with side lengths $a, b$ and hypotenuse length $c$, the unique line incident to the vertex at the right angle that meets the hypotenuse in a right angle. (N.B.: this construction is specific to Euclidean geometry.) This bisects the hypotenuse into two line segments, of length $c_1, c_2$ say with $c_1 + c_2 = c$. (A picture is given [here](http://en.wikipedia.org/wiki/Pythagorean_theorem#Proof_using_similar_triangles).) Moreover the given right triangle is thereby bisected into two right triangles, and due to the fact that the sum of the interior angles is always $180^\circ$, it may be seen that the two right triangles are similar (i.e., in proportion) to the given right triangle. Therefore, if one has side lengths $h, c_1$ and hypotenuse $a$ say, and the other has side lengths $h, c_2$ and hypotenuse $b$, we may derive proportionality equations

$$c_1/a = a/c, \qquad c_2/b = b/c,$$

from which we may derive $a^2 + b^2 = c_1 c + c_2 c = (c_1 + c_2)c = c^2$.

[^3]: The proof which appears in Book 1 of the _[[Elements]]_ looks somewhat more elaborate, but the proof based on similar triangles perhaps had to await the development of a theory of proportion, set out in Book 5.

## Modern points of view

One modern point of view on the Pythagorean theorem is the following:

Without loss of generality, we may consider the context of a 2-dimensional [[Euclidean space]] (the theorem being trivial in lower dimensions, and Euclidean structure on a higher-dimensional space just amounting to compatible Euclidean structure on each of its 2-dimensional subspaces). Over such a space, we consider the symmetric bilinear product taking two vectors to the oriented area of the parallelogram formed by the first and the quarter-turn of the second. (The quarter-turn operator is only canonical up to negation, but it doesn't matter for the canonicality of this product, interpreting its codomain suitably). Vectors are perpendicular just in case this product of them is zero; furthermore, the quadratic form corresponding to this product takes a vector to the area of the square formed upon it, which is to say, its squared length. Thus, we are in fact using the notions of perpendicularity and length which arise from an [[inner product space]], and the Pythagorean Theorem amounts to a special case of the simple algebraic identity $(a+b)^2=a^2+b^2+2 a b$ (which more generally is the [[Law of Cosines]]).

That the Pythagorean Theorem holds in an inner product space is trivial; all that matters is establishing that one is, in fact, working in an inner product space. What it takes to establish this depends on what one is starting from (e.g., one might just as well axiomatize Euclidean geometry as the study of inner product spaces of particular dimension…), but the above approach might fairly be construed as teasing out the inner product structure inherent in geometry as an interested amateur would already be inclined to think of it.

## References

Named after [[Pythagoras]].

* {#Maor07} Eli Maor, _The Pythagorean Theorem: A 4000-year History_, Princeton University Press, 2007 ([doi:10.2307/j.ctvh9w0ks](https://doi.org/10.2307/j.ctvh9w0ks), [jstor:j.ctvh9w0ks](https://www.jstor.org/stable/j.ctvh9w0ks))

[[!redirects Pythagorean theorem]]
[[!redirects Pythagorean Theorem]]
[[!redirects Pythagoras theorem]]
[[!redirects Pythagoras Theorem]]


Pythagorean triple

A __Pythagorean triple__ is an [[integer|integral]] solution $(a, b, c)$ to the [[Diophantine equation]] $a^2 + b^2 = c^2$. For example, $(3, 4, 5)$ is a solution.

Dividing through by $c^2$, the solution set may be described in terms of _[[rational number|rational]]_ solutions to $x^2 + y^2 = 1$. This is a smooth [[conic]] and can be parametrized by the [[projective line]] $\mathbb{P}^1(\mathbb{Q})$ by means of [[stereographic projection]]. Specifically, by stereographically projecting from the point $(-1, 0)$ on the conic to the projective line $x = 1$ (in the [[projective space|projective plane]]), each rational solution $(r, s)$ is taken to a point with rational coordinate $t = \frac{2 s}{1 + r}$. In the other direction, given a rational point $(1, t)$, the line connecting this to $(-1, 0)$ indeed intersects the conic in two rational points (or a double point if $t = \infty$), viz. $(-1, 0)$ and $(r = \frac{4 - t^2}{4 + t^2},\; s = \frac{4 t}{4 + t^2})$, or more recognizably, $(\frac{1 - t'^2}{1 + t'^2},\; \frac{2 t'}{1 + t'^2})$ if $t' = t/2$.

In this way all rational solutions are accounted for. Putting $t' = \frac{u}{v}$ for integers $u, v$, this shows that any integral solution to $a^2 + b^2 = c^2$ is of the form $a = v^2 - u^2$, $b = 2 v u$, $c = v^2 + u^2$ (up to reordering $a$ and $b$, obviously).

[[!redirects Pythagorean triple]]
[[!redirects Pythagorean triples]]


Python

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## Idea

A [[programming language]].

## Related entries

* [[DisCoPy]]

## References

* [Python home page](https://www.python.org/)

* Wikipedia, *Python (programming language)*


Pólya enumeration theorem

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## Idea

The *Pólya enumeration theorem* is a particular [[theorem]] in [[enumerative combinatorics]]; it serves as a basis of methods often called Pólya's enumeration (counting) theory. It is especially used in counting of graphs, including often in applications like chemistry.

The Pólya "theory" can be redone in terms of Joyal's [[species]].

## References

* wikipedia [Pólya enumeration theorem](http://en.wikipedia.org/wiki/P%C3%B3lya_enumeration_theorem)

* G. Pólya, R. C. Read, _Combinatorial enumeration of groups, graphs, and chemical compounds_, Springer-Verlag

The theory can also be rederived from state sum
identities in certain positive [[TQFTs]]:

* [[Markus Banagl]], *Positive topological quantum field theories*, Quantum Topology **6** 4 (2015) 609-706 [[arxiv/1303.4276](http://arxiv.org/abs/1303.4276), [doi:10.4171/qt/71](https://doi.org/10.4171/qt/71)]

category: combinatorics


Q Sharp

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## Idea

"$Q#$" -- a [[quantum programming language]].

## References

* *[Azure Quantum](https://learn.microsoft.com/en-us/azure/quantum/)*:

  * [What are the Q# programming language and the Quantum Development Kit?](https://learn.microsoft.com/en-us/azure/quantum/overview-what-is-qsharp-and-qdk)

  * [The Q# quantum programming language user guide](https://learn.microsoft.com/en-us/azure/quantum/user-guide/)

*  Kartik Singhal, Kesha Hietala, Sarah Marshall, Robert Rand, *Q# as a Quantum Algorithmic Language*, Proceedings of *Quantum Physics and Logic* (2022) [[arXiv:2206.03532](https://arxiv.org/abs/2206.03532)]

See also:

* Wikipedia, *[Q Sharp](https://en.wikipedia.org/wiki/Q_Sharp)*


Q-algebra

[[!redirects Q-algebras]]
[[!redirects rational algebra]]
[[!redirects rational algebras]]

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## Idea ##

Similar to how [[rational vector spaces]] could be defined without using the [[rational numbers]], as a [[torsion-free]] [[divisible group]], there is a definition of an [[associative]] [[unital]] rational [[associative algebras|algebra]] without using the rational numbers.

## Definition ##

By the [[universal property]] of the [[ring]] of [[integers]], every ring $R$ has a [[ring homomorphism]] $h:\mathbb{Z} \to R$ from the integers to $R$ which lands in the [[center]] of $R$, and there is an injection $i:\mathbb{Z}_+ \to \mathbb{Z}$ from the positive integers to the integers.

A ring $R$ is a **rational algebra** or **$\mathbb{Q}$-algebra** if there is a function $j:\mathbb{Z}_+ \to R$ such that for all positive integers $a\in\mathbb{Z}_+$ and elements $b \in R$, $h(i(a)) \cdot j(a) = 1$ and  $j(a) \cdot b = b \cdot j(a)$.

The [[rational numbers]] $\mathbb{Q}$ are the initial $\mathbb{Q}$-algebra. As a result, every $\mathbb{Q}$-algebra $R$ has a [[ring homomorphism]] $h:\mathbb{Q}\to R$, which corresponds to the definition of $\mathbb{Q}$-algebra in terms of ring homomorphisms.

## Examples ##

* Every [[ordered field]] is a $\mathbb{Q}$-algebra.

## See also ##

* [[rational vector space]]
* [[rational numbers]]
* [[ordered field]]
* [[associative unital algebra]]