profinite space

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

#Contents#
* table of contents
{:toc}

## Idea

A _profinite set_ is a [[pro-object]] in [[FinSet]]. By [[Stone duality]] these are [[equivalence|equivalent]] to [[Stone spaces]] and thus are often called _profinite spaces_. So these are [[compact space|compact]] [[Hausdorff space|Hausdorff]] [[connected space|totally disconnected]] [[topological spaces]].

These are precisely the spaces which are small [[cofiltered limits]] of finite [[discrete spaces]], and moreover (as a consequence of [[Stone duality]]) the [[category]] of Stone spaces is [[equivalence of categories|equivalent]] to the category $pro(FinSet)$ of [[pro-objects]] in [[FinSet]] and finite sets sit $FinSet\hookrightarrow pro(FinSet)$ as finite discrete spaces.  This is especially common when talking about [[profinite groups]] and related topics.

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

An [[internal group]] in the category of Stone spaces /  profinite spaces and continuous maps is a [[profinite group]].

=--

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

Just as the term '[[space]]' is used by some schools of [[algebraic topology|algebraic topologists]] as a synonym for [[simplicial set]], so 'profinite space' is sometimes used as meaning a '[[simplicial object]] in the category of compact and totally disconnected topological spaces', i.e. in the other terminology a 'simplicial profinite space'. This is further complicated by the question of whether or not pro(finite simplicial sets)
and simplicial profinite spaces are the same thing.

=--

## Properties

### Profinite reflection

* [[profinite reflection]]

## Related concepts

* [[extremally disconnected profinite set]]

* [[profinite homotopy type]]

* [[pro-homotopy theory]]

## References

The primary meaning (as Stone space) is used in sources on profinite groups, for which see the entries _[[Stone space]]_, _[[profinite group]]_.

* Abolfazl Tarizadeh, _On the category of profinite spaces as a reflective subcategory_ ([arXiv:1207.5963](http://arxiv.org/abs/1207.5963))

Discussion of [[homotopy theory]] of pro(finite) [[simplicial sets]] is in

* [[G. Quick]], _Profinite homotopy theory_, Documenta Mathematica, 13,
(2008), 585–612, (Archiv [0803.4082](http://arxiv.org/abs/0803.4082))

* [[Jacob Lurie]], _Profinite spaces_ ([pdf](http://www-math.mit.edu/~lurie/917notes/Lecture26.pdf))

[[!redirects Profinite space]]
[[!redirects profinite spaces]]
[[!redirects profinite set]]
[[!redirects profinite sets]]


profunctor

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

#Contents#
* table of content
{:toc}

## Idea

The concept of _profunctor_ is a generalization of the concept of [[functor]] in much the same way that the concept of [[bimodule]] generalizes that of [[associative algebra|algebra]] [[homomorphism]] (in fact, this may be understood as a special case of enriched profunctors).

## Definition ##

If $C$ and $D$ are [[categories]], then a **profunctor** from $C$ to $D$ is a [[functor]] of the form

$$
  H_F \colon D^{op}\times C \to Set
  \,.
$$

Such a  profunctor is usually written as $F\colon  C $⇸$ D$. For $d \in D$ and $c \in C$ the set $H_F(d,c)$ is also called the set of [[heteromorphisms]] from $d$ to $c$.

Every [[functor]] $f\colon C\to D$ induces two profunctors $D(1,f)\colon C $⇸$ D$ and $D(f,1)\colon D $⇸$ C$, defined by $D(1,f)(d,c) = D(d,f(c))$ and $D(f,1)(c,d) = D(f(c),d)$. (Here $D(-,-)$ denotes the [[hom functor]] of $D$ and $1$ denotes the identity functor on the respective category.) Since this construction may be thought of as the [[adjunct]] of the composition of $f$ with the [[Yoneda embedding]] $C \stackrel{f}{\longrightarrow} D \stackrel{Yoneda}{\longrightarrow} [D^{op},Set]$, these profunctors are called *representable* (or sometimes one of them is called *corepresentable*) and this way profunctors subsume and generalize ordinary functors.

In particular the identity profunctor $Id \colon  C $⇸$ C$ is represented by the identity functor and hence is given by the [[hom-functor]] $C(-,-) : C^{op} \times C \to Set$ itself.

The notion generalizes to many other kinds of categories.  For instance, if $C$ and $D$ are [[enriched category|enriched]] over some symmetric [[closed monoidal category]] $V$, then a profunctor from $C$ to $D$ is a $V$-functor $D^{op} \otimes C\to V$.  If they are [[internal categories]], then a profunctor $C $⇸$ D$ is an [[internal diagram]] on $D^{op}\times C$, and so on.  There are also other equivalent definitions in each case; see below.

A profunctor is also sometimes called a **[[bimodule|(bi)module]]** or a **distributor** or a **relator** or a **correspondence**, though the latter word is also used for a [[span]].  The term "module" tends to be common in Australia, especially in the enriched case; here the intuition is that for one-object $V$-categories, i.e. monoids in $V$, profunctors really are the same as [[bimodules]] between such monoids in the usual sense.  "Profunctor" is perhaps more common in the Set-based and internal cases (but is also used in the enriched case); here the intuition is that a profunctor is a generalization of a functor, via the construction of "representable" profunctors.  Jean Bénabou, who invented the term and originally used "profunctor," now prefers "distributor," which is supposed to carry the intuition that a distribut**or** generalizes a funct**or** in a similar way to how a [[distribution]] generalizes a [[function]].

Note that the convention that a profunctor is a functor $D^{op}\times C \to Set$ is not universal; some authors reverse $C$ and $D$ and/or put the "op" on the other one.  See the discussion below.

## The bicategory of profunctors

Profunctors are composed by using a [[coend]] to "trace out" the middle variable.  Specifically, for profunctors $F : C $⇸$ D$ and $G : D $⇸$ E$, their composite $G \circ F: C $⇸$ E$ is defined to be

$$
  (G \circ F)(e,c) := \int^{d \in D} F(d,c)\otimes G(e,d)
  \,.
$$

This yields a [[bicategory]] in which

* [[objects]] are [[small categories]],

* [[morphism]]s are profunctors with the above composition, and

* [[2-morphisms]] are [[natural transformation]]s.

This bicategory is variously denoted [[Prof]], $Mod$, or $Dist$, according to one's chosen name for profunctors.  In the $V$-enriched case, it is written $V Prof$ or $V Mod$ or $V Dist$.

The construction of the "representable" profunctors $D(1,f)$ and $D(f,1)$ from a functor $f\colon C\to D$ yield two identity-on-objects functors $Cat \to Prof$ and $Cat^{op}\to Prof$.  Moreover, it is easy to check that $D(1,f) \vdash D(f,1)$ in the bicategory $Prof$; thus $Cat\to Prof$ is a [[proarrow equipment]] in the sense of Wood (in fact, the prototypical one).  This same fact can also be expressed by defining a (pseudo) [[double category]] in which functors and profunctors are the two kinds of arrows; the construction of representable profunctors is then given by [[companion]]s and [[conjoint]]s in this double category, which make it a [[framed bicategory]], hence an equivalent representation of a proarrow equipment.

The [[full sub-2-category]] $Prof_{rep}$ on representable profunctors is equivalent to $Cat_{ana}$, the 2-category of [[anafunctor]]s. See there for more details.

## Alternative definitions ##

### In terms of colimit-preserving functors on presheaf categories {#FuncsOnPresheaves}

A basic fact (e.g. Kashiwara, Schapira, [[Categories and Sheaves]], corollary 2.7.4, page 63) is that for $A$ a [[cocomplete category]], [[cocontinuous functor|colimit-preserving functors]] from [[presheaf|presheaves]] on some small category $C$ to $A$ are canonically equivalent to functors from $C$ to $A$: we have an equivalence of [[functor category|functor categories]]

$$
 Cocont(PSh(C),A)
 \simeq
  Func(C,A)
  \,.
$$

This may be thought of as a consequence of the [[co-Yoneda lemma]] (and hence, of course, of the [[Yoneda lemma]]) which says that every presheaf is a colimit over [[representable functor|representables]], i.e. over objects in the image of the [[Yoneda embedding]] $Y : C \to PSh(C)$. This immediately implies that a colimit-preserving functor on $PSh(C)$ is already determined by its restriction along $Y$ to $C$.

Now, profunctors $D^{op} \otimes C \to V$ are [[adjunct]] to functors $C \to [D^{op}, V] \simeq PSh(D)$. Hence by the above, profunctors are equivalent to colimit-preserving functors

$$
  PSh(C) \to PSh(D)
  \,.
$$

Indeed, there is an equivalence of bicategories between $V Prof$ and the 2-category of categories and colimit-preserving functors and natural transformations between their presheaf categories.  Note that the latter is a [[strict 2-category]] which can thus serve as a "natural" strictification of $V Prof$ (e.g. [Cattani 1999, Prop. 4.2.4](#Cattani99)).

From this perspective, the representable profunctor induced by an ordinary $V$-functor $f : C \to D$ is
the [[adjunct]] of the postcomposition
$$
  C \stackrel{f}{\to} D \stackrel{Y}{\to} [D^{op},V]
$$
with the [[Yoneda embedding]] under the Hom-adjunction.

The formulation of profunctors as colimit-preserving functors on presheaf categories plays a big role also in the context of [[(∞,1)-categories]]. A [[presentable (∞,1)-category]] is one equivalent to a [[localization]] of some [[(∞,1)-category of (∞,1)-presheaves]] (i.e. some [[reflective (∞,1)-subcategory]] of the latter). The collection of all [[presentable (∞,1)-categories]] and colimit-preserving [[(∞,1)-functors]] betweem them forms the [[symmetric monoidal (∞,1)-category of presentable (∞,1)-categories]], whose [[tensor product]] is the "bilinear" tensor product coming from interpreting colimit-preserving functors as "linear" (reading: colimit $\sim$ sum).

This $(\infty,1)$-category $Pr^L$ therefore is an $(\infty,1)$-analog of $Set\text{-}Mod$. In [[geometric ∞-function theory]] one finds (see section 4 there) that morphisms in $Pr^L$ encode the "correspondence operations" such as Fourier-Mukai and its generalizations. See in that context also the examples below.

### In terms of two-sided discrete fibrations

Recall that a functor $D^{op}\to Set$ can equivalently be described as a [[discrete fibration|discrete]] [[(Grothendieck) fibration]], and similarly a functor $C\to Set$ can be described as a discrete opfibration.  Thus, a profunctor $D^{op}\times C\to Set$ could be described by a discrete opfibration over $D^{op}\times C$, or a discrete fibration over $D\times C^{op}$, but there is also a more directly "two-sided" fibrational description.  A *[[two-sided fibration]]* from $C$ to $D$ is a functor $E\to C\times D$ which is a fibration over $D$ and an opfibration over $C$ in a compatible way.  Such a fibration represents a pseudofunctor $D^{op}\times C\to Cat$, and hence if it is discrete it represents a profunctor $D^{op}\times C\to Set$.

This definition/characterization of profunctors works for internal categories as well, but *not* for enriched ones.  It is sometimes called the [[graph of a profunctor]] (although this is sometimes also used for the other fibrational representations mentioned above).

### In terms of two-sided codiscrete cofibrations

Yet another way of representing profunctors is via their [[collages]], also called [[cograph of a profunctor|cographs]].  The collage of a profunctor $H\colon C$⇸$ D$ is, in particular, a category $\bar{H}$ equipped with functors $C\to \bar{H}$ and $D\to\bar{H}$ which are [[fully faithful functor|fully faithful]] and jointly bijective on objects.

In fact, the objects of the [[undercategory]] $(C\sqcup D)/Cat$ which are collages of profunctors $C$⇸$D$ can be characterized, up to equivalence, as the two-sided [[codiscrete cofibrations]], i.e. the two-sided discrete fibrations in $Cat^{op}$.  In simpler and more explicit language, these are the categories $M$ which contain $C$ and $D$ as disjoint full subcategories which are jointly-[[wide subcategory|wide]] (i.e. together contain all the objects), and such that there are no morphisms from an object of $C$ to an object of $D$.  Equivalently, they are the categories which admit a functor to the [[interval category]] $I$ such that $D$ is the fiber over $0$ and $C$ is the fiber over $1$.

When viewing a profunctor $H\colon C$⇸$D$ in this way, one may sometimes speak of elements of $H(d,c)$ as *heteromorphisms* from $d$ to $c$, since they are morphisms in the category $\bar{H}$ and can be "composed" with morphisms of $C$ and $D$ (this corresponds to the "action" of $C$ and $D$ on $H$ in the other formulations), but they go between objects of two different categories (namely $C$ and $D$).

This characterization works just as well in both the internal and enriched case.  Perhaps surprisingly, it also tends to give the "right" notion of profunctor starting with many other, even more exotic, 2-categories.  However, it is trickier to figure out how to define the composite of profunctors viewed as codiscrete cofibrations; see [[codiscrete cofibration]].

### Comparing fibrations and cofibrations

If $C \overset{g}{\to} \bar{H} \overset{f}{\leftarrow} D$ is a codiscrete cofibration representing a profunctor $H$ from $C$ to $D$, then the two-sided discrete fibration representing the same profunctor can be obtained as the [[comma category]] $(f\downarrow g)$ with its two projections to $C$ and $D$.

Dually, if $C \overset{p}{\leftarrow} E \overset{q}{\to} D$ is a two-sided discrete fibration representing a profunctor from $C$ to $D$, then the codiscrete cofibration representing the same profunctor can be obtained as the [[cocomma object]] $(q\uparrow p)$ with the two inclusions of $C$ and $D$.

In fact, in any 2-category with comma and cocomma objects, we have an [[adjunction]]
$$ cocomma: Span(C,D) \;\rightleftarrows\; Cospan(C,D) : comma. $$
One can show that comma objects are always discrete fibrations, and dually cocomma objects are always codiscrete cofibrations.  In $Cat$ and other similar 2-categories, this adjunction is [[idempotent adjunction|idempotent]] and restricts to an equivalence between the categories of discrete fibrations and codiscrete cofibrations (both of which are of course equivalent to the category of profunctors from $C$ to $D$).  This is a two-sided version of the [[Grothendieck construction]].

### In terms of spans

A profunctor $F$ between $C$ and $D$ can also be viewed as a span $C_0 \leftarrow F_1 \rightarrow D_0$ that is compatible with composition in $C$ and $D$ in an appropriate way. This formulation falls out from the definition of a category as a [[monad]] in the [[bicategory]] (more generally [[double category]]) [[Span]] and defining a profunctor to be a [[module over a monad|bimodule]]. The formulation in terms of double categories can be used to produce the appropriate notion of both enriched and internal profunctors.

## Examples ##

* Recall that a one-object [[Vect]]-[[enriched category]] is just an [[algebra]], while a general [[Vect]]-[[enriched category]] is an [[algebroid]]. The full sub-bicategory of $Vect\Mod$ on one-object $Vect$-enriched categories is the familiar category of [[algebras]], [[bimodules]] and bimodule homomorphisms.  In this case, the "representable" profunctors correspond to the way in which every morphism $A \to B$ of [[algebras]] induces the $A$-$B$ bimodule which as a vector space is $B$ with obvious right $B$ action and left $A$-action induced by first mapping $A$ to $B$ via $f$ and then using multiplication in $B$.

* The full sub-bicategory of $Set Prof$ on [[discrete category|discrete categories]] is the bicategory of sets, [[spans]] of sets and morphisms of spans:
  $$
    Set\Mod_{disc} \simeq Span(Set)
    \,.
  $$

  In particular a [[relation]] between sets is a special case of this. From this point of view the 2-category [[Prof]] of profunctors is a [[categorification]] of the category [[Rel]] of sets and relations.

  Moreover, the representable profunctor between discrete category induced by a [[function]] $f : C \to D$ of sets is the span
  $$
  \array{
    && C
    \\
    & {}^{Id}\swarrow && \searrow^{f}
    \\
    C &&&& D
  }
  \,.
  $$

* Similarly, the full sub-bicategory of internal profunctors in $S = (Set^{op}, \times)$ on "discrete categories" is the bicategory of [[cospans]]
  $$
  Set^{op}\Mod_{disc} \simeq Span(Set^{op}) = Cospan(Set)
  \,.
  $$

* for [[enriched category|enrichment]] over a [[category of chain complexes]] an [[enriched category]] is a [[dg-category]] and a profunctor is now a dg-[[bimodule]] of dg-categories. This appears notably in the definition of [[noncommutative motives]].

## Properties
 {#Properties}

* If a [[functor]] represents a given profunctor, then the action of the functor on morphisms is determined by the action of the profunctor and the representation isomorphism. For details see at _[[representability determines functoriality]]_

* If $\mathcal{D}$ is [[Cauchy complete category|Cauchy complete]], then the profunctors that correspond to functors (via Yoneda, which here looks like $F\mapsto F_*$, with $F_*(d,c)=\mathcal{D}(d,F c)$) are exactly those that admit right adjoints.  In general, right adjoint profunctors correspond to functors into the [[Cauchy complete category#InOrdinaryCatTheoryByProfunctors|Cauchy completion]] of their codomain.

## Related entries ##

* [[module]]

* [[2-vector space]]

* [[double profunctor]]

* [[nucleus of a profunctor]]

* [[Mealy morphism]]

## References ##

Original articles:

* [[Maren Justesen]], *Bikategorien af Profunktorer*, Aarhus 1968 ([pdf](https://ncatlab.org/nlab/files/Justesen_Profunktoren.pdf))

* [[Marta Bunge]], Chapter 3 of: _Categories of Set-Valued Functors_, University of Pennsylvania, 1966

  > based on suggestions by [[Bill Lawvere]]

* [[Michel André]], _Categories of Functors and Adjoint Functors_, Batelle Institute at Geneva, 1964, see also _Categories of Functors and Adjoint Functors_,American Journal of Mathematics, Vol. 88, No. 3 (Jul., 1966), pp. 529-543

* {#Benabou73} [[Jean Bénabou]], *Les distributeurs*, Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée,  rapport **33** (1973) [[pdf](http://www.entretemps.asso.fr/maths/DistributeursLouvain.pdf), [[Benabou-LesDistributeurs.pdf:file]]]

  > (including discussion of [[enriched category|enriched]] and [[internal category|internal]] profunctors)

Some of these ideas were exposed at Oberwolfach in 1966. There are extant notes taken by [[Anders Kock]] of a talk by [[Bill Lawvere]], but there is only a passing mention of 'generalised functors' (what are now called profunctors) and their 'generalised matrix multiplication'.

Texbook account:

* [[Francis Borceux]], Sec. 7.7 _[[Handbook of Categorical Algebra]] I_, Cambridge UP 1994.

Lecture notes:

* {#Benabou00} [[Jean Bénabou]] (notes by [[Thomas Streicher]]), _Distributors at work_, (2000), [pdf](http://www.mathematik.tu-darmstadt.de/~streicher/FIBR/DiWo.pdf)

Exposition:

* [[John Baez]], _Re: Klein 2-Geometry VII_ ([blog](http://golem.ph.utexas.edu/category/2006/11/klein_2geometry_vii.html#c005985))

A nice example of profunctors between Lawvere [[metric spaces]] can be found in [this comment](http://golem.ph.utexas.edu/category/2009/11/equipments.html#c029633).

The following classic paper is a good appetizer

* [[William Lawvere]], _Metric Spaces, Generalized Logic, and Closed Categories_ , Rendiconti di Seminario matématico e fisico di Milano XLIII (1973) pp.135-166. Reprinted in TAC Reprints no.1 (2002) pp.1-37. ([tac](http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html))

See the [[joyalscatlab:HomePage|Joyal's CatLab]] for the theory of [[Set]]-valued distributors:

* [[André Joyal]], _[[joyalscatlab:Distributors and barrels]]_ .

Internal profunctors are considered in

* [[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977. (Dover reprint Minneola 2014; sect. 2.4)

* [[Peter Johnstone]], _Sketches of an [[Elephant]] I_, Oxford UP 2002. (pp.359-367)

The relation to [[locally presentable categories]] is almost explicit in:

* {#Cattani99} [[Gian Luca Cattani]], *Presheaf Models for Concurrency*, PhD thesis from BRICS, University of Aarhus 1999 ([pdf](http://www.brics.dk/DS/99/1/BRICS-DS-99-1.pdf), [[Cattani_PresheafModels.pdf:file]])

Profunctors play an important in categorical [[shape theory]]. The original source is

* [[Dominique Bourn]], [[Jean-Marc Cordier]], _Distributeurs et théorie de la forme_, Cah. Top. Géom. Diff. Cat. **21** no.2 (1980) pp.161-189. ([numdam:CTGDC_1980__21_2_161_0](http://www.numdam.org/item/CTGDC_1980__21_2_161_0), [pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1980__21_2/CTGDC_1980__21_2_161_0/CTGDC_1980__21_2_161_0.pdf))

The material together with a general discussion of profunctors is also available in English in the reprinted monograph

* J.-M. Cordier , [[Tim Porter]], _Shape Theory: Categorical Methods of Approximation_ , (1989), Mathematics and its Applications, Ellis Horwood. Reprinted Dover (2008).

Discussion of simplicial enriched profunctors in the context of strong shape theory in

* [[Michael Batanin]], _Categorical Strong Shape Theory_, Cah. Top. Géom. Diff. Cat. **38** no.1 (1997) pp.3-66. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1997__38_1/CTGDC_1997__38_1_3_0/CTGDC_1997__38_1_3_0.pdf))

Overview and application in [[computer science]]:

* [[Gian Luca Cattani]], G. Winskel, _Profunctors, Open Maps, and Bisimulation_ , BRICS Report **04-22** (2004). ([pdf](http://www.brics.dk/RS/04/22/BRICS-RS-04-22.pdf))

The common generalization of [[bimodules]] and [[spans]] in terms of profunctors has been discussed on the blog at

* [[John Baez]], _Bimodules versus spans_ ([blog](http://golem.ph.utexas.edu/category/2008/08/bimodules_versus_spans.html))

## Notation ## {#notation}

Profunctors are often notated with a slashed or barred arrow, as in $C$⇸$D$, which is U+21F8 in Unicode.  It is not always obvious how to draw this character, so here are some hints.

* On the nLab (or anywhere that accepts SGML character entities, including raw HTML on the web), it can be found using a Unicode entity:

      ⇸

* In LaTeX, one can use `\nrightarrow` (producing '$\nrightarrow$') in a pinch, but a nice-looking extensible barred arrow command `\xslashedrightarrow` can also be produced with the following preamble code (modified from amsmath's `\xrightarrow`).  It requires the packages `amsmath` and `mathtools` to be loaded.

      <nowiki>\makeatletter
      \def\slashedarrowfill@#1#2#3#4#5{%
        $\m@th\thickmuskip0mu\medmuskip\thickmuskip\thinmuskip\thickmuskip
        \relax#5#1\mkern-7mu%
        \cleaders\hbox{$#5\mkern-2mu#2\mkern-2mu$}\hfill
        \mathclap{#3}\mathclap{#2}%
        \cleaders\hbox{$#5\mkern-2mu#2\mkern-2mu$}\hfill
        \mkern-7mu#4$%
      }
      \def\rightslashedarrowfill@{%
        \slashedarrowfill@\relbar\relbar\mapstochar\rightarrow}
      \newcommand\xslashedrightarrow[2][]{%
        \ext@arrow 0055{\rightslashedarrowfill@}{#1}{#2}}
      \makeatother
      </nowiki>

  The command `\xslashedrightarrow` can then be used with one required argument and one optional argument, just like `\xrightarrow`.  A version taking no arguments can of course be defined with

      \def\slashedrightarrow{\xslashedrightarrow{}}

  A simpler barred arrow taking no arguments can be created with

      \def\slashedrightarrow{\relbar\joinrel\mapstochar\joinrel\rightarrow}

* In Xypic, a barred arrow (to the right, in this example) can be produced with

      \ar[r]|-@{|}

[[!redirects profunctors]]
[[!redirects distributor]]
[[!redirects distributors]]
[[!redirects relator]]
[[!redirects relators]]
[[!redirects representable profunctor]]
[[!redirects representable profunctors]]

[[!redirects internal profunctor]]
[[!redirects internal profunctors]]

[[!redirects enriched profunctor]]
[[!redirects enriched profunctors]]


profunctor > history

< [[profunctor]]
[[!redirects profunctor -- history]]


program

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

#Contents#
* table of contents
{:toc}

## In type theory

In [[computer science]] a _program_ is a construction of a [[term]] of a certain [[type]] (the output of the program).

Under [[computational trinitarianism]] this is essentially the same as a [[proof]] in [[type theory]]  of the [[proposition]] presented by the type (see at _[[propositions as types]]_ and  _[[proofs as programs]]_).

## Related concepts

* [[algorithm]]

* [[programming language]]

* [[theory]]

* [[proposition]]/[[type]] ([[propositions as types]])

* [[definition]]/[[proof]]/**program** ([[proofs as programs]])

* [[theorem]], [[axiom]]

[[!redirects programs]]

[[!redirects computer program]]
[[!redirects computer programs]]


programming language

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

#Contents#
* table of contents
{:toc}

## Idea

A formal language ([[theory]]) for expressing [[programs]]. Used and studied in [[computer science]].

## Examples

### Types of programming languages

* [[imperative programming]]

  * [[object-oriented programming]]

* [[functional programming]]

* [[quantum programming language]]

* [[logic programming]]

### Specific examples

* **general**

  * [[Python]]

* **[[linear logic]]

  * [[Rust]]

* **[[higher order logic]]**:

  * [[Isabelle]]

* **plain [[type theory]]**:

  * [[Haskell]]

  * [[Scala]]

* **[[dependent type theory]]**:

  * [[Coq]], [[Agda]], [[Lean]], [[NuPRL]], ....

* **[[quantum computation]]**

  * [[QML]]

  * [[Quipper]]

* [[Mathematica]]

## Related concepts

* [[pseudocode]]

* [[computation]]

* [[computational trinitarianism]]

* [[monad in computer science]], [[algebraic side effect]]

* [[proof management system]]

* [[computational topology]]

* [[list of mathematics software]]

* [[domain specific programming language]], [[domain specific embedded programming language]]

* [[quantum programming language]]

## References

Textbook accounts:

With an eye towards [[operational semantics|operational]] or [[denotational semantics|denotational]] [[semantics]]:

* [[Carl A. Gunter]], *Semantics of Programming Languages -- Structures and Techniques*, MIT Press (1992) [[ISBN:9780262570954](https://mitpress.mit.edu/9780262570954/semantics-of-programming-languages/)]

* [[Glynn Winskel]], *The Formal Semantics of Programming Languages*, MIT Press (1993) [[ISBN:9780262731034](https://mitpress.mit.edu/9780262731034/the-formal-semantics-of-programming-languages/), [pdf](https://www.cin.ufpe.br/~if721/intranet/TheFormalSemanticsofProgrammingLanguages.pdf)]

* [[Kenneth Slonneger]], [[Barry Kurtz]]: *Formal Syntax and Semantics of Programming Languages*, Addison-Wesley (1995) [[webpage](https://homepage.divms.uiowa.edu/~slonnegr/plf/Book/), [pdf](https://doc.lagout.org/science/Artificial%20Intelligence/Natural%20Language%20Processing/Formal%20Syntax%20and%20Semantics%20of%20Programming%20Languages%20-%20Kenneth%20Slonneger.pdf)]

With an eye towards ([[data type|data-]])[[type theory]]:

* [[Carl A. Gunter]], *The Semantics of Types in Programming Languages*, in: *[[Handbook of Logic in Computer Science]]* Vol 3: *Semantic structures*, Oxford University Press (1995) [[ISBN:9780198537625](https://global.oup.com/academic/product/handbook-of-logic-in-computer-science-9780198537625), [pdf](http://seclab.illinois.edu/wp-content/uploads/2011/04/Gunter94.pdf)]

* [[Robert Harper]]: *[[Practical Foundations for Programming Languages]]*, Cambridge University Press (2016) [[ISBN:9781107150300](http://www.cambridge.org/us/academic/subjects/computer-science/programming-languages-and-applied-logic/practical-foundations-programming-languages-2nd-edition?format=HB), [pdf](https://www.cs.cmu.edu/~rwh/pfpl/2nded.pdf)]

On [[category theory]] in [[computer science]]/[[programming languages]] (such as for [[monads in computer science]]):

* [[Bartosz Milewski]] (compiled by Igal Tabachnik), *Category Theory for Programmers*, Blurb (2019) [[pdf](https://github.com/hmemcpy/milewski-ctfp-pdf/releases/download/v1.3.0/category-theory-for-programmers.pdf), [github](https://github.com/hmemcpy/milewski-ctfp-pdf), [webpage](https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/), [ISBN:9780464243878](https://www.blurb.com/b/9621951-category-theory-for-programmers-new-edition-hardco)]

Use of [[adjunctions]] in programming languages:

* {#Hinze12} Ralf Hinze, _Generic Programming with Adjunctions_,  In: J. Gibbons  (ed.) _Generic and Indexed Programming_ Lecture Notes in Computer Science, vol 7470. Springer 2012 ([pdf](http://www.cs.ox.ac.uk/ralf.hinze/LN.pdf), [slides](http://www.cs.ox.ac.uk/ralf.hinze/SSGIP10/Slides.pdf) [doi:10.1007/978-3-642-32202-0_2](https://doi.org/10.1007/978-3-642-32202-0_2))

* Jeremy Gibbons, Fritz Henglein, Ralf Hinze, Nicolas Wu, _Relational Algebra by Way of Adjunctions_, Proceedings of the ACM on Programming Languages archive Volume 2 Issue ICFP, September 2018 Article No. 86  ([pdf](https://www.cs.ox.ac.uk/jeremy.gibbons/publications/reladj.pdf), [doi:10.1145/3236781](https://dl.acm.org/citation.cfm?doid=3243631.3236781))

category: software, applications

[[!redirects programming languages]]

[[!redirects programming language theory]]


programming theory > history

< [[programming theory]]


progress graph

## Progress graphs

# Contents
* automatic table of contents goes here
{:toc}
## Idea.
**Progress graphs** are a simple model for a certain aspect of concurrency theory in Computer Science. The basic idea is to give a description of what can happen when several processes are modifying shared resources.  Given a shared resource, $a$, we consider it as a semaphore which controls its behaviour with respect to processes. (This is a bit like the counter or token used in some old single track railway systems to prevent two trains being on the same stretch of line at the same time.) We are not worried exactly how the resource is used merely the 'locking' and 'unlocking' of the access to that resource.  For instance, $a$ may be a shared variable, so a semaphore is used to ensure that only one process can write on it at a time. (Editing pages in the n-Lab is another example!)

Our system will have $n$ deterministic sequential processes $Q_1, \ldots, Q_n$ Each will be abstracted as a sequence of 'locks' (denoted $P$) or 'unlocks' (denoted $V$) applied to shared resources (denoted $a$ with suffices and superfixes to distinguish them), so $Q_i = R^1a^1_i.R^2a^2_i\ldots R^{n_i}a^{n_i}_i$, as so one. Here each $R$ is an occurrence of a $P$ or a $V$.

There is a natural way to understand the possible behaviours of the concurrent execution of these processes. We associate to each process a different coordinate  direction in the topological space, $\mathbb{R}^n$. The state of the system corresponds to a point in $\mathbb{R}^n$ whose $i^{th}$ coordinate describes the state or  'local time' of the  $i^{th}$ process.

We assume that each process starts at (local time) 0 and finishes at (local time) 1; the $P$ and $V$ actions correspond to sequences of real numbers between 0 and 1, which reflect the order of the $P$s and $V$s.  The initial state is $(0,\ldots,0)$ and the final state $(1,\ldots, 1)$.

To look at a particular example more closely, we suppose $n = 2$ and the two processes look like $T_1= P a.P b.V b.V a$ and $T_2 = P b.P a.V a.V b$.  This gives rise to the two dimensional progress graph below:

<svg
   xmlns:dc="http://purl.org/dc/elements/1.1/"
   xmlns:cc="http://creativecommons.org/ns#"
   xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
   xmlns:svg="http://www.w3.org/2000/svg"
   xmlns="http://www.w3.org/2000/svg"
   xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
   xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
   width="401.625"
   height="401.625"
   id="svg2"
   sodipodi:version="0.32"
   inkscape:version="0.46"
   sodipodi:docname="swiss-flag.svg"
   inkscape:output_extension="org.inkscape.output.svg.inkscape"
   version="1.0">
  <defs
     id="defs4">
    <inkscape:perspective
       sodipodi:type="inkscape:persp3d"
       inkscape:vp_x="0 : 526.18109 : 1"
       inkscape:vp_y="0 : 1000 : 0"
       inkscape:vp_z="744.09448 : 526.18109 : 1"
       inkscape:persp3d-origin="372.04724 : 350.78739 : 1"
       id="perspective10" />
  </defs>
  <sodipodi:namedview
     id="base"
     pagecolor="#ffffff"
     bordercolor="#666666"
     borderopacity="1.0"
     gridtolerance="10000"
     guidetolerance="10"
     objecttolerance="10"
     inkscape:pageopacity="0.0"
     inkscape:pageshadow="2"
     inkscape:zoom="0.35355339"
     inkscape:cx="246.70904"
     inkscape:cy="508.22482"
     inkscape:document-units="px"
     inkscape:current-layer="layer1"
     showgrid="true"
     inkscape:window-width="663"
     inkscape:window-height="701"
     inkscape:window-x="0"
     inkscape:window-y="22">
    <inkscape:grid
       type="xygrid"
       id="grid2383"
       visible="true"
       enabled="true" />
  </sodipodi:namedview>
  <metadata
     id="metadata7">
    <rdf:RDF>
      <cc:Work
         rdf:about="">
        <dc:format>image/svg+xml</dc:format>
        <dc:type
           rdf:resource="http://purl.org/dc/dcmitype/StillImage" />
      </cc:Work>
    </rdf:RDF>
  </metadata>
  <g
     inkscape:label="Layer 1"
     inkscape:groupmode="layer"
     id="layer1"
     transform="translate(-79.25,-91.612183)">
    <rect
       style="fill:none;stroke:#050803;stroke-width:1.5;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
       id="rect2385"
       width="400"
       height="400"
       x="80.125"
       y="92.487183" />
    <path
       style="fill:none;stroke:#050803;stroke-width:1;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
       d="M 240,172.36218 L 320,172.36218 L 320,252.36218 L 400,252.36218 L 400,332.36218 L 320,332.36218 L 320,412.36218 L 240,412.36218 L 240,332.36218 L 160,332.36218 L 160,252.36218 L 240,252.36218 L 240,172.36218 z"
       id="path2381" />
    <path
       style="fill:#000000;fill-opacity:0.52999998;stroke:#050803;stroke-width:2.82842708;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
       d="M 241.61787,371.37325 L 241.61787,331.77527 L 202.01989,331.77527 L 162.42191,331.77527 L 162.42191,293.5915 L 162.42191,255.40773 L 202.01989,255.40773 L 241.61787,255.40773 L 241.61787,214.39554 L 241.61787,173.38335 L 279.80164,173.38335 L 317.9854,173.38335 L 317.9854,214.39554 L 317.9854,255.40773 L 357.58338,255.40773 L 397.18136,255.40773 L 397.18136,293.5915 L 397.18136,331.77527 L 357.58338,331.77527 L 317.9854,331.77527 L 317.9854,371.37325 L 317.9854,410.97123 L 279.80164,410.97123 L 241.61787,410.97123 L 241.61787,371.37325 z"
       id="path2383" />
    <path
       style="fill:none;fill-opacity:0.52999998;stroke:#050803;stroke-width:1.5;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
       d="M 80,492.36218 L 160,172.36218 L 480,92.362183 L 478.00418,102.01067 L 400,412.36218 L 80,492.36218 z"
       id="path3155" />
  </g>
</svg>

The shaded area in the picture represents those states that correspond to 'mutual exclusion'. Suppose we look at a state $(x,y)$ with $Pb\lt x\lt Vb$, but $Pb\lt y\lt Vb$.  In such a state, both $T_1$ and $T_2$ have acquired $b$ and not yet relinquished it, which is impossible since $b$ can only be viewed by at most on process at any time.  Such states are in the **forbidden region**.

##Formalising $PV$-programs##
The PV language was introduced in 1968 by [[E. W. Dijkstra]] as an example
of a toy language allowing concurrent execution of sequential processes .
The PV language offers only two instructions called ${P}$ and ${V}$ as abbreviations for the Dutch terms _Prolaag_, short for _probeer te verlagen_, literally "try to reduce,"  and _Verhogen_ ("increase").

Let $S$ be a set whose elements
are called the _semaphores_. Each semaphore _s_ is associated with an _arity_ that
is an integer $\alpha_s\geq 2$. We suppose that for each integer $\alpha  \geq 2$, there exist
infinitely many semaphores whose arity is $\alpha$. The only instructions are ${P}(s)$
and ${V}(s)$, where $s$ is some semaphore. The terms, $\mathbb{P}$, of the language, called _processes_, are the
finite sequences of instructions. When $\mathbb{P}$ is a process and $j$, an integer less or
equal to the length of $\mathbb{P}$, we denote by $\mathbb{P}(j )$ the $j^{th}$ instruction of the process,
in particular $\mathbb{P}(1)$ is the first instruction.

We will separate the instructions of a process by a dot, mostly in order to make
them easier to read. For example we have the processes

$${P}(a).{V}(a)$$

$$P(a).P(b).V(a).V(b)$$
 as we discussed more informally earlier.

##Formalising $PV$-programs and their semantics##

+--{: .un_defn}
######Definition######
A **PV program** is a finite sequence of processes separated by the operator
$\mid$, which should be read _run concurrently with_.
=--

For instance,  ${P}(a).{V}(a) | {P}(a).{V}(a) $
is an example of a PV program made of two copies of a process, whilst
$${P}(a).{P}(b).{V}(a).{V}(b) | {P}(b).{P}(a).{V}(b).{V}(a)$$
is an example made of two distinct processes.

(To be continued.)

##References##

This entry is partially adapted from the treatment of the idea in the following paper:

 *  [[Eric Goubault]], [[Some geometric perspectives in concurrency theory]],

and from

* [[Emmanuel Haucourt]], [Concurrency and Directed Algebraic Topology](http://www.lix.polytechnique.fr/Labo/Emmanuel.Haucourt/uploads/lectures-cdat.pdf), Notes École Polytechique, Jan.  2010.


progroup

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

#Contents#
* table of contents
{:toc}

## Definition

A **progroup** is a [[pro-object]] in the [[category]] [[Grp]] of [[groups]].  In other words, it is a formal [[cofiltered limit]] of groups.

## Surjective progroups versus localic groups

Of course, the category [[Grp]] is [[complete category|complete]], but in general a progroup represented by some cofiltered diagram of groups is not equivalent to the actual limit of that diagram in $Grp$.  However, [[profinite groups]] (i.e. cofiltered systems of *finite* groups) can be identified with actual limits of finite groups if we take those limits, not in $Grp$, but in the larger category $TopGrp$ of [[topological groups]].  The resulting topological groups are precisely those with [[Stone space|Stone]] topologies.

This is not true for pro-systems of non-finite groups, even if we restrict to those with surjective transition maps.  The following counterexample is due to Higman and Stone, and is reproduced in ([Moerdijk](#Moerdijk)).  Let $\omega_1$ be the set of countable ordinals, with the reverse of its usual ordering, and for $\alpha\in\omega_1$ let $S_\alpha$ be the set of strictly increasing functions $[0,\alpha]\to \mathbb{R}$.  For $\alpha\lt \beta$, let $S_\beta \to S_\alpha$ be the restriction.  Then each such transition map is surjective, but the inverse limit is empty.  The sets $S_\alpha$ are not groups, but if we take the free [[vector space]] on each of them, we obtain a nontrivial pro-group with surjective transition maps whose limit in $Grp$, hence also in $TopGrp$, is trivial.

However, we do get an embedding on pro-groups with surjective transition maps if instead of [[Top]] we take the limit in the category [[Loc]] of [[locales]].

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

The following are equivalent for a [[localic group]] $G$:
1. $G$ is a cofiltered limit of [[discrete group]]s (considered as discrete localic groups)
1. $G$ is a cofiltered limit of discrete groups with [[surjection|surjective]] transition maps.
1. The [[open subset|open]] [[normal subgroup]]s of $G$ form a [[neighborhood]] [[topological base|base]] at the identity $e\in G$.
=--
+--{: .proof}
###### Proof

This can be found in ([Moerdijk](#Moerdijk)).

=--

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

A localic group with these properties is called **prodiscrete**.

=--

We may as well assume that any surjective progroup is indexed on a directed [[poset]].  If $(G_i)_{i\in I}$ is such an inverse system, then the localic group $G=\lim_i G_i$ is presented by the following [[posite]].  The elements of the underlying poset are pairs $(x,i)$ where $x\in G_i$, with $(x,i)\le (y,j)$ when $i\le j$ and $f_{ij}(x)=y$.  The coverings are given as follows: for any $j$, the element $(x,i)$ is covered by the family of all $(z,k)$ such that $k\le j$ and $(z,k)\le (x,i)$.

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

A **surjective progroup**, also called a **strict progroup**, is a progroup whose cofiltered diagram consists of [[surjection]]s.

=--

One can show that a progroup is isomorphic to a surjective one, in the category of pro-groups, if and only if it satisfies the **Mittag-Leffler condition**: for each $G_i$ the images of the functions $G_j\to G_i$ are eventually constant.

By a fundamental fact about [[locales]], if $G$ is prodiscrete and represented as the limit of a system with surjective transition maps, then the legs $G\to G_i$ of the limiting cone are also surjective (i.e. they are represented by injective [[frame]] homomorphisms).  This is false for limits of topological spaces.

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

The category of prodiscrete localic groups is equivalent to the category of surjective progroups.

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

In view of the [above proposition](#EquivalentCharacterizations) it suffices to show that for surjective progroups $(G_i)$ and $(H_j)$, with prodiscrete localic limits $G$ and $H$, we have
$$Hom_{LocGrp}(G,H) \cong \lim_j \colim_i Hom_{Grp}(G_i,H_j).$$
But since $H = \lim_j H_j$, we have $Hom_{LocGrp}(G,H) \cong \lim_j Hom_{LocGrp}(G,H_j)$.  Thus it suffices to show that any map from $G$ to a [[discrete group]] $K$ (such as $H_j$) factors through some essentially unique $G_i$.

But if $f\colon G\to K$ is such a map, then $ker(f)$ is an open normal subgroup of $G$.  And if $p_i\colon G\to G_i$ are the projections, then the [[kernel]]s $ker(p_i)$ are a neighborhood base at $e$, so we have $ker(p_i)\subseteq ker(f)$ for some $i$, hence $f$ factors through $G/ker(p_i)$.  Finally, this last is isomorphic to $G_i$, since $p_i\colon G\to G_i$ is an open surjection of locales.

=--

Any [[localic group]] $G$ has a [[classifying topos]] consisting of continuous $G$-sets, i.e. discrete locales with a $G$-action.  In general, the resulting [[functor]]
$$ LocGrp \to Topos $$
is not an [[embedding]] into [[Topos]], but it can be shown to be so when restricted to prodiscrete localic groups.  One can also characterize the toposes that are sheaves on a prodiscrete localic group as the [[Galois topos|Galois toposes]].

Most of these results have corresponding facts for pro-[[groupoids]] and prodiscrete localic groupoids.  However, in full generality, the category of (even surjective) pro-groupoids does not embed into localic groupoids, since the category of [[pro-sets]] (= categorically discrete pro-groupoids) does not embed into locales (= categorically discrete localic groupoids).

## References

* [[Ieke Moerdijk]], *Prodiscrete groups and Galois toposes*
{#Moerdijk}

[[!redirects progroups]]
[[!redirects pro-group]]
[[!redirects pro-groups]]
[[!redirects prodiscrete localic group]]
[[!redirects prodiscrete localic groups]]


projection

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

# Contents
* table of contents
{: toc}

## Definition

### General

Generally in [[category theory]] a _projection_ is one of the canonical [[morphisms]] $p_i$ out of a (categorical) [[product]]:

$$
  p_i \colon \prod_j X_j \to X_i
  \,.
$$

or, more generally out of a [[limit]]

$$
  p_i \colon \underset{\leftarrow_j}{\lim} X_j \to X_i
  \,.
$$

Hence a projection is a component of a [[limit|limiting]] _[[cone]]_ over a given [[diagram]].

In fact, in older literature the [[filtered category|filtered diagrams]] of spaces or algebraic systems (usually in fact indexed by a codirected set) were called [[projective systems]] (or [[inverse systems]]).

Dually, for colimits the corresponding maps in the opposite direction are sometimes caled _[[coprojection]]s_.

### In linear algebra

In [[linear algebra]] an [[idempotent]] [[linear operator]] $P:V\to V$ is called a projection onto its [[image]]. See at _[[projector]]_.

  In [[functional analysis]], one sometimes requires additionally that this idempotent is in fact [[self-adjoint operator|self-adjoint]]; or one can use the slightly different terminology _[[projection operator]]_.

  This relates to the previous notion as follows: the existence of the [[projector]] $P \colon V \to V$ canonically induces a decomposition of $V$ as a [[direct sum]] $V \simeq ker(V) \oplus im(V)$ and in terms of this $P$ is the [[composition]]

  $$
    P \colon V \simeq im(V) \oplus ker(V)\to im(V) \hookrightarrow V
  $$

  of the _projection_ (in the above sense of maps out of [[products]]) out of the [[direct sum]] $im(V) \oplus ker(V) \simeq im(V) \times ker(V)$ followed by the [[subobject]] inclusion of $im(V)$. Hence:

  _A projector is a projection followed by an inclusion_.

## Related concepts

A different concept of a similar name is _[[projection formula]]_.

[[!redirects projection]]
[[!redirects projections]]

[[!redirects product projection]]
[[!redirects product projections]]
[[!redirects projection morphism]]
[[!redirects projection morphisms]]

[[!redirects projection map]]
[[!redirects projection maps]]


projection formula

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

#Contents#
* table of contents
{:toc}

## Idea

Given an [[adjoint pair]] of [[functors]] $(f^\ast \dashv f_\ast)$ or $(f_! \dashv f^\ast)$ between two [[monoidal categories]] such that $f^\ast$ is a [[strong monoidal functor]], then the _projection morphism_ is the canonical [[natural transformation]] of the form

$$
  B \otimes f_\ast A \longrightarrow f_\ast (f^\ast B \otimes A)
$$

or

$$
  f_! (f^\ast B \otimes A) \longrightarrow  B \otimes f_! A
$$

respectively. If these morphisms are [[equivalences]] then one often calls them the **projection formula** or the **reciprocity** relation.

## Examples

Examples include the [[six operations]] setup in [[Grothendieck context]] and [[Wirthmüller context]], respectively (in a [[transfer context]] both pushforward maps satisfy their projection formula). In particular in [[representation theory]] and in [[formal logic]] reciprocity is also called _[[Frobenius reciprocity]]_, see there for more.

For more examples see also at MO _[Where do all the projection formulas come from?](http://mathoverflow.net/q/67228/381)_

## Related concepts

* [[monoidal adjunction]]

* [[dependent linear type theory]]

* [[Frobenius reciprocity]], [[Beck-Chevalley condition]]

## References

Discussion in a context of [[formal logic]]:

* [[Bill Lawvere]], p. 6 in: _[[Equality in hyperdoctrines and comprehension schema as an adjoint functor]]_, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970) 1-14 [[pdf](https://ncatlab.org/nlab/files/LawvereComprehension.pdf)]

Discussion in [[real cohomology]] for [[integration of differential forms]]:

* [[Raoul Bott]], [[Loring Tu]], Prop. 6.15 in: _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Mathematics 82, Springer 1982 ([doi:10.1007/978-1-4757-3951-0](https://link.springer.com/book/10.1007/978-1-4757-3951-0))

Discussion in the context of [[Grothendieck's yoga of six functors]]:

* [[Martin Gallauer]], p. 11 of *An introduction to six-functor formalism*, lecture at *[The Six-Functor Formalism and Motivic Homotopy Theory](https://sites.google.com/view/summer-school-motivic/home)*, Università degli Studi di Milano (Sept. 2021) [[arXiv:2112.10456](https://arxiv.org/abs/2112.10456), [pdf](https://homepages.warwick.ac.uk/staff/Martin.Gallauer/docs/m6ff.pdf)]

A general abstract account is in

* H. Fausk, P. Hu, [[Peter May]],  _Isomorphisms between left and right adjoints_, Theory and Applications of Categories , Vol. 11, 2003, No. 4, pp 107-131. ([TAC](http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html), [pdf](http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf))
 {#May05}

For the [[Grothendieck context]] of [[quasicoherent sheaves]] in [[E-infinity geometry]] the projection formula appears as remark 1.3.14 in

* [[Jacob Lurie]], _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_

See also

* MathOverflow, _[Where do all the projection formulas come from?](http://mathoverflow.net/q/67228/381)_

* MathOverflow, _[Ubiquity of the push-pull formula](http://mathoverflow.net/q/18799/381)_

[[!redirects reciprocity]]

[[!redirects projection formulas]]


projection measure

#Contents#
* table of contents
{:toc}

## Idea

Projection (or: projection-valued) measures are operator-valued [[measures]] of a special type. They appear for example in the theory of [[reproducing kernel]] [[Hilbert spaces]], [[coherent states]] and the foundations of [[quantum mechanics]]. A projection measure is used to parametrize a complete family of [[projection operator]]s by subsets of some parameter space.

Given a set $X$ and some $\sigma$-algebra $B$ of subsets of $X$, with $X\in B$, and a complex Hilbert space $H$, a map $P: B\to End H$ is called a __projection-(valued) measure__ on $B$ with values in $End H$ if

* all operators in the image are selfadjoint $P(A) = P(A)^*$

* $P(A_1\cap A_2) = P(A_1) P(A_2)$ for all $A_1,A_2\in B$ where the product is the composition of the operators

* $P(A_1\cup A_2) = P(A_1)+P(A_2)$ for all $A_1,A_2\in B$ such that $A_1\cap A_2 = \emptyset$

* if $A_n\to A$, in the sense of coinciding upper and lower limit of sets, $A= \cap_n \cup_{k\geq n} A_k = \cup_n \cap_{k\geq n} A_k$, then $P(A_n)\to P(A)$ in the strong  [[operator topology]]. (note: check if strong)

Typical example is that $(X,\tau)$ is a [[topological space]] and $B$ is the $\sigma$-[[sigma-algebra|algebra]] $\mathcal{B}(X)$ of [[Borel set|Borel subsets]] of $X$.

## References

* [[Gerald B. Folland]], *A course in abstract harmonic analysis*, Studies in Advanced Mathematics. CRC Press (1995) [[pdf](https://sv.20file.org/up1/1415_0.pdf), [gBooks](http://books.google.com/books?hl=en&lr=&id=0VwYZI1DypUC)]

* A. A. Kirillov, A. D. Gvišiani, Теоремы и задачи функционального анализа (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988
[[!redirects projective measure]]
[[!redirects projection-valued measure]]


projective Banach space

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

#Contents#
* table of contents
{:toc}

Very incomplete!

(I wasn't sure how to title this page, so the name just mentions projective Banach(able) spaces, even though I hope to eventually get onto injective Banach(able) spaces.)

## Definition

Let $P$ be a [[Banach space]] and let $C$ be a strictly positive constant. We say $P$ is **C-projective** or **projective with constant $C$** if, whenever $Y$ is a Banach space, $X$ is a closed subspace of $Y$, and $f:P\to Y/X$ is a continuous linear map, there exists a continuous linear map $g:P\to X$ with $\Vert g \Vert \leq C\Vert f\Vert$ and $qg=f$, where $q:Y\to Y/X$ is the canonical quotient map.

Semadeni (see below for reference) gives a slightly different definition. Slightly paraphrased, it reads as follows: $F$ is a $Ban_1$-projective space if, for any [[short linear map]] $\phi: F\to H$ and any short linear map $\pi: G\to H$ which satisfies $\phi(Ball(G))=Ball(H)$, there exists a short linear map $\psi:F\to G$ such that $\phi=\psi\pi$. Here, $Ball(\cdot)$ denotes the closed unit ball of a given Banach space.

The condition that $\phi(Ball(G))=Ball(H)$ is stronger than requiring the canonical induced map $G/\ker\phi \to H$ to be an isometric isomorphism. For instance, let us work momentarily with real Banach spaces, and consider the map $\phi:c_0 \to {\mathbb R}$ defined by $\phi((x_n)) = \sum_{n=1}^\infty x_n/2^n$, where $\mathbb R$ is normed by saying that $1$ has norm $1$. Clearly this is a surjective, short linear map, and one can check that the induced map from $c_0/\ker(\phi)$ to $\mathbb R$ is an isometry. But a little thought shows that $1\notin \phi(Ball(c_0))$.

## Examples

Let $S$ be any [[set]] and let $\ell^1(S)$ be the [[free functor|free]] Banach space on $S$.
That is: let $Bang$ denote the category of [[Banach space]]s and short linear maps, and let $Ball$ be the functor $Bang\to Set$ which sends a Banach space $X$ to its closed unit ball $Ball(X)$. Then $\ell^1$ is [[left adjoint]] to $Ball$.

In particular, for any set $S$ and Banach space $V$,
$$ Bang(\ell^1(S), V) \cong Set(S, Ball(V)) $$
(This is the more precise version of the statement "bounded linear maps from $\ell^1(S)$ to $V$ correspond to bounded $V$-valued functions on $S$.)

...

Then $\ell^1(S)$ is a $C$-projective Banach space for any $C\gt1$. (When $S$ is uncountable this definitely needs [[Choice]]; I guess that when $S$ is countably infinite this can be relaxed to [[countable choice]].)

>David Roberts: Or even [[dependent choice]]?

...

It turns out that every projective Banach(able) space is [[isomorphic]] as a Banach(able) space to $\ell^1(S)$ for some set $S$. In the separable case this is more or less a consequence of a standard machine which applies more generally and goes by the name of "block bases plus Pelczynski decomposition". The non-separable case was thrashed out a bit later by Köthe (citation needed).

YC has often wondered if this result has any meaningful kinship with the [[Nielsen-Schreier theorem]]. He usually ends up deciding "probably not".

## Comments on attempts at equivalent formulations ##

In an [[abelian category]] $C$ one sometimes (often?) sees the following definition of a [[projective object]]: an object $P$ is projective in $C$ if the [[hom-functor]] $C(P,\cdot)$ is (right) [[exact functor|exact]]. (I think it is always left exact?). In an abelian category this is the same as preserving [[epimorphisms]], but not in general.

> Urs: yes, because the hom-functor preserves all limits.

Now the [[categories]] of Banach spaces and Banachable spaces each possess notions of [[zero object]], [[kernel]] and [[cokernel]]. So we could define exactness in terms of these. But now 'exactness' in this sense of a [[short exact sequence]] of Banach(able) spaces does not imply exactness of the underlying short exact sequence of [[vector spaces]]!

Put another way: as we (will) have defined it above, projectivity of a Banach space $P$ is different from asking that the functor $\operatorname{hom}(P,\cdot)$ preserves epimorphisms...

+--{: .query}
[[David Roberts]]: I suppose it's about asking what you want to do with said projective objects - projective resolutions? Calculate some sort of cohomology? Know that every epimorphism (or analogous map - perhaps the quotient map of a ses?) to a projective object splits?

[[Yemon Choi]]: Good question, David. The answer is "sort of" - my original background is in Hochschild cohomology of Banach algebras, so I am more concerned about Banach modules (over a fixed Banach algebra) that are projective/injective/flat _in some appropriate sense_ ... Now if one looks up Hochschild coho in something like Weibel's book, defined for R-bimodules where R is a k-algebra, then the theory works much better if all short exact sequences in k-bimod split. This is fine, with Choice, if k is a field (i.e. we are working with vector spaces and usual algebras) but does not hold in general; and really what makes things work is that every object of Vect is projective in the presence of Choice, i.e. you always have a linear section.

Now to get back to your question: it has been found from experience that the Banach version of Hochschild coho is more likely to behave like the classical case when your algebra and module both have underlying Banach space "toplinearly isomorphic" to $\ell^1$, and maybe even $L^1$ is OK in some circumstances. What seems to be going on here is that $\ell^1$ is a projective Banach space and $L^1$ a flat Banach space... hence my interest, hence this page.

=--

## Properties

### Projective in which category? ##

### Köthe's theorem?

## Terminology: "metrically projective" and "metrically injective"

## Injective Banach spaces ##

Theorem of Kelley-Nachbin and others?

## References

* Z. Semadeni, _Banach spaces of continuous functions, Vol. 1._ Warsaw, 1971


projective bundle

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

#Contents#
* table of contents
{:toc}

## Idea

The concept of _projective bundle_ is the generalization of that of _[[projective space]]_ from [[vector spaces]] to [[vector bundles]].

## Definition
  {#Definition}

For $\pi_E \colon E \to X$ a [[topological vector bundle]] over some [[topological field]] $k$, write $E \setminus X \subset E$ for its [[complement]] of the [[zero section]], regarded with its [[subspace topology]].

Then its _projective bundle_ is the [[fiber bundle]] $P(\pi_E) \,\colon\, P(E) \to X$ whose total space is the [[quotient topological space]] of $E \setminus X$ by the [[equivalence relation]]

$$
  \left(
    v_1 \sim v_2
  \right)
   \;\Leftrightarrow\;
  \left(
    (\pi(v_1) = \pi(v_2))
    \,\text{and}\,
    \left(
      \underset{c \in k \setminus \{0\}}{\exists}
      ( v_2 = c v_1)
    \right)
  \right)
$$

hence

$$
  P(E)
    \coloneqq
  (E \setminus X)/\sim
  \,,
$$

and whose bundle projection is

$$
  \array{
    P(E) &\longrightarrow& X
    \\
    [v] &\mapsto& \pi(v)
  }
  \,.
$$

## Properties

### Splitting
 {#Splitting}

The [[pullback bundle]] of a vector bundle $E$ to its own projective bundle

$$
  \array{
    P( \pi_E )^\ast E
    &\longrightarrow&
    E
    \\
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow {}^{\mathrlap{\pi}}
    \\
    P(E) &\underset{P(\pi)}{\longrightarrow}& X
  }
$$

naturally contains a [[line bundle]], whose [[point-set topology|point-set]] definition is

$$
  \mathcal{L}
  \;\coloneqq\;
  \big\{
    \left.
    ( \ell, v )
    \,\in\,
    P(E) \times_X E
    \;\right\vert\;
    v \in \ell
  \big\}
  \;\subset\;
  P( \pi_E )^\ast E
  \,.
$$

Hence the pullback splits as the [[direct sum of vector bundles]] of this $\mathcal{L}$ and its [[orthogonal complement]] (for any choice of [[inner product on vector bundles|inner product]]):

$$
  (P(\pi_E))^\ast (E)
  \;\simeq\;
  {
    \color{blue}
    \mathcal{L}
  }
    \oplus
  \big(
    P(\pi)^\ast(E)
  \big)/\mathcal{L}
  \,.
$$

This process continues: Pulling back further to the projective bundle of $\mathcal{L}^{\perp}$ yields

$$
  (P(\pi_{\mathcal{L}^\perp}))^\ast (P(\pi_E))^\ast (E)
  \;\simeq\;
  {
    \color{blue}
    \mathcal{L}'
  }
    \oplus
  (P(\pi_{\mathcal{L}^\perp}))^\ast
  \big(
    {\color{blue}
      \mathcal{L}
    }
  \big)
  \oplus
  \big(
    (P(\pi_{\mathcal{L}^\perp}))^\ast P(\pi)^\ast(E)
  \big)/\big( P(\pi_{\mathcal{L}^\perp}))^\ast \mathcal{L} \oplus \mathcal{L}' \big)
  \,,
$$

for some line bundle $\mathcal{L}'$

This way, eventually one finds a [[pullback bundle]] of $E$ which is entirely a [[direct sum of vector bundles|direct sum]] of [[line bundles]].

(e.g. [Wirthmüller 12, p. 15 (17 of 67)](#Wirthmuller12))

This is one incarnation of the _[[splitting principle]]_.

## Examples

### Complex-projective bundle of quaternionic tautological line bundle
 {#ComplexProjectiveBundleOfQuaternionicTautologicalLineBundle}

+-- {: .num_prop #ComplexProjectiveBundleOfQuaternTautologicalLineBundleIsComplexProjSpace}
###### Proposition
**([[complex numbers|complex]] [[projective bundle]] of [[quaternionic vector bundle|quaternionic]] [[tautological line bundle]] is [[complex projective space]])**

For $n \in \mathbb{N}$, the [[projective bundle]] of the [[rank of a vector bundle|rank]]=2 [[complex vector bundle]] underlying the [[quaternionic line bundle|quaternionic]] ([[dual vector bundle|dual]]) [[tautological line bundle]] $\mathcal{L}_{{}_{\mathbb{H}P^n}}$ over [[quaternionic projective space]] $\mathbb{H}P^n$ is the [[complex projective space]] $\mathbb{C}P^{2n+1}$ equipped with the map that sends complex lines to the quaternionic lines that they span:

$$
  \array{
    P_{\mathbb{C}}
    \big(
      \mathcal{L}^\ast_{{}_{\!\!\!\mathbb{H}P^n}}
    \big)
    &&
      \simeq
    &&
    \mathbb{C}P^{2n+1}
    \\
    &
    \searrow
    &&
    \swarrow_{
      \mathrlap{
        v \cdot \mathbb{C}^\times
        \,\mapsto\,
        v \cdot \mathbb{H}^\times
      }
    }
    \\
    && \mathbb{H}P^n
  }
$$

=--

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

We compute as follows (showing this for the dual tautological bundle just for definiteness of notation):

$$
  \begin{aligned}
    P_{\mathbb{C}}
    \big(
      \mathcal{L}^\ast_{\!\!\!\mathbb{H}P^n}
    \big)
    & = \;
    \Big(
      \mathcal{L}_{{}_{\mathbb{H}P^n}}
      \setminus
      \big(
        \mathbb{H}P^n \times \{0\}
      \big)
    \Big)
    \big/
    \mathbb{C}^\times
    \\
    & = \;
    \Big(
      \big(
        \mathbb{H}^{n+1} \setminus \{0\}
      \big)
      \underset
      {
        \mathbb{H}^\times
      }{
        \times
      }
      \big(
        \mathbb{H} \setminus \{0\}
      \big)
    \Big)
    \big/
    \mathbb{C}^\times
    \\
    &
    \simeq\;
    \Big(
      \big(
        \mathbb{H}^{n+1} \setminus \{0\}
      \big)
      \times
      \{1\}
    \Big)
    \big/
    \mathbb{C}^\times
    \\
    & \simeq
    \big(
      \mathbb{C}^{2n+2}
    \big)
    \big/
    \mathbb{C}^\times
    \\
    & \simeq \;
    \mathbb{C}P^{2n+1}
    \,.
  \end{aligned}
$$

Here:

1. The first line is the definition of the complex projective bundle ([here](#Definition));

1. the second line inserts the definition of the tautological quaternionic line bundle ([here](tautological+line+bundle#BareDefinition));

1. the third line observes that, being away from its [[zero section]], we have unique representatives of the elements in its defining [[quotient space]] whose fiber component is the unit $1 \in \mathbb{C} \subset \mathbb{H}$;

1. the fourth line identifies the evident underlying $\mathbb{C}^\times$[[G-space|-space]];

1. the fifth line is the definition of complex projective space.

This shows that the total space is as claimed.

Moreover, since the projection map is the [[quotient space]] projection from this by the full $\mathbb{H}^\times$-action, the same computation with $(-)/\mathbb{C}^\times$ replaced by $(-)/\mathbb{H}^\times$ shows that the bundle map is as claimed.

=--

## Related entries

* [[Thom space]]

* [[fundamental product theorem in topological K-theory]]

* [[projective unitary group]]

* [[twisted K-theory]], [[twisted equivariant K-theory]]

## References

* {#Wirthmuller12} [[Klaus Wirthmüller]], section 5 of: _Vector bundles and K-theory_, 2012 ([[wirthmueller-vector-bundles-and-k-theory.pdf:file]])

[[!redirects projective bundles]]


projective cover

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

#Contents#
* table of contents
{:toc}

## Definition

For $N \in \mathcal{A}$ an [[object]], a _projective cover_ is a morphism
$\tilde N \to N$ out of a [[projective object]] which is a _[[superfluous epimorphism]]_.

## Related concepts

* [[projective object]], [[projective presentation]], **projective cover**, [[projective resolution]]

  * [[projective module]]

* [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

  * [[injective module]]

* [[free object]], [[free resolution]]

  * [[free module]]

* flat object, [[flat resolution]]

  * [[flat module]]

## References

* U. Shukla, _On the projective cover of a module and related results_, Pacific J. Math. Volume 12, Number 2 (1962), 709-717 ([Euclid](http://projecteuclid.org/euclid.pjm/1103036506))

* Wikipedia, _[Projective cover](http://en.wikipedia.org/wiki/Projective_cover)_

[[!redirects projective covers]]


projective dimension

We here discuss homological dimensions of rings and abelian categories: projective, injective and global dimensions.

## Definitions

If $C$ is an [[abelian category]] (usually assumed with enough projectives), and $M$ an object in $C$, then its __projective dimension__ $pd(M)$ is the minimal integer $n$ (if it exists) such that there is a resolution of $M$ by projective modules of length $n$, that is the exact sequence of the form
$$
0\to P_n\to P_{n-1}\to\cdots\to P_1\to P_0\to M\to 0
$$
where $P_n,\ldots,P_1,P_0$ are [[projective object]]s in $C$.

An important identity involving the projective dimension of a module of finite projective dimension over a commutative Noetherian local ring is the [[Auslander-Buchsbaum formula]].

Dually, we define the injective dimension of $M$.

Id $R$ is a ring then both the category of all right $R$-modules and the category of all left $R$-modules have enough projectives. The left (right) __global dimension__ of $R$ is the supremum of projective dimensions of all left (right) $R$-modules. More generally, a __global dimension of an abelian category__ $C$ may be defined as
$$
gl.dim(C) := sup\{ i\geq 0\,|\, \exists M,N\in C,\, Ext^i(M,N) = 0\}
$$
Thus the left (right) global dimension of a ring $R$ is the global dimension of the category of left (right) $R$-modules.

See also [[global dimension theorem]].

category: algebra

[[!redirects injective dimension]]
[[!redirects global dimension]]


projective G-space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A projective $G$-space is a [[projective space|projective]] [[topological G-space]].

## Definition

Let $G$ be a [[finite group]] (or maybe a [[compact Lie group]]) and let $V$ be a $G$-[[linear representation]] over some [[topological field|topological]] [[ground field]] $k$.

Then the corresponding _projective $G$-space_ is the [[quotient space]] of the [[complement]] of the origin in (the [[Euclidean space]] underlying) $V$ by the given [[action]] of the [[group of units]] of $k$ (from the $k$-[[vector space]]-[[mathematical structure|structure]] on $V$):

$$
  k P(V)
  \;:=\;
  \big(
    V \setminus \{0\}
  \big) / k^\times
$$

and equipped with the residual $G$-[[action]] on $V$ (which passes to the [[quotient space]] since it commutes with the $k$-action, by linearity).

## Examples

### Ordinary projective spaces

If $G = 1$ is the [[trivial group]], then

$$
  k P(k^{n+1})
  \;=\;
  k P^n
$$

is ordinary (non-equivariant) [[projective space]] of [[dimension]] $n$ over $k$.

### Representation spheres

+-- {: .num_prop #OneDimensionalRepresentationSpheresAsProjectiveLine}
###### Proposition
**(1-dimensional [[representation spheres]] are [[projective G-spaces]])**

If $\mathbf{1}_V \,\in\, G Representations_k$ is 1-dimensional over the given [[ground field]] $k$, [[stereographic projection]] identifies the [[representation sphere]] of $V$ with the [[projective G-space]] over $k$ of $\mathbf{1}_V \oplus \mathbf{1}$:

$$
  \array{
    V^{cpt}
    &
      \longrightarrow
    &
    k P
   \big(
     \mathbf{1}_V \oplus \mathbf{1}
   \big)
    \\
    v &\mapsto&
    \left\{
    \array{
      [v,1] &\vert& v \in V
      \\
      [1,0] &\vert& v = \infty
    }
    \right.
  }
$$

=--

(e.g. [Atiyah 68, Sec. 4](#Atiyah68), [Greenlees 01, 9.C](#Greenlees01))

Prop. \ref{OneDimensionalRepresentationSpheresAsProjectiveLine} underlies the concept of [[equivariant complex oriented cohomology theory]].

### Infinite projective space

+-- {: .num_defn #InfiniteProjectiveGSpace}
###### Definition
**([[infinite complex projective G-space]])**

For $G$ an [[abelian group|abelian]] [[compact Lie group]], let

\[
  \label{GUniverseForInfiniteComplexProjectiveSpace}
  \mathcal{U}_G
  \;\coloneqq\;
  \underset{k \in \mathbb{N}}{\bigoplus}
  \underset{\mathbf{1}_V \in R(G)}{\bigoplus}
  \mathbf{1}_V
\]

be the [[G-universe]] being the infinite [[direct sum]] of all [[complex numbers|complex]] 1-[[dimension|dimensional]] [[linear representations]] of $G$, regarded as a [[topological G-space]] with [[topological space|toplogy]] the [[colimit]] of its [[finite dimensional vector space|finite-dimensional]] [[linear subspaces]].

Then the _[[infinite complex projective G-space]]_ is the [[colimit]]

$$
  P\big(
    \mathcal{U}_G
  \big)
  \;\coloneqq\;
  \underset{
    \underset{
      {
        V \subset \mathcal{U}_G
      }
      \atop
      {
         dim(V) \lt \infty
      }
    }{\longrightarrow}
  }{\lim}
  P\big( V \big)
$$

of the [[projective G-spaces]] for all the [[finite dimensional vector space|finite-dimensional]] $G$-[[linear representations]] inside the [[G-universe]] (eq:GUniverseForInfiniteComplexProjectiveSpace).

=--

(e.g. [Greenlees 01, Sec. 9.2](#Greenlees01))

## Properties

### Equivariant complex K-theory

+-- {: .num_prop #EquivariantKTheoryOfProjectiveGSpace}
###### Proposition
**([[equivariant K-theory of projective G-space]])**

For $G$ an [[abelian group]] [[compact Lie group]], let

$$
  \underset{i}{\oplus}
  \,
  \mathbf{1}_{V_i}
  \;\;
  \in
  \;\;
  G Representations_{\mathbb{C}}^{fin}
$$

be a [[finite dimensional vector space|finite-dimensional]] [[direct sum]] of [[complex numbers|complex]] 1-[[dimension|dimensional]] [[linear representations]].

The $G$-[[equivariant K-theory]] [[cohomology ring|ring]] $K_G(-)$ of the corresponding [[projective G-space]] $P(-)$ is the following [[quotient ring]] of the [[polynomial ring]] in one [[variable]] $L$ over the complex [[representation ring]] $R(G)$ of $G$:

$$
  K_G
  \Big(
    P
    \big(
      \underset{i}{\oplus}
      \,
      \mathbf{1}_{V_i}
    \big)
  \Big)
  \;\;
    \simeq
  \;\;
  R(G)
  \big[
    L
  \big]
  \big/
  \underset{i}{\prod}
  \big(
    1 - 1_{{}_{V_i}} L
  \big)
  \,,
$$

where

* $L \,=\, \big[ \mathcal{L}_{ \underset{i}{\oplus} \mathbf{1}_{V_i} }\big]$ is the K-theory class of the [[tautological equivariant line bundle]] on the given [[projective G-space]]

* $ 1_{{}_{V_i}}  L \;=\; \big[  \mathbf{1}_{V_i} \boxtimes \mathcal{L}_{ \underset{i}{\oplus} \mathbf{1}_{V_i} } \big]$ is the class of its [[external tensor product]] of [[equivariant vector bundles]] with the given [[linear representation]].

=--

([Greenlees 01, p. 248 (24 of 39)](#Greenlees01))

## Related concepts

* [[tautological equivariant line bundle]]

## References

Discussion in the context of [[Bott periodicity]] in [[equivariant K-theory]]:

* {#Atiyah68} [[Michael Atiyah]], _Bott periodicity and the index of elliptic operators_, The Quarterly Journal of Mathematics, Volume 19, Issue 1, 1968, Pages 113–140 ([doi:10.1093/qmath/19.1.113](https://doi.org/10.1093/qmath/19.1.113))

Discussion in the context of [[equivariant complex oriented cohomology theory]]:

* {#Greenlees01} [[John Greenlees]], Section 9.A of _Equivariant formal group laws and complex oriented cohomology theories_, Homology Homotopy Appl. Volume 3, Number 2 (2001), 225-263 ([euclid:hha/1139840255](https://projecteuclid.org/euclid.hha/1139840255))

[[!redirects projective G-spaces]]


projective general linear group

[[!redirects projective linear group]]

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

#Contents#
* table of contents
{:toc}

## Definition

The _projective general linear group_ $PGL(n)$ (in some [[dimension]] $n$ and over some [[coefficients]]) is the [[quotient]] of the [[general linear group]] $GL(n)$ by its [[center of a group|center]].

## Examples

* $PGL(2)$ has a canonical [[action]] on the [[upper half plane]] and as such is equivalent to the [[modular group]] of [[Möbius transformations]]

## Related concepts

* [[projective special linear group]]

* [[projective space]]

* [[projective unitary group]]

* [[special linear group]]

* [[Fuchsian group]]

* [[Kleinian group]]

* [[projective representation]]

* [[cross ratio]]

## References

* Wikipedia, _[Projective linear group](https://en.wikipedia.org/wiki/Projective_linear_group)_

[[!redirects projective general linear groups]]

[[!redirects projective transformation]]
[[!redirects projective transformations]]


projective geometry

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related entries

* [[projective space]]

  * [[projective line]]

  * [[projective variety]]

  * [[quasiprojective variety]]

* [[synthetic projective geometry]]

* [[conic section]]

* [[cross ratio]]

## References

* [[Emil Artin]], §II in: *[[Geometric Algebra]]*, Wiley 1957 (1988) [[ISBN:978-1-118-16454-9](https://www.wiley.com/en-ae/Geometric+Algebra-p-9781118164549), Wikipedia entry, [ark:/13960/t4nk37034](https://archive.org/details/geometricalgebra033556mbp/page/n5/mode/2upa)]

See also:

* Wikipedia, _[Projective geometry](http://en.wikipedia.org/wiki/Projective_geometry)_

category: geometry

[[!redirects projective geometries]]


projective limit

+-- {: .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

Generally, a __projective limit__ is the same thing as a [[limit]].  (Similarly, an [[inductive limit]] is the same thing as a [[colimit]].)  In this context, a __projective system__ is the same thing as a [[diagram]], and a __projective cone__ is the same thing as a [[cone]].

However, many authors restrict this terminology to limits over [[codirected sets]] (or [[cofiltered categories]]), especially the codirected set $(\mathbb{N},\geq)$ of [[natural numbers]]; see [[codirected limit]] (or [[cofiltered limit]]) for discussion of this case if you think that it may be what you want.

The dual concept is _[[inductive limit]]_.

## Related concepts

* [[pro-object]], [[ind-object]]

* [[inverse limit]], another terminology for limit

* [[derived limit functor]]

[[!redirects projective limit]]
[[!redirects projective limits]]

[[!redirects projective system]]
[[!redirects projective systems]]

[[!redirects projective cone]]
[[!redirects projective cones]]


projective line

# Projective lines

* table of contents
{: toc}

## Idea

A projective line is a [[projective space]] of [[dimension]] 1.

## Analytic projective lines

If $k$ is a field, the projective line over $k$ is typically denoted $\mathbb{P}^1(k)$.  Set-theoretically it is a disjoint union $k \sqcup \{\infty\}$ where each $a \in k$ has homogeneous coordinates $[a, 1]$ and $\infty$ has homogeneous coordinates $[1, 0]$.

The classical case of a projective line is over the complex numbers $\mathbb{C}$, where $\mathbb{P}^1(\mathbb{C})$ is also known as the [[Riemann sphere]]. A [[meromorphic function]] on $\mathbb{C}$ may be naturally interpreted as a [[holomorphic function]] $\mathbb{C} \to \mathbb{P}^1(\mathbb{C})$.

In particular, a [[rational function]] $p/q \in \mathbb{C}(z)$ may be interpreted as a holomorphic function $[p/q]: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$; concretely, if $p(z), q(z)$ are relatively prime and of degrees $m, n$ respectively, then we may homogenize by setting $p(x, y) \coloneqq y^m p(x/y)$ and $q(x, y) \coloneqq y^n q(x/y)$, and define $[p/q]$ by the mapping on homogeneous coordinates $[x, y] \mapsto [p(x, y), q(x, y)]$. In fact, there is a bijective correspondence between such holomorphic endomaps on $\mathbb{P}^1(\mathbb{C})$ and rational functions on $\mathbb{C}$ (well, almost: the constant holomorphic map valued at $\infty$ corresponds to the illegitimate "rational function" $1/0$).

## Synthetic projective lines

It is possible to define a [[synthetic mathematics|synthetic]]/[[axiom|axiomatic]] notion of "projective line", somewhat analogously to the synthetic definition of [[projective plane]].  It is less obvious how to do this, since there is no relation of "incidence" inside a projective line.  One approach, due to [(Buekenhout)](#Buekenhout74), is to axiomatize the collection of "central collineations" of a projective line.

In general, a *central collineation* of an $n$-dimensional [[projective space]] $\pi$ (with $n\ge 2$) is an automorphism $\sigma$ of $\pi$ such that

1. there exists a point $O$, called the *center*, such that $\sigma$ fixes every line through $O$ (setwise, i.e.\ it sends every point on that line to a point on the same line), and
2. there exists a hyperplane $H$ (an $(n-1)$-dimensional subspace), called the *axis*, which $\sigma$ fixes pointwise (i.e.\ it sends every point on $H$ to itself).

If $\pi$ has dimension $\ge 3$, so that it has nontrivial sub-projective-spaces of dimension $\ge 2$, then the restriction of a central collineation of $\pi$ to any such sub-projective-space containing the center $O$ and not contained in the axis $H$ is again a central collineation.  Conversely, every central collineation of any subspace of such a $\pi$ is induced from a central collineation of $\pi$ itself (see for instance [Beutelspacher-Rosenbaum, Theorem 3.1.10](#BeutelspacherRosenbaum)).  The latter fact uses Desargues' theorem, but this is true since $\pi$ must be of dimension $\ge 3$ to have any nontrivial sub-projective-spaces.

Indeed, by a theorem of Baer ([Beutelspacher-Rosenbaum, Theorem 3.1.8](#BeutelspacherRosenbaum)), whenever $\pi$ (of dimension $\ge 2$) is Desarguesian, a central collineation is uniquely determined by its axis, its center, and one more pair of corresponding points.  Thus, given $O$ and $H$, the central collineations with center $O$ and axis $H$ act freely and transitively on $\pi \setminus (H\cup \{O\})$.

Of course, when $\pi$ is of dimension $\ge 2$, before we can talk about central collineations, we need to already know what the "hyperplanes" are.  However, in the hypothetical 1-dimensional case, hyperplanes are just points, so that the center and axis are both points, and we can imagine giving structure to $\pi$ by *axiomatizing* its central collineations instead of *defining* them.  This is done by the following definition, due to Buekenhout ([paper](#Buekenhout74), [book (chapter 6)](#BuekenhoutCohen)).

+-- {: .num_defn}
###### Definition
**(Buekenhout)**
A **projective line** is a set $\ell$ of cardinality $\ge 3$ together with the following.

1. For each $p,q\in \ell$, a group $\Lambda(p,q)$ whose elements are called *central collineations with center $p$ and axis $q$*.  Note that $q$ need not be distinct from $p$.
1. $\Lambda(p,q)$ acts on $\ell$ fixing $p$ and $q$, and if an element of it fixes a point $r \notin \{p,q\}$, then it is the identity.  In particular, the action is [[faithful action|faithful]], and therefore embeds each $\Lambda(p,q)$ in the [[permutation group]] $Aut(\ell)$.
1. If $p\neq q$, then $\Lambda(p,q)$ and $\Lambda(q,p)$ commute with each other.
1. For any $\sigma\in\Lambda(r,s)$ and any $p,q$, we have $\sigma \Lambda(p,q) \sigma^{-1} = \Lambda(\sigma(p),\sigma(q))$.
1. The composite of two collineations with center $p$ (possibly with different axes) is again a collineation with center $p$, and dually the composite of two collineations with axis $q$ (possibly with different centers) is again a collineation with axis $q$.  Thus we have two groups $\Lambda(p) = \bigcup_q \Lambda(p,q)$ and $\Lambda^\vee(q) = \bigcup_p\Lambda(p,q)$.

A projective line is **Desarguesian** if in addition $\Lambda(p,q)$ acts [[transitive action|transitively]] on $\ell \setminus \{p,q\}$.  In other words, if $r,s\in \ell\setminus \{p,q\}$, there is a (necessarily unique) $\sigma\in\Lambda(p,q)$ with $\sigma(r)=s$.
=--

### Examples

#### Lines in projective planes

We saw above that when $\pi$ is of dimension $\ge 3$, then every central collineation of a subspace (of dimension $\ge 2$) is induced by some central collineation of $\pi$.  Even though this required Desargues' theorem to prove, which might not be true in a projective plane (dimension $2$), we can still take the point of view that every "central collineation" of a line in a projective plane ought to be induced by a central collineation of the plane itself.

This yields the following construction: Given a projective plane $\pi$ (not necessarily Desarguesian), and a line $\ell$ in $\pi$, for any $p,q\in \ell$ define $\Lambda(p,q)$ to be the set of permutations of $\ell$ that are the restriction to $\ell$ of some central collineation of $\pi$ with center $p$ and axis containing $q$.  It is straightforward to verify that this makes $\ell$ into a "projective line" in the above sense.

#### Trivial examples

There are, however, plenty of projective lines not arising from projective planes.  For instance, we might set $\Lambda(p,q) = 1$ for all $p,q$.

#### Analytic examples

Let $k$ be a [[division ring]] and $V$ a 2-dimensional right [[vector space]] over $k$.  Then $\mathbb{P}(V)$ has the structure of a Desarguesian projective line, where

* if $p\neq q$, then $\Lambda(p,q)$ is the image in $PGL(V)$ of those automorphisms in $GL(V)$ fixing $q$ pointwise and $p$ setwise.
* if $p= q$, then $\Lambda(p,q)$ is the image in $PGL(V)$ of those automorphisms $\alpha\in GL(V)$ such that $\alpha(p)=p$ and $\alpha(x)-x\in p$ for all $x\in V$.

Conversely, every Desarguesian projective line arises from a division ring in this way.  Fix three points $0,1,\infty \in \ell$ and define

* $k=\ell\setminus\{\infty\}$.
* $a+b = t_b(a)$, where $t_b$ is the unique element of $\Lambda(\infty,\infty)$ with $t_b(0)=b$.
* $a b = \lambda_a(b)$, where $\lambda_b$ is the unique element of $\Lambda(0,\infty)$ with $t_a(1)=a$.

It follows that every Desarguesian projective line can be embedded into a Desarguesian projective plane, and indeed a projective space of any dimension.  See [Buekenhout-Cohen, Chapter 6](#BuekenhoutCohen) for details.

## References

* {#BeutelspacherRosenbaum} Albrecht Beutelspacher and Ute Rosenbaum, *Projective Geometry: from foundations to applications*.  Cambridge University Press, 1998. ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/beutel.pdf))

* {#Buekenhout74} Francis Buekenhout, *Foundations of one Dimensional Projective Geometry based on Perspectivities*.  Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 43 (1975) 21-29. doi:[10.1007/BF02995931](https://doi.org/10.1007/BF02995931)

* {#BuekenhoutCohen} Francis Buekenhout and Arjeh M. Cohen, *Diagram Geometry: Related to Classical Groups and Buildings*.  Springer, 2013, doi:[10.1007/978-3-642-34453-4](https://doi.org/10.1007/978-3-642-34453-4) ([author pdf](http://www.win.tue.nl/~amc/buek/))

[[!redirects projective lines]]


projective manifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

A [[manifold]] that is a [[projective variety]].

[[!redirects projective manifolds]]


projective module

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

# Contents
* table of contents
{: toc}

## Definition

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

For $R$ a [[ring]], a _projective $R$-module_ is a [[projective object]] in the [[category]] $R$[[Mod]] of $R$-[[modules]].

Hence an $R$-module $N$ is _projective_ precisely if for all [[diagrams]] of $R$-[[module]] [[homomorphisms]] of the form

$$
  \array{
    && A
    \\
    & & \downarrow^{\mathrlap{epi}}
    \\
    N &\underset{f}{\longrightarrow}& B
  }
$$

there exists a [[lift]], hence a morphism $N \overset{\phi}{\to} A$ making a [[commuting diagram]] of the form

$$
  \array{
    && A
    \\
    & {}^{\mathllap{\phi}}\nearrow & \downarrow^{\mathrlap{epi}}
    \\
    N &\underset{f}{\longrightarrow}& B
  }
  \,.
$$

=--

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

An $R$-module $N$ is projective (def. \ref{ProjectiveModule}) precisely if the [[hom functor]]

$$
  Hom_{R Mod}(N, - ) : R Mod \to Ab
$$

(out of $N$) is an [[exact functor]].

=--

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

The hom-functor in question is a [[left exact functor]] for all $N$, hence we need to show that it is a [[right exact functor]] precisely if $N$ is projective.

That $Hom_R(N,-)$ is right exact means equivalently that for

$$
  0
    \to
  A
    \overset{i}{\longrightarrow}
  B
    \overset{p}{\longrightarrow}
  C
    \to
  0
  \,,
$$

any [[short exact sequence]],
hence for $p$ any [[epimorphism]] and $i$ its [[kernel]] inclusion,  then $Hom_R(N,p)$ is an epimorphism, hence that for any element $f \in Hom_R(N,C)$,

$$
  \array{
    && B
    \\
    && \downarrow^{\mathrlap{p}}
    \\
    N &\underset{f}{\longrightarrow}& C
  }
$$

there exists $\phi \colon N \to B$ such that $f =  Hom_R(N,p)(\phi) \coloneqq p \circ \phi$, hence that

$$
  \array{
    && B
    \\
    &{}^{\mathllap{\phi}}\nearrow& \downarrow^{\mathrlap{p}}
    \\
    N &\underset{f}{\longrightarrow}& C
  }
  \,.
$$

This is manifestly the condition that $N$ is projective.

=--

## Properties

### Existence of enough projective modules

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

Assuming the [[axiom of choice]],
a [[free module]] $N \simeq R^{(S)}$ is projective.

=--

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

Explicitly: if $S \in Set$ and $F(S) = R^{(S)}$ is the [[free module]] on $S$, then a module homomorphism $F(S) \to N$ is specified equivalently by a [[function]] $f : S \to U(N)$ from $S$ to the underlying set of $N$, which can be thought of as specifying the images of the unit elements in $R^{(S)} \simeq \oplus_{s \in S} R$ of the ${\vert S\vert}$ copies of $R$.

Accordingly then for $\tilde N \to N$ an epimorphism, the underlying function $U(\tilde N) \to U(N)$ is an epimorphism, and the [[axiom of choice]] in [[Set]] says that we have all lifts $\tilde f$ in

$$
  \array{
     && U(\tilde N)
     \\
     & {}^{\tilde f} \nearrow & \downarrow
     \\
     S &\stackrel{f}{\to}& U(N)
  }
  \,.
$$

By [[adjunction]] these are equivalently lifts of module homomorphisms

$$
  \array{
     && \tilde N
     \\
     & \nearrow & \downarrow
     \\
     R^{(S)} &\stackrel{}{\to}& N
  }
  \,.
$$

=--

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

Assuming the [[axiom of choice]], the category $R$[[Mod]] has _[enough projectives](projective%20object#EnoughProjectives)_: for every $R$-[[module]] $N$ there exists an [[epimorphism]] $\tilde N \to N$ where $\tilde N$ is a projective module.

=--

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

Let $F(U(N))$ be the [[free module]] on the [[set]] $U(N)$ underlying $N$. By lemma \ref{FreeModulesAreProjective} this is a projective module.

The [[counit of an adjunction|counit]]

$$
  \epsilon : F(U(N)) \to N
$$

of the [[free functor|free]]/[[forgetful functor|forgetful]]-[[adjunction]] $(F \dashv U)$ is an [[epimorphism]].

=--

Actually, the full axiom of choice is not necessary here; it is enough to have the [[presentation axiom]], which states the [[category of sets]] has enough projectives (whereas the axiom of choice states that *every* set is projective).  Then we can replace $U(N)$ above by a projective set $P \twoheadrightarrow U(N)$, giving an epimorphism $F(P) \twoheadrightarrow F(U(N)) \twoheadrightarrow N$ (and $F(P)$ is projective).

### Explicit characterizations

We discuss the more explicit characterization of projective modules
as [[direct sum|direct summands]] of [[free modules]].

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

If $N \in R Mod$ is a [[direct sum|direct summand]] of a [[free module]], hence if there is $N' \in R Mod$ and $S \in Set$ such that

$$
  R^{(S)} \simeq N \oplus N'
  \,,
$$

then $N$ is a projective module.

=--

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

Let $\tilde K \to K$ be a surjective homomorphism of modules and
$f : N \to K$ a homomorphism. We need to show that there is a lift $\tilde f$ in

$$
  \array{
    && \tilde K
    \\
    & {}^{\mathllap{\tilde f}}\nearrow & \downarrow
    \\
    N &\stackrel{f}{\to}& K
  }
  \,.
$$

By definition of [[direct sum]] we can factor the [[identity]] on $N$ as

$$
  id_N : N \to N \oplus N' \to N
  \,.
$$

Since $N \oplus N'$ is free by assumption, and hence projective by lemma \ref{FreeModulesAreProjective}, there is a lift $\hat f$ in

$$
  \array{
    && && \tilde K
    \\
    && & {}^{\mathllap{\hat f}}\nearrow & \downarrow
    \\
    N &\to& N \oplus N'  &\to& K
  }
  \,.
$$

Hence $\tilde f : N \to N \oplus N' \stackrel{\hat f}{\to} \tilde K$ is a lift of $f$.

=--

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

An $R$-module $N$ is projective precisely if it is the [[direct summand]] of a [[free module]].

=--

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

By lemma \ref{DirectSummandOfFreeIsProjective} if $N$ is a direct summand then it is projective. So we need to show the converse.

Let $F(U(N))$ be the [[free module]] on the [[set]] $U(N)$ underlying $N$ as in the proof of prop. \ref{FreeForgetfulCounitEpimorphism}. The [[counit of an adjunction|counit]]

$$
  \epsilon : F(U(N)) \to N
$$

of the [[free functor|free]]/[[forgetful functor|forgetful]]-[[adjunction]] $(F \dashv U)$ is an [[epimorphism]]. Thefore if $N$ is projective, there is a [[section]] $s$ of $\epsilon$. This exhibits $N$ as a direct summand of $F(U(N))$.

=--

This proposition is often stated more explicitly as the existence of a [[dual basis]], see there.

In some cases this can be further strengthened:

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

If the [[ring]] $R$ is a [[principal ideal domain]] (in particular $R = \mathbb{Z}$ the [[integers]]), then every projective $R$-module is free.

=--

The details are discussed at _[pid - Structure theory of modules](http://ncatlab.org/nlab/show/principal+ideal+domain#StructureTheoryOfModules)_.

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

For an $R$-module $P$, the following statements are equivalent:

1. $P$ is finite locally free in that there exists a partition $1 = \sum_i f_i \in R$ such that the localized modules $P[f_i^{-1}]$ are finite free modules over $R[f_i^{-1}]$.

1. $P$ is finitely generated and projective.

1. $P$ is a [[dualizable object]] in the category of $R$-[[modules]] (equipped with the tensor product as monoidal structure).

1. There exist elements $x_1,\ldots,x_n \in P$ and linear forms $\vartheta_1,\ldots,\vartheta_n \in Hom(P,R)$ such that $x = \sum_i \vartheta_i(x) x_i$ for all $x \in P$.

=--

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

The equivalence of 2., 3., and 4. is mostly formal. For the equivalence with 1., see [this math.SE discussion](http://math.stackexchange.com/questions/16814/finitely-generated-projective-modules-are-locally-free) for good references. Note that the equivalences are true without assuming that $R$ is [[Noetherian ring|Noetherian]] or that $P$ satisfies some finiteness condition.
=--

### Relation to projective resolutions of chain complexes
 {#RelationToProjectiveResolution}

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

For $N \in R Mod$ a **[[projective resolution]]** of $N$ is a [[chain complex]] $(Q N)_\bullet \in Ch_\bullet(R Mod)$ equipped with a [[chain map]]

$$
  Q N \to N
$$

(with $N$ regarded as a complex concentrated in degree 0) such that

1. this morphism is a [[quasi-isomorphism]] (this is what makes it a [[resolution]]), which is equivalent to

   $$ \cdots \to (Q N)_1 \to (Q N)_0 \to N $$

   being an [[exact sequence]];

1.  all whose entries $(Q N)_n$ are projective modules.

=--

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

This means precisely that $Q N \to N$ is an [[cofibrant resolution]] with respect to the standard [[model structure on chain complexes]] (see [here](model%20structure%20on%20chain%20complexes#StandardQuillenOnBounded)) for which the fibrations are the positive-degreewise epimorphisms. Notice that in this model structure every object is fibrant, so that cofibrant resolutions are the only resolutions that need to be considered.

=--

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

Every $R$-module has a projective resolution.

=--

See at _[[projective resolution]]_.

## Examples

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

Assuming the [[axiom of choice]], then by the [[basis theorem]] every module over a [[field]] is a [[free module]] and hence in particular every module over a field is a projective module (by prop. \ref{DirectSummandOfFreeIsProjective}).

=--

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

If $R$ is the [[integers]] $\mathbb{Z}$, or a [[field]] $k$, or a [[division ring]], then every projective $R$-module is already a free $R$-module.

=--

## Related concepts

* [[Schanuel's lemma]]

* [[projective object]], [[projective presentation]], [[projective cover]], [[projective resolution]]

* [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

  * [[injective module]]

* [[free object]], [[free resolution]]

  * [[free module]], [[locally free module]]

* flat object, [[flat resolution]]

  * [[flat module]]

* [[free module]] $\Rightarrow$ **projective module** $\Rightarrow$ [[flat module]] $\Rightarrow$ [[torsion-free module]]

* [[Serre-Swan theorem]]

* [[Quillen-Suslin theorem]]

## References

Early discussion:

* {#CartanEilenberg56} [[Henri Cartan]], [[Samuel Eilenberg]], *[[Homological Algebra]]*, Princeton Univ. Press (1956), Princeton Mathematical Series **19** (1999) [[ISBN:9780691049915](https://press.princeton.edu/books/paperback/9780691049915/homological-algebra-pms-19-volume-19), [doi:10.1515/9781400883844](https://doi.org/10.1515/9781400883844), [pdf](http://www.math.stonybrook.edu/~mmovshev/BOOKS/homologicalalgeb033541mbp.pdf)]

Lecture notes include

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_, section 2.2

* _Projective modules, Presentations and resolutions_ ([pdf](http://livetoad.org/Courses/Documents/8875/Notes/presentations_and_resolutions.pdf))

* Thomas Lam, chapter 6 ([pdf](http://www.math.lsa.umich.edu/~tfylam/Math221/6.pdf))

Original articles include

* Irving Kaplansky, _Projective modules_, The Annals of Mathematics
Second Series, Vol. 68, No. 2 (Sep., 1958), pp. 372-377 ([JSTOR]( http://www.jstor.org/stable/1970252))

[[!redirects projective modules]]


projective morphism of schemes

## References

* [[The Stacks Project]], _[28.43. Projective morphisms](http://stacks.math.columbia.edu/tag/01W7)_

[[!redirects projective morphism of schemes]]