Fraïssé limit

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### Context
#### Model theory
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[[!include model theory - contents]]
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# Contents
* table of contents
{: toc}

## Idea

Given a collection of finite(ly-generated) [[structure|first-order structures]] which formally resemble collections of finite(ly-generated) substructures found inside an infinite structure, there are additional assumptions on such collections we can make which ensure that they can be amalgamated into an infinite structure which somehow captures the "generic theory" of the collection we started with.

## Definition

A class of finitely-generated structures closed under substructures and isomorphisms such that:

1. (Amalgamation) every [[span]] of embeddings (not necessarily elementary) admits a [[cocone]], and

2. (Joint embedding) Every pair of structures admits a [[cocone]]

is called a _Fraïssé class_.

A _Fraïssé limit_ $\underset{\rightarrow}{\operatorname{Flim}}(\mathcal{C})$ of a Fraïssé class $\mathcal{C}$ is the unique (existence and uniqueness was Fraïssé's theorem) countable _ultrahomogeneous_ (every isomorphism of finitely-generated substructures extends to an automorphism of $M$) structure into which every member of $\mathcal{C}$ embeds.

## Constructing Fraïssé limits

[Will fill this in after I learn how to typeset diagrams]

## Examples

- The [[DLO|countable dense linear order]] is the Fraisse limit of the class of finite linear orders.

- The [[countable random graph]] is the Fraisse limit of the class of finite graphs.

- The [[Henson graph]] (the generic triangle-free graph) is the Fraisse limit of the class of finite triangle-free graphs.

- A modification ([[Hrushovski construction|Hrushovski constructions]]) to the Fraïssé construction was used by [[Ehud Hrushovski]] to produce a counterexample to [[Zilber's trichotomy conjecture]] classifying the possible [[matroid|combinatorial geometry]] on a [[strongly minimal]] set.

## Remarks

- Olivia Caramello remarks in her topos-theoretic Fraïssé paper (linked below) that the [[classifying topos]] of an [[omega-categorical]] structure $A$ presentable as a Fraïssé limit is precisely the [[category of G-sets|category of continuous Aut(A)]]-sets, where the automorphism group of $A$ is equipped with the topology of pointwise convergence.

- It is interesting when a structure is presentable as both a Fraïssé limit and (an elementary submodel of) an ultraproduct of finite structures. In more suggestive terminology, this means that the _generic_ and the _almost-sure_ (one way of obtaining the almost-sure theory of a collection of finite structures is to take the theory of their ultraproduct) theories of the underlying collection of finite structures coincide. For example, the countable random graph above satisfies a zero-one law.

- Recently, Zilber has been trying to study [[field with one element|F_1-geometry]] by 'fusing' the algebraic closures of finite fields together with a Hrushovski construction and studying a nonstandard extension ([[ultrapower]]) of the result.

## Related concepts

[[omega-categorical structure]]

[[homogeneous structure]]

## References

- David Corfield, [Fraïssé limits](https://golem.ph.utexas.edu/category/2009/11/fraisse_limits.html)

- Hodges, [A shorter model theory](http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521587136)

- Olivia Caramello, [Fraïssé's construction from a topos-theoretic perspective](https://arxiv.org/abs/0805.2778)

[[!redirects Fraïssé limit]]
[[!redirects Fraïssé limits]]
[[!redirects Fraisse limit]]
[[!redirects Fraisse limits]]


Frechet-Uryson space

# Frechet--Urysohn spaces
* tic
{: toc}

## Idea

A Frechet--Urysohn (or Frechet--Uryson) space is a [[topological space]] in which the [[closed subspace|closure]] of a [[subspace]] may be described using only [[sequences]].

## Definition

Recall that, given any subset $A$ of *any* topological space, a point $x$ belongs to the closure of $A$ if and only if $x$ is a limit point of at least one [[net]] whose elements belong to $A$.

A topological space is __Frechet--Uryson__ (or __Frechet--Urysohn__) if a point $x$ of the closure of any given subset $A$ of $X$ is a limit point of at least one [[sequence]] whose elements belong to $A$.

## Related countability properties

[[!include topology - countability axioms]]

[[!redirects Frechet-Uryson space ]]
[[!redirects Frechet–Uryson space]]
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[[!redirects Frechet-Uryson spaces]]
[[!redirects Frechet–Uryson spaces]]
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[[!redirects Frechet–Urysohn spaces]]
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Frechet-Uryson space > history

< [[Frechet-Uryson space]]
[[!redirects Frechet-Uryson space  -- history]]
[[!redirects Frechet-Uryson space -- history]]


Fred Diamond

* [institute page](https://www.kcl.ac.uk/people/fred-diamond)

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

## Selected writings

On the [[modularity theorem]]:

* {#BreuilConradDiamondTaylor2001} [[Christophe Breuil]], [[Brian Conrad]], [[Fred Diamond]], [[Richard Taylor]], *On the modularity of elliptic curves over $\mathbf{Q}$: Wild 3-adic exercises*, Journal of the American Mathematical Society **14** 4 (2001) 843–939 $[$[doi:10.1090/S0894-0347-01-00370-8](https://doi.org/10.1090/S0894-0347-01-00370-8)$]$

On [[modular forms]]:

* [[Fred Diamond]], [[Jerry Shurman]], *A First Course in Modular Forms*, GTM **228**, Springer (2005) $[$[doi:10.1007/978-0-387-27226-9](https://doi.org/10.1007/978-0-387-27226-9), ISBN-13: 978-0387232294$]$

category: people


Fred Jegerlehner

* [webpage](http://www-com.physik.hu-berlin.de/~fjeger/)

## selected writings

Discussion of [[Higgs inflation]] with emphasis on relation to the [near-criticality of the Higgs field](Higgs+field#Curiously):

* [[Fred Jegerlehner]], _The hierarchy problem of the electroweak Standard Model revisited_ ([arXiv:1305.6652](https://arxiv.org/abs/1305.6652))

* [[Fred Jegerlehner]], _Higgs inflation and the cosmological constant_, Acta Phys.Polon. B45 (2014) 1215-1254 ([arXiv:1402.3738](https://arxiv.org/abs/1402.3738))

* [[Fred Jegerlehner]], _About the role of the Higgs boson in the evolution of the early universe_ ([arXiv:1406.3658](https://arxiv.org/abs/1406.3658))

* [[Fred Jegerlehner]], _The hierarchy problem and the cosmological constant problem in the Standard Model_ ([arXiv:1503.00809](https://arxiv.org/abs/1503.00809))

* [[Fred Jegerlehner]], _The Hierarchy Problem and the Cosmological Constant Problem Revisited -- A new view on the SM of particle physics_ ([arXiv:1812.03863](https://arxiv.org/abs/1812.03863))

Discussion of the [[anomalous magnetic moment]] of the [[muon]]:

* [[Friedrich Jegerlehner]], _The Anomalous Magnetic Moment of the Muon_, Springer Tracts in Modern Physics 226, Springer-Verlag Berlin Heidelberg, 2008

* {#Jegerlehner18a} [[Fred Jegerlehner]], _The Muon g-2 in Progress_, Acta Physica Polonica 2018 ([doi:10.5506/APhysPolB.49.1157](https://arxiv.org/ct?url=https%3A%2F%2Fdx.doi.org%2F10.5506%252FAPhysPolB.49.1157&v=01073571), [arXiv:1804.07409](https://arxiv.org/abs/1804.07409))

* {#Jegerlehner18b} [[Fred Jegerlehner]], _The Role of Mesons in Muon $g-2$_ ([arXiv:1809.07413](https://arxiv.org/abs/1809.07413))

## related $n$Lab entries

* [[Higgs inflation]]

* [[anomalous magnetic moment]]

category:people

[[!redirects Friedrich Jegerlehner]]


Fred Linton

* [webpage](http://tlvp.net/~fej.math.wes/)

## Selected writings

On [[monads]] in [[universal algebra]] and ([[cohomology|co-]])[[homology]]-theory:

* [[Fred Linton]], §7 in: *An outline of functorial semantics*, in *[[Seminar on Triples and Categorical Homology Theory]]*, Lecture Notes in Mathematics **80**, Springer (1969) 7-52 [[doi:10.1007/BFb0083080](https://doi.org/10.1007/BFb0083080)]

in:

* [[H. Applegate]], [[M. Barr]], [[J. Beck]], [[F. W. Lawvere]], [[F. E. J. Linton]], [[E. Manes]], [[M. Tierney]], [[F. Ulmer]]: _[[Seminar on Triples and Categorical Homology Theory]]_, ETH 1966/67, edited by [[Beno Eckmann]] and [[Myles Tierney]], **[[LNM 80]]**, Springer (1969), reprinted as: Reprints in Theory and Applications of Categories **18** (2008) 1-303 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

On ([[reflexive coequalizer|reflexive]]) [[coequalizers]] in [[categories]] of [[algebras over a monad]]:

* [[Fred Linton]], *Coequalizers in categories of algebras*, in: *Seminar on Triples and Categorical Homology Theory*, Lecture Notes in Mathematics **80** (1969) 75-90 [[doi:10.1007/BFb0083082](https://doi.org/10.1007/BFb0083082)]

category: people

[[!redirects F. E. J. Linton]]


Fred Richman

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=6518)

## Selected writings

On the [[fundamental theorem of algebra]] in [[constructive mathematics]]:

* [[Fred Richman]], *The fundamental theorem of algebra: a constructive development without choice*. Pacific Journal of Mathematics **196** 1 (2000) 213–230 [[doi:10.2140/pjm.2000.196.213](http://dx.doi.org/10.2140/pjm.2000.196.213), [pdf](https://msp.org/pjm/2000/196-1/pjm-v196-n1-p10-p.pdf)]

On [[constructive mathematics]]:

* [[Fred Richman]], *Constructive Mathematics without Choice*,
in: *Reuniting the Antipodes -- Constructive and Nonstandard Views of the Continuum*, Synthese Library **306**, Springer (2001) 199-206 [[doi:10.1007/978-94-015-9757-9_17](https://doi.org/10.1007/978-94-015-9757-9_17)]

On the [[real numbers]] in [[constructive mathematics]]:

* [[Fred Richman]], *Real numbers and other completions*, Mathematical Logic Quarterly **54** 1 (2008) 98-108 [[doi:10.1002/malq.200710024](https://onlinelibrary.wiley.com/doi/10.1002/malq.200710024)]

On algebraic functions on the [[real numbers]]:

* {#Richman12} [[Fred Richman]], *Algebraic functions, calculus style*. Communications in Algebra, Volume 40, Issue 7, July 2012, Pages 2671-2683 [[doi:10.1080/00927872.2011.584337](https://doi.org/10.1080/00927872.2011.584337)]

On a [[division]] [[algorithm]] for [[polynomials]] over [[reduced local rings]]:

* [[Fred Richman]], *A division algorithm*. Journal of Algebra and Its Applications, Vol. 04, No. 04, pp. 441-449 (2005) [[doi:10.1142/S0219498805001289](https://doi.org/10.1142/S0219498805001289)]

On [[weak countable choice]]:

* [[Fred Richman]], [[Douglas Bridges]], Peter Schuster, *A weak countable choice principle*. Proceedings of the American Mathematical Society 128(9):2749-2752, March 2000. [[doi:10.1090/S0002-9939-00-05327-2](http://dx.doi.org/10.1090/S0002-9939-00-05327-2)]

On whether 0.999... equals 1:

* [[Fred Richman]], *Is 0.999… = 1?*. Mathematics Magazine, Volume 72, Issue 5, 1999, Pages 396-400, [[doi:10.1080/0025570X.1999.11996777](https://doi.org/10.1080/0025570X.1999.11996777)]

On [[Laurent series]] of [[real numbers]]:

* [[Fred Richman]], *Laurent series over $\mathbb{R}$*. Communications in Algebra, Volume 48, Issue 5, 11 Jan 2020 Pages 1982-1984 [[doi:10.1080/00927872.2019.1710166](https://doi.org/10.1080/00927872.2019.1710166)]

On [[polynomials]] and [[linear transformations]]:

* [[Fred Richman]], *Polynomials and linear transformations*. Linear Algebra and its Applications, Volume 131, 1 April 1990, Pages 131-137. [doi:10.1016/0024-3795(90)90379-Q]

category: people


Fred van Oystaeyen

__Fred van Oystaeyen__ is an [[algebraist]] and one of the pioneers of [[noncommutative algebraic geometry]]; he is a Professor at Antwerp. He is especially well known for his work on graded and filtered rings, localization theory for noncommutative rings and the study of Brauer groups and generalizations.

* [home page](http://www.ua.ac.be/main.aspx?c=fred.vanoystaeyen)

## Selected writings

Introducing the notion of [[separable functors]]:

* {#NăstăsescuVanDen BerghVanOystaeyen89} [[Constantin Năstăsescu]], [[Michel van den Bergh]], [[F. van Oystaeyen]], _Separable functors applied to graded rings_, J. Algebra __123__ (1989) 397-413 [doi:10.1016/0021-8693(89)90053-7]

On [[non-commutative algebraic geometry]]:

* [[Stefaan Caenepeel]], [[Alain Verschoren]], (Preface for) *Noncommutative Rings and Geometry*, Proceedings of the conference held in Almería, September 17–22, 2007, in honour of Freddy Van Oystaeyen, Algebras and Representation Theory (2009) 12:101–102, [doi](http://dx.doi.org/10.1007/s10468-009-9162-5), [pdf](http://www.springerlink.com/content/jq374737w8l34481/fulltext.pdf)

* F. van Oystaeyen, [[Alain Verschoren]], Non-commutative algebraic geometry (Springer LNM 887, 1981)

category: people

[[!redirects Fred Van Oystaeyen]]
[[!redirects F. van Oystaeyen]]
[[!redirects F. Van Oystaeyen]]
[[!redirects Freddy Van Oystaeyen]]


Frederic Bohnenblust

* [Wikipedia entry](https://de.wikipedia.org/wiki/Frederic_Bohnenblust) (deutsch)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=7491)

## Selected writings

Early discussion of the [[braid group]]:

* [[Frederic Bohnenblust]], *The Algebraical Braid Group*, Annals of Mathematics Second Series **48** 1 (1947) 127-136 [[doi:10.2307/1969219](https://doi.org/10.2307/1969219)]

category: people

[[!redirects H. Frederic Bohnenblust]]


Frederic Fitch

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

## Selected writings

On [[mathematical logic]]:

* {#Fitch52} [[Frederic Fitch]], *Symbolic Logic: An introduction*, Ronald Press, New York 1952

On [[Curry's paradox]] and the [[axiom of full comprehension]]:

* {#Fitch69} [[Frederic Fitch]], *A method for avoiding the Curry paradox*, in *Essays in Honor of Carl G. Hempel*, Reidel, Dordrecht, Holland 1969, pp. 255--265 ([doi:10.1007/978-94-017-1466-2](https://link.springer.com/book/10.1007/978-94-017-1466-2))

category: people

[[!redirects Frederic Brenton Fitch]]
[[!redirects Frederic B. Fitch]]


Frederic Shultz

* [webpage](http://palmer.wellesley.edu/~fshultz/)

## Related entries

* [[Alfsen-Shultz theorem]]

category: people


Frederick R. Cohen

> not to be confused with *[[Ralph Cohen]]*.

***

* [personal page](http://web.math.rochester.edu/people/faculty/cohf/)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=6654)

## Selected writings

On [[configuration spaces of points]]:

* {#Cohen09} [[Frederick R. Cohen]], *Introduction to configuration spaces and their applications*, in *Braids* Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore **19** (2009)  183-261  $[$[doi:10.1142/9789814291415_0003](https://doi.org/10.1142/9789814291415_0003), [pdf](https://www.mimuw.edu.pl/~sjack/prosem/Cohen_Singapore.final.24.december.2008.pdf), [doi:https://doi.org/10.1142/7550 ](https://doi.org/10.1142/7550 )$]$

On the [[real cohomology]] of (the [[Fulton-MacPherson compactification]] of) [[configuration spaces of points|configuration spaces of ordered points]] in [[Euclidean space]]:

* {#Cohen73} [[Fred Cohen]], _Cohomology of braid spaces_, Bull. Amer. Math. Soc. Volume 79, Number 4 (1973), 763-766 ([euclid:1183534761](https://projecteuclid.org/euclid.bams/1183534761))

* {#Cohen76} [[Fred Cohen]], _The homology of $C_{n+1}$-Spaces, $n \geq 0$_, In: _The Homology of Iterated Loop Spaces_, Lecture Notes in Mathematics, vol 533. Springer 1976([doi:10.1007/BFb0080467](https://doi.org/10.1007/BFb0080467))

On [[G-structure]] for $G = Br_\infty$ the infinite [[braid group]]:

* [[Frederick R. Cohen]], *Braid orientations and bundles with flat connections*, Inventiones mathematicae **46** (1978) 99–110 [[doi:10.1007/BF01393249](https://doi.org/10.1007/BF01393249)]

On the [[ordinary cohomology]] of [[configuration spaces of points]] (such as in [[surfaces]] and [[knot complements]]):

* [[Carl-Friedrich Bödigheimer]], [[Fred Cohen]], L. Taylor, _On the homology of configuration spaces_, Topology Vol. 28 No. 1, p. 111-123 1989 ([pdf](https://core.ac.uk/download/pdf/82124359.pdf))

On the [[ordinary homology]] of [[loop spaces]] of [[configuration spaces of points]] and their relation to [[Vassiliev invariants]]:

* [[Fred Cohen]], [[Samuel Gitler]], _Loop spaces of configuration spaces, braid-like groups, and knots_, In: Jaume Aguadé, Carles Broto, [[Carles Casacuberta]]  (eds.) _Cohomological Methods in Homotopy Theory_. Progress in Mathematics, vol 196. Birkhäuser, Basel ([doi:10.1007/978-3-0348-8312-2_7](https://doi.org/10.1007/978-3-0348-8312-2_7))

* [[Fred Cohen]], [[Samuel Gitler]], _On loop spaces of configuration spaces_, Trans. Amer. Math. Soc. __354__ (2002), no. 5, 1705–1748, ([jstor:2693715](https://www.jstor.org/stable/2693715), [MR2002m:55020](http://www.ams.org/mathscinet-getitem?mr=1881013))

On [[braid group representations]] seen in the [[topological K-theory]] of the [[classifying space]] of the [[braid group]]:

* [[Alejandro Adem]], [[Daniel C. Cohen]], [[Frederick R. Cohen]], *On representations and K-theory of the braid groups*, Math. Ann. **326** (2003) 515-542 ([arXiv:math/0110138](https://arxiv.org/abs/math/0110138), [doi:10.1007/s00208-003-0435-8](https://doi.org/10.1007/s00208-003-0435-8))

On [[homotopy of rational maps]] and [[moduli spaces of monopoles]] related to [[braid groups]]:

* [[Fred Cohen]], [[Ralph Cohen]], [[B. M. Mann]], [[R. J. Milgram]], _The topology of rational functions and divisors of surfaces_, Acta Math (1991) 166: 163 ([doi:10.1007/BF02398886](https://doi.org/10.1007/BF02398886))

* [[Ralph Cohen]], [[John D. S. Jones]],  _Monopoles, braid groups, and the Dirac operator_, Comm. Math. Phys. Volume 158, Number 2 (1993), 241-266 ([euclid:cmp/1104254240](https://projecteuclid.org/euclid.cmp/1104254240))

On [[mapping spaces]] of [[group homomorphisms]] from [[finitely generated group|finitely generated]] [[discrete groups]] to [[Lie groups]]:

* [[Frederick R. Cohen]], [[Mentor Stafa]], *A survey on spaces of homomorphisms to Lie groups*,  In: Callegaro F., Cohen F., De Concini C., Feichtner E., Gaiffi G., Salvetti M. (eds.) *Configuration Spaces*, INdAM Series, vol 14, Springer 2016 ([arXiv:1412.5668](https://arxiv.org/abs/1412.5668), [doi:10.1007/978-3-319-31580-5_15](https://doi.org/10.1007/978-3-319-31580-5_15))

category: people

[[!redirects Fred Cohen]]
[[!redirects Frederick Ronald Cohen]]


Frederik Denef

__Frederik Denef__ is a theoretical physicist at Leuven, Belgium. Most of his research is related to [[string theory]], including its connections to [[cosmology]], [[black holes]], [[wall crossing]],  the formal analogy of [[holography]] in [[condensed matter physics]] and generally [[mathematical physics]].

* [webpage](http://www.kuleuven.be/wieiswie/en/person/00009261), [publications](https://lirias.kuleuven.be/cv?u=U0009261)

## Selected writings

On [[string theory vacua]]:

* {#Denef08} [[Frederik Denef]], _Les Houches Lectures on Constructing String Vacua_, in: _[[String theory and the real world]]_ ([arXiv:0803.1194](http://arxiv.org/abs/0803.1194), [spire:780946](https://inspirehep.net/literature/780946))

category: people


Fredholm determinant

_Fredholm determinant_ is a generalization of a determinant of a finite-dimensional matrix to a class of operators on Banach spaces which differ from identity by a trace class operator or by an appropriate analogue in more abstract context (there are appropriate determinants on certain Banach ideals). It is often considered as an analytic function of a perturbation parameter $\lambda$. The calculations with Fredholm determinants have applications in operator theory, [[random matrix]] theory, integrable models etc.

In his original setup, Fredholm attached the determinant
$$
d(\lambda) = \sum_{n=0}^{\infty}
\frac{\lambda^n}{n!}\int_a^b\int_a^b\ldots \int_a^b det((K(x_k, x_l))_{k,l=1}^n) d x_1 \cdots d x_n
$$
to the Fredholm integral equation of the second kind
$$
u(x) + \lambda \int^b_a K(x,y) u(y) d y  = f(x), \,\,\,x\in (a,b),
$$
and $d$ is an [[entire function]] of $\lambda$ such that $d(\lambda) \neq 0$ iff the integral equation has a unique solution.

## References

#### General and mathematical

* wikipedia [Fredholm determinant](http://en.wikipedia.org/wiki/Fredholm_determinant)
* Albrecht Pietsch, _Operator ideals_,  North-Holland Mathematical Library 20; _Eigenvalues and s-Numbers_,  Cambridge Studies in Advanced Mathematics 13; _History of Banach spaces and linear operators_, sec 6.5.2
* [[Alexander Grothendieck]], _La théorie de Fredholm_, Bulletin de la Société Mathématique de France __84__ (1956), p. 319-384, [numdam](http://www.numdam.org/item?id=BSMF_1956__84__319_0)
* Fritz Gesztesy, Yuri Latushkin, Konstantin A. Makarov, _Evans functions, Jost functions and Fredholm determinants, Arch. Rational Mech. Anal. 186 (2007) 361–421,[doi](http://dx.doi.org/10.1007/s00205-007-0071-7) [pdf](http://www.math.missouri.edu/~yuri/preprints/EJF.pdf)
* Barry Simon, _Notes on infinite determinants of Hilbert space operators_, Advances in Math. __24__ (1977), no. 3, 244–273 [MR482328](http://www.ams.org/mathscinet-getitem?mr=482328) doi</doi>
* Folkmar Bornemann, _On the numerical evaluation of Fredholm determinants_, Math. Comp. 79 (2010), no. 270, 871–915 [MR2600548](http://www.ams.org/mathscinet-getitem?mr=2600548)
[pdf](http://www-m3.ma.tum.de/foswiki/pub/M3/Allgemeines/FolkmarBornemannPublications/Fredholm.pdf) [doi](http://dx.doi.org/10.1090/S0025-5718-09-02280-7)

#### Physics applications

* F. J. Dyson, _Fredholm determinants and inverse scattering problems_, Comm. Math. Phys. 47, 171–183 (1976) [MR406201](http://www.ams.org/mathscinet-getitem?mr=406201) [euclid](http://projecteuclid.org/euclid.cmp/1103899727)
* C. A. Tracy, H. Widom, _Fredholm determinants, differential equations and matrix models_, Commun. Math. Phys. __163__, 33–72 (1994) [MR1277933](http://www.ams.org/mathscinet-getitem?mr=1277933) [euclid](http://projecteuclid.org/getRecord?id=euclid.cmp/1104270379)
* J. Harnad, Alexander R. Its, _Integrable Fredholm operators and dual isomonodromic deformations_,
Comm. Math. Phys. 226 (2002), no. 3, 497–530 [MR1896879](http://www.ams.org/mathscinet-getitem?mr=1896879) [doi](http://dx.doi.org/10.1007/s002200200614)
* M. Adler, [[M. Cafasso]], P. van Moerbeke, _Nonlinear PDEs for Fredholm determinants arising from string equations_, [arxiv/1207.6341](http://arxiv.org/abs/1207.6341)
* M. Bertola, [[M. Cafasso]], _Fredholm determinants and pole-free solutions to the noncommutative Painlevé II equation_, Comm. Math. Phys. __309__ (2012), no. 3, 793–833 [MR2885610](http://www.ams.org/mathscinet-getitem?mr=2885610) [doi](http://dx.doi.org/10.1007/s00220-011-1383-x)
* Michio Jimbo, Tetsuji Miwa, Yasuko Môri, Mikio Sato,
_Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent_, Phys. D 1 (1980), no. 1, 80–158 [MR573370](http://www.ams.org/mathscinet-getitem?mr=573370)

category: functional analysis
[[!redirects Fredholm determinants]]


Fredholm index

+-- {: .rightHandSide}
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### Context
#### Index theory
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[[!include index theory - contents]]
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#### Operator algebra
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[[!include AQFT and operator algebra contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A [[Fredholm operator]] $F \,\colon\, \mathcal{H} \longrightarrow \mathcal{H}$ by definition has [[finite-dimensional vector space|finite-dimensional]] [[kernel]] $ker(F)$ and [[cokernel]] $coker(F)$. Its *[[index]]* is the difference between these [[dimension of a vector space|dimensions]]:

$$
  ind(F)
  \;\coloneqq\;
  dim\big(ker(F)\big)
  \,-\,
  dim\big(coker(F)\big)
  \,.
$$

## Examples

* [charged vacua of free Dirac field in Coulomb background](Dirac+field#FreeDiracFieldInCoulombBackground)

## Related concepts

* [[index]]

* [[index of a Dirac operator]]

## References

Textbook account:

* [[William Arveson]], Section 3.4 of: *A Short Course on Spectral Theory*, Graduate Texts in Mathematics **209** $[$[doi:10.1007/b97227](https://link.springer.com/book/10.1007/b97227)$]$

[[!redirects Fredholm indices]]


Fredholm module

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### Context
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $A$ a [[C-star algebra]], an _odd Fredholm module_ over $A$ is a [[representation of a C-star-algebra|representation]] $\pi : A \to \mathcal{B}(\mathcal{H})$ of $A$ on a [[Hilbert space]] $\mathcal{H}$ together with a [[Fredholm operator]] $F$ on $\mathcal{H}$ such that $[F,\pi(a)] \in \mathcal{K}(\mathcal{H})$ for all $a \in A$.

(...)

(Here $\mathcal{B}(-)$ denotes [[bounded operators]] and $\mathcal{K}(-)$ denotes [[compact operators]]).

A Fredholm module defines a class in [[K-homology]], hence in [[KK-theory]].

## Related concepts

* [[unbounded Fredholm module]] ([[spectral triple]])

* [[K-homology]]

* [[KK-theory]]

* [[Dirac operator]]

[[!redirects Fredholm modules]]


Fredholm operator

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

# Contents
* table of contents
{:toc}

## Definition

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

A [[continuous function|continuous]] [[linear operator]] $F \colon B_1\to B_2$ between [[Banach spaces]] is **Fredholm** if it has [[finite set|finite]] [[dimension|dimensional]] [[kernel]] and finite dimensional [[cokernel]].

=--

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

The difference between the [[dimension of a vector space|dimensions]] of the kernel and the cokernel of a Fredholm operator $F$ is called its *[[index]]* (the *[[Fredholm index]]*)

$$
  ind F \coloneqq dim (ker F) - dim (coker F) = dim (ker F) - codim (im F)
  \,.
$$

=--

A standard equivalent characterization of Fredholm operators is the following:

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

A **parametrix** of a [[bounded operator|bounded]] [[linear operator]] $F \colon \mathcal{H}_1 \to \mathcal{H}_2$ is a reverse operator $P \colon \mathcal{H}_2 \to \mathcal{H}_1$ which is an "[[inverse]] up to [[compact operators]]", i.e. such that $F \circ P - id_{\mathcal{H}_2}$ and $P \circ F - id_{\mathcal{H}_1}$ are both [[compact operators]].

=--

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

A [[bounded operator|bounded]] [[linear operator]] $ F \colon B_1\to B_2$ between [[Banach spaces]] is Fredholm, def. \ref{FredholmOperator} precisely it is has a parametrix, def. \ref{Parametrix}.

=--

## Examples

* [[elliptic operator|Elliptic operators]] on [[compact topological space|compact]] [[manifolds]] are naturally Fredholm, when understood between the appropriate [[Sobolev spaces]].

* [charged vacua of free Dirac field in Coulomb background](Dirac+field#FreeDiracFieldInCoulombBackground) are characterized by Fredholm operators

## Properties

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

The [[image]] (range) of a Fredholm operator is closed.

=--

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

The subspace $Fred(B_1,B_2)\subset B(B_1,B_2)$ of Fredholm operators in the space of bounded linear operators with the norm topology is open.

=--

  In other words, given a Fredholm operator $F$, there exists $\epsilon\gt 0$ such that every bounded linear operator $G$ satisfying $\| G-F\|\lt \epsilon$ is Fredholm. Fredholm operators on a fixed separable Hilbert space $H = B_1 = B_2$ form a [[semigroup]] with respect to the composition and the index is a morphism of semigroups: $ind F G = ind F + ind G$.

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

The space $Fred$ of all Fredholm operators on an (infinite dimensional) [[separable Hilbert space]] is a model for the [[classifying space]] of degree-0 [[topological K-theory]].

=--

(...)

## Generalizations

Fredholm operators generalize to Fredholm complexes. A finite [[chain complex]]

$$
0 \to C_0 \stackrel{d_0}\to C_1\stackrel{d_1}\to C_2 \ldots C_n\to 0
$$

of [[Banach space]]s and bounded operators is said to be a **Fredholm complex** if the images $d_i$ are closed and the [[chain homology]] of the complex is finite dimensional. The [[Euler characteristic]] (the alternative sum of the dimensions of the homology groups) is then called the __index__ of the Fredholm complex. Each Fredholm operator can be considered as a Fredholm complex concentrated at zero. Each Fredholm complex produces a Fredholm operator from the sum of the even- to the sum of the odd-numbered spaces in the complex.

One can consider *Fredholm almost complexes*, where $d_i \circ d_{i-1}$ is not zero but the image of that operator is compact. Out of every Fredholm almost complex one can make a complex by correcting the differentials by compact perturbation terms. [[elliptic complex|Elliptic complexes]] give examples of Fredholm complexes.

## Related concepts

* [[Fredholm module]],

* [[Fredholm determinant]],

* [[Dirac operator]],

* [[Toeplitz operator]]

* [[K-theory]], [[KK-theory]]

* [[index theory]]

## References

Textbook accounts:

* [[William Arveson]], Section 3.3 of: *A Short Course on Spectral Theory*, Graduate Texts in Mathematics **209**, Springer (2002) $[$[doi:10.1007/b97227](https://link.springer.com/book/10.1007/b97227)$]$

Discussion of the space of Fredholm operators as the [[classifying space]] for [[topological K-theory]]:

* {#Atiyah67}  [[Michael Atiyah]], Appendix of: _K-theory_, Harvard Lecture 1964 (notes by D. W. Anderson), Benjamin 1967 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf), [[AtiyahKTheory.pdf:file]])

See also:

* Wikipedia, *[Fredholm operator](http://en.wikipedia.org/wiki/Fredholm_operator)*

* A. S. Mishchenko, Векторные расслоения и их применения (Vector bundles and their applications), Nauka, Moscow, 1984. 208 pp. MR87f:55010

* S. Rempel, B-W. Schulze, _Index theory of elliptic boundary problems_, Akademie--Verlag, Berlin, 1982.

* [[Lars Hörmander]], _The analysis of linear partial differential operators. III. Pseudo-differential operators_, 1994, reprinted 2007.

* Pietro Aiena, _Fredholm and local spectral theory, with applications to multipliers_, [book page](http://www.springer.com/mathematics/analysis/book/978-1-4020-1830-5)

* [[Otgonbayar Uuye]], _A simple proof of the Fredholm Alternative_, [arxiv/1011.2933](http://arxiv.org/abs/1011.2933)

* [[Alexander Grothendieck]], _La théorie de Fredholm_, Bulletin de la Société Mathématique de France __84__ (1956), p. 319-384, [numdam](http://www.numdam.org/item?id=BSMF_1956__84__319_0)

* Marina Prokhorova, _Spectral Sections_, [arXiv:2008.04672](https://arxiv.org/abs/2008.04672).

* Marina Prokhorova, _Spaces of unbounded Fredholm operators. I. Homotopy equivalences_, [arXiv:2110.14359](https://arxiv.org/abs/2110.14359).

* Marina Prokhorova, _The continuity properties of discrete-spectrum families of Fredholm operators_, [arXiv:2201.09869](https://arxiv.org/abs/2201.09869).

* Marina Prokhorova, _From graph to Riesz continuity_, [arXiv:2202.03337](https://arxiv.org/abs/2202.03337).

For Fredholm complexes, see

* [[Graeme Segal]], _Fredholm complexes_, Quarterly Journal of Mathematics 21:4 (1970), 385–402.  [doi](http://dx.doi.org/10.1093/qmath/21.4.385).

category: analysis

[[!redirects Fredholm complex]]
[[!redirects Fredholm operators]]
[[!redirects parametrix]]


Fredrik Forsberg

[[!redirects Fredrik Nordvall Forsberg]]

* [webpage](https://personal.cis.strath.ac.uk/fredrik.nordvall-forsberg/)

## Selected writings

On [[quotient types]] and [[higher inductive types]]:

* [[Thorsten Altenkirch]], [[Paolo Capriotti]], [[Gabe Dijkstra]], [[Nicolai Kraus]], [[Fredrik Forsberg]], *Quotient inductive-inductive types*, Foundations of Software Science and Computation Structures (FoSSaCS 2018), Lecture Notes in Computer Science **10803** (2018) [[doi:10.1007/978-3-319-89366-2_16](https://doi.org/10.1007/978-3-319-89366-2_16), [arXiv:1612.02346](https://arxiv.org/abs/1612.02346)]

On [[ordinals]] in [[type theory]] and [[marked extensional well-founded orders]]:

* {#deJongEtAl23} [[Tom de Jong]], [[Nicolai Kraus]], [[Fredrik Nordvall Forsberg]], [[Chuangjie Xu]], *Set-Theoretic and Type-Theoretic Ordinals Coincide* [[arXiv:2301.10696](https://arxiv.org/abs/2301.10696)]

## Related entries

* [[inductive-inductive type]]

category: people

[[!redirects Fredrik Nordvall Forsberg]]


Fredrik Meyer

* [personal page](https://fredrikmeyer.net/uio-math/)

## Selected writings

On [[algebraic stacks]]:

* [[Fredrik Meyer]], *Notes on algebraic stacks* (2013) [[pdf](https://fredrikmeyer.net/uio-math/algstacks.pdf), [[Meyer_AlgebraicStacks.pdf:file]]]

category: people


free abelian group

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

# Contents
* table of contents
{: toc}

## Idea

The _free abelian group_ $\mathbb{Z}[S]$ on a [[set]] $S$ is the [[abelian group]] whose elements are _formal $\mathbb{Z}$-[[linear combinations]]_ of elements of $S$.

## Definition
 {#Definition}

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

Let

$$
  U \colon Ab \longrightarrow Set
$$

be the [[forgetful functor]] from the category [[Ab]] of abelian groups, to the category [[Set]] of sets. This has a [[left adjoint]] [[free construction]]:

$$
  \mathbb{Z}[-] \colon Set \longrightarrow Ab
  \,.
$$

This is the _free abelian group functor_. For $S \in $ [[Set]], the **free abelian group** $\mathbb{Z}[S] \in $ [[Ab]] is the [[free object]] on $S$ with respect to this [[free-forgetful adjunction]].

=--

Of course, this notion is meant to be invariant under isomorphism: it doesn't depend on the left adjoint chosen. Thus, if a functor of the form $\hom_{Set}(S, U-): Ab \to Set$ is [[representable functor|representable]] by
an abelian group $A$, then we may say $A$ is a free abelian group on $S$. A specific choice of isomorphism

$$\hom_{Ab}(A, -) \cong \hom_{Set}(S, U-)$$

corresponds, via the [[Yoneda lemma]], to a function $S \to U A$ which exhibits $S$, or rather its image under this function, as a specific [[basis]] of $A$. If $A$ is so equipped with such a universal arrow $S \to U A$, then it is harmless to call $A$ "the" free abelian group on $S$.

Explicit descriptions of free abelian groups are discussed [below](#Properties).

## Properties
 {#Properties}

### In terms of formal linear combinations

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

A **formal linear combination** of elements of a set $S$ is
a [[function]]

$$
  a : S \to \mathbb{Z}
$$
such that only finitely many of the values $a_s \in \mathbb{Z}$ are non-zero.

Identifying an element $s \in S$ with the function
$S \to \mathbb{Z}$ which sends $s$ to $1 \in \mathbb{Z}$ and all other elements to 0, this is written as

$$
  a = \sum_{s \in S} a_s \cdot s
  \,.
$$

In this expression one calls $a_s \in \mathbb{Z}$ the [[coefficient]] of $s$ in the formal linear combination.

=--

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

Definition \ref{FormalLinearCombination} of formal linear combinations makes sense with  [[coefficients]] in any [[abelian group]] $A$, not necessarily the
integers.

$$
  A[S] \coloneqq \mathbb{Z}[S] \otimes A
  \,.
$$

=--

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

For $S \in $ [[Set]], the __group of formal linear combinations__ $\mathbb{Z}[S]$ is the
[[group]] whose underlying [[set]] is that of formal linear combinations,
def. \ref{FormalLinearCombination}, and whose group operation is
the pointwise addition in $\mathbb{Z}$:

$$
  (\sum_{s \in S} a_s \cdot s)
  +
  (\sum_{s \in S} b_s \cdot s)
  =
  \sum_{s \in S} (a_s + b_s) \cdot s
  \,.
$$

=--

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

The free abelian group on $S \in Set$ is, up to [[isomorphism]], the
group of [[formal linear combinations]], def. \ref{GroupOfFormalLinearCombinations}, of elements of $S$.

=--

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

For $S$ a set, the free abelian group $\mathbb{Z}[S]$ is the [[direct sum]] in [[Ab]] of ${|S|}$-copies of $\mathbb{Z}$ with itself:

$$
  \mathbb{Z}[S] \simeq \oplus_{s \in S} \mathbb{Z}
  \,.
$$

=--

### Basic properties

\begin{proposition}\label{FreeAbelianGroupFunctorSendsProductsToTensorProducts}
  The free abelian group of a [[Cartesian product]]
  $S \times Z$ of [[sets]] $S, T \,\in\, Sets$
  is [[natural isomorphism|naturally isomorphic]] to
  the [[tensor product of abelian groups|tensor product of]]
  the free abelian groups of the factors:
  $$
    \mathbb{Z}[S \times T]
    \;\simeq\;
    \mathbb{Z}[S] \otimes \mathbb{Z}[T]
    \,.
  $$
  In other words, as a [[functor]] from [[Set]] to [[Ab]]
  $$
    \mathbb{Z}[-]
    \;\colon\;
    Set \longrightarrow Ab
  $$
  this is a [[strong monoidal functor]].
\end{proposition}
This follows, for instance, from the above expression (Prop. \ref{FreeAbelianGroupIsGroupOfFormalLinearCombinations}) of free abelian groups as groups of formal linear combinations.

\begin{remark}
\label{FreeSimplicialAbelianGroupConstrIsStrongMonoidal}
  In [[homotopy theory]]/[[algebraic topology]], the free abelian group construction is frequently applied degreewise to a [[simplicial set]] $S_\bullet$ which is (or is [[simplicial weak homotopy equivalence|weakly equivalent to]]) the [[singular simplicial complex]] of a [[topological space]] --- because the resulting [[chain complex]] (under the [[Dold-Kan correspondence]]) then computes the [[ordinary homology|ordinary]] [[singular homology]] of that space.
But since

1. the [[product of simplicial sets]] is degreewise that in [[Set]],

1. the [tensor product of simplicial abelian groups](monoidal+Dold-Kan+correspondence#MonoidalCategoryStructures) is degreewise the [[tensor product of abelian groups]]

the above Prop. \ref{FreeAbelianGroupFunctorSendsProductsToTensorProducts} implies that also the functor
$$
 \mathbb{Z}[-]
  \;\colon\;
  sSet \longrightarrow sAb
$$
from [[sSet]] to [[sAb]] is [[strong monoidal functor|strong monoidal]].
\end{remark}

### Subgroups
 {#Subgroups}

\bgein{proposition}
\label{SubgroupsOfFreeAbelianGroupsAreFree}
Assuming the [[axiom of choice]], then every [[subgroup]] of a free abelian group (def. \ref{FreeAbelianGroup}) is itself a free abelian group.
\end{proposition}
(e.g. [Lang 02, Appendix 2 §2, page 880](#Lang02)) For a full **proof** see at _[[principal ideal domain]]_ [this theorem](principal+ideal+domain#free).

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

Prop. \ref{SubgroupsOfFreeAbelianGroupsAreFree} implies that (assuming [[axiom of choice|AC]]) every abelian group admits a [[free resolution]] of length 2, hence with trivial [[syzygies]]. See [there](free+resolution#AbelianGroupHasFreeResolutionOfLength2).

=--

## Examples

* The free abelian group on the [[singular simplicial complex]] of a [[topological space]] $X$ consists of the [[singular chains]] on $X$.

* For $R$ a ring and $S$ a set, the [[tensor product of abelian groups]]
  $\mathbb{Z}[S] \otimes R$ is the [[free module]] over $R$ on the [[basis]] $S$. If $R = k$ is a [[field]], then this is the [[vector space]] over $k$ with basis $S$.

* For $R$ a ring, the [[tensor product of abelian groups]]  $\mathbb{Z}[\mathbb{N}]\otimes R$ is the abelian group underlying the [[ring]] of [[polynomials]] over $R$.

## Related concepts

* [[free group]]

* [[free commutative monoid]]

* [[free simplicial abelian group]]

* [[solid abelian group]]

## References

Textbook accounts:

* {#Lang02} [[Serge Lang]], _Algebra_, Graduate Texts in Mathematics 211 (Revised third ed.), Springer. 2002

[[!redirects free abelian group]]
[[!redirects free abelian groups]]

[[!redirects free abelian group functor]]
[[!redirects free abelian group functors]]


free action

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

#Contents#
* table of contents
{:toc}

## Definition

### In sets
 {#InSets}

\begin{definition}
\label{FreeGroupActionInComponents}
A [[group action]]

$$
  (-)\cdot(-) \;\colon\; G \times X \to X
$$

of a [[group]] $G$ on a [[set]] $X$ is called _free_ if for every $x \in X$,  the [[equation]] $g \cdot x = x$ [[implication|implies]] $g = \mathrm{e}$ (the [[neutral element]]), hence if only the action of the [[neutral element]] has [[fixed points]]

\[
  \label{FreenessConditionInComponents}
  \underset{x \in X}{\exists}
  \;\;
  g \cdot x \,=\, x
  \;\;\;\;\;\;\;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;\;\;\;\;\;\;
  g \,=\, \mathrm{e}
  \,.
\]

Equivalently, an action is free if and only if for any [[pair]] of elements $x,y \in X$, there is _at most one_ group element $g \in G$ such that $g \cdot x = y$.

\end{definition}

\begin{remark}
Beware the similarity to and difference of free actions with [[effective group action|effective action]]: a free action is effective, but an effective action need not be free.
\end{remark}

\begin{remark}
A free action that is also [[transitive action|transitive]] is called _[[regular action|regular]]_ (at least after passing to its [[linear representation|linear]] [[permutation representation]]).
\end{remark}

\begin{proposition}
\label{FreeGroupActionViaMonomorphicShearMap}
A group action is free according to Def. \ref{FreeGroupActionInComponents} iff its [[shear map]] is a [[monomorphism]]:

\[
  \label{FreeActionWitnessedByMonomorphicShearMap}
  G \curvearrowright X
  \;
  \text{is free}
  \;\;\;\;\;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;\;\;\;\;\;
  \array{
    G \times X
    &
    \xhookrightarrow{\;(pr_2, \cdot)\;}
    &
    X \times X
    \\
    (g, x)
    &
    \mapsto
    &
    \big( x, g \cdot x \big)
    \,.
  }
\]
\end{proposition}
\begin{proof}
  In one direction, assume that the
action is free in the sense of Def. \ref{FreeGroupActionInComponents}.
Then given $(x,g\cdot x) = (x', g' \cdot x')$
it follows that $x = x'$ and thus that $g \cdot x = g' \cdot x$, which by (eq:FreenessConditionInComponents) implies that $g = g'$. This implication means that the shear map is injective.

In the other direction, assume that the shear map is injective. Then given $g \cdot x = x$, hence equivalently $(x, g \cdot x) = (x, \mathrm{e} \cdot x)$ it follows that ($x = x$ and) $g = \mathrm{e}$. This is the required implication (eq:FreenessConditionInComponents).
\end{proof}

### In categories with finite products

In the form (eq:FreeActionWitnessedByMonomorphicShearMap) the definition of free action makes sense more generally for [[action objects]] [[internalization|internal]] to any ambient [[category]] (with [[finite products]]):

\begin{definition}
\label{FreeInternalActions}
  Let $\mathcal{C}$ be a [[category]] with [[finite products]], let $G \,\in\, Grp(\mathcal{C})$ be an [[internal group]] and $G \curvearrowright X \,\in\, G Act(\mathcal{C}) $ an [[internal action]]:

$$
  G \times X \xrightarrow{\;\; rho \;\;} X
  \,.
$$

Then this is a *free action* if its [[shear map]] is a [[monomorphism]]  in $\mathcal{C}$:
\[
  \label{FreeInternalAction}
  \rho \; \text{is free}
  \;\;\;\;
  \Leftrightarrow
  \;\;\;\;
  G \times X \xhookrightarrow{\; (\rho, pr_2)\;} X \times X
  \,.
\]
\end{definition}

### In an $\infty$-topos
 {#DefinitionInAnInfinityTopos}

Def. \ref{FreeInternalActions} also makes sense in the generality of [[(infinity,1)-category theory|$(\infty,1)$-category theory]], with [[monomorphisms]] understood to be [[n-truncated object in an (infinity,1)-category|(-1)-truncated morphisms]].

\begin{definition}
\label{FreeInfinityAction}
**(free $\infty$-action in an $\infty$-topos)**
\linebreak
Let

* $\mathbf{H}$ be an [[(infinity,1)-topos|$\infty$-topos]];

* $G \,\in\, Grp(\mathbf{H})$ a [[group object in an (infinity,1)-category|group object]] (a [[geometric homotopy theory|geometric]] [[infinity-group|$\infty$-group]]);

* $G \curvearrowright X \,\in\, G Act(\mathbf{H})$ an [[infinity-action|$\infty$-action]] of $G$ on $X$.

we say that the $\infty$-action is *free*, if its [[shear map]] $shear_1$ is [[(-1)-truncated]]:

\begin{tikzcd}
      X \times G
      \ar[rr, "\rho"]
      \ar[dd, "\mathrm{pr}_1"{left}]
      \ar[dr, hook, dashed, "\mathrm{shear}_1"{sloped}]
      &&
      X
      \ar[dr, -, shift left=1pt]
      \ar[dr, -, shift right=1pt]
      \ar[dd, ->>]
      \\
      &
      X \times X
      \ar[rr, crossing over, "\mathrm{pr}_1"{pos=.17}]
      \ar[ddrr, phantom, "{\mbox{\tiny (pb)}}"{pos=.1}]
      &&
      X
      \ar[dd]
      \\
      X
      \ar[dr, -, shift left=1pt]
      \ar[dr, -, shift right=1pt]
      \ar[rr, ->>]
      &&
      X /\!\!/ G
      \ar[dr]
      \\
      &
      X \ar[rr]
      \ar[from=uu, crossing over, "\mathrm{pr}_2"{left, pos=.25}]
      &&
      \ast
\end{tikzcd}

\end{definition}

\begin{lemma}
\label{HigherShearMapsOfFreeInfinityActionsAreMinusOneTruncated}
**(higher shear maps of free $\infty$-actions are $(-1)$-truncated)**
\linebreak
  If an $\infty$-action $G \curvearrowright X$ is free according to Def. \ref{FreeInfinityAction} and $X$ is [[inhabited object|inhabited]], $X \twoheadrightarrow \ast$, then also all its higher shear maps $shear_n$ are [[(-1)-truncated]]:

\begin{tikzcd}
      {}
      \ar[d, -, shift left=15pt, dotted]
      \ar[from=d, -, shift left=10pt, dotted]
      \ar[d, -, shift left=5pt, dotted]
      \ar[from=d, -, dotted]
      \ar[d, -, shift right=5pt, dotted]
      \ar[from=d, -, shift right=10pt, dotted]
      \ar[d, -, shift right=15pt, dotted]
      &
      {}
      \ar[d, -, shift left=15pt, dotted]
      \ar[from=d, -, shift left=10pt, dotted]
      \ar[d, -, shift left=5pt, dotted]
      \ar[from=d, -, dotted]
      \ar[d, -, shift right=5pt, dotted]
      \ar[from=d, -, shift right=10pt, dotted]
      \ar[d, -, shift right=15pt, dotted]
      \\
      X
        \times
      {G} \times {G}
      \ar[d, shift left=10pt, hook]
      \ar[from=d, shift left=5pt]
      \ar[d]
      \ar[from=d, shift right=5pt]
      \ar[d, shift right=10pt]
      \ar[r, hook, "\mathrm{shear}_2"]
      &
      X \times X \times X
      \ar[d, shift left=10pt]
      \ar[from=d, shift left=5pt]
      \ar[d]
      \ar[from=d, shift right=5pt]
      \ar[d, shift right=10pt]
      \\
      X
        \times
      {G}
      \ar[d, shift left=5pt, hook]
      \ar[from=d]
      \ar[d, shift right=5pt]
      \ar[r, hook, "\mathrm{shear}_1"]
      &
      X \times X
      \ar[d, shift left=5pt]
      \ar[from=d]
      \ar[d, shift right=5pt]
      \\
      X
      \ar[r, -, shift right=1pt]
      \ar[r, -, shift left=1pt, "\mathrm{shear}_0"{above}]
      \ar[d, ->>]
      &
      X
      \ar[d, ->>]
      \\
      X /\!\!/ G
      \ar[r]
      &
      \ast
\end{tikzcd}

\end{lemma}
\begin{proof}
  Notice that for $X \twoheadrightarrow \mathcal{X}$ any [[effective epimorphism in an (infinity,1)-category|effective epimorphism]] in the $\infty$-topos $\mathbf{H}$, we have for $n \in \mathbb{N}_+$ the following [[homotopy pullback]] [[pasting diagram]]:

\begin{tikzcd}[column sep=30pt]
    &&
    X^{\times^{n+2}_{\mathcal{X}}}
    \ar[dl]
    \ar[dd, phantom, "\mbox{\tiny (pb)}"]
    \ar[dr]
    \\
    &
    X^{\times^{n+1}_{\mathcal{X}}}
    \ar[dl]
    \ar[dr]
    \ar[dd, phantom, "\mbox{\tiny (pb)}"]
    &&
    X^{\times^2_{\mathcal{X}}}
    \ar[dl]
    \ar[dr]
    \ar[dd, phantom, "\mbox{\tiny (pb)}"]
    \\
    X^{\times^n_{\mathcal{X}}}
    \ar[dr]
    &&
    X
    \ar[dl]
    \ar[dr]
    &&
    X
    \ar[dl]
    \\
    &
    \mathcal{X}
    &&
    \mathcal{X}
\end{tikzcd}
Here the two bottom squares are homotopy Cartesian by definition of [[Cech nerves]], and the top square is homotopy Cartesian since the Cech nerve is a [[groupoid object in an (infinity,1)-category|groupoid object]] (see also at *[[groupoid objects in an (∞,1)-topos are effective]]*) which satisfies the groupoidal [[Segal conditions]] (by [this Def.](groupoid+object+in+an+infinity1-category#GroupoidObjectsViaGroupoidalSegalCondition)).

But this implies, for all $n \geq 1$, that the $n+1$st shear map is the [[homotopy fiber product]] in the [[arrow category]] $\mathbf{H}^{\Delta[1]}$ of the 1st with the $n$th shear map over the [[identity morphism]] on $X$:

\begin{tikzcd}
    X \times G^{\times^{n+1}}
    \ar[rr]
    \ar[dr, hook, dashed, "\mathrm{shear}_{n+1}"{sloped, below}]
    \ar[dd]
    &&
    X \times G
    \ar[dr, hook, "\mathrm{shear}_1"{sloped}]
    \ar[dd, "\mathrm{pr}_1"{left, pos=.3}]
    \\
    &
    X^{\times^{n+2}}
    \ar[rr, crossing over]
    \ar[ddrr, phantom, "\mbox{\tiny (pb)}"{pos=.15}]
    &&
    X \times X
    \ar[dd, "\mathrm{pr}_1"{left, pos=.3}]
    \\
    X \times G^{\times^n}
    \ar[rr]
    \ar[dr, hook, "\mathrm{shear}_n"{sloped}]
    &&
    X
    \ar[dr, -, shift left=1pt]
    \ar[dr, -, shift right=1pt]
    \\
    &
    X^{\times^{n+1}}
    \ar[rr]
    \ar[from=uu, crossing over]
    &&
    X
\end{tikzcd}

Now the claim follows by [[induction]] from the fact that [[(-1)-truncated]] morphisms are the right class in an [[orthogonal factorization system in an (infinity,1)-category|orthogonal factorization system]] (namely the
[[(n-connected, n-truncated) factorization system]] for $n = -1$) and such classes of morphisms are closed under all [[(infinity,1)-limits|$\infty$-limits]], in particular under [[homotopy pullbacks]], in the [[arrow category]] (by [this Prop.](orthogonal+factorization+system+in+an+infinity1-category#RightClassStableUnderLimitsInArrowCategory)).
\end{proof}

## Examples

* Any [[group]] $G$ acts freely on itself by multiplication $\cdot \colon G \times G \to G$, which is called the (left) [[regular representation]] of $G$.

* An action of $\mathbb{Z}/2\mathbb{Z}$ on a set $X$ corresponds to an arbitrary [[involution]] $i \colon X \to X$, but the action is free just in case $i$ is a _[[fixed point]]-free_ involution.

* There is a rich structure in the classification of [[free group actions on n-spheres]], see there for more.

* For any set $X$ equipped with a [[transitive action]] $* : G \times X \to X$, the [[automorphism group]] $Aut_G(X)$ of $G$-equivariant [[automorphisms]] of $X$ (i.e., [[bijections]] $\phi : X \to X$ commuting with the action of $G$) acts freely on $X$.  In particular, suppose $\phi \in Aut_G(X)$ is such that $\phi(x) = x$ for some $x\in X$, and let $y\in X$ be arbitrary.  By the assumption that $G$ acts transitively, there is a $g \in G$ such that $y = g*x$. But then $G$-equivariance implies that $\phi(y) = \phi(g*x) = g*\phi(x) = g*x = y$. Since this holds for all $y\in Y$, $\phi$ must be equal to the identity $\phi = id_X$, and therefore $Aut_G(X)$ acts freely on $X$.

* A [[combinatorial species]] $F : \mathbb{P} \to Set$ is said to be _flat_ if all of the actions $S_n \times F(n) \to F(n)$ are free (see [[Combinatorial species and tree-like structures]]). For example, the species of [[linear orders]] is flat.

## Properties

### Homotopy colimits of free actions
 {#HomotopyColimitsOfFreeActions}

\begin{proposition}
\label{HomotopyQuotientsOfFreeActionsAreQuotientsOfZeroTruncatedAction}
**([[homotopy quotients]] of [[free actions|free]] [[infinity-action|$\infty$-action]] are plain [[quotients]] of [[0-truncated]] [[group actions]])**
\linebreak
Let

* $\mathbf{H}$ be an [[(infinity,1)-topos|$\infty$-topos]];

* $G \,\in\, Grp(\mathbf{H})$ a [[group object in an (infinity,1)-category|group object]] (a [[geometric homotopy theory|geometric]] [[infinity-group|$\infty$-group]]);

* $X \,\in\, \mathbf{H}$ an [[inhabited object]], $X \twoheadrightarrow \ast$,

* $G \curvearrowright X \,\in\, A Act(\mathbf{H})$ an [[infinity-action|$\infty$-action]] of $G$ on $X$.

If this is a free $\infty$-action (Def. \ref{FreeInfinityAction}) then the [[homotopy quotient]]

\[
  \label{HomotopyQuotientAsInfinityColimit}
  X
    \!\sslash\! G
   \,=\,
   \underset{\underset{[n] \in \Delta^{op}}{\longrightarrow}}{\lim}
  X \times G^{\times_n}
  \;\;\;
  \in
  \;
  \mathbf{H}
\]

is [[equivalence in an (infinity,1)-category|equivalent]] to the plain [[quotient]], namely to the [[coequalizer]]

\[
  \label{QuotientAsCoequalizer}
  \tau_0(X)
    /
  \tau_0(G)
   \,\coloneqq\,
  coeq
  \big(
    \tau_0(X) \times \tau_0(G)
    \overset{\phantom{---}}{\rightrightarrows}
    \tau_0(X)
  \big)
  \;\;\;
  \in
  \;
  \mathbf{H}_0
\]

of [[0-truncated]] objects

\[
  \label{ZeroTruncationReflectionInProofOfFreeHomotopyQuotients}
\]
\begin{tikzcd}
  \mathbf{H}_0
  \ar[r, hook, shift right=7pt, "i_0"{below}]
  &
  \mathbf{H}
  \,,
  \ar[
    l,
    shift right=7pt,
    "{\mathclap{\times}}"{description, pos=0},
    "{\tau_0}"{above}
  ]
  \ar[l, phantom, "{\scalebox{.7}{$\bot$}}"]
\end{tikzcd}

in that

\[
  \label{EquivalenceOfHomotopyQuotientOfFreeActionToPlainQuotient}
  X \!\sslash\! G
  \;\;
  \simeq
  \;\;
  i_0\big( \tau_0(X)/\tau_0(G) \big)
  \;\;\;
  \in
  \;
  \mathbf{H}_0
  \xhookrightarrow{i_0}
  \mathbf{H}
  \,.
\]

In particular, if both $X$ and $G$ are already [[0-truncated]], then the action of $G$ on $X$ is free iff it is free in the [[1-category]] theoretic sense (eq:FreeInternalAction), and then the homotopy quotient coincides with their ordinary quotient.
\end{proposition}

\begin{proof}
First we show that $X \!\sslash\! G$ is [[0-truncated]], in that for every $U \,\in\, \mathbf{H}$ and every [[infinity-group|
$\infty$-group]] $K$ in the [[inverse image]] $Grpd_\infty \xrightarrow{ LConst } \mathbf{H}$ of the [[terminal geometric morphism]],
every morphism into it out of $U \times \mathbf{B}K$  factors through the [[projection]] onto $U$:

\begin{tikzcd}
  U \times \mathbf{B}K
  \ar[rr, "\forall"]
  \ar[d, ->>, "\mathrm{pr}_1"{left}]
  &&
  X /\!\!/ G
  \ar[d]
  \\
  U
  \ar[urr, dashed, "\exists"]
  \ar[rr, -, shift left=1pt]
  \ar[rr, -, shift right=1pt]
  &&
  \ast
\end{tikzcd}

Here $\ast$ denotes the [[terminal object]], which we adjoin, without changing the situation, to bring out the form of the [[lifting problem]].

Incidentally, the left morphism above is an [[effective epimorphism in an (infinity,1)-category|effective epimorphism]] as shown, hence is [[(-1)-connected]]. Threfore, *if* the right vertical morphism $X \!\sslash\! G \xrightarrow{\;\;} \ast$ were [[(-1)-truncated]] (hence if $X \!\sslash\! G$ were [[subterminal object|subterminal]]), then the [[(n-connected, n-truncated) factorization system]] would imply the required [[lift]]. While this would-be argument fails, as $X \!\sslash\! G$ is in general far from being subterminal, the following argument observes that with a suitable choice of [[atlases]] for all four [[infinity-stack|$\infty$-stacks]], their [[groupoid object in an (infinity,1)-category|groupoid objects]] do form a [[lifting problem]] to which the [[(n-connected, n-truncated) factorization system]] does apply:

So consider extending the above square diagram to a square of [[augmented simplicial object|augmented]] [[simplicial objects]] by considering  [[atlas|atlases]] and their [[Cech nerves]], as shown by the following solid arrows:
\begin{tikzcd}[row sep=30pt]
    U \times K \times K
    \ar[dd, shift left=12pt, start anchor={[yshift=-4pt]}]
    \ar[from=dd, shift left=6pt]
    \ar[dd]
    \ar[from=dd, shift right=6pt]
    \ar[dd, shift right=12pt]
    \ar[rr]
    \ar[dr, ->>]
    &&
    X \times G \times G
    \ar[dd, shift left=12pt]
    \ar[from=dd, shift left=6pt]
    \ar[dd]
    \ar[from=dd, shift right=6pt]
    \ar[dd, shift right=12pt, start anchor={[yshift=-6pt]}]
    \ar[dr, hook]
    \\
    &
    U
    \ar[rr, crossing over]
    \ar[ur, dashed]
    &&
    X \times X \times X
    \ar[dd, shift left=12pt]
    \ar[from=dd, shift left=6pt]
    \ar[dd]
    \ar[from=dd, shift right=6pt]
    \ar[dd, shift right=12pt]
    \\
    U \times K
    \ar[rr]
    \ar[dr, ->>]
    \ar[dd, shift left=6pt]
    \ar[from=dd]
    \ar[dd, shift right=6pt]
    &&
    X \times G
    \ar[dr, hook]
    \ar[dd, shift left=6pt]
    \ar[from=dd]
    \ar[dd, shift right=6pt]
    \\
    &
    U
    \ar[rr, crossing over]
    \ar[from=uu, shift left=12pt, crossing over, end anchor={[yshift=7pt]}]
    \ar[uu, shift left=6pt, crossing over, start anchor={[yshift=3pt]}]
    \ar[from=uu, crossing over]
    \ar[uu, shift right=6pt, crossing over, start anchor={[yshift=3pt]}]
    \ar[from=uu, shift right=12pt, crossing over, end anchor={[yshift=7pt]}]
    \ar[ur, dashed]
    &&
    X \times X
    \ar[dd, shift left=6pt]
    \ar[from=dd]
    \ar[dd, shift right=6pt]
    \\
    U
    \ar[rr]
    \ar[dd, ->>]
    \ar[dr, -, shift left=1pt]
    \ar[dr, -, shift right=1pt]
    &&
    X
    \ar[dd, ->>]
    \ar[dr, -, shift left=1pt]
    \ar[dr, -, shift right=1pt]
    \\
    &
    U
    \ar[rr, crossing over]
    \ar[from=uu, shift left=6pt, crossing over]
    \ar[uu, crossing over]
    \ar[from=uu, shift right=6pt, crossing over]
    \ar[ur, dashed]
    &&
    X
    \ar[dd, ->>]
    \\
    U \times \mathbf{B}K
    \ar[rr]
    \ar[dr]
       &&
    X /\!\!/ G
    \ar[dr]
    \\
    &
    U
    \ar[rr]
    \ar[from=uu, ->>, crossing over]
    \ar[ur, dashed]
    &&
    \ast
\end{tikzcd}
{#ObservingTheConnectedTruncatedLiftingPropblems}
Observe that all the upper horizontal squares in this diagram have, as indicated:

1. a [[(-1)-connected]] morphisms $\twoheadrightarrow$ on the left, since the [[underlying]] [[infinity-groupoid|$\infty$-groupoid]] $K \in Grpd_\infty$ of every [[discrete infinity-groupoid|discrete $\infty$-group]] is [[inhabited object|inhabited]] and since $LConst$, being an [[inverse image]]-functor, is a [[lex functor|lex]] [[left adjoint]] and hence [[preserved limit|preserves]] [[Cech nerves]] and [[(infinity,1)-colimits|$\infty$-colimits]] and hence [[effective epimorphism in an (infinity,1)-category|effective epimorphisms]].

1. a [[(-1)-truncated]] morphism $\hookrightarrow$ on the right, by Lemma \ref{HigherShearMapsOfFreeInfinityActionsAreMinusOneTruncated}.

Since [[n-connected object in an (infinity,1)-topos|$n$-connected]]/[[n-truncated object in an (infinity,1)-category|$n$-truncated]] morphisms in  [[(infinity,1)-categories of (infinity,1)-presheaves|$\infty$-categories of $\infty$-presheaves]] (here: of [[simplicial objects]] in $\mathbf{H}$) are detected objectwise (since they are characterized by [[categorical homotopy groups in an (infinity,1)-topos|categorical homotopy groups]]),  this means that the entire square diagram of [[simplicial objects]] (i.e. disregarding the bottom square) has a [[(-1)-connected]] morphism on the left and a [[(-1)-truncated]] morphism in the right.  Therefore, the [[(n-connected, n-truncated) factorization system]] implies that there exist compatible dashed [[lifts]] filling all the upper squares, as shown.

But then taking the [[(infinity,1)-colimits|$\infty$-colimit]] over [[simplicial objects]] and using that [[groupoid objects in an (infinity,1)-topos are effective|groupoid objects in an $\infty$-topos are effective]], recovers the bottom square, but now also equipped with a dashed [[lift]]. This is the claimed factorization which shows that $X \!\sslash\! G$ is [[0-truncated]];

\[
  \label{ZeroTruncationOfHomotopyQuotientOfFreeAction}
  i_0
  \circ
  \tau_0
  \big(
    X \!\sslash\! G
  \big)
  \;\;
  \simeq
  \;\;
  X \!\sslash\! G
  \,.
\]

To conclude the proof of (eq:EquivalenceOfHomotopyQuotientOfFreeActionToPlainQuotient), use that $\tau_0$ is a [[left adjoint]] (eq:ZeroTruncationReflectionInProofOfFreeHomotopyQuotients), [[left adjoints preserve colimits|hence preserves $\infty$-colimits]], and also [[preserved limit|preserves]] [[products]], i.e. [[homotopy products]] (by [this Prop.](n-truncated+object+of+an+infinity1-category#nTruncationInToposPreservesFiniteProducts)). Here this implies that:

$$
  \begin{array}{lll}
    X \!\sslash\! G
    &
    \;\simeq\;
    i_0
    \circ
    \tau_0
    \big(
      X \!\sslash\! G
    \big)
    &
    \text{ (eq:ZeroTruncationOfHomotopyQuotientOfFreeAction) }
    \\
    &
    \;\simeq\;
    i_0
    \circ
    \tau_0
    \Big(
      \underset{\underset{ [n] \in \Delta^{op} }{\longrightarrow}}{\lim}
      \,
      X \times G^{\times_n}
    \Big)
    &
    \text{ (eq:HomotopyQuotientAsInfinityColimit) }
    \\
    & \;\simeq\;
    i_0
    \Big(
    \underset{\underset{ [n] \in \Delta^{op} }{\longrightarrow}}{\lim}
    \,
    \tau_0(X) \times \big(\tau_0(G)\big)^{\times_n}
    \Big)
    &
    \text{ (eq:ZeroTruncationReflectionInProofOfFreeHomotopyQuotients) }
    \\
    & \;\simeq\;
    i_0
    \Big(
      coeq
      \big(
        \tau_0(X) \times \tau_0(G)
        \overset{\phantom{---}}{\rightrightarrows}
        \tau_0(X)
      \big)
    \Big)
    &
    \text{ (eq:InclusionOfParallelPairIntoOppositeSimplexCategoryIsFinal) }
    \\
    & \;=\;
    i_0
    \big(
      \tau_0(X)
      /
      \tau_0(G)
    \big)
    &
    \text{ (eq:QuotientAsCoequalizer). }
  \end{array}
$$

In the second but last line we used (from [this Example](final+functor#FirstPairOfFaceMapsFinalInOppositeSimplexCategory)) that the inclusion of the [[diagram]] consisting of a [[pair]] of [[parallel morphisms]] into the [[opposite category|opposite]] of the [[simplex category]] is [[final functor]]
\[
  \label{InclusionOfParallelPairIntoOppositeSimplexCategoryIsFinal}
    \left(
      [1]
      \underoverset
        {d_0}
        {d_1}
        {\rightrightarrows}
      [0]
    \right)
    \;\xhookrightarrow{\phantom{---}}\;
    \Delta^{op}
    \;\;\;\;\;\;
    \text{is final}
    \,,
\]
meaning that the [[colimit]] over a [[simplicial object]] in a [[1-category]] is equivalently the [[coequalizer]] of the first two [[face maps]].
\end{proof}

## Related concepts

* [[fixed point]]

* [[free group actions on n-spheres]]

* [[homogeneous space]]

* [[regular action]], [[torsor]], [[principal bundle]]

* [[universal principal bundle]]

* [[transitive action]], [[effective action]], [[free action]], [[semi-free action]], [[regular action]],

* [[proper action]], [[properly discontinuous action]]

[[!redirects free actions]]


free bicompletion

+-- {: .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

Given a [[small category]] $S$, we may consider freely adjoining (small) [[colimits]] to $S$, giving the [[free cocompletion]] of $S$. Concretely, this is the [[category of presheaves]] on $S$. Dually, we may consider freely adjoining (small) [[limits]] to $S$, giving the [[free completion]] of $S$. Concretely, this is the [[category of copresheaves]] on $S$.

Alternatively, we may consider freely adjoining both (small) limits *and* colimits to $S$ simultaneously. (Note that this means there will be no [[distributivity of limits and colimits]].) This is known as the **free bicompletion** of $S$.

Unlike the [[free completion]] or [[free cocompletion]], it is not known whether a concrete description of the free bicompletion of a small category exists.

Abstractly, the free bicompletion construction is the [[coproduct]] of the two [[2-monads]] associated to the [[free completion]] and the [[free cocompletion]].

## Examples

* The [[Cauchy completion]] of a category is its free bicompletion under absolute limits and absolute colimits.

* The free bicompletion of a category $C$ under an [[initial object]] and [[terminal object]] is given by adjoining two objects $\bot$ and $\top$ and defining:
$$
  C(\bot, \bot) = 1 \qquad
  C(\bot, c) = 1 \qquad
  C(c, \top) = 1 \qquad
  C(\bot, \top) = 1 \qquad
  C(\top, \top) = 1 \qquad
$$
The composition laws are uniquely determined. (This is equivalently the result of completing $C$ under a terminal object, and then cocompleting under an initial object.)

## Related concepts

* [[free cocompletion]], [[free completion]]

* [[Cauchy completion]]

## References

* [[André Joyal]]. _Free bicomplete categories_. Mathematical Reports of the Academy of Sciences 17.5 (1995): 219-224. ([link](https://mathreports.ca/article/free-bicomplete-categories/))

* [[André Joyal]]. _Free bicompletion of enriched categories_. Mathematical Reports of the Academy of Sciences 17.5 (1995): 213-218. ([link](https://mathreports.ca/article/free-bicomplete-categories/))

The above references do not contain proofs. For a sketch of the existence of free bicompletions of categories, see [this MathOverflow question](https://math.stackexchange.com/questions/1686774/can-we-simultaneously-freely-adjoin-both-limits-and-colimits-to-a-category).

The free bicompletion of a category $C$ under nonempty [[products]] and nonempty [[coproducts]] is characterised as the category of nonempty $C$-valued contractible [[coherence spaces]] and $C$-maximal maps in:

* [[Hongde Hu]], _Contractible coherence spaces and maximal maps_, Electronic Notes in Theoretical Computer Science 20 (1999): 309-319. ([link](https://core.ac.uk/download/pdf/82113252.pdf))

A different approach for discrete categories is described in:

* [[Dominic J. D. Hughes]], _A canonical graphical syntax for non-empty finite products and sums_, Technical report (2002).

The free completion of a category $C$ under [[products]], [[coproducts]], and a [[zero object]] is characterised as the category of **contractible $C$-coherence spaces** in:

* [[Hongde Hu]] and [[André Joyal]], _Coherence completions of categories and their enriched softness_, Electronic Notes in Theoretical Computer Science 6 (1997): 174-190.

* [[Hongde Hu]] and [[André Joyal]], _Coherence completions of categories_, Theoretical Computer Science 227.1-2 (1999): 153-184.

[[!redirects free bicompletion]]


free cartesian category

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

# Contents
* table of contents
{: toc}

In this article, we explore [[cartesian monoidal category|cartesian monoidal categories]] as a [[doctrine]] on [[Cat]], by analyzing the structure of [[free construction|free]] cartesian categories.

It is convenient to discuss the case of categories with arbitrary small [[products]] (or arbitrary small [[coproducts]]) first, because there we get some very pleasant [[adjoint functors|adjunctions]] to work with. Then, we may cut down to finite products and derive some possibly unexpected bonuses involving [[finite limits]].

## Free product-completions of sets

We start by constructing the [[free construction|free]] category with small [[products]] [[generators and relations|generated]] by a [[set]] $S$. This is,  by definition, a category with small products, $Prod(S)$, equipped with a [[function]] $i \,\colon\, S \to Ob(Prod(S))$ with the following [[universal property]]: given a category $C$ with small products and a function $f \,\colon\, S \to Ob(C)$, there exists up to (unique) [[isomorphism]] a unique product-preserving functor $\Pi \,\colon\, Prod(S) \to C$ such that $f = Ob(\Pi) \circ i$.

The [[slice category]] $Set/S$ is [[small-complete category|small-cocomplete]]; therefore $(Set/S)^{op}$ has small products. There is an evident function $i: S \to Ob((Set/S)^{op})$ which takes $s \in S$ to $s: 1 \to S$. This can also be regarded as a [[Yoneda embedding]] under the equivalence $Set/S \simeq Set^S$.

Each object $n: N \to S$ of $(Set/S)^{op}$ can be regarded as a formal product $\prod_{s \in S} x_{s}^{n_s}$ where $S$ indexes a set of variables $x_s$ and $n_s = n^{-1}(s)$ is the (cardinality of the) fiber over $S$. Therefore the following theorem comes as no surprise.

+-- {: .num_theorem}
###### Theorem
$(Set/S)^{op}$ together with $i: S \to (Set/S)^{op}$ is the free category with small products generated by $S$.
=--

+-- {: .proof}
###### Proof
Let $C$ be a category with small products. Given $f: S \to Ob(C)$, the functor $\Pi: (Set/S)^{op} \to C$ is defined on the object-level: $n: N \to S$ is sent to $\prod_{s \in S} f(s)^{n_s}$ in $C$. In addition, a morphism from $m: M \to S$ to $n: N \to S$ in $(Set/S)^{op}$ is a function $\phi: N \to M$ that satisfies $m \circ \phi = n$; this in turn consists of a collection of functions $\phi_s: n_s \to m_s$ indexed by elements $s \in S$, and each such function induces a morphism

$$f(s)^{\phi_s}: f(s)^{m_s} \to f(s)^{n_s}$$

in $C$, uniquely specified by declaring the $i^{th}$ component of this morphism for $i \in n_s$ to be given by

$$f(s)^{m_s} \stackrel{proj_{\phi_s(i)}}{\to} f(s)$$

Given a morphism $\phi$ of $(Set/S)^{op}$, define $\Pi(\phi)$ by

$$\Pi(\phi) = \langle f(s)^{\phi_s} \rangle_{s \in S}: \prod_{s \in S}  f(s)^{m_s} \to \prod_{s \in S} f(s)^{n_s}.$$

The functor $\Pi$ is the unique product-preserving functor $(Set/S)^{op} \to C$ (up to isomorphism) that extends $f: S \to Ob(C)$.
=--

Given categories with products $C$ and $D$, let $\mathbf{Prod}(C, D)$ denote the category of product-preserving functors $F: C \to D$ and natural transformations between them. By the preceding theorem, there is an equivalence

$$\mathbf{Prod}((Set/S)^{op}, C) \simeq C^S$$

where the right side consists of functors $S \to C$ (where $S$ is regarded as a discrete category) and natural transformations between them.

\begin{corollary}
The category $\mathbf{Prod}((Set/S)^{op}, Set)$ is isomorphic to $Set/S$.
\end{corollary}

\begin{proof}
Because $\mathbf{Prod}((Set/S)^{op}, Set) \simeq Set^S \simeq Set/S$.
\end{proof}

Let $C$ be a [[locally small category]] with [[products]]. There is a [[Yoneda embedding]]

$$
  y \;\colon\; C^{op} \to \mathbf{Prod}(C, Set)
$$

As we just saw, the Yoneda embedding $y: Set/S \to \mathbf{Prod}((Set/S)^{op}, Set)$ is an equivalence. We may arrange that the reverse equivalence $\mathbf{Prod}((Set/S)^{op}, Set) \to Set/S$ is right adjoint to $y$, and so we denote it as $y^*$.

+-- {: .num_theorem}
###### Theorem
If $C$ is a locally small category with small products and $f: S \to C$ is a functor, the coproduct-preserving functor $\Pi^{op}: Set/S \to C^{op}$ has a right adjoint given by the composite

$$C^{op} \stackrel{y}\to \mathbf{Prod}(C, Set) \stackrel{\mathbf{Prod}(\Pi, Set)}{\to} \mathbf{Prod}((Set/S)^{op}, Set) \stackrel{y^*}{\to} Set/S$$
=--

+-- {: .proof}
###### Proof
Let $g: X \to S$ be an object of $Set/S$ and $c$ an object of $C$. We have the following natural bijections between sets of morphisms:

$$\array{
\Pi^{op}(g: X \to S) \to c & & in C^{op}\\
c \to \Pi(g: X \to S) & & in C\\
y(g: X \to S) \to \hom_C(c, \Pi-) & & in \mathbf{Prod}((Set/S)^{op}, Set)\\
(g: X \to S) \to y^* \hom_C(c, \Pi-) & & in Set/S
}$$
where we get to the third line by the Yoneda lemma. This proves the theorem.
=--

+-- {: .num_corollary}
###### Corollary
Under the hypotheses of the previous theorem, the product-preserving functor $\Pi: (Set/S)^{op} \to C$ is a right adjoint. In particular, it
preserves _all_ limits, not just products.
=--

## Free product-completions of categories

It will be convenient to consider first the coproduct-cocompletion; then the product-completion is obtained by dualizing.

### Explicit construction

Let $C$ be a small category. We may construct the free category with _co_products generated by $C$ directly as follows:

* The objects of $Coprod(C)$ are those of $Set/C_0$, where $C_0$ is the set of objects.

* A morphism (say from $f: X \to C_0$ to $g: Y \to C_0$) is given by a function $\phi: X \to Y$ together with an $X$-indexed collection of morphisms of $C$ of the form $\Phi_x: f(x) \to g(\phi(x))$.

The data of a morphism may be exhibited as a 2-cell of the form

$$\array{
X & \stackrel{\phi}{\to} & Y \\
\mathllap{1_X} \downarrow & \stackrel{\Phi}{\Rightarrow} & \downarrow \mathrlap{g} \\
X & \underset{f}{\to} & C
}$$

Coproducts of objects in $Coprod(C)$ are computed just as they are in $Set/C_0$. Each $f: X \to C_0$ is then a formal coproduct $\sum_x f(x)$, and a morphism $(\phi, \Phi): \sum_{x \in X} f(x) \to \sum_{y \in Y} g(y)$ is uniquely determined by its restrictions along the coprojections $i_x: f(x) \to \sum_x f(x)$, which in turn are given by composites

$$f(x) \stackrel{\Phi_x}{\to} g(y) \stackrel{i_y)}{\to} \sum_{y \in Y} g(y)$$

where $y = \phi(x)$. The proof of the universal property of $Coprod(C)$ as coproduct cocompletion is then obvious.

### As Kleisli construction

Another light on this construction comes from seeing the [[free coproduct-cocompletion]] $Coprod(C)$ as sitting inside the [[free cocompletion]]  $Set^{C^{op}}$ (the [[category of presheaves]]). First, we have a [[Yoneda embedding|Yoneda mapping]]

$$C_0 \to Set^{C^{op}} \;\colon\; c \mapsto \hom(-, c)$$

This induces a coproduct-[[preserved colimit|preserving]] [[functor]] $Set/C_0 \to Set^{C^{op}}$, which takes an object $g: Y \to C_0$ to the presheaf

$$\sum_{y \in Y} \hom_C(-, g(y))$$

By the preceding theorem, this functor has a right adjoint. The right adjoint is simply described: it is the evident underlying functor

$$Set^{C^{op}} \to Set/C_0$$

taking a presheaf $F: C^{op} \to Set$ to the $C_0$-indexed set where the fiber over $c \in C_0$ is $F(c)$.

+-- {: .proof}
######Proof
As is well-known, this underlying functor $Set^{C^{op}} \to Set/C_0$ is monadic; the monad $M_C$ takes an object $g: Y \to C_0$ to the object whose fiber over $c$ is the set

$$\sum_{y \in Y} \hom_C(c, g(y))$$

In other words, the left adjoint to the underlying functor is the coproduct-preserving functor that takes $g: Y \to C_0$ to

$$\sum_{y \in Y} \hom(-, g(y))$$

as described above.
=--

Notice that $M_C(g)$ is given by the top horizontal composite

$$\array{
P & \to & C_1 & \stackrel{\dom}{\to} & C_0 \\
\downarrow & & \downarrow \mathrlap{\cod} & & \\
Y & \underset{g}{\to} & C_0 & &
}$$

where the square is a pullback.

+-- {: .num_theorem #Kleisli}
###### Theorem
The free coproduct-cocompletion of $C$ is the Kleisli category of the monad $M_C: Set/C_0 \to Set/C_0$.
=--

+-- {: .proof}
###### Proof
Let $f: X \to C_0$ and $g: Y \to C_0$ be two objects of $Kl(M_C)$. A morphism from $f$ to $g$ in $Kl(M_C)$ is a morphism from $f$ to $M_C(g)$ in $Set/C_0$, in other words a map $h: X \to P$ making the following diagram commute:

$$\array{
X & \stackrel{1_X}{\to} & X & & \\
h \downarrow & & & \searrow \mathrlap{f} \\
P & \to & C_1 & \stackrel{\dom}{\to} & C_0 \\
\downarrow & & \downarrow \mathrlap{\cod} & & \\
Y & \underset{g}{\to} & C_0 & &
}$$

Since $P$ is the pullback, the map $h$ is given by maps $\phi: X \to Y$, $\Phi: X \to C_1$ with $\Phi(x): f(x) \to g(\phi(x))$ for each $x \in X$. But this is how a morphism from $f$ to $g$ was described in the explicit construction of $Coprod(C)$.
=--

#### Example: $Subset$ {#Subset}

We calculate the free coproduct completion of the [[interval category]] $\mathbf{2} = (0 \to 1)$, and record some properties of this category.

According to theorem \ref{Kleisli}, this is the Kleisli category of the monad on $Set/\{0, 1\}$ corresponding to the monadic functor $Set^{\mathbf{2}^{op}} \to Set/\{0, 1\}$. This monad takes a pair of sets $(X_1, X_0)$ to $(X_1, X_0 + X_1)$; the algebra structure of the free algebra $(X_1, X_0 + X_1)$ is the structure of presheaf over $\mathbf{2}$ given by the inclusion $X_1 \hookrightarrow X_0 + X_1$.

Thus the [[free coproduct completion]] of $\mathbf{2}$ is equivalent to the category $Subset$ who objects are pairs of sets $(A, X)$ where $A$ is a subset of $X$; morphisms $(A, X) \to (B, Y)$ are functions $f: X \to Y$ which carry $A$ into $B$.

This category has many pleasant properties, largely summarized by the fact that it is a Grothendieck [[quasitopos]]. Indeed, it can be described as the category of $\neg \neg$-separated presheaves on $\mathbf{2}$ (so that it is a reflective subcategory of $Set^{\mathbf{2}^{op}}$). The reflection takes an object $f: X \to Y$ to the pair $(Im(f), Y)$.

In particular, $Subset$ is [[complete category|complete]] and [[cocomplete category|cocomplete]] and [[cartesian closed category|cartesian closed]]. Limits are formed as they are in $Set^{\mathbf{2}^{op}}$ (that is, objectwise). As are exponentials, since $Subset$ is actually an [[exponential ideal]] in $Set^{\mathbf{2}^{op}}$. Naturally, [[coproducts]] are formed as they are in the [[presheaf category]] since $Subset$ is the coproduct completion; coequalizers are constructed by forming them in the presheaf category and then reflecting back into $Subset$.

Categories [[enriched category|enriched]] in $Subset$ are also called [[M-categories]].

### Algebraic theory of a category

The free product-completion of $C$ is obtained by dualization:

$$Prod(C) = Coprod(C^{op})^{op} = Kl(M_{C^{op}})^{op}$$

Here $M = M_{C^{op}}$ is the monad on $Set/C_0$ whose algebras are functors $C \to Set$. This functor category is the category of models of a simple multi-sorted [[algebraic theory]], where the set of sorts is $C_0$ and each morphism $f: c \to d$ is regarded as an unary operation. According to the entry [[algebraic theory]], the corresponding Lawvere theory is just $Kl(M)^{op}$, in other words the product-completion of $C$. Indeed, the category of models of $Kl(M)^{op}$ is

$$\mathbf{Prod}(Kl(M)^{op}, Set) \simeq \mathbf{Prod}(Prod(C), Set) \simeq Set^C$$

by the universal property of $Prod(C)$.

### KZ property

There is no foundational difficulty in forming the free small coproduct-cocompletion of any locally small category $C$ (any more than forming the free cocompletion with respect to small limits). An object of $Coprod(C)$ is a set $X$ together with a function $f: X \to C_0$. A morphism from $f: X \to C_0$ to $g: Y \to C_0$ consists of a function $\phi: X \to Y$ together with an $X$-indexed collection of morphisms $\Phi_x: f(x) \to g(\phi(x))$. The hom-sets are clearly small.

To be continued...

## Free cartesian category on a signature

There are a number of ways of explaining the same material from a more categorical point of view. Whatever the explanation is, the point is to describe the free category with finite products generated from a multigraph $F$ over $S$. We denote this free category with products as $Term(S, F)$.

As before, the objects of $Term(S, F)$ are product types: elements of $S^\ast$. Morphisms from a type $T: [m] \to S$ to a type $T': [n] \to S$ are, in the language of the preceding section, $n$-tuples of normal terms $(t_1, \ldots, t_n)$ where each $t_i$ has arity $T \to T'(i)$. Composition is effected by term substitution. The cartesian product structure is essentially given by concatenating (juxtaposing) lists of sorts and terms.

It is hopefully more or less clear how all of this works in the term syntax approach. Despite this, we believe that the bureaucracy of handling variables in the term syntax is something of a hack; from one point of view (closely related to string diagrams), some of it is actually unnecessary.

For example, the input strings play the role of variables declared in the context, but the difference is that they do not need "variable names" -- they only need to be labeled by an appropriate sort, for type-checking purposes. This trick effectively eliminates the need for rules of $\alpha$-conversion. As we will see, one can also effect a neat division of labor between the business of variable declarations and the business of "pure substitutions"; moreover, this division clarifies the precise entry point of the particular doctrine we are working in (the doctrine of finite product categories).

To begin, recall the following abstract definition:

+-- {: .num_defn}
###### Definition
Let $M$ be a cartesian monad acting on a finitely complete category $C$. An **$M$-span** in $C$ from $c$ to $d$ is a span of the form

$$M c \stackrel{f}{\leftarrow} x \stackrel{g}{\to} d$$

$M$-spans are composed by consideration of a pullback

$$\array{
& & & & & & x \circ y & & & & \\
& & & & & \swarrow & & \searrow & & & \\
& & & & M y & & & & x & & \\
& & & ^\mathllap{M h} \swarrow & & \searrow ^\mathrlap{M k} & & ^\mathllap{f} \swarrow & & \searrow ^\mathrlap{g} & \\
M b & \stackrel{m_b}{\leftarrow} & M M b & & & & M c & & & & d
}$$

where $m: M M \to M$ is the multiplication on the monad $M$. Under this composition, the $M$-spans are 1-cells of a bicategory $M$-$Span$.
=--

In the case where $M$ is the free monoid monad acting on $Set$, an $M$-span from a set $S$ to $S$ is the same as a multigraph over $S$. A monad on $S$ in the bicategory $M$-$Span$ is a **multicategory** over $S$. We are especially interested in the free multicategory generated from a multigraph over $S$.

The free multicategory construction has other names and descriptions. We could also call it the free nonpermutative $S$-sorted operad generated by a set of $S$-typed function symbols. The apex of its underlying span, together with its map to $S$, can also be referred to as the initial algebra for the polynomial endofunctor $P$ on $Set/S$ which takes an $S$-indexed set $X_s$ to

$$P(X)_s = \sum_{T \in S^\ast} \sum_{f: T \to s} \prod_{s' \in S} X_{s'}^{T_{s'}}$$

($f \in F$). However it is named, the underlying multigraph of the free multicategory generated by a multigraph $F$ can be described as

$$\array{
 & & Tree(F) & & \\
 & ^\mathllap{in} \swarrow & & \searrow ^\mathrlap{out} & \\
S^\ast & & & & S
}$$

where $Tree(F)$ is the set of $F$-labeled planar trees. This means that

* Nodes of a planar tree are labeled by elements $f: T \to s$ of $F$;

* Edges of that planar tree are labeled by sorts $s$, such that the list of labels of incoming edges at a node $f: T \to s$ is the word $T$, and the outgoing edge is labeled $s$.

The list of labels of "leaves" of an $F$-labeled tree $t$ (edges that are not outgoing edges of any node) gives an element $in(t) \in S^\ast$, and the label of the "root" edge gives an element $out(t) \in S$. Notice that $F$-labeled trees have obvious string diagram representations.

Next, any multicategory generates a (strict monoidal) category. In the present instance, we denote this category as $Pro(S, F)$ (we write "Pro" by analogy with "prop" -- whereas props are used for the doctrine of symmetric monoidal categories, pros are used for monoidal categories). The objects of $Pro(S, F)$ are elements of $S^\ast$. The morphisms of $Pro(S, F)$ could be described as "$F$-labeled forests". Formally, the underlying span of $Pro(S, F)$ is

$$S^\ast \stackrel{m_S}{\leftarrow} S^{\ast\ast} \stackrel{in^\ast}{\leftarrow} Tree(F)^\ast \stackrel{out^\ast}{\to} S^\ast$$

The monoidal product on forests is simply juxtaposition. The composition of forests is the obvious one, plugging in roots of one forest for leaves of another.

The construction $Pro(S, F)$ formalizes what we meant earlier by "pure substitutions of terms", and lives in the doctrine of (strict) monoidal categories. To switch over to the doctrine of categories with finite cartesian products, we apply a simple trick. Let $Pro(S, F) \circ Prod(S)$ be the composite of the two spans

$$\array{
 & & Prod(S) & & & & Pro(S, F) & & \\
 & ^\mathllap{dom} \swarrow & & \searrow ^\mathrlap{cod} & & ^\mathllap{in} \swarrow & & \searrow ^\mathrlap{out} & \\
S^\ast & & & & S^\ast & & & & S^\ast
}$$

Then, in the first place, $Pro(S, F) \circ Prod(S)$ carries a canonical category structure. The reason is that, viewing $Prod(S)$ and $Pro(S, F)$ as monads in the bicategory of spans, there is a canonical distributive law

$$\theta: Prod(S) \circ Pro(S, F) \to Pro(S, F) \circ Prod(S)$$

(here $Pro(S, F)$ can be replaced by any pro over $S$). The idea is that the distributive law $\theta$ maps an element of $Prod(S) \circ Pro(S, F)$, which is a pair of arrows

$$T \stackrel{f}{\to} T' \stackrel{d}{\to} T''$$

with $f$ an arrow of $Pro(S, F)$ and $d$ an arrow of $Prod(S)$, to a pair of arrows in $Pro(S, F) \circ Prod(S)$ whose precise form is dictated by a naturality requirement in $d$. For example, if $d = \delta_T': T' \to T' \otimes T'$, then

$$\theta(f, \delta_T') = (T \stackrel{\delta_T}{\to} T \otimes T \stackrel{f \otimes f}{\to} T' \otimes T')$$

Using this distributive law, the composite of the monads $Prod(S)$ and $Pro(S, F)$ is another monad in the bicategory of spans, and therefore a category with set of objects $S^\ast$.

The same trick works for other doctrines over the doctrine of monoidal categories. For example, if $Perm(S)$ is the free symmetric (strict) monoidal category generated by $S$, regarded as a span from $S^\ast$ to $S^\ast$, then there is a distributive law $Perm(S) \circ Pro(S, F) \to Pro(S, F) \circ Perm(S)$. Similarly with "symmetric monoidal" replaced with "braided monoidal".

+-- {: .num_defn}
###### Definition
$Term(S, F) \coloneqq Pro(S, F) \circ Prod(S)$, regarded as a cartesian category with product structure inherited from $Prod(S)$, meaning that
$$Prod(S) \cong id_{S^\ast} \circ Prod(S) \stackrel{u \circ 1}{\to} Pro(S, F) \circ Prod(S)$$
is a product-preserving functor, where $u$ is the unit of the monad $Pro(S, F)$.
=--

More explicitly, the tensor product $\bigotimes$ on $Term(S, F)$ is a cartesian product provided that there are natural transformations $\delta: id \to \otimes \Delta$, $\varepsilon: id \to e !$ which endow each object with a cocommutative comonoid structure. But the naturality follows from the definition of the distributive law, and the cocommutative comonoid axioms already hold in $Prod(S)$.

Note that this abstract description of $Term(S, F)$ is _identical_ to that given by the syntax of normal terms. The "bureaucracy of variables" is here organized into two departments, $Prod(S)$ and $Pro(S, F)$, each having its own individual categorical structure, which interact via the distributive law. (This is the "division of labor" we were talking about, where each arrow of $Term(S, F)$ is factorized into a generalized diagonal map followed by an $F$-labeled forest.)

Another name for $Term(S, F)$ is "term algebra", and yet another name for it is "the free $S$-sorted Lawvere theory generated by a set of $S$-sorted operation symbols $F$".

+-- {: .num_theorem}
###### Theorem
$Term(S, F)$ is the free category with products generated by the multigraph $F$ over $S$.
=--

+-- {: .proof}
###### Proof
$Tree(S, F)$ is the free multicategory generated by the multigraph $F$ over $S$, and $Pro(S, F)$ is the free monoidal category generated by the multicategory $Tree(S, F)$. Therefore $Pro(S, F)$ is the free monoidal category generated by the multigraph $F$. It remains to show that the monoidal inclusion
$$Pro(S, F) \cong Pro(S, F) \circ id_{S^\ast} \stackrel{1 \circ u}{\to} Pro(S, F) \circ Prod(S)$$
is universal among monoidal functors $X: Pro(S, F) \to C$ to cartesian monoidal categories $C$.

There is of course a product-preserving functor $Prod(S) \to C$ compatible with the restriction $S^\ast \hookrightarrow Pro(S, F) \stackrel{X}{\to} C$. At the level of spans, this gives a composable pair of span morphisms

$$\array{
S^\ast & \stackrel{\dom}{\leftarrow} & Prod(S) & \stackrel{\cod}{\to} & S^\ast & \stackrel{in}{\leftarrow} & Pro(S, F) & \stackrel{out}{\to} & S^\ast \\
\downarrow & & \downarrow & & \downarrow & & \downarrow & & \downarrow \\
C_0 & \underset{d_0}{\leftarrow} & C_1 & \underset{d_1}{\to} & C_0 & \underset{d_0}{\leftarrow} & C_1 & \underset{d_1}{\to} & C_0
}$$
which we then compose with the span morphism $m: C_1 \circ C_1 \to C_1$ given by composition in $C$:

$$Pro(S, F) \circ Prod(S) \to C_1 \circ C_1 \stackrel{m}{\to} C_1$$

This gives a morphism between underlying spans, $Term(S, F) \to C$. This is functorial (i.e., is a morphism of monads in $Span$) because both $Prod(S) \to C$ and $Pro(S, F) \to C$ are functorial, and also because the compositional equalities enforced by the distributive law $\theta$ are preserved: they are taken to equalities expressed by naturality of diagonal maps and projection maps in $C$. The functor $Term(S, F) \to C$ is product-preserving, because $Prod(S) \to C$ is product-preserving. The uniqueness of the product-preserving extension $Term(S, F) \to C$ is clear since the subcategories $Prod(S)$ and $Pro(S, F)$ together generate $Term(S, F)$.
=--

## Related concepts

* [[free coproduct completion]]

[[!redirects free cartesian category]]
[[!redirects free cartesian categories]]

[[!redirects free product completion]]
[[!redirects free product completions]]


free category

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

#Contents#
* table of contents
{:toc}

## Idea

The _free category_ on a "set of arrows", hence on a [[directed graph]], is the ([[strict category|strict]]) [[category]] whose:

* [[objects]] are the [[vertices]] of the graph,

* [[morphisms]] are the [[tuples]] of [[composition|composable]] [[edges]], hence the morphisms [[free construction|freely generated]] from these (directed) edges.

  These morphisms may also be thought of as the "paths" traced out in the directed graph by traversing stepwise along its  [[edges]], whence free categories are also called *[[path categories]]*.

{#MorePrecisely} More formally, there is a [[forgetful functor]] $U_{Cat}$ from the [[1-category]] [[Cat|$Cat^{strct}_{smll}$]] of [[small categories|small]] [[strict categories]] (and [[functors]] between them) to that of [[directed graphs]] (assigning the [[underlying]] graph of morphisms, i.e. forgetting the [[composition]]-operation); and the free category construction is its [[left adjoint]] [[free construction]] $F_{Cat}$:

$$
  Cat^{strct}_{smll}
  \underoverset
    {\underset{U_{Cat}}{\longrightarrow}}
    {\overset{F_{Cat}}{\longleftarrow}}
    {\;\;\bot\;\;\;}
  DiGrph
$$

{#FreeGroupoid} Often this is considered in [[composition]] with the "localization" functor that constructs the [[free groupoid]] on a given category:

$$
  Grpd^{strct}_{smll}
  \underoverset
    {\underset{U_{Grpd}}{\longrightarrow}}
    {\overset{F_{Grpd}}{\longleftarrow}}
    {\;\;\bot\;\;\;}
  Cat^{strct}_{smll}
  \underoverset
    {\underset{U_{Cat}}{\longrightarrow}}
    {\overset{F_{Cat}}{\longleftarrow}}
    {\;\;\bot\;\;\;}
  DiGrph
$$

(which relates to the [[Milnor construction]], see [Goerss & Jardine (2009), V.7](Dwyer-Kan+loop+groupoid#GoerssJardine09))

## Related concepts

* [[free groupoid]], [[core groupoid]]

* [[free diagram]]

* [[free monoid]], [[free operad]]

* [[free topos]]

* a [[quiver representation]] is equivalently a [[functor]] on the free category of the given [[quiver]]

## References
 {#References}

Some introductions to [[category theory]] have
a discussion of free categories, for instance:

* {#BarrWells} [[Michael Barr]], [[Charles Wells]], §2.6.16 in: *Category theory for computing science*, Prentice-Hall International Series in Computer Science (1995); reprinted in: Reprints in Theory and Applications of Categories **22** (2012) 1-538 [[pdf](http://www.math.mcgill.ca/barr/papers/ctcs.pdf), [tac:tr22](http://www.tac.mta.ca/tac/reprints/articles/22/tr22abs.html)]

* {#MacLane} [[Saunders MacLane]], Section II.7 (pp. 48) in: *[[Categories for the Working Mathematician]]*, Springer (1998) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

On path categories of [[context-free grammars]]:

* {#Latch} Dana Latch, _The connection between the fundamental groupoid and a unification algorithm for syntactil algebras (extended abstract)_ , Cah. Top. Géom. Diff. Cat. **XXXII** (1991).  ([link](http://www.numdam.org/item?id=CTGDC_1991__32_3_203_0))

* {#Walters1} [[Bob Walters]], _The free category with products on a multigraph_ , JPAA **62** (1989). ([link](http://www.sciencedirect.com/science/article/pii/0022404989901527))

* {#Walters2} [[Bob Walters]], _A note on context-free languages_ , JPAA **62** (1989). ([link](http://www.sciencedirect.com/science/article/pii/0022404989901515))

Since the perspective on categories as 'graphs with operations' is closely related to their 'logico-deductive' side, it figures prominently in the work of [[Jim Lambek]] and [[Phil Scott]] (cf. the references at [[free topos]]).

[[!redirects free categories]]

[[!redirects free category on a quiver]]
[[!redirects free categories on quivers]]