Jónsson-Tarski algebra

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[[!redirects Jónsson-Tarski object]]
[[!redirects Jonsson-Tarski object]]
[[!redirects idempotent object]]
[[!redirects Cantor algebra]]

##Idea

A _Jónsson-Tarski algebra_ is a set that looks like two copies of itself. Since historically the classical examples of these occured in the cardinal arithmetics of [[Georg Cantor]], they are also known as _Cantor algebras_.

## Definition

A **Jónsson-Tarski algebra**, also called a [[Georg Cantor|Cantor]] algebra, is a [[set]] $A$ together with an [[isomorphism]] $A\cong A\times A$.

More generally, an object $A$ in a [[symmetric monoidal category]] $\mathcal{M}$ together with an isomorphism $\alpha:A\otimes A\rightarrow A$ is called a **Jónsson-Tarski object**, or an _idempotent_ object (Fiore&Leinster 2010).

In another possible direction for generalization, one defines a **Jónsson-Tarski n-algebra** as a set $X$ together with an [[isomorphism]] $X\overset{\simeq}{\to}X^n$ (cf. Smirnov 1971, Higman 1974).[^fine]

[^fine]: A profunctorial variation on this theme has been proposed by Leinster (2007). See at [[Jónsson-Tarski topos]] for some details.

## Properties

* Clearly (at least in [[classical mathematics]]), any Jónsson-Tarski algebra is either [[empty set|empty]], a [[singleton]], or [[infinite set|infinite]].

* The structure of a Jónsson-Tarski algebra can be described by an [[algebraic theory]], with one binary operation $\mu$ and two unary operations $\lambda$ and $\rho$ such that $\mu(\lambda(x),\rho(x)) = x$, $\lambda(\mu(x,y))=x$, and $\rho(\mu(x,y))=y$.

* Any two Jónsson-Tarski algebras freely generated from finite non empty sets are isomorphic. It was this property they owe their introduction to (Jónsson&Tarski 1956,1961).

* Just like in the category $Grp$ of [[group|groups]], subalgebras of free algebras are free themselves (cf. this [Stackexchange question](#Stack)).

* The category of Jónsson-Tarski algebras is a [[topos]], the so called [[Jónsson-Tarski topos]] $\mathcal{J}_2$, and hence is an example for a [[variety]] that is also a topos (cf. Johnstone 1985).

* The [[Thompson Group]] F is the group of order-preserving automorphisms of the free Jónsson-Tarski algebra on one generator (cf. Fiore-Leinster 2010).

##Related concepts

* [[Jónsson-Tarski topos]]

* [[Thompson group]]

* [[idempotent]]

* [[self-similarity]]

##Links

* Wikipedia, [Jónsson-Tarski algebra](https://en.wikipedia.org/wiki/J%C3%B3nsson%E2%80%93Tarski_algebra)

* {#Stack}Stackexchange: [Jónsson-Tarski algebra form a Schreier variety](https://math.stackexchange.com/questions/3544667/do-j%C3%B3nsson-tarski-algebras-form-a-schreier-variety)

##References

* K. S. Brown, _Finiteness Properties of Groups_ , JPAA **44** (1987) pp.45-75.

* J. Dubeau, _Jónsson Jónsson-Tarski algebras_ , arXiv:2202.02460 (2022).  ([abstract](https://arxiv.org/abs/2202.02460v1))

* J. Dudek, A. W. Marczak, _On Cantor Identities_ , Algebra Universalis
**68** (2012) pp.237–247.

* [[Marcelo Fiore]], [[Tom Leinster]], _An abstract characterization of Thompson's group F_ , arXiv.math/0508617 (2010). ([pdf](http://arxiv.org/pdf/math/0508617v2.pdf))

* R. Freese, J. B. Nation, _Free Jónsson-Tarski algebras_ , ms. 2020. ([pdf](http://www.math.hawaii.edu/~jb/Jonsson-Tarski_algebras.pdf))

* G. Higman, _Finitely presented infinite simple groups_ , Notes on Pure Mathematics **8** (1974) Australian National University Canberra.

* P. Hines, _The Categorical Theory of Self-Similarity_ , TAC **6** no.3 (1999). ([abstract](http://www.tac.mta.ca/tac/volumes/6/n3/6-03abs.html))

* [[Peter Johnstone]], _When is a Variety a Topos?_ , Algebra Universalis **21** (1985) pp.198-212.

* [[Peter Johnstone]], _Collapsed Toposes and Cartesian Closed Varieties_ , JA **129** (1990) pp.446-480.

* B. Jónsson, [[A. Tarski]] , _Two General Theorems Concerning Free Algebras_ , Bull. Amer. Math. Soc. **62** p.554. ([pdf](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-62/issue-6/The-summer-meeting-in-Seattle/bams/1183521094.full))

* B. Jónsson, [[A. Tarski]] , _On Two Properties of Free Algebras_ , Math. Scand. **9** (1961) pp.95-101. ([pdf](http://ojs.statsbiblioteket.dk/index.php/math/article/viewFile/10627/8648))

* [[Tom Leinster]], _Jónsson-Tarski toposes_, Talk Nice 2007.  ([slides](http://www.maths.ed.ac.uk/~tl/nice/jt.pdf))

* A. K. Rumjancev, _An independent basis for the quasi-identities of a free Cantor algebra_ , Algebra and Logic **16** (1977) pp.119-129.

* D. M. Smirnov, _Cantor algebras with a single generator I_ , Algebra and
Logic **10** (1971) pp.40-49.

* D. M. Smirnov, _Cantor algebras with a single generator II_ , Algebra and
Logic **12** (1973) pp.399-404.

* D. M. Smirnov, _Bases and automorphisms of free Cantor algebras of finite rank_ , Algebra and Logic **13** (1974) pp.17-33.

* S. Swierczkowski, _On isomorphic free algebras_ , Fund. Math. **50**  (1961) pp.35–44.

[[!redirects Jonsson-Tarski algebras]]
[[!redirects Jonsson-Tarski algebra]]
[[!redirects Jónsson-Tarski algebras]]
[[!redirects Cantor algebra]]
[[!redirects Cantor algebras]]


Jónsson-Tarski topos

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# Contents
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##Idea
The _Jónsson-Tarski topos_ is 'the topos analogue of the [[Cantor space]]'.[^fine]

[^fine]: Quote from Bunge&Funk (2006, p.183).

##Definition
The **Jónsson-Tarski topos** $\mathcal{J}_2$ is the category of [[Jónsson-Tarski algebras]] considered as topos, i.e. its objects are sets $X$ together with an isomorphism $X\to X\times X$ and morphisms are functions that commute with the structure isomorphisms.

##Properties

* $\mathcal{J}_2$ is an example of an [[variety of algebras]] that is also a topos (cf. Johnstone 1985).

* $\mathcal{J}_2=Sh(M_2,J)$ where $M_2$ is the free monoid on two generators $a, b$ and $J$ is the [[coverage]] whose only covering family on the unique object $\cdot$ of $M_2$ is $\{a:\cdot\rightarrow \cdot ,b:\cdot\rightarrow\cdot\}$. So $\mathcal{J}_2$ is in a fact a [[Grothendieck topos]].

* $M_2$, as a free monoid, is cancellative, hence all-monic and, accordingly, $\mathcal{J}_2$ is an [[étendue]] ([[Peter Freyd]]). It is discussed from this [[étendue|perspective]] as a [[petit topos]] for _labeled graphs_ in (Lawvere 1989).

* Actually, Freyd observed that $\mathcal{J}_2/F(1)\cong Sh(2^\mathbb{N})$ with $F(1)$ the free Jónsson-Tarski algebra on one generator and $2^\mathbb{N}$ the [[Cantor space]] - this motivates the above quote from Bunge&Funk: $\mathcal{J}_2$ looks locally like (the sheaf topos on) $2^\mathbb{N}$ ! $Sh(2^\mathbb{N})$ classifies ' subsets of $\mathbb{N}$ ' in the sense that for cocomplete $\mathcal{E}$ geometric morphisms $\mathcal{E}\to Sh(2^\mathbb{N})$ correspond to morphisms $\Delta(\mathbb{N})\to\Delta(2)$ in $\mathcal{E}$, with $\Delta$ the [[constant sheaf functor]] (cf. [Mac Lane& Moerdijk](#MacLaneMoerdijk), ex.VIII.10, pp.470-71, ex.X.6, pp.572-73).

##Generalizations

The idea to consider generalizations of $\mathcal{J}_2$ seemed to have appeared first in the context of work on étendues (Rosenthal 1981, Lawvere 1989).

###Rosenthal's approach

K. Rosenthal (1981) starts from two basic facts about étendues, namely that $Set^{\mathcal{C}^{op}}$ is an [[étendue]] iff all morphisms in $\mathcal{C}$ are monic, and that $Sh(\mathcal{E},J)$ is an étendue if $\mathcal{E}$ is an étendue. His goal is to construct étendues
from an all-monic $\mathcal{C}$ by defining a topology $J$ on $Set^{\mathcal{C}^{op}}$ from a functor $H:\mathcal{C}\to Set$ satisfying:

1. $H(f)$ is monic ,and

2. if $x\in Im(H(f))\cap Im(H(g))$ then there is $k\in\mathcal{C}$ with $k\leq f$ and $k\leq g$ such that $x\in Im(H(k))$.

Now given $X\in\mathcal{C}$ and a sieve $B\in\Omega(X)$ define a sieve

$j_X(B):=\{\quad f\in\mathcal{C}| cod(f)=X\quad\wedge\quad Im(H(f))\subseteq\bigcup _{g\in B} Im(H(g))\quad\}.$

The resulting map $j:\Omega\to\Omega$ is a topology.

For $\mathcal{C}^{op}=\mathcal{M}_2$, the free monoid on two generators, and $H(X)=2^\mathbb{N}$, the functor constantly the Cantor set, this yields $\mathcal{J}_2$.

The generalization to $\mathcal{C}^{op}=\mathcal{M}_\infty$, the free monoid on countably infinite many generators, and the [[Baire space of sequences|Baire space]] $\mathbb{B}=\mathbb{N}^\mathbb{N}$ exhibits the **infinite Jónsson-Tarski topos** $\mathcal{J}_\infty$, i.e. the category of sets $A$ with an isomorphism to $A^\mathbb{N}$, as $Sh(\mathbb{N}^\mathbb{N})$ locally.

### Jónsson-Tarski toposes via profunctors

Work on a categorical concept of self-similarity led T. Leinster (2007) to another generalization of the Jónsson-Tarski topos. He starts from the observation that any profunctor $M:\mathcal{A} ⇸\mathcal{B}$ comes with an adjunction $-\otimes M\dashv [M,-]:Set^{\mathcal{A}^{op}}\to Set^{\mathcal{B}^{op}}$ where the left adjoint stems from profunctor composition and the right adjoint is defined for presheaves $X\in Set^{\mathcal{B}^{op}}$ and $a\in \mathcal{A}$ as $[ M,X ] (a) := Nat(M(-,a),X)$ i.e. the set of natural transformations from $M(-,a)$ to $X$ or, in other words, $Hom(M(-,a),X)$ in $Set^{\mathcal{B}^{op}}$.

For an endoprofunctor $M:\mathcal{A} ⇸\mathcal{A}$ define a **Jónsson-Tarski M-algebra** as a pair $(X,\xi)$ where $X\in Set^{\mathcal{A}^{op}}$ and $\xi$ a natural isomorphism $X\overset{\simeq}{\to} [M,X]$. The resulting category $\mathcal{J}_M$ is a topos since a site $(\mathcal{A}_M,J)$ can be constructed from $\mathcal{A}$ by adjoining new arrows $b\overset{m}{\to} a$ for each $b,a\in\mathcal{A}$ and $m\in M(b,a)$ with covers $J(a)$ the set of these arrows (cf. Worrell 2002, Leinster 2007). Furthermore, $\mathcal{J}_M$ is monadic over $Set^{\mathcal{A}^{op}}$.

The classical Jónsson-Tarski topos $\mathcal{J}_2$ arises from this process by taking $M=\mathbf{2}:\mathbf{1} ⇸\mathbf{1}$ assigning to the unique object $(\bullet,\bullet)$ of the category $\mathbf{1}\times\mathbf{1}$ the two element set $\{\star,\ast\}$. A presheaf on $\mathbf{1}$ is just a set $X$ whence $[\mathbf{2}, X](\bullet)$ is just the set of all maps $\{\star,\ast\}\to X$ i. e. $X\times X$. The site for it is the free category generated by the graph with one node $\bullet$ and two edges $\star,\ast$ i. e. the free monoid on two generators and with coverage $\{\star,\ast\}$.

## Related entries

* [[Jónsson-Tarski algebra]]

* [[Cantor space]]

##Link

Some further properties of $\mathcal{J}_2$ are discussed in the answer to this

* [MO-question](https://mathoverflow.net/questions/396041/images-of-complemented-subobjects-in-hyperconnected-toposes-over-boolean-bases/396043?r=SearchResults&s=1|33.5942#396043)

##References

* [[Marta Bunge]], [[Jonathon Funk]], _Singular Coverings of Toposes_ , Springer LNM vol. 1890, Heidelberg 2006.

* [[Peter Johnstone]], _When is a Variety a Topos?_ , Algebra Universalis **21** (1985) pp.198-212.

* [[Peter Johnstone]], _Collapsed Toposes as Bitopological Spaces_ , pp.19-35 in _Categorical Topology_ , World Scientific Singapore 1989.

* [[Peter Johnstone]], _Collapsed Toposes and Cartesian Closed Varieties_ , JA **129** (1990) pp.446-480.

* [[Peter Johnstone]], _[[Sketches of an Elephant]] vol. I_ , Oxford UP 2002. (sec. A2.1, p.80)

* [[Peter Johnstone|P. T. Johnstone]], A. J. Power, T. Tsujishita, H. Watanabe, J. Worrell, _The structure of categories of coalgebras_ , Theoret. Comp. Sci **260** (2001) pp.87-117.

* [[F. William Lawvere]], _Qualitative Distinctions between some Toposes of Generalized Graphs_ , Cont. Math. **92** (1989) pp.261-299.

* [[Tom Leinster]], _Jónsson-Tarski toposes_, Talk Nice 2007.  ([slides](http://www.maths.ed.ac.uk/~tl/nice/jt.pdf))

* [[Saunders Mac Lane|S. Mac Lane]], [[Ieke Moerdijk|I. Moerdijk]], _Sheaves in Geometry and Logic_ , Springer Heidelberg 1994.{#MacLaneMoerdijk} (pp.470-471)

* [[Kimmo I. Rosenthal]], _Étendues and Categories with Monic Maps_ , JPAA **22** (1981) pp.193-212.

* James Worrell, _A Note on Coalgebras and Presheaves_ , Electronic Notes in Theoretical Computer Science **65** no.3 (2003) pp.1-10.

[[!redirects Jonsson-Tarski topos]]
[[!redirects Jonsson-Tarski Topos]]


Jörg Rachen

member of the [[Planck Collaboration]] 2015

* [webpage](https://www.astro.ru.nl/wiki/general/people/joerg_rachen)

## selected Publications

* _Bayesian Classification of Astronomical Objects -- and what is behind it_ ([arXiv:1302.2429](http://arxiv.org/abs/1302.2429))

[[!redirects Jorg Rachen]]
[[!redirects Joerg Rachen]]


Jörg Teschner

* [webpage](http://wwwiexp.desy.de/sfb676/researchers/teschner/)

## Selected writings

On [[2d CFT]]:

* [[Jörg Teschner]], _A guide to two-dimensional conformal field theory_ ([arXiv:1708.00680](http://arxiv.org/abs/1708.00680))

On [[non-perturbative effects]] and [[Stokes phenomena]] in [[topological string]] [[string scattering amplitude|amplitudes]]:

* Murad Alim, Arpan Saha, [[Jörg Teschner]], Iván Tulli, *Mathematical structures of non-perturbative topological string theory: from GW to DT invariants*,  Communications in Mathematical Physics  __399__ (2023) 1039--1101 ([arXiv:2109.06878](https://arxiv.org/abs/2109.06878) [doi](https://doi.org/10.1007/s00220-022-04571-y))

category: people


Jørgen Andersen

* [Webpage](http://pure.au.dk/portal/en/andersen@qgm.au.dk)

* [ReNewQuantum](https://renewquantum.eu/)

## Selected writings

Proof that the values of a [[modular functor]] at [[genus of a surface|genus]]=0 (ie. the [[conformal blocks]] on the punctured [[Riemann sphere]]) determine the full modular functor:

* [[Jørgen Ellegaard Andersen]], [[Kenji Ueno]], *Modular functors are determined by their genus zero data*, Quantum Topology **3** 3/4 (2012) 255-291 $[$[doi:10.4171/QT/29](https://doi.org/10.4171/qt/29)$]$

On [[Reshetikhin-Turaev theory]] producing [[modular functors]] from [[modular tensor categories]]:

* [[Jørgen Ellegaard Andersen]], William Petersen, *Construction of Modular Functors from Modular Tensor Categories* ([arXiv:1602.04907](https://arxiv.org/abs/1602.04907))

## Related entries

* [[Chern-Simons theory]]

* [[Reshetikhin-Turaev model]], [[Turaev-Viro model]]

* [[topological recursion]]

category: people

[[!redirects Jorgen Andersen]]

[[!redirects Jørgen Ellegaard Andersen]]
[[!redirects Jorgen Ellegaard Andersen]]


Jørgen Tornehave

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=12017)

## Selected writings

On [[infinite loop spaces]]:

* [[Ib Madsen]], [[Victor Snaith]], [[Jørgen Tornehave]], *Infinite loop maps in geometric topology*, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 81, Issue 3, (1977)([doi:10.1017/S0305004100053482](https://doi.org/10.1017/S0305004100053482))

On [[E-infinity spaces|$E_\infty$-spaces]] and [[E-infinity ring spectra|$E_\infty$-ring sectra]]:

* [[Peter May]] with contributions by [[Frank Quinn]], [[Nigel Ray]] and [[Jørgen Tornehave]], _$E_\infty$-Ring spaces and $E_\infty$ ring spectra_, Lecture Notes in Mathematics 577, Springer 1977 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf), [cds:1690879](https://cds.cern.ch/record/1690879))

category: people

[[!redirects Jorgen Tornehave]]


Jürg Fröhlich

* [Wikipedia entry](http://de.wikipedia.org/wiki/J%C3%BCrg_Fr%C3%B6hlich)

## Selected writings

On the [[Jones polynomial]] via [[Chern-Simons theory]]:

* [[Jürg Fröhlich]], C. King, _The Chern-Simons theory and knot polynomials_, Comm. Math. Phys. Volume 126, Number 1 (1989), 167-199 ([euclid.cmp/1104179728](https://projecteuclid.org/euclid.cmp/1104179728))

On [[anyons]] and [[braid group statistics]] via [[algebraic quantum field theory]]:

* {#FroehlichMarchetti88} [[Jürg Fröhlich]], [[Pieralberto Marchetti]], *Quantum field theory of anyons*, Lett. Math. Phys. **16** (1988) 347–358 (reprinted in [Wilczek 1990, p. 202-213](#Wilczek90)) $[$[doi:10.1007/BF00402043](https://doi.org/10.1007/BF00402043)$]$

* [[Jürg Fröhlich]], [[Fabrizio Gabbiani]], *Braid statistics in local quantum theory*, Reviews in Mathematical Physics, **2** 03 (1990) 251-353 $[$[doi:10.1142/S0129055X90000107](https://doi.org/10.1142/S0129055X90000107)$]$

and making explicit the role of the [[configuration space of points]]:

* {#FroehlichGabbianiMarchetti90} [[Jürg Fröhlich]], [[Fabrizio Gabbiani]], [[Pieralberto Marchetti]], *Braid statistics in three-dimensional local quantum field theory*, in: H.C. Lee (ed.) *Physics, Geometry and Topology* NATO ASI Series, **238** Springer (1990)  $[$[doi:10.1007/978-1-4615-3802-8_2](https://doi.org/10.1007/978-1-4615-3802-8_2), [[FroehlichGabbianiMarchetti-BraidStatistics.pdf:file]]$]$

On [[non-perturbative quantum field theory]]:

* {#Froehlich92} [[Jürg Fröhlich]], _Non-perturbative quantum field theory -- Mathematical Aspects and Applications_, Advanced Series in Mathematical Physics, World Scientific 1992 ([doi:10.1142/1245](https://doi.org/10.1142/1245))

On [[supersymmetric quantum mechanics]] from the point of view of [[spectral geometry]] ("[[noncommutative geometry]]"):

* {#FGR96} [[Jürg Fröhlich]], Oiver Grandjean, [[Andreas Recknagel]], _Supersymmetric quantum theory and (non-commutative) differential geometry_, Commun.Math.Phys. 193 (1998) 527-594 ([arXiv:hep-th/9612205](http://arxiv.org/abs/hep-th/9612205))

and with an eye towards the [[superstring]] via [[2-spectral triples]]:

* {#FGR97} [[Jürg Fröhlich]], Oliver Grandjean, [[Andreas Recknagel]], _Supersymmetric quantum theory, non-commutative geometry, and gravitation_  Lecture Notes Les Houches (1995) ([arXiv:hep-th/9706132](http://arxiv.org/abs/hep-th/9706132)).

On [[QFT with defects|defects]] and [[duality in physics|duality]] ("[[generalized global symmetry]]") in [[2d CFT]] (via the [[FRS theorem on 2d rational CFT]]):

* [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], *Kramers-Wannier duality from conformal defects*, Phys. Rev. Lett. **93** (2004) 070601 [[doi:10.1103/PhysRevLett.93.070601](https://doi.org/10.1103/PhysRevLett.93.070601), [arXiv:cond-mat/0404051](http://arxiv.org/abs/cond-mat/0404051)]

* [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], *Duality and defects in rational conformal field theory*, Nucl. Phys. B **763** (2007) 354-430 [[arXiv:hep-th/0607247](http://arxiv.org/abs/hep-th/0607247), [doi:10.1016/j.nuclphysb.2006.11.017](https://doi.org/10.1016/j.nuclphysb.2006.11.017)]

On [[quantum probability theory]] and the [[operator algebra]]-foundations for [[quantum physics]]:

* {#FroehlichSchubnel15} [[Jürg Fröhlich]],  B. Schubnel, _Quantum Probability Theory and the Foundations of Quantum Mechanics_. In: Blanchard P., Fröhlich J. (eds.) _The Message of Quantum Science_. Lecture Notes in Physics, vol 899. Springer 2015 ([arXiv:1310.1484](https://arxiv.org/abs/1310.1484), [doi:10.1007/978-3-662-46422-9_7](https://doi.org/10.1007/978-3-662-46422-9_7))

* {#Froehlich} [[Jürg Fröhlich]], _The structure of quantum theory_, Chapter 6 in _The quest for laws and structure_, EMS 2016  ([doi](https://www.researchgate.net/publication/308595814_The_Quest_for_Laws_and_Structure), [doi:10.4171/164-1/8](https://www.ems-ph.org/books/show_abstract.php?proj_nr=207&vol=1&rank=8))

On [[quantum measurement]]:

* [[Jürg Fröhlich]], Zhou Gang, *On the Evolution of States in a Quantum-Mechanical Model of Experiments* [[arXiv:2212.02599](https://arxiv.org/abs/2212.02599)]

On [[topological phases of matter]] in [[condensed matter physics]] with focus on the role of [[quantum anomalies]]:

* [[Jürg Fröhlich]], *Gauge Invariance and Anomalies in Condensed Matter Physics*, Journal of Mathematical Physics [[arXiv:2303.14741](https://arxiv.org/abs/2303.14741)]

## Quotes
 {#Quotes}

From [Fröhlich 92, p. 11](#Froehlich92), on laying foundations for [[perturbative string theory]] via rigorous formulation of [[2d SCFT]] in terms of [[conformal nets]]/[[2-spectral triples]] or similar:

> I still have hopes, perhaps romantic ones, that [[string theory]], or something inspired by it, will come back to life again. I believe it is interesting to attempt to formulate string theory in an "invariant" way, quite like it is useful to formulate geometry in a coordinate-independent way. One might, for example, start with a family $\mathcal{F}$, of [[von Neumann algebra factor|hyperfinite type $III_1$ von Neumann algebras]] -- to be a little technical -- [[conformal net|indexed by]] intervals of the circle with non-empty complement (or of the super-circle). It may pay to formulate the starting point using the language of [[sheaves]]. $[...]$ This structure determines a [[braided monoidal category|braided monoidal]] [[C*-category]] with unit, ...; briefly, a quantum theory. From a combination of such [[tensor categories]] (left and right movers) one would attempt to reconstruct (symmetries of) _physical space-time_. String amplitudes would correspond to [[morphisms|arrows]] ([[intertwiners]]) of the [[tensor category]]. $[...]$ it would provide a general way of thinking about string theory that does not presuppose knowing the target space-time of the theory.

## Related $n$Lab entries

* [[FRS formalism]]

* [[non-perturbative quantum field theory]]

category: people

[[!redirects Juerg Froehlich]]


Jürg Gasser

* [webpage](http://www.gasser.itp.unibe.ch/)

## Selected writings

On [[chiral perturbation theory]]:

* [[Jürg Gasser]], [[Heinrich Leutwyler]], _Quark masses_, Physics Reports Volume 87, Issue 3, July 1982, Pages 77-169 (doi:10.1016/0370-1573(82)90035-7)

* [[Jürg Gasser]], [[Heinrich Leutwyler]], _Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark_, Nucl. Phys. B 250 (1985) 465-516, ([spire:200027](https://inspirehep.net/literature/200027), doi:10.1016/0550-3213(85)90492-4))

* [[Jürg Gasser]], [[Heinrich Leutwyler]], _Chiral perturbation theory to one loop_, Annals of Physics
Volume 158, Issue 1, November 1984, Pages 142-210 (doi:10.1016/0003-4916(84)90242-2)

* [[Gerhard Ecker]], [[Jürg Gasser]], [[Antonio Pich]], E. DeRafael, Sec 3 and Appendix A in: _The role of resonances in chiral perturbation theory_, Nuclear Physics B
Volume 321, Issue 2, 24 July 1989, Pages 311-342 (doi:10.1016/0550-3213(89)90346-5)

* [[Jürg Gasser]], _The QCD Vacuum and Chiral Symmetry_, In: Vautherin D., Lenz F., Negele J.W. (eds.) _Hadrons and Hadronic Matter_, NATO ASI Series (Series B: Physics), vol 228. Springer 1990 ([doi:10.1007/978-1-4684-1336-6_4](https://doi.org/10.1007/978-1-4684-1336-6_4))

category: people


Jürgen Berndt

* [webpage](https://nms.kcl.ac.uk/juergen.berndt/)

## Selected writings

On the [[Arnold-Kuiper-Massey theorem]] and its further generalization to the [[13-sphere]] being an [[Sp(1)]]-quotient of the [[octonionic projective plane]]:

* {#AtiyahBerndt02} [[Michael Atiyah]], [[Jürgen Berndt]], *Projective planes, Severi varieties and spheres*, in:  Surv. Differ. Geom. VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck (International Press, Somerville, MA, 2003) 1-27 ([arXiv:math/0206135](https://arxiv.org/abs/math/0206135), [doi:10.4310/SDG.2003.v8.n1.a1](https://dx.doi.org/10.4310/SDG.2003.v8.n1.a1))

category: people


Jürgen Fuchs

* [webpage](http://www.ingvet.kau.se/juerfuch/)

## Selected writings

The [[FRS-theorem on rational 2d CFT]]:

* [[Ingo Runkel]], [[Jens Fjelstad]], [[Jürgen Fuchs]], [[Christoph Schweigert]], *Topological and conformal field theory as Frobenius algebras*, in: *Categories in Algebra, Geometry and Mathematical Physics*, Contemp. Math. **431** (2007) 225-248 $[$[doi:10.1090/conm/431](http://dx.doi.org/10.1090/conm/431), [arXiv:math/0512076](https://arxiv.org/abs/math/0512076)$]$

On [[QFT with defects|defects]] and [[duality in physics|duality]] ("[[generalized global symmetry]]") in [[2d CFT]] (via the [[FRS theorem on 2d rational CFT]]):

* [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], *Kramers-Wannier duality from conformal defects*, Phys. Rev. Lett. **93** (2004) 070601 [[doi:10.1103/PhysRevLett.93.070601](https://doi.org/10.1103/PhysRevLett.93.070601), [arXiv:cond-mat/0404051](http://arxiv.org/abs/cond-mat/0404051)]

* [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], *Duality and defects in rational conformal field theory*, Nucl. Phys. B **763** (2007) 354-430 [[arXiv:hep-th/0607247](http://arxiv.org/abs/hep-th/0607247), [doi:10.1016/j.nuclphysb.2006.11.017](https://doi.org/10.1016/j.nuclphysb.2006.11.017)]

On [[QFTs with defects]] via [[higher algebra|higher]] [[categorical algebra]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Alessandro Valentino]], *Bicategories for boundary conditions and for surface defects in 3-d TFT*, Commun. Math. Phys. **321**  (2013) 543–575 [[arXiv:1203.4568](https://arxiv.org/abs/1203.4568), [doi:10.1007/s00220-013-1723-0](https://doi.org/10.1007/s00220-013-1723-0)]

On potential generalization to [[logarithmic CFT]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], *Full Logarithmic Conformal Field theory - an Attempt at a Status Report*, Proc. of *[[Higher Structures in M-Theory 2018]]*, Fortschr. Phys.  **67** 8-9 (2019)  $[$[arXiv:1903.02838](https://arxiv.org/abs/1903.02838), [doi:10.1002/prop.201910018](https://doi.org/10.1002/prop.201910018)$]$

Constructing [[correlators]] in [[rational conformal field theory]] via [[string net models]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Yang Yang]], *String-Net Construction of RCFT Correlators*, Springer Briefs in Mathematical Physics **45**, Springer (2023) [[doi:10.1007/978-3-031-14682-4](https://doi.org/10.1007/978-3-031-14682-4), [arXiv:2112.12708](https://arxiv.org/abs/2112.12708)]

Constructing [[modular functors]] from [[pivotal category|pivotal]] [[bicategories]] using [[string net models]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Yang Yang]], *String-net models for pivotal bicategories* [[arXiv:2302.01468](https://arxiv.org/abs/2302.01468)]

Survey of [[algebra|algebraic]] [[structures]] in [[2d conformal field theory]] ([[vertex operator algebras]], [[conformal blocks]], [[modular tensor category|modular]] [[fusion categories]], [[modular functors]], [[FRS theorem]], [[string-net models]]):

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], *Algebraic structures in two-dimensional conformal field theory*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]

## Related entries

* [[CFT]]

* [[FFRS formalism]]

category: people


Jürgen Jost

Jürgen Jost is an applied mathematician at Max Planck for Mathematics in Natural Science in Leipzig, studying geometry, the calculus of variations and geometric analysis, dynamical systems, cognitive and neural systems, theoretical and mathematical biology, economic and social systems, history of science and philosophy, graphs and networks and system theory.

* [publications](http://www.mis.mpg.de/de/jjost/publikationen)

## Selected writings

On the [[Polyakov action]] and [[bosonic strings]]:

* [[Jürgen Jost]], *Bosonic Strings: A Mathematical Treatment*, AMS/IP Stud. Adv. Math. **21** (2001) [[ISBBN:978-0-8218-4336-9](https://bookstore.ams.org/view?ProductCode=AMSIP/21.S.B), [spire:1388134](https://inspirehep.net/literature/1388134)]

On [[Riemannian geometry]] and [[geometric analysis]]:

* [[Jürgen Jost]], *Riemannian Geometry and Geometric Analysis*, Springer (2017) [[doi:10.1007/978-3-319-61860-9](https://doi.org/10.1007/978-3-319-61860-9)]

[[!redirects Juergen Jost]]


Jürgen Moser

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

## related $n$Lab entries

* [[CR manifold]]

category: people


Jürgen Neukirch

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

## Selected writings

On [[cohomology]] of [[number fields]]:

* {#NSW08} [[Jürgen Neukirch]], Alexander Schmidt, Kay Wingberg, *Cohomology of Number Fields*, Springer Grundlehren der mathematischen Wissenschaften **323**, Springer 2008 ([doi:10.1007/978-3-540-37889-1](https://link.springer.com/book/10.1007/978-3-540-37889-1), [webpage](https://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/))

## Related $n$Lab entries

* [[algebraic number theory]]

* [[genus of a number field]]

* [[crossed homomorphism]]

category: people


Jędrzej Śniatycki

* [institute page](https://contacts.ucalgary.ca/info/math/profiles/101-152972)

* [GoogleScholar page](https://scholar.google.ca/citations?user=4aR1HasAAAAJ&hl=en)

## Selected writings

On [[geometric quantization]]:

* {#Śniatycki75} [[Jędrzej Śniatycki]], *Wave functions relative to a real polarization*, Internat. J. Theoret. Phys., **14** 4 (1975) 277-288 [[doi:10.1007/BF01807689)](https://doi.org/10.1007/BF01807689)]

* {#Śniatycki80} [[Jędrzej Śniatycki]], *Geometric Quantization and Quantum Mechanics*, Applied Mathematical Sciences **30**, Springer-Verlag (1980) [[doi:10.1007/978-1-4612-6066-0](https://doi.org/10.1007/978-1-4612-6066-0)]

* {#Śniatycki16} [[Jędrzej Śniatycki]], *Lectures on Geometric Quantization*, in *Geom. Integrability & Quantization* (2016) 95-129 [[doi:10.7546/giq-17-2016-95-129](https://projecteuclid.org/ebooks/geometry-integrability-and-quantization-proceedings-series/Lectures-on-Geometric-Quantization/chapter/Lectures-on-Geometric-Quantization/10.7546/giq-17-2016-95-129)]

## Related entries

* [[Bohr-Sommerfeld quantization]]

* [[Weinstein symplectic category]]

category: people

[[!redirects Jedrzej Sniatycki]]


K modal logic

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

# Contents #
* table of contents
{:toc}

## Idea

The flavor of _[[modal logic]]_ called $K$ is [[propositional logic]] equipped with a single [[modality]] usually written "$\Box$" subject to the rules that for all [[propositions]] $p, q \colon Prop$ we have

* $\Box K \colon \Box(p \to q) \to (\Box p \to \Box q)$ ([[K modal logic]])

Often one adds to this the following further axioms

* $\Box T \colon \Box p \to p$ ([[T modal logic]])

* $\Box 4 \colon \Box p \to \Box \Box p$. ([[S4 modal logic]]).

$K$ is the basic [[epistemic logic]].

## Properties

### Axiomatisation

* (Taut)  All (instances of ) propositional tautologies.

*   For each $i = 1,\ldots, m$, the axiom, ($K_i$):

$$(K_i\phi \wedge K_i(\phi \to \psi))\to K_i\psi.$$

### Derivation rules

* (MP)

$$\frac{\phi \quad \phi\to \psi}{\psi} \quad$$

 (i.e. _modus ponens_);

* (Generalisation)

$$\frac{\phi}{K_i\phi}.$$

The second deduction rule corresponds to the idea that _if a statement has been proved, then it is known to all 'agents'_.

### Normality

This logic is the smallest [[normal modal logic]].

## Semantics

The semantics of $K_{(m)}$ is just the [[Kripke semantics]] of this context, so a [[frame (modal logic)|frame]], $\mathfrak{F}$ is just a set, $W$  of [[possible worlds]] with $m$ [[relations]] $R_i$. A [[geometric models for modal logics|model]], $\mathfrak{M} = (\mathfrak{F},V)$, is a frame in that sense together with a valuation, $V \colon Prop \to \mathcal{P}(W)$, and the satisfaction relation is as described in [[geometric models for modal logics]] with just the difference implied by the fact that that page correspond to the use of $\Diamond_i = M_i$ whilst this uses $K_i$.  This means that

* $\mathfrak{M},w \models K_i \phi$ if and only if, for all $v \in W$ such that $ R_i w v$, $\mathfrak{M},v \models \phi$.

## Related concepts

[[!include flavors of modal logics and relations -- table]]

* [[modal logic]], [[epistemic modal logic]]

* [[necessity and possibility]]

* [[T modal logic]]

* [[S4 modal logic]]

* [[S5 modal logic]]

* [[axiom K (modal logic)]]

## References

The notation "K" for a modal operator meant to express [[epistemic modality]] originates with:

* [[K. Jaakko J. Hintikka]], *Knowledge and belief: An introduction to the logic of the two notions*, Cornell University Press (1962) [[ark:/13960/t9k437s65](https://archive.org/details/knowledgebeliefi00hint_0)]

[[!redirects the logic K(m)]]
[[!redirects The logic K(m)]]
[[!redirects The logic $K_{(m)}$]]
[[!redirects K(m)]]


K(n)-local stable homotopy theory

+-- {: .rightHandSide}
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###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[stable homotopy theory]] of [[local spectra]], local with respect to [[Morava K-theory]] $K(n)$.

## Properties

### Bilimits

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

Say that an [[∞-groupoid]] is _[[groupoid cardinality|strictly tame]]_ or _of [[finite type]]_ ([Hopkins-Lurie 14, def. 4.4.1](#HopkinsLurie14)) or maybe better is a _[[homotopy type with finite homotopy groups|truncated homotopy type with finite homotopy groups]]_ if it has only finitely many nontrivial [[homotopy groups]] each of which is furthermore a [[finite group]].

=--

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

For $F \colon X \to Sp_{K(n)}$ a strictly tame [[diagram]], def. \ref{StrictlyTame}, of $K(n)$-local spectra, then its [[(∞,1)-limit]] and [[(∞,1)-colimit]] agree in that the canonical comparison map is an [[equivalence in an (∞,1)-category|equivalence]]

$$
  \underset{\longrightarrow}{\lim} F
    \stackrel{\simeq}{\longrightarrow}
  \underset{\longleftarrow}{\lim} F
  \,.
$$

=--

This is ([Hopkins-Lurie 14, theorem 0.0.2](#HopkinsLurie14)).

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

So in particular $K(n)$-local spectra have _[[biproducts]]_, called _[[0-semiadditivity]]_ in ([Hopkins-Lurie 14, prop. 4.4.9](#HopkinsLurie14).

=--

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

For $X$ pointed [[connected]], hence $X \simeq B G$ the [[delooping]] of an [[∞-group]], a [[diagram]] in prop. \ref{HopkinsLurieTheorem} exhibits an [[∞-action]] of $G$ on some $K(n)$-local spectrum, the [[(∞,1)-colimit]] produces the [[homotopy quotient]] of the [[∞-action]] and the [[(∞,1)-limit]] the [[homotopy invariants]]. In this case ([Hovey-Sadofsky 96](#HoveySadofsky96)) show that the comparison map exhibits the homotopy invariant as the $K(n)$-[[localization of a spectrum|localization]] of the homotopy coinvariants. This in particular means that the comparison map is a $K(n)$-local equivalence, which is the statement of prop. \ref{HopkinsLurieTheorem}.

=--

### Logarithms of twists of generalized cohomology

Let $E$ be an [[E-∞ ring]] and write $GL_1(E)$ for its [[abelian ∞-group|abelian]] [[∞-group of units]]  and $gl_1(E)$ for the corresponding [[connective spectrum]].

Via the [[Bousfield-Kuhn functor]] there are [[natural equivalences]] between the $K(n)$-localizations of $gl_1(E)$ and $E$ itself.

$$
  L_{K(n)} gl_1(E) \simeq L_{K(n)} E
  \,.
$$

Composed with the [[Bousfield localization of spectra|localization map]] itself, this yields [[logarithmic cohomology operations]]

$$
  gl_1(E) \longrightarrow L_{K(n)} gl_1E \stackrel{\simeq}{\to} L_{K(n)}E
  \,.
$$

## Related concepts

* [[chromatic homotopy theory]]

* [[Bousfield-Kuhn functor]]

* [[logarithmic cohomology operation]]

* [[Bousfield localization of spectra]]

* [[KU-local stable homotopy theory]]

## References

* {#HoveySadofsky96} [[Mark Hovey]], H. Sadofsky, _Tate cohomology lowers chromatic Bouseld classes_ Proceedings of the AMS 124, 1996, 3579-3585.

* {#HopkinsLurie14} [[Michael Hopkins]], [[Jacob Lurie]], _[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]_ (2014)

Some basics of [[Morava K-theory|K(1)]]-local [[E-∞ rings]] are in

* [[Michael Hopkins]], _$K(1)$-local $E_\infty$-Ring spectra_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/knlocal.pdf))

[[!redirects K(n)-local spectrum]]
[[!redirects K(n)-local spectra]]

[[!redirects K(n)-localization]]
[[!redirects K(n)-localizations]]


K* meson

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[vector meson]]:

The [[Lorentz group|Lorentz]] [[vector representation|vector]] and [[isospin]] [[fundamental representation|doublet]], [[bound state|bound]] from [[up-quark|up]]/[[down-quarks]] in the [[first generation of fundamental particles]] and a [[strange quark]] in the [[second generation of fundamental particles]].  Hence a [[heavy meson]] with respect to 2-flavor [[chiral perturbation theory]].

Together with the [[kaons]], which are the corresponding [[scalar mesons]], these are also called the _[[K-mesons]]_ (alongide [[D-mesons]] and [[B-mesons]], etc.)

\begin{imagefromfile}
    "file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": 0,
        "right": 10,
        "bottom": 0,
        "left": 20
    }
\end{imagefromfile}

[[!include flavors of fundamental particles -- table]]

## Properties

### Fusion from kaon and pion

Fusion from [[kaon]] and [[pion]]:

[Ozaki-Nagahiro-Hosaka 08, p. 4](#OzakiNagahiroHosaka08)

[ICMB 16](#ICMB16)

### WZW term

The [[WZW term]] [[interaction]] with the [[photon]] and [[kaon]]:

[Nam-Choi-Hosaka-Kim 06 (1)](#NamChoiHosakaKim06),
[Ozaki-Nagahiro-Hosaka 08, (3), (9)](#OzakiNagahiroHosaka08)

## Related concepts

* [[kaon]]

## References

### General

* {#NamChoiHosakaKim06} Seung-il Nam, Ki-Seok Choi, Atsushi Hosaka, Hyun-Chul Kim, _Test of the reaction mechanism for $\gamma N \to K \Lambda(1520)$ using the polarized photon_ ([arXiv:hep-ph/0611213](https://arxiv.org/abs/hep-ph/0611213))

* {#ICMB16} Andrej Ilner, Daniel Cabrera, Christina Markert, Elena Bratkovskaya, _$K^\ast$ vector meson resonances dynamics in heavy-ion collisions_, Phys. Rev. C 95, 014903 (2017) ([arXiv:1609.02778](https://arxiv.org/abs/1609.02778), [doi:10.1103/PhysRevC.95.014903](https://doi.org/10.1103/PhysRevC.95.014903))

On [[kaon]]-[[K*-meson]]-[[photon]] [[interaction]] via the [[WZW term]]:

* S. Ozaki, H. Nagahiro, [[Atsushi Hosaka]], Equations (3) and (9) in: _Magnetic interaction induced by the anomaly in kaon-photoproductions_, Physics Letters B Volume 665, Issue 4, 24 July 2008, Pages 178-181 ([arXiv:0710.5581](https://arxiv.org/abs/0710.5581), [doi:10.1016/j.physletb.2008.06.020](https://doi.org/10.1016/j.physletb.2008.06.020))

[[!include Skyrme hadrodynamics with heavy mesons -- references]]

[[!redirects K* mesons]]

[[!redirects K*-meson]]
[[!redirects K*-mesons]]

[[!redirects K-star meson]]
[[!redirects K-star mesons]]

[[!redirects K∗ meson]]
[[!redirects K∗ mesons]]

[[!redirects K∗-meson]]
[[!redirects K∗-mesons]]


k-ary factorization system

+-- {: .rightHandSide}
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###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
=--
=--

# $k$-ary factorisation systems
* table of contents
{: toc}

## Idea

A $k$-ary factorization system is a generalization of (binary) [[orthogonal factorization systems]] and [[ternary factorization systems]] to factorizations into a string of $k$ morphisms.

## Definition

For $k \gt 0$ a [[natural number]] and $C$ a [[category]] (or $\infty$-[[infinity-category|category]]), a __$k$-ary factorisation system__ on $C$ consists of $(k - 1)$ factorisation systems $(E_i, M_i)$ (for $0 \lt i \lt k$) on $C$, such that

*  $M_i \subseteq M_{i + 1}$ whenever this is meaningful (equivalently, $E_i \subseteq E_{i - 1}$).

We can extend this to include two other factorisation systems, one for $i = 0$ and one for $i = k$:

*  $M_0$ consists of only [[isomorphisms]]/[[equivalences]] (equivalently, $E_0$ consists of all morphisms), and
*  $M_k$ consists of all morphisms (equivalently, $E_k$ consists of only isomorphisms/equivalences).

Given a $k$-ary factorisation system, the ([[coimage|co]])[[image]] of $(E_i,M_i)$ is the __$i$-(co)image__ of the entire $k$-ary factorisation system.

Note that every (higher) category has a unique $1$-ary factorisation system, since no structure at all is required.  We also say that a [[groupoid]] (or $\infty$-[[infinity-groupoid|groupoid]]) has a (necessarily unique) $0$-ary factorisation system; this makes sense since we have $M_0 = M_k$ (and $E_0 = E_k$) in that case.  A [[discrete category]] has a (necessarily unique) $(-1)$-ary factorisation system.

A $k$-ary factorisation system may also be called a __$k$-step factorisation system__ or a __$(k+1)$-stage factorisation system__.  You can see why if you count the basic morphisms (steps) and objects (stages) that $k - 1$ overlapping factorisation systems produce from a morphism.

## Infinitary factorisation systems

Here is an incomplete attempt at a general definition:

Fix any [[ordinal number]] (or [[opposite poset|opposite]] thereof, or any [[poset]], really) $\alpha$.  Then an __$\alpha$-stage factorisation system__ (in an ambient $\infty$-category $C$) consists of an $\alpha$-indexed family of factorisation systems $(E_i, M_i)$ in $C$ such that:

*  $M_i \subseteq M_j$ whenever $i \leq j$ (equivalently, $E_i \supseteq E_j$ whenever $i \leq j$),
*  each morphism $f\colon X \to Y$ is both the [[inverse limit]] $\underset{i \to \infty}\lim \im_i f$ in the [[slice category]] $C/Y$ and the [[direct limit]] $\underset{i \to -\infty}\colim \coim_i f$ in the [[coslice category]] $X/C$, and
*  for each $f\colon X \to Y$, $\id_Y$ is $\underset{i \to -\infty}\colim \im_i f$ and $\id_X$ is $\underset{i \to \infty}\lim \coim_i f$.

This seems to be correct whenever $\alpha$ really is either an ordinal or the opposite thereof, as well as some other posets such as $\omega^op + \omega$ (which is the poset of [[integers]]), but it seems to be missing something for (for example) $\omega + \omega^op$.  Notice that, when $\alpha$ is *both* an ordinal and the opposite thereof, we recover the above definition of an $(alpha-1)$-ary factorisation system.

## Examples

* For $k = 3$ one speaks of a _[[ternary factorization system]]_. See there for more examples

* In an [[(∞,1)-topos]] the [[(epi, mono) factorization system]] in a [[topos]] splits up to an $\infty$-ary factorization system consisting of the [[(n-epi, n-mono) factorization systems]] (the [[n-image]]-factorization) for all $n \in \mathbb{N}$. This is called the _[[Postnikov system]]_.

## References

* [Cafe discussion](http://golem.ph.utexas.edu/category/2010/07/ternary_factorization_systems.html) mainly on the ternary version

* [Forum discussion](https://nforum.ncatlab.org/discussion/1629) including the k-ary case, even when k is infinite

[[!redirects k-ary factorization system]]
[[!redirects k-ary factorization systems]]
[[!redirects k-ary factorisation system]]
[[!redirects k-ary factorisation systems]]
[[!redirects n-ary factorization system]]
[[!redirects n-ary factorization systems]]
[[!redirects n-ary factorisation system]]
[[!redirects n-ary factorisation systems]]
[[!redirects alpha-ary factorization system]]
[[!redirects alpha-ary factorization systems]]
[[!redirects alpha-ary factorisation system]]
[[!redirects alpha-ary factorisation systems]]

[[!redirects k-fold factorization system]]
[[!redirects k-fold factorization systems]]
[[!redirects k-fold factorisation system]]
[[!redirects k-fold factorisation systems]]
[[!redirects n-fold factorization system]]
[[!redirects n-fold factorization systems]]
[[!redirects n-fold factorisation system]]
[[!redirects n-fold factorisation systems]]
[[!redirects alpha-fold factorization system]]
[[!redirects alpha-fold factorization systems]]
[[!redirects alpha-fold factorisation system]]
[[!redirects alpha-fold factorisation systems]]

[[!redirects k-step factorization system]]
[[!redirects k-step factorization systems]]
[[!redirects k-step factorisation system]]
[[!redirects k-step factorisation systems]]
[[!redirects n-step factorization system]]
[[!redirects n-step factorization systems]]
[[!redirects n-step factorisation system]]
[[!redirects n-step factorisation systems]]
[[!redirects alpha-step factorization system]]
[[!redirects alpha-step factorization systems]]
[[!redirects alpha-step factorisation system]]
[[!redirects alpha-step factorisation systems]]

[[!redirects k-stage factorization system]]
[[!redirects k-stage factorization systems]]
[[!redirects k-stage factorisation system]]
[[!redirects k-stage factorisation systems]]
[[!redirects n-stage factorization system]]
[[!redirects n-stage factorization systems]]
[[!redirects n-stage factorisation system]]
[[!redirects n-stage factorisation systems]]
[[!redirects alpha-stage factorization system]]
[[!redirects alpha-stage factorization systems]]
[[!redirects alpha-stage factorisation system]]
[[!redirects alpha-stage factorisation systems]]

[[!redirects infinitary factorization system]]
[[!redirects infinitary factorization systems]]


K-cohomology

see at _[[K-theory]]_


K-homology

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

#Contents#
* table of contents
{:toc}

## Idea

[[K-theory]] as a [[generalized homology theory]].

In terms of [[KK-theory]], the $K$-homology of a [[C*-algebra]] $A$ is $KK(A,\mathbb{C})$.

## Presentations

There are various useful ways to present K-homology classes.

### By geometric cycles

See at _[[Baum-Douglas geometric cycle]]_.

### By Fredholm modules and Dirac operators

Let $(X,g)$ be a [[Riemannian manifold]]. Let

$$
  C_\tau(X) \coloneqq \Gamma_0(\wedge^\bullet T^\ast X)
$$

be the algebra of continuous [[sections]] of the [[exterior bundle]] [[vanishing at infinity]], and let

$$
  \mathcal{H} \coloneqq L^2(\wedge^\bullet T^\ast X)
$$

be the space of [[square integrable function|square integrable]] [[sections]] of the [[exterior bundle]]. Write $\mathcal{D} = d + d^\ast$ for the [[Kähler-Dirac operator]] and $\mathcal{F} = \mathcal{D} (1 + \mathcal{D}^2)^{-1/2}$. Then $(\mathcal{H}, \mathcal{F})$ is a [[Fredholm module|Fredholm]]-[[Hilbert module]] which hence represents an element

$$
  [d_X + d^\ast_X] \in KK(C_\tau(X), \mathbb{C})
  \,.
$$

Now assume that $X$ carries a [[spin^c structure]]. Then there exists a [[vector bundle]] $S \to X$ such that $C_\tau(X) = End(S)$ and hence a [[Morita equivalence]] $C_\tau(X) \simeq_{Morita} C_0(X)$ with the algebra of [[continuous functions]] [[vanishing at infinity]].

Let then

$$
  H \coloneqq L^2(S)
$$

and write $D$ for the [[Spin^c Dirac operator]] and $F \coloneqq D (1+ D^2)^{-1/2}$.

Then under the above Morita equivalence these two [[Fredholm module|Fredholm]]-[[Hilbert modules]] represent the same element in K-homology

$$
  [d_X + d^\ast_X] = [D] \in KK(C_0(X), \mathbb{C})
  \,.
$$

## Related concepts

* [[K-cohomology]]

* [[Baum-Douglas geometric cycles]]

## References

The "geometric model" of K-homology (cf. *[[Baum-Douglas geometric cycle]]*) is due to:

* [[Paul Baum]], R. Douglas, *K-homology and index theory*, in: R. Kadison (ed.), *Operator Algebras and Applications, Proceedings of Symposia in Pure Math. **38** AMS (1982) 117-173 [[ams:pspum-38-1](https://bookstore.ams.org/pspum-38-1)]

* [[Paul Baum]], R. Douglas. _Index theory, bordism, and K-homology_, Contemp. Math. **10** (1982) 1-31

Analytic formulation via [[KK-theory]]:

* {#KK} [[Gennady Kasparov]], _Equivariant KK-theory and the Novikov conjecture_, Inventiones Mathematicae, **91** (1988) 147 [[doi:10.1007/BF01404917](https://doi.org/10.1007/BF01404917), [web](http://adsabs.harvard.edu/abs/1988InMat..91..147K)]

An expository account of the analytic model:

* [[Nigel Higson]], [[John Roe]], _Analytic K-homology_, Oxford Mathematical Monographs. Oxford University Press, Oxford (2000) [[ISBN: 9780198511762](https://global.oup.com/academic/product/analytic-k-homology-9780198511762?cc=nl&lang=en&)]

Twisted K-homology:

* [[Christopher Douglas]], _On the Twisted K-Homology of Simple Lie Groups_, Topology 45 (2006), 955-988 ([arXiv:math/0402082](https://arxiv.org/abs/math/0402082))

[[!redirects K-homology theory]]


k-invariant

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

#Contents#
* table of contents
{:toc}

## Idea

The  [[characteristic classes]] that classify the stages of a [[Postnikov tower]] (or dually: [[Whitehead tower]]) are called _Postnikov invariants_ or _k-invariants_.

## Examples

(...)

## References

* Emery Thomas, _Postnikov invariants and higher order cohomology operations_ ([JSTOR](http://www.jstor.org/stable/1970439))

Of [[H-spaces]]:

* Arlettaz, Pointet-Tischler, _Postnikov invariants of H-spaces_ ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm161/fm16112.pdf))

[[!redirects k-invariants]]

[[!redirects k invariant]]
[[!redirects k invariants]]

[[!redirects K-invariant]]
[[!redirects K-invariants]]

[[!redirects K invariant]]
[[!redirects K invariants]]

[[!redirects Postnikov invariant]]
[[!redirects Postnikov invariants]]


K-matrix model

#Contents#
* table of contents
{:toc}

## Idea

Towards a [[matrix model]] for [[D-brane]] ([[D-instanton]]) dynamics in [[string theory]] which takes the [[K-theory classification of D-brane charge]] into acount [[Asakawa, Sugimoto & Terashima (2001)](#AsakawaSugimotoTerashima01)]

## Related concepts

[[!include brane matrix models -- content]]

## References

* {#AsakawaSugimotoTerashima01} [[Tsuguhiko Asakawa]], [[Shigeki Sugimoto]], [[Seiji Terashima]], *D-branes, Matrix Theory and K-homology*, JHEP 0203 (2002) 034 [[arXiv:hep-th/0108085](https://arxiv.org/abs/hep-th/0108085), [doi:10.1088/1126-6708/2002/03/034](https://doi.org/10.1088/1126-6708/2002/03/034)]

[[!redirects K-matrix models]]


k-monoidal infinity-group

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

#Contents#
* table of contents
{:toc}

## Idea

An ordinary [[group]] is either an [[abelian group]] or not. For an [[∞-group]] there is an infinite tower of notions ranging from completely non-abelian to completely abelian.

## Related concepts

[[!include k-monoidal table]]

## References

Section 5 of

* [[Jacob Lurie]], _[[Higher Algebra]]_

[[!redirects k-monoidal ∞-group]]


k-monoidal table

| | [[(∞,1)-operad]] | [[∞-algebra over an (∞,1)-operad|∞-algebra]] | grouplike version | in [[Top]] | generally |
|--|----------------|--------------------------------|----|--------|------|
| | [[A-∞ operad]]  | [[A-∞ algebra]] | [[∞-group]] | [[A-∞ space]], e.g. [[loop space]] | [[loop space object]] |
| | [[E-k operad]] | [[E-k algebra]] | [[k-monoidal ∞-group]] | [[iterated loop space]] | [[iterated loop space object]] |
|  | [[E-∞ operad]] | [[E-∞ algebra]] | [[abelian ∞-group]] | [[E-∞ space]], if grouplike: [[infinite loop space]] $\simeq$ [[∞-space]] | [[infinite loop space object]] |
|  |  |  |  | $\simeq$ [[connective spectrum]] | $\simeq$ [[connective spectrum object]] |
| **[[stabilization]]** |  |  |  |  [[spectrum]] | [[spectrum object]] |

* [[looping and delooping]], [[stabilization hypothesis]]


k-morphism

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###Context###
#### Higher category theory
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[[!include higher category theory - contents]]
=--
=--
=--

# Higher morphisms ($k$-morphisms)
* table of contents
{: toc}

## Idea

In an $n$-[[n-category|category]], or most generally an $\infty$-[[infinity-category|category]], there are many levels of [[morphism]], parametrised by [[natural numbers]].  Those at level $k$ are called __$k$-morphisms__ or __$k$-cells__.

* a [[0-morphism]] is an [[object]]

* a 1-morphism is a [[morphism]]

* next are [[2-morphism]]s

* and so on

## Definition

All notions of higher category have $k$-morphisms, but the [[geometric shape for higher categories|shapes]] may depend on the model (or theory) employed.

For a simplicially based [[geometric model of higher categories]], i.e., simplicial sets subject to some filler conditions, the $k$-morphisms are literally $k$-cells in the sense of a [[simplicial set]]. This applies for example to [[quasi-category|quasi-categories]], weak $n$-categories in the sense of Street, and the weak complicial sets of Verity. In other geometric models, based not on simplices but on other shapes such as [[opetope|opetopes]] (Baez-Dolan), [[multitope|multitopes]] (Hermida-Makkai-Power), or $n$-[[Theta category|disks]] (Joyal), a higher category is a presheaf

$$X: Shapes^{op} \to Set$$

again subject to some filler conditions, and in each case $k$-morphisms are elements of $X(\sigma)$ where $\sigma$ is a shape of dimension $k$. Still other shapes (e.g., [[cubical category|cubes]]) are possible (see also [[n-fold category]]).

Many notions of algebraic higher category, such as those due to Batanin, Leinster, Penon, and Trimble, are algebras over certain monads acting on globular sets (such as those induced by [[globular operad|globular operads]]), so that each higher category $X$ has an underlying [[globular set]] $U(X)$. In that case, the $k$-morphisms are the $k$-cells of $U(X)$. In such globularly based definitions, every $k$-morphism $f$ has a $(k-1)$-morphism $\sigma f$ as its [[source]] and a $(k-1)$-morphism $\tau f$ as its [[target]], and the source $(k-2)$-morphisms $\sigma \sigma f$ and $\sigma \tau f$ must be the same, as must the target $(k-2)$-morphisms $\tau \sigma f$ and $\tau \tau f$.

A $1$-morphism may simply be called a [[morphism]]; a $0$-morphism is an [[object]].

For the purposes of [[negative thinking]], it may be useful to recognise that every $\infty$-category has a $(-1)$-morphism, which is the source and target of every object.  (In the geometric picture, this comes as the $(-1)$-simplex of an [[augmented simplicial set]].)

+-- {: .query}
>every (non-empty? -David R) $\infty$-category

I think every.  Up to equivalence, a $k$-morphism in $C$ is given by a functor from the oriented $k$-simplex to $C$.  As the $(-1)$-simplex is empty, there is a unique such functor for every $C$; thus every $C$ has a unique $(-1)$-morphism.

Also note that every $k$-morphism has $k + 1$ identity $(k+1)$-morphisms, which just happen to all be the same (which can be made a result of the [[Eckmann–Hilton argument]]).  Thus, the $(-1)$-morphism has $0$ identity $0$-morphisms, so we don\'t need any object.  (This confused me once.)

---[[Toby Bartels]]
=--

## Related concepts

* [[object]], [[0-morphism]]

* [[morphism]]

* [[2-morphism]]

* [[3-morphism]]

* **k-morphism**

[[!redirects higher morphism]]
[[!redirects higher morphisms]]
[[!redirects k-morphism]]
[[!redirects k-morphisms]]
[[!redirects j-morphism]]
[[!redirects j-morphisms]]
[[!redirects n-morphism]]
[[!redirects n-morphisms]]
[[!redirects m-morphism]]
[[!redirects m-morphisms]]
[[!redirects r-morphism]]
[[!redirects r-morphisms]]
[[!redirects s-morphism]]
[[!redirects s-morphisms]]


K-motive

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###Context###
#### Motivic cohomology
+--{: .hide}
[[!include motivic cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The definition of [[pure motive|pure]]/[[mixed motive|mixed]] [[motive]] as a [[correspondence]] (of suitable [[smooth schemes]]) equipped with an [[algebraic cycle]] on its tip may be generalized to that of a correspondence carrying instead a cycle in [[algebraic K-theory]] ([Gillet-Soulé 96](#GilletSoule96), [Gillet-Soulé 09](#GilletSoule09)).

These might be called _Gillet-Soulé motives_ ([Tabuada 13, section 5.3](#Tabuada13)) or just _K-motives_.

Via the algebraic [[Chern character]] these K-motives map to rational pure [[Chow motives]]  ([Tabuada 13, p. 9](#Tabuada13)). Moreover, they faithfully embed into [[noncommutative motives]] ([Tabuada 13, p. 10](#Tabuada13)).

## Related concepts

* [[bivariant algebraic K-theory]]

* [[KK-theory]]

## References

* {#GilletSoule96} Henri Gillet and Christophe Soulé, _Descent, motives and K-theory_, J. Reine Angew. Math. 478 (1996), 127–176. ([arXiv:9507013](http://arxiv.org/abs/alg-geom/9507013))

* {#GilletSoule09} Henri Gillet and Christophe Soulé, _Motivic weight complexes for arithmetic varieties_ J. Algebra 322 (2009), no. 9, 3088–3141.

* {#Tabuada13} [[Gonçalo Tabuada]], _Chow motives versus noncommutative motives_, Journal of Noncommutative Geometry __7__:3, 2013, pp. 767–786, [arXiv:1103.0200](http://arxiv.org/abs/1103.0200), [doi](http://dx.doi.org/10.4171/JNCG/134)

* [[Grigory Garkusha]], Ivan Panin, _K-motives of algebraic varieties_, [arxiv/1108.0375](http://arxiv.org/abs/1108.0375); _The triangulated category of K-motives $DK(k)$_, [arxiv/1302.2163](http://arxiv.org/abs/1302.2163)

[[!redirects K-motives]]


K-orientation

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

#Contents#
* table of contents
{:toc}

## Idea

A _K-orientation_ is an [[orientation in generalized cohomology]] for [[K-theory]] (typically: [[topological K-theory]]).

## Definition

### In topological K-theory

#### K-Orientation of oriented manifolds

For ordinary [[topological K-theory]] a [[smooth function]] between [[smooth manifolds]] $f \colon X \to Y$ is K-oriented if $T X \oplus f^\ast(T Y)$ has a [[spin^c structure]].

In this case there is an [[Umkehr map]]

$$
  f_! \colon K^\bullet(X) \to K^{\bullet + dim(Y) - dim(X)}(Y)
  \,.
$$

#### The universal Atiyah-Bott-Shapiro orientation
 {#UniversalAtiyahBottShapiroOrientation}

There is a universal [[orientation in generalized cohomology]] of

* [[complex K-theory]]$\,$[[KU]] over [[spin^c structures]];

* [[KO]] over [[spin structure]]

hence [[E-infinity ring]] homomorphisms out of the [[Thom spectrum]] [[MSpin]]

 [[MSpin]] $\longrightarrow$ [[KO]]

and

[[MSpin^c|MSpinc]] $\longrightarrow$ [[KU]]

These are (both) referred to as the _Atiyah-Bott-Shapiro orientation_ (after [Atiyah-Bott-Shapiro 64](#AtiyahBottShapiro64)); the $E_\infty$-structure is due to ([Joachim 04](#Joachim04)).

The [[genus]] induced by $M Spin \to KO$ is the [[A-hat genus]], that induced by $M Spin^c \to KU$ is the [[Todd genus]].

{#OfCourse} Of course, $KU$ is also a [[complex oriented cohomology theory]] and as such canonically comes with a [[homomorphism]] of [[homotopy-commutative ring spectra]] $MU \to KU$ ([this general Prop.](complex+oriented+cohomology+theory#ComplexOrientationIsMUAlgebraStructure), see at [spin^c structure -- from almost complex structure](spin%5Ec+structure#FromAlmostComplexStructures)).

On this level there is also the [[real oriented cohomology|real orientation]] of [[real K-theory]] given by a map

$$
  M \mathbb{R} \longrightarrow K \mathbb{R}
$$

from [[MR-theory]] ([Kriz 01, (3.20)](#Kriz01)).

#### The universal Conner-Floyd orientation
 {#ConnerFloydOrientation}

There are multiplicative [[natural transformations]] of [[multiplicative cohomology theories]], hence homomorphisms of [[homotopy-commutative ring spectra]]

[[MU]] $\longrightarrow$ [[KU]]

and

[[MSU]] $\longrightarrow$ [[KO]]

which are compatible in that the [[diagram]] formed by the evident vertical morphisms

\begin{xymatrix@C=15pt}
  M S\mathrm{U}
  \ar[rr]
  \ar[d]
  &&
  K\mathrm{O}
  \ar[d]
  \\
  M \mathrm{U}
  \ar[rr]
  &&
  K \mathrm{U}
\end{xymatrix}

[[commuting diagram|commutes]].

This _[[Conner-Floyd orientation]]_ is originally due to [Conner-Floyd 66, Section 5](#ConnerFloyd66).

The morphism [[MU]]$\to$[[KU]] reflects the [[complex oriented cohomology theory|complex orientation]] of [[KU]] via the [[universal complex orientation on MU]].

### In KK-theory

In the context of [[KK-theory]], a morphism $f \colon A \to B$ of [[C*-algebra]], a _K-orientation_ of this morphism is a map

$$
  f! \in KK_d(B,A)
$$

such that (...). The corresponding [[fiber integration]]/[[Gysin map]] on [[operator K-theory]] is then the postcomposition operation

$$
  f_! \colon K_\bullet(B) \simeq KK_\bullet(\mathbb{C}, B) \stackrel{f! \circ (-)}{\to} KK_\bullet(\mathbb{C}, A) \simeq K_{\bullet + d}(A)
  \,.
$$

([BMRS 07](#BMRS07))

## Properties

### As the relation between cobordisms cohomology and K-theory

The Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$ induces an equivalence

$$
  M Spin^c_\bullet(-)\otimes_{M Spin^c_\bullet} KU_\bullet \simeq KU_\bullet(-)
$$

and $M Spin \to KO$ similarly induces

$$
  M Spin_\bullet(-)\otimes_{M Spin_\bullet} KO_\bullet \simeq KO_\bullet(-)
$$

This is due to ([Hopkins-Hovey 92](#HopkinsHovey92)), a variation of the [[Conner-Floyd isomorphism]]. See at _[[cobordism theory determining homology theory]]_ for more.

## Related concepts

* [[spin^c structure]]

* [[Conner-Floyd isomorphism]], [[cobordism theory determining homology theory]]

* [[Grothendieck-Riemann-Roch theorem]]

[[!include genera and partition functions - table]]

## References

### In topological K-theory

The [[MSpin]]/[[MSpin^c|MSpinc]]-orientation of [[KO]]/[[KU]] [[topological K-theory]] is attributed to

* {#AtiyahBottShapiro64} [[Michael Atiyah]], [[Raoul Bott]], [[Arnold Shapiro]], _Clifford modules_, Topology Volume 3, Supplement 1, July 1964, Pages 3-38 (doi:10.1016/0040-9383(64)90003-5, [pdf](http://dell5.ma.utexas.edu/users/dafr/Index/ABS.pdf))

The [[MSU]]/[[MU]]-orientation of [[KO]]/[[KU]] ([[Conner-Floyd orientation]]) is originally due to

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Section 5 of: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

That ABS-orientation $M Spin^c\to KU$  extendes to a homomorphism of [[E-infinity rings]] is due to

* {#Joachim04} [[Michael Joachim]], _Higher coherences for equivariant K-theory_, in _Structured ring spectra_, volume 315 of London Math. Soc. Lecture Note Ser., pages 87–114. Cambridge Univ. Press, Cambridge, 2004 ([pdf](http://mat.uab.es/~aguade/bcat02/slides/joachim.pdf))

Discussion for [[real K-theory]] includes

* {#Kriz01} [[Igor Kriz]], _Real-oriented homotopy theory and an analogue of the
Adams-Novikov spectral sequence_, Topology 40 (2001) 317-399 ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/hukriz.pdf))

The discussion of [[cobordism theory determining homology theory]] for the K-orientation is due to

* {#HopkinsHovey92} [[Michael Hopkins]], [[Mark Hovey]], _Spin cobordism determines real K-theory_, Mathematische Zeitschrift 210.1 (1992): 181-196. ([[HopkinsHoveyCobordismK.pdf:file]])

### In KK-theory

Discussion in [[noncommutative topology]]/[[KK-theory]] is in

* [[Alain Connes]], , [[Georges Skandalis]], _The longitudinal index theorem for foliations_. Publ. Res. Inst. Math. Sci. 20,
no. 6, 1139–1183 (1984)
 {#ConnesSkandalis84}

* Jacek Brodzki, [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], _Noncommutative correspondences, duality and D-branes in bivariant K-theory_, Adv. Theor. Math. Phys.13:497-552,2009 ([arXiv:0708.2648](http://arxiv.org/abs/0708.2648))
 {#BMRS07}

[[!redirects K-orientations]]

[[!redirects KU-orientation]]
[[!redirects KU-orientations]]

[[!redirects orientation in K-theory]]
[[!redirects orientations in K-theory]]

[[!redirects K-theory orientation]]
[[!redirects K-theory orientations]]

[[!redirects Conner-Floyd orientation]]
[[!redirects Conner-Floyd orientations]]

[[!redirects Atiyah-Bott-Shapiro orientation]]
[[!redirects Atiyah-Bott-Shapiro orientations]]

[[!redirects Spin orientation of KO]]
[[!redirects Spin^c orientation of KU]]


k-ring of functions on a k-functor

> see _[[ring of functions]]_


k-simply connected n-category

# $k$-simply connected $n$-categories
* table of contents
{: toc}

## Definition

An $n$-[[n-category|category]] is __$k$-simply connected__ (or just __$k$-connected__) if any two parallel $j$-morphisms are [[equivalence|equivalent]] for $j \leq k$.  You may interpret this definition as weakly or strictly as you like, by starting with weak or strict notions of $n$-category.  Note that we include the case $j = -1$ to mean that the $n$-category is [[inhabited set|inhabited]] when $k \geq -1$.

## Special cases

Thus:

*  A $(-2)$-simply connected $n$-category is simply an $n$-category;
*  A $(-1)$-simply connected $n$-category is an [[inhabited set|inhabited]] $n$-category.
*  A $0$-simply connected $n$-category is a [[connected category|connected]] $n$-category: an inhabited $n$-category in which all objects are equivalent;
*  A $1$-simply connected $n$-category is a [[simply connected category|simply connected]] $n$-category: an inhabited $n$-category in which all objects and morphisms are equivalent;
*  etc.

The [[delooping hypothesis]] says that a $k$-[[k-tuply monoidal n-category|tuply monoidal]] $n$-category is the same thing as a [[pointed object|pointed]] $(k-1)$-simply connected $(n+k)$-category; more generally, a $(j+k)$-tuply monoidal, $i$-simply connected $n$-category is the same as a $j$-tuply monoidal, $(i+k)$-simply connected $(n+k)$-category.

One might want a stricter definition for $n$-categories, but this is certainly correct for $n$-[[n-groupoid|groupoid]]s.  Indeed, we can say that an $\infty$-[[infinity-groupoid|groupoid]] $X$ is $k$-simply connected if and only if its [[fundamental n-groupoid]] $\Pi_n(X)$ is trivial for $n \leq k$.  In particular, an $\infty$-simply connected $\infty$-groupoid is [[contractible space|contractible]].

## Discussion

This discussion on terminology occurred when the page was at [[k-tuply connected n-category]].

+-- {: .query}
[[Mike Shulman|Mike]]: I'm pretty sure that in algebraic topology, 1-connected means simply connected, and 0-connected means connected.  So your definitions make a space 0-connected when its fundamental $\infty$-groupoid is 1-tuply connected.  I would prefer that we retain the topologists' numbering and call this a 0-connected $\infty$-groupoid (the 'tuply' sounds weird to me for connectedness), with the off-by-one shift happening in the delooping: the $k$-fold delooping of a $k$-tuply monoidal $n$-category would be a $(k-1)$-connected $(n+k)$-category.

_Toby_:  If the topologists have a system for this, then we should probably use it, even if it is a poor system.  (I don\'t suppose that any topologists say '$1$-simply connected' instead?)  The previous usage on the Lab was inconsistent, although I didn\'t check whether that inconsistency was all my fault.

In general, a $1$-foo should always be the same as a foo, as with $1$-category, $1$-poset, $1$-groupoid, $1$-group, $1$-stuff, etc.  Trying to match the 'right' dimension isn\'t going to work consistently and runs into too many other existing terms.  I\'m only sorry that the topologists did it otherwise here.

(Not to mention that most homotopy-theoretic dimension counting is off by $1$ anyway.  But that\'s another topic, indeed a topic that always making a $1$-foo a foo will safely avoid.)

I have no opinion about 'tuply'; it just came naturally to me since I was thinking about $k$-tuple monoidality.

[[Mike Shulman|Mike]]: Yeah, I agree that in general it is better if a 1-foo is the same as an unadorned foo, but not everyone has adhered to that.  [Wikipedia](http://en.wikipedia.org/wiki/N-connected) agrees with me about the meaning of $k$-connected in topology.

_Toby_:  All right, I\'ll change the numbering and move the page.  But what do you think about '$k$-simply connected'?  This doesn\'t seem to have any conflicts, although search for '$m$-simply connected' to see what might be potential conflicts ---or might actually agree with me (I haven\'t been able to check).
=--

[[!redirects k-simply connected n-category]]
[[!redirects k-connected n-category]]
[[!redirects k-tuply connected n-category]]


k-surjective functor

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

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

## Idea

The notion of _$k$-surjective functor_ is the continuation of the sequence of notions

* [[essentially surjective functor]]

* essentially surjective and [[full functor]]

* essentially surjective and [[full and faithful functor]]

from [[category theory]] to an infinite sequence of notions in [[higher category theory]].

Roughly, a functor $F : C \to D$ between [[∞-categories]] $C$ and $D$ is _$k$-surjective_ if for each boundary of a [[k-morphism]]s in $C$, each $k$-morphism between the image of that boundary in $D$ is in the image of $F$.

## Generalization to $\infty$-categories

### $k$-Surjectivity

For the moment, this here describes the notion for _globular_ models of $\infty$-categories. See below for the simplicial reformulation.

An $\omega$-functor $f : C \to D$ between $\infty$-[[infinity-category|categories]] is
0-surjective if $f_0 : C_0 \to D_0$ is an epimorphism.

For $k \in \mathbb{N}$, $k \geq 1$ the functor is _$k$-surjective_
if the universal morphism
$$
  C_k \to P_k
$$
to the pullback $P_k$ in
$$
  \array{
     P_k
     &\to&
     D_k
     \\
     \downarrow && \downarrow^{s \times t}
     \\
     C_{k-1} \times C_{k-1}
     &\stackrel{F_{k-1} \times F_{k-1}}{\to}&
     D_{k-1} \times D_{k-1}
  }
$$
coming from the commutativity of the square
$$
  \array{
     C_k
     &\stackrel{f_{k}}{\to}&
     D_k
     \\
     \downarrow^{s \times t} && \downarrow^{s \times t}
     \\
     C_{k-1} \times C_{k-1}
     &\stackrel{F_{k-1} \times F_{k-1}}{\to}&
     D_{k-1} \times D_{k-1}
  }
$$
(which commutes due to the functoriality axioms of $f$) is an [[epimorphism]].

If you interpret $C_k$ and $P_k$ as sets and take 'epimorphism' in a strict sense (the sense in [[Set]], a [[surjection]]), then you have a __strictly $k$-surjective functor__.  But if you interpret $C_k$ and $P_k$ as $\infty$-categories or $\infty$-groupoids and take 'epimorphism' in a weak sense (the [[homotopy limit|homotopy]] sense from $\infty$-[[infinity-Grpd|Grpd]]), then you have an __essentially $k$-surjective functor__; equivalently, project $C_k$ and $P_k$ to $\omega$-[[equivalence]]-classes before testing surjectivity.  A functor is essentially $k$-surjective if and only if it is equivalent to some strictly $k$-surjective functor, so essential $k$-surjectivity is the notion that respects the [[principle of equivalence]].

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

For $C$ and $D$ [[categories]] we have

 1. $f$ is (essentially) $0$-surjective $\Leftrightarrow$
    $f$ is [[essentially surjective functor|(essentially) surjective on objects]];
 2. $f$ is (essentially) $1$-surjective $\Leftrightarrow$ $f$ is [[full functor|full]];
 3. $f$ is (essentially) $2$-surjective $\Leftrightarrow$ $f$ is [[faithful functor|faithful]];
 4. $f$ is always $3$-surjective.
=--

### In terms of lifting diagrams {#Lifting}

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

An $\omega$-functor $f : C \to D$ is $k$-surjective for $k \in \mathbb{N}$ precisely if it has the right [[weak factorization system|lifting property]] with respect to the inclusion $\partial G_{k} \to G_k$ of the boundary of the $k$-[[globe]] into the $k$-[[globe]].

$$
  \array{
    \partial G_k &\to& C
    \\
    \downarrow &{}^{\exists}\nearrow& \downarrow^f
    \\
    G_k &\to& D
  }
  \,.
$$
=--

One recognizes the similarity to situation for [[geometric definition of higher category]]. A morphism $f : C \to D$ of [[simplicial set]]s is an acyclic fibration with respect to the [[model structure on simplicial sets]] if it all diagrams

$$
  \array{
    \partial \Delta[k] &\to& C
    \\
    \downarrow && \downarrow^f
    \\
    \Delta[k] &\to& D
  }
$$

have a lift

$$
  \array{
    \partial \Delta[k] &\to& C
    \\
    \downarrow &\nearrow& \downarrow^f
    \\
    \Delta[k] &\to& D
  }
$$

for all $k$, where now $\Delta[k]$ is the $k$-[[simplex]].

### Weak equivalences, acyclic fibrations and hypercovers##

With respect to the [[folk model structure]]  on $\omega$-categories an $\omega$-functor is

* an [[acyclic fibration]] if it is $k$-surjective for all $k \in \mathbb{N}$;

* a [[weak equivalence]] if it is essentially $k$-surjective for all $k \in \mathbb{N}$.
See also [[equivalence of categories]].

###Remarks

All this has close analogs in other models of higher structures, in particular in the context of simplicial sets: an acyclic fibration in the standard [[model structure on simplicial sets]] is a morphism $X \to Y$ for which all diagrams

$$
  \array{
    \partial \Delta[n] &\to& X
    \\
    \downarrow && \downarrow
    \\
    \Delta[n] &\to& Y
  }
$$

have a lift

$$
  \array{
    \partial \Delta[n] &\to& X
    \\
    \downarrow &\nearrow& \downarrow
    \\
    \Delta[n] &\to& Y
  }
  \,.
$$

This is precisely in simplicial language the condition formulated above in globular language.

## Literature

The general idea of $k$-surjectivity is described around [definition 4](http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=17) of

* Baez-Shulman, [[Lectures on n-Categories and Cohomology]]
([arXiv](http://arxiv.org/abs/math.CT/0608420))

The concrete discussion in the context of [[strict omega-category|strict omega-categories]] is in

* Yves Lafont, Francois Métayer, Krzysztof Worytkiewicz, _A folk model structure on $\omega$-cat_ ([arXiv](http://arxiv.org/abs/0712.0617)).

For the analogous discussion for simplicial sets see

* [[model structure on simplicial sets]]

and references given there.

[[!redirects k-surjectivity]]


k-surjectivity > history

< [[k-surjectivity]]
[[!redirects k-surjectivity -- history]]


K-theory

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> The way I first visualized a K-group was as a group of "classes of objects" of an abelian (or more generally, additive) category, such as coherent sheaves on an algebraic variety, or vector bundles, etc. I would presumably have called this group $C(X)$ ($X$ being a variety or any other kind of "space"), $C$ the initial letter of 'class', but my past in functional analysis may have prevented this, as $C(X)$ designates also the space of continuous functions on $X$ (when $X$ is a topological space). Thus, I reverted to $K$ instead of $C$, since my mother tongue is German, Class = Klasse (in German), and the sounds corresponding to $C$ and $K$ are the same.

> ([[Alexander Grothendieck]], [letter to Magurn, 1985](Alexander+Grothendieck#Bak87))

## Idea ##

Given a [[stable (∞,1)-category]] $C$, its naive [[decategorification]] (its set of objects modulo the relation of [[equivalence]]) naturally inherits the structure of an abelian [[monoid]] from the [[biproducts]] in $C$.  That is, we define $[a]+[b] = [a\oplus b]$.  We can then further pass to the [[Grothendieck group]], adding formal additive inverses to get an abelian [[group]].

Note that such a biproduct sits in a [[split exact sequence|split]] (co)[[fibration sequence]] $a\to a\oplus b \to b$.  Often we would like to obtain an analogous additivity result for *arbitrary* (co)fibration sequences (a.k.a. [[distinguished triangles]]).  This requires imposing the [[generators and relations|relation]] $[c] = [a] + [b]$ for any fibration sequence $a\to c\to b$, thereby passing to a quotient abelian group called $K_0(C)$, the **K-group** or [[Grothendieck group]] of $C$; see the latter entry for more details.

The "K" is chosen by Grothendieck for the German word _Klasse_ for "class". The K-group of $C$ is the group of equivalence classes of $C$: it is a group due to the existence of a notion of exact sequences in $C$.

K-theory starts with the study of these K-groups and their higher analogues $K_n(C)$, collectively denoted $K(C)$. Sometimes the K-groups themselves are called "K-theory". One would say for instance: "$K(C)$ is the K-theory of $C$."

More generally, there is a [[symmetric monoidal (infinity,1)-category|symmetric groupal ∞-groupoid]] $\mathbf{K}(C)$  -- i.e. a connective [[spectrum]] -- in between the [[decategorification]] from $C$ to $K(C)$, for which $K_0(C)$ is the set of [[simplicial homotopy group|connected components]]

$$
  C \mapsto \mathbf{K}(C) \to \pi_0 \mathbf{K}(C) = K_0(C)
  \,.
$$

and more generally $K_n(C) = \pi_n \mathbf{K}(C)$.

In nice cases this is the degree 0 part of a non-connective [[spectrum]] which is then the **K-theory spectrum** of $C$.
This is also called the **Waldhausen K-theory** of $C$.

## Special cases and models ##

Much of the literature on K-theory discusses constructions that _model_ the above abstract setup in terms of [[model category|model categories]], or just their [[homotopy category|homotopy categories]], often of the [[derived category|derived catgeories]] type and then often expressed in terms of the [[abelian category]] or more generally [[Quillen exact category]] from which the derived category is derived.

Only a subset of the structure on a [[model category]] is necessary in order to conveniently extract the K-groups of the [[presentable (infinity,1)-category|presented]] [[stable (∞,1)-category]]. For that reason the axioms of a [[Waldhausen category]] have been devised to provide just the necessary convenient prerequisites to compute the K-groups of the [[(∞,1)-category]] [[presentable (infinity,1)-category|presented]] by the underlying [[homotopical category]].

* In particular, the K-group associated to the [[stable (∞,1)-category]] $Ch^b(A)$ of _bounded_ [[chain complex]]es in an [[abelian category]] or [[exact category]] $A$ is often called the K-group of $A$ itself and just denoted

  $$
    K(A) := K(Ch^b(A))
    \,.
  $$

  Most explicit constructions of K-theory spectra start with the data of an [[exact category]], such as notably Quillen's [[Q-construction]] and the [[Waldhausen S-construction]].

* In particular if the exact category $A$ is that of [[vector bundle]]s on a [[topological space]] $X$

  $$
    A = VectBund(X)
  $$

  the corresponding K-group is degree 0 [[topological K-theory]]. This was the original of the notion and the term K-theory.

## Definition ##

Recall that given a [[(∞,1)-category]] $C$, we may regard it as a [[complete Segal space]] $C_{\bullet,\bullet}$, a [[bisimplicial object|bisimplicial set]]. For instance if $C$ is originally given as a [[quasicategory]] then

$$
  C_{\bullet,\bullet} : [n],[m] \mapsto
  Core(Func(\Delta^n,C))_{m}
  \,,
$$

where $Core(Func(\Delta^n,C))$ denotes the maximal [[Kan complex]] inside the [[(∞,1)-category of (∞,1)-functors]] from $\Delta^n$ to $C$.

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

For $\mathcal{C}$ an [[(∞,1)-category]] and $n \in \mathbb{N}$, write $Gap(\Delta^n, \mathcal{C})$ for the full sub-$\infty$-category on $Func(Arr(\Delta^n),\mathcal{C} )$ on those objects $F$ for which

* the [[diagonal]] $F(n,n)$ is inhabited by [[zero objects]], for all $n$;

* all [[diagrams]] of the form

  $$
    \array{
      F(i,j) &\to& F(i,k)
      \\
      \downarrow && \downarrow
      \\
      F(j,j) &\to& F(j,k)
    }
  $$

  is an [[(∞,1)-pushout]].

=--

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

Let $C$ be a [[stable (∞,1)-category]]. Then its **Waldhausen K-theory**

$$
  \mathbf{K}(C) := \underset{\rightarrow}{\lim}_n Core(Gap(C^{\Delta^n}))
$$

is the [[geometric realization of simplicial topological spaces|geometric realization]] of/[[homotopy colimit]] of the degreewise [[core]] of the  $Gap$, def. \ref{Gap},  of the corresponding [[complete Segal space]] (as a simplicial diagram of $\infty$-groupoids).

=--

This is [remark 11.4](http://arxiv.org/PS_cache/math/pdf/0608/0608228v5.pdf#page=43) in [[stable (infinity,1)-category|StCat]]. See also [Blumberg-Gepner-Tabuada, section 7](#BGT).

This construction is also conjectured in the last section of Toen-Vezzosi's _A remark on K-theory_ .

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

In the case that $C$ is the [[simplicial localization]] of a [[Waldhausen category]] $\bar C$ the explicit way to obtain this is the [[Waldhausen S-construction]].

=--

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

It should be true that with this definition we have an isomorphism of groups

$$
  K(C) \simeq \pi_0 \mathbf{K}(C)
  \,.
$$

=--

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

This Waldhausen/hocolim-construction gives the _connective_ K-theory, taking values in [[connective spectra]]. The [[universal construction|universal]] completion to functor that sends [[homotopy cofibers]] of [[stable (infinity,1)-categories]] to homotopy cofibers of spectra is the corresponding unconnective $\mathbb{K}$-functor.

=--

There is a universal characterization of the construction of the $\mathbb{K}$-theory spectrum of a stable $(\infty,1)$-category $A$:

there is an $(\infty,1)$-functor

$$
  U : (\infty,1)StabCat \to N
$$

to a stable $(\infty,1)$-category which is universal with the property that it respects colimits and exact sequences in a suitable way. Given any stable $(\infty,1)$-category $A$, its (connective or non-connective, depending on details) algebraic K-theory spectrum is the hom-object

$$
  K(A) \simeq Hom(U(Sp), U(A))
  \,,
$$

where $Sp$ denotes the stable $(\infty,1)$-category of compact [[spectra]]. ([BGT](#BGT))

## Related entries ##

* [[Grothendieck group]]

* [[K-theory spectrum]]

* [[topological K-theory]]

  * [[fiber integration in K-theory]]

* [[Tate K-theory]]

* [[K-orientation]]

* [[algebraic K-theory]]

  * [[K-theory of a permutative category]]

  * [[K-theory of a bipermutative category]]

  * [[differential algebraic K-theory]]

  * [[noncommutative motive]]

* [[Waldhausen K-theory of a dg-category]]

* [[theorem of the heart]]

* [[groupoid K-theory]]

* [[operator K-theory]]

  * [[equivariant operator K-theory]]

  * [[KK-theory]]

  * [[E-theory]]

* [[equivariant K-theory]]

* [[Karoubi K-theory]]

* [[Morava K-theory]]

* [[L-theory]]

* [[twisted K-theory]]

* [[differential K-theory]]

* [[K-theory and physics]]

## References ##

It was in

* [[Bertrand Toen]], Gabriele Vezzosi, _A remark on K-theory and $S$-categories_ ([arXiv](http://arxiv.org/PS_cache/math/pdf/0210/0210125v2.pdf)).

that it was proven that the the [[Waldhausen S-construction]] of the [[K-theory spectrum]] depends precisely on the [[simplicial localization]] of the [[Waldhausen category]], i.e. of the [[(∞,1)-category]] that it presents.

In view of this remark 11.4 in

* [[Jacob Lurie]], [[stable (infinity,1)-category|Stable ∞-Categories]] .

interprets the construction of the K-theory spectrum as a natural operation of
[[stable (infinity,1)-category|stable (∞,1)-categories]], as described above.

The universal property of the $(\infty,1)$-categorical definition is studied in

* {#BGT} [[Andrew Blumberg]], [[David Gepner]], [[Goncalo Tabuada]], _A universal characterization of higher algebraic K-theory_ ([arXiv:1001.2282](http://arxiv.org/abs/1001.2282)).

The standard constructions of K-theory spectra from [[Quillen exact categories]] are discussed in detail in chapter 1 of

* [[Eric Friedlander]], [[Daniel Grayson]], _Handbook of K-theory_, Springer Verlag .

A useful introduction to the definition and computation of K-groups (with a little on K-spectra) is

* Charles Weibel, _The K-book: An introduction to algebraic K-theory_ ([web](http://www.math.rutgers.edu/~weibel/Kbook.html))

[[!redirects Waldhausen K-theory]]


K-theory and physics

There are many applications of [[K-theory]] to [[physics]], including old ones. However, more recently (late 1990s) it has been realized that some version of [[K-theory]] classifies [[RR-field|D-brane charges]].

See  wapedia ,  wikipedia.


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#### Solid state physics
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#### Cohomology
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#Contents#
* table of contents
{:toc}

## Idea

The [[phases of matter|phases]] of [[condensed matter physics|quantum materials]] with [[gapped Hamiltonian]] admit a
"[[topology|topological]]" (really: [[homotopy theory|homotopy theoretical]]) classification by [[twisted K-theory|twisted]] [[equivariant K-theory|equivariant]] [[topological K-theory]], whence one speaks of _[[topological phases of matter]]_, _[[topological states of matter]]_, _[[topological order]]_, _[[topological insulators]]_, and the like.

## Properties

### Holographic relation to K-theory classification of D-brane charge

Under [[AdS/CFT duality in solid state physics]] the [[K-theory classification of topological phases of matter]] corresponds to the [[K-theory classification of D-brane charge]] ([Ryu-Takayanagi 10a](#RyuTakayanagi10a), [Ryu-Takayanagi 10v](#RyuTakayanagi10b))

## Related concepts

* [[K-theory classification of D-brane charge]]

## References

[[!include topological phases of matter via K-theory -- references]]

[[!redirects K-theory classification of topological phases]]


K-Theory for Operator Algebras

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This entry collects links for the book

* [[Bruce Blackadar]],

  _K-Theory for Operator Algebras_

  Cambridge University Press 1986, second ed. 1999

  [doi:10.1007/978-1-4613-9572-0](https://link.springer.com/book/10.1007/978-1-4613-9572-0)

  [pdf](http://wolfweb.unr.edu/homepage/bruceb/Book6.pdf)

on [[operator algebra]], [[operator K-theory]] and [[KK-theory]].

#Contents#
* table of contents
{:toc}

## Chapter I. Introduction to K-theory

### 1. Survey of topological K-theory

* [[topological K-theory]]

### 2. Overview of operator K-theory

* [[operator K-theory]]

## Chapter II. Preliminaries

### 3. Local Banach algebras and inductive limits

* [[Banach algebra]]

### 4. Idempotents and Equivalence

* [[idempotent]]

## Chapter III. $K_0$-theory and order

### 5. Basic $K_0$-theory

(...)

### 6. Order structure on $K_0$

(...)

### 7. Theore of AF algebras

(...)

## Chapter IV. $K_1$-theory and Bott periodicity

### 8. Higher K-groups

(...)

### 9. Bott periodictiy

* [[Bott periodicity]]

## Chapter V. $K$-theory of crossed products

### 10. The Pimsner-Voiculescu exact sequence and Connes' Thom isomorphism

* [[crossed product C*-algebra]]

* [[Thom isomorphism]]

### 11. Equivariant K-theory

* [[equivariant K-theory]]

## Chapter VI. More preliminaries

### 12. Multiplier algebras

* [[multiplier algebra]]

### 13. Hilbert modules

* [[Hilbert module]]

### 14. Graded $C^\ast$-algebra

* [[graded C*-algebra]]

## Chapter VII. Theory of extensions

### 15. Basic theory of extensions

(...)

### 16. Brown-Douglas-Fillmore theory and other applications

(...)

## Chapter VIII. KK-theory

### 17. Basic theory

* [[KK-theory]]

### 18. The intersection product

(...)

### 19. Further structure in KK-theory

(...)

### 20. Equivariant KK-theory

* [[equivariant KK-theory]]

## Chapter IX. Further topics

### 21. Homology and Cohomology Theories on $C^\ast$-algebras

(...)

### 22. Axiomatic K-theory

* [[bootstrap category]]

### 23. Universal coefficient theorems and Künneth theorems

* [[universal coefficient theorem]]

* [[Künneth theorem]]

### 24. Survey of applications to geometry and topology

(...)

### 25. E-theory

* [[E-theory]]

category: reference


K-theory of a bipermutative category

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## Idea

While a [[permutative category]] already induces a [[symmetric spectrum]], [[Brown representability theorem|representing]] the corresponding [[K-theory of a permutative category]], for a [[bipermutative category]] this naturally carries the additional structure of an [[E-infinity ring spectrum]]. The [[generalized (Eilenberg-Steenrod) cohomology]] [[Brown representability theorem|represented]] by this spectrum is called the _([[algebraic K-theory|algebraic]]) [[K-theory]]_ of $C$ and it is hence a [[multiplicative cohomology theory]].

## Related concepts

* [[K-theory of a permutative category]]

* [[K-theory of a symmetric monoidal (∞,1)-category]]

## References

* [[Peter May]], _$E_\infty$ Ring Spaces and $E_\infty$ Ring spectra_, Springer lectures notes in mathematics, Vol. 533, (1977) ([pdf](http://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf)) chaper VI

* [[Anthony Elmendorf]], [[Michael Mandell]], _Rings, modules and algebras in infinite loop space theory_, K-Theory 0680 ([web](http://www.math.uiuc.edu/K-theory/0680/), [pdf](http://www.math.uiuc.edu/K-theory/0680/RMAsubmit.pdf))

* [[Anthony Elmendorf]], [[Michael Mandell]], _Permutative categories, multicategories, and algebraic K-theory_ ([arXiv:0710.0082](http://arxiv.org/abs/0710.0082))

[[!redirects K-theories of bipermutative categories]]