univalent dagger category

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

#Contents#
* table of contents
{:toc}

## Idea ##

When working in [[intensional type theory]] such as [[homotopy type theory]], the notion of [[category]] has both a [[precategory]] version and a [[univalent category]] version.  The same is true for other categorical structures, in particular for [[dagger-categories]] we have $\dagger$-precategories and univalent $\dagger$-categories.

## Definition ##

+--{: .un_defn}
###### Definition
A __$\dagger$-precategory__ is defined in type theory using the "family of functions" definition of [[dagger-category]].  Thus, it consists of

* A [[precategory]] $C$, with type of objects $Ob(C)$ and hom-sets $Hom : Ob(C) \to Ob(C) \to Set$.

* For all $A,B:Ob(C)$, a function $(-)^\dagger : Hom(A,B) \to Hom(B,A)$

* For every $A \in Ob(C)$, $(1_A)^\dagger = 1_A$.

* For every $A,B \in Ob(C)$ and every $f \in Hom_C(A,B)$ and $g \in Hom_C(B,C)$, $(g \circ f)^\dagger = f^\dagger \circ g^\dagger$.

* For every $A,B \in Ob(C)$ and every $f \in Hom_C(A,B)$, $((f)^\dagger)^\dagger = f$.
=--

Note that as for any precategory, there is no [[h-level]] restriction on the type $Ob(C)$.  As usual in a $\dagger$-category, we define an isomorphism in a $\dagger$-precategory to be **unitary** if $f^{-1} = f^\dagger$.

Let $(a \cong^\dagger b)$ denote the type of unitary isomorphisms.  The identity morphism is unitary, so there is a function
$$idtouiso : (a=b) \to (a \cong^\dagger b)$$
defined by [induction on identity](identity+type#InTermsOfInductiveTypes).

+--{: .un_defn}
###### Definition
A $\dagger$-precategory is **univalent** (or **saturated**) if $idtouiso$ is an equivalence for all $a,b:Ob(C)$.
=--

We may denote the inverse of $idtouiso$ by $uisotoid$.

## Examples ##

* Every [[univalent groupoid]] is a univalent dagger category with $f^\dagger = f^{-1}$.

* There is a $\dagger$-precategory [[Rel]] of [[h-sets]] and [[relations]] (i.e. [[Prop]]-valued binary functions), where $f^\dagger$ is the opposite relation of a relation $f$.  Assuming the global [[univalence axiom]], this $\dagger$-category is univalent because an invertible relation is necessarily a function, and hence a bijection.

* There is a $\dagger$-category $Hilb\mathbb{R}$ of Hilbert spaces over the real numbers and linear maps, where $f^\dagger$ is the adjoint linear map of $f$.  This is also univalent, assuming the univalence axiom.

Indeed, if the univalence axiom holds, then most naturally-occurring $\dagger$-categories can be proven to be univalent.

## See also ##

* [[univalence]]

* [[dagger category]]

* [[univalent setoid]]

* [[univalent groupoid]]

* [[poset]]

* [[univalent category]]

## References ##

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013): Section 9.7.

* [[Benedikt Ahrens]], [[Paige North]], [[Mike Shulman]], and Dimitris Tsementzis, *The Univalence Principle*, [arxiv](https://arxiv.org/abs/2102.06275) (2021): Example 12.1.

[[!redirects univalent dagger category]]
[[!redirects univalent dagger categories]]
[[!redirects saturated dagger category]]
[[!redirects saturated dagger categories]]
[[!redirects dagger precategory]]
[[!redirects dagger precategories]]

[[!redirects univalent dagger-category]]
[[!redirects univalent dagger-categories]]
[[!redirects saturated dagger-category]]
[[!redirects saturated dagger-categories]]
[[!redirects dagger-precategory]]
[[!redirects dagger-precategories]]

[[!redirects dagger category in homotopy type theory]]
[[!redirects dagger categories in homotopy type theory]]


univalent foundations for mathematics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _univalent foundations for mathematics_ refers to any [[foundations of mathematics|foundational system]] in which [[universes]] are [[object classifiers]], or, equivalently, satisfy the [[univalence axiom]].

Specifically, this applies to [[mathematical foundations]] in
(a) [[dependent type theory]], or in (b) [[topos theory]],
respectively, which -- apart from exactly such fine-print on the existence and nature of [[universes]] -- correspond to each other under the [[syntax-semantics duality|syntax/semantics]]-[[relation between type theory and category theory]].

Here, _univalent foundations_ is in contrast to other (earlier) approaches of laying foundations in [[type theories]] or [[toposes]], that have (only) a [[universe of propositions]] or (only) a [[subobject classifier]], respectively, hence where only a truncated sub-class of all [[types]]/[[objects]] is reflected in the [[universe]].

The basic idea of univalent foundations is, simply, to allow for universes that reflect _all_ [[types]]/[[objects]]. Or rather -- to avoid the notorious [[inconsistencies]] of the form of [[Russell's paradox]]/[[Girard's paradox]] -- universes that reflect all those [[types]]/[[objects]] that are "small" compared to the universe. It is this technical subtlety -- due to which a [[type]]/[[object]] may not be reflected in a given universe (namely if it is too large) but if it is, then it is so essentially _uni_quely -- which motivates the term "_uni_valent".

The term "univalent foundations" as such was coined by [Voevodsky 11](#Voevodsky11), much popularized with the textbook [UFIAS 12](#UFIAS12), and is usually taken to refer to [[mathematical foundations]] based on [[Martin-Löf dependent type theory]], or similar, with the [[univalence axiom]] imposed on the [[type universe]]. But the existence of (the [[categorical semantics]] of) such "univalent type universes" in higher analogues of [[toposes]] ("[[(∞,1)-toposes|∞-toposes]]") was observed (and published) earlier in [Lurie 09, Sec. 6.1.6](#Lurie09), there called _[[object classifiers]]_ and attributed to yet earlier private conversation with [[Charles Rezk]].

That both concepts,

1. [[univalence axiom|univalent]]$\,$[[type universes|universes]] in [[Martin-Löf dependent type theory|type theory]]

1. [[object classifiers]] in ([[elementary (∞,1)-topos|elementary]]) [[(∞,1)-toposes|∞-toposes]]

do indeed correspond to each other was understood from the beginning (see [Shulman 12a](#Shulman12a), [lecture 3, slide 3](http://home.sandiego.edu/~shulman/hottminicourse2012/03models.pdf#page=3), for the statement); based on proofs in special cases ([Shulman 12b](#Shulman12b)). Full proof appeared in [Shulman 19](#Shulman19).

Concretely, what makes univalent foundations univalent is the condition, respectively,

1. that there exists a [[type universe]] $Type$ (also often denoted $\mathcal{U}$) which _reflects types and their equivalences_, in that for any two types $A,B$ _in_ that universe, them being [[equivalence in homotopy type theory|equivalent as types]] is equivalent to them being equivalent ("[[identity type|equal]]") as [[terms]] of type $Type$;

   the usual formal [[syntax]] for this statement of the [[univalence axiom]] is the famous expression

   $$
     \underset{
       \color{blue}
       \text{equivalence as types}
     }{
       (A \simeq B)
     }
     \underset{
       \color{blue}
       \;\;\text{univalence}\;\;
     }{
       \simeq
     }
     \underset{
       \color{blue}
       \text{equivalence as terms}
       \atop \color{blue}
       \text{in the type universe}
     }{
       (A = B)
     }
   $$

1. that there exists an [[object classifier]] $Object$ which _reflects objects and their equivalences_, in that for any two objects $A,B$ them being [[equivalence in an (infinity,1)-category|equivalent as objects]]
is equivalent to their [[classifying maps]] to $Object$ being equivalent ([[homotopy|homotopic]]).

### In higher topos theory

See at _[[object classifier]]_ (in [[(∞,1)-toposes]]).

### In homotopy type theory

Voevodsky's univalent foundations ([UFIAS 12](#UFIAS12)) are an extension of [[Martin-Löf's dependent type theory]] obtained by adding:

* the [[univalence axiom]],

* [[propositional truncations]].

Notice that:

* [[function extensionality]] is implied by the [[univalence axiom]], so it is not necessary to assume separately;

* [[propositional resizing]] is sometimes also included as an axiom;

* the [[UniMath]] library also uses _type in type_, i.e., the [[axiom]] that the [[type universe]] contains itself. Strictly speaking, this makes the system [[inconsistency|inconsistent]], due to [[Girard's paradox]]. But the idea is that in practice the inconsistencies are readily avoided, so that the axiom serves as a convenient hack.

* The [[axiom of choice]] is consistent with the univalent foundations of mathematics and may be assumed separately.

## Overview of the basic concepts
 {#OverviewOfTheBasicConcepts}

A map $f:A\to B$ is an [[equivalence]] if it has a [[section]] and a [[retraction]]. The [[univalence axiom]] asserts that for any two types $A,B:\mathcal{U}$, the canonical map $(A=B)\to (A\simeq B)$ obtained via the induction principle of the [[identity type]], is an equivalence.

In the univalent foundations, one can distinguish between [[property]] and [[structure]] (in general, [[higher structure]]). Being an equivalence, for instance, should be a property. There is a hierarchy of [[homotopy level|homotopy levels]] that starts at the level of [[contractible type|contractible types]]. The next level consists of types of which each identity type is contractible. Such types are called [[propositions]]. The fact that being an equivalence is a property is proven by showing that $\mathsf{is{-}equiv}(f)$ is a proposition in the above sense.

Types of which the identity types are propositions are called sets. Many familiar types, such as the types of [[natural numbers]], [[integers]], and [[real numbers]] are sets in this sense, and in many subjects of algebra, such as [[groups]], [[rings]], [[modules]], and so on, it is assumed that the underlying types are sets. The [[axiom of choice]] is also restricted to sets, because there are surjective maps between other types that provably do not split. For instance, the [[double cover]] of the [[circle]] has no section, while each of its fibers is inhabited by two points. We say that sets are of [[homotopy level]] $0$. A type is said to be of homotopy level $n+1$ if its identity types are of homotopy level $n$.

By the univalence axiom, we often have nice characterizations of "the type of a certain kind of thing". For instance, the type of $n$-dimensional [[vector spaces]] (over a [[discrete topology|discrete]] [[field]]) is the [[classifying space]] of the [[general linear group]] $GL(n)$. Similarly, the type of $n$-element sets is the classifying space of the [[symmetric group]] $\Sigma_n$, and the type of [[cyclic order|cyclically ordered]] $n$-element sets is the classifying space of the [[cyclic group]] $\mathrm{C}_n$. Such observations follow from the fact that the loop space of the type of objects of a certain kind is equivalent to the type of automorphisms of that kind.

## Related entries

* [[univalence axiom]]

* [[UniMath project]]

* [[modal homotopy type theory]]

* [[higher structures]]

## References

### In higher topos theory

* {#Lurie09} [[Jacob Lurie]], Section 6.1.6 of: _[[Higher Topos Theory]]_ (attributed there to [[Charles Rezk]])

### In type theory

#### Texts

Comprehensive introduction motivated form the concept of [[symmetry]]:

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: *[[Symmetry]]* (2021) $[$[pdf](https://unimath.github.io/SymmetryBook/book.pdf)$]$

See also:

* _[Univalent foundations webpage](http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations.html)_

* {#Voevodsky11} [[Vladimir Voevodsky]],  _Univalent foundations_, talk at IAS 2011 ([pdf](https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2011_UPenn.pdf))

* {#UFIAS12} [[UF-IAS-2012|Univalent Foundations Project]], _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_ ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf), [[Planet Math|PM]] [wiki version](http://planetmath.org/node/87534))

* {#Corfield20} [[David Corfield]], _Modal homotopy type theory_, Oxford University Press 2020 ([ISBN: 9780198853404](https://global.oup.com/academic/product/modal-homotopy-type-theory-9780198853404)) doi:[10.1093/oso/9780198853404.001.0001](https://doi.org/10.1093/oso/9780198853404.001.0001)

#### Formalization libraries

* [UniMath](https://github.com/UniMath/UniMath)

* [Coq HoTT library](https://github.com/HoTT/HoTT)

* [Agda HoTT library](https://github.com/HoTT/HoTT-Agda)

* [[Martin Escardo]], _Introduction to Univalent Foundations of Mathematics with Agda_. ([ArXiv](https://arxiv.org/abs/1911.00580), [web](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/index.html), [GitHub](https://github.com/martinescardo/HoTT-UF-Agda-Lecture-Notes))

* [[Egbert Rijke]], [[Introduction to Homotopy Type Theory]] ([GitHub](https://github.com/HoTT-Intro/Agda))

### Relation between higher topos theory and univalent type theory

* {#Shulman12a} [[Michael Shulman]], _Minicourse on Homotopy Type Theory_ 2012 ([webpage](http://home.sandiego.edu/~shulman/hottminicourse2012/))

* {#Shulman12b} [[Michael Shulman]], _The univalence axiom for inverse diagrams_ ([arXiv:1203.3253](http://arxiv.org/abs/1203.3253)).

* {#Shulman19} [[Michael Shulman]], _All $(\infty,1)$-toposes have strict univalent universes_ ([arXiv:1904.07004](https://arxiv.org/abs/1904.07004)).

[[!redirects univalent foundations of mathematics]]

[[!redirects univalent foundations]]
[[!redirects Univalent Foundations]]

[[!redirects Univalent Foundations for Mathematics]]

[[!redirects univalent mathematics]]


univalent groupoid

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

#Contents#
* table of contents
{:toc}

## Definition ##

In [[homotopy type theory]], a **univalent groupoid** or **saturated groupoid** is a [[univalent dagger category]] $A$ where for all objects $a \colon A$ and $b \colon A$ and morphisms $f \colon \mathrm{Hom}_A(a, b)$, $f \circ f^\dagger = \mathrm{id}_B$ and $f^\dagger \circ f = \mathrm{id}_A$. Equivalently, it is a univalent dagger category where every morphism is a [[unitary morphism]].

Equivalently, a univalent or saturated groupoid is a [[precategory]] $A$ where for all objects $a:A$ and $b:A$ there is an equivalence between the [[identity type]] $a =_A b$ and the type $\mathrm{Hom}_A(a, b)$:

$$a:A, b:A \vdash \mathrm{disc}(a, b):\mathrm{isEquiv}(\mathrm{idToHom}(a, b))$$

where

$$
  \mathrm{idToHom}(a, b)
  \;\colon\;
  (a =_A b)
  \;
  \to
  \;
  \mathrm{Hom}_A(a, b)
$$

$$
  \mathrm{idToHom}(x, y)
  \big(
    \mathrm{refl}_A(x)
  \big)
    \;\coloneqq\;
  \mathrm{id}_A(x)
$$

The second definition implies that the only morphisms in $A$ are [[isomorphisms]], making $A$ a [[groupoid]], and thus the Rezk completeness condition holds in $A$, making $A$ [[univalent category|univalent]].

## See also ##

* [[univalence]]

* [[groupoid]]

* [[univalent dagger category]]

* [[discrete Segal type]]

[[!redirects univalent groupoids]]
[[!redirects saturated groupoid]]
[[!redirects saturated groupoids]]


univalent reflexive graph

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

\tableofcontents

## Definition

In [[dependent type theory]], a *[[reflexive graph]]* is a [[type]] $A$ with a [[type family]] $x:A, y:A \vdash R(x, y)$ and a [[family of functions]] $x:A, y:A \vdash \mathrm{idtofam}(x, y):(x =_A y) \to R(x, y)$.

Such a [[reflexive graph]] is a **univalent reflexive graph** if the following conditions hold, which are all equivalent by the [[fundamental theorem of identity types]]:

1. For each $x:A$ the type of elements $y:A$ such that $R(x, y)$ is a [[contractible type]].
$$x:A \vdash \mathrm{ua}(x):\mathrm{isContr}\left(\sum_{y:A} R(x, y)\right)$$

1. There is a family of equivalences
$$x:A, y:A \vdash \mathrm{ua}(x, y):(x =_A y) \simeq R(x, y)$$

1. $R(x, y)$ is an [[identity system]].

1. For each $x:A$ and $y:A$, the function $\mathrm{idtofam}(x, y)$ is an [[equivalence of types]]
$$x:A, y:A, \vdash \mathrm{ua}(x, y):\mathrm{isEquiv}(\mathrm{idtofam}(x, y))$$

1. $\mathrm{idtofam}(x, y)$ has a [[retraction]]
$$x:A, y:A \vdash \mathrm{ua}(x, y):R(x, y) \to (x =_A y)$$
$$x:A, y:A, r:R(x, y) \vdash G(x, y):\mathrm{idtofam}(x, y, \mathrm{ua}(x, y, r)) =_{R(x, y)} r$$

1. That $R(x, y)$ with the function $\mathrm{idtofam}(x, y)$ satisfies the [[universal property]] of the [[unary sum]] of $x =_A y$.

## Examples

There are many examples of univalent reflexive graphs in mathematics. These include [[posets]], [[T0-spaces]], [[metric spaces]], [[univalent categories]], [[univalent groupoids]],  [[univalent dagger categories]], [[univalent bicategories]], [[univalent setoids]], and [[univalent universes]].

## See also

* [[reflexive graph]]

* [[fundamental theorem of identity types]]

## References

The definitions and proofs are generalized to general univalent reflexive graphs from the various definitions of univalent universes found in the [[homotopy type theory]] literature:

* [[UF-IAS-2012|Univalent Foundations Project]], p. 4 & Sec. 2.10 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013)  [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

* {#Rijke22} [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

* [[Dan Licata]], *weak univalence with "beta" implies full univalence* ([web](https://groups.google.com/forum/#!msg/homotopytypetheory/j2KBIvDw53s/YTDK4D0NFQAJ))

[[!redirects univalent reflexive graph]]
[[!redirects univalent reflexive graphs]]


univalent setoid

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

=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

There are two ways to show that two objects of a [[presetoid]] are identified with each other: by way of the [[identity type]], and by way of the type of [[isomorphisms]] in the [[core]] [[pregroupoid]] of a presetoid. Univalent setoids are precisely the presetoids which the two notions of identifying objects with each other coincide.

## Definition

For each object $a:Ob(A)$ and $b:Ob(A)$ of a presetoid $A$, there is a function
$$\mathrm{idtoiso}(a, b):(a =_A b) \to (a \cong_A b)$$
from the [[identity type]] to the type of isomorphisms in the [[core]] [[pregroupoid]] $\mathrm{Core}(A)$ of $A$. A presetoid is a **univalent setoid** or a **saturated setoid** if for each object $a:Ob(A)$ and $b:Ob(A)$ the function
$$\mathrm{idtoiso}(a, b):(a =_A b) \simeq (a \cong_A b)$$
is an equivalence of types.

Since every type of morphisms in the core of the presetoid is a [[h-set]], this means that every univalent setoid is a [[h-groupoid]].

## Examples

* Every thin univalent setoid is a [[set]] (i.e. an [[h-set]]).
* [[univalent dagger categories]]
* [[h-groupoids]] (i.e. univalent groupoids)

## See also

* [[setoid]]
* [[presetoid]]
* [[quotient set]]
* [[structure identity principle]]

[[!redirects univalent setoids]]
[[!redirects saturated setoid]]
[[!redirects saturated setoids]]


univalent type theory

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

#Contents#
* table of contents
{:toc}

## Definition ##

A __univalent type theory__ is a [[dependent type theory]] with [[type universes]] and [[identity types]] such that all universes satisfy the [[univalence axiom]].

Univalent type theories usually have additional types such as the [[empty type]], the [[unit type]], the [[booleans]], and the [[natural numbers type]], making it into a [[Martin-Löf dependent type theory]].

## See also ##

* [[Martin-Löf dependent type theory]]

* [[proof theoretic strength of univalent type theory plus HITs]]

* [[homotopy type theory]]

* [[univalent foundations]]

* [[UniMath]]

## References ##

* Auke Booij, _Analysis in Univalent Type Theory_, ([slides](https://www.cs.bham.ac.uk/~abb538/slides/2018-02-darmstadt.pdf)) ([thesis](https://www.cs.bham.ac.uk/~abb538/thesis-first_submission.pdf))

* [[Martín Hötzel Escardó]] and [Cory Knapp](https://github.com/cmknapp), _Partial Elements and Recursion via Dominances in Univalent Type Theory_, ([LIPIcs:2017:7682](https://drops.dagstuhl.de/opus/volltexte/2017/7682/))

[[!redirects univalent type theories]]

[[!redirects univalent type universe]]
[[!redirects univalent type universes]]


universal algebra

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

#Contents#
* table of contents
{:toc}

## Idea

Universal algebra (also _[[categorical algebra]]_) is the study of [[algebraic theories]] and their models or algebras.  Whereas abstract algebra studies groups, rings, modules and so on --- that is, models of particular theories --- universal algebra is about algebraic or __equational__ theories in general.

Traditionally, the subject studies models of algebraic theories in the [[category]] of [[Set|sets]].  The category-theoretic approach abstracts the traditional notions, to study models in more general categories.  There are several ways of doing this, such as by using [[monads]], [[Lawvere theories]], or [[type theory]].

As with the category-theoretic understanding of many other branches of mathematics, the advantage of doing things this way is not so much the obtaining of new results as the __unification__ of many previously disparate points of view.  Examples might include how a [[Hopf algebra]] is the same thing as a model in a category of vector spaces of the theory of groups, or how computational side-effects in the theory of programming languages may be understood in terms of [[free algebras]].

## Related entries

* [[algebraic theory]]

* [[Lawvere theory]]

* [[monad]]

* [[operad]]

* [[algebra over a Lawvere theory]]

* [[algebra over a monad]]

* [[algebra over an operad]]

* [[Eilenberg-Moore category]]

* [[universal algebra in a monoidal category]]

* [[bar construction]]

* [[generalized ring]]

* [[Omega-group]]

* [[PRO]],

## References
 {#References}

* [[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)]

* [[R. F. C. Walters]], *A categorical approach to universal algebra*, Ph.D. Thesis (1970) [[anu:1885/133321](https://openresearch-repository.anu.edu.au/handle/1885/133321)]

* {#MacLane71} [[Saunders MacLane]], §VI of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5**  Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Martin Hyland]], [[John Power]], _The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads_, Electronic Notes in Theor. Comp. Sci. **172** (2007) 437-458 [[doi:10.1016/j.entcs.2007.02.019](https://doi.org/10.1016/j.entcs.2007.02.019), [preprint](https://www.dpmms.cam.ac.uk/~martin/Research/Publications/2007/hp07.pdf)]

* [[Max Kelly]] and [[John Power]], _Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads_, JPAA 89, 1993.

* {#BaiGuoLoday11} Chengming Bai, Li Guo, [[Jean-Louis Loday]], _[Proceedings of the China-France Summer Institute on Operads and Universal Algebra](http://andromeda.rutgers.edu/~liguo/OUA10/book.htm)_, World Scientific 2011 ([web](http://andromeda.rutgers.edu/~liguo/OUA10/operadua.html))

A short introductory/overview note:

* [[Anthony Voutas]], *The basic theory of monads and their connection to universal algebra*, (2012) [[pdf](https://voutasaur.us/monad-algebra.pdf), [[Voutas-Monads.pdf:file]]]

* {#keml-diagrams} [[Andreas Nuyts]], _Understanding Universal Algebra Using Kleisli-Eilenberg-Moore-Lawvere Diagrams_, (2022) [[pdf](https://anuyts.github.io/files/keml-diagrams.pdf)]

Discussion via [[sheaves]] of quotient algebras:

* [[Michael Johnson]] and Shu-Hao Sun, _Remarks on representations of universal algebras by sheaves of quotient algebras_, Proceedings of the 1991 Summer Category Theory Meeting, 1992. (pdf)

[[!redirects universal algebras]]


universal algebra in a monoidal category

{:myproof: .proof style="margin-left:2em;"}
{:mynumdef: .num_defn style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;"}
{:forklaring: .un_remark style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

+-- {: forklaring}
### Initial Purpose of Page ###
The ideas on this page are things that I, [[Andrew Stacey]], want to use.  I suspect that they are well-known and well-developed in the literature but have no idea how to track them down.  So I'm expounding on them here in the hope that some kindly soul will fill in the gaps for me.
=--

+-- {: .query}
So should we merge this with [[PRO]] now?  ---Toby

Yes.  However, not knowing what is standard and what isn't, I'm not sure I could do that merge.  The stuff on this page is what I want to know about so I don't want to lose any of it in the merge. ---Andrew

Understood.  I\'m not sure how to do it either, but I\'ll think about it.  ---Toby

I should say that if any of this _doesn't_ fit in the merge then I'm happy for it to find a home elsewhere.  My main purpose in scribbling this lot is to find out what is and isn't known so I don't have to reprove the known stuff just to make use of it. ---Andrew

Now I want to wait until Todd figures out what you really want to do here.  (^_^)  ---Toby
=--

### The Basic Idea ###

The idea is that one tries to "do" [[universal algebra]] in a [[monoidal category]] by using the monoidal products in place of the standard cartesian ones.  The extent to which this is possible is closely related to the extent to which the monoidal product resembles the cartesian product of sets.

### In More Detail ###

Let us consider only finitary theories; that is, with [[Lawvere theories]].  In categorical language, recall that a Lawvere theory is  a [[category]] $T$ with finite cartesian products in which every object is isomorphic to a finite cartesian power of a distinguished object.  A model for that theory in some category $C$ is then a product-preserving functor $T \to C$.  In a [[monoidal category]], we can ask for this to be a _monoidal_ functor.

The problem with this naive definition is that as the product in $T$ **is** the cartesian product, it comes along with a whole heap of extra structure that may not be strictly needed.  We therefore need to weaken the definition of a Lawvere theory.

+-- {: mynumdef #monlawth}
###### Definition
A **monoidal Lawvere theory** is a monoidal category $(T, \otimes, I)$ in which every object is isomorphic to a finite monoidal product of some distinguished object.
=--

It will be important for later to note what is missing between a __monoidal__ Lawvere theory and an ordinary one.  The missing components are:

1. No projection morphisms; e.g. $x^n \to x$
2. No diagonal morphisms; e.g. $x \to x^n$
3. No symmetry morphisms; e.g. $x^2 \to x^2$

With this definition, we can define a model of such a theory in a monoidal category.

+-- {: mynumdef #model}
###### Definition
Let $(C, \otimes, I)$ be a [[monoidal category]].  Let $(T, \otimes, I)$ be a  monoidal Lawvere theory.  A **monoidal $T$-object** in $(C,\otimes, I)$ is a monoidal functor $T \to C$.
=--

### The Trade ###

Any Lawvere theory is automatically a monoidal Lawvere theory and so one can consider monoidal models of an ordinary Lawvere theory in a monoidal category.  However, as ordinary Lawvere theories have all the extra structure of projections, diagonals, and symmetry, the underlying object (i.e. the image of the distinguished object in the Lawvere theory) needs to have them as well.  Indeed, if one were looking for candidates for such models one would be well-advised to look in the subcategories of objects which have such structure.  Let us therefore give these names.

+-- {: mynumdef #cartcats}
###### Definition
Let $(C, \otimes, I)$ be a monoidal category.  We define the following full subcategories of $C$.

1. $C^{\operatorname{sym}}$ is the largest **symmetric** monoidal subcategory.  That is, it is the full subcategory where the switcheroo functor is naturally isomorphic to the identity.
2. $C^{\operatorname{co}}$ is the subcategory of commutative, associative, unital **comonoids**.  Note that we take morphisms in $C$, not morphisms of comonoids.  These are the objects that come equiped with diagonal and projection morphisms.
=--

+--{.query}

[[Todd Trimble|Todd]]: Sorry, I don't know what you mean by the switcheroo functor for a monoidal category.

There is what is called the [[center]] of a monoidal category, which is a braided monoidal category. The center of the center should be symmetric monoidal (although I'd have to think this through again carefully). Comonoids in the center or center of center form a monoidal category, and I'm thinking something like this would be a natural habitat to consider.

[[Andrew Stacey|Andrew]]: Knowing that none of my terminology would be right, I let myself get carried away there.  Apologies.  I mean the functor that swaps the order of the product, namely that takes $A \otimes B$ to $B \otimes A$.  (Actually, now I come to think about it I'm not convinced that such must exist in an arbitrary monoidal category.)  What I mean is that if I have a monoidal category then even if it isn't symmetric, there may still be an object that is itself symmetric.  I guess I don't need the category of such to be symmetric monoidal and I don't need it to commute with products of other objects, just with itself.  So I could still have a commutative monoid in an arbitrary monoidal category.

Or maybe I can't.  Now I write it, it seems a lot more tenuous than when I thought it.  It'd be useful to get that clear.

[[Todd Trimble|Todd]]: Not sure I can add anything else useful right now. I don't think there's a general switcheroo functor. In some sense a monoidal functor $Fin^{op} \to M$ to a monoidal category $M$ ought to simulate a cocommutative comonoid in $M$, but I'm not sure how useful an observation this is for you.

Is there a specific setting you have in mind where what you want to do is unproblematic?

[[Andrew Stacey|Andrew]]: Yes.  I want to do it in the category of models in set of a [[commutative algebraic theory]].  Over there, it doesn't say anything about the monoidal structure being symmetric but I have an inkling that it is.  However, not every object is a coalgebra.

(As a little background: I'm returning to some stuff that I thought about a while ago and am trying to finish off, but I don't remember all of what I thought when I originally thought about it which is why I'm not as clear as I would like to be)

_Toby_:  Even in a symmetric monoidal category, I don\'t know how to define a functor that takes $A \otimes B$ to $B \otimes A$.  Of course, there is a functor $C \times C \to C$ that takes $(A,B)$ to $B \otimes A$, but that doesn\'t help as far as I can see.

There\'s an inherent problem with considering the largest (or any) symmetric subcategory of a given monoidal category, which is that being symmetric is not a property but a structure on a monoidal category.  So it\'s more like looking for the largest commutative submonoid of a given *set* than looking for the largest commutative submonoid of a given monoid.

Even if that worked, I wouldn\'t understand the full subcategory of CAU comonoids, since again being a CAU comonoid is not a property but a structure on an object in a given symmetric monoidal category.  This is like asking for the full subcategory of $Set$ consisting of groups; it doesn\'t parse.  At best, I would understand this as the full subcategory of sets that have at least one group structure, but you\'re no longer keeping track of that structure (and in fact, at least assuming the axiom of choice, this is just $Set \setminus \{\empty\}$, which has nothing much to do with groups at all).

[[Andrew Stacey|Andrew]]: Okay, let me be absolutely specific here.  I was trying to get at the general story first, but that's the wrong way round for _here_ (it's the right way round for the paper, but that's once I've understood what the general story is!).

This is tied up with [[Tall-Wraith monoids]].  If you start with a set and take the free ring on that set, then that's again a biring (co-ring object in rings).  Now start with abelian groups instead.  If you take the free ring on an abelian group, then that's not a biring, but still has a certain amount of co-structure because a ring is a monoid in $(Ab, \otimes, \mathbb{Z})$.  To make it a biring, I need to start with a coalgebra - that's enough to supply the missing pieces of the co-structure.

*  _Toby_:  OK, this makes sense.  In general, you start with a comonoid in your monoidal category; a comonoid in $(Ab,\otimes)$ is a coalgebra, while a comonoid in $(Set,\times)$ is simply a set (by an argument that applies whenever the monoidal category is cartesian).

Furthermore, if I wish to interpret the resulting biring as representing a functor from abelian groups to rings, then I can only do so on the subcategory of coalgebras - but I need to take abelian group homs not coalgebra homs.

*  _Toby_:  H\'m, this is odd.  I still don\'t know exactly what the 'subcategory' of coalgebras is, since being a coalgebra is not a property of an abelian group.  Since the forgetful functor from coalgebras to abelian groups is faithful, you can make it a literal [[subcategory]] up to equivalence, then consider the corresponding full subcategory; or (equivalently) just take the category whose objects are coalgebras and whose morphisms are homomorphisms of the underlying abelian groups.  (In general, this concept is the [[full image]] of a functor; here we are taking the full image of the forgetful functor from coalgebras to abelian groups.)

   But I don\'t understand why you need to use this instead of simply the category of coalgebras.  After all, if you have a functor from the full image, then you certainly have a functor from the category of coalgebras by composing with the inclusion functor of coalgebras into the full image.  On the other hand, it seems very strange that you can define this functor on arbitrary abelian group homomorphisms.  Applying this to abelian group isomorphisms, you are basically saying that, if you start with two different coalgebra structures on a given abelian group, the resulting birings will be isomorphic.  Is that true?

   Andrew --- Okay, I still wasn't being completely clear.  I _do_ mean the category of coalgebras.  But the _functor_ is constructed by taking the composition of the functor from coalgebras to abelian groups and _then_ taking the hom-functor _in abelian groups_ with a biring.  So if $C$ is a coalgebra and $B$ is a biring then I want to consider $Hom_{Ab}(C,B)$.  That's again a ring and it's functorial for _coalgebra_ morphisms.

   Phew!  "Oh what a tangeled web we weave, when first we practise to understand category theory."

   _Toby_:  OK, so when you wrote 'I need to take abelian group homs not coalgebra homs', you weren\'t still specifying the source of the functor but were instead specifying how that biring *represents* the functor.  Now it makes sense!

This all generalises quite nicely: replace abelian groups by a [[commutative algebraic theory]] and replace rings (rather, replace monoids) by a [[PRO]].  Then the PRO-objects in the commutative algebraic theory will again be an algebraic theory and the free functor from the commutative theory to the other one has lots of nice properties.

But this _felt_ like it was a specialisation of an even more general story.  My mistake was to try to guess the general story first rather than tell you the special case.

(Not that I regard this particularly as a _mistake_!  It was enough to learn about [[PRO|PROs]] and it's all part of learning how to ask good questions.)

_Todd_: Okay, so to pick up the thread again, here is a preliminary observation. Suppose we have an operad $P$ (giving rise to a prop $Pr(P)$) and a symmetric monoidal closed category $K$ (such as the category of abelian groups, or the category of algebras of a commutative monad on $Set$), then a $P$-algebra is essentially the same as a symmetric monoidal functor $A: Pr(P) \to K$, and we can define a $P$-coalgebra to be a symmetric monoidal functor $C: Pr(P)^{op} \to K$. Let $A_1$ and $C_1$ denote the underlying $K$-objects (= values of $A$ and $C$ at tensor-generating object 1 of $Pr(P)$), and let $K(-, -)$ denote internal hom in $K$. Then we can define a $P$-algebra structure with underlying object $K(C_1, A_1)$ whose operations

$$\widehat{\theta}: K(C_1, A_1)^{\otimes n} \to K(C_1, A_1)$$

are obtained as composites

$$K(C_1, A_1)^{\otimes n} \to K(C_{1}^{\otimes n}, A_{1}^{\otimes n}) \overset{K(C(\theta), A(\theta))}{\to} K(C_1, A_1)$$

where the first map is a canonical map available in any symmetric monoidal closed category. If we denote this $P$-algebra by $K(C, A)$, then this gives us an [[actegory]] action

$$K(-, -): Coalg_P^{op} \times Alg_P \to Alg_P$$

and in this framework one could go on to consider $P$-bialgebras, etc. I think this is what you were getting at or had observed yourself; Barr touches upon this sort of thing in the paper I mentioned over at [[CommCoalg]] or whatever we're calling it now. I don't know how well it generalizes to general props or pros.

=--

Thus to be able to consider *arbitrary* Lawvere theories we need to be in the intersection of $C^{\operatorname{sym}}$ and $C^{\operatorname{co}}$.  However, there are many Lawvere theories that do not need such structure.  Accordingly we can interpose a finer classification on Lawvere theories according to whether or not they need the symmetric or comonoid structure.

If one is given a presentation of a Lawvere theory in terms of operations and identities, it is quite simple to identify what level of structure it requires.  The key is to look at the identities and express them in terms of elements (this, of course, violates the [[principle of equivalence]]).  An identity consists of two ways of applying operations starting with an ordered list of _distinct_ elements.

1. If an identity involves changing the order of the elements, it needs the symmetric structure.
2. If an identity involves duplicating an element, it needs the diagonal structure.
3. If an identity involves ignoring an element, it needs the projection structure.

### A Worked Example ###

The passage from the theory of unital, associative [[monoids]] to that of [[abelian groups]] is a good illustration of all of this.  The Lawvere theory of unital, associative monoids has a presentation with 2 operations: one zeroary, $1$, and one binary, $\mu$, together with the identities:

1. Associativity
   $$
   \mu(a,\mu(b,c)) = \mu(\mu(a,b),c)
   $$
2. Unit
   $$
   \mu(a,1) = a, \qquad \mu(1,a) = a
   $$

None of these identities involve swaps, repetitions, or omissions and so this presentation is also of a monoidal Lawvere theory (note that $1$ is an operation not an element).

Now let us add in an inverse.  That is, a unary operation $\iota$ with the identities:

1. Inverses
   $$
   \mu(\iota(a),a) = 1, \qquad = \mu(a, \iota(a)) = 1
   $$

These involve both a duplication and an omission.  The duplication is on the left-hand side in each case, the omission on the right.  Note that the starting point for each is the ordered list $(a)$.  Therefore a **group** object in a monoidal category must lie in the subcategory of comonoids.

Let us forget the inverse for a moment and add in the commutativity identity instead.  This is the identity:

1. Commutativity
   $$
   \mu(a,b) = \mu(b,a)
   $$

This obviously involves a swap, and thus requires a _symmetric_ object as its model.

Finally, for an abelian group object we need both: symmetry and comonoid.

### Representing Functors ###

One of the applications of Lawvere theories is to the theory of representing objects.  Under mild conditions, a model of a Lawvere theory is equivalent to a lift of a representable contravariant functor $C \to \operatorname{Set}$ to a functor $C \to \operatorname{Set}T$, where $\operatorname{Set}T$ is the category of models of $T$ in $\operatorname{Set}$.  However, except for very simple theories (i.e. those with no binary or higher operations), this does not carry over to the monoidal case because the construction of the operations on the Hom sets involves extensive use of diagonal morphisms.

However, where it does work is when one restricts to the subcategory of comonoid objects _with comonoid-structure preserving morphisms_.  That is, if $C^{\operatorname{co}}_{c}$ denotes the subcategory of $C$ of comonoids with such morphisms then a model $G$ for $T$ in $(C,\otimes,I)$ represents a functor $C^{\operatorname{co}}_{c} \to \operatorname{Set}T$ via

$$
C^{\operatorname{co}}_c \to C \overset{Hom(-,G)}{\to} \operatorname{Set}T
$$

### Questions ###

Apart from the obvious question of finding the literature behind all of this (and thus getting the terminology correct), certain other questions spring to mind.  Well, okay, one question springs to mind.

1. Suppose that I have two monoidal Lawvere theories, $T_1$ and $T_2$, that are equivalent as ordinary Lawvere theories.  Are they equivalent as monoidal Lawvere theories?  More concretely, suppose that I have two presentations for the same Lawvere theory in terms of operations and identities and both define monoidal Lawvere theories (so neither has identities involving repetition, omission, or exchanging[^minute]), are the **monoidal** Lawvere theories equivalent?

_Todd_: My previous answer (which has been deleted) was based on a too-hasty reading of the question. I'll try to set aside some time to find a better answer.

_Todd_: The time set aside turned out to be time last night when I had trouble falling asleep. I believe the answer is no. If for example $T_1$ is the pro for comonoids and $T_2$ is the initial pro (so equivalent to the natural numbers as discrete monoidal category), then the "Lawverifications" of those pros ought to be equivalent, both being equivalent to the initial Lawvere theory $FinSet^{op}$. At the semantic level, this boils down to the easy observation that in a cartesian monoidal category, there is a unique comonoid structure on every object.

###References###

S. Lack, Composing PROPS, [Theory Appl. Categ.](http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html) 13 (2004), No. 9, 147--163.

[^minute]: I keep wanting to write "repetition, hesitation, or deviation".


universal bundle

* [[universal principal bundle]]

* [[generalized universal bundle]]

* [[Milnor construction]]


universal central extension

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

#Contents#
* table of contents
{:toc}

## Definition

For any [[algebra|algebraic]] object, such as an [[associative algebra]], [[group]], [[Lie algebra]], etc. such that there is a notion of [[central extension]], the _universal_ central extension is, if it exists, the [[initial object]] in the [[category]] of central extensions of that object.

## Examples

For $G$ a [[discrete group]] which is [[perfect group|perfect]], the _universal [[central extension]]_ of $G$ is the essentially unique [[group]] $K$ that is a [[Schur covering group]] of $G$.

## Properties

* The universal central extension of a perfect group is also perfect.

* The universal central extension of a perfect group is a Schur-trivial group, and hence a superperfect group (superperfect means that it's perfect and Schur-trivial).

* The universal central extension operator is idempotent, i.e., the universal central extension of the universal central extension is the universal central extension. This follows directly from the universal central extension being a Schur-trivial group.

## Related concepts

* [[maximal central extension]]

* [[universal higher central extension]]

## References

### General

Discussion in [[semi-abelian categories]]:

* Jose Casas, Tim Van der Linden, _A relative theory of universal central extensions_ ([arXiv:0908.3762](https://arxiv.org/abs/0908.3762))

### For discrete groups

* [[Groupprops]], _[Universal central extension](http://groupprops.subwiki.org/wiki/Universal_central_extension)_

### For Lie groups

* [[Karl-Hermann Neeb]], _Universal Central Extensions of Lie Groups_,  Acta Applicandae Mathematicae (2002) 73: 175 (doi:10.1023/A:1019743224737)

### For super Lie algebras

For [[super Lie algebras]]

* Erhard Neher, _An introduction to universal central extensions of Lie superalgebras_ ([[NeherUniversalCentralExtension.pdf:file]])

[[!redirects universal central extension]]


universal characteristic class

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

#Contents#
* table of contents
{:toc}

## Idea

The [[characteristic classes]] of [[universal principal ∞-bundles]] are called _universal characteristic classes_. Every other characteristic class arises by pullback from a universal characteristic class.

See _[[characteristic class]]_ for more.

## Related concepts

* [[cohomology operation]]

* [[universal Chern-Simons n-bundle]]

[[!include extended prequantum field theory - table]]

## References

Textbook accounts include

* {#Kochman96} [[Stanley Kochman]], section 2.3 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

[[!redirects universal characteristic classes]]

[[!redirects universal characteristic map]]
[[!redirects universal characteristic maps]]


universal Chern-Simons circle 7-bundle with connection

[[!redirects Chern-Simons circle 7-bundle with connection]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Chern-Simons circle 7-bundle_ is the [[circle n-bundle with connection|circle 7-bundle with connection]]  classified by the cocycle in degree-8 [[ordinary differential cohomology]] that is canonically associated to a [[string group]]-[[principal 2-bundle]] with [[connection on a 2-bundle|connection]].

The [[characteristic class]] called the second fractional [[Pontryagin class]] $\frac{1}{6}p_2 : \mathcal{B}String \to \mathcal{B}^8 \mathbb{Z}$ in [[Top]] on the [[classifying space]] of the [[string group]] has a smooth lift to the smooth second fractional Pontryagin class

$$
  \frac{1}{6} \mathbf{p}_2 : \mathbf{B}String \to \mathbf{B}^7 U(1)
$$

in $\mathbf{H} :=$ [[∞LieGrpd]], mapping from the [[delooping]] [[∞-Lie groupoid]] of the [[string Lie 2-group]] to that of the  circle Lie 7-group. This is the Lie integration  of the degree 7 [[∞-Lie algebra cocycle]] $\mu_7 : \mathfrak{string} \to b^6 \mathbb{R}$ on the [[string Lie 2-algebra]] which classified the [[fivebrane Lie 6-algebra]].

Therefore, by [[∞-Chern-Weil theory]], there is a refinement of this morphism to [[connection on an ∞-bundle|∞-bundles with connection]]

$$
  \frac{1}{6}\hat \mathbf{p} : \mathbf{B}String_{conn} \to
  \mathbf{B}^7 U(1)_{conn}
$$

hence on [[cocycle]] [[∞-groupoid]]s

$$
  \frac{1}{6} \hat \mathbf{p} : \mathbf{H}_{conn}(X,\mathbf{B}String)
  \to \mathbf{H}_{diff}^8(X)
$$

a map from [[string Lie 2-group]]-[[principal 2-bundle]]s with [[connection on a 2-bundle|connection]] to [[circle n-bundle with connection|circle 7-bundles with connection]], hence degree 8 [[ordinary differential cohomology]].

For $(P,\nabla)$ a String-principal 2-bundle, we call the image $\frac{1}{6}\hat\mathbf{p}(\nabla) \in \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))$ its **Chern-Simons circle 7-bundle with connection**.

This is a differential refinement of the [[twisted cohomology|obstruction]] to lifting $P$ to a [[fivebrane Lie 6-group]]-bundle.

By construction, the [[curvature]] 8-form of $\hat \mathbf{c}(\nabla)$ is the [[curvature characteristic form]] $\langle F_\nabla \wedge F_\nabla \wedge F_\nabla \wedge F_\nabla\rangle$ of $\nabla$ and accordingly the 7-form connection on $\hat \mathbf{c}(\nabla)$ is locally a [[Chern-Simons form]] $CS(\nabla)$ of $\nabla$.

Therefore the [[higher parallel transport]] induced by $\frac{1}{6}\hat \mathbf{p}_2(\nabla)$ over 7-dimensional volumes $\phi : \Sigma \to X$ is the [[action functional]] of degree-7 [[∞-Chern-Simons theory]].  This is the analog of the way the [[Chern-Simons circle 3-bundle]] arises from Spin-principal bundles.

## Construction

Using the discussion at [[∞-Chern-Weil theory]] and in direct analogy to the construction of the [[Chern-Simons circle 3-bundle]] we can model the [[(∞,1)-functor]]

$$
  \mathbf{H}_{conn}(X, \mathbf{B}String)
  \to
  \mathbf{H}_{conn}(X, \mathbf{B}^7 U(1))
$$

by postcomposition with the [[∞-anafunctor]]

$$
  \array{
    \exp(\mathfrak{string})_{conn}
    &\stackrel{\exp(\mu_7)_{conn}}{\to}&
    \exp(b^6 \mathbb{R})_{conn}
    \\
    \downarrow && \downarrow
    \\
    \mathbf{cosk}_7 \exp(\mathfrak{string})_{conn}
    &\to&
    \mathbf{B}^7 U(1)_{conn}
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}String_{conn}
  }
$$

where $\mu_7 : \mathfrak{string} \to b^6 \mathbb{R}$ is the 7-cocycle that classifies the [[fivebrane Lie 6-algebra]].

For

$$
  \array{
     C(U) &\stackrel{g}{\to}& \mathbf{B}String_{conn}
     \\
     \downarrow^{\mathrlap{\simeq}}
     \\
     X
  }
$$

an [[∞-anafunctor]] modelling a [[cocycle]] for a [[string 2-group]]-[[principal 2-bundle]] with [[connection on a 2-bundle]] the $\infty$-anafunctor composition

$$
  \array{
    && \exp(\mathfrak{string})_{conn}
    &\stackrel{\exp(\mu_7)_{conn}}{\to}&
    \exp(b^6 \mathbb{R})_{conn}
    \\
    && \downarrow && \downarrow
    \\
    C(V)
    &\stackrel{\hat g}{\to}&
    \mathbf{cosk}_7 \exp(\mathfrak{string})_{conn}
    &\to&
    \mathbf{B}^7 U(1)_{conn}
    \\
    \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}}
    \\
    C(U) &\stackrel{g}{\to}& \mathbf{B}String_{conn}
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
$$

produces a lift of the transition functions $g$ to $\mathbf{cosk}_7 \exp(\mathfrak{string})$. The string-cocycle is itself in first degree a collection of paths in $G$, in second a collection of surfaces with labels in $U(1)$. That lift corresponds to further resolving this to families

$$
  U_{i_1} \cap \cdots U_{i_k} \times \Delta^k \to G
$$

up to $k = 7$. That this is indeed always possible is the statement about [[Lie integration]] that $\mathbf{cosk}_7  \exp(\mathfrak{string}) \stackrel{\simeq}{\to} \mathbf{B}String$ is a weak equivalence, which in turn is due to the fact that the next nonvanishing [[homotopy group]] of $G = SO(n)$ after $\pi_3$ is $\pi_7$.

The above composite [[∞-anafunctor]] is manifestly a degree 8-cocycle in [[Cech cohomology|Cech]]-[[Deligne cohomology]] given by

$$
  \left(
     CS_7(\sigma_i^* A)
     \,,\,
     \int_{\Delta^1} g_{i j}^*CS_7(A)
     \,,\,
     \int_{\Delta^2} g_{i j k}^*CS_7(A)
     \,,\,
     \int_{\Delta^3} \hat g_{i j k l}^*CS_7(A)
     \,,\,
     \int_{\Delta^5} \hat g_{i j k l m}^*CS_7(A)
     \,,\,
     \int_{\Delta^6} \hat g_{i j k l m n}^*CS_7(A)
     \,,\,
     \int_{\Delta^7} \hat g_{i j k l m n o}^* \mu(A)
  \right)
  \,,
$$

where $A$ is a connection form on the total space of the $Spin(n)$-[[principal bundle]] that the string bundle itself is lifted from and $CS_7$ is the [[Chern-Simons element]] in degree 7 defining the [[fivebrane Lie 6-algebra]].

(...)

## Applications

* [[differential fivebrane structure]]

* [[dual heterotic string theory]]

## Related concepts

* [[universal Chern-Simons n-bundle]]

  * [[Chern-Simons circle 3-bundle]]

  * **Chern-Simons circle 7-bundle**

The CS 7-bundle serves as the [[extended Lagrangian]] for a _[[7d Chern-Simons theory]]_. See there for more.

## References

The CS 7-bundle as an [[circle n-bundle with connection|circle 7-bundle with connection]] on the [[smooth infinity-groupoid|smooth]] [[moduli infinity-stack]] of [[string 2-group]]-[[connection on a 2-bundle|2-connections]] has been constructed in

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Hisham Sati]], _[[schreiber:Cech cocycles for differential characteristic classes]]_

and identified as part of the [[11-dimensional supergravity]] Chern-Simons terms after [[KK-reduction]] on $S^4$ to [[7-dimensional supergravity]] (for [[AdS/CFT]] duality with the [[M5-brane]] [[worldvolume]] [[6d (2,0)-superconformal QFT|6d (2,0)-superfonformal]] [[schreiber:∞-Wess-Zumino-Witten theory]]) in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:7d Chern-Simons theory and the 5-brane]]_

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The moduli 3-stack of the C-field]]_

[[!redirects Chern-Simons circle 7-bundle]]

[[!redirects Chern-Simons circle 7-bundles]]

[[!redirects universal Chern-Simons circle 7-connection]]


universal Chern-Simons line 3-bundle

[[!redirects universal Chern-Simons circle 3-bundle]]
[[!redirects Chern-Simons circle 3-bundle]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Chern-Simons circle 3-bundle_ is the [[circle n-bundle with connection]]  classified by the cocycle in degree 4 [[ordinary differential cohomology]] that is canonically associated to a $G$-[[principal bundle]] with [[connection on a bundle|connection]].

For $G$ a [[Lie group]] and $c : \mathcal{B}G \to K(\mathbb{Z},4)$ a cocycle for a degree 4 [[characteristic class]] in [[integral cohomology]] and $X$ a [[smooth manifold]], [[Chern-Weil theory]] provides a morphism (the refined [[Chern-Weil homomorphism]])

$$
  \hat c : G Bund_\nabla(X) \to H_{diff}^4(X)
$$

from $G$-[[principal bundle]]s with [[connection on a bundle|connection]] $\nabla$ to degree 4 [[ordinary differential cohomology]]. The cocycles on the right may be thought of as

* [[circle n-bundle with connection|circle 3-bundles with connection]] $\hat c(\nabla)$;

* [[bundle gerbe|bundle 2-gerbe]]s with connection.

By construction, the [[curvature]] 4-form of $\hat c(\nabla)$ is the [[curvature characteristic form]] $\langle F_\nabla \wedge F_\nabla\rangle$ of $\nabla$ and accordingly the 3-form connection on $\hat c(\nabla)$ is locally a [[Chern-Simons form]] $CS(\nabla)$ of $\nabla$.

Accordingly, the [[higher parallel transport]] induced by $\hat c(\nabla)$ over 3-dimensional manifolds $\phi : \Sigma \to X$ is the [[action functional]] of the [[quantum field theory]] called [[Chern-Simons theory]]. In this form it appears for instance as the [[gauge field]] called the [[supergravity C-field]] in certain [[supergravity]] theories. In particular, if (with due care) one takes $\nabla$ to be the _universal connection on the $G$-[[universal principal bundle]]_ over a smooth version of $B G$, then $\hat c(\nabla)$ is the background [[gauge field]] for bare [[Chern-Simons theory]].

Therefore this structure $\hat c(\nabla)$ has become known as the **Chern-Simons 2-gerbe**  of $\nabla$. We may also think of it as the _Chern-Simons [[circle n-bundle with connection|circle 3-bundle]]_ .

At least for simply connected $G$ one may enhance the assignment $\nabla \mapsto \hat c(\nabla)$ to a morphism of [[∞-groupoid]]s

$$
  \hat \mathbf{c} : \mathbf{H}_conn(X,\mathbf{B}G)
  \to \mathbf{H}_{diff}(X,\mathbf{B}^3 U(1))
  \,,
$$

where on the left we have the [[groupoid]] of smooth $G$-principal bundles with connection on $X$, and on the right the 3-groupoid of circle 3-bundles with connection. The [[homotopy fiber]]s of this morphism over a trivial circle 3-bundle with connection are 3-groupoids whose objects are naturally identified with pairs consisting of a connection $\nabla$ on a $G$-bundle and a _trivialization_ of its corresponding Chern-Simons 3-bundle. This in particular implies a trivialization of the underlying cocycle in degree 4 [[integral cohomology]] and therefore defines a [[string structure]]. One calls these homotopy fibers therefore [[differential string structure]]s.

## Constructions

### In differential geometry -- For Lie groups

#### In Cech-Deligne cohomology {#InCechDelineCohomology}

In ([Brylinski-McLaughlin I](#GeomConstructionFirst)) there is spelled out an explicit construction of $\hat c(\nabla)$ for given $\nabla$ in [[Cech cohomology|Cech]]-[[Deligne cohomology]]. This is a special case of the general construction presented in ([Brylinski-McLaughlin II](#CechCocyclesForCharClasses)).

In this section here we review this explicit cocycle construction. In the [next section](#InInfChernWeil) we discuss a systematic way to derive this construction.

Assume that $G$ is a [[simply connected]] compact simple [[Lie group]], such as the [[spin group]], and take the characteristic class $c$ to be that whose [[transgression]] to $G$ has as image in [[de Rham cohomology]] the de Rham class of the normalized canonical [[Lie algebra cohomology|Lie algebra cocycle]] $\mu \in CE(\mathfrak{g})$.

For $P \to X$ a $G$-bundle with connection $\nabla$, there exists an [[open cover]] $\{U_i \to X\}$ such that we have a [[Cech cohomology]] cocycle for $P$ given by a smooth transition function

$$
  g_{i,j} : U_i \cap U_j \to G
$$

satisfying on $U_i \cap U_j \cap U_k$ the cocycle condition $g_{i j} \cdot g_{j k} = g_{i k}$.

Since $G$ is assumed connected and [[simply connected]] and since for every Lie group the second [[homotopy group]] is trivial we have that the first nonvanishing homotopy group of $G$ is the third one.

Therefore we can always find (possibly after refining the cover) a lift of this cocycle as follows:

* on double intersections, choose smooth functions

  $$
    \hat g_{i j} : (U_i \cap U_j) \times \Delta^1 \to G
  $$

  such that $\hat g_{i j}(x,0 ) = e$ is the identity in $g$, and such that such that $\hat g_{i j}(x,1) = g_{i j}(x)$;

* on triple intersections, choose smooth functions

  $$
    \hat g_{i j k} : (U_i \cap U_j \cap U_k) \times \Delta^2 \to G
  $$

  that cobound these paths in the evident way

  $$
    \array{
      && g_{i j}
      \\
      & {}^{\mathllap{\hat g_{i j}}}\nearrow
      &\Downarrow^{\mathrlap{\hat g_{i j k}}}&
      \searrow^{g_{i j} \cdot \mathrlap{\hat g_{j k}}}
      \\
      e &&\underset{\hat g_{i k}}{\to}&& g_{i k}
    }.
  $$

  (This can be done because $\pi_1(G) = *$.)

* on quadruple intersection choose smooth functions

  $$
    \hat g_{i j k l} : (U_i \cap U_j \cap U_k \cap U_l) \times \Delta^3  \to G
  $$

  such that these 3-balls fill the evident tetrahedra.

  (This can be done because $\pi_2(G) = 0$.)

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

The [[Cech cohomology]] cocycle with coefficients in $\mathbf{B}^3 \mathbb{R}/\mathbb{Z}$ which is given by

$$

  c(g)_{i j k l} := \int_{\Delta^3} \hat g_{i j k l}^* \mu
  \;\;\;\;\; mod \mathbb{Z}
$$

is well defined and represents the class $c(P) \in H^4(X)$ in [[integral cohomology]].

Moreover, this refines to the cocycle in Cech-[[Deligne cohomology]] that is given by

$$
  \left(
     CS(\sigma_i^* A)
     \,,\;\;
     \int_{\Delta^1} CS((P\cdot \hat g_{i j})^* A)
     \,,\;\;
     \int_{\Delta^2} CS((P\cdot \hat g_{i j k})^* A)
     \,,\;\;
     \int_{\Delta^3} \hat g_{i j k l}^* \mu
     \;\;\; mod \mathbb{Z}
  \right)
  \,,
$$

where

* $A \in \Omega^1(P,\mathfrak{g})$ is the incarnation of the connection $\nabla$ as an [[Ehresmann connection]] given by a glbally defined 1-form on the total space $P \to X$ of the bundle;

* $\sigma_i : U_i \to P$ are the local [[section]]s of $P \to X$ that induce the original Cech cocycle $(g_{i j} := \sigma_i \cdot \sigma_j^{-1})$;

* $P \cdot \hat g_{i j} : \Delta^1 \to P$ is given by the right [[action]] of $G$ on $P$ and analogously for the other terms;

* $CS(...)$ denotes the [[Chern-Simons form]] of the given $\mathfrak{g}$-valued 1-form.

=--

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

First notice that this is indeed well-defined: by compactness and simplicty of $G$ we have $\pi_3(G) = \mathbb{Z}$. By assumption on $\mu \in \Omega^3(G)$, for any map $f : S^3 \to G$, we have $\int_{S^3} f^*\mu \in \mathbb{Z} \subset \mathbb{R}$. This implies that $c(g)$ is indeed a Cech cocycle.

Then the proof is effectively just the observation that the given collection of differential forms indeed does refine this to a Cech-cocycle with coefficients in the Deligne complex, and that therefore we can read off the image of the integral cohomology class $[c(g)]$ in de Rham cohomology from the curvature 4-form of this Deligne cocycle. That is by construction $\langle F_\nabla \wedge F_\nabla \rangle \in \Omega^4_{cl}(X)$, which by [[Chern-Weil theory]] is indeed the image of the claimed integral class.

=--

So the only mystery about this construction is really: where does it come from? Apart from making this clever Ansatz and checking that it works, can one somehow systematically derive this construction? This we shall try to answer the section [below](#InInfChernWeil).

#### In $\infty$-Chern-Weil theory {#InInfChernWeil}

The above Cech-Deligne cocycle construction of $\hat c(\nabla)$ may be understood as a special case of the general construction of Chern-Weil homomorphisms by the methods discussed at [[∞-Chern-Weil theory]].

We briefly recall the general approach and then spell out the details.

The basic ingredients in [[∞-Chern-Weil theory]] that give the refinement of a [[characteristic class]] to a morphism

$$
  \mathbf{H}_{conn}(X,\mathbf{B}G) \to \mathbf{H}_{diff}(X, \mathbf{B}^k U(1))
$$

from the $G$-principal bundles with connection to [[ordinary differential cohomology]] are these:

1. for a given [[Lie algebra]] $\mathfrak{g}$ the realization of the corresponding [[Lie group]] as a truncation of the simplicial presheaf

   $$
     (U,[n]) \mapsto \{\Omega^\bullet(U\times \Delta^n) \leftarrow CE(\mathfrak{g})\}
   $$

   (see [[Lie integration]]);

1. the observation that, up to subtleties with the truncation, a [[Lie algebra cohomology|Lie algebra cocycle]]

   $$
     CE(\mathfrak{g}) \leftarrow CE(b^{k-1}\mathbb{R}) : \mu
   $$

   induces therefore an integrated cocycle $\mathbf{B}G \to \mathbf{B}^k U(1)$;

1. the observation that this is lifted to connections and differential refinement by

   1. thickening the simplicial presheaf for $\mathbf{B}G$ to

      $$
        (U,[n]) \mapsto
        \left\{
          \array{
              C^\infty(U) \otimes \Omega^\bullet(\Delta^n)
              &\stackrel{}{\leftarrow}&
              CE(\mathfrak{g})
              \\
              \uparrow && \uparrow
              \\
              \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^n)
              &\stackrel{A}{\leftarrow}&
              W(\mathfrak{g})
          }
        \right\}
      $$

   1. thickening the Lie algebra cocycle by its Chern-Simons element

$$
        \array{
           CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R})
           \\
           \uparrow && \uparrow
           \\
           W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}&
           W(b^{k-1} \mathbb{R})
 }
      $$

   and then postcomposing with that. Note that the above diagram is part of a larger diagram involving the invariant polynomial $\langle-\rangle$ for $\mu$ and exhibiting the Chern-Simons element as a transgression element between these two:

$$
        \array{
           CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R})
           \\
           \uparrow && \uparrow
           \\
           W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}&
           W(b^{k-1} \mathbb{R})
\\
 \uparrow && \uparrow
           \\
           inv(\mathfrak{g}) &\stackrel{\langle-\rangle}{\leftarrow}&
           inv(b^{k-1} \mathbb{R})
        }
      $$
Also note that $inv(b^{k-1} \mathbb{R})\cong CE(b^k \mathbb{R})$.

For the case at hand, let $\mathfrak{g}$ be a [[semisimple Lie algebra]], $\langle -\rangle : CE(b^3\mathbb{R})\to W(\mathfrak{g})$ its canonical [[Killing form]] [[invariant polynomial]], $\mu = \langle -,[-,-]\rangle: CE(b^2\mathbb{R})\to CE(\mathfrak{g})$ the corresponding [[Lie algebra cohomology|Lie algebra cocycle]], $cs: W(b^2\mathbb{R})\to W(\mathfrak{g})$ the Chern-Simons elements exhibiting the transgression between the two, $G$ the [[simply connected]] [[Lie group]] [[Lie integration|integrating]] it.

First consider the bare cocycle for the Chern-Simons circle 3-bundle as the Lie integration of the cocycle  $\mu$.

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

Consider the [[simplicial presheaf]]

$$
  \exp(\mathfrak{g}) : (U,[n])
   \mapsto
  \{C^\infty(U) \otimes \Omega^\bullet(\Delta^n) \leftarrow CE(\mathfrak{g})\}
  \,,
$$

where here and in what follows differential forms $\omega$ on simplices are taken to have _sitting instants_ in that for all $k \in \mathbb{N}$ there exists for every $k$-face of $\Delta^n$ an open neighbourhood such that $\omega$ restricted to that open neighbourhood is constant in the direction perpendicular to the boundary.

=--

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

The canonical map

$$
  \mathbf{cosk}_3 \exp(\mathfrak{g}) \to \mathbf{B}G
$$

from the 3-[[coskeleton]] of $\exp(\mathfrak{g})$ to the [[delooping]] of the simply connected Lie group $G$ which is given on 1-morphisms by [[higher parallel transport]] is an equivalence ( in the [[model structure on simplicial presheaves]] $[CartSp^{op}, sSet]_{proj}$).

=--

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

Use that a $\mathfrak{g}$-valued 1-form on the interval is canonically identified with a based path in $G$. Then use that for $k \leq 2$ we have $\pi_k(G) = 0$. See [[Lie integration]] for more.

=--

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

There is a commuting diagram

$$
  \array{
    \exp(\mathfrak{g}) &\stackrel{\exp(\mu)}{\to}& \exp(b^2 \mathbb{R})
    \\
    \downarrow && \downarrow^{\mathrlap{\int_{\Delta^\bullet}}}
    \\
    \mathbf{cosk}_3 \exp(\mathfrak{g}) &\stackrel{}{\to}&
    \mathbf{B}^3 \mathbb{R}/\mathbb{Z}
  }
  \,,
$$

where the right vertical morphism is the composite of the equivalence

$$
  \int_{\Delta^\bullet}  : \exp(b^2 \mathbb{R}) \stackrel{\simeq}{\to} \mathbf{B}^3 \mathbb{R}
$$

discussed at Integration to Line n-groups with the evident quotient $\mathbf{B}^3 \mathbb{R} \to \mathbf{B}^3 \mathbb{R}/\mathbb{Z}$, where the copy of $\mathbb{Z}$ in $\mathbb{R}$ is the lattice of periods of $\mu$ over 3-spheres in $G$.

=--

The top morphism sends a $U$-family of 3-morphisms $\Omega^\bullet(U \times \Delta^3) \stackrel{A}{\leftarrow} CE(\mathfrak{g})$ -- which we may  think of as a $U$-family of based 3-balls $\Sigma : U \times \Delta^3 \to G$ -- to the family of 3-forms

$$
  \Omega^\bullet(U \times \Delta^k)_{vert} \stackrel{A}{\leftarrow} CE(\mathfrak{g})
  \stackrel{\mu}{\leftarrow}
  CE(b^2 \mathbb{R})
  :
  \mu^*(A)
  \,.
$$
which we may think as a family of closed 3-forms

$$
\mu^*(A)\in \Omega^\bullet(U \times \Delta^k)_{vert}
$$

The right vertical morphism sends this to the [[fiber integration]]

$$
  U \mapsto \int_{\Delta^3} \mu^*(A) \;\;\; \in C^\infty(U;\mathbb{R})
$$

and regards the result then modulo $\mathbb{Z}$. That this indeed gives a morphism down at the bottom is the statement that for a 4-morphism in $\mathbf{cosk}_3 \exp(\mathfrak{g})$ -- which is a 3-sphere $V : S^3 \to G$ -- we have that $\int_{S^3} V^* \mu^*(A) = 0 \;mod\; \mathbb{Z}$, which is true by the fact that we take $\mathbb{Z}$ to be precisely generated by these periods. (Alternatively we can assume $\mu$ to be  normalized such that it generates the image in deRham cohomology of $H^3(G,\mathbb{Z}) \simeq \mathbb{Z}$.)

We shall by slight abuse of notation write $\exp(\mu)$ also for the morphism $\mathbf{cosk}_3 \exp(\mathfrak{g}) \to \mathbf{B}^3 \mathbb{R}/\mathbb{Z}$.

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

For $\{U_i \to X\}$ a [[cover]] and $C(U) \in [CartSp^{op}, sSet]_{proj}$ the corresponding [[Cech nerve]] we have that

* a morphism $g : C(U) \to \mathbf{B}G$ is precisely a Cech 1-cocycle with values in $G$;

* a lift

  $$
    \array{
       && \mathbf{cosk}_3 \exp(\mathfrak{g})
       \\
       & {}^{\mathllap{\hat g}}\nearrow & \downarrow^{\mathrlap{\simeq}}
       \\
       C(U) &\stackrel{g}{\to}& \mathbf{B}G
    }
  $$

  is precisely a lift of this cocycle to a system of paths, triangles and tetrahedra in $G$, as [above](#InCechDelineCohomology).

=--

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

Write $\exp(\mathfrak{g})_{diff}$ for the [[simplicial presheaf]]

$$
  (U,[n]) \mapsto
  \left\{
    \array{
      \Omega^\bullet(U \times \Delta^k)_{vert}
      &\stackrel{}{\leftarrow}& CE(\mathfrak{g})
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U \times \Delta^k)
      &\leftarrow&
      W(\mathfrak{g})
    }
  \right\}
  \,.
$$

=--

Its 3-[[coskeleton]] $\mathbf{cosk}_3 \exp(\mathfrak{g} \to inn(\mathfrak{g}))$ is the coefficient for $G$-principal bundles with [[pseudo-connection]] adapted to the model $\mathbf{cosk}_3 \exp(\mathfrak{g})$ for $\mathbf{B}G$.

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

Pseudo-connections $\hat \nabla$

$$
  \array{
    && \mathbf{cosk}_3 \exp(\mathfrak{g})_{diff}
    \\
    &{}^{\mathllap{\hat \nabla}}\nearrow& \downarrow^{\mathrlap{\simeq}}
    \\
    && \mathbf{cosk}_3 \exp(\mathfrak{g})
    \\
    & {}^{\mathllap{\hat g}}\nearrow & \downarrow^{\mathrlap{\simeq}}
    \\
    C(U) &\stackrel{g}{\to}& \mathbf{B}G
  }
$$

which are _genuine_ $\infty$-connections in that their curvature components have no leg along the simplicial directions are in bijection with ordinary connections on the $G$-bundle given by $C(U) \to \mathbf{B}G$.

=--

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

On single patches $\hat \nabla$ is a collection of $\mathfrak{g}$-valued 1-forms $A_i \in \Omega^1(U_i, \mathfrak{g})$.

On double intersection it is a collection

$$
  \hat A_{i j} = A_{i j} + \lambda_{i j} \in
  \Omega^1(U_i \cap U_j, \mathfrak{g})\otimes C^\infty(\Delta^1) \oplus
   C^\infty(U)\otimes \Omega^1(\Delta^1, \mathfrak{g})
  \subset \Omega^1(U_i \cap U_j  \times \Delta^1, \mathfrak{g})
$$

whose restriction $\lambda_{i j}$ to $\Delta^1$ is the given path $\hat g_{i j}$ that is being covered. The condition that the curvature of $\hat A_{i j}$ has no component in the simplicial direction is the [[differential equation]]

$$
  \frac{\partial}{\partial t} A_{i j} = d_U (\lambda_{i j})_t + [(\lambda_{i j})_t, A_{i j}]
 \,.
$$

This differential equation has a unique solution for the boundary condition $A_{i j}(0) = A_i$ given by

$$
  A_{i j}(t) = \hat g_{i j}(t)^{-1}(A_i + d)\hat g_{i j}(t)
  \,.
$$

(To see this, use the formulas from [[parallel transport]]. If we assume just for notational simplicity that we are dealing with a [[matrix Lie algebra]] then we have $\frac{\partial}{\partial t} \hat g_{i j} = \hat g_{i j} \cdot \lambda$ (by definition) and using that the claim follows.)

In particular this implies  the forms on single patches satisfy the ordinary cocycle relation

$$
  A_{i j}(1) = A_j = g_{i j}^{-1}(A_i + d)g_{i j}
$$

for connections.

Similarly there are differential equations on 2-simplices and 3-simplices with unique solutions.

=--

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

Pasting postcomposition with the diagram

$$
  \array{
     CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^2 \mathbb{R})
     \\
     \uparrow && \uparrow
     \\
     W(\mathfrak{g}) &\stackrel{(cs,\langle \rangle)}{\leftarrow}&
     W(b^2 \mathbb{R})
  }
$$

induces a morphism $\exp(\mathfrak{g})_{diff} \to \exp(b^2\mathbb{R})_{diff}$ and we obtain a commuting diagram

$$
  \array{
     \exp(\mathfrak{g})_{diff} &\to& \exp(b^2\mathbb{R})_{diff}
     \\
     \downarrow && \downarrow
     \\
     \mathbf{cosk}_3\exp(\mathfrak{g})_{diff} &\to&
     \mathbf{B}^3 U(1)_{chn,diff}
  }
$$

that covers the corresponding diagram we had before.

=--

Here we are using the object $\mathbf{B}^3 U(1)_{ch,diff}$ described in detail at [[circle n-bundle with connection]].

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

The deeper reason for this construction is that the zig-zag composite

$$
  \mathbf{B}G
  \stackrel{\simeq}{\leftarrow}
  \mathbf{cosk}_3\exp(\mathfrak{g})_{diff}
  \to
  \mathbf{B}^3 U(1)_{diff,chn}
  \to
  \mathbf{\flat}_{dR}\mathbf{B}^4 U(1)_{chn}
$$

of morphisms of simplicial presheaves models the intrinsically defined morphism

$$
  \mathbf{B}G \to \mathbf{\flat}_{dR}\mathbf{B}^4 U(1)
$$

in the [[(∞,1)-topos]] [[Smooth∞Grpd]].

=--

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

The outer composite morphism

$$
  \array{
    && \mathbf{cosk}_3 \exp(\mathfrak{g})_{diff}
    &\stackrel{\exp((cs,\langle -\rangle))}{\to}&
    \mathbf{B}^3 U(1)_{diff}
    \\
    &{}^{\mathllap{\hat \nabla}}\nearrow& \downarrow^{\mathrlap{\simeq}}
    \\
    && \mathbf{cosk}_3 \exp(\mathfrak{g})
    \\
    & {}^{\mathllap{\hat g}}\nearrow & \downarrow^{\mathrlap{\simeq}}
    \\
    C(U) &\stackrel{g}{\to}& \mathbf{B}G
  }
$$

is precisely the Cech-Deligne cocycle

$$
  (CS(A_i), \int_{\Delta^1} CS(\hat A_{i j} ), \int_{\Delta^2} CS(\hat A_{i j k}), \int_{\Delta^3} \mu(A_{i j k l}))
  \,.
$$

This is is exactly equal to the cocycle discussed [above](#InCechDelineCohomology).

=--

Notice by the way that this construction also serves as a manifest proof that this collection of data indeed does constitute a Deligne cocycle.

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

This is a matter of plugging the above pieces into each other. For instance, on double intersections we have that the 3-form $CS(\hat A_{i j})$ is the image of the degree 3-generator on $W(b^2 \mathbb{R})$ under the composite

$$
  \Omega^\bullet(U \times \Delta^k)_{vert}
  \stackrel{\hat A_{i j}}{\leftarrow}
  W(\mathfrak{g})
  \stackrel{(cs,\langle -\rangle)}{\leftarrow}
  W(b^2 \mathbb{R})
  \,.
$$

The remaining fiber integration is then that exhibiting the equivalence of simplicial differential forms

$$
  \int_{\Delta^\bullet} :
  \mathbf{\flat}_{dR} \mathbf{B}^3 U(1)_{diff,simp}
    \stackrel{\simeq}{\to}
  \mathbf{\flat}_{dR} \mathbf{B}^3 U(1)_{diff,chn}
$$

that is described in some detail at Circle n-bundle with connection -- models from ∞-Lie integration.

=--

#### As a bundle 2-gerbe

We indicate (for the moment) the way the Chern-Simons 3-bundle is realized as a [[bundle 2-gerbe]] (for instance in [CJMS](#CJMS) and [Waldorf CS](#WaldorfCS)) .

One first constructs the canonical [[bundle gerbe]] $\mathcal{G} \to G$ on the Lie group and notices that (more or less implicitly by recourse to its [[delooping]] 2-gerbe on $\mathbf{B}G$) that this has a _multiplicative structure_ .

Using this we see that for $P \to X$ any $G$-[[principal bundle]] and $P^{[2]} : = P \times_X P \to P \times G$ the principality isomorphism, the pullback of $\mathcal{G}$ along

$$
  f : P^{[2]} \to P \times G \stackrel{p_2}{\to} G
$$

serves to provide the diagram

$$
  \array{
     f^* \mathcal{G}
     \\
     \downarrow
     \\
     P^{[2]} &\stackrel{\to}{\to}& P
     \\
     && \downarrow
     \\
     && X
  }
$$

on which the pullback of the multiplicative structure on $\mathcal{G}$ induces the structure of a bundle 2-gerbe, in that we get morphisms of bundle gerbes

$$
  \mu : \pi_0^{\ast} f^{\ast} \mathcal{G} \otimes \pi_2^{\ast} f^{\ast} \mathcal{G} \to \pi_2^* f^* \mathcal{G}
$$

that are associative up to a higher coherent morphisms, etc.

### In complex analytic geometry -- For complex Lie groups

(...) ([Brylinski 00](#Brylinski00)) (...)

### In arithmetic geometry -- For reductive groups
 {#ForReductiveAlgebraicGroups}

For [[reductive algebraic group]] $G$ there is no sensible element in $H^3(\mathbf{B}G, \mathbb{G}_m)$, but there is the following.

Write $K_2(R)$ for the degree-2 [[algebraic K-theory]] [[group]] of a [[commutative ring]]  (e.g. [Isely 05, section 4](algebraic+K-theory#Isely05)) and write $\mathbf{K}_2$ for corresponding [[abelian sheaf]] on the suitable [[etale site]] (e.g. [Deligne-Brylinski 01, page 6](#DeligneBrylinski01)).

Then

$$
  H^4(B G_{\mathbb{C}}, \mathbb{Z})
  \simeq
  H^2(\mathbf{B}G, \mathbf{K}_2)
$$

This is ([HKLV 98, theorem 4.11](#HKLV98)) also ([Deligne-Brylinski 01](#DeligneBrylinski01)), going back to ([Bloch 80](#Bloch80)). See also ([Kapranov 00, (2.1)](#Kapranov00), [MO discussion](http://mathoverflow.net/a/24862/381)).

Notice that over $\mathbb{C}$ the [[Beilinson regulator]] $c_{1,2}$ (e.g. [Brylinski 94, theorem, 3.3](#Brylinski94)) relates

$$
  H^1(X, \mathbf{K}_2) \longrightarrow H^2(X, \mathbb{G}_m)
  \,.
$$

## Related concepts

* The [[higher parallel transport]] of a Chern-Simons circle 3-bundle is the [[action functional]] for [[Chern-Simons theory]].

* For the case that $G = O$ is the [[orthogonal group]] and $X \to \mathbf{B}O$ the classifying map of the [[tangent bundle]] of $X$, a trivialization of the corresponding Chern-Simons 3-bundle is a [[string structure]] on $X$. A trivialization of the Chern-Simons 3-bundle _with connection_ is a [[differential string structure]] on $X$.

  For products of torus groups and the [[cup product]] class the same construction yields the [[T-duality 2-group]]

* [[universal Chern-Simons n-bundle]]

  * **Chern-Simons circle 3-bundle**

  * [[Chern-Simons circle 7-bundle]]

The CS 3-bundle 3-connection is the [[extended Lagrangian]] for ordinary $G$-[[Chern-Simons theory]]. See there for more.

* [[supergravity C-field]]

* [[Yang monopole]]

## References

### For Lie groups

#### The Cech-Deligne cohomology realization

As cocycles in [[Cech cohomology|Cech]]-[[Deligne cohomology]] the Chern-Simons 2-gerbe has been constructed explicitly in

* {#GeomConstructionFirst} [[Jean-Luc Brylinski]] and Dennis McLaughlin, _A geometric construction of the first Pontryagin class_ (1993) ([pdf](http://www.math.uni-hamburg.de/home/schreiber/Brylinski-McLaughlin-I.pdf))

as a special case of the general construction in

* [[Jean-Luc Brylinski]] and Dennis McLaughlin, _Cech cocycles for characteristic classes_ ,  Communications in Mathematical Phiysics, Volume 178, Number 1, ([Springer](http://www.springerlink.com/content/762g1m76jp6425x5/))
{#CechCocyclesForCharClasses}

#### The 2-gerbe realization {#2GerbeReferences}

Conceived as a genuine [[gerbe]] the Chern-Simons 2-gerbe appears in

* [[Jean-Luc Brylinski]] and Dennis McLaughlin, _The geometry of degree-4 characteristic classes and of line bundles on loop spaces II_ ([pdf](http://www.math.uni-hamburg.de/home/schreiber/Brylinski-McLaughlin-Duke-II.pdf)).

Among the first references to apply specifically [[bundle gerbe]] technology to this construction is

* [[Alan Carey]], Stuart Johnson, [[Michael Murray]], [[Danny Stevenson]] and [[Bai-Ling Wang]], _Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories_ Communications in Mathematical Physics, 259 (3). (2005) ([arXiv]())
{#CJMS}

This was later refined in

* [[Konrad Waldorf]], _Multiplicative Bundle Gerbes with Connection_ ([arXiv:0804.4835](http://arxiv.org/abs/0804.4835))

Here are some slides from talks:

* [[Konrad Waldorf]], _Multiplicative gerbes and Chern-Simons theory_ ([pdf](http://www.konradwaldorf.de/docs/bonn.pdf))
{#WaldorfCS}

* [[Krzysztof Gawedzki]], _Wess-Zumino-Witten and Chern-Simons theories for non-simply connected Lie groups_ ([pdf](http://dftuz.unizar.es/ftzar/activities/highenergy09_talks/gawedzki.pdf))

The full Chern-Simons circle 3-connection on the full [[moduli stack]] of $G$-[[principal connections]] $\mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$ was then constructed in

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech Cocycles for Differential characteristic Classes]]_, Advances in Theoretical and Mathematical Phyiscs, Volume 16 Issue 1 (2012) ([arXiv:1011.4735](http://arxiv.org/abs/1011.4735))

Exposition of this and further developments are in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_

### For complex reductive groups
 {#ReferencesForComplexReductiveGroups}

Discussion in [[complex analytic geometry]] of multiplicative Chern-Simons [[holomorphic line 2-bundles]] is in

* [[Jean-Luc Brylinski]], around theorem 5.4.10 (p. 226-227) of _Loop spaces and characteristic classes_, Birkhäuser

* {#Brylinski00} [[Jean-Luc Brylinski]], _Gerbes on complex reductive Lie groups_ ([arXiv:math/0002158](http://arxiv.org/abs/math/0002158))

The relevant [[Beilinson regulator]] is discussed also in

* {#Brylinski94} [[Jean-Luc Brylinski]], _Holomorphic gerbes and the Beilinson regulator_, Astérisque 226 (1994): 145-174 ([[Brylinski94.pdf:file]])

### For reductive algebraic groups

For the case of [[reductive algebraic groups]]:

* {#Bloch80} [[Spencer Bloch]], _The dilogarithm and extensions of Lie algebras_, Algebraic K-Theory Evanston 1980 Lecture Notes in Mathematics Volume 854, 1981, pp 1-23 ([publisher](http://link.springer.com/chapter/10.1007%2FBFb0089515))

* {#HKLV98} [[Hélène Esnault]], [[Bruno Kahn]], [[Marc Levine]], [[Eckart Viehweg]], _The Arason invariant and mod 2 algebraic cycles_,  J. Amer. Math. Soc. 11 (1998), 73-118  ([pdf](https://www.uni-due.de/~bm0032/publ/Arason.pdf),[publisher page](http://www.ams.org/journals/jams/1998-11-01/S0894-0347-98-00248-3/))

* {#Kapranov00} [[Mikhail Kapranov]], _The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups_ ([arXiv.math/0001005](http://arxiv.org/abs/math/0001005))

* {#DeligneBrylinski01} [[Pierre Deligne]], [[Jean-Luc Brylinski]], _Central extensions of reductive groups by $K_2$_, Publications Mathématiques de l'IHÉS 2001 ([web](http://publications.ias.edu/node/424), [pdf](http://publications.ias.edu/sites/default/files/78_ReductiveGroupsK2.pdf))

* [MO discussion](http://mathoverflow.net/a/24862/381)

[[!redirects Chern-Simons bundle 2-gerbe]]
[[!redirects Chern-Simons bundle 2-gerbes]]

[[!redirects Chern-Simons 2-gerbe]]
[[!redirects Chern-Simons 2-gerbes]]

[[!redirects Chern-Simons circle 3-bundle with connection]]
[[!redirects Chern-Simons circle 3-bundles with connection]]

[[!redirects universal Chern-Simons circle 3-connection]]
[[!redirects universal Chern-Simons circle 3-connections]]

[[!redirects universal Chern-Simons 3-connection]]
[[!redirects universal Chern-Simons 3-connections]]

[[!redirects universal Chern-Simons circle 3-bundle with connection]]
[[!redirects universal Chern-Simons circle 3-bundles with connection]]

[[!redirects Chern-Simons line 3-bundle]]
[[!redirects Chern-Simons line 3-bundles]]


universal Chern-Simons n-bundle

#Contents#
* table of contents
{:toc}

## Idea

[[principal infinity-bundle]] on [[moduli infinity-stack]] modulated by [[universal characteristic map]]

$$
  \array{
    P
    \\
    \downarrow
    \\
    \mathbf{B}G
    &\stackrel{\mathbf{c}}{\to}&
    \mathbf{B}^n U(1)
  }
$$

## Examples

* [[Chern-Simons circle 3-bundle]]

* [[Chern-Simons circle 7-bundle]]

## Related concepts

[[!include gauge field - table]]

[[!redirects universal Chern-Simons n-bundles]]


universal class

\tableofcontents

## Idea

In [[class theory]], a universal class is a "[[class]] of all [[sets]]".

## Definition

### In material class theory

to be defined...

### In structural class theory

A **universal class** is a [[class]] $U$ such that for all classes $C$, there is a [[monic]] class [[map]] $U \hookrightarrow C$.

There should also be a definition from a hypothetical "[[division allegory]] with class structure" rather than a [[category with class structure]].

## See also

* [[class theory]]

* [[category with class structure]]

* [[universe]]

## References

* [[Steve Awodey]]. *Notes on algebraic set theory*, Notes for lectures given at the Summer School on Topos Theory, Haute-Bodeux, Belgium. May 29 to June 5, 2005. Carnegie Mellon University Technical Report No. CMU-PHIL-170. June 2005. ([pdf](https://www.phil.cmu.edu/projects/ast/Papers/bnotes.pdf))

[[!redirects class of all sets]]


universal closure operator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

If one considers a [[hyperdoctrine]] of [[subobject lattices]], hence a collection of them parameterized over a [[category of contexts]] and equipped with [[pullback]]/[[substitution]]/[[context extension]], then a _universal closure operator_ or _[[modality]]_ is a [[closure operator]] acting on each of the slices and being compatible with the pullback operation. A hyperdoctrine equipped with such an operator is sometimes called a _[[modal hyperdoctrine]]_.

If there is moreover a [[subobject classifier]] $\Omega$ (hence if that hyperdoctrine is the collection of [[slice categories]] of a [[topos]]) then a universal closure operator is represented by a single [[monad]]/[[comonad]] on that subobject classifier, $\Diamond \colon \Omega \to \Omega$, see the discussion [below](DefinitionForReflectiveSubcategories).

Instead of [[subobjects]] one may consider more generally closures acting on the full [[slice categories]]. This promotes the corresponding _[[modal logic]]_ to _[[modal type theory]]_.  Unlike in the posetal case, such a monad will not automatically be [[idempotent monad|idempotent]], but often one requires this explicitly.

## Definition

### For reflective subcategories
 {#DefinitionForReflectiveSubcategories}

Let $Sh_j(\mathcal{C}) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} \mathcal{E}$ be a [[reflective subcategory]] of a [[topos]] $\mathcal{E}$.

Here we discuss explicit translations between the structure given by the [[localization|reflector]] $L$ and the corresponding closure operator $j \colon \Omega \to \Omega$ (the [[Lawvere-Tierney topology|Lawvere-Tierney operator]]) in a way that makes the relation to [[modal type theory]] and [[monads (in computer science)]] most manifest.

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

Given a reflector $\sharp : \mathcal{E} \stackrel{L}{\to} Sh_j(\mathcal{E}) \hookrightarrow \mathcal{E}$, define for each object $X \in \mathcal{E}$ a **closure operator**,
being a [[functor]] on the [[poset of subobjects]] of $X$

$$
  c_L : Sub(X) \to Sub(X)
  \,,
$$

by sending any [[monomorphism]] $A \hookrightarrow X$ classified by a [[characteristic function]] $\chi_A : X \to \Omega$ to the [[pullback]] $c_L(A)$ in

$$
  \array{
    c_L(A) &\to& \sharp A
    \\
    \downarrow && \downarrow
    \\
    X &\to& \sharp X
  }
  \,,
$$

where $X \to \sharp X$ is the [[unit of an adjunction|adjunction unit]].

=--

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

This is well defined. Moreover, $c_L$
commutes with [[pullback]] ([[change of base]]).

=--

This appears as ([Johnstone, lemma A4.3.2](#Johnstone)).

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

A family of functors $Sub(X) \to Sub(X)$ for all objects $X$ that commutes with [[change of base]] is called a **universal closure operation**.

=--

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

Given a [[left exact functor|left exact]] reflector $\sharp$ as above with induced closure operation $c_L$, the corresponding Lawvere-Tierney operator $j : \Omega \to \Omega$ is given as the composite

$$
  j : \Omega \to \sharp \Omega \stackrel{\chi_{\sharp true}}{\to} \Omega
  \,,
$$

where

* $\Omega \to \sharp \Omega$ is the [[unit of an adjunction|adjunction unit]];

* $\chi_{\sharp true} : \sharp \Omega \to \Omega$ is the [[characteristic function]] of the result of applying $\sharp$ to the [[subobject classifier|universal subobject]]

  $$
    (* \stackrel{\sharp true}{\hookrightarrow} \sharp \Omega)
     :=
    \sharp (* \stackrel{true}{\hookrightarrow} \Omega)
  $$

  (which is again a [[monomorphism]] since $\sharp$ preserves [[pullbacks]]).

=--

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

For $A \hookrightarrow X$ any [[subobject]] with [[characteristic function]] $\chi_A : X \to \Omega$, we need to show that we have a [[pullback]] [[diagram]]

$$
  \array{
    c_L(A) &\to& &\to& &\to& *
    \\
    \downarrow && && && \downarrow
    \\
    X &\stackrel{\chi_A}{\to}& \Omega
    &\stackrel{}{\to}&
    \sharp \Omega
    &\stackrel{}{\to}&
    \Omega
  }
  \,.
$$

The pullback along the rightmost morphism is by definition $# * \to \sharp \Omega$

$$
  \array{
    c_L(A) &\to& &\to& # * = * &\to& *
    \\
    \downarrow && && \downarrow && \downarrow
    \\
    X &\stackrel{\chi_A}{\to}& \Omega
    &\stackrel{}{\to}&
    \sharp \Omega
    &\stackrel{}{\to}&
    \Omega
  }
  \,.
$$

Moreover, by the [[natural transformation|naturality]] of the [[unit of an adjunction|adjunction unit]] we have a [[commuting diagram]]

$$
  \array{
    X &\to& \sharp X
    \\
    {}^{\mathllap{\chi_A}}\downarrow && \downarrow^{\mathrlap{\sharp \chi_A}}
    \\
    \Omega &\to& \sharp \Omega
  }
  \,.
$$

Using this in the remaining bottom morphism of our would-be pullback square we find that equivalently

$$
  \array{
    c_L(A) &\to& &\to& # * = * &\to& *
    \\
    \downarrow && && \downarrow && \downarrow
    \\
    X &\stackrel{}{\to}& \sharp X
    &\stackrel{\sharp \chi_A}{\to}&
    \sharp \Omega
    &\stackrel{}{\to}&
    \Omega
  }
$$

needs to be a pullback diagram. Since $\sharp$ preserves pullbacks we have that also the middle square in

$$
  \array{
    c_L(A) &\to& \sharp A &\to& # * = * &\to& *
    \\
    \downarrow && \downarrow && \downarrow && \downarrow
    \\
    X &\stackrel{}{\to}& \sharp X
    &\stackrel{\sharp \chi_A}{\to}&
    \sharp \Omega
    &\stackrel{}{\to}&
    \Omega
  }
$$

is a pullback. But then also the left square is a pullback, by def. \ref{ClosureOperatorOfReflection}. This finally means, by the [[pasting law]], that also the total rectangle is a pullback.

=--

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

Equivalently, by the [[pasting law]], we have that $j : \Omega \to \Omega$ is the [[characteristic function]] of the $L$-closure, def. \ref{ClosureOperatorOfReflection}, of the universal subobject $* \to \Omega$, because we have a [[pasting diagram]] of [[pullback]] squares

$$
  \array{
    c_L(*) &\to& \sharp * = * &\to & *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \Omega &\to& \sharp \Omega &\stackrel{\chi_{\sharp true}}{\to} & \Omega
  }
  \,.
$$

In this form the statement appears in the proof of ([Johnstone, Theorem A4.3.9](#Johnstone)).

=--

## Examples

* In a [[topos]] a universal closure operator as [above](#DefinitionForReflectiveSubcategories) is also called a  _[[Lawvere-Tierney topology]]_.

* For a [[local topos]] there are the closure operators _[[flat modality]] $\dashv$ [[sharp modality]]_.

## Related concepts

* [[modal type theory]]

## References

Section A4.4 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
 {#Johnstone}

[[!redirects universal closure operator]]
[[!redirects universal closure operators]]
[[!redirects universal closure operation]]
[[!redirects universal closure operations]]


universal coacting bialgebra

There are two closely related constructions, due Yuri Manin, for finitely generated quadratic algebras, and due Tambara, for finite dimensional algebras.

## Tambara's universal coacting bialgebra

If $A$ is a finite dimensional (associative unital) $k$-algebra, and $D$ the functor $D\mapsto D\otimes A$ where $D$ is a $k$-algebras has a left adjoint $a(A,-)$.
$$
Hom_{k-alg}(B,A\otimes D) \cong Hom_{k-alg}(a(A,B),D)
$$
where $A,D$ are arbitrary $k$-algebras.
$a(A,A)$ has a canonical structure of a coalgebra, making it into a $k$-bialgebra, the universal coacting bialgebra.

* [[Daisuke Tambara]], _The coendomorphism bialgebra of an algebra_, J. Fac. Sci. Univ. Tokyo Sect. IA Math, __37__, 425-456, 1990 [pdf](https://repository.dl.itc.u-tokyo.ac.jp/record/39399/files/jfs370210.pdf)

Tambara's construction is dual to the universal [[measuring coalgebra]] of Sweedler.

## Manin's universal coacting bialgebra

In a similar way to above, one utilizes the adjunction between inner hom and $!$ functor for quadratic algebras.

* [[Yu. I. Manin]], _Quantum groups and non-commutative geometry_, CRM, Montreal 1988.

See also [[quantum linear group]].

## Generalizations and analogues

* A. L. Agore, A. S. Gordienko, [[Joost Vercruysse]], _V -universal Hopf algebras (co)acting on Ω-algebras_
Commun. Contemp. Math. 25 (2023), 2150095.
* A. L. Agore, _Functors between representation categories. Universal modules_, [arXiv:2301.03051](https://arxiv.org/abs/2301.03051)

category: algebra

[[!redirects Tambara's universal coacting bialgebra]]