BHK interpretation

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### Context
#### Type theory
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#### Constructivism, Realizability, Computability
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[[!include constructivism - contents]]
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# Contents
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{: toc}

## Idea
 {#Idea}

In [[constructive mathematics]] (originally: [[intuitionistic logic]]), the *BHK interpretation* of [[logical connectives]] (due to [Kolmogorov (1932, p. 59)](#Kolmogorov32), [Troelstra (1969, §2)](#Troelstra69), see the comments on attribution [below](#Attribution)) is such that the resulting [[proposition]] is regarded as [[true]] only if it is possible to [[constructive mathematics|construct]] a [[proof]] of its [[assertion]].

For instance, to assert a [[logical conjunction]] ("and") or a [[universal quantification]] ("for all") is taken to mean to provide a proof of all the instances.

Dually but more notably, to assert a [[logical disjunction]] ("or") or an [[existential quantification]] ("exists") is taken to mean to prove one of the instances, so that there is no intuitionistic existence statement without construction of an example (the "disjunction property", see [there](intuitionistic+logic#DisjunctionProperty)).

This constructive interpretation of logical truth is the crux of the rejection of the [[principle of excluded middle]] in [[intuitionism]]/[[constructive mathematics]], for it implies that to prove $P \vee (\not P)$ (which may superficially/classically seem tautologous) one must prove $P$ or one must prove $\not P$ --- but neither proof may be known (e.g. if $P$ = [[Riemann hypothesis]]).

(Here the classical mathematician is regarded as "idealistic" in their assumption that either case must hold, even if it is impossible to tell which one.)

In short this means that a proposition is regarded a [[true]] if there is an [[algorithm]], hence a [[computable function]],  to realize its [[proof]] whence one also speaks of the *[[realizability]] interpretation*.

Closely related to the point of being synonymous is the paradigm of *[[propositions as types]]* and *[[proofs as programs]]*, also known as the *[[Curry-Howard correspondence]]* in [[type theory]].

Indeed, the fully formal version of the BHK interpretation may be understood as being the [[inference rules]], specifically the [[term introduction rules]], of [[intuitionistic type theory]] (as amplified in [Girard (1989, §2)](#Girard89) and [Martin-Löf (1996, Lec 3)](#MartinLöf96)).

## Historical versions
 {#HistoricalVersions}

The BHK interpretation evolved from a rather informal statement in [Kolmogorov (1932, p. 59)](#Kolmogorov32), over a more pronounced but still informal statement due to [Troelstra (1969, p. 5)](#Troelstra69), to the modern [[inference rules]] in [[intuitionistic type theory]] due to [Martin-Löf 75](Martin-Löf+dependent+type+theory#MartinLof75) (gentle exposition in [Martin-Löf (1996, Lec 3)](#MartinLöf96)):

\linebreak

From [Kolmogorov (1932, p. 59)](#Kolmogorov32):

\begin{imagefromfile}
    "file_name": "KolmogorovIntroducingBHK.jpg",
    "width": 540,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

From [Heyting (1956, p. 97)](#Heyting56):

\begin{imagefromfile}
    "file_name": "HeytingIntroducingBHK.jpg",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

From [Troelstra (1969, p. 5)](#Troelstra69):

\begin{imagefromfile}
    "file_name": "Troelstra-IntroducingBHKInterp.jpg",
    "width": 660,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

From [Troelstra (1977, p. 977)](#Troelstra77):

\begin{imagefromfile}
    "file_name": "Troelstra-AttributingBHK.jpg",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

{#FromTroelstraVanDalen88} From [Troelstra & van Dalen (1988)](#TroelstraVanDalen88):

\begin{imagefromfile}
    "file_name": "Troelstra-VanDalen-BHKInterpretation.jpg",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

{#FromBridges} From [Bridges (1999), p. 96](#Bridges99):

\begin{imagefromfile}
    "file_name": "Bridges-IntuitInterpOfConnectives.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

The analogous discussion for [[inference rules]] in [[intuitionistic type theory]] is then given in [Martin-Löf (1975)](#MartinLof75),
reviewed in [Girard (1989, §2)](#Girard89) and with more emphasis in [Martin-Löf (1996, Lec 3)](#MartinLöf96).

## Attribution
 {#Attribution}

The idea of the interpretation is clearly expressed in [Kolmogorov (1932, p. 59)](#Kolmogorov32), though rather briefly and in unusual terminology: Instead of *propositions*, Kolmogorov speaks of *Aufgaben* (Deutsch for "tasks", but here in the sense used in math classes where it means "exercises" or "mathematical problems") in order to amplify the [[constructive mathematics|constructive]] aspects (he expands it out to *Konstruktionsaufgabe*, meaning "construction task").

While [[Arend Heyting]] may have been speaking of the idea of the "proof interpretation" of propositions since the 1930s, the first recognizable statement of what later was called the BHK interpretation is given in [Heyting (1956, §7.1.1)](#Heyting56).

The statement of the interpretation close to its modern formulation (but clearly of the same conceptual content as previously expressed in [Kolmogorov (1932, p. 59)](#Kolmogorov32) and [Heyting (1956, §7.1.1)](#Heyting56)) was then made explicit in [Troelstra (1969, §2)](#Troelstra69), where it is not attributed to anyone but presented as if the author's invention.

Later, [Troelstra (1977, §2)](#Troelstra77) credits the idea to "Brouwer-Heyting-*Kreisel* (BHK)", not mentioning Kolmogorov --- indeed [Kreisel (1965)](#Kreisel65) is the only reference that [Troelstra (1969, §2)](#Troelstra69) makes explicit use of.
(While this is the origin of the term "BHK interpretation", the adjustment/correction of its expanded version by exchanging "*K*reisel" for "*K*olmogorov" seems to be due to [Troelstra (1990, §2.1)](#Troelstra90).)

Still later, [Girard (1989, §1.2.2)](#Girard89) presents the same idea as "Heyting's semantics" (not mentioning anyone else, nor actually citing Heyting).

More discussion of this history is in *[SEP: "The Development of Intuitionistic Logic"](https://plato.stanford.edu/entries/intuitionistic-logic-development/)*.

Notice that Brouwer never explicitly formulated any interpretation of the kind that [Troelstra (1977, §2)](#Troelstra77) attributes to him, and in fact remained against all formalism his entire life. (His role in this history is to provide motivation and inspiration for Heyting and Kolmogorov.) Moreover, [Escardo & Xu (2015)](#EscardoXu15) have shown that Brouwer's famous intuitionistic theorem "all functions $\mathbb{N}^{\mathbb{N}} \to \mathbb{N}$ are continuous" is actually inconsistent under a literal version of this interpretation (i.e. without including [[propositional truncation]]).

Thus, perhaps it should be called the "Heyting-Kolmogorov" interpretation.

## Related concepts

* [[realizability topos]]

* [[computational trinitarianism]]

## References

Original articles on [[intuitionism]] and, eventually, on [[intuitionistic logic]]:

* [[Arend Heyting]], *Die intuitionistische Grundlegung der Mathematik*, Erkenntnis **2** (1931) 106-115 [[jsotr:20011630](https://www.jstor.org/stable/20011630), [pdf](http://www.psiquadrat.de/downloads/heyting1931.pdf)]

* {#Kolmogorov32} [[Andrey Kolmogorov]], *Zur Deutung der intuitionistischen Logik*, Math. Z. **35** (1932) 58-65 [[doi:10.1007/BF01186549](https://link.springer.com/article/10.1007/BF01186549)]

* [[Hans Freudenthal]], *Zur intuitionistischen Deutung logischer Formeln*, Comp. Math. **4** (1937) 112-116 [[numdam:CM_1937__4__112_0](http://www.numdam.org/item/?id=CM_1937__4__112_0)]

* [[Arend Heyting]], *Bemerkungen zu dem Aufsatz von Herrn Freudenthal "Zur intuitionistischen Deutung logischer Formeln"*, Comp. Math. **4** (1937) 117-118 [[doi:CM_1937__4__117_0](http://www.numdam.org/item/?id=CM_1937__4__117_0)]

* [[L. E. J. Brouwer]], *Points and Spaces*, Canadian Journal of Mathematics **6** (1954) 1-17 [[doi:10.4153/CJM-1954-001-9](https://doi.org/10.4153/CJM-1954-001-9)]

Early monographs:

* {#Heyting56} [[Arend Heyting]], *Intuitionism: An introduction*, Studies in Logic and the Foundations of Mathematics, North-Holland (1956, 1971) [[ISBN:978-0720422399]()]

* {#Kreisel65} [[Georg Kreisel]], Section 2 of: *Mathematical Logic*,  in T. Saaty et al. (ed.), *Lectures on Modern Mathematics III*, Wiley New York (1965) 95-195

* {#TroelstraVanDalen88} [[Anne Sjerp Troelstra]], [[Dirk van Dalen]], Section 3.1 of: *Constructivism in Mathematics -- An introduction*, Vol 1, Studies in Logic and the Foundations of Mathematics **121**, North Holland (1988) [[ISBN:9780444702661](https://www.elsevier.com/books/constructivism-in-mathematics-vol-1/troelstra/978-0-444-70266-1)]

Early historical review:

* {#Troelstra90} [[Anne Sjerp Troelstra]], *On the Early History of Intuitionistic Logic*, in *Mathematical Logic*, Springer, (1990) [[doi:10.1007/978-1-4613-0609-2_1](https://doi.org/10.1007/978-1-4613-0609-2_1)]

Recognizable formulation of the so-called "BHK interretation" first appears in:

*

 [Kolmogorov (1932, p. 59)](#Kolmogorov32)

(who however speaks not of propositions but of *Aufgaben*, i.e. "tasks", here in the sense of: "mathematical problems")

* [Heyting (1956, §7.1.1)](#Heyting56)

(who is maybe the first to speak of the "meaning of logical connectives") and then

* {#Troelstra69} [[Anne Sjerp Troelstra]], §2 of: *Principles of Intuitionism*, Lecture Notes in Mathematics **95** Springer Heidelberg (1969)  [[doi:10.1007/BFb0080643](https://link.springer.com/book/10.1007/BFb0080643)]

(where it is presented as if the author's invention) and then recalled in

* {#Troelstra77} [[Anne Sjerp Troelstra]], §2 of: *Aspects of Constructive Mathematics*, Studies in Logic and the Foundations of Mathematics **90** 973-1052 (1977) [doi:10.1016/S0049-237X(08)71127-3]

(where the same is now called the "Brouwer-Heyting-*Kreisel (BHK)* explanation" --- not mentioning Kolmogorov).

and later recalled in the context of [[constructive analysis]]:

* [[Douglas Bridges]], p. 96 of: *Constructive mathematics: a foundation for computable analysis*, Theoretical Computer Science **219** 1–2 (1999) 95-109 [doi:10.1016/S0304-3975(98)00285-0]

This lead to the formulation of [[intuitionistic type theory]] in

* {#MartinLof75} [[Per Martin-Löf]], _An intuitionistic theory of types: predicative part_, in: H. E. Rose, J. C. Shepherdson (eds.), *Logic Colloquium '73, Proceedings of the Logic Colloquium*, Studies in Logic and the Foundations of Mathematics **80**, Elsevier (1975) 73-118 [doi:10.1016/S0049-237X(08)71945-1, [CiteSeer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.926)]

reviewed in

* {#Girard89} [[Jean-Yves Girard]] (translated and with appendiced by [[Paul Taylor]] and [[Yves Lafont]]), §1.2.2 of: *Proofs and Types*, Cambridge University Press (1989) [[ISBN:978-0-521-37181-0](), [webpage](http://www.paultaylor.eu/stable/Proofs+Types.html), [pdf](https://www.paultaylor.eu/stable/prot.pdf)]

(where it is called *Heyting semantics*) and then in

* {#MartinLöf96} [[Per Martin-Löf]], Lecture 3 of: *On the Meanings of the Logical Constants and the Justifications of the Logical Laws*, Nordic Journal of Philosophical Logic, **1** 1 (1996) 11-60 [[pdf](http://docenti.lett.unisi.it/files/4/1/1/6/martinlof4.pdf), [[MartinLofOnTheMeaning96.pdf:file]]]

See also:

* Wikipedia, _[BHK interpretation](http://en.wikipedia.org/wiki/BHK_interpretation)_

* Stanford Encyclopedia of Philosophy, *[The Development of Intuitionistic Logic](https://plato.stanford.edu/entries/intuitionistic-logic-development/#ProoInte)*

* Wouter Pieter Stekelenburg, _Realizability Categories_, ([arXiv:1301.2134](http://arxiv.org/abs/1301.2134)).

* {#EscardoXu15} [[Martin Escardo]], Chuangjie Xu, _The inconsistency of a Brouwerian continuity principle with the Curry--Howard interpretation_, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015), Leibniz International Proceedings in Informatics (LIPIcs) *38* (2015) [[doi:10.4230/LIPIcs.TLCA.2015.153](https://doi.org/10.4230/LIPIcs.TLCA.2015.153), [pdf](http://www.cs.bham.ac.uk/%7Emhe/papers/escardo-xu-inconsistency-continuity.pdf)]

* E. G. F. Díez, _Five observations concerning the intended meaning of the intuitionistic logical constants_ , J. Phil. Logic **29** no. 4 (2000) pp.409–424 . ([preprint](https://webs.um.es/picazo/miwiki/lib/exe/fetch.php?id=inicio&cache=cache&media=five_observations_2000_jour_of_phil_logic_29_4_pp409_424.pdf))

Links to many papers on realizability and related topics may be found [here](http://www.staff.science.uu.nl/~ooste110/realizability.html).

For a comment see also

* [[Robert Harper]], _Extensionality, Intensionality, and Brouwer's Dictum_ ([web](http://existentialtype.wordpress.com/2012/08/11/extensionality-intensionality-and-brouwers-dictum/))

[[!redirects Brouwer-Heyting-Kolmogorov interpretation]]

[[!redirects Brouwer Heyting Kolmogorov interpretation]]
[[!redirects Brouwer-Heyting-Kolmogorov interpretation]]
[[!redirects Brouwer–Heyting–Kolmogorov interpretation]]
[[!redirects BHK interpretation]]

[[!redirects realizability interpretation]]
[[!redirects realisability interpretation]]

[[!redirects BHK correspondence]]


bi-

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#### Category theory
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#### Mathematics
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[[!include mathematicscontents]]
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# bi-
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{: toc}

## Meanings

In [[category theory]],
the prefix "bi-" is used in several ways with different (and not always compatible) meanings.

### Weak 2-dimensional

Here, "bi-" is used to denote a weak 2-dimensional analogue of a 1-categorical concept.

Examples:

- [[bicategory]]
- [[biadjunction]]
- [[bilimit]]

In some cases, the prefix "pseudo-" may be used instead, such as [[pseudoadjunction]] rather than [[biadjunction]]. However, be warned that in some cases, these (unfortunately) refer to different concepts: e.g. a [[pseudolimit]] is a [[bilimit]], but the converse is not always true.

### X and also co-X

Here, "bi-X" is used to denote a structure that is equipped with both X-[[structure]] and [[formal duality|dual]] co-X-structure (for some definition X) that *do not necessarily coincide*.

Examples:

- [[bicartesian category]]

- [[bireflective subcategory]]

- [[bifibrant objects]]

- [[bifibration]]

An unambiguous alternative is to simply be explicit about both structures being assumed.

### Compatible X and co-X

Here, "bi-X" is used to denote a structure that is equipped with X-structure and co-X structure (not coincident) that verify some compatibility.

Examples:

- [[bimonoid]]
- [[Hopf monad|bimonad]]

In many cases, these concepts have different, unambiguous names (such as [[Hopf monad]]) that may be used instead.

### Coincident X and co-X

Here, "bi-X" is used to denote a structure that is equipped with X-structure and co-X-structure that coincides. This is a particularly common special case of the meaning above.

Examples:

- [[bilimit]]
- [[biproduct]]

An alternative, unambiguous prefix that can be used for this meaning is "ambi-".

### Left-X and also right-X

In some settings without symmetry, "bi-X" is occasionally used to denote having X-structure on the left and also X-structure on the right (not coincident).

Examples:

- **biclosed**
- [[bimodule]]
- [[bicomodule]]

In such settings, it is unambiguous to simply elide the prefix "bi-". E.g. one can simply write "[[closed category|closed]]" to mean left- and right-closed.

Note that this is not the same as "X and co-X" (except in the [[delooping]]): a "biclosed category" with this terminology would be one that is [[closed category|closed]] and [[coclosed category|coclosed]].

Some compatibility may be required between the left and right X-structures.

### Two compatible X structures

A less common usage is the prefix "bi-" to mean two structures of the same kind that interact in some way.

Examples:

- **bimonoidal category** (i.e. [[rig category]])

However, these examples tend to have alternative, more descriptive terminology.

## Related pages

- [[locally]]

category: disambiguation


bi-brane

#Contents#
* table of contents
{:toc}

## Idea

A _bi-brane_ is to a [[QFT with defects|defect]] in a [[FQFT]] as a [[brane]] is to a boundary condition.

The term "bi-brane" was apparently introduced in

* Jürgen Fuchs, Christoph Schweigert, Konrad Waldorf, _Bi-branes: Target Space Geometry for World Sheet topological Defects_ ([arXiv](http://arxiv.org/abs/hep-th/0703145))

The description below approaches the concept in a slightly more abstract context.

The notion of  [[brane]] and [[bi-brane]] can be made very abstract, but to get the main idea it is useful to start with considering what is usually called the _geometric case_.

Recall for instance from

* Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard J. Szabo, _D-Branes, RR-Fields and Duality on Noncommutative Manifolds_ ([arXiv](http://arxiv.org/abs/hep-th/0607020), [blog](http://golem.ph.utexas.edu/string/archives/000879.html))

that a geometric _brane_ on some space $X$ and a bundle gerbe $\mathcal{G}$, regarded as a special kind of 2-vector bundle, on $X$ is

* a map $\iota : Q \to X$

* a morphism $\sigma : 1 \to \iota^* \mathcal{G}$
from the trivial 2-vector bundle on $Q$
into the pullback of $\mathcal{G}$ to $Q$ --
this morphism is called a _gerbe module_ or _twisted vector bundle_.

If we write this more diagrammatically using the classifying (fiber-assigning) cocycle $g : X \to 2 Vect$ of $\mathcal{G}$, then this data of a brane is
a transformation

$$
 \array{
  && Q
  \\
  & \swarrow && \searrow^{\iota}
  \\
  pt &&\stackrel{\sigma}{\Rightarrow}&& X
  \\
  & \searrow && \swarrow_{g}
  \\
  && 2 Vect
 }
  \,.
$$

Conceived in this form the notion has an obvious generalizations:

let $X$ and $Y$ be two possibly different spaces with two possibly different 2-vector bundles on them, classified by cocycles $g_1$ and $g_2$, then a bi-brane for this situation is

* a span $ x \stackrel{\iota_1}{\leftarrow} Q
  \stackrel{\iota_2}{\rightarrow} Y$;

* and a transformation between the two pulled back bundles

$$
 \array{
  && Q
  \\
  & {}^{\iota_1}\swarrow && \searrow^{\iota_2}
  \\
  X &&\stackrel{\bibrane}{\Rightarrow}&& Y
  \\
  & {}_{g_1}\searrow &&& \swarrow_{g_2}
  \\
  && 2 Vect
 }
  \,.
$$

The description of branes in the above diagrammatic form was first given in

* Urs Schreiber, [[quant2states.pdf:file]], talk at Fields Institute workshop _Higher categories and their applications_ (2007)

and described in more detail in

* Urs Schreiber, Konrad Waldorf, _Connections on Nonabelian Gerbes and their Holonomy_ ([arXiv](http://arxiv.org/abs/0808.1923) [blog](http://golem.ph.utexas.edu/category/2008/08/connections_on_nonabelian_gerb.html)).

The generalization to bi-branes is developed at

* [[schreiber:Nonabelian cocycles and their quantum symmetries]].

This is very closely related to the spans appearing in

* [[geometric function theory]].

The relation is discussed a bit at [this blog entry](http://golem.ph.utexas.edu/category/2009/01/benzvi_on_geometric_function_t.html#c021321).

At least some aspects of the concept have more or less implicitly been considered before, notably in the context of  [[topological T-duality]].  A translation of the construction in topological T-duality to the above diagrammatic formulation was originally given [here](http://golem.ph.utexas.edu/category/2007/02/qft_of_charged_nparticle_tdual.html).

The interpretation of T-duality in terms of bi-branes is discussed in more detail in

* Gor Sarkissian, Christoph Schweigert, _Some remarks on defects and T-duality_ ([arXiv](http://arxiv.org/abs/0810.3159))

##Bi-branes motivated from 2d CFT##

[[2-spectral triple|Recall]] that from a 2-dimensional [[conformal field theory|CFT]] one induces a (generalized) _target space_ geometry in generalization of how a [[spectral triple]] induces such a generalized geometry.

From category-algebraic considerations one obtains [[defect line]]s in 2-d [[conformal field theory|CFT]], which are encoded by bimodules as [brane]s are encoded by modules. **Bi-branes** are the answer to the question: "What is the target space structure corresponding to defect lines in the 2d CFT"?

For certain 2-d CFTs based on current algebras the bi-branes corresponding to certain defect lines in these theories have been introduced and discussed in

* Jürgen Fuchs, Christoph Schweigert, Konrad Waldorf, _Bi-branes: Target Space Geometry for World Sheet topological Defects_ ([arXiv](http://arxiv.org/abs/hep-th/0703145))

##In WZW theories##

In the above article it is found that, just as symmetric conformal branes in WZW models, whose target space is a Lie group $G$, correspond to
submanifolds of $G$ given by conjugacy classes in $G$, bi-branes in WZW model correspond to [[span]]s or _correspondences_
$$
  \array{
     && B
     \\
     & \swarrow &&\searrow
      \\
     G &&&& G
  }
$$
where $B$ is a [[higher group character|biconjugacy class]] of $G$.

## Related concepts

* [[motivic quantization]]

[[!include field theory with boundaries and defects - table]]

## References

* [[Joost Nuiten]], _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_

[[!redirects bi-branes]]


bi-branes > history

< [[bi-branes]]
[[!redirects bi-branes -- history]]


bi-Heyting topos

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### Context
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#### Topos Theory
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[[!include topos theory - contents]]
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# Contents
* automatic table of contents goes here
{:toc}

[[!redirects bi-Heyting toposes]]
[[!redirects biHeyting topos]]
[[!redirects Bi-Heyting topos]]
[[!redirects Bi-Heyting Topos]]
[[!redirects bi-Heyting Topos]]
[[!redirects bi-heyting topos]]

##Idea

A **bi-Heyting topos** is a [[topos]] whose lattices of subobjects carry the structure of a [[co-Heyting algebra]] in addition to their normal [[Heyting algebra]] structure and thereby permit to model bi-intuitionistic logic.

##Definition

A [[topos]] $\mathcal{E}$ such that the lattice $sub(X)$ of subobjects is a [[co-Heyting algebra|bi-Heyting algebra]] for every object $X\in\mathcal{E}$ is called a [[bi-Heyting topos]].

## Examples

* [[Boolean toposes]] are bi-Heyting since their subobject lattices are Boolean algebras which are self-dual Heyting algebras.

* Presheaf toposes and their [[essential subtoposes]].

##Properties

+-- {: .num_prop}
###### Proposition
A Grothendieck topos is bi-Heyting when finite unions distribute over arbitrary intersections:
$$
S\cup (\bigcap_{i\in I} T_i)=\bigcap_{i\in I} (S\cup T_i)\quad .
$$
=--

**Proof.** Because this distributivity amounts to the preservation of limits by $T\vee$\_ and implies by the [[adjoint functor theorem]] the existence of a left adjoint \_$\backslash T$ which provides the [[co-Heyting algebra]] structure. $\qed$

+-- {: .num_prop}
###### Proposition
Let $\mathcal{E}$ be a Grothendieck topos with generating set $\{G_i | i\in I\}$. Then $\mathcal{E}$ is bi- Heyting precisely iff the lattice of subobjects of $G_i$ is a bi-Heyting algebra for all $G_i$.
=--

cf. [Borceux-Bourn-Johnstone](#BBJ04) p.350.

+-- {: .num_prop #surjective_essential}
###### Proposition
Let $f:\mathcal{E}\to\mathcal{F}$ be a surjective [[essential geometric morphism]] between Grothendieck toposes. If $\mathcal{E}$ is bi-Heyting so is $\mathcal{F}$.
=--

cf. [Borceux-Bourn-Johnstone](#BBJ04) p.351.

+-- {: .num_prop}
###### Proposition
Let $\mathcal{C}$ be a small category. Then $Set^{\mathcal{C}^{op}}$ is bi-Heyting.
=--

**Proof.**  Let $|\mathcal{C}|$ be the discrete category on the objects of $\mathcal{C}$. The inclusion $i:|\mathcal{C}|\hookrightarrow \mathcal{C}$ induces a surjective essential geometric morphism $Set^{|\mathcal{C}|^{op}}\to Set^{\mathcal{C}^{op}}$ and the first of these is Booelan hence bi-Heyting. By prop. \ref{surjective_essential} follows the claim. $\qed$

##Remark

* In bi-Heyting toposes the co-Heyting algebra operations are generally not preserved by [[inverse image functor|inverse image functors]], so that the co-Heyting logical operators like [[co-Heyting negation]] or [[co-Heyting boundary]] are subject to _[[de re and de dicto]]_ effects.

##Related entries

* [[co-Heyting algebra]]
* [[co-Heyting negation]]
* [[co-Heyting boundary]]
* [[bitopological space]]
* [[semi-abelian category]]
* [[mereology]]

##References

Bi-Heyting toposes are explicitly defined in

* {#BBJ04} [[Francis Borceux]], [[Dominique Bourn]], [[Peter Johnstone]], _Initial Normal Covers in Bi-Heyting Toposes_, Arch. Math. **42** (2006) pp.335-356. ([pdf](https://www.emis.de/journals/AM/06-4/johnston.pdf))

But their logical possibilities were exploited well before this e.g. in the work of Lawvere, Reyes and collaborators:

* {#Law86} [[William Lawvere]], _Introduction_ , pp.1-16 in Lawvere, Schanuel (eds.), _Categories in Continuum Physics_, Springer LNM **1174** 1986.

* {#Law91a} [[William Lawvere]], _Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes_ , pp.279-281 in A. Carboni, M. Pedicchio, G. Rosolini (eds.) , _[[Como|Category Theory - Proceedings of the International Conference held in Como 1990]]_, LNM **1488** Springer Heidelberg 1991.

* {#Reyes}[[Gonzalo E. Reyes]], Houman Zolfaghari,  _Bi-Heyting Algebras, Toposes and Modalities_, J. Phi. Logic **25** (1996) pp.25-43.


bi-infinite sequence

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## Definition

A **bi-infinite sequence** or **doubly infinite sequence** in a [[type]] $A$ is a [[function]] $f:\mathbb{Z} \to A$ from the [[integers]] to $A$.

## See also

* [[integers]]
* [[sequence]]
* [[doubly infinite series]]
* [[sequential spectrum type]]

## References

* Wikipedia, [Bi-infinite sequence](https://en.wikipedia.org/wiki/Bi-infinite_sequence)

[[!redirects doubly infinite sequence]]


bi-initial object

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## Idea

Bi-initial objects are the [[bicategory|bicategorical]] analogues of [[initial object|initial objects]] in categories.

## Definition

In a [[bicategory]] $A$, an [[object]] $L \in A$ is **bi-initial** (or **biinitial**) when for all $X \in A$, there is an [[equivalence of categories]] between $A(L,X)$ and the [[terminal category]] with a single object and single morphism $\mathbb{1}$.

## Related concepts

* [[initial object]]

* [[bi-terminal object]]

* [[initial object in an (infinity,1)-category|initial object in an $(\infty,1)$-category]]

## References

This concept appears among others in:

* Tslil Clingman, Lyne Moser, *2-limits and 2-terminal objects are too different* ([arXiv:2004.01313](https://arxiv.org/abs/2004.01313))

[[!redirects bi-initial objects]]

[[!redirects biinitial object]]
[[!redirects biinitial objects]]

[[!redirects bi-initial]]
[[!redirects biinitial]]

[[!redirects initial object in a bicategory]]
[[!redirects initial object in a 2-category]]


bi-invertible morphism

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## Idea

In [[(n,1)-categories]] and [[(infinity,1)-categories]], the usual definition of [[invertible morphism]] is too strict and would violate the [[principle of equivalence]]. There are a few options to resolve this issue: one is by using [[weak inverses]]; but another is by using bi-invertible morphisms, which are a generalization of the notion of bi-invertible function in [[type theory]].

## Definition

A morphism $f:Mor(A, B)$ in a [[(n,1)-category]] or [[(infinity,1)-category]] is **bi-invertible** if there are morphisms $\mathrm{sec}(f):Mor(B, A)$ and $\mathrm{ret}(f):Mor(B, A)$ such that $\mathrm{sec}(f)$ is a [[section]] of $f$ and $\mathrm{ret}(f)$ is a [[retraction]] of $f$, with section and retraction defined using [[path space objects]] rather than strict equality.

## Examples

An [[integers object]] in an [[(n,1)-category]] $C$ with a [[terminal object]] is defined as the (homotopy) initial object $\mathbb{Z}$ with a [[global element]] $0:Mor(1, \mathbb{Z})$ and a bi-invertible [[endomorphism]] $\mathrm{succ}:Mor(\mathbb{Z}, \mathbb{Z})$ with section $\mathrm{pred}_1:Mor(\mathbb{Z}, \mathbb{Z})$ and retraction $\mathrm{pred}_2:Mor(\mathbb{Z}, \mathbb{Z})$. This corresponds to a definition of the [[integers]] [[higher inductive type]] in the [[internal logic]] of an [[(n,1)-category]].

## See also

* [[invertible morphism]]

* [[weak inverse]]

* [[equivalence of types]]

## References

For the integers object as defined with a bi-invertible morphism in the internal logic of a [[(infinity,1)-category]] see

* [[Thorsten Altenkirch]], [[Luis Scoccola]], *The Integers as a Higher Inductive Type*, [arXiv](https://arxiv.org/abs/2007.00167)

[[!redirects bi-invertible morphism]]
[[!redirects bi-invertible morphisms]]


bi-pointed object

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## Definition

In evident variation of the notion of *[[pointed objects]]*, a **bi-pointed object** in a [[category]] $V$ with [[terminal object]] $pt$ is a [[co-span]] from $pt$ to itself, i.e. a [[diagram]] of this form:

$$
  \array{
    && S
    \\
    & {}^{\sigma_S}\nearrow && \nwarrow^{\tau_S}
    \\
    pt &&&& pt
  }
  \,.
$$

Similarly, a [[pointed object]] in a category with [[initial object]] $\varnothing$ and terminal object $pt$ may be regarded as a co-span from $\varnothing$ to $pt$.

If $V$ has in addition binary [[coproducts]] then a bi-pointed object in $V$ is the same as a [[co-span]] from $\varnothing$  to the [[coproduct]] $pt \sqcup pt$.

## Examples

* The [[subobject classifier]] $\Omega$ and the [[Sierpinski space]] $\mathbb{S}$ in the category of [[choice object|choice sets]] are bi-pointed objects.

* In general, any non-degenerate subobject classifier in a topos or pretopos is a bi-pointed object.

* Any [[interval object]] is a bi-pointed object with a [[2-morphism]] connecting the two global elements.

* The [[boolean domain]] is the [[initial object|initial]] bi-pointed object in [[Set]].

## Closed structure

From the [[bicategory]] structure on [[co-span]]s in $V$ bi-pointed objects in $V$ naturally inherit the structure of a monoidal category

$$
  BiPointed(V) = End_{CoSpan(V)}(pt)
  \,.
$$

Assume that the terminal object $pt$ is the tensor unit in $V$.

Then moreover, following the construction of the $V$-internal hom of [[pointed object]]s and being a special case of that of [[co-span]]s in $V$, there is an internal [[hom-object]]
${}_{pt}[X,Y]_{pt} \in Obj(V)$
of bipointed objects $X$ and $Y$ defined as the pullback

$$
\array{
  {}_{pt}[X,Y]_{pt}
   & \longrightarrow &
  pt \sqcup pt
  \\
  \Big\downarrow
    &&
  \Big\downarrow\mathrlap{{}^{\sigma_Y \sqcup \tau_Y}}
  \\
  [X,Y]
  &
  \underset
    {\sigma_X^* \times \tau_X^*}
    {\longrightarrow}
  &
  [pt \sqcup pt,Y].}
$$

Here the [[map]]
$pt \sqcup pt \stackrel{\sigma_Y \sqcup \tau_Y}{\to}
[pt \sqcup pt,Y]$
is [[adjunct]] to
$\pt \otimes (pt \sqcup pt) \to pt \sqcup pt
\stackrel{\sigma_Y \sqcup \tau_Y}{\to} Y$.

This $V$-object ${}_{pt}[X,Y]_{pt}$ is itself naturally bi-pointed with the bi-point $pt \sqcup pt \to {}_pt[X,Y]_{pt}$ given by the morphism induced from the above pullback diagram by the commuting diagram

$$
  \array{
     pt \sqcup pt
     &
       \overset{Id}{\longrightarrow}&
     pt \sqcup pt
     \\
     \Big\downarrow\mathrlap{{}^{\sigma_X \sqcup \sigma_X}}
     &&
     \Big\downarrow\mathrlap{{}^{\sigma_Y \sqcup \tau_Y}}
     \\
     [X,Y]
     &
     \underset
       {\sigma_X^* \times \sigma_Y^*}
       {\longrightarrow}
     &
     [pt \sqcup pt, Y],
  }

$$

where the morphism
$pt \sqcup pt \stackrel{\sigma_X \sqcup \sigma_X}{\to}
[X,Y]
$ is [[adjunct]] to
$
  X \otimes (pt \sqcup pt)
  \to
  pt \otimes (pt \sqcup pt)
  \simeq pt \sqcup pt
  \stackrel{\sigma_Y \sqcup \tau_Y}{\to}
  Y
$.

## See also

* [[pointed object]]

* [[boolean domain object]]

* [[bi-pointed type]]

[[!redirects bi-pointed object]]
[[!redirects bi-pointed objects]]

[[!redirects bi-pointed set]]
[[!redirects bi-pointed sets]]


bi-pointed type

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## Idea

_Bi-pointed types_ are [[axiom|axiomatizations]] of the [[bi-pointed objects]] in the context of [[homotopy type theory]].

## Definition

A **bi-pointed type** is a type $A$ with two points $a:A$ and $b:A$. Equivalently, it is a type $A$ with a function $\mathrm{Bool} \to A$. Examples include the [[interval type]] and the [[function type]] of the [[natural numbers type]].

## See also

* [[bi-pointed object]]

[[!redirects bi-pointed type]]


bi-terminal object

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## Idea

Bi-terminal objects are the [[bicategory|bicategorical]] analogues of [[terminal object|terminal objects]] in categories.

## Definition

In a [[bicategory]] $A$, an object $L \in A$ is **bi-terminal** (or **biterminal**) when for all $X \in A$, there is an [[equivalence of categories]] between $A(X,L)$ and the [[terminal category]] $\mathbb{1}$ with a single object and single morphism.

## Related concepts

* [[terminal object]]

* [[bi-initial object]]

* [[terminal object in an (infinity,1)-category|terminal object in an $(\infty,1)$-category]]

## References

This concept appears among others in:

* Tslil Clingman, Lyne Moser, *2-limits and 2-terminal objects are too different* ([arXiv:2004.01313](https://arxiv.org/abs/2004.01313)).

[[!redirects bi-terminal objects]]

[[!redirects biterminal object]]
[[!redirects biterminal objects]]

[[!redirects bi-terminal]]
[[!redirects biterminal]]

[[!redirects terminal object in a bicategory]]
[[!redirects terminal object in a 2-category]]


biactegory

Let $C$ be a [[monoidal category]] (in fact one can easily modify all statements to generalize all statements here to [[bicategories]]). One can consider several variants of the [[2-category]] of all categories with monoidal [[action]] of $C$, (co)lax [[monoidal functors]] and their transformations. A category with an action of $C$ is sometimes called a __$C$-[[actegory]]__. The word 'module category' over $C$ is also used, especially when the category acted upon is in addition also [[additive category|additive]], like the examples in [[representation theory]].

If a $C$-actegory is a [[categorification]] of a [[module]], then for two monoidal categories $C$ and $D$, we should categorify a [[bimodule]], which we call __$C$-$D$-biactegory__. The two actions on a usual bimodule commute; for biactegories the commuting is up to certain coherence laws, which are in fact the expression of an invertible [[distributive law]] between the two monoidal actions. The tensor product of biactegories can be defined (here the invertibility of the distributive law is needed) as a [[2-limit|bicoequalizer]] of a certain diagram.

For very basic outline see section 2 in

* [[Zoran Škoda]], Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183–-202, [arXiv:0811.4770](https://arxiv.org/abs/0811.4770).

and partial writeup

* Zoran Škoda, _Biactegories_, 2006, [pdf](https://www2.irb.hr/korisnici/zskoda/biact.pdf))

In a language of "module categories", a different treatment is now available in

* Justin Greenough, Monoidal 2-structure of bimodule categories, [arxiv:0911.4979](https://arxiv.org/abs/0911.4979).

It is one of the notions discussed in the review

* Matteo Capucci, Bruno Gavranović, _Actegories for the working amthematician_, [arXiv:2203.16351](https://arxiv.org/abs/2203.16351)


biaction

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## Idea

A ternary [[function]] which simultaneously exhibits an [[action]] on a [[set]] from both the left and the right side.

Sets with biactions are the [[bimodule objects]] [[internalization|internal to]] [[Set]].

## Definition

Given a [[set]] $S$ and [[monoids]] $(M, e_M, \mu_M)$ and $(N, e_N, \mu_N)$, a **$M$-$N$-biaction** or **two-sided action** is a ternary function $\alpha:M \times S \times N \to S$ such that

* for all $s \in S$, $\alpha(e_M, s, e_N) = s$

* for all $s \in S$, $a \in M$, $b \in M$, $c \in N$, and $d \in N$, $\alpha\big(a, \alpha(b, s, c), d\big) = \alpha\big(\mu_M(a, b), s, \mu_N(c, d)\big)$

## Left and right actions

The [[left action|left $M$-action]] is defined as

$$\alpha_M(a, s) \coloneqq \alpha(a, s, e_N)$$

for all $a \in M$ and $s \in S$. It is a left action because

$$\alpha_M(e_M, s) = \alpha(e_M, s, e_N) = s$$

$$\alpha_M\big(a, \alpha_L(b, s)\big) = \alpha\big(a, \alpha(b, s, e_N), e_N\big) = \alpha\big(\mu_M(a, b), s, \mu_N(e_N, e_N)\big) = \alpha\big(\mu_M(a, b), s, e_N\big) = \alpha_M\big(\mu_M(a, b), s\big)$$

The [[right action|right $N$-action]] is defined as

$$\alpha_N(s, c) \coloneqq \alpha(e_M, s, c)$$

for all $c \in N$ and $s \in S$. It is a right action because

$$\alpha_N(s, e_N) = \alpha(e_M, s, e_N) = s$$

$$\alpha_N\big(\alpha_N(s, c), d\big) = \alpha\big(e_M, \alpha(e_M, s, c), d\big) = \alpha\big(\mu_M(e_M, e_M), s, \mu_N(c, d)\big) = \alpha\big(e_M, s, \mu_N(c, d)\big) = \alpha_N\big(s, \mu_N(c, d)\big)$$

The left $M$-action and right $N$-action satisfy the following identity:

* for all $s \in S$, $a \in M$ and $c \in N$, $\alpha_M\big(a, \alpha_N(s, c)\big) = \alpha_N\big(\alpha_M(a, s), c\big)$.

This is because when expanded out, the identity becomes:

$$\alpha\big(a, \alpha(e_M, s, c), e_N\big) = \alpha\big(e_M, \alpha(a, s, e_N), c\big)$$

$$\alpha\big(\mu_M(a, e_M), s, \mu_N(c, e_N)\big) = \alpha\big(\mu_M(e_M, a), s, \mu_N(e_N, c)\big)$$

$$\alpha(a, s, c) = \alpha(a, s, c)$$

## See also

* [[action]]

* [[bimodule]]

[[!redirects biactions]]

[[!redirects two-sided action]]
[[!redirects two-sided actions]]


biadjoint pair

Moved to [[ambidextrous adjunction]]. To be removed.


biadjunction

> This page is about the generalisation of an [[adjunction]] to [[bicategories]]. For a functor $L$ which is both a left and a right adjoint to a functor $R$, see [[ambidextrous adjunction]].

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## Idea

A _biadjunction_ is the "maximally weak" kind of [[2-adjunction]]: a [[categorification|higher generalization]] of the notion of [[adjunction]] from [[category theory]] to [[2-category theory]], and specifically to [[bicategories]] (or more generally internally in a [[tricategory]].  See [[2-adjunction]] for other kinds of 2-adjunction.

## Definition

### Incoherent versions

Given (possibly weak) [[2-categories]], $A$ and $C$, and (possibly weak) [[2-functors]] $F:A\to C$ and $U:C\to A$, a __biadjunction__ is given by specifying for each object $a$ in $A$ and each object $c$ in $C$ an [[equivalence of categories]] $C(F a,c)\cong A(a,U c)$, which is [[pseudonatural transformation|pseudonatural]] both in $a$ and in $c$.

By the [[Yoneda lemma for bicategories]], this is equivalent to giving pseudonatural transformations $\eta : Id_A \to U F$ and $\varepsilon : F U \to Id_C$ satisfying the [[triangle identities]] up to invertible [[modifications]].  This latter definition can be internalized in any (weak) [[3-category]], such as a [[Gray-category]] or a [[tricategory]].

### Coherent versions

Both the "equivalence of hom-categories" definition and the "unit and counit" definition have stronger "coherent" versions:  We can ask the equivalences $C(F a,c)\cong A(a,U c)$ to be [[adjoint equivalences]], or for the "triangulator" modifications to satisfy the [[lax 2-adjunction#definition|swallowtail equations]].  These two conditions are roughly equivalent, and any "incoherent" biadjunction can be improved to a coherent one by altering one of the triangulators; see [Gurski](#Gurski12), [Riehl-Verity](#RV15), [Pstrągowski](#Pstrągowski14), [Riehl-Shulman](#RS17), and [this proof](http://globular.science/1512.006) in [[Globular]].

## Related concepts

* [[adjoint functor]], [[adjoint triple]], [[adjoint quadruple]]

* [[proadjoint]], [[Hopf adjunction]]

* [[2-adjunction]]

  **biadjunction**, [[lax 2-adjunction]], [[pseudoadjunction]]

* [[pseudoadjunction]]

* [[biequivalence]], [[biadjoint biequivalence]]

* [[adjoint (infinity,1)-functor]]

## References

* [[John Gray]], _[[Gray-adjointness-for-2-categories|Formal category theory: Adjointness for 2-categories]]_, Lecture Notes in Mathematics __391__, Springer, Berlin, 1974.

* [[Thomas M. Fiore]], _Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory_, Memoirs of the American Mathematical Society __182__ (2006), no. 860. 171 pages, [MR2007f:18006](http://www.ams.org/mathscinet-getitem?mr=2007f:18006),  [math.CT/0408298](http://arxiv.org/abs/math.CT/0408298)

* [[Nick Gurski]], _Biequivalences in tricategories_, Theory and applications of categories 2012, [journal web site](http://www.tac.mta.ca/tac/volumes/26/14/26-14abs.html), [arxiv](https://arxiv.org/abs/1102.0979)
{#Gurski12}

* [[Emily Riehl]] and [[Dominic Verity]], _Homotopy coherent adjunctions and the formal theory of monads_, 2015, [arxiv](https://arxiv.org/abs/1310.8279)
{#RV15}

* Piotr Pstrągowski, _On dualizable objects in monoidal bicategories, framed surfaces and the Cobordism Hypothesis_, 2014 [arxiv](https://arxiv.org/abs/1411.6691)
{#Pstrągowski14}

* [[Emily Riehl]] and [[Mike Shulman]], _A type theory for synthetic ∞-categories_, 2017, [arxiv](https://arxiv.org/abs/1705.07442)
{#RS17}

[[!redirects biadjunctions]]
[[!redirects biadjoint]]
[[!redirects biadjoints]]


bialgebra

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# Bialgebra
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## Idea

A _bialgebra_ (or bi[[gebra]]) is both an [[algebra]] and a [[coalgebra]], where the operations of either one are [[homomorphisms]] for the other. A bialgebra structure on an [[associative algebra]] is precisely such as to make its [[category of modules]] into a [[monoidal category]] equipped with a [[fiber functor]].

A bialgebra is one of the ingredients in the concept of [[Hopf algebra]].

## Definition

A **bialgebra** is a [[monoid]] [[internalization|in]] the category of [[coalgebra|coalgebras]].  Equivalently, it is a [[comonoid]] [[internalization|in]] the category of [[algebra|algebras]].  Equivalently, it is a monoid in the category of comonoids in [[Vect]] --- or equivalently, a comonoid in the category of monoids in [[Vect]].

More generally, a **[[bimonoid]]** in a monoidal category $M$ is a monoid in the category of comonoids in $M$ --- or equivalently, a comonoid in the category of monoids in $M$.  So, a bialgebra is a bimonoid in $Vect$.

## Properties

### Relation to sesquialgebras

Bialgebras are precisely those [[sesquialgebras]] $A$ whose product $A \otimes A$-$A$-[[bimodule]] is induced from an algebra [[homomorphism]] $A \to A \otimes A$ and whose unit $k$-$A$ bimodule is induced from an algebra homomorphism $A \to k$.

### Tannaka duality and categories of modules
 {#TannakDuality}

The structure of a bialgebra on an [[associative algebra]] equips its [[category of modules]] with the structure of a [[monoidal category]] and a monoidal [[fiber functor]]. In fact that construction is an equivalence. This is the statement of [[Tannaka duality]] for bialgebras. For instance ([Bakke](#Bakke))

[[!include structure on algebras and their module categories - table]]

## Examples

Notions of bialgebra with further structure notably include _[[Hopf algebras]]_ and their variants.

## Related concepts

* [[bimonoid]]

* [[Gerstenhaber-Schack cohomology]], [[bialgebra cocycle]], [[weak bialgebra]], [[bialgebroid]]

* [[Hopf algebra]]

* [[quasitriangular bialgebra]]

* [[sesquialgebra]]

* [[algebra]], [[module]]

* **bialgebra**,  [[2-module]]

* [[trialgebra]], [[3-module]]

## References

[[Tannaka duality]] for bialgebras

* {#Bakke} Tørris Koløen Bakke, _Hopf algebras and monoidal categories_ (2007) ([pdf](http://munin.uit.no/bitstream/handle/10037/1084/finalthesis.pdf;jsessionid=C0D15EADBDC35E93D95D2DD090411004?sequence=1))

On [[bialgebras]] in [[locally presentable categories]]:

* [[Friedrich Ulmer]]. *Bialgebras in locally presentable categories*, University of Wuppertal preprint (1977) [[pdf](/nlab/files/Bialgebras_in_locally_presentable_categories.pdf)]

[[!redirects bialgebra]]
[[!redirects bialgebras]]
[[!redirects bigebra]]
[[!redirects bigebras]]


bialgebra cocycle

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## Idea

[[Shahn Majid]] has introduced a notion of __bialgebra cocycles__ which as special cases comprise [[group cohomology|group cocycles]], nonabelian Drinfel'd 2-cocycle and 3-cocycle, abelian [[Lie algebra cohomology]] and so on.

Besides this case, by "bialgebra cohomology" many authors in the literature mean the abelian cohomology ([[Ext|Ext-groups]]) in certain category of "tetramodules" over a fixed bialgebra, which will be in $n$Lab referred as [[Gerstenhaber-Schack cohomology]].

## Definition

Let $(B,\mu,\eta,\Delta,\epsilon)$ be a $k$-[[bialgebra]].
Denote $\Delta_i : B^{\otimes n}\to B^{\otimes (n+1)} := \id_B^{\otimes (i-1)}\otimes\Delta\otimes\id_B^{\otimes(n-i+1)}$, for $i = 1,\ldots, n$, and $\Delta_0 := 1_B\otimes \id_B^{\otimes n}$, $\Delta_{n+1} := \id_B^{\otimes n}\otimes 1_B$. Notice that for the compositions $\Delta_i\circ\Delta_j = \Delta_{j+1}\circ\Delta_i$ for $i\leq j$.

Let $\chi$ be an invertible element of $B^{\otimes n}$. We define the __coboundary__ $\partial\chi$ by

$$\partial \chi = (\prod_{i=0}^{i \mathrm{ even}} \Delta_i\chi) (\prod_{i=1}^{i \mathrm{ odd}} \Delta_i \chi^{-1})$$

This formula is symbolically also written as $\partial\chi = (\partial_+\chi)(\partial_-\chi^{-1})$.

An invertible $\chi\in B^{\otimes n}$ is an __$n$-cocycle__ if $\partial\chi = 1$. The cocycle $\chi$ is __counital__ if
for all $i$, $\epsilon_i\chi=1$ where $\epsilon_i =\id_B^{\otimes i-1}\otimes\epsilon\otimes\id_B^{\otimes n-i}$.

## Examples

### Low dimensions

$\chi\in H$ is a 1-cocycle iff it is invertible and [[grouplike element|grouplike]] i.e. $\Delta\chi=\chi\otimes\chi$ (in particular it is counital).  A 2-cocycle is an invertible element $\chi\in H^{\otimes 2}$ satisfying

$$
(1\otimes\chi)(id\otimes\Delta)\chi = (\chi\otimes 1)(\Delta\otimes id)\chi,
$$
which is counital if $(\epsilon\otimes id)\chi = (id\otimes\epsilon)\chi = 1$ (in fact it is enough to require one out of these two counitality conditions). Counital 2-cocycle is hence the famous [[Drinfel'd twist]].

The 3-cocycle condition for $\phi\in H^{\otimes 3}$ reads:

$$
(1\otimes\phi)((id\otimes\Delta\otimes id)\phi)(\phi\otimes 1) = ((id\otimes id\otimes\Delta)\phi)((\Delta\otimes id\otimes id)\phi)
$$

A counital 3-cocycle is the famous __Drinfel'd associator__ appearing in CFT and quantum group theory. The coherence for monoidal structures can be twisted with the help of Drinfel'd associator; Hopf algebras reconstructing them appear then as quasi-Hopf algebras where the comultiplication is associative only up to twisting by a 3-cocycle in $H$.

### For particular Hopf algebras

If $G$ is a finite group and $H=k(G)$ is the
[[Hopf algebra]] of $k$-valued functions on the group, then we recover the usual notions: e.g. the 2-cocycle is a function $\chi:G\times G\to k$ satisfying the cocycle condition

$$
\chi(b,c)\chi(a,b c) = \chi(a,b)\chi(a b,c)
$$
and the condition for a 3-cocycle $\phi:G\times G\times G\to k$ is

$$
\phi(b,c,d)\phi(a,b c,d)\phi(a,b,c) = \phi(a,b,c d)\phi(a b,c,d)
$$

$n$-cocycles can be in low dimensions twisted by $(n-1)$-cochains (I think it is in this context not know for hi dimensions), what gives an equivalence relation:

For example, if $\chi\in H\otimes H$ is a counital 2-cocycle, and $\partial\gamma\in H$ a counital coboundary, then

$$ \chi^\gamma =  (\partial_+\gamma)\chi(\partial_-\gamma^{-1})= (\gamma\otimes\gamma)\chi\Delta\gamma^{-1} $$

is another 2-cocycle in $H\otimes H$. In particular, if $\chi = 1$ we obtain that $\partial\gamma$ is a cocycle (that is every 2-coboundary is a cocycle).

## A dual theory

In addition to cocycles "in" $H$ as above, Majid introduced a dual version -- cocycles on $H$. The usual Lie algebra cohomology $H^n(L,k)$, where $L$ is a $k$-Lie algebra, is a special case of that dual construction.

Instead of $\Delta_i$ one uses multiplications $\cdot_i$ defined analogously ($\cdot_i$ is the multiplication in $i$-th place for $1\leq i\leq n$ and $\psi\circ\cdot_0 =\epsilon\otimes\psi$, $\psi\circ\cdot_{n+1} = \psi\otimes\epsilon$).
An $n$-cochain on $H$ is a linear functional $\psi:H^{\otimes n}\to k$, invertible in the [[convolution algebra]]. An $n$-cochain $\psi$ on $H$ is a __coboundary__ if

$$
\partial\psi = (\prod_{i=0}^{\mathrm{even}}\psi\circ \cdot_i))(\prod_{i=1}^{\mathrm{odd}}\psi^{-1}\circ\cdot_i)
$$

If $\psi\in H$ then this condition reads

$$
(\partial\psi)(a\otimes b) = \sum \psi(b_{(1)})\psi(a_{(1)})\psi^{-1}(a_{(2)}b_{(2)})
$$

and, for $\psi\in H\otimes H$, the condition is

$$
(\partial\psi)(a\otimes b\otimes c) = \sum \psi(b_{(1)}\otimes c_{(1)})\psi(a_{(1)}\otimes b_{(2)}c_{(2)})\psi^{-1}(a_{(2)}\otimes b_{(3)}c_{(3)})\psi^{-1}(a_{(3)}b_{(4)})
$$

If one looks at the [[group algebra]] $kG$ of a finite group then the cocycle conditions above can be obtained by a Hopf algebraic version of the $k$-linear extension of the cocycle conditions for the group cohomology in the form appearing in Schreier's theory of extensions.

However for all $n$ the Lie algebra cohomology also appears as a special case.

(to be completed later)

## References

* [[Shahn Majid]], _Cross product quantisation, nonabelian cohomology and twisting of Hopf algebras_, in H.-D. Doebner, V.K. Dobrev, A.G. Ushveridze, eds., Generalized symmetries in Physics. World Sci. (1994) 13-41; ([arXiv:hep.th/9311184](https://arxiv.org/abs/hep-th/9311184))

* [[Shahn Majid]], _Foundations of quantum group theory_, Cambridge UP

[[!redirects bialgebra cocycles]]


bialgebra ideal


bialgebroid

## Idea

A bialgebroid may be viewed as a multiobject generalization of a concept of a [[bialgebra]], or a possibly noncommutative generalization of a space-algebra dual version of the concept of an internal category in spaces.

#### Nomenclature

This entry is about "associative" bialgebroid, see also the different concept of a [[Lie bialgebroid]].

#### Motivation in Tannakian formalism

When a monoidal category has a [[fiber functor]] to a category of vector spaces over a field, one tries to "reconstruct" the category as the category of representations of the endomorphism object of a fiber functor. One often does not have a fiber functor to vector spaces but only to bimodules over some base algebra $A$. Sometimes in such cases, the object of endomorphisms of the fiber functor form a bialgebroid over $A$ and the category is equivalent to the category of representations of that bialgebroid.

## Definition

### Via monoidal categories

Given a unital (possibly noncommutative) ring $R$ an __$R$-bialgebroid__ is an $R$-$R$-bimodule $H$ (object of ${}_R \mathcal{M}_R$) equipped with a structure of a comonoid in ${}_R \mathcal{M}_R$ (i.e. an $R$-coring) and of a monoid in ${}_{R^e}\mathcal{M}_{R^e}$ (i.e. an $R^e$-ring), where $R^e = R^{op}\otimes R$ is the enveloping ring of $R$; and the structures of a monoid and a comonoid satisfy certain compatibility conditions. These compatibility conditions are equivalent to the fact that the monad ${}_{\otimes_{R^e}} H : \mathcal{M}_{R^e}\to \mathcal{M}_{R^e}$ is [[opmonoidal monad|opmonoidal]]. The category of $R$-comodules is by definition the category of comodules over the underlying $R$-coring.

### An explicit definition

If $A$ is an [[associative algebra]] over some [[ground field]] $k$, then a left associative $A$-bialgebroid is another associative $k$-algebra $H$ together with the following additional maps:
an algebra map $\alpha:A\to H$ called the source map,
an algebra map $\beta:A^{op}\to H$ called the target map, so that the elements of the images of $\alpha$ and $\beta$ commute in $H$, therefore inducing an $A$-bimodule structure on $H$ via the rule $a.h.b = \alpha(a)\beta(b) h$ for $a,b\in A, h\in H$; an $A$-bimodule morphism $\Delta:H\to H\otimes_A H$ which is required to be a counital coassociative comultiplication on $H$ in the monoidal category of $A$-bimodules with monoidal product $\otimes_A$. The map $H\otimes A\ni h\otimes a\mapsto \epsilon(h\alpha(a))\in $ must be a left action extending the multiplication $A\otimes A\to A$ along $\alpha\otimes id_A$. Furthermore, a compatibility between the comultiplication $\Delta$ and multiplications on $H$ and on $H\otimes H$ is required. For a noncommutative $L$ the tensor square $H\otimes_A$ is not an algebra, hence asking for a bialgebra-like compatibility that $\Delta:H\to H\otimes_A H$ is a morphism of $k$-algebras does not make sense. Instead, one requires that $H\otimes_A H$ has a $k$-subspace $T$ which contains the image of $\Delta$ and has a well-defined multiplication induced from its preimage under the projection from the usual tensor square algebra $H\otimes H$. Then one requires that the [[corestriction]] $\Delta|^T :H\to T$ is a homomorphism of unital algebras. Under these conditions, one can make a canonical choice for $T$, namely the so called Takeuchi's product $H\times_A H\subset H\otimes_A H$, which always inherits an associative multiplication along the projection from $H\otimes H$.

### Via $A\otimes A^{op}$-rings

All modules and morphisms will be over a fixed ground commutative ring $k$.

A __left $A$-bialgebroid__ is
an $A\otimes_k A^{op}$-[[ring]] $(H,\mu_H,\eta)$,
together with the $A$-bimodule map "comultiplication" $\Delta : H\to H\otimes_A H$, which is coassociative and counital with a counit $\epsilon$, such that

(i) the $A$-bimodule structure used on $H$ is $a.h.a':= s(a)t(a')h$, where $s := \eta(-\otimes 1_A):A\to H$ and $t:=\eta(1_A\otimes -):A^{op}\to H$ are the algebra maps induced by the unit $\eta$ of the $A\otimes A^{op}$-ring $H$

(ii) the coproduct $\Delta : H\to H\otimes_A H$ corestricts to the [[Takeuchi product]] and the corestriction $\Delta : H\to H\times_A H$
is a $k$-algebra map, where the Takeuchi product $H\times_A H$ has a multiplication induced factorwise

(iii) $\epsilon$ is a left [[character]] on the $A$-ring $(H,\mu_H,s)$.

Notice that $H\otimes_A H$ is in general not an algebra, just an  $A$-bimodule. That is why (ii) is needed. An equivalent condition to (ii) is the following: the formula $h.(\sum_i k_i \otimes l_i) = \sum_i h_{(1)}\cdot k_i \otimes h_{(2)} \cdot l_i$ defines a well-defined action of $H$ on $H\otimes_A H$.

The definition of a __right $A$-bialgebroid__ differs by the $A$-bimodule structure on $H$ given instead by $a.h.a':= h s(a')t(a)$ and the counit $\epsilon$ is a _right_ character on the $A$-coring $(H,\mu_H,t)$ ($t$ and $s$ can be interchanged in the last requirement).

## Literature

Related notions: [[Hopf algebroid]]

#### Commutative case

The commutative case is rather classical. See for example the appendix to

* Douglas C. Ravenel, _Complex cobordism and stable homotopy  groups of spheres_, Pure and Applied Mathematics __121__. Academic Press Inc., Orlando, FL, 1986.

#### Noncommutative case

The first version of a bialgebroid over a noncommutative base was more narrow:

* M. Sweedler, _Groups of simple algebras_, Publ. IHES 44:79–189, 1974, [numdam](http://www.numdam.org/item?id=PMIHES_1974__44__79_0)

A modern generality, but in different early formalism, is due to Takeuchi (who was motivated to generalize the results from the Sweedler's paper), under the name of $\times_A$-bialgebra (as it involves the $\times_A$-product, nowdays called [[Takeuchi product]]):

* M. Takeuchi, _Groups of algebras over $A \times \bar{A}$, J. Math. Soc. Japan __29__, 459–492, 1977, [MR0506407](http://www.ams.org/mathscinet-getitem?mr=0506407), [euclid](http://projecteuclid.org/euclid.jmsj/1240432948)

Lu introduces the name bialgebroid for a structure which is equivalent to the Takeuchi's $\times_A$-bialgebra (though differently axiomatized there):

*  Jiang-Hua Lu, _Hopf algebroids and quantum groupoids_, Int. J. Math. __7__, 1 (1996) pp. 47-70, [q-alg/9505024](http://arxiv.org/abs/q-alg/9505024), [MR95e:16037](http://www.ams.org/mathscinet-getitem?mr=95e:16037), [doi](http://dx.doi.org/10.1142/S0129167X96000050)

Modern treatments are in

* [[Gabriella Böhm]], _Internal bialgebroids, entwining structures and corings_, [math.QA/0311244](http://arxiv.org/abs/math/0311244), in: Algebraic structures and their representations, 207–226,  Contemp. Math. __376__, Amer. Math. Soc. 2005.

* [[G. Böhm]], _Hopf algebroids_, (a chapter of) Handbook of algebra, [arxiv:math.RA/0805.3806](http://arxiv.org/abs/0805.3806)
* [[Kornél Szlachányi]], _The monoidal Eilenberg–Moore construction and bialgebroids_, J. Pure Appl. Algebra __182__, no. 2–3 (2003) 287–315; _Fiber functors, monoidal sites and Tannaka duality for bialgebroids_, [arxiv/0907.1578](http://arxiv.org/abs/0907.1578)
* [[T. Brzeziński]], G. Militaru, _Bialgebroids, $\times_{R}$-bialgebras and duality_, J. Algebra 251: 279-294, 2002, [math.QA/0012164](http://arxiv.org/abs/math/0012164)
* J. Donin, A. Mudrov, _Quantum groupoids and dynamical categories_,
J. Algebra __296__ (2006), no. 2, 348–384, [math.QA/0311316](http://arxiv.org/abs/math/0311316), [MR2007b:17022](http://www.ams.org/mathscinet-getitem?mr=2201046), [doi](http://dx.doi.org/10.1016/j.jalgebra.2006.01.001); MPIM-2004-21, [dvi](http://www.mpim-bonn.mpg.de/preblob/1921) with hyperlinks, [ps](http://www.mpim-bonn.mpg.de/preblob/1922)

There is also a notion of quasibialgebroid, where the coassociativity is weakened by a [[bialgebra cocycle|bialgebroid 3-cocycle]]. See also [[Hopf algebroid]].

[[!redirects bialgebroids]]
[[!redirects associative bialgebroid]]


Bianca Cerchiai

* [InSpire page](https://inspirehep.net/authors/1014175)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=107587)

## Selected writings

On [[magic supergravity]]:

* {#CacciatoriCerchiaiMarrani12} Sergio L. Cacciatori, [[Bianca Cerchiai]], [[Alessio Marrani]], _Squaring the Magic_ [[arXiv:1208.6153](http://arxiv.org/abs/1208.6153)]

* {#CacciatoriCerchiaiMarrani2013} Sergio L. Cacciatori, [[Bianca Cerchiai]], [[Alessio Marrani]], _Magic coset decompositions_, Adv. Theor. Math. Phys. **17** 5 (2013) 1077-1128 [[doi:10.4310/ATMP.2013.v17.n5.a4](https://doi.org/10.4310/ATMP.2013.v17.n5.a4)]

Usage of full [[supergravity]] (retaining the [[gravitino]]) for [[AdS-CMT duality]], with application to [[graphene]]-like systems:

* [[Laura Andrianopoli]], [[Bianca L. Cerchiai]], [[Riccardo D'Auria]], [[Mario Trigiante]], *Unconventional Supersymmetry at the Boundary of $AdS_4$ Supergravity*,  J. High Energ. Phys. **2018** 7 (2018) [[arXiv:1801.08081](https://arxiv.org/abs/1801.08081), doi:10.1007/JHEP04(2018)007]

* [[Laura Andrianopoli]], [[Bianca L. Cerchiai]], [[Riccardo D'Auria]], A. Gallerati, R. Noris, [[Mario Trigiante]], [[Jorge Zanelli]], *$\mathcal{N}$-Extended $D=4$ Supergravity, Unconventional SUSY and Graphene*, JHEP 01 (2020) 084 [[arXiv:1910.03508](https://arxiv.org/abs/1910.03508), doi:10.1007/JHEP01(2020)084]

Exposition:

* [[Bianca L. Cerchiai]],  *Supergravity in a pencil*, [talk at](M-Theory+and+Mathematics#Cerchiai2020) *[[M-Theory and Mathematics]] [2020](M-Theory+and+Mathematics#2020)*, NYU Abu Dhabi [[[CerchiaiSlidesAtMTheoryAndMathematics2020.pdf:file]], video: [YT](https://youtu.be/xE7TmwyqqaU)]

* [[Bianca L. Cerchiai]], *Holography, Supergravity and Graphene*, talk at *106th online SIF Congress* (2020) [[pdf](https://agenda.infn.it/event/23656/contributions/120378/attachments/75347/96340/cerchiai_sif2020.pdf), [[Cerciai-SIF2020.pdf:file]]]

On [[SU(N)]]-[[Skyrmions]]:

* Pedro D. Alvarez, Sergio L. Cacciatori, Fabrizio Canfora, [[Bianca Cerchiai]], _Analytic $SU(N)$ Skyrmions at finite Baryon density_ ([arXiv:2005.11301](https://arxiv.org/abs/2005.11301))

category: people

[[!redirects Bianca Letizia Cerchiai]]
[[!redirects Bianca L. Cerchiai]]


Bianchi identity

+-- {: .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#
* automatic table of contents goes here
{:toc}

## Idea

The _Bianchi identity_ is a [[differential equation]] satisfied by [[curvature]] data.

It can be thought of as generalizing the equation $d (d A) = 0$ for a real-valued [[differential form|1-form]] $A$ to higher degree and nonabelian forms.

Generally it applies to the [[curvature]] of [[∞-Lie algebroid valued differential forms]].

## Definition

### For 2-form curvatures

Let $U$ be a [[smooth manifold]].

For $A \in \Omega^1(U)$ a differential 1-form, its [[curvature]] 2-form is the de Rham differential $F_A = d A$. The Bianchi identity in this case is the equation

$$
  d F = 0
  \,.
$$

More generally, for $\mathfrak{g}$ an arbitrary [[Lie algebra]] and $A \in \Omega^1(U,\mathfrak{g})$ a [[Lie-algebra valued 1-form]], its [[curvature]] is the 2-form $F_A = d A + [A \wedge A]$. The Bianchi identity in this case is the equation

$$
  d F_A + [A\wedge F_A] = 0
$$

satisfied by these curvature 2-forms.

### Reformulation in terms of Weil algebras

We may reformulate the above identities as follows.

For $\mathfrak{g}$ a [[Lie algebra]] we have naturally associated two [[dg-algebra | dg-algebras]]: the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ and the [[Weil algebra]] $W(\mathfrak{g})$.

The dg-algebra morphisms

$$
  \Omega^\bullet(U) \leftarrow W(\mathfrak{g}) : (A,F_A)
$$

are precisely in bijection with [[Lie-algebra valued 1-form | Lie-algebra valued 1-forms]] as follows: the [[Weil algebra]] is of the form

$$
  W(\mathfrak{g}) = \wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1]), d_{W(\mathfrak{g})}
$$

with one copy of $\mathfrak{g}^*$ in degree 1, the other in degree 2. By the [[free construction|free]] nature of the Weil algebra, dg-algebra morphisms $\Omega^\bullet(U) \leftarrow W(\mathfrak{g})$ are in bijection to their underlying morphisms of vector spaces of generators

$$
  \Omega^1(U) \leftarrow \mathfrak{g}^* : A
  \,.
$$

This identifies the 1-form $A \in \Omega^1(U,\mathfrak{g})$. This extends uniquely to a morphism of dg-algebras and thereby fixes the image of the shifted generators

$$
  \Omega^2(U) \leftarrow \mathfrak{g}^*[1] : F_A
  \,.
$$

The _Bianchi identity_ is precisely the statement that these linear maps, extended to morphisms of graded algebra, are compatible with the differentials and hence do constitute [[dg-algebra]] morphisms.

Concretely, if $\{t^a\}$ is a dual basis for $\mathfrak{g}^*$ and $\{r^a\}$ the corresponding dual basis for $\mathfrak{g}^*[1]$ and $\{C^a{}_{b c}\}$ the structure constants of the Lie bracket $[-,-]$ on $\mathfrak{g}$, then the differential $d_{W(\mathfrak{g})}$ on the Weil algebra is defined on generators by

$$
  d_{W(\mathfrak{g})} t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a
$$

and

$$
  d_{W(\mathfrak{g})} r^a = C^a{}_{b c} t^b \wedge r^c
  \,.
$$

The image of $t^a$ under $\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) : (A,F_A)$ is the component $A^a$. The image of $r^a$ is therefore, by respect for the differential on $t^a$

$$
  r^a \mapsto (F_A)^a := d A^a + \frac{1}{2}C^a{}_{b c} A^b \wedge A^c
  \,.
$$

Respect for the differential on $r^a$ then implies

$$
  d (F_A)^a + C^a{}_{b c} A^a \wedge (F_A)^c = 0
  \,.
$$

This is the Bianchi identity.

### For curvature of $\infty$-Lie algebra valued forms.

Let now $\mathfrak{g}$ be an arbitrary [[∞-Lie-algebra]] and $W(\mathfrak{g})$ its [[Weil algebra]]. Then a collection of [[∞-Lie algebra valued differential forms]] is a [[dg-algebra]] morphism

$$
  \Omega^\bullet(U) \leftarrow W(\mathfrak{g}) : A
  ,.
$$

It [[curvature]] is the composite of morphism of [[graded vector space]]

$$
  \Omega^\bullet(U)
  \stackrel{A}{\leftarrow}
  W(\mathfrak{g})
  \stackrel{F_{(-)}}{\leftarrow}
  \mathfrak{g}^*[1]
  :
  F_A
  \,.
$$

Since $A$ is a [[homomorphism]] of [[dg-algebra | dg-algebras]], this satisfies

$$
  d_{dR} F_A  + A(d_{W(\mathfrak{g})}(-)) = 0
  \,.
$$

This identity is the **Bianchi identity for $\infty$-Lie algebra valued forms**.

## Related concepts

* [[Gauss law]]

* [[∞-Lie algebroid valued differential forms]]

* [[connection on an ∞-bundle]]

* [[curvature]]

* [[adjusted Weil algebra]]

* [[curvature characteristic form]]

* [[Chern-Simons form]]

## References

Named after *[[Luigi Bianchi]]*.

The Bianchi identity for [[∞-Lie algebroid valued differential forms]] is discussed in

* SSSI (web)

[[!redirects Bianchi identities]]


Biao Lian

* [Institute page](https://phy.princeton.edu/people/biao-lian)

## Selected writings

On [[Chern-Simons theory]] potentially applicable to a form of [[Berry curvature]] on the [[Brillouin torus]] of 3d [[topological semimetals]], with nodal lines playing the role of [[Wilson lines]]:

* [[Biao Lian]], [[Cumrun Vafa]], [[Farzan Vafa]], [[Shou-Cheng Zhang]], *Chern-Simons theory and Wilson loops in the Brillouin zone*, Phys. Rev. B **95** (2017) 094512 [[doi:10.1103/PhysRevB.95.094512](https://doi.org/10.1103/PhysRevB.95.094512)]

On [[Shoucheng Zhang]]:

* [[Biao Lian]] et al (eds). *Memorial Volume for Shoucheng Zhang*, World Scientific (2021) [[doi:10.1142/12146](https://doi.org/10.1142/12146)]

category: people


biased definition

# Biased definitions
* table of contents
{: toc}

## Idea

Often in mathematics, when requiring some structure/operation/property/... to exist at every [[finite set|finite]] [[arity]], it suffices to require only the binary ($2$-ary) and nullary ($0$-ary) forms, from which the others follow.  A definition in which only these are required is called __biased__.

For example, in defining a [[category]], one could use an "unbiased" definition in which [[composites]] of all [[list|finite sequences]] of [[morphisms]] are directly postulated, with corresponding [[associativity]] laws, but it suffices to require only binary composites and nullary composites (i.e., [[identity morphisms]]) and some particular corresponding associativity laws.

As a special case of this, we have perhaps the prototypical example of a binary/nullary pair sufficing to generate all finite instances of some structure: the [[natural numbers]] themselves (the arities of the operations that we are considering) are the [[free monoid]] on one generator, and thus are freely associatively produced from that one generator (aka, $1$) using only binary and nullary addition.

Sometimes it is too easy to write a definition that involves only the binary condition; writing an unbiased definition can make it easier to notice the corresponding nullary condition.  Compare when things are [[too simple to be simple]].

## When a nullary operation does not exist

Sometimes a nullary operation does not exist but one still wants to decompose a n-ary operation into binary operations. For example, consider [[real number|the reals]], $\mathbb{R}$, as an [[lattice|unbounded lattice]] ([[top|top]], $\top$, and [[bottom|bottom]], $\bot$, do not exist) where

 $\wedge =$ [[product]] $=$ [[meet]] $= infimum = min$

and

 $\vee =$ [[coproduct]] $=$ [[join]] $= supremum = max$.

Here $\bigwedge(\{\})$ does not exist while

 $\bigwedge(\{a\}) = a$

 $\bigwedge(\{a,b\}) = a \wedge b$

 $\bigwedge(\{a,b,c\}) = a \wedge (b \wedge c) = (a \wedge b) \wedge c = a \wedge b \wedge c$.

One approach is to compute in the [[extended real number|extended reals]], $\mathbb{R}_{\pm \infty}$ ($\mathbb{R}$ enlarged with $+\infty$ and $-\infty$.) Here

 $\top = +\infty =$ [[terminal object]]

and

 $\bot = -\infty =$ [[initial object]].

In $\mathbb{R}_{\pm \infty}$ we have the nullary $\wedge() = +\infty$ which gives:

 $\bigwedge(\{\}) = +\infty$

 $\bigwedge(\{a\}) = \bigwedge(\{a, +\infty\}) = a$.

Another approach is to define a special scheme for composition for when a nullary operator does not exist that instead uses a unary operator that is an identity map (or factored through one).

This approach generalizes to any [[semigroup]] or to any category with binary ([[coproduct|co]]) [[products]].  A yet more general context (possibly not fully worked out) would be a [[binary operation]] in any [[associative operad]].

[[!redirects binary/nullary pair]]
[[!redirects binary/nullary pairs]]
[[!redirects binary-nullary pair]]
[[!redirects binary-nullary pairs]]
[[!redirects binary nullary pair]]
[[!redirects binary nullary pairs]]

[[!redirects bias]]
[[!redirects biased]]
[[!redirects unbiased]]
[[!redirects biased definition]]
[[!redirects biased definitions]]
[[!redirects unbiased definition]]
[[!redirects unbiased definitions]]