multi-adjoint

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# Multi-adjoints

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## Idea

A **multi-adjoint** is like an [[adjoint functor]] but sends each [[object]] to a family of objects rather than a single one.

Under mild conditions it is equivalent to the notion of [[parametric right adjoint]].

Every multiadjunction induces a [[multimonad]].

## Definition

\begin{defn}\label{MultiAdjoints}
Let $R \colon D\to C$ be a [[functor]].  We say $R$ has a **left multi-adjoint** if for each $x \in C$ there is a [[family]] of [[morphisms]] $\{\eta_{x,i} \colon x \to R L(x,i)\}_{i\in I(x)}$ such that for any morphism $f  \colon x\to R y$ there exists a unique [[pair]] of an index $i\in I(x)$ and a morphism $g \colon L(x, i) \to y$ such that $f = R g \circ \eta_{x,i}$.
\end{defn}

\begin{remark}
Of course, if each set $I(x)$ is always a [[singleton]], then Def. \ref{MultiAdjoints} reduces to the notion of an actual [[left adjoint]].  On the other hand, if we remove the uniqueness requirement, this reduces to the [[solution set condition]].  Thus, a multi-adjoint is "halfway between" the solution-set condition and the actual existence of an adjoint.
\end{remark}

\begin{remark}
If in Def. \ref{MultiAdjoints} we weaken the notion of [[adjunction]] in the opposite way, by removing the uniqueness requirement but keeping $I$ a singleton (or equivalently add to the solution-set condition that $I$ is a singleton), we obtain the notion of a [[weak adjoint]].
\end{remark}

\begin{theorem}\label{HomCharacterization}
Having a left multi-adjoint can be equivalently characterized as follows:

$R : D \to C$ has a left multi-adjoint $(I, L)$ with $I : Obj(C) \to Set$ and $L : (x : Obj(C)) \to I(x) \to Obj(D)$ if for all $x$ and $y$, there is an isomorphism $\alpha : \Sigma(i \in I(x)).Hom_D(L(x, i), y) \cong Hom_C(x, Ry)$, natural in $y$.
\end{theorem}
\begin{proof}
We first prove the Hom-characterization from the definition. Given $(i, \phi : L(x, i) \to y)$, we get $\alpha(\phi) := R\phi \circ \eta_{x, i} : x \to Ry$. The definition demands that this is an isomorphism. To see naturality, let $\chi : y \to z$.
Then
\[
        R\chi \circ \alpha(i, \phi) = R\chi \circ R\phi \circ \eta_{x,i} = \alpha(i, \chi \circ \phi).
\]

Now let $\alpha$ be given. Then we define $\eta_{x, i} = \alpha(i, id) : x \to RL(x, i)$.
By naturality, $\alpha(i, \phi) = R\phi \circ \eta_{x, i}$. Then invertibility of $\alpha$ proves the condition in the definition.
\end{proof}

\linebreak

{#FamOp} For the following theorem, let $Fam(D)$ be the category

* whose objects are [[indexed sets]] of objects of $D$, hence pairs $(I, h)$ where $I$ is a [[set]] and $h \colon I \to Obj(D)$,

* whose morphisms $(f, \phi) \colon (I, h) \to (I', h')$ consist of a function $f \colon I' \to I$ (note the contravariance) and $\phi: (i \in I') \to Hom_D\Big(h\big(f(i)\big), h'(i)\Big)$.

Define $J : D \to Fam(D) : d \mapsto (\{*\}, * \mapsto d)$.

Hence $Fam(D)$ is the free [[cartesian monoidal category]] over $D$, with $J$ the unit of the free cartesian monoidal category monad.

(This is dual to the [[free coproduct completion]], an example of a [[Grothendieck construction]], see [here](free+coproduct+completion#AsAGrothendieckConstruction).)

\begin{theorem}
A multi-adjoint to $R \col D \to C$ is a functor $K : C \to Fam(D)$ such that $R$ is the [[relative adjoint functor | $J$-right adjoint to $K$]], meaning that
\[
        \alpha : Hom_{Fam(D)}(K c, J d) \cong Hom_{C}(c, R d),
\]
naturally in $c$ and $d$.
\end{theorem}
\begin{proof}
We use the characterization in theorem \ref{HomCharacterization}.

First of all, note that the type of the object part of $K$ is the same as the type of $(I, L)$; let us identify these.
Next, note that $Hom_{Fam(D)}(K c, J d) \cong \Sigma(i \in I(x)).Hom_D(L(c, i), d)$.

So to show that the characterization here implies the definition, we simply need to forget functoriality of $K$ and naturality in $c$.

Conversely, assume that the object part of $K$ is given. Then we construct a morphism part such that $\alpha$ is natural in $c$.

Take $\phi : Hom_C(c, c')$. We want $K\phi : Hom_{Fam(D)}(Kc, Kc')$, i.e.
\[
        (i' \in I(c')) \to \Sigma(i \in I(c)).Hom_D(L(c, i), L(c', i')),
\]
i.e.
\[
        (i' \in I(c')) \to Hom_{Fam(D)}(Kc, JL(c', i'))
\]
so we can take $K(\phi, i') = \alpha^{-1}(\eta_{c', i'} \circ \phi)$. Denote the components of $K\phi$ as $I\phi : I(c') \to I(c)$ and $L\phi : (i' \in I(i')) \to Hom_D(L(c, I(\phi, i')), L(c', i'))$.

* This preserves the identity: $K(id) = \lambda i.\alpha^{-1}(\eta_{x, i} \circ id) = \lambda i.(i, id)$.

* This preserves composition. Let $\phi : Hom_C(c, c')$ and $\chi : Hom_C(c', c'')$. With some abuse of notation:

  $K \chi \circ K \phi = (I\phi \circ I\chi, \lambda i''.L(\chi, i'') \circ L(\phi, I(\chi, i'')))$

  $= \lambda i''.(I(\phi, I(\chi, i'')), L(\chi, i'') \circ L(\phi, I(\chi, i'')))$

  $= \lambda i''.JL(\chi, i'') \circ (I(\phi, I(\chi, i'')), L(\phi, I(\chi, i'')))$

  $= \lambda i''.JL(\chi, i'') \circ K(\phi, I(\chi, i''))$

  $= \lambda i''.JL(\chi, i'') \circ \alpha^{-1}(\eta_{c', I(\chi, i'')} \circ \phi)$

  $= \lambda i''.\alpha^{-1}(RL(\chi, i'') \circ \alpha(I(\chi, i''), id_{c'}) \circ \phi)$

  $= \lambda i''.\alpha^{-1}(\alpha(I(\chi, i''), L(\chi, i'')) \circ \phi)$

  $= \lambda i''.\alpha^{-1}(\alpha(K(\chi, i'')) \circ \phi)$

  $= \lambda i''.\alpha^{-1}(\eta_{c'', i''} \circ \chi \circ \phi)$.

Now we prove that $\alpha$ is natural in $c$. Let $\psi : Hom_{Fam(D)}(Kc', Jd)$ and $\phi : Hom_{C}(c, c')$.
Note that $\psi$ is essentially of the form $(i', \chi)$ for $i' \in I(c')$ and $\chi \in \Hom_D(L(c', i), d)$.
With more abuse of notation:

$\alpha(\psi \circ K\phi) = \alpha((i', \chi) \circ K\phi) = \alpha(I(\phi, i'), \chi \circ L(\phi, i'))$

$= \alpha(J\chi \circ K(\phi, i')) = \alpha(J\chi \circ \alpha^{-1}(\eta_{c', i'} \circ \phi))$

$= R\chi \circ \eta_{c', i'} \circ \phi = \alpha(J\chi \circ (i', id_{c'})) \circ \phi = \alpha(i', \chi) \circ \phi = \alpha(\psi) \circ \phi$.
\end{proof}

\begin{proposition}
A functor has a left multiadjoint if and only if it factors (essentially uniquely) as a right adjoint functor followed by a discrete fibration.
\end{proposition}

Since both [[right adjoint]] functors and [[discrete fibrations]] have left multiadjoints, one direction follows from the closure of functors having left multiadjoints under composition.

In the other direction, every functor $R$ factors into a [[final functor]] followed by a [[discrete fibration]] via the [[comprehensive factorisation]]. If $R$ has a left multiadjoint, the final functor will in addition be right adjoint.

For a complete proof, see ([Diers 80, Proposition 1.1](#Diers1980)).

## Examples

\begin{example}
Every [[parametric right adjoint]] $R : D \to C$ with [[locally small category|locally small]] codomain has a left multi-adjoint, where

* $I(c) = Hom_C(c, R\top)$,
* $L$ is the left adjoint to $R^{/\top} : D \to C/R\top : d \mapsto (Rd, R())$.

Conversely, if $D$ has a [[terminal object]] and $R \colon D\to C$ has a left multi-adjoint, then it is a parametric right adjoint. Indeed, using the characterization in theorem \ref{HomCharacterization}, we see immediately that
\[
        Hom_C(c, R\top) \cong \Sigma(i \in I(c)).Hom_D(L(c, i), \top) \cong I(c).
\]
Moreover, by naturality in $d$, for any morphism $\phi : Hom_C(c, Rd)$, the first component of $\alpha^{-1}(\phi)$ is fully determined by $R() \circ \phi : Hom_C(c, R\top)$.
\end{example}

\begin{example}\label{MultiInitialObjectsInFields}
The [[category]] [[Field]] of [[fields]] has a multi-initial object, i.e. the functor $Fld \to 1$ has a left multi-adjoint (Def. \ref{MultiAdjoints}).  This consists of all the "[[prime fields]]", $\mathbb{Q}$ and $\mathbb{F}_p$ for some [[prime number]] $p$.
\end{example}

\begin{example}
The forgetful functor from the category of commutative [[local rings]] and local homomorphisms to the category of sets has a left multi-adjoint.
\end{example}

## Diers spectrum

The notion of a multiadjoint functor is used to define the [[Diers spectrum]].
See there for more details.

## Related pages

* [[multirepresentable functor]]

## References

The notion of multi-adjoints is credited by [Osmond 20a](#Osmond2020a), [20b](#Osmond2020a) to:

* [[Yves Diers]], _Catégories localisables_, PhD thesis. Paris 6 et Centre universitaire de Valenciennes et du Hainaut Cambrésis (1977) [[[Categories localisables.pdf:file]]]

* [[Yves Diers]], _Catégories localement multiprésentables_, Archiv der Mathematik 34.1 (1980), pp. 344–356.

* [[Yves Diers]], _Une construction universelle des spectres, topologies spectrales et faisceaux structuraux_, Communication in Algebra Volume 12, Issue 17-18 (1984) ([doi:10.1080/00927878408823101](https://doi.org/10.1080/00927878408823101))

(in discussion of the notion of [[spectrum (geometry)|spectra in geometry]]).

See also:

* {#Diers1980} [[Yves Diers]], _Multimonads and multimonadic categories_, Journal of Pure and Applied Algebra 17 (1980) 153-170 ([pdf](https://core.ac.uk/download/pdf/82552386.pdf))

* {#Osmond2020a} [[Axel Osmond]], _On Diers theory of Spectrum I: Stable functors and right multi-adjoints_, ([arXiv:2012.00853](https://arxiv.org/abs/2012.00853))

* {#Osmond2020b} [[Axel Osmond]], _On Diers theory of Spectrum II: Geometries and dualities_, ([arXiv:2012.02167](https://arxiv.org/abs/2012.02167))

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multi-cospan

The concept [[duality|dual]] to [[multispan]]. See there for details.

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[[!redirects multicospan]]
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multi-trace operator

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#Contents#
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## Idea

A _multi trace observable_ in a [[gauge theory]] is a [[polynomial]] in [[single trace operators]] (see there for more background).

## Properties

### Under AdS/CFT correspondence

Under the [[AdS/CFT correspondence]], [[single trace observables]] in the gauge theory correspond to single particle/string excitations on the [[gravity]]-side, while multi-trace observables correspond to multi-particle/string excitations ([Liu 98, p. 6 (7 of 39)](#Liu98), [Andrianapoli-Ferrara 99, p. 13](#AndrianapoliFerrara99), [Chalmers-Schalm 00, Section 7](#ChalmersSchalm00), [Aharony-Gubser-Maldacena-Ooguri-Oz 99, p. 75](#AharonyGubserMaldacenaOoguriOz9))

The [[asymptotic boundary|asymptotic]] [[boundary conditions]] for [[field (physics)|fields]] $\Phi$ on the AdS-side

$$
  \Phi(\vec x, r)
  \overset{r \to 0}
  \sum_i
  \alpha_i (\vec x)
  r^{d -\lambda_i}
  +
  \cdots
$$

that correspond to multi-trace observables $W(\{\mathcal{O}_i\})$ have [[coefficients]] given by the [[derivative]] of the multi-trace [[polynomial]] by its single-trace variables $\mathcal{O}_i$:

$$
  \alpha_i
  =
  \partial_{\mathcal{O}_i}
  W
  (\{\langle\mathcal{O}_i\rangle\})
$$

([Witten 01, Section 3](#Witten01))

## Examples

[[!include chord diagrams as multi-trace observables in BMN matrix model -- example]]

## References

Discussion of [[multi-trace operators]] in [[super Yang-Mills theory]] and of their [[AdS-CFT duality|AdS-CFT dual]] [[gravity]]/[[string theory]] incarnation:

* [[Tom Banks]], [[Michael Douglas]], [[Gary Horowitz]], [[Emil Martinec]], _AdS Dynamics from Conformal Field Theory_ ([arXiv:hep-th/9808016](https://arxiv.org/abs/hep-th/9808016), [spire:474214](http://inspirehep.net/record/474214))

* {#Liu98} [[Hong Liu]], _Scattering in Anti-de Sitter Space and Operator Product Expansion_, Phys. Rev. D 60, 106005 (1999) ([arXiv:hep-th/9811152](https://arxiv.org/abs/hep-th/9811152))

* {#AndrianapoliFerrara99} [[Laura Andrianopoli]], [[Sergio Ferrara]], _On short and long $SU(2,2/4)$ multiplets in the AdS/CFT correspondence_, Lett. Math.Phys. 48 (1999) 145-161 ([arXiv:hep-th/9812067](https://arxiv.org/abs/hep-th/9812067))

* {#ChalmersSchalm00} Gordon Chalmers, Koenraad Schalm, _Holographic Normal Ordering and Multi-particle States in the AdS/CFT Correspondence_, Phys. Rev. D61:046001, 2000 ([arXiv:hep-th/9901144](https://arxiv.org/abs/hep-th/9901144))

* {#AharonyGubserMaldacenaOoguriOz99} [[Ofer Aharony]], [[Steven Gubser]], [[Juan Maldacena]], [[Hirosi Ooguri]], [[Yaron Oz]], _Large $N$ Field Theories, String Theory and Gravity_, Phys. Rept. 323:183-386, 2000 ([arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111))

* Massimo Bianchi, Stefano Kovacs, Giancarlo Rossi, Yassen S. Stanev, _On the logarithmic behaviour in $\mathcal{N}=4$ SYM theory_, JHEP 9908 (1999) 020 ([arXiv:hep-th/9906188](https://arxiv.org/abs/hep-th/9906188))

* Witold Skiba, _Correlators of Short Multi-Trace Operators in $\mathcal{N}=4$ Supersymmetric Yang-Mills_, Phys. Rev. D 60, 105038 (1999) ([arXiv:hep-th/9907088](https://arxiv.org/abs/hep-th/9907088))

* [[Eric D'Hoker]], [[Daniel Freedman]], [[Samir Mathur]], A. Matusis, [[Leonardo Rastelli]], _Extremal Correlators in the AdS/CFT Correspondence_, in: _[[The many faces of the superworld]]_ ([arXiv:hep-th/9908160](https://arxiv.org/abs/hep-th/9908160))

* [[Gleb Arutyunov]], [[Sergey Frolov]], A. C. Petkou, _Perturbative and instanton corrections to the OPE of CPOs in $\mathcal{N}=4$ $SYM_4$_, Nucl. Phys. B602:238-260, 2001; Erratum-ibid. B609:540, 2001 ([arXiv:hep-th/0010137](https://arxiv.org/abs/hep-th/0010137))

* [[Ofer Aharony]], [[Micha Berkooz]], [[Eva Silverstein]], _Multiple-Trace Operators and Non-Local String Theories_, JHEP 0108:006, 2001 ([arXiv:hep-th/0105309](https://arxiv.org/abs/hep-th/0105309))

* [[Ofer Aharony]], [[Micha Berkooz]], [[Eva Silverstein]], _Non-local string theories on $AdS_3$ times $S^3$ and stable non-supersymmetric backgrounds_, Phys. Rev. D65:106007, 2002 ([arXiv:hep-th/0112178](https://arxiv.org/abs/hep-th/0112178))

* L. Hoffmann, L. Mesref, A. Meziane, W. Rühl, _Multi-trace quasi-primary fields of $\mathcal{N}=4$ $SYM_4$ from AdS $n$-point functions_, Nucl. Phys. B641 (2002) 188-222 ([arXiv:hep-th/0112191](https://arxiv.org/abs/hep-th/0112191))

* {#Witten01} [[Edward Witten]], _Multi-Trace Operators, Boundary Conditions, And AdS/CFT Correspondence_ ([arXiv:hep-th/0112258](https://arxiv.org/abs/hep-th/0112258))

* [[Steven Gubser]], Indrajit Mitra, _Double-trace operators and one-loop vacuum energy in AdS/CFT_, Phys. Rev. D67 (2003) 064018 ([arXiv:hep-th/0210093](https://arxiv.org/abs/hep-th/0210093))

* [[Vijay Balasubramanian]], [[Jan de Boer]], [[Bo Feng]], [[Yang-Hui He]], Min-xin Huang, Vishnu Jejjala, Asad Naqvi, _Multi-Trace Superpotentials vs. Matrix Models_, Commun. Math. Phys. 242:361-392, 2003 ([arXiv:hep-th/0212082](https://arxiv.org/abs/hep-th/0212082))

* [[Steven Gubser]], [[Igor Klebanov]], _A universal result on central charges in the presence of double-trace deformations_, Nucl. Phys. B656 (2003) 23-36 ([arXiv:hep-th/0212138](https://arxiv.org/abs/hep-th/0212138))

* P. J. Heslop, [[Paul Howe]], _Aspects of $\mathcal{N}$=4 SYM_, JHEP 0401 (2004) 058 ([arXiv:hep-th/0307210](https://arxiv.org/abs/hep-th/0307210))

* Thomas Hartman, [[Leonardo Rastelli]], _Double-Trace Deformations, Mixed Boundary Conditions and Functional Determinants in AdS/CFT_, JHEP 0801:019, 2008 ([arXiv:hep-th/0602106](https://arxiv.org/abs/hep-th/0602106))

Textbook account:

* {#HartnollLucasSachdev16} [[Sean Hartnoll]], [[Andrew Lucas]], [[Subir Sachdev]], Section 1.7.3 of: _Holographic quantum matter_, MIT Press 2018 ([arXiv:1612.07324](https://arxiv.org/abs/1612.07324), [publisher](https://mitpress.ublish.com/book/holographic-quantum-matter))

  (with an eye towards [[AdS/CFT in condensed matter physics]])

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multi-valued function

# Multi-valued functions
* table of contents
{: toc}

## Idea

A _multi-valued function_ $f: A \to B$ is like a [[function]] from $A$ to $B$ except that there may be more than one possible value $f(x)$ for a given element $x$ of $A$.  (Compare a [[partial function]], where $f(x)$ may not exist at all.)

In older literature (into the 20th century, especially in analysis), functions were often considered to be multi-valued by default, requiring one to specify a _singe-valued_ function otherwise.  As [[set theory|set-theoretic formalisation]] spread, this intuition became difficult to maintain, and the modern concept of function must be single-valued.  If you want multi-valued functions, then you can get them in terms of single-valued functions as below.

## Definitions

### With power sets

Given sets $A$ and $B$, a **multi-valued function** $f$ from $A$ to $B$ is a function $f$ from $A$ to the [[power set]] $\mathcal{P}(B)$ such that for each element $x$ in $A$ the subset $f(x)$ of $B$ is [[inhabited set|inhabited]]. By [[uncurrying]] the function one gets an [[entire relation]].

### Without power sets

Given [[set]]s $A$ and $B$, a __multi-valued function__ $f$ from a $A$ to $B$ is a [[span]]
$$ \array {
  &              & D \\
  & \swarrow_\pi &   & \searrow^f \\
A &              &   &            & B \\
} $$
of single-valued functions, where $\pi: D \to A$ is a [[surjection]].  (This condition can be dropped to define a multi-valued [[partial function]], which is simply a span.)

We will call $A$ and $B$ the _[[source]]_ and _[[target]]_ of $f$ as usual; then we call $D$ the __domain__ of $f$ and $\pi: D \to A $ the __projection__ of the domain onto the source.  By abuse of notation, the multi-valued function $f$ is conflated with the (single-valued) function $f: D \to B$.

Often one can assume that the induced function $D \to A \times B$ is an [[injection]]; in that case, a multi-valued function is the same as an [[entire relation]].  On the other hand, if you\'re considering all of the multi-valued functions for a given $D$, then this restriction is not really appropriate.

We consider two multi-valued functions (with the same given source and target) to be __equal__ if there is a [[bijection]] between their domains that makes the obvious diagrams commute.

## Examples

* In 19th-century analysis, one considered the square-root function, the [[logarithm]], and so forth to be multi-valued functions of complex numbers.  We now understand this in terms of [[Riemann surface]]s; the domain $D$ above is a Riemann surface.  (Notice that the logarithm is actually a multi-valued *partial* function from $\mathbf{C}$ to $\mathbf{C}$, although it is a multi-valued total function on $\mathbf{C} \setminus \{0\}$.)

* On [[anyon]]-[[wavefunctions]] as multi-valued functions on a [[configuration space of points]]: see [there](braid+group+statistics#AsAFictitiousAharonovBohmEffect)

## Related entries

* [[selection theorem]]

* [[universal cover]]

* [[multivalued partial function]]

## References

See also:

* Wikipedia, *[Multivalued function](https://en.wikipedia.org/wiki/Multivalued_function)*

[[!redirects multivalued function]]
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multiagent system

[[!redirects knowledge in a multiagent system]]
# Multiagent systems
* table of contents
{:toc}

##Idea
To quote from a book on the subject:

>"Multiagent systems are a new paradigm for understanding and building distributed systems, where it is assumed that the computational components are autonomous: able to control their own behaviour in the furtherance of their own goals." ([Wooldridge09](#Wooldridge09))

The area is a very wide one and this entry only deals with some generalities especially links between local 'knowledge' and more global forms and the models of this that involve the use of various types of [[modal logic]].

##Introduction

In many studies of [[distributed system]]s, a multiagent model is
used.

* An agent is a processor, sensor or finite state machine,
interconnected by a communication network with other 'agents'.

Typically each agent has a local state that is a function of its
initial state, the messages received from other agents,
observations of the external environment and possible internal
actions.  It has become customary when using formal models of
distributed systems to use modal [[epistemic logics]] as one of the
tools for studying the knowledge of such systems. The basic such
logic for handling a system with $n$-agents is one known as
[[the logic S5(m)|S5(n)]]. Unless the system is very simple the actual logic will be
an extension of that basic one, that is, it may have more axioms.

For instance, the way the various agents are connected influences
the logic in subtle ways. Suppose that agent 1 sends all its
information immediately to agents 2 and 3, then if we denote by
$K_i \phi$, the statement that agent $i$ 'knows' proposition
$\phi$, we clearly expect within the logic of that system that
we have as an axiom:

$$K_1 \phi \Rightarrow K_2\phi \wedge K_3 \phi,$$

in other words, if 1 knows something then we have that both 2 and 3 know it.

The logic $S5_n$ is obtained from ordinary [[propositional logic]] by
adding 'knowledge operators', $K_i$, as above. (In the literature
the notation $K_i\phi$ is often replaced by $\Box_i\phi$.)  It
models a community of *ideal* knowledge agents who have
the properties of

* veridical knowledge (everything they know is true),

* positive introspection (they know what they know)

and

*  negative introspection (they know what they do not know).

These properties are reflected in the axiom system for the logic.  For more on this see   the entry [[the logic S5(m)|S5(n)]].

##Models for these logics

The classical models for multimodal logics, and for $S5_n$ and its extensions in particular, are combinatorial models known as [[Kripke frames]] and, for $S5_n$, Kripke equivalence frames.  These consist of a set $W$, called the _set of possible worlds_, and $n$-equivalence relations $\sim_i$, one for each agent.  The interpretation of $\sim_i$ is that if $w_1$, $w_2$ are two possible worlds and $w_1\sim_i w_2$, then agent $i$ cannot tell these two worlds apart.

##Interpreted systems

 Fagin, Halpern, Moses and Vardi, in various combinations, have put forward a simpler combinatorial model known as an _interpreted system_.  These have the same formal expressive power as Kripke frames, but are nearer the intuition of interacting agents than is the more abstract Kripke model.

 As before one has a set, $A = \{1,2, \ldots, n\}$, of agents, and now one assumes each agent $i$ can be in any state of a set $L_i$ of local states.  In addition one assumes given a set $L_e$ of possible states of the `environment'.  More formally:

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

A set of global states (SGS) for an interpreted system is a subset $S$ of the product $L_e \times L_1 \times \ldots \times L_n$ with each $L_e$, $L_i$ non-empty.  If $S = L_e \times L_1 \times \ldots \times L_n$, then the SGS is called a _hypercube_.
=--

##References

###Wikipedia

* [Wikipedia article](http://en.wikipedia.org/wiki/Multi-agent_system)

###Books

* {#Wooldridge09} [[Michael Wooldridge]], _An Introduction to MultiAgent Systems_, Wiley, 2009

###Articles

 * [[A. Lomuscio]], [[M. D. Ryan]], _An algorithmic approach to knowledge evolution_, Artificial Intelligence for Engineering Design, Analysis and Manufacturing (AIEDAM), Vol. 13, No. 2 (Special issue on Temporal Logic in Engineering), 23pp, 1999.

* [[A. Lomuscio]], R. van der Meyden, [[M. D. Ryan]], _Knowledge in multi-agent systems: initial configurations and broadcast_, ACM Transactions on Computational Logic (TOCL), Volume 1, Issue 2, pp. 247- 284. ACM Press. October 2000.

category: computer science


multicategory

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##  Idea

Recall that a [[category]] consists of a collection of [[morphisms]] each having a single [[object]] as source or input, and a single object as target or output, together with laws for composition and identity obeying associativity and identity axioms. A _multicategory_ is like a category, except that one allows multiple inputs and a single output.

Another term for _multicategory_ is _coloured [[operad]]_.

## Definition

### In components

A **multicategory** $C$ consists of

* A collection of _objects_, $C_0$.
* A collection of _multimorphisms_, $C_1$.
* A source map $s: C_1 \to (C_0)*$ to the collection of finite, possibly empty [[list]]s of objects (where $(C_0)*$ is the [[free monoid]] generated by $C_0$), and a target map $t: C_1 \to C_0$. We write $f: c_1, \ldots, c_n \to c$ to indicate the source and target of a multimorphism $f$.
* Identity and composition laws. The identity law is a map $1_{-}: C_0 \to C_1$ where $1_c: c \to c$. The composition law assigns, to each $f: c_1, \ldots, c_n \to c$ together with an $n$-tuple $\langle f_i: \vec{c}_i \to c_i: i = 1, \ldots, n \rangle$, a composite
$$f \circ (f_1, \ldots, f_n): \vec{c}_1, \ldots, \vec{c}_n \to c$$
where the source is obtained by concatenating lists in the evident way.

These operations are subject to [[associativity]] and [[identity]] axioms which the reader can probably figure out, but see for example ([Leinster, page 35 ff.](#Leinster)), for details.

Many people (especially non-category theorists) use the word _multicategory_ or the word _colored [[operad]]_ to mean what we would call a _[[symmetric multicategory]]_ / _[[symmetric operad]]_. These are multicategories equipped with an [[action]] of the [[symmetric group]] $S_n$ on the multimorphisms $c_1, \ldots, c_n \to c$ such that the composition is equivariant with respect to these actions.

### In terms of cartesian monads

An efficient abstract method for defining multicategories and related structures is through the formalism of [[cartesian monads]]. For ordinary categories, one uses the identity monad on [[Set]]; for ordinary multicategories, one uses the [[free monoid]] monad $(-)*: Set \to Set$.  This is a special case of the yet more general notion of [[generalized multicategory]].

We summarize here how the theory applies to the case of a [[cartesian monad]] $T$ on a category with [[pullbacks]]; see [[generalized multicategory]] for the fully general context.

* First, a **$T$-span** from $X$ to $Y$ is a [[span]] $p$ from $T X$ to $Y$, that is, a diagram
$$T X \stackrel{p_1}{\leftarrow} P \stackrel{p_2}{\to} Y$$
A $T$-span is often written as $p: X ⇸ Y$.

When $T$ is the free monoid monad on $Set$, a $T$-span from $X$ to itself is called a _multigraph_ on $X$.

* $T$-spans are the 1-cells of a [[bicategory]]. A 2-cell between $T$-spans $e, f: X ⇸ Y$ is a 2-cell between ordinary spans from $T X$ to $Y$. To horizontally compose $T$-spans $e: X ⇸ Y$ and $f: Y ⇸ Z$, take the ordinary span composite of
$$(T X \stackrel{m X}{\leftarrow} T^2 X \stackrel{T e_1}{\leftarrow} T E \stackrel{T e_2}{\to} T Y) \circ (T Y \stackrel{f_1}{\leftarrow} F \stackrel{f_2}{\to} Z)$$
where $m: T^2 \to T$ is the monad multiplication. The identity $T$-span from $X$ to itself is the span
$$T X \stackrel{u X}{\leftarrow} X \stackrel{1_X}{\to} X$$
where $u: I \to T$ is the monad unit. The verification of the bicategory axioms uses the cartesianness of $T$ in concert with the corresponding axioms on the bicategory of spans.

* A _$T$-multicategory_ is defined to be a [[monad]] in the bicategory of $T$-spans.

When $T$ is the free monoid monad on sets, then a $T$-multicategory is a multicategory as defined above.  For more examples and generalizations, see [[generalized multicategory]].

## Properties

### Relation to operads

A [[nonpermutative operad|nonpermutative]] (or Stasheff-) [[operad]] in [[Set]] may be defined as an ordinary multicategory with exactly one object.  Likewise, a [[symmetric operad]] in any [[symmetric monoidal category]] $V$ is equivalent to a $V$-[[enriched category|enriched]] multicategory with one object.

More generally, the notion of _multi-colored [[planar operad]]_ is equivalent to that of multicategory, and the notion of _multi-colored [[symmetric operad]]_ is equivalent to that of [[symmetric multicategory]].

Fully generally, for each cartesian monad $T$, there is a corresponding notion of $T$-operad, namely a $T$-multicategory whose underlying $T$-span has the form $1 ⇸ 1$.

For example, in Batanin's approach to (weak) $\infty$-[[infinity-category|categories]], a [[globular operad]] is a $T$-operad, where $T$ is the free (strict) $\omega$-[[strict omega-category|category]] monad on the category of [[globular set]]s.

Ordinary (permutative/symmetric) operads, and their generalization to [[symmetric multicategory|symmetric multicategories]], can also be treated in the framework of [[generalized multicategories]], but they require a framework more general than that of cartesian monads.

### Relation to monoidal categories

There is a [[faithful functor]] from [[monoidal categories]] to [[multicategories]], given by forming [[representable multicategories|represented multicategories]].  Conversely, to any multicategory $C$ there is an associated (strict) monoidal category $F(C)$, whose objects (respectively, arrows) are [[lists]] of objects (respectively, arrows) of $C$, and where the tensor product in $F(C)$ is given by concatenation.

### Exponentiability

A multicategory is [[exponentiable object|exponentiable]] if and only if it is [[promonoidal category|promonoidal]] (Proposition 2.8 of [Pisani 2014](#Pisani14)). In particular, [[representable multicategories]] and [[sequential multicategories]] are exponentiable. This gives an abstract construction of the [[Day convolution]] tensor product on $[C, D]$ for any [[promonoidal category]] $C$ and any [[monoidal category]] $D$: it is precisely the (representable) multicategory structure on the functor multicategory $D^C$.

## Examples and special cases

* [[permutative categories]] with multi-linear functors between them form a multicategory [[PermCat]]

See also the examples at _[[operad]]_.

## Related concepts

* [[multimorphism]]

* [[multifunctor]]

* [[polycategory]]

* [[properad]]

* [[fibration of multicategories]]

* [[symmetric multicategory]]

* [[cartesian multicategory]]

* [[sequential multicategory]]

* [[generalized multicategory]], [[operad]]

## References

* [[Tom Leinster]], _Higher operads, higher categories_, London Math. Soc. Lec. Note Series __298__, [math.CT/0305049](http://arxiv.org/abs/math.CT/0305049)
 {#Leinster}

* {#Pisani14} [[Claudio Pisani]], _Sequential multicategories_, Theory and Applications of Categories 29.19 (2014), [arXiv:1402.0253](https://arxiv.org/abs/1402.0253)

[[!redirects multicategories]]

[[!redirects pseudomonoidal category]]


multifunction

#Contents#
* table of contents
{:toc}

## Idea

A _multifunction_ is a [[function]] of several [[variables]], hence equivalently an ordinary [[function]] on a [[cartesian product]].

[[set|Sets]] with multifunctions as [[morphisms]] between them form a [[multicategory]].

## Related concepts

* [[multifunctor]]

[[!redirects multifunction]]
[[!redirects multifunctions]]
[[!redirects multiary function]]
[[!redirects multiary functions]]
[[!redirects multi-ary function]]
[[!redirects multi-ary functions]]
[[!redirects function of several variables]]
[[!redirects functions of several variables]]


multifunctor

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## Idea

By a *multifunctor* one may mean:

1. a "functor of several variables", i.e., the [[categorification]] of the notion of a *[[multifunction]]*)

1. a *[[morphism of multicategories]]*.

## Functor of several variables

There are at least two ways to generalize the notion of a [[functor]] to the case where its [[domain]] may an [[n-tuple]] of [[categories]]:

1. [jointly functorial maps](#JointlyFunctorialMaps)

1. [separately functorial maps](#SeparatelyFunctorialMaps)

which we discuss in turn.

In all of the following,

* $D$ denotes a [[category]] which serves as the [[codomain]];

* $C_1, \cdots, C_n$ denotes an [[n-tuple|$n$-tuple]] of categories serving as the multi-[[domain]];

* ${\vert -\vert}$ denotes passage to the [[class]] of [[objects]] of a ([[strict category|strict]]) category.

### Jointly functorial maps
 {#JointlyFunctorialMaps}

A **jointly functorial** [[map]] from $C_1, \cdots, C_n$ to $D$ consists of:

1. a [[multifunction]] on [[objects]]

   $$
     F
       \,\colon\,
    \big(
      {|C_1|}, \ldots, {\vert C_n\vert}
    \big)
      \longrightarrow
     {|D|}
   $$

2. For all [[n-tuples]] of [[morphisms]]

   $$
     f_i \,\in\, C_i(a_i,b_i),
     \;\;\;\;
     1 \leq i \leq n
   $$

   a morphism of the form

   $$
     F(f_1,\ldots, f_n)
      \,\colon\,
     F(a_1,\ldots, a_n) \to F(b_1,\ldots, b_n)
   $$

   in $D$.

such that:

1. [[identity morphisms]] are preserved, in that

   $$
    F(id_{a_1}, \ldots, id_{a_n})
    \;=\;
    id_{F(a_1, \ldots, a_n)}
   $$

4. [[composition]] is respected, in thay

   $$
     F(f_1 \circ g_1,\ldots, f_n \circ g_n)
     \;=\;
     F(f_1,\ldots, f_n) \circ F(g_1,\ldots, g_n)
     \,.
   $$

Such a jointly functorial map is the same as an ordinary [[functor]] out of the [[product category]] of the $n$-tuple of domain categories:

$$
  C_1 \times \cdots \times C_n \longrightarrow D
  \,.
$$

In the case $n = 1$ this is an ordinary functor, while for $n = 2$ this is a "[[bifunctor]]". And if one understands multifunctions of [[zero]] arguments as functions out of the empty product of domain categories, which is the [[terminal category]], then for $n = 0$ this is just a choice of [[object]] of $D$.

### Separately functorial maps
 {#SeparatelyFunctorialMaps}

On the other hand, rather than requiring an "action" on morphisms from each domain category simultaneously, one may want to require an action of each domain category *separately*, which we could call *separately functorial*. I.e., a separately functorial map $\big(C_1,\ldots, C_n\big) \to D$ consists of:

1. A [[multifunction]] of objects

   $$
     F
      \colon
     \big(
       {|C_1|},\ldots, {\vert C_n\vert}
     \big)
       \longrightarrow
     D
   $$

2. Such that for each domain category $C_i$, and objects $a_1,\ldots\widehat{a_i},\ldots, a_n$, the map $F(a_1,\ldots,\widehat {a_i},\ldots, a_n) \colon C_i \to D$ extends to a functor from $C_i$ to $D$.

This definition is instead equivalent to an ordinary functor out of the [[funny tensor product]] of the domain categories

$$
  C_1 \Box \cdots \Box C_n \longrightarrow D
  \,.
$$

For $n = 0$ and $n = 1$ this definition coincides with that of jointly functorial maps [above](#JointlyFunctorialMaps), bu for $n \geq 2$ arguments it is different.

For more see also at *[funny tensor product -- Separate functoriality](funny+tensor+product#SeparateFunctoriality)*

### Relation Between Joint and Separate Functoriality

Every jointly functorial map defines a separately functorial map by defining the action as
$$F(a_1,\ldots,\widehat {a_i},\ldots, a_n)(f_i) = F(id,\ldots,f_i,\ldots,id)$$

On the other hand, there is not way to extend an arbitrary separately functorial map in this way to be jointly functorial. We can attempt to define

$$F(f_1,\ldots) = F(\widehat{a_1},\ldots)(f_1) \circ F(a_1,\widehat{a_2},\ldots)(f_2) \circ \cdots$$
But note that for $n\geq 2$ this involves an arbitrary choice of which order to compose these actions, and this will only be a jointly functorial action if all such choices are equal. This can be stated as saying for any pair $i\leq j\leq n$ that
$$F(a_1,\ldots,\widehat{a_i},\ldots a_n)(f_i) \circ F(a_1,\ldots,\widehat{a_j},\ldots,a_n)(f_j) = F(a_1,\ldots,\widehat{a_j},\ldots a_n)(f_j) \circ F(a_1,\ldots,\widehat{a_i},\ldots,a_n)(f_i)$$

## Related concepts

* [[two-variable adjunction]]

## References

On "functors of several variables":

* [[eom]]: *[Multi-functor](https://encyclopediaofmath.org/wiki/Multi-functor)*

[[!redirects multifunctors]]

[[!redirects multi-functor]]
[[!redirects multi-functors]]


multigrade operator

The material previously on this page is now located at the [Online Encyclopedia of Integer Sequences Wiki](https://oeis.org/wiki/Multigrade_operator).

[[!redirects multigrade operators]]


multigraph

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* table of contents
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## Idea

In [[graph theory]] a _multigraph_ a particular type of [[graph]]. A multigraph is a [[set]] of [[vertices]] and for each unordered pair of distinct vertices a set of [[edges]] between these.

The terminology is used in distinction to

1. _[[simple graphs]]_, which are multigraphs with at most one edge between any unordered pair of distinct vertices;

1. _[[pseudographs]]_, which are like multigraphs but admitted to have "[[loops]]" in the sense of [[edges]] between a vertex and itself.

Beware that the terminology is not completely consistent across different authors. Some authors may allows [[loops]] when they speak of multigraphs.

## Definition

More formally this means that a multigraph is

1. a [[set]] $V$ ("of [[vertices]]");

1. a [[set]] $E$ ("of [[edges]]");

1. a [[function]] $E \to \left\{ {\,\atop \,} \{v_1, v_2\} = \{v_2, v_1\} \;\vert\; v_1, v_2 \in V \,,\; v_1 \neq v_2 {\, \atop \,} \right\}$ (sending any [[edge]] to the unordered pair of distinct [[vertices]] that it goes between).

Specifically a [[finite graph|finite]] multigraph is, after choosing a [[linear order]] on its set of vertices, the same as

1. a [[natural number]] $v \in \mathbb{N}$ (the number of [[vertices]]);

1. for each $i \lt j \in \{1, \cdots, v\}$ a natural number $e_{i,j} \in \mathbb{N}$ (the number of [[edges]] between the $i$th and the $j$th vertex)

## Examples

* In [[perturbative quantum field theory]] the [[graphs]] underlying [[Feynman diagrams]] are [[finite graph|finite]] [[multigraphs]] (e.g. [Borinsky](#Borinsky)).

## References

Discussion with an eye towards [[Feynman diagrams]] includes

* {#Borinsky} Michael Borinsky, section 2.1.1 of _Algorithmization of the Hopf algebra of Feynman graphs_ ([pdf](http://www2.mathematik.hu-berlin.de/~kreimer/wp-content/uploads/BorinskyMaster.pdf))

See also

* Wikipedia, _[Multigraph](https://en.wikipedia.org/wiki/Multigraph)_

[[!redirects multigraphs]]

[[!redirects finite multigraph]]
[[!redirects finite multigraphs]]


multilimit

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## Idea

A **multilimit** of a diagram $F\colon D\to C$ in a [[category]] $C$ is a (small) set $L$ of [[cones]] over $F$ such that any cone over $F$ factors uniquely through a unique element of $L$.  Of course, [[limits]] are precisely multilimits for which $L$ is a [[singleton]].

For example, the [[category]] [[Field]] of [[fields]] has no [[initial object]], but it has a multi-initial-object, namely the family of all [[prime field]]s (in fact, this category is [[locally multipresentable]]).

## Related entries

* [[multi-adjoint]]
* [[weak multilimit]]

[[!redirects multicolimit]]
[[!redirects multi-limit]]
[[!redirects multi-colimit]]
[[!redirects multilimits]]
[[!redirects multicolimits]]


multilinear algebra

Multilinear algebra studies [[multilinear map | multilinear maps]] and constructions.  These are usually formulated as [[linear-algebra]] constructions on [[tensor products]] (especially [[tensor powers]]) of [[vector spaces]], and their [[linear subspace|subspaces]] and [[quotient vector space|quotient spaces]] that have some symmetry with respect to a natural action of a [[symmetric group]], for example the [[exterior power|exterior]] and [[symmetric power|symmetric]] powers of vector spaces, and generalizations for [[vector bundles]], [[coherent sheaves]], etc., and also those related to constructions relating other Ferrers--[[Young tableaux]], not only symmetrizers and antisymmetrizers.

Related entries include [[Young diagram]], [[tensor algebra]], [[symmetric algebra]], [[exterior algebra]], [[Schur functor]], [[tensor]], etc.

[[!redirects multilinear algebra]]
[[!redirects multilinear-algebra]]
[[!redirects multilinear algebras]]
[[!redirects multilinear-algebras]]
[[!redirects multilinear algebraic]]
[[!redirects multilinear-algebraic]]


multimonad

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# Multi-adjoints

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## Idea

A **multimonad** is to a [[multiadjunction]] what a [[monad]] is to an [[adjunction]].

(Note that this is **not** related to monads on [[multicategories]].)

## Definition

\begin{defn}
A **multimonad** on a category $C$ comprises a [[presheaf]] on $C$ together with a monad on its [[category of elements]].
\end{defn}

## Multimonadic categories

Multi-monadic categories on $Set$ can be characterized in the following way: they are regular, with connected limits, with coequalizers of coequalizable pairs, their equivalence relations are effective, their forgetful functors preserve coequalizers of equivalence relations and reflect isomorphisms. Unlike  monadic categories they need not have products. Examples include local rings, fields, inner spaces, locally compact spaces, locally compact groups, and complete ordered sets. ([Diers 80, p.153](#Diers1980))

The forgetful functor from a multimonadic category [[created limit|creates]] [[connected limits]].

## References

* {#Diers1980} [[Yves Diers]], _Multimonads and multimonadic categories_, Journal of Pure and Applied Algebra 17 (1980) 153-170 ([pdf](https://core.ac.uk/download/pdf/82552386.pdf))


multimorphism

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## Idea

A _multimorphism_ is a [[morphism]] $A_1, \cdots, A_n \to B$ in a [[multicategory]].

## Related concepts

* [[binary function]], [[bilinear map]], [[multilinear map]]

* [[binary morphism]], **multimorphism**

* [[bifunctor]], [[multifunctor]]

[[!redirects multimorphism]]
[[!redirects multimorphisms]]


multinomial coefficient

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## Idea

For $n \in \mathbb{N}_0$ a [[natural number]] or zero and $(k_i \in \mathbb{N}_0)_{i = 1}^r$ with $\underset{i}{\sum} k_i = k$, the corresponding _multinomial  coefficient_

$$
  \left(
     n
     \atop
     {
       k_1, k_2, \cdots, k_r
     }
  \right)
  \;\coloneqq\;
  \frac{
     n!
  }{
    k_1 ! \, k_2 ! \, \cdots \, k_r
  }
  \;\in\;
  \mathbb{N}
$$

is the quotient of the [[factorial]] of $n$ by the [[multiplication]] of the factorials of the $k_i$.

This number is the number of ways of drawing $k_1$ elements and then $k_2$ elements and so forth from $n$ elements, all in an unordered way.

For $r = 2$ this is the _[[binomial coefficient]]_.

## References

See also

* Wikipedia, _[Multinomial theorem](https://en.wikipedia.org/wiki/Multinomial_theorem)_

[[!redirects multinomial coefficients]]


multiple zeta values

#Contents#
* table of contents
{:toc}

## Idea

A _multiple zeta value_ (MZV) is like the value of a [[zeta function]] in more than one [[variable]].

Where [[zeta functions]] appear in [[physics]] as expressions for [[vacuum amplitudes]], so multiple zeta functions appear in expressions for more general [[scattering amplitudes]]. The intricate combinatorics of these becomes often more tractable when re-expressing them as _[[motivic multiple zeta values]]_.

## Definition

The standard multiple zeta values are the values of the _multiple zeta function_ $\zeta$ of $k$ [[variables]] $s_k$ defined by

$$
  \zeta(s_1, \cdots , s_k)
  =
  \underset{n_1 \gt n_2 \gt \cdots \gt n_k\gt 0}{\sum}
  n_1^{-s_1} \cdots n_k^{-s_k}
  \,.
$$

For $k = 1$, hence for a single argument, this reduces to the [[Riemann zeta function]]

## Related concepts

* [[zeta function]]

* [[motivic multiple zeta values]]

* [[polylogarithm]]

## References

* Leila Schneps, _Survey of the theory of multiple zeta values_ (2011) ([pdf](http://www.maths.bris.ac.uk/events/meetings/uploads/4159LeilaSchnepsII.pdf))

[[!redirects multiple zeta function]]
[[!redirects multiple zeta functions]]


multiplication

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## Idea

The binary operation in a [[monoid]] like a [[group]] or [[ring]] is called _multiplication_. Sometimes an [[action]] on a [[monoid]] is also called _multiplication_

## Related concepts

* [[sum]]

* [[arithmetic]]

* [[multiplicative cohomology theory]]

## Related concepts

* The [[categorification]] of the operation of multiplication in a [[monoid]] is the _[[tensor product]]_ in a _[[monoidal category]]_.

[[!redirects multiplications]]


multiplicative cohomology theory

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## Idea

A [[cohomology theory]] $E$ is called _multiplicative_ if each [[graded module|graded]] [[abelian group|abelian]] $E$-[[cohomology group]] $E^\bullet(X)$ is compatibly equipped with the structure of a [[graded ring]].

The corresponding [[Brown representability theorem|representing]] [[spectra]] are [[ring spectra]].

## Definition

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

Let $E_1, E_2, E_3$ be three unreduced [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology theories]] ([def.](generalized+cohomology+theory#GeneralizedCohomologyTheory)). A **pairing of cohomology theories**

$$
  \mu \;\colon\; E_1 \Box E_2 \longrightarrow E_3
$$

is a [[natural transformation]] (of functors on $(Top_{CW}^{\hookrightarrow}\times Top_{CW}^{\hookrightarrow})^{op} $) of the form

$$
  \mu_{n_1,n_2}
   \;\colon\;
  E_1^{n_1}(X,A)
    \otimes
  E_2^{n_2}(Y,B)
    \longrightarrow
  E_3^{n_1 + n_2}(X\times Y \;,\; A\times Y \cup X \times B)
$$

such that this is compatible with the connecting homomorphisms $\delta_i$ of $E_i$, in that the following are [[commuting squares]]

$$
  \array{
    E_1^{n_1}(A)
      \otimes
    E_2^{n_2}(Y,B)
      &\overset{\delta_1 \otimes id_2}{\longrightarrow}&
    E_1^{n_1+1}(X,A) \otimes E_2^{n_2}(Y,B)
    \\
    {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1+1, n_2}}}
    \\
   \underoverset
     {E_3^{n_1 + n_2}(A \times Y  \cup  X \times B , X \times B)}
     {E_3^{n_1 + n_2}(A \times Y, A \times B)}
     {\simeq}
     &\overset{\delta_3}{\longrightarrow}&
     E_3^{n_1 + n_2+ 1}(X \times Y, A \times B)
  }
$$

and

$$
  \array{
    E_1^{n_1}(X,A)
      \otimes
    E_2^{n_2}(B)
      &\overset{(-1)^{n_1} id_1 \otimes \delta_2}{\longrightarrow}&
    E_1^{n_1+1}(X,A) \otimes E_2^{n_2}(Y,B)
    \\
    {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{n_1, n_2 + 1}}}
    \\
   \underoverset
     {E_3^{n_1 + n_2}(A \times Y \cup X \times B , A \times Y)}
     {E_3^{n_1 + n_2}(X \times B, A \times B)}
     {\simeq}
     &\overset{\delta_3}{\longrightarrow}&
     E_3^{n_1 + n_2+ 1}(X \times Y, A \times B)
  }
  \,,
$$

where the isomorphisms in the bottom left are the [excision isomorphisms](generalized+cohomology+theory##excision).

=--

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

An (unreduced) **multiplicative cohomology theory** is an unreduced [[generalized cohomology theory]] theory $E$ ([def.](generalized+cohomology+theory#GeneralizedCohomologyTheory)) equipped with

1. (external multiplication) a pairing (def. \ref{PairingOfUnreducedCohomologyTheories}) of the form $\mu \;\colon\; E \Box E  \longrightarrow E$;

1. (unit) an element $1 \in E^0(\ast)$

such that

1. ([[associativity]]) $\mu \circ (id \otimes \mu) = \mu \circ (\mu \otimes id)$;

2. ([[unitality]]) $\mu(1\otimes x) = \mu(x \otimes 1) = x$ for all $x \in E^n(X,A)$.

The mulitplicative cohomology theory is called **commutative** (often considered by default) if in addition

* **(graded commutativity)**

  $$
    \array{
      E^{n_1}(X,A) \otimes E^{n_2}(Y,B)
      &\overset{(u \otimes v) \mapsto (-1)^{n_1 n_2} (v \otimes u) }{\longrightarrow}&
      E^{n_2}(Y,B) \otimes E^{n_1}(X,A)
      \\
      {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2}}}
      \\
      E^{n_1 + n_2}( X \times Y , A \times Y \cup X \times B)
      &\underset{(switch_{(X,A), (Y,B)})^\ast}{\longrightarrow}&
      E^{n_1 + n_2}( Y \times X , B \times X \cup Y \times A)
    }
    \,.
  $$

{#InternalMultiplicationOfMultiplicativeCohomologyTheory} Given a multiplicative cohomology theory $(E, \mu, 1)$, its **[[cup product]]** is the composite of the above external multiplication with pullback along the [[diagonal]] maps $\Delta_{(X,A)} \colon (X,A) \longrightarrow (X\times X, A \times X \cup X \times A)$;

$$
  (-) \cup (-)
   \;\colon\;
  E^{n_1}(X,A)
   \otimes
  E^{n_2}(X,A)
    \overset{\mu_{n_1,n_2}}{\longrightarrow}
  E^{n_1 + n_2}( X \times X, \; A \times X \cup X \times A)
    \overset{\Delta^\ast_{(X,A)}}{\longrightarrow}
  E^{n_1 + n_2}(X, \; A \cup B)
  \,.
$$

=--

e.g. ([Tamaki-Kono 06, II.6](#TamakiKono06))

## Properties

### Ring and module structure on cohomology groups
 {#RingAndModuleStructureOnCohomologyGroups}

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

Let $(E,\mu,1)$ be a multiplicative cohomology theory, def. \ref{MultiplicativeCohomologyTheory}. Then

1. For every space $X$ the [cup product](#InternalMultiplicationOfMultiplicativeCohomologyTheory) gives $E^\bullet(X)$ the structure of a $\mathbb{Z}$-[[graded ring]], which is graded-commutative if $(E,\mu,1)$ is commutative.

1. For every pair $(X,A)$ the external multiplication $\mu$ gives $E^\bullet(X,A)$ the structure of a left and right [[module]] over the graded ring $E^\bullet(\ast)$.

1. All pullback morphisms respect the left and right action of $E^\bullet(\ast)$ and the connecting homomorphisms respect the right action and the left action up to multiplication by $(-1)^{n_1}$

=--

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

Regarding the third point:

For pullback maps this is the [[natural transformation|naturality]] of the external product: let $f \colon (X,A) \longrightarrow (Y,B)$ be a morphism in $Top_{CW}^{\hookrightarrow}$ then naturality says that the following square commutes:

$$
  \array{
    E^{n_1}(\ast) \otimes E^{n_2}(Y,B)
    &\overset{\mu_{n_1,n_2}}{\longrightarrow}&
    E^{n_1 + n_2}(Y, B)
    \\
    {}^{\mathllap{(id,f^\ast)}}\downarrow && \downarrow^{\mathrlap{f^\ast}}
    \\
    E^{n_1}(\ast) \otimes E^{n_2}(X,A)
    &\overset{\mu_{n_1,n_2}}{\longrightarrow}&
    E^{n_1 + n_2}(Y,B)
  }
  \,.
$$

For connecting homomorphisms this is the (graded) commutativity of the squares in def. \ref{MultiplicativeCohomologyTheory}:

$$
  \array{
    E^{n_1}(\ast)\otimes E^{n_2}(A)
      &\overset{(-1)^{n_1} (id, \delta)}{\longrightarrow}&
    E^{n_1}(\ast) \otimes E^{n_2 + 2}(X)
    \\
    {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2}}}
    \\
    E^{n_1 + n_2}(A)
      &\overset{\delta}{\longrightarrow}&
     E_3^{n_1 + n_2+ 1}(X,B)
  }
  \,.
$$

=--

### Multiplicative Atiyah-Hirzebruch spectral sequences

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

Given a multiplicative cohomology theory $(A,\mu,1)$ (def. \ref{MultiplicativeCohomologyTheory}), then for every [[Serre fibration]] $X \to B$ the corresponding [[Atiyah-Hirzebruch spectral sequence]] inherits the structure of a [[multiplicative spectral sequence]].

=--

A proof of prop. \ref{AHSSForMultiplicativeCohomologyTheoryIsMultiplicative} via [[Cartan-Eilenberg systems]] is given at _[[multiplicative spectral sequence]]_. A proof arguing via [[Brown representability theorem|representing]] [[ring spectra]] is in ([Kochman 96, prop. 4.2.9](#Kochman96)).

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

Given a multiplicative cohomology theory $(A,\mu,1)$ (def. \ref{MultiplicativeCohomologyTheory}), then for every [[Serre fibration]] $X \to B$ all the differentials in the corresponding [[Atiyah-Hirzebruch spectral sequence]]

$$
  H^\bullet(B,A^\bullet(F))
  \;\Rightarrow\;
  A^\bullet(X)
$$

are linear over $A^\bullet(\ast)$.

=--

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

By construction ([here](Atiyah–Hirzebruch+spectral+sequence#ConstructionByFilteringTheBaseSpace)) the differentials are those induced by the [[exact couple]]

$$
  \array{
    \underset{s,t}{\prod} A^{s+t}(X_{s})
    &&
      \stackrel{}{\longrightarrow}
    &&
    \underset{s,t}{\prod} A^{s+t}(X_{s})
    \\
    & \nwarrow && \swarrow
    \\
    && \underset{s,t}{\prod} A^{s+t}(X_{s}, X_{s-1})
  }
  \;\;\;\;\;\;\;
  \left(
      \array{
        A^{s+t}(X_s) & \longrightarrow & A^{s+t}(X_{s-1})
        \\
        \uparrow && \downarrow_{\mathrlap{\delta}}
        \\
        A^{s+t}(X_s, X_{s-1}) && A^{s+t+1}(X_{s}, X_{s-1})
      }
  \right)
  \,.
$$

consisting of the pullback homomorphisms and the connecting homomorphisms of $A$.

By the nature of spectral sequences induced from exact couples ([this prop.](exact+couple#CohomologicalSpectralSequenceOfAnExactCouple)) its differentials on page $r$ are the composites of one pullback homomorphism, the preimage of $(r-1)$ pullback homomorphisms, and one connecting homomorphism of $A$. Hence the statement follows with prop. \ref{RingAndModuleStructureOnCohomologyGroupsOfMultiplicativeCohomplogyTheory}.

=--

### Brown representability by ring spectra
 {#BrownRepresentabilityByRingSpectra}

A _multiplicative structure_ on a [[generalized (Eilenberg-Steenrod) cohomology]] theory is the structure of a [[ring spectrum]] on the [[spectrum]] that [[Brown representability theorem|represents]] it.

(e.g. [Tamaki-Kono 06, appendix C](#TamakiKono06), for more details see [here](suspension+spectrum#SmashMonoidalDiagonals)).

In particular every [[E-∞ ring]] is a [[ring spectrum]], hence represents a multiplicative cohomology theory, but the converse is in general false.

## Examples

* [[ordinary cohomology]] with [[coefficients]] in a [[ring]], in particular [[integral cohomology]]

* [[topological K-theory]], [[K-theory]]

* [[cobordism cohomology theory]]

* [[tmf]]

* etc....

## Related entries

* [[Kronecker pairing]]

* [[universal coefficient theorem]], [[module spectrum|module cohomology theories]], ...

See also

* [[complex oriented cohomology theory]], [[chromatic homotopy theory]]

* [[Thom isomorphism]]

* [[orientation in generalized cohomology]]

* [[fiber integration]]

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

* [[multiplicative spectral sequence]]

## References

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], p. 23 (30 of 120) in: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

* [[John Michael Boardman]], Section 3 (p. 10-11) of: _Stable Operations in Generalized Cohomology_ [[pdf](https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf), [[Boardman-StableOperations.pdf:file]]] in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford 1995 ([doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

* {#TamakiKono06} [[Dai Tamaki]], [[Akira Kono]], chapter 2.6 of _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([[GeneralizedCohomology.pdf:file]], [ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

* [[Jacob Lurie]], _[[A Survey of Elliptic Cohomology - cohomology theories]]_

See also

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

[[!redirects multiplicative cohomology theories]]

[[!redirects multiplicative cohomology]]


multiplicative conjunction

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

# Contents
* table of contents
{: toc}

## Idea

In [[linear logic]]/[[linear type theory]], one of the two linear versions of [[logical conjunction]] is called the _multiplicative conjunction_ and usually denoted "$\otimes$".  The analogous connective in [[relevance logic]] is sometimes called the _intensional conjunction_ or the _fusion_ or _cotenability_, and sometimes denoted "$\circ$".

The [[categorical semantics]] of the multiplicative conjunction is as the [[tensor product]] with respect to a ([[symmetric monoidal category|symmetric]]) [[monoidal category]] structure on the collection of [[types]].

There are various [[fragments]] of [[linear logic]] that contain the multiplicative conjunction.  For instance, _[[multiplicative linear logic]]_ (MLL) contains $\otimes$ along with its unit $\mathbf{1}$ and also the [[multiplicative disjunction]] $\parr$ and its unit $\bot$ and the linear [[negation]] $(-)^\bot$.  While _multiplicative intuitionistic linear logic_ (MILL)  contains $\otimes$ and $\mathbf{1}$ along with the linear implication $\multimap$.

## Related concepts

* [[exponential conjunction]]

* [[linear implication]]

[[!include logic symbols -- table]]

## References

Eg.

* {#MihályiNovitzká13} Daniel Mihályi, Valerie Novitzká, p. 3, 8 of: *What about Linear Logic in Computer Science?*, Acta Polytechnica Hungarica **10** 4 (2013) 147-160 [[pdf](http://acta.uni-obuda.hu/Mihalyi_Novitzka_42.pdf), [[MihalyiNovitzka-LinearLogic.pdf:file]]]

For more see the references at *[[linear logic]]*.

[[!redirects multiplicative conjunction]]
[[!redirects multiplicative conjunctions]]
[[!redirects intensional conjunction]]
[[!redirects intensional conjunctions]]
[[!redirects fusion conjunction]]
[[!redirects fusion conjunctions]]
[[!redirects cotenability conjunction]]
[[!redirects cotenability conjunctions]]
[[!redirects cotenability]]

[[!redirects linear conjunction]]
[[!redirects linear conjunctions]]


multiplicative disjunction

# Multiplicative disjunction

* table of contents
{: toc}

## Idea

In [[linear logic]]/[[linear type theory]], and also sometimes in other [[substructural logics]] such as [[relevance logic]], the **multiplicative disjunction** (also called **negative disjunction**, **intensional disjunction**, or **fission**) is a [[logical connective]] that is [[dual]] to the [[multiplicative conjunction]].  In linear logic it is often written following Girard as $A\invamp B$, but a multitude of other symbols have also been used for it, including $\oplus,\bullet,\odot,\Box,\sharp,*,\parallel$.

It is probably the rarest of the binary connectives of linear logic to appear semantically.  The additive conjunction and additive disjunction are categorically [[products]] and [[coproducts]], while the [[multiplicative conjunction]] is the [[tensor product]] of a [[monoidal category]] and the [[linear implication]] is the [[internal-hom]] of a [[closed category]]; but the multiplicative disjunction only appears nontrivially (i.e. without coinciding with one of the others) in a [[star-autonomous category]] (or more generally a [[linearly distributive category]]) that is not [[compact closed category|compact]].  Syntactically, this means that it mainly appears when the multiplicative connectives are "classical", i.e. the multiplicative [[negation]] is involutive.

It is probably also the hardest to understand intuitively of any of the linear logic connectives.  We include several intuitive explanations below.

## Definition

In terms of the [[multiplicative conjunction]] $\otimes$ and an involutive multiplicative [[negation]] $\neg$, the multiplicative disjunction can be defined as

$$ A \invamp B \coloneqq \neg (\neg A \otimes \neg B) $$

Categorically, this means that it is obtained from the tensor product in a [[star-autonomous category]] by applying the self-duality.

When classical linear logic is expressed with multiple-conclusion [[sequent calculus]], its left and right rules are

$$ \frac{\Gamma \vdash \Delta,A,B}{\Gamma \vdash \Delta, A\invamp B}  \qquad
\frac{\Gamma,A \vdash \Delta \quad \Gamma',B\vdash \Delta'}{\Gamma,\Gamma',A\invamp B \vdash \Delta,\Delta'}$$

## Explanations

Here we collect some possible ways to understand $\invamp$ intuitively.

### In terms of resources

When linear logic is interpreted as tracking the usage of resources, we can interpret $A\invamp B$ as saying that only one of $A$ or $B$ will happen, but we don't know which, so we have to allocate enough resources to deal with each of them separately.  For example, a military general who knows that he will be attacked from either the east or the north must divide his forces between both fronts, even if the attack will only come from one direction; thus what he has is "attack from east $\invamp$ attack from north".  (If he had spies out that would be able to inform him in advance where the attack was coming from so that he could concentrate his forces there, then he would instead have an additive disjunction "attack from east $\oplus$ attack from north".)

### As a delayed or nondeterministic choice

Relatedly, we can interpret $A\invamp B$ as a "delayed choice" $A$ and $B$.  That is, if I give you $A\invamp B$, then it might appear at first that what I gave you is $A$, but as you proceed to make use of $A$ I might change my mind and decide that really I wanted to give you $B$ instead.  This can also be regarded as a "nondeterministic" choice, so that your responses to being given $A$ or $B$ have to proceed in parallel.

In particular, note that in classical linear logic it is the multiplicative disjunction, not the additive one, that is "classical" in the sense that it satisfies [[excluded middle]]:

$$ \vdash A \invamp \neg A$$

We don't have $A\oplus \neg A$ because, as in [[constructive mathematics]], we can't decide which of $A$ or $\neg A$ might hold.  But we can say $A \invamp \neg A$ because that allows us to delay making a choice until we have enough information.  This is similar to how constructively we can say $\neg\neg(A\vee \neg A)$ by a "continuation-passing" argument (see [[excluded middle]] for details).

### In game semantics

Imagine we play two parallel games of chess, where person A is playing white in one game and black in the other.  If both players play the "copycat strategy", i.e. wait for the other one to open in one game and "copy" their moves to the other game, then we have a deadlock.  In order for two simultaneous games to proceed, one player has to play

$$ Chess \otimes \neg Chess$$

and the other

$$ Chess \invamp \neg Chess$$

Here $Chess$ is the game from the perspective of white, so that the negation $\neg Chess$ is the perspective of black, and $\otimes$ denotes the [[multiplicative conjunction]].  Note that both players are playing both games --- although $\invamp$ is a disjunction, it has certain "conjunctive properties".  But the person playing $ Chess \invamp \neg Chess$  is in the stronger position, because they can play copycat in this situation, i.e. they can wait for the other to move, whereas the person playing $Chess \otimes \neg Chess$ has to play as soon as it's their turn in one of the games.

Another, stronger interpretation is that in $A\otimes B$ we can not use any information from A to play B, and vice versa, so A and B are played by two non-communicating threads.  So in a sense, linear logic is a type system for concurrent processes that allows to avoid dead-locks.

### Disentangling additivity from disjunctive syllogism

The basic rules for proving and using a disjunction $A\vee B$ in [[intuitionistic logic]] are *additivity* and *proof by cases*.

* If we can prove $A$, we can prove $A\vee B$.  Dually, if we can prove $B$, we can prove $A\vee B$.
* If we have $A\vee B$, then we can split any proof into two cases, assuming $A$ in one and $B$ in the other.

These are also a correct set of rules for disjunction in [[classical logic]], but an equivalent pair of rules is based instead on the [[disjunctive syllogism]]:

* If assuming $\neg A$ we can prove $B$, then we can prove $A\vee B$.  Dually, if assuming $\neg B$ we can prove $A$, we can prove $A\vee B$.
* If we have $A\vee B$ and also $\neg A$, then we can conclude $B$.  Dually, if we have $A\vee B$ and also $\neg B$, then we can conclude $A$.

In linear logic, these two sets of rules are no longer equivalent: additivity and proof by cases characterize the [[additive disjunction]], while the disjunctive syllogism rules characterize the multiplicative disjunction.

In particular, note that the traditional argument for [[ex contradictione quodlibet]] depends fundamentally on having a single disjunction that obeys both additivity and the disjunctive syllogism.  (More simply, linear logic also distinguishes between the additive [[falsity]] --- the unit for the additive disjunction --- which satisfies [[ex falso quodlibet]], and the multiplicative falsity --- the unit for the multiplicative disjunction --- which doesn't.)  Thus, one might expect linear logic to be a [[paraconsistent logic]], and indeed it is (in an appropriate sense).

## Related concepts

[[!include logic symbols -- table]]

[[!redirects multiplicative disjunction]]
[[!redirects multiplicative disjunctions]]
[[!redirects intensional disjunction]]
[[!redirects intensional disjunctions]]
[[!redirects negative disjunction]]
[[!redirects negative disjunctions]]
[[!redirects fission]]


multiplicative group of integers modulo n

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

#Contents#
* table of contents
{:toc}

## Definition

For $n \in \mathbb{N}$, the [[group of units]] of the [[ring]] of [[integers modulo n]]

$$
  \left(
    \mathbb{Z}/n
  \right)^\times
$$

is typically called the _multiplicative group of integers modulo $n$_.

This consists of all those elements $k \in \mathbb{Z}/n$ which are represented by [[coprime integers]] to $n$. It is typically denoted $\mathbb{Z}/n ^\ast$.

## Euler totient function

The cardinality ${|(\mathbb{Z}/n)^\times|}$ is often denoted $\phi(n)$ (after [[Leonhard Euler]]), and $\phi$ is known as the *Euler totient function*.

The function $\phi$ is *multiplicative* in the standard number theory sense: $\phi(m n) = \phi(m)\phi(n)$ is $m$ and $n$ are coprime. This is a corollary of the [[Chinese remainder theorem]], which asserts that the canonical ring map

$$\mathbb{Z}/m n \to \mathbb{Z}/m \times \mathbb{Z}/n$$

is an isomorphism if $m, n$ are coprime. Thus, if $n = p_1^{r_1} \ldots p_k^{r_k}$ is the prime factorization of $n$, we have

$$\phi(n) = \phi(p_1^{r_1}) \ldots \phi(p_k^{r_k}).$$

Furthermore, as $\mathbb{Z}/p^r$ is a [[local ring]] with maximal ideal $p(\mathbb{Z}/p^r)$, the cardinality of the group of units is

$$\phi(p^r) = p^r - p^{r-1} = p^{r-1}(p-1).$$

## Related concepts

* [[integers modulo n]]

* [[cyclic group]]

* [[Q/Z]]

## External links

* Wikipedia, _[Multiplicative group of integers modulo n](https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n)_

* Wikipedia, *[Euler's totient function](https://en.wikipedia.org/wiki/Euler%27s_totient_function)*

[[!redirects multiplicative groups of integers modulo n]]

[[!redirects modulo]]

[[!redirects Euler totient function]]
[[!redirects Euler's totient function]]


multiplicative group scheme

## Reminder

Let $G$ be a commutative $k$-group functor (in cases of interest this is a finite flat commutative [[group scheme]]). Then the *Cartier dual* $D(G)$ of $G$ is defined by

$$D(G)(R):=Gr_R(G\otimes_k R,\mu_R)$$

where $\mu_k$ denotes the group scheme assigning to a ring its multiplicative group $R^\times$ consisting of the invertible elements of $R$.

This definition deserves the name [[duality]] since we have

$$hom(G,D(H))=hom(H,D(G))=hom(G\times H,\mu_k)$$

## Discussion

The notion of a *multipliciative group scheme* generalizes

+-- {: .num_defn}
###### Definition and Remark
A group scheme is called *multiplicative group scheme* if the following equivalent conditions are satisfied:

1. $G\otimes_k k_s$ is [[diagonalizable group scheme|diagonalizable]].

1. $G\otimes_k K$ is diagonalizable for a field $K\in M_k$.

1. $G$ is the Cartier dual of an étale $k$-group.

1. $\hat D(G)$ is an étale $k$-formal group.

1. $Gr_k(G,\alpha_k)=0$

1. (If $p\neq 0)$, $V_G$ is an epimorphism

1. (If $p\neq 0)$, $V_G$ is an isomorphism
=--

+-- {: .num_remark}
###### Remark
Let $G_const$ denote a [[constant group scheme]], let $E$ be an [[étale group scheme]]. Then we have the following [[Cartier duality|cartier duals]]:

1. $D(G_const)$ is [[diagonalizable group scheme|diagonalizable]].

1. $D(E)$ is [[multiplicative group scheme|multiplicative]]
=--


multiplicative spectral sequence

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

#Contents#
* table of contents
{:toc}

## Idea

A [[spectral sequence]] is called _multiplicative_ or a _spectral ring_ if there is a bi-[[graded algebra]] structure on each page such that the differentials act as graded [[derivations]] of total degree 1.

For example the [[Serre spectral sequence|Serre]]-[[Atiyah-Hirzebruch spectral sequence]] with [[coefficients]] in a [[ring spectrum]].

## Constructions

### From spectral products on Cartan-Eilenberg systems

The following gives sufficient conditions for a [[Cartan-Eilenberg spectral sequence]] to be multiplicative. This is due to ([Douady 58](#Douady58)). The following is taken from ([Goette 15a](#Goette15)).

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

Let $(H,\eta,\partial)$, $(H',\eta',\partial')$ und $(H'',\eta'',\partial'')$ be [[Cartan-Eilenberg systems]]. A **spectral product** $\mu\colon(H',\partial')\times(H'',\partial'')\to(H,\partial)$ is a sequence of [[homomorphisms]]

$$
  \mu_r\colon H'(m,m+r)\otimes H''(n,n+r)\to H(m+n,m+n+r)
$$

such that for all $m$, $n$, $r\ge 1$, the following two [[commuting diagram|diagrams commute]]:

$$
  \array{
    H'(m, m+r) \otimes H''(n, n+r)
      &\stackrel{\mu_r}{\longrightarrow}&
    H(m+n, m+n+r)
    \\
    \downarrow^{\mathrlap{\eta' \oplus \eta''}} && \downarrow^{\mathrlap{\eta}}
    \\
    H'(m, m+1) \otimes H''(n, n+1)
     &\stackrel{\mu_1}{\longrightarrow}&
    H(m+n, m+n+1)
  }
$$

and

$$
  \array{
    H'(m, m+r) \otimes H''(n, n+r) &\stackrel{\mu_r}{\longrightarrow}& H(m+n,m+n+r)
    \\
    \downarrow^{\mathrlap{\partial' \otimes \eta'' \oplus \eta' \otimes \partial''}}
    &&
    \downarrow^{\mathrlap{\partial}}
    \\
    H'(m+r, m+r+1) \otimes H''(n,n+1)
    \\
    \oplus     &\stackrel{\mu_1 + \mu_1}{\longrightarrow}&
    H_{p+q-1}(m+n+r, m+n+r+1)
    \\
    H'(m,m+1) \otimes H''(n+r, n+r+1)
  }
  \,.
$$

=--

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

The first diagram in def. \ref{SpectralProduct} is weaker than in ([Douady 58](#Douady58)). The second may be read as a [[Leibniz rule]].

=--

Write $E$ for the [[Cartan-Eilenberg spectral sequence]] induced from the Cartan-Eilenberg system $H$.

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

A spectral product $\mu\colon(H',\partial')\times(H'',\partial'')\to(H,\partial)$ as in def. \ref{SpectralProduct} induces products

$$
  \mu^r\colon E^{\prime r}_m\otimes E^{\prime\prime r}_n\to E^r_{m+n}\;,
$$

such that

1. $\mu^1=\mu_1$

2. $d^r_{m+n}\circ\mu^r=\mu^r\circ(d^{\prime r}_m\otimes\mathrm{id})\pm\mu^r\circ(\mathrm{id}\circ d^{\prime\prime r}_n)$,

3. $\mu^{r+1}$ is induced by $\mu^r$.

=--

([Goette 15a](#Goette15), following [Douady 58, theorem II](#Douady58)).

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

Assume by [[induction]] that $\mu^r$ is induced by $\mu_1$. In particular,
$$Z^{\prime r}_m\otimes Z^{\prime\prime r}_n\stackrel{\mu_1}\to Z^r_{m+n}\;,$$
$$B^{\prime r}_m\otimes Z^{\prime\prime r}_n\stackrel{\mu_1}\to B^r_{m+n}\;,$$
$$Z^{\prime r}_m\otimes B^{\prime\prime r}_n\stackrel{\mu_1}\to B^r_{m+n}\;.$$
This is clear for $r=1$ if we put $\mu^1=\mu_1$ because $E^1_p=Z^1_p=H(p,p+1)$
and $B^1_p=0$.

Let $[a]\in Z^{\prime r}_m$, $[b]\in Z^{\prime\prime r}_n$ be represented by $a=\eta'(a_0)\in H'(m,m+1)$, $b=\eta''(b_0)\in H''(n,n+1)$ with $a_0\in H'(m,m+r)$, $b_0\in H''(n,n+r)$.
Using the first diagram and the construction of $d^r_{m+n}$, we conclude that

$$
  (d^r_{m+n}\circ\mu^r)([a]\otimes[b])=d^r_{m+n}[\mu_1(a\otimes b)]=d^r_{m+n}[\eta(\mu_r(a_0\otimes b_0))]=(\partial\circ\mu_r)(a_0\otimes b_0)
 \;.
$$

From the second diagram, we get

$$
  (\partial\circ\mu_r)(a_0\otimes b_0)=\mu_1(\partial'a_0\otimes\eta''b_0)\pm\mu_1(\eta'a_0\otimes\partial''b_0)=\mu^r(d^{\prime r}_m[a]\otimes[b])\pm\mu^r([a]\otimes d^{\prime\prime r}_n[b])
\;.
$$

This proves the Leibniz rule (2).

From the Leibniz rule and the facts that $\ker(d^r_p)=Z^{r+1}_p/B^r_p$ and $\mathrm{im}(d^r_p)=B^{r+1}_p/B^r_p$, we conclude that $\mu^r$ induces a product on $E^{r+1}_p\cong\ker(d^r_p)/\mathrm{im}(d^r_p)$, which proves (3). Because $\mu^r$ is induced by $\mu_1$, so is $\mu^{r+1}$, and we can continue the induction.

=--

## Examples

### AHSS for multiplicative cohomology
 {#AHSSForMultiplicativeCohomology}

We discuss that the multiplicative structure on the cohomology [[Serre spectral sequence|Serre]]-[[Atiyah-Hirzebruch spectral sequence]] for [[multiplicative cohomology theory|multiplicative]] [[generalized cohomology]]. This is taken from ([Goette 15b](#Goette15)).

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

For $\pi\colon X\to B$ a [[Serre fibration]] over a [[CW-complex]] $B$.  And for $(\tilde h^\bullet,\delta,\wedge)$ a [[multiplicative cohomology theory|multiplicative]] [[reduced cohomology|reduced]]  [[generalized (Eilenberg-Steenrod) cohomology]] theory, define a [[Cartan-Eilenberg system]] $(H,\eta,\partial)$ by

$$
  H(p,q)=\tilde h^\bullet(X^{q-1}/X^{p-1})
$$

(where $X^k=\pi^{-1}(B^k)$) for $p\le q$ with the obvious maps $\eta\colon H(p',q')\to H(p,q)$ for $p\le p'$, $q\le q'$.

The [[Cartan-Eilenberg spectral sequence]] of this Cartan-Eilenberg system is the [[Serre spectral sequence|Serre]]-[[Atiyah-Hirzebruch spectral sequence]].

=--

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

The spectral product $\mu\colon(H,\eta,\partial)\times(H,\eta,\partial)\to(H,\eta,\partial)$,
def. \ref{SpectralProduct}, on the Cartan-Eilenberg system of def. \ref{CESystemForMultiplicativeGeneralizedCohomology}
is that given by the following morphism

$$
  \begin{aligned}
  F_{m,n,r}
    & \colon
    (X\wedge X)^{m+n+r-1}/(X\wedge X)^{m+n-1}
    \cong
    \bigcup_{a+b=m+n+r-1}(X^a\wedge X^b) / \bigcup_{c+d=m+n-1}(X^c\wedge X^d)
    \\
    &
    \twoheadrightarrow
    \bigcup_{a+b=m+n+r-1}
     (X^a\wedge X^b)/(\bigcup_{a=0}^m(X^{a-1}\wedge X^{m+n+r-a})
    \cup\bigcup_{b=0}^n(X^{m+n+r-b}\wedge X^{b-1})
    \\
    &
    \cong \bigcup_{a=m+1}^{m+r}(X^{a-1}\wedge X^{m+n+r-a})
    /
    \bigl(X^{m+r-1}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r-1}\bigr)
    \\
    & \hookrightarrow
    X^{m+r-1}\wedge X^{n+r-1}/(X^{m+r-1}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r-1})
    \\
    & \cong (X^{m+r-1}/X^{m-1})\wedge(X^{n+r-1}/X^{n-1})
  \;.
  \end{aligned}
$$

Together with the [[diagonal]] map $\Delta$, for $r\ge 1$, we define

$$
  \begin{aligned}
    \mu_r
     & \colon
    H(m,m+r)\otimes H(n,n+r)
    \\
    & \cong\tilde h(X^{m+r-1}/X^{m-1})\otimes\tilde h(X^{n+r-1}/X^{n-1})
    \\
    &\stackrel\wedge\longrightarrow\tilde h\bigl((X^{m+r-1}/X^{m-1})\wedge(X^{n+r-1}/X^{n-1})\bigr)
    \\
    &\stackrel{F_{m,n,r}^*}\longrightarrow\tilde h\bigl((X\wedge X)^{m+n+r-1}/(X\wedge X)^{m+n-1}\bigr)
    \\
    &\stackrel{\Delta_X^*}\longrightarrow\tilde h(X^{m+n+r-1}/X^{m+n-1})=H(m+n,m+n+r)\;.
\end{aligned}

=--

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

With def. \ref{SpectralProductForAHSS}, then
for all $m$, $n$, $r\ge 1$, the following [[commuting diagram|diagram commutes]]

$$
  \array{
    H(m,m+1) \otimes H(n,n+1) &\stackrel{\mu_1}{\longrightarrow}& H(m+n, m+n+1)
    \\
    \uparrow^{\mathrlap{\eta \oplus \eta}}
    &&
    \uparrow^{\mathrlap{\eta}}
    \\
    H(m,m+r) \otimes H(n,n+r) &\stackrel{\mu_r}{\longrightarrow}& H(m+n,m+n+r)
    \\
    \downarrow^{\mathrlap{\partial \otimes \eta \oplus \eta \otimes \partial}}
    &&
    \downarrow^{\mathrlap{\partial}}
    \\
    H(m+r, m+r+1) \otimes H(n,n+1)
    \\
    \oplus &\stackrel{\mu_1 \pm \mu_1}{\longrightarrow}&
       H(m+n+r, m+n+r+1)
    \\
    H(m,m+1) \otimes H(n+r, n+r+1)
  }
  \,.
$$

Hence by prop. \ref{MultiplicativeStructureFromSpectralProduct}
the spectral product of def. \ref{SpectralProductForAHSS}
defines a mutliplicative structure on the Serre-WhiteheadAtiyah-Hirzebruch spectral sequence for multiplicative generalizted cohomology.

=--

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

The upper square commutes because the maps $F_{m,n,r}$ are [[natural transformations]].
For the lower square, we consider the boundary morphism $\delta$ of the triple

$$
  \begin{aligned}
    (&
     X^{m+r}\wedge X^{n+r-1}\cup X^{m+r-1}\wedge X^{n+r},
    \\
    &
    X^{m+r}\wedge X^{n-1}\cup X^{m+r-1}\wedge X^{n+r-1}\cup X^{m-1}\wedge X^{n+r},
    \\
    & X^{m+r}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r})
  \end{aligned}
  \;.
$$

The following diagram commutes:

$$
  \array{
    \tilde h^{-p}(X^{m+r-1}/ X^{m-1})
    \otimes
    \tilde h^{-q}(X^{n+r-1}/X^{n-1})
    &\stackrel{\wedge}{\longrightarrow}&
    \tilde h^{-p-q}((X^{m+r-1}/X^{m-1}) \wedge (X^{n+r-1}/X^{n-1})
    \\
    \downarrow^{\mathrlap{\delta \wedge id \oplus id \wedge \delta}}
    &&
    \downarrow^{\mathrlap{\delta}}
    \\
    \tilde h(1-p)(X^{m+r}/X^{m+r-1}) \otimes \tilde h^{-q}(X^{n+r-1}/X^{n-1})
    &&
    \tilde h^{1-p-q}((X^{m+r}/X^{m+r-1}) \wedge (X^{n+r-1}/X^{n-1}))
    \\
    \oplus
    &\stackrel{\wedge \oplus \wedge}{\longrightarrow}&
    \\
    \tilde h^{-p}(X^{m+r-1}/X^{m-1}) \otimes \tilde h^{1-q}(X^{n+r}/X^{n+r-1})
    &&
    \tilde h^{1-p-q}( (X^{m+r-1}/ X^{m-1}) \wedge (X^{n+r}/ X^{n+r-1}) )
  }
  \,.
$$

By extend this diagram to the right using the maps $F_{m,n,r}$ once
concludes that the lower square above also commutes.

=--

## Related concepts

* [[multiplicative cohomology theory]]

## References

* {#Douady58} [[Adrien Douady]], _La suite spectrale d'Adams : structure multiplicative_ Séminaire Henri Cartan, 11 no. 2 (1958-1959), Exp. No. 19, 13 p ([Numdam](http://www.numdam.org/item?id=SHC_1958-1959__11_2_A10_0))

* {#Gray80} Brayton Gray, _Products in the Atiyah-Hirzebruch spectral sequence and the calculation of $M SO_\ast$, Trans. Amer. Math. Soc. 260 (1980), 475-483 ([web](http://www.ams.org/journals/tran/1980-260-02/S0002-9947-1980-0574793-9/))

* {#Kochmann96} [[Stanley Kochmann]], prop. 4.2.9 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#McCleary00} [[John McCleary]], section 2.3 in _A User's Guide to Spectral Sequences_, Cambridge University Press (2000)

* {#Dugger03a} [[Daniel Dugger]], _Multiplicative structures on homotopy spectral sequences I_ ([arXiv:math/0305173](http://de.arxiv.org/abs/math/0305173))

* {#Dugger03b} [[Daniel Dugger]], _Multiplicative structures on homotopy spectral sequences II_ ([arXiv:math/0305187](http://de.arxiv.org/abs/math/0305187))

* {#Goette15} [[Sebastian Goette]], [MO comment a](http://mathoverflow.net/a/231235/381), [MO comment b](http://mathoverflow.net/a/231236/381) Feb 15, 2015

[[!redirects multiplicative spectral sequences]]


multiplicative subset

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

#Contents#
* table of contents
{:toc}

## Definition

For $R$ a [[commutative ring]], a [[subset]] $S \hookrightarrow R$ of its set of elements is called _multiplicative_ if it is closed under multiplication "$\cdot$"in $R$, hence if $s_1,s_2 \in S \hookrightarrow R$ implies that $s_1 \cdot s_2 \in S \hookrightarrow R$, or equivalently, that $S$ forms a [[subobject|sub]][[semigroup]] of $R$. Usually one also demands that the [[unit]] element is in the set $S$, or equivalently, that $S$ forms a [[submonoid]] of $R$.

Every two-sided [[filter of a ring]] in a commutative ring is a multiplicative subset.

The [[localization of a commutative ring]] at a multiplicative subset exists, denoted $S^{-1}R$ or $R[S^{-1}]$.

## See also

* [[filter of a ring]]
* [[multiplicative subset of cancellative elements]]

## References

* [[The Stacks Project]], def. 10.9.1

[[!redirects multiplicative subsets]]
[[!redirects multiplicative set]]
[[!redirects multiplicative sets]]


multiplicative system

## In a ring

A **multiplicative system** in a [[ring]] $R$ is a set $S\subset R$ which is a [[monoid|submonoid]] of the multiplicative monoid of $R$, i.e. $1\in S$ and if $a,b\in S$ then $a b\in S$.

If $R$ is [[commutative ring|commutative]], then a multiplicative system is precisely what is necessary to construct the [[localization]] $R[S^{-1}]$ as formal 'fractions' in $R$.  If $R$ is an [[integral domain]], and $S=R\setminus \{0\}$, then $R[S^{-1}]$ is its [[field]] of fractions.

If $R$ is not commutative, then one generally needs extra conditions, such as the [[Ore condition]], in order to construct a localization. Still, if the Ore condition is required, we usually state this and call it a 2-sided Ore set, a denominator set or alike.

## In a category

The term 'multiplicative system' is often sometimes used for a set of morphisms in a category admitting a [[calculus of fractions]] hence it satisfies the Ore conditions.

category: algebra


multiplicative unitary

[[!redirects pentagon relation]]
[[!redirects multiplicative unitaries]]
[[!redirects fundamental operator]]

### Idea

While the following idea is originally in operator setup and with an involution, consider the following. Let $H$ be a finite-dimensional vector space.
Consider the invertible operator $W : H\otimes H \to H\otimes H$ satisfying the pentagon identity

$$
W_{1 2} W_{1 3} W_{2 3} = W_{2 3} W_{1 2}
$$
in the space of linear endomorphisms of $H\otimes H\otimes H$. Then the formula

$$
\Delta(h) = W (h\otimes 1) W^{-1}
$$

define a coassociative coproduct on $H$. Usually we replace the structure of the coproduct with knowing $W$, which can be easier to define in infinite-dimensional analogues when the coproduct needs to take values in some hard to manage completions.

for all $h\in H$. For finite-dimensional Hopf algebras $W(g\otimes h) = g_{(1)}\otimes g_{(2)} h$ and $W^{-1}(g\otimes h) = g_{(1)}\otimes (S g_{(2)}) h$ and then we can reproduce the antipode via the formula

$$
S h = (\epsilon\otimes id)\circ W^{-1}(h\otimes - )
$$

We can also make a discussion in terms of the dual space $H^*$. Then the coproduct on $H^*$ which is dual to the product on $H$ is also obtained from $W$ by the formula

$$
\Delta_{H^*}(\psi) = W^{-1} (1\otimes\psi) W
$$

### Literature and further directions

In the setup of operator algebras, the multiplicative unitaries were introduced as so called Kac-Takesaki operator. Following some ideas on noncommutative extensions of Pontrjagin duality (in Tannaka-Krein spirit) by [[George Kac|George's Kac]] and also M. Takesaki, Lecture Notes in Mathematics. 247, Berlin: Springer; 1972. pp. 665–785. The followup work of Baaj and Skandalis introduced two more fundamental axioms, regularity and irreducibility, important in $C^*$-algebraic setup.

* Saad Baaj, Georges Skandalis, _Unitaires multiplicatifs et dualité pour les produits croisés de $C^*$-algèbre_, Annales scientifiques de l'École Normale Supérieure __26__:4 (1993) 425-488  [numdam](https://doi.org/10.24033/asens.1677); _Transformations pentagonales (Pentagonal transformations)_, Comptes Rendus de l'Académie des Sciences, I - Mathematics __327__:7 (1998) 623-628, <a: href="https://doi.org/10.1016/S0764-4442(99)80090-1">doi</a>

* Saad Baaj, Étienne Blanchard, Georges Skandalis, _Unitaires multiplicatifs en dimension finieet leurs sous-objets_, Annales de l’institut Fourier, tome  49, no4 (1999), p. 1305-1344 [numdam](http://www.numdam.org/item?id=AIF_1999__49_4_1305_0)

The monograph on [[Kac algebra]]s is

* M. Enock, J. M. Schwartz, _Kac algebras and duality of locally compact groups_,  Springer-Verlag, 1992, , x+257 pp. [gBooks](http://books.google.com/books/about/Kac_algebras_and_duality_of_locally_comp.html?id=U6e6aD1gj3oC), [MR94e:46001](https://mathscinet.ams.org/mathscinet-getitem?mr=1215933)

and a more recent view of duality between Hopf algebra approach and an approach to quantum groups via multiplicative unitaries is in the book

* Thomas Timmermann, _An invitation to quantum groups and duality: From Hopf algebras to multiplicative unitaries and beyond_, Europ. Math. Soc. 2008.

Introduction to Ch. 7 says in Timmerman's book says

> Multiplicative unitaries are fundamental to the theory of quantum groups in the setting of $C^*$-algebras and [[von Neumann algebra]]s, and to generalizations of [[Pontrjagin duality]]. Roughly, a multiplicative unitary is one single map that encodes all structure maps of a quantum group and of its generalized Pontrjagin dual simultaneously.

Woronowicz has introduced managaeble multiplicative unitaries

* [[S. L. Woronowicz]], _From multiplicative unitaries to quantum groups_, Internat. J. Math. __7__ (1996), 127–149.

It is useful to look at the survey

* Johan Kustermans, Stefaan Vaes, _The operator algebra approach to quantum groups_,
Proc Natl Acad Sci USA 97(2): 547–552 (2000) [doi](https://doi.org/10.1073/pnas.97.2.547)

* A. Van Daele, S. Van Keer, _The Yang-Baxter and pentagon equation_, Compositio Mathematica __91__:2 (1994)  201-221 [numdam](http://www.numdam.org/item/CM_1994__91_2_201_0)

The categorical background of the pentagon equation has been studied in

* [[Ross Street]], _Fusion operators and cocycloids in
monoidal categories_, Applied Categorical Structures 6: 177–191 (1998) [doi](https://doi.org/10.1023/A:1008655911796)

A finite dimensional version is reformulated in section 3 of

* [[Shahn Majid]], _Quantum random walks and time reversal_ [doi](https://doi.org/10.1142/S0217751X93001818)

and reprinted in Majid's, _Foundations of quantum group theory_, 1995, as Theorem 1.7.4. Majid has stated in this finite-dimensional case, ideas about [[quantum group Fourier transform]] (see there and Majid's book). This has been used in

* [[Laurent Freidel]], [[Shahn Majid]], _Noncommutative harmonic analysis, sampling theory and the Duflo map in $2+1$ quantum gravity_, Classical Quantum Gravity __25__ (2008) no. 4, 045006 [MR2009f:83058](https://mathscinet.ams.org/mathscinet-getitem?mr=2388191), [doi](https://doi.org/10.1088/0264-9381/25/4/045006)

More categorical treatment and relation to [[Hopf-Galois extension]]s is in

* A. A. Davydov, _Pentagon equation and matrix bialgebras_, Commun. Alg. __29__(6), 2627–2650 (2001) [doi](https://doi.org/10.1081/agb-100002412)

In the language of finite-dimensional Heisenberg doubles see also the treatment of fundamental operator in

* [[Gigel Militaru]], _Heisenberg double, pentagon equation, structure and classification of finite-dimensional Hopf algebras_, J. London Math. Soc. (2) 69 (2004) 44--64 [doi](https://doi.org/10.1112/S0024610703004897)

Kashaev has explained the pentagon relations for quantum [[dilogarithm]] as coming from the pentagon for the canonical element in the double.

* R.M. Kashaev, _Heisenberg double and the pentagon relation_, St. Petersburg Math. J. 8 (1997) 585– 592 [q-alg/9503005](http://arxiv.org/abs/q-alg/9503005).

The multiplicative unitary representing quantum dilogarithm has been studied analytically, in the disguise of a quantum exponential, in relation to the construction of "noncompact quantum ax+b group" in

* [[S. L. Woronowicz]], _Quantum exponential function_, Reviews in Mathematical Physics __12__:06,  873-920 (2000) [doi](https://doi.org/10.1142/S0129055X00000344)

The formalism is also in

* [[S. L. Woronowicz]], From multiplicative unitaries to quantum groups, Internat. J. Math. 7(1) (1996) 127–149 [doi](https://doi.org//10.1142/S0129167X96000086), MR 1369908
* P. M. Sołtan, S. L. Woronowicz, _From multiplicative unitaries to quantum groups. II_, J. Funct. Anal
252(1), 42–67 (2007) [doi](https://doi.org/10.1016/j.jfa.2007.07.006)
* [[Ralf Meyer]], Sutano Roy, [[S. L. Woronowicz]], _Quantum group-twisted tensor products of $C^*$-algebras_, Int. J. __25__:02, 1450019 (2014) [doi](https://doi.org/10.1142/S0129167X14500190); _Semidirect products of $C^*$-quantum groups: multiplicative unitaries approach_, Commun. Math. Phys. __351__, 249–282 (2017) [doi](https://doi.org/10.1007/s00220-016-2727-3)
* [[Ralf Meyer]], Sutanu Roy, _Braided multiplicative unitaries as regular objects_, Adv. Stud. Pure Math., Operator Algebras and Mathematical Physics, M. Izumi, Y. Kawahigashi, M. Kotani, H. Matui, N. Ozawa, eds. (Tokyo: Mathematical Society of Japan, 2019) 153 - 178 [doi](https://doi.org/10.2969/aspm/08010153)

The role of quantum torus is here quite clear; later treatments of more general quantum [[dilogarithm]]s influenced by Kontsevich-Soibelman work on [[wall crossing]] and related Goncharov's cluster varieties quantization also witness the appearance of the similar quantum torus.


multiplicatively cancellable rig

[[!redirects multiplicatively cancellable semi-ring]]
[[!redirects multiplicatively cancellable semi-ring]]

## Definition

In [[classical mathematics]], a multiplicatively cancellable [[rig]] is a rig $R$ such that each $(r \neq 0) \in R$ is multiplicatively cancellable.

An element $r \in R$ where $R$ is a rig is left multiplicatively cancellable iff for every $s,t \in R$ we have that $(r.s = r.t) \Rightarrow (s = t)$ and it is right multiplicatively cancellable iff for every $s,t \in R$ we have that $(s.r = t.r) \Rightarrow (s = t)$. An element $r \in R$ is multiplicatively cancellable iff it is left multiplicatively cancellable and right multiplicatively cancellable.

If $R$ is a commutative rig, an element is left multiplicatively cancellable iff it is right multiplicatively cancellable.

### In constructive mathematics

[[regular element of a commutative ring|Like in the case for integral domains]] and [[fields]], the definition above bifurcates into multiple definitions in [[constructive mathematics]].

definitions to be ported over

\begin{definition}
A residue multiplicatively cancellable [[rig]] is a rig $R$ such that for all $r \in R$, if $r$ is not multiplicatively cancellable, then $r = 0$.
\end{definition}

\begin{definition}
A Heyting multiplicatively cancellable [[rig]] is a rig $R$ with a [[tight apartness relation]] $\#$ such that each $(r \# 0) \in R$ is multiplicatively cancellable.
\end{definition}

\begin{definition}
A discrete multiplicatively cancellable [[rig]] is a rig $R$ such that each $r \in R$ is zero [[xor]] multiplicatively cancellable.
\end{definition}


multiplicity free fusion category

## Idea

A multiplicity free fusion category is a [[fusion category]] in which all of the fusion coefficients are $0$ or $1$.

That is, a fusion category $\mathcal{C}$ is multiplicity free if for all simple objects $A,B$ we have that $A\otimes B$ is a direct sum of simple objects, each isomorphism class appearing with multiplicity at most $1$.

Equivalently, this means that the spaces $\text{Hom}(A\otimes B,C)$ and $\text{Hom}(C,A\otimes B)$ are at most one dimensional for all triples of simple objects $A,B,C$.

## Related concepts

* [[6j symbols]]


multiplier algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Noncommutative geometry
+-- {: .hide}
[[!include noncommutative geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Given a nonunital [[C*-algebra]] $A$, the _multiplier algebra_ $M(A)$ is the maximal unital extension of $A$ in which $A$ is an [[essential ideal]] (an ideal having nonzero intersection with any other proper ideal). In the [[commutative ring|commutative]] case, every unital commutative extension in which $A$ is an essential ideal corresponds to a [[compactification]] of the [[spectrum]] $Spec(A)$; $M(A)$ is in that case the $C^*$-algebra of functions on the [[Stone–Čech compactification]] $\beta(Spec(A))$.

## Definition

The __multiplier algebra__ $M(A)$ of a (not necessarily unital) $C^*$-algebra $A$ is the $C^*$-algebra satisfying the following [[universal property]]:
for any $C^*$-algebra $B$ containing $A$ as an ideal, there exists a unique $*$-homomorphism $\phi:B\to M(A)$ such that $\phi$ extends the identity homomorphism on $A$ and
$\phi(A^\perp)=\{0\}$.

The multiplier algebra $M(A)$ can be realized as the set of 2-sided multipliers in the [[enveloping von Neumann algebra]] of $A$. If $A\subset B$, $b\in B$ is a multiplier (operator) for $A$ if $\forall a\in A$, $b a\in A$ and $a b\in A$.

Of course, if $A$ is already unital, then $M(A) = A$.

## Related concepts

There are also larger [[local multiplier algebra]]s $M_{loc}(A)$.

## Literature

* [[eom]]: Gert K. Pedersen, [multiplier algebra](http://www.encyclopediaofmath.org/index.php/Multipliers-of-C*-algebras)

* wikipedia: [multiplier algebra](http://en.wikipedia.org/wiki/Multiplier_algebra)

* Ch.A. Akemann, G.K. Pedersen, J. Tomiyama, _Multipliers of $C^*$-algebras_, J. Funct. Anal. __13__ (1973) 277–301 [MR470685](http://www.ams.org/mathscinet-getitem?mr=470685)

* Paul Skoufranis, _An introduction to multiplier algebras_, ([pdf](http://pskoufra.info.yorku.ca/files/2016/07/Multiplier-Algebras.pdf))

* Corran Webster, _On unbounded operators affiliated with $C^*$-algebras_, J. Operator Theory __51__ (2004) 237-244 [dvi](http://corranwebster.com/Research/unbounded.dvi) [pdf](http://corranwebster.com/Research/unbounded.pdf) [ps](http://corranwebster.com/Research/unbounded.ps)

> We show that the multipliers of Pedersen's ideal of a $C^*$-algebra $A$ correspond to the densely defined operators on $A$ which are affiliated with $A$ in the sense defined by Woronowicz, and whose domains contain Pedersen's ideal. We also extend the theory of q-continuity developed by Akemann to unbounded operators and show that these operators correspond to self-adjoint operators affiliated with $A$.

* S. L. Woronowicz, _$C^*$-algebras generated by unbounded elements_, [pdf](http://www.fuw.edu.pl/~slworono/PDF-y/GENER.pdf); _Unbounded operators in the context of $C^*$-algebras_, slides [pdf](http://www.siue.edu/MATH/BA2009/ABSTRACTS/lecture-Woronowicz.pdf)

* [[Bruce Blackadar]], _[[K-Theory for Operator Algebras]]_, Cambridge University Press 1998.

There is also a notion of multiplier Hopf algebra and even multiplier bimonoid and Hopf monoid:

* Gabriella Böhm, Stephen Lack, _Multiplier Hopf monoids_, [arxiv/1511.03806](http://arxiv.org/abs/1511.03806); _A category of multiplier bimonoids_, [arxiv/1509.07171](http://arxiv.org/abs/1509.07171)

category: operator algebras, noncommutative geometry, analysis

[[!redirects multiplier algebra]]
[[!redirects multiplier algebras]]


multireflexive subcategory

## Idea

[…]

## Definition

A __multireflexive subcategory__ is a [[subcategory]] $\iota\colon A\to B$
such that the functor $\iota$ has a [[left multiadjoint]].

## Examples

See at [[Diers spectrum]].

## References

* [[Yves Diers]], _Some spectra relative to functors_.  Journal of Pure and Applied Algebra 22:1 (1981), 57–74.  [doi][1]

[1]: https://doi.org/10.1016/0022-4049(81)90082-7


multirepresentable functor

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#### Category theory
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#### Yoneda lemma
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#Contents#
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## Definition

A [[functor]] is **multirepresentable** if it is a [[coproduct]] of [[representable functors]].

## Properties

A functor is multirepresentable if and only if each [[connected component]] of its [[category of elements]] has an [[initial object]]: this is Proposition 2.1.2 of [Diers 1977](#Diers77).

## Related pages

* [[representable functor]]

* [[multiadjoint]]

## References

* {#Diers77} [[Yves Diers]], _Catégories localisables_, PhD thesis. Paris 6 et Centre universitaire de Valenciennes et du Hainaut Cambrésis (1977) [[[Categories localisables.pdf:file]]]

Multirepresentable presheaves are called **familially representable** in:

* [[Aurelio Carboni]], [[Peter Johnstone]], _Connected limits, familial representability and Artin glueing_ , Mathematical Structures in Computer Science **5** (1995) pp.441-459. ([pdf](http://journals.cambridge.org/article_S0960129500001183))
{#CJ}

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MultiSet

$MultiSet$ is the [[category]] of [[multisets]].

## References

* [Mathematics of Multisets](http://obelix.ee.duth.gr/~apostolo/Articles/MathMSet.pdf), Apostolos Syropoulos

* [Categorical Models of Multisets](http://obelix.ee.duth.gr/~apostolo/Articles/mset.pdf), Apostolos Syropoulos

category: category

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