semidirect product group

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## Definitions

If a [[group]] $G$ [[action|acts]] on a group $\Gamma$ (on the left, say) by group [[automorphism]]

$$
  \rho : G \to Aut(\Gamma)
  \,,
$$

then there is a __semidirect product__ group $\Gamma \rtimes \, G$ whose underlying set is the [[Cartesian product]] $\Gamma \times  G$ but whose multiplication is twisted by $\rho$:

$$
  (\delta,h)(\gamma,g)= (\delta \rho(h)(\gamma) , h g)
$$

for $\delta, \gamma \in \Gamma,\; h,g \in G$.  For the rest of this page, $^h \gamma \coloneqq \rho(h)(\gamma)$ denotes the result of acting with $h$ on the left on $\gamma$.

If the twist is trivial, then this reduces to just the [[direct product group]] construction, whence the name.

There is a [[projection]] morphism $p:\Gamma \rtimes \, G \to G$ ,
$(\gamma, g) \to g$. A [[section]] $s$ of this can be identified with a
[[derivation]] $d$, i.e. $d$  satisfies $d(h g) = (d h) \,^h (d g)$.

### Interior semidirect products

Let $H$ be any group.  A decomposition of $H$ as an __internal__ semidirect product consists of a [[normal subgroup]] $\Gamma$ and a [[subgroup]] $G$, such that every element of $H$ can be written uniquely in the form $\gamma g$, for $\gamma \in \Gamma$ and $g \in G$.

The internal and external concepts are equivalent.  In particular, any (external) semidirect product $\Gamma \rtimes G$ is an internal semidirect product of the [[images]] of $\Gamma$ and $G$ in it.

### Right semidirect products

The definitions above are not symmetric in left and right; since the first definition begins with a left action, we may call it a *left* semidirect product.  Then a __right__ semidirect product is given by an action on the right, or internally by the requirement that every element can be written in the form $g \gamma$.

However, right and left semidirect products are equivalent.  Essentially, this is because any left action $(h,g) \mapsto {}^h{g}$ defines a right action $(g,h) \mapsto g^h \coloneqq {}^{h^{-1}}g$ and vice versa.

### As a left adjoint

Consider the [[category]] $GrpActions$ with

* [[objects]] $(G,\Gamma,\rho)$ where $G$ and $\Gamma$ are [[groups]] and $\rho: G \to \Aut(\Gamma)$ is a [[group homomorphisms]], and whose

* [[morphisms]] $(G,\Gamma,\rho) \to (G',\Gamma',\rho')$ are $G$-[[equivariant]] pairs of morphisms $f: G \to G'$ and $h: \Gamma \to \Gamma'$, i.e. such that $h(\rho(g)(\gamma)) = \rho'(f(g))(h(\gamma))$ for all $g \in G$ and $\gamma \in \Gamma$.

There is a [[forgetful functor]] $U: Arr(Grp) \to GrpActions$ from the [[arrow category]] of [[Grp]], sending a [[group homomorphism]] $f\colon G \to \Gamma$ to $(G,\Gamma,\rho)$ where $\rho: G \to Aut(\Gamma)$ is given by conjugation, i.e. $\rho(g)(\gamma) \coloneqq f(g) \gamma f(g)^{-1}$.

Now, its [[left adjoint functor]] $GrpActions \to Arr(Grp)$ maps $(G,\Gamma,\rho)$ to the [[inclusion]] $G \hookrightarrow \Gamma \rtimes_\rho G$.

### As a Grothendieck construction

Writing $\mathbb{B} G$ for the [[category]] with a single [[object]] $\ast$ and the [[group]] $G$ as its [[hom set]] (i.e. the [[delooping]] [[groupoid]] of $G$), define a [[functor]] $F \colon \mathbb{B}G \to$ [[Cat]] to send that single object to the delooping groupoid of $\Gamma$, i.e. $* \mapsto \mathbb{B}\Gamma$ and to send the morphisms $G \to Aut(\Gamma)$ according to the given [[action]] of $G$ on $\Gamma$.

Then the delooping of the semidirect product group $\Gamma \rtimes G$ arises as the [[Grothendieck construction]] of this functor:

$$
  \mathbb{B}( \Gamma \rtimes G)
  \;\simeq\;
  \int_{\mathbb{B}G}F
$$

### Semidirect products of groupoids

It is useful to generalise this to the case $\Gamma$ is a [[groupoid]].
This occurs if for example $\Gamma = \pi_1 X$ where $X$ is a (left)
$G$-space.

So if $X=Ob(\Gamma)$, then $\Gamma \rtimes \, G$ has object set $X$ and
a morphism $y \to x$ is a pair $(\gamma,g)$ such that $\gamma: y \to
g x$ in $\Gamma$. The composition law is then given again by

$$(\delta,h)(\gamma,g)= (\delta \, ^h \gamma, h g) $$

if  $(\delta, h): z \to y$, so that $\delta: z \to h y$ in $\Gamma$.

If $\Gamma$ is a discrete groupoid, and so identified with $X$, then
we get $X \rtimes \, G$ which is the [[action groupoid]] of the action. In
this case the projection $p: X \rtimes \, G \to G$ is a covering
morphism of groupoids, i.e.  any $g \in G$ has a unique lifting with
given initial point. Note that if $Y \to X $ is a covering map of
spaces, then the induced morphism of fundamental groupoids is a
covering morphism of groupoids. If $q: H \to \pi_1 X$ is a covering
morphism of groupoids, and $X$ admits a universal covering map, then
there is a topology on $Y=Ob(H)$ such that $H \cong \pi_1 Y$. In
this way, the category of covering maps of $X$ is equivalent to the
category of covering morphisms of $\pi_1 X$.

The utility of the more general construction is that there is notion
of orbit groupoid $\Gamma //G$ (identify any $\gamma$ and $^g
\gamma$) and it is a theorem that the orbit groupoid is isomorphic to
the quotient groupoid

$$ (\Gamma \rtimes \, G)/N$$

where $N$ is the [[normal closure]] in $\Gamma \rtimes \, G$ of all
elements $(1_x,g)$. Details are in the book reference below (but the
conventions are not quite the same).

## Properties

### As split group extensions

Semidirect product groups $A \rtimes_\rho G$ are precisely the split [[group extensions]] of $G$ by $A$. See at _[group extension -- split extensions and semidirect product groups](group+extension#SplitExtensionsAndSemidirectProductGroups)_.

## Examples

### The automorphisms on the circle group

For $U(1) = \mathbb{R}/\mathbb{Z}$ the [[circle group]], the [[automorphism group]] is

$$
  Aut(U(1)) \simeq \mathbb{Z}_2
  \,,
$$

where the nontrivial element in $\mathbb{Z}_2$ acts on $\mathbb{R}$ by multiplication with $-1$. Write $\rho_{aut} : U(1) \times \mathbb{Z}_2 \to U(1)$ for the automorphism [[action]]. The corresponding semidirect product group is the [[group extension]]

$$
  U(1) \stackrel{}{\hookrightarrow} U(1) \rtimes_{\rho_{aut}} \mathbb{Z}_2
  \to \mathbb{Z}_2
$$

where the group operation is given by

$$
  (c_1 \; mod \; \mathbb{Z}, \sigma_1)
  \cdot
  (c_2\; mod \; \mathbb{Z}, \sigma_2)
  =
  (c_1 + \sigma_1(c_2) \; mod \; \mathbb{Z}, \sigma_1 + \sigma_2)
  \,.
$$

## Related concepts

* [[direct product of groups]]

* [[free product of groups]]

* [[central product of groups]]

* [[semidirect product Lie algebra]]

* [[equivariant group]]

* [[crossed homomorphism]]

## References

A general survey is in

* Wikipedia, _[Semidirect product](http://en.wikipedia.org/wiki/Semi-direct_product)_

Lecture notes include

* Patrick Morandi, _Semidirect products_ ([pdf](http://sierra.nmsu.edu/morandi/notes/semidirect.pdf))

Relevant textbooks include

* [[R. Brown]], _Topology and groupoids_, Booksurge 2006.

* [[P. J. Higgins]] and J. Taylor,  _The Fundamental Groupoid and Homotopy Crossed Complex of an Orbit Space_, in K.H. Kamps et al., ed., Category Theory: Proceedings Gummersbach 1981, Springer LNM
  962  (1982) 115--122.

[[!redirects semidirect product groups]]


semidirect product Lie algebra

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## Idea

Consider a [[Lie algebra action]] of a [[Lie algebra]] $\mathfrak{g}$ on Lie algebra $\mathfrak{a}$, hence a Lie algebra [[homomorphism]]

$$
  \rho
    \;\colon\;
  \mathfrak{g}
    \longrightarrow
   \mathfrak{der}(\mathfrak{a})
$$

to the [[derivation Lie algebra]] of $\mathfrak{a}$.

There is a [[Lie algebra extension]] of $\mathfrak{g}$ by $\mathfrak{a}$ whose [[underlying]] [[vector space]] is the [[direct sum]] ([[product]] in [[Vect]])

$$
  \hat \mathfrak{g}
  \;=\;
  \mathfrak{g} \oplus \mathfrak{a}
$$

and whose [[Lie bracket]] is given by the formula

$$
  \big[(x_1,t_1), (x_2,t_2)\big]
  \;=\;
  \big(
    [x_1,x_2],
    \;
    [t_1,t_2] + \rho(x_1)(t_2) - \rho(x_2)(t_1)
  \big)
  \,.
$$

This is the _semidirect product_ or *semidirect sum* of $\mathfrak{g}$ with $\mathfrak{a}$.

## Related concepts

* [[Lie algebra extension]]

* [[semidirect product group]]

## References

* Tadeusz Ostrowski, *A note on semidirect sum of Lie algebras*, Discussiones Mathematicae - General Algebra and Applications (2013) **33** 2 (2013) 233-247 [[eudml:270203](https://eudml.org/doc/270203)]

* Mohammad Reza Alemi, Farshid Saeedi, *Derivation algebra of semi-direct sum of Lie algebras*, Asian-European Journal of Mathematics **15** 04 2250063 (2022) [[doi:10.1142/S1793557122500632](https://doi.org/10.1142/S1793557122500632)]

See also:

* Wikipedia, *[Lie algebra extension -- Construction](https://en.wikipedia.org/wiki/Lie_algebra_extension#Construction)*

[[!redirects semidirect product Lie algebras]]
[[!redirects semi-direct product Lie algebra]]
[[!redirects semi-direct product Lie algebras]]

[[!redirects semidirect sum Lie algebra]]
[[!redirects semidirect sum Lie algebras]]

[[!redirects semi-direct sum Lie algebra]]
[[!redirects semi-direct sum Lie algebras]]


semifree dga

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## Definition

A [[differential graded algebra]] is __semifree__ (or semi-free) if the underlying [[graded algebra]] is free: if after forgetting the differential, it is isomorphic as a graded algebra to a (polynomial) [[tensor algebra]] of some ([[super vector space|super]])[[graded vector space]].

A [[differential graded-commutative algebra]] is __semifree__ (or semi-free) if the underlying graded-commutative algebra is free: if after forgetting the differential, it is isomorphic as a graded-commutative algebra to a [[Grassmann algebra]] of some [[graded vector space]] .

## Roiter's theorem

Roiter's theorem

* A. V. Roiter, _Matrix problems and representations of BOCS's_; in Lec. Notes. Math. 831, 288--324 (1980)

says: semi-free differential graded algebras are in bijective correspondence with [[coring]]s with a [[grouplike element]]:

to an $A$-coring $(C,\Delta, A)$ with a grouplike element $g$ associate its [[Amitsur complex]] with underlying graded module $T_A(\Omega^1 A)=\oplus_{n=0}^\infty (\Omega^1 A)^{\otimes_A n}$ where $\Omega^1=ker\,\epsilon$
and differential linearly extending the formulas $d a = g a - a g$ for $a\in A$ and
$$
d c = g\otimes c + (-1)^n c\otimes g +\sum_{i=1}^n (-1)^i c_1\otimes\ldots\otimes c_{i-1}\otimes\Delta(c_i)\otimes c_{i+1}\otimes\ldots\otimes c_n
$$
for $c=c_1\otimes_A\ldots\otimes_A c_n\in (ker\,\epsilon)^{\otimes_A n}$;

conversely, to a semi-free dga $\Omega^\bullet A$ one associates the $A$-coring $A g\oplus\Omega^1 A$ where $g$ isa new group-like indeterminate; this is by definition a direct sum of left $A$-modules with a right $A$-module structure given by
$$
(a g +\omega)a' := a a' g + a d a'+\omega a'.
$$
In other words, we want the commutator $[g,a']=d\omega'$.
We obtain an $A$-bimodule. The coproduct on $Ag\oplus\Omega^1 A$ is $\Delta(a g)=a g\otimes g$ and $\Delta(\omega)=
g\otimes\omega+\omega\otimes g- d\omega$.
The two operations are mutual inverses (see [lectures](http://www.newton.ac.uk/programmes/NCG/seminars/080411301.pdf) by Brzezinski or the arxiv version [math/0608170](http://arxiv.org/abs/math/0608170)).

Moreover [[connection for coring|flat connections]] for a semi-free dga are in $1$-$1$ correspondence with the comodules over the corresponding coring with a group-like element.

## Relation to Lie $\infty$-algebroids

One can identify semifree [[differential graded algebra]]s in non-negative degree with Chevalley--Eilenberg algebras of (degreewise finite dimensional) [[Lie infinity-algebroid]]s

At least when the algebra in degree $0$ is of the form $C^\infty(X)$ for some space $X$, which then is the space of objects of the [[Lie infinity-algebroid]]. But if it is a more general algebra in degree $0$ one can think of a suitably generalized $L_\infty$-algebroid, for instance with a noncommutative space of objects. This generalizes the step from [[Lie algebroid]]s to Lie--Rinehart pairs.

## Terminology

Sometimes semi-free DGAs are called _quasi-free_, but this is in collision with the terminology about formal smoothness of noncommutative algebras, i.e. quasi-free algebras in the sense of Cuntz and Quillen (and with extensions to homological smootheness of dg-algebras by Kontsevich).

## Related concepts

* [[Sullivan algebra]]

* [[L-infinity algebra]], [[super L-infinity algebra]], "[[FDA]]"

[[!redirects semifree dgas]]
[[!redirects semi-free dga]]
[[!redirects semi-free dgas]]
[[!redirects semi-free differential graded algebra]]
[[!redirects semi-free differential graded algebras]]
[[!redirects semifree differential graded algebra]]
[[!redirects semifree differential graded algebras]]

[[!redirects quasifree dgas]]
[[!redirects quasi-free dga]]
[[!redirects quasi-free dgas]]
[[!redirects quasi-free differential graded algebra]]
[[!redirects quasi-free differential graded algebras]]
[[!redirects quasifree differential graded algebra]]
[[!redirects quasifree differential graded algebras]]

[[!redirects semifree dg-algebra]]
[[!redirects semifree dg-algebras]]

[[!redirects semifree dgc-algebra]]
[[!redirects semifree dgc-algebras]]

[[!redirects semi-free dg-algebra]]
[[!redirects semi-free dg-algebras]]

[[!redirects semi-free dgc-algebra]]
[[!redirects semi-free dgc-algebras]]

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[[!redirects semifree differential graded-commutative algebras]]


semifunctor

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# Semifunctors
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## Idea

A _semifunctor_ is a [[homomorphism]] between [[semicategories]], like a [[functor]] is a homomorphims between [[categories]].

## Definition

A **semifunctor** $F$ from a [[semicategory]] $C$ to a semicategory $D$ is a map sending each [[object]] $x \in C$ to an object $F(x) \in D$ and each [[morphism]] $f : x \to y$ in $C$ to morphism $F(f) : F(x) \to F(y)$ in $D$, such that

* $F$ preserves [[composition]]: $F(g\circ f) = F(g)\circ F(f)$ whenever the left-hand side is well-defined.

If $C$ is a [[category]], then $F$ need *not* preserve its [[identity morphisms]], but the composition axiom does require that it send them to [[idempotents]] in $D$.

## Relation to Profunctors

In [[Rel]], the bicategory of relations, adjoint pairs of relations are equivalent to functions (see [[principle of unique choice]]). Even with the axiom of choice, the analogous theorem for [[Prof]], the bicategory of profunctors, says that an adjoint pair of profunctors is equivalent, not necessarily to a functor, but instead a semifunctor. If the codomain of the semifunctor is [[Cauchy complete]] this is equivalent to a functor, which recovers the usual formulation of this theorem.

## Examples

A mapping $F$ of a category into another category that sends $id_X$ to a nontrivial idempotent endomorphism of $F(X)$ is a semifunctor but not a functor.

More generally, recall from [[semicategory]] that the forgetful functor $U \colon Cat \to Semicat$ has a [[right adjoint]] $G$, which sends a semicategory to its category of idempotents, or [[Karoubi envelope]].  Thus to give a semifunctor from a category $C$ to a (semi)category $D$ is the same as giving a functor from $C$ to the Karoubi envelope $\bar D$ of $D$ (but beware that this correspondence does not hold for [[natural transformations]]).

Semifunctors between categories arise in the study of non-extensional type theories such as [[lambda calculus]] with $\beta$ equivalence, but not $\eta$ equivalence for the function type (see [Hayashi 85](#H85)).
Whereas with both $\beta, \eta$, a function type $A \to B$ can be described as a right adjoint functor to $A \times -$, the non-extensional function type can be described by weakening the requirement to the right adjoint being merely a semifunctor.

## Related concepts

* [[semicategory]]

* [[semi-simplicial set]]

* [[Karoubi envelope]]

## References

An early reference for semifunctors (there called "weak functors") is:

- B. Elkins and J. A. Zilber. _Categories of actions and Morita equivalence_, The Rocky Mountain Journal of Mathematics **6** 2 (1976): 199-225.

The terminology *semifunctors* originates in

* {#Hayashi1985} [[Susumu Hayashi]], Def. 1.1 in: *Adjunction of semifunctors: categorical structures in nonextensional lambda calculus*, Theoretical Computer Science **41** (1985) 95-104 [doi:10.1016/0304-3975(85)90062-3]

whose author proceeds to discuss their [[adjoint pairs]] in the context of [[categorical semantics]] for a version of the [[lambda calculus]].

[[!redirects semifunctor]]
[[!redirects semifunctors]]

[[!redirects semi-functor]]
[[!redirects semi-functors]]


semigroup

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# Contents
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## Idea

A semigroup is like a [[monoid]] where there might not be an [[identity element]].

The term "semigroup" is standard, but _semi-monoid_ would be more systematic.

## Definition

A __semigroup__ is, equivalently,

*  a [[set]] equipped with an [[associativity|associative]] [[binary operation]].

*  an [[associative magma]];

*  the [[hom-set]] of a [[semicategory]] with a single [[object]].

## Properties

Some semigroups happen to be [[monoid]]s; even then, a semigroup [[homomorphism]] might not be a monoid homomorphism (because it might not preserve the identity element).  Nevertheless, semigroup [[isomorphisms]] must be monoid isomorphisms.  Thus, the identity element of a monoid forms a [[property-like structure]] on the underlying semigroup.

This should be contrasted with the phenomenon that a semigroup homomorphism between two semigroups that happens to be [[group]]s does, in fact, happen to be a group homomorphism, since in this special case one can show a semigroup homomorphism must preserve the identity and inverses.

As a monoid is a [[category]] with one object, so a semigroup is a [[semicategory]] with one object.

Any [[small category]] $\mathcal{C}$ can be thought of as a semigroup by defining $S = \text{Mor}(\mathcal{C})\cup \{0\}$ and taking $f*g = f \circ g$ for any composable morphisms $f, g$, and $f*g = 0$ otherwise. Then the semigroup $(S, *)$ fully describes $\mathcal{C}$. This type of semigroup is a [[weakly reductive semigroup]].

Generalizing this, any [[category]] can be thought of as a semigroup which isn't necessarily defined on a set.

## Attitudes toward semigroups

Some mathematicians consider semigroups to be a case of [[centipede mathematics]].  [[category theory|Category theorists]] sometimes look with scorn on semigroups, because unlike a monoid, a semigroup is not an example of a [[category]].

However, a semigroup can be promoted to a monoid by adjoining a new element and decreeing it to be the identity.  This gives a [[fully faithful functor]] from the category of semigroups to the category of monoids.  So, a semigroup can actually be seen as a monoid with extra property.

+-- {: .un_example}
###### Exercise

Describe this property.
=--

On the other hand, analysts run across semigroups often in the wild, and don\'t always want to add formal identities just to turn them into monoids.

Another variant with strong links with category theory is that of [[inverse semigroups]], which [[Charles Ehresmann]] showed were closely related to [[ordered groupoid]]s. Inverse semigroups naturally occur when considering partial symmetries of an object.

## Examples

* A left or right [[ideal in a monoid|ideal of a monoid ]] $M$ is a subsemigroup of $M$ and is only a submonoid if it contains the unit in which case it is $M$  itself. A monoid $M$ induces the [[topos]] of its right [[action]]s on sets - its right [[M-Set]] $= Set^{M^op}$. The set of all of $M$'s right ideals corresponds to the elements of  the [[subobject classifier|truth value object]], $\Omega$, of this topos. The analogous construction holds for left M-Sets  $= Set^{M}$ .

* The natural numbers with a binary operation $(-) +_n (-):\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ inductively defined as $0 +_n 0 = n$, $S(x) +_n y = S(x +_n y)$, and $x +_n S(y) = S(x +_n y)$ for all $x,y:\mathbb{N}$ is a commutative semigroup for all $n:\mathbb{N}$.

## Internalization

We can [[internalization|internalize]] the concept of semigroup in any [[monoidal category]] (or even [[multicategory]]) $V$ to get a __semigroup object__ in $V$.

## Related topics

* [[magma]] (nonassociative version)

* [[commutative semigroup]] (commutative version)

* [[invertible semigroup]] (invertible version)

* [[monoid]] (unital version)

*  There is a strong connection between semigroups and finite automata; see e.g. [[Krohn-Rhodes theory]].

* [[inverse semigroups]]

* [[associative quasigroup]]

* [[cancellative semigroup]]

* [[weakly reductive semigroup]]

* [[one-parameter semigroup]]

* [[n-ary semigroup]]

* [[semiring]]

* [[semi-category]], [[semi-simplicial set]], [[semi-Segal space]]

* [[rectangular band]]

[[!include oidification - table]]

## References

On the history of the notion:

* {#Hollings09} Christopher Hollings, *The Early Development of the Algebraic Theory of Semigroups*, Archive for History of Exact Sciences **63** (2009) 497–536 [[doi:10.1007/s00407-009-0044-3](https://doi.org/10.1007/s00407-009-0044-3)]

Semicategories and semigroups are mentioned for instance

* W. Dale Garraway, Section 2 in _Sheaves for an involutive quantaloid_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 46 no. 4 (2005), p. 243-274  ([numdam](http://www.numdam.org/item?id=CTGDC_2005__46_4_243_0))

Discussion in the context of [[Lie theory]]:

* [[Joachim Hilgert]], [[Karl-Hermann Neeb]], *Lie Semigroups and their Applications*. Lecture Notes in Mathematics **1552**, Springer 1993 ([doi:10.1007/BFb0084640](https://link.springer.com/book/10.1007/BFb0084640))

[[!redirects semigroup]]
[[!redirects semigroups]]
[[!redirects semigroup object]]
[[!redirects semigroup objects]]

[[!redirects semi-group]]
[[!redirects semi-groups]]
[[!redirects semi-monoid]]
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[[!redirects semimonoid]]
[[!redirects semimonoids]]

[[!redirects Lie semigroup]]
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semigroups > history

< [[semigroups]]
[[!redirects semigroups -- history]]


semiinfinite cohomology

__Semiinfinite cohomology__ (or semi-infinite cohomology) is a mathematical counterpart of BRST reduction. The name corresponds to working with resolutions involving certain subspaces of infinite-dimensional Fock representations which correspond to half-dimensional splittings.

In mathematics it is introduced by [[Boris Feigin]].

* [[B. Feigin]], D. Fuchs, _Verma modules over the Virasoro algebra_, Topology (Leningrad, 1982), p. 230–245, Lecture Notes in Math. __1060__, Springer, Berlin, 1984.

* A. Pressley, G. Segal, _Loop groups_, Oxford Math. Monographs, 1986.

* Leonid Positselski, _Homological algebra of semimodules and semicontramodules. Semi-infinite homological algebra of associative algebraic structures_, [arxiv/0708.3398](http://arxiv.org/abs/0708.3398)

* A.V. Stoyanovsky, B.L. Feigin, _Функциональные модели представлений алгебр токов и полубесконечные клетки Шуберта_, Функц. анализ и его прил., 1994, __28__ (1), с. 68–90, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=626&what=fullt&option_lang=rus); Engl. transl.: _Functional models for representations of current algebras and semi-infinite Schubert cells_, Funct. Anal. Appl. __28__ (1), 55–72 (1994), [doi](http://dx.doi.org/10.1007/BF01079010)

* S. Arkhipov, _Semi-infinite cohomology of associative algebras and bar-duality_, Internat. Math. Research Notices 1997, __17__, 833–863. [q-alg/9602013](http://arxiv.org/abs/q-alg/9602013)

* S. Arkhipov, _Semi-infinite cohomology of Tate Lie algebras_, Moscow Math. Journ. __2__, n.1, 35–40, 2002, [math.QA/0003015](http://arxiv.org/abs/math.QA/0003015)

* A. Sevostyanov. Semi-infinite cohomology and Hecke algebras. Advances Math. 159, n.1, p. 83–141, 2001, [math.RA/0004139](http://arxiv.org/abs/math.RA/0004139)

* Boris Feigin, Edward Frenkel, _Semi-infinite Weil complex and the Virasoro algebra_, Comm. Math. Phys. __137__:3 (1991), 617-639 [euclid](http://projecteuclid.org/euclid.cmp/1104202744)

[[!redirects semi-infinite cohomology]]


SemiLat

__$Semi Lat$__ is the [[category]] whose [[objects]] are [[semilattices]] and whose [[morphisms]] are semilattice [[homomorphisms]], that is [[functions]] which preserve finitary [[joins]] (equivalently, binary joins and the [[bottom element]]).

We mention joins above as if the objects are join-semilattices; one can just as well consider them meet-semilattices and say that the homomorphisms preserve finitary [[meets]] (including the [[top element]]).  For $Semi Lat$ in itself, this is purely a difference in notational convention. We can avoid this choice by saying $Semi Lat$ is the category whose objects are [[idempotent]] [[commutative monoids]] and whose morphisms are monoid homomorphisms.    However, the choice corresponds to using either of two [[inclusion functor]]s representing $Semi Lat$ as a [[subcategory]] of [[Pos]].

There is a forgetful functor from $Semi Lat$ to $Set$, and its left adjoint sends any [[set]] $X$ to the __[[free object|free]] semilattice__ on that set: namely, the __finite [[powerset]]__ $\mathcal{P}_{fin}(X)$ of $X$, i.e. the set of [[finite set|finite]] subsets of $X$, thought of as a join-semilattice.  (Note that this exists even in [[predicative mathematics]], as long as we are allowed to define sets by [[recursion]] over [[natural numbers]], although you have to construct it by general nonsense instead of as a subset of the full powerset.  For purposes of [[constructive mathematics]], by 'finite' we mean [[K-finite set|Kuratowski finite]].)

$Semi Lat$ is given by a finitary [[variety of algebras]], or equivalently by a [[Lawvere theory]], so has all the usual properties of such categories.

category: category

[[!redirects SemiLat]]
[[!redirects Semi Lat]]
[[!redirects SLat]]
[[!redirects S Lat]]

[[!redirects free semilattice]]
[[!redirects free meet semilattice]]
[[!redirects free meet-semilattice]]
[[!redirects free join semilattice]]
[[!redirects free join-semilattice]]
[[!redirects finite powerset]]
[[!redirects finite power set]]


semilattice

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#### $(0,1)$-Category theory
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#Contents#
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## Definition

### General

A **join-semilattice** is a [[poset]] which admits all [[finite set|finite]] [[join|joins]], or equivalently which admits a bottom element $\bot$ and binary joins $a\vee b$.  If we think of a poset as a category, a join-semilattice is the same as a poset with finite colimits, or equivalently, a poset with finite coproducts.

In a join-semilattice the binary join $\vee$ is commutative, associative, has $\bot$ as a unit, and is **idempotent**:  $a\vee a =a$.  And in fact, given any commutative and idempotent [[monoid]] $(A,\vee,\bot)$, we can define $a\le b$ to mean $a \vee b = b$ to make it into a poset with finite joins; thus we have an equivalent algebraic definition of a join-semilattice.

Dually, a **meet-semilattice** is a poset which admits all finite [[meet|meets]], including a top element $\top$ and binary meets $\wedge$.  Once again $\wedge$ is commutative, associative, unital for $\top$, and idempotent.  Once again we can recover the order from it, but this time defining $a\le b$ to mean $a \wedge b = a$.  If we think of a poset as a category, a meet-semilattice is the same as a poset with finite limits, or equivalently, a poset with finite products.

If we treat join- and meet-semilattices purely algebraically there is no difference: they are both just idempotent commutative monoids.   The difference comes in how we define the order starting the commutative monoid structure, or in the notation we use (just as we distinguish additive and multiplicative groups notationally).

Since the opposite of a join-semilattice is a meet-semilattice, it would be possible to take one as standard and call the other a _cosemilattice_ (compare [[direction|directed]] and [[codirection|codirected]] sets), but this may not have ever been done.

If a poset is both a join- *and* a meet-semilattice, then we call it a [[lattice]].

### Bounded semilattices and semipseudolattices
 {#BoundedAndPseudo}

Traditionally, a semilattice need have only finite [[inhabited set|inhabited]] meets/joins; that is, it need not have a top/bottom element.  Algebraically, this means that a semilattice need not be a monoid, but is any commutative idempotent [[semigroup]].

One might call a semilattice that does have a top/bottom element a __bounded semilattice__; the problem with this is that a [[bounded poset]] already means a poset that has both top *and* bottom elements, whereas here we really only want to require one.

Another approach is to define a semilattice, as above, to require a top/bottom element and then use the term __pseudosemilattice__ or __semipseudolattice__ to allow for the possibility that it might not.

See [[lattice]] for more discussion of this issue.

### Semilattice homomorphisms

A homomorphism of join-semilattices $f: A \to B$ is a function that preserves finite joins, or equivalently:
$$ f(x \vee y) = f(x) \vee f(y),\; f(\bot) = \bot .$$
Note that such a homomorphism is necessarily a [[monotone function]], but the converse fails.  Thus, a semilattice is a poset with [[property-like structure]].

A homomorphism of meet-semilattices is defined in an analogous (i.e., dual) way.  In what follows we take join-semilattices as the default, but all results apply to meet-semilattices with slight modifications.

### The category of semilattices

Semilattices and semilattice homomorphims form a [[concrete category]] [[SemiLat]].  By the remarks above, this is equivalent to the category of commutative idempotent monoids.  Since these are algebras of a [[Lawvere theory]], or equivalently a [[finitary monad]] on $Set$, the category $Semi Lat$ has all the properties that [[algebraic category|finitary monadic categories]] enjoy.

The poset $\{F,T\}$ where $F \le T$ becomes a commutative [[rig]] with $\vee$ and meet $\wedge$ as addition and multiplication, respectively; let us call this rig $Bool$.    We can define a module of a rig much as we define a module of a ring, but with the module's underlying abelian group generalized to be a commutative monoid (thus eliminating the need for negatives).   The underlying commutative monoid of a $Bool$-module is idempotent because, writing addition in this monoid as $+$, we have $a + a = (T \vee T)a = T a = a$.  Conversely, any idempotent commutative monoid becomes a $Bool$-module in a unique way.  Thus, the category $Semi Lat$ is equivalent to the category of $Bool$-modules.

## Properties

### The free join-semilattice on a poset

There is a [[forgetful functor]]
$$
U \colon SemiLat \to Poset
$$

This has a [[left adjoint]]

$$
F \colon Poset \to SemiLat
$$

where for any poset $P$, the join-semilattice $F(P)$ is the poset of finitely generated downsets of $P$, ordered by inclusion.   Here a **downset** of a poset $P$ is a subset $S \subseteq P$ such that

$$ s \in S, s' \le s \quad  \implies \quad s' \in S. $$

This set of all downsets in $P$, say $\hat{P}$, is ordered by inclusion, and it's a [[suplattice]]: any union of downsets is a downset.  There's an embedding of $P$ in  $\hat{P}$ that sends each $p \in P$ to its **principal** downset $\{s \in P : \; s \le p \}$.  A downset is **finitely generated** if it is the union of finitely many principal downsets.  (To give a finitely generated downset is to give a finite [[antichain]], and so the free join-semilattice is sometimes described equivalently as the set of finite antichains in $P$ equipped with a certain partial order.)

To understand this description of the free join-semilattice on a poset, some [[enriched category theory]] is useful.  A [[preorder]] is the same as  a $Bool$-enriched category, where now $Bool$ stands for the monoidal category with two objects $F$, $T$ and one nontrivial morphism $F \implies T$, its monoidal structure being "and".   The downsets of a poset $P$ correspond in a one-to-one way with order-preserving maps $f \colon P^{op} \to Bool$, but in terms of enriched category theory these are precisely the $Bool$-enriched functors $f \colon P^{op} \to Bool$, just as presheaves on a category $C$ are functors $f \colon C^{op} \to Set$.   Thus, the embedding $y \colon P \to \hat{P}$ that sends each $p \in P$ to its principal downset is the $Bool$-enriched version of the Yoneda embedding.   So, just as the category of presheaves on a category $C$ is the [[free cocompletion|free cocomplete category]] on $C$, $\hat{P}$ is the free cocomplete $Bool$-enriched category on $P$.   But a cocomplete $Bool$-enriched category that happens to be a poset (instead of a mere preorder) is the same as a suplattice.  Thus, the free suplattice on $P$ is $\hat{P}$.

Similarly, the free join-semilattice on a poset $P$ is the $Bool$-enriched analogue of the free _finitely_ cocomplete category on a category $C$, since objects in this category are presheaves that are finite colimits of representables.

## Examples

* [[semilattice of commutative subalgebras]]

* [[suplattice]]

## Related concepts

* [[band]]

* [[enriched poset]]

* [[semilattice object]]

* [[join-semilattice object]]

* [[meet-semilattice object]]

[[!redirects meet-semilattice]]
[[!redirects join-semilattice]]
[[!redirects meet semilattice]]
[[!redirects join semilattice]]
[[!redirects cosemilattice]]
[[!redirects bounded semilattice]]
[[!redirects pseudosemilattice]]
[[!redirects semipseudolattice]]
[[!redirects semilattices]]
[[!redirects meet-semilattices]]
[[!redirects join-semilattices]]
[[!redirects meet semilattices]]
[[!redirects join semilattices]]
[[!redirects cosemilattices]]
[[!redirects bounded semilattices]]
[[!redirects pseudosemilattices]]
[[!redirects semipseudolattices]]


semilattice object

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#### Algebra
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#### Category theory
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# Contents
* table of contents
{: toc}

## Idea

The notion of a *semilattice object* is the generalization of that of *[[semilattice]]* as one passes from the ambient [[category of sets]] [[internalization|into]] more general ambient [[categories]] with suitable properties.

## Definition

In a [[symmetric monoidal category]] [[monoidal category with diagonals|with diagonals]] $(C, \otimes, I, \Delta_{(-)})$, a **semilattice object** is a [[commutative monoid object]] $(M, \mu, \eta)$ such that for every morphism $a:I \to M$, $\mu \circ \Delta_M \circ a = a$, where $\Delta_M$ is the [[diagonal morphism]] of $M$.

## See also

* [[commutative monoid object]]

* [[join-semilattice object]]

* [[meet-semilattice object]]

* [[semilattice]]

[[!redirects semilattice objects]]


semilinear map

A semilinear map (older term: semilinear transformation) from a left vector space $V$ over a division ring $k$ to a left vector space $W$ over a division ring $k'$ is an additive map $f: V\to W$ of the abelian groups of vectors together with a homomorphism of rings $\alpha : k\to k'$ such that the following weaker form of homogeneity holds:

$$
f (\lambda v) = \alpha(\lambda) f(v), \,\,\,\,\,a\in k, v\in V
$$

This notion is very much used in the classical affine and projective geometry over division rings, for example in the standard definition of a homotethy.

[[!redirects semilinear transformation]]


semilinearity

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* table of contents
{:toc}

## Idea

The definition of semilinearity first appeared in Parikh’s theorem [Parikh (1961)](#Parikh61) and has then gained significance in [[linguistics]].

## Definition

A [[formal grammar]] is semilinear if the languages it generates are [[regular language|regular]] when forgetting the order.

## References

* {#Parikh61} Rohit J Parikh. Language generating devices. Quarterly Progress Report, 60:199–212, 1961.
* Jens Michaelis and Marcus Kracht. Semilinearity as a syntactic invariant. In Jaime G. Carbonell, J¨org
Siekmann, G. Goos, J. Hartmanis, J. van Leeuwen, and Christian Retor´e, editors, Logical Aspects of
Computational Linguistics, volume 1328, pages 329–345. Springer Berlin Heidelberg, Berlin, Heidelberg, 1997.


semimodel category

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## Idea

Introduced by [Hovey 1998](#Hovey98), the notion of *semimodel categories* s a relaxation of that of [[model categories]] which allows for a largely similar theory.

The notion of a [[weak model category]] and [[premodel category]]
relaxes the definition even further.

## Definition

(See [Hovey 98](#Hovey98), Theorem 3.3.)

A __left semimodel category__ is a [[relative category]]
equipped with a class of [[cofibrations]] and [[fibrations]]
such that [[weak equivalences]] are closed under [[retracts]]
and the [[2-out-of-3 property]],
[[cofibrations]] have a [[left lifting property]] with respect
to [[trivial fibrations]],
[[trivial cofibrations]] with [[cofibrant]] source
have a [[left lifting property]] with respect to [[fibrations]],
and morphisms with [[cofibrant]] source
can be factored as a [[cofibration]] followed by a [[fibration]],
either one of which can be further made trivial.

A __right semimodel category__ is defined by passing to the [[opposite category]].

## Examples

\begin{example}
**([[semimodel structure on semisimplicial sets]])**
There exists a right semi-model structure on the category of [[semi-simplicial sets]] ([Rooduijn 2018](#Rooduijn2018)).
\end{example}

## References

The definition is due to:

* {#Hovey98} [[Mark Hovey]], _Monoidal model categories_ ([arXiv:math/9803002](https://arxiv.org/abs/math/9803002))

The example of the [[semimodel structure on semisimplicial sets]]:

* {#Rooduijn2018} Jan Rooduijn, *A right semimodel structure on semisimplicial sets*, Amsterdam 2018 ([pdf](https://eprints.illc.uva.nl/id/eprint/1663/1/MoL-2018-34.text.pdf), [mol:4787](https://msclogic.illc.uva.nl/theses/archive/publication/4787/A-right-semimodel-structure-on-semisimplicial-sets))

[[!redirects semimodel categories]]

[[!redirects semi-model category]]
[[!redirects semi-model categories]]

[[!redirects semimodel structure]]
[[!redirects semimodel structures]]
[[!redirects semi-model structure]]
[[!redirects semi-model structures]]


seminar on factorization algebras

## Seminar on Factorization Algebras at Goettingen: 2014 Fall - 2015 Winter ##
***

**Place: Sitzungssaal,  [Page in studip](https://www.studip.uni-goettingen.de/dispatch.php/course/overview?cid=a829cf0bce827ed6bc8a6ae2413e96bf) for people at Gottingen, and its [Google group](https://groups.google.com/forum/?hl=en#!forum/factorizationalgebras).

* table of contents
{: toc}

### Objectives ###
{#Objectives}

The aim of this seminar is bi-fold: to learn factorization algebra, and to understand perturbative quantum field theory.

Quantum field theory is an amazing generating machine to obtain interesting mathematical objects (observables), and to relate different parts of math together. However, the mathematical formulation for full quantum field theory seems to be a quite formidable task. Nonetheless, interesting mathematical frameworks have been constructed and developed all the time. For example, topological field theories, (rational) conformal field theories, string field theories, mirror symmetries, Sieberg-Witten theory and Donaldson theory, and integrable systems. A common theme, behind those examples, is to capture the topological, algebraic or geometric invariants in various settings. Perhaps, this verifies why quantum field theory should be interesting to mathematicians.

It turns out, the very theme of looking for invariants also shows up in the development of factorization algebras. And more is true. The various axioms one would like to have in constructing a field theory, are naturally encoded in the definition of a factorization algebra. In this picture, it is quite natural to describe the classical and quantum observables in a coherent picture, and quantization is expected to be a kind of deformation. We are going to see how to materialize this during our seminar. Due to the deep impact from field theory to mathematics, this also means that we are going to revisit relevant mathematics from this new point of view, which includes, the theory of operads, Poisson geometry, vertex algebra, general and differential cohomology, index theorems, higher geometry, and dg-categories.

Factorization algebra is broadly connected to different branches in mathematics, and attracts algebraists, geometers and tologists in general. It has both the abstract, higher point of view, and the down-to-earth calculation/applications. Thus it is a good arena to train our professional skills. We hope this seminar help us to

* review some important classical contents from the modern perspective,
* build up a common language to relate to each other's research topics,
* contribute to this exciting area from our own expertise.

### Prerequisite ###
{#Prerequisite}

Graduate-level courses on abstract algebra, differential geometry and topology. No knowledge on classical/quantum field theory is assumed. It would be good to have in mind some possible connections/applications of factorization algebras in your research topics. Or, if you are not sure at this point, feel free to raise up a discussion among the participants.

### Plan of Talks ###
{#Plan}

We initiated the first several talks, which give a gentle introduction to the topic. The future talks will be the on more specific developments.

* Oct. 21, 2014: Organization meeting. Why factorization algebra is interesting, and a brief account on the introduction chapter of Costello-Gwilliam (CG).

* Nov. 4, 2014: Prefactorization algebra by Chenchang. Definition, relation to $E_n$ operad theory (TFT) and functorial field theories ($\mathbf G$-FT). Associative algebras in quantum mechanics. CG: Chap 1, 3.

* Nov. 11, 2014: Continuation of Chenchang's prefactorization algebra: a description of quantum observables in free field theory through divergence operator, and a construction of $\mathcal{U}(\mathfrak{g})$. CG: Chap 2-3.

* Nov. 18, 2014: Examples of prefactorization algebras from field theories by Thomas. A finite dimensional analogy --- the divergence complex of a measure, Koszul resolution of the derived critical locus, and the relation to the Chevalley-Eilenberg complex of Heisenberg Lie algebra.  CG: Chap 4.1-4.2.

* Nov. 25, 2014: Continuation of free field theory by Thomas. Elliptic complexes, the homotopy equivalence between smooth and distributional sections, free field theories: classical observables and Poisson structure, quantum observables and the BD structure. CG: Chap 4.

* Dec. 2, 2014: An introduction to operads by Malte. $End$-operad, the definitions of (unital) operads, and algebras over operads. LV, F.

* Dec. 9, 2014: (1) A computation of 1d massive free scalars: quantum observables and Weyl algebra by Dorothea. CG: Chap 4.3. (2) $E_n$ operads and examples of $E_\infty$ operad by Malte. LV, F.

* Dec. 16, 2014: A talk on T-dualities by Bei.

* Jan. 6, 2015: Factorization algebra by Dennis. Weiss cover and Ran space, descent condition with respect to three topologies (a la Beilinson-Drinfeld, Lurie and Costello-Gwilliam), (co)sheafification vs glue, a recast of $E_n$-algebras, locally constant factorization algebras and $E_n$-algebras are equivalent (as $(\infty,1)$-categories). Lurie: Chap 5, CG: Chap 6, Ginot: Chap 4.

* Jan. 13, 2015: Guest lecture by Owen Gwilliam. The general picture of QFT: classical field theory is about critical loci, while quantization involves integration over the critical loci plus that along the "normal directions". To deal with the singular critical loci, one considers the derived deformation theory. A family version of it leads to $L_\infty$ spaces. Derived loop spaces as an example. GG.

* Jan. 20, 2015: Quantum BV formalism by Dennis. Kontsevich quantization of Poisson algebras (operadically $P_1\to E_1$), Etingof-Kazhdan quantization of Lie bialgebras ($P_2\to E_2$), an operadic description of the BV formalism, which quantizes a $P_0$ stricture into a $BD$(, or $E_0$) structure. CG: Chap 8,13.

In the following meetings we shall look into various applications and comparisons relevant to other parts of mathematics.

* Jan. 27, 2015: Quantum BV formalism in AQFT by Dorothea. QFT in globally hyperbolic spacetime: free theory as an example, the functorial description, the classical observables. The quantum description involves three types of product: the star product, the (naive) pointwise product and the time-ordered product. Time-ordering operator relates the last two, while the connection to the star product is given by the casual structure. FR1, FR2.

* Feb. 3, 2015: Factorization algebras and Goodwillie Calculus by Dmitri. The Eilenberg-Steenrod axioms give a collection of (homology) functors from spaces to chain complexes that are represented by their values at a point. With a slight generalization, one considers functors from the category $Man_n$ of manifolds with embeddings to a general symmetric monoidal $\infty$-category $\mathcal{D}$ subject to similarly defined axioms. There is an equivalence between those functors and the little-disk algebras valued in $\mathcal{D}$ induced by the evaluation map, whose inverse is given by factorization homology. Given one such functor $A: Man_n\to \mathcal{D}$, one defines $k$-th Taylor (co)tower $P_k A$ via the left Kan extension from $Disk_n^{\leq k}$ to $Man_n$, and $A$ is analytic if $P_\infty A\to A$ is an equivalence, which encodes the factorization descent condition. Homotopy fibers $D_\bullet A$ of the (co)tower $P_\bullet A$ at each stage can be defined. In the homotopy context for Taylor towers $P_\bullet \mathcal{F}$, the results of Goodwillie allow one to describe $D_\bullet \mathcal{F}$ quite explicitly. Fra.

### Discussion Sessions ###
{#Discussion}

We launched a discussion program, as certain kind of continuation of the factorization algebra seminar during the break. The aim is to promote relaxing and enlightening discussions on factorization algebra and nearby areas among people at Goettingen.

For each session, there will be one-hour long presentation given by the leader/speaker (so to "tell the story"), s/he may also bring up some points to discuss during the tea break and the extra discussion hours. If people are interested, we could also go for joint dinner later that day.

* Session 1 @ Sitzungssaal

| February 23       |    Leader                        |    Topic                   |
| ------------- | --------------------| ---------------- |
| 12:40 - 13:40   |    Xiaoyi Cui    | Geometric context for quantum BV formalism.                                 |
| 13:40 - 14:10   |    tea time                                                     ||
| 14:10 - 15:10   |    Dmitri Pavlov              |    $1\vert 1$-dimensional field theories and K-theory.      |
| 15:10 - 18:00[^1]   |    tea time                                               ||

* Session 2 @ Sitzungssaal

| March 5 [^2]            |    Leader                        |    Topic                   |
| ------------- | --------------------| ---------------- |
| 13:00 - 14:00   |    Jan Jitse Venselaar     |     Factorization algebras for noncommutative geometry.                           |
| 14:00 - 14:30   |    tea time                                                     ||
| 15:00 - 16:00   |    Vadim Alekseev         |     Groups, operator algebras and ergodic theory.               |
| 16:00 - 18:00[^1]   |    tea time                                               ||

* Session 3 @ Sitzungssaal

| March 19           |    Leader                        |    Topic                   |
| ------------- | --------------------| ---------------- |
| 13:00 - 14:00   |    David Buecher            |       Holomorphic field theories and vertex algebras                       |
| 14:00 - 14:30   |    tea time                                                     ||
| 14:30 - 15:30   |    Dmitri Pavlov               |        Bundle n-gerbes with connection as field theories.                        |
| 15:30 - 18:00[^1]   |    tea time                                               ||

[^1]: The room has been reserved till 18:00 to allow for further discussion.
[^2]: Please note that the second talk will start at 15:00.

### References ###
{#References}

The main relation/application to quantum field theory is given by the following book, which also contains a careful introduction to the notion of factorization algebra.

* [CG] K. Costello and O. Gwilliam, Factorization algebras in quantum field theory.

An out-dated version is available on Costello's webpage. Participants can also join our mailing list to get the most up-dated version.

A more mathematical introduction to factorization algebra is given in

* [Ginot] G. Ginot, Notes on factorization algebras, factorization homology and applications, arXiv:1307.5213[math.AT].

A much more elaborated treatment (using slightly different languages from the formers):

* [Lurie] J. Lurie, Higher Algebra. [pdf](http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf).

The introduction to operads is based on

* [LV] J.-L. Loday, B. Vallette: Algebraic Operads. [Pdf available here](http://math.unice.fr/~brunov/Operads.html).

For $E_\infty$ operads, see

* [F] B. Fresse: Introduction to $E_n$-operads and little discs operads. [pdf](http://www.newton.ac.uk/files/seminar/20130305140015152-153497.pdf).

and the material here:

* B. Fresse: Homotopy of Operads and Grothendieck-Teichmueller Groups. [Book project homepage](http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/).

Next here are references with more specific applications.

For the perturbative quantum field theory, and for the intuition of BV formalism, see the following monograph.

* K. Costello, Renormalization and effective field theory, Surveys and monographs, American Mathematical Society, 2011.

For the application to topological manifold, see

* [Fra] J. Francis, Factorization homology of topological manifolds, arXiv:1206.5522[math.AT].

* D. Ayala, J. Francis and H. Tanaka, Structured singular manifolds and factorization homology, arXiv:1206.5164[math.AT].

On $L_\infty$ space, see

* [GG] R. Grady and O. Gwilliam, L-infinity spaces and derived loop spaces, arXiv:1404.5426[math.AG].

On Algebraic Quantum Field Theory, see

* [FR1] K. Fredenhagen and K. Rejzner, Batalin-Vilkovisky formalism in the functional approach to classical field theory, Commun.Math.Phys. 314 (2012) 93-127, arXiv:1101.5112[math-ph].

* [FR2] K. Fredenhagen and K. Rejzner, Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, Commun.Math.Phys. 317 (2013) 697-725, arXiv:1110.5232[math-ph].


seminar on Grothendieck-Teichmuller theory

We meet every Monday from 12:15 until 14:00 in Sitzungzimmer.  To enroll, visit [Studip](https://www.studip.uni-goettingen.de/dispatch.php/course/overview?cid=e830fe84232de54d1e504a031fa62ae7)

This semester, we'll try to study again the theory about the symmetries in deformation quantization. We start with reviewing Kontsevich's original paper and its physics background, then go into the algebraic/topologic profile of this theory. Hopefully, we may land in associators and their symmetries in the middle of the semester.

A very tentative plan:

***

*27.04: Chenchang on Cattaneo-Felder's approach to Kontsevich's quantization formula, reference  arXiv:math/9902090.

*04.05: Chenchang on Kontsevich's quantization, reference  arXiv:q-alg/9709040

*11.05: no meeting

*18.05: Malte on Algebraic/topological profile of deformation quantization, e.g. Deligne conjecture, reference

*26.05 (note different meeting time!): Xiaoyi on homotopy BV algebra and formality conjecture, reference 	[arXiv:math/0002116](http://lanl.arxiv.org/abs/math/0002116)

*01.06: David on Etingof-Kazhdan theory on deformation quantization on Lie bialgebras, reference

*08.06: Dennis on arithmetric topology.


This page compiles pointers related to:

* [[Beno Eckmann]] and [[Myles Tierney]] (eds.)

  with contributions by

  [[H. Applegate]], [[M. Barr]], [[J. Beck]], [[F. W. Lawvere]], [[F. E. J. Linton]], [[E. Manes]], [[M. Tierney]], [[F. Ulmer]]:

  \linebreak

  **Seminar on Triples and Categorical Homology Theory**

  \linebreak

  Lecture Notes in Mathematics **80**

  Springer (1969)

  \linebreak

  reprinted as:

  Reprints in Theory and Applications of Categories __18__ (2008) 1-303

  [TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)

  \linebreak

  MSC 2000: 18C05, 18C15, 18E25, 18G10

on [[monads]] in [[universal algebra]] and ([[cohomology|co]]-)[[homology]]-theory.

\linebreak

**Contents**

On [[functorial semantics]] (and the [[codensity monad]]):

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

On [[distributive laws]]:

* [[Jon Beck]], *Distributive Laws*, in: *[[Seminar on Triples and Categorical Homology Theory]]*, ETH 1966/67, Lecture Notes in Mathemativs, Springer (1969), Reprints in Theory and Applications of Categories **18** (2008) 1-303 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

On equational [[doctrines]] (and introducing the notion of [[Frobenius monads]]):

* {#Lawvere69} [[William Lawvere]], *Ordinal sums and equational doctrines*, in *[[Seminar on Triples and Categorical Homology Theory]]* LNM **80** (1969) 141-155 [[doi:10.1007/BFb0083085](https://doi.org/10.1007/BFb0083085)]

On [[comonads]] (such as dual to the [[codensity monad]]):

* [[Harry Applegate]], [[Myles Tierney]], *Categories with models*, in: [[Beno Eckmann]] (ed.) *[[Seminar on Triples and Categorical Homology Theory]]* Lecture Notes in Mathematics, **80**, Springer  (1969) 156-244   [[doi:10.1007/BFb0083086](https://doi.org/10.1007/BFb0083086), [pdf](https://link.springer.com/content/pdf/10.1007/BFb0083086)]

On [[monadic cohomology]], or rather [[homology]] via [[comonad|comonadic]] [[canonical resolutions]]:

* [[Michael Barr]], [[Jon Beck]], _Homology and Standard Constructions_, in: _[[Seminar on Triples and Categorical Homology Theory]]_, Lecture Notes in Maths. **80**, Springer (1969), Reprints in Theory and Applications of Categories __18__ (2008) 186-248 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

On [[Hochschild cohomology]] in relation to [[monadic cohomology]] of [[algebras]]:

* [[Michael Barr]], *Cohomology and obstructions: commutative algebras*, in: _[[Seminar on Triples and Categorical Homology Theory]]_, Lecture Notes in Maths. **80**, Springer (1969), Reprints in Theory and Applications of Categories __18__ (2008) 357–374 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf), [pdf](https://www.math.mcgill.ca/barr/papers/obstruct.pdf)]

(...)

category: reference

[[!redirects Seminar on Triples]]
[[!redirects LNM 80]]


seminormal representation

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## Idea

(...)

## Related concepts

* [[regular representation]]

* [[representation theory of the symmetric group]]

* [[Jucys-Murphy element]]

* [[Gelfand-Tsetlin basis]]

## References

Due to:

* [[Alfred Young]], *On quantitative substitutional analysis VI*, Proc. London Math. Soc. (2) **31** (1931), 253-289. Reprinted in: *The Collected Papers of Alfred Young, 1873-1940*, Mathematical Expositions 21, Univ. of Toronto Press, Toronto and Buffalo, 1977

Further discussion in the context of [[Jucys-Murphy elements]]:

* {#Murphy81} [[G. E. Murphy]], *A new construction of Young's seminormal representation of the symmetric groups*, Journal of Algebra Volume 69, Issue 2, April 1981, Pages 287-297 (doi:10.1016/0021-8693(81)90205-2)

* {#Murphy83} [[G. E. Murphy]], *The idempotents of the symmetric group and Nakayama's conjecture*, Journal of Algebra Volume 81, Issue 1, March 1983, Pages 258-265 ([pdf](https://core.ac.uk/download/pdf/82692868.pdf))

* {#Murphy92} [[G. E. Murphy]], *On the representation theory of the symmetric groups and associated Hecke algebras*, J. Algebra 152 (1992) 492–513 ([pdf](https://core.ac.uk/download/pdf/82422274.pdf), doi:10.1016/0021-8693(92)90045-N))

On the [[representation theory of the symmetric group]] via the [[Gelfand-Tsetlin basis]]/[[seminormal representation]]:

* {#VershikOkounkov04} [[Anatoly Vershik]], [[Andrei Okounkov]], *A New Approach to the Representation Theory of the Symmetric Groups*,
Part I: Selecta Mathematica, New Series **2**, 581-605 ([arXiv:math/0503040](https://arxiv.org/abs/math/0503040), [doi:10.1007/BF02433451](https://doi.org/10.1007/BF02433451));
Part II (incorporates Part I in revised and improved form):
Russian version: [Записки научных семинаров ПОМИ 307 (2004), 57–98](ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v307/p057.ps.gz) (Zapiski nauchnyh seminarov POMI 307 (2004), 57–98);
English version: Journal of Mathematical Sciences 131 (2005), 5471–5494 ([doi:10.1007/s10958-005-0421-7](https://doi.org/10.1007/s10958-005-0421-7)).

[[!redirects Young's seminormal representation]]

[[!redirects seminormal basis]]
[[!redirects seminormal bases]]


semiorthogonal decomposition

* [[Dmitri Orlov]], _Smooth and proper noncommutative schemes and gluing of DG categories_, Adv. Math. __302__ (2016) 59-105 [doi](https://doi.org/10.1016/j.aim.2016.07.014) [arXiv:1402.7364](https://arxiv.org/abs/1402.7364v1)
* [[Alexei Bondal]], Dmitri Orlov, _Semiorthogonal decomposition for algebraic varieties_, arxiv:[alg-geom/9506012](https://arxiv.org/abs/alg-geom/9506012)
* [[Alexander Kuznetsov]], _Base change for semiorthogonal decompositions_, Compositio Mathematica __147__:03 (2011) 852--876 [arXiv:0711.1734](https://arxiv.org/abs/0711.1734), [doi](https://doi.org/10.1112/S0010437X10005166)
* [[Alexander Kuznetsov]], _Semiorthogonal decompositions in families_, [arXiv:2111.00527](https://arxiv.org/abs/2111.00527)
* [[Valery A. Lunts]], Olaf M. Schnürer, _Matrix factorizations and semi-orthogonal decompositions for blowing-ups_, J. Noncommut. Geom. __10__:3 (2016) 907--979 [doi](https://10.4171/JNCG/252)

category: algebraic geometry, triangulated categories

[[!redirects semi-orthogonal decomposition]]


semiotic equivalence relation

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

[[!redirects semiotic congruence]]
[[!redirects semiotic congruences]]
[[!redirects semiotic equation]]
[[!redirects semiotic equations]]
[[!redirects semiotic equivalence]]
[[!redirects semiotic equivalence class]]
[[!redirects semiotic equivalence classes]]
[[!redirects semiotic equivalence relations]]
[[!redirects semiotic partition]]
[[!redirects semiotic partitions]]


semiotic information

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


semiotics

\tableofcontents

## Idea

__Semiotics__ is a general science about systems of signs, e.g. within culture, law, [[linguistics]] etc. It is closely related to [[semantics]] because a sign involves a correspondence between its form/expression (signifying) and its external pointer/meaning (signified). It also studies mechanisms how these semiotic systems appear, develop and disappear in real world (e.g. in communities).

## Related entries

* [[syntax]], [[semantics]]

* [[linguistics]]

## References

* [[Umberto Eco]], _A theory of semiotics_, Advances in Semiotics, Indiana University Press 1976
* [[Umberto Eco]], _Semiotics and the philosophy of language_, Indiana University Press 1984

See also:

* Wikipedia, *[Semiotics](https://en.m.wikipedia.org/wiki/Semiotics)*


semipresheaf

#Definition#

A **semipresheaf** on a [[semicategory]] $C$ is a [[semifunctor]]

$$
 F : C^{op} \to Set
$$

from the opposite semicategory $C^{op}$ of $C$ to the category [[Set]] of [[set]]s.

The **category of semipresheaves** on $C$, usually denoted $[C^{op},Set]_{semi}$, or just $[C^{op},Set]$ when the context is clear has:

* semifunctors $F : C^{op} \to Set$ as objects;

* [[natural transformation|natural transformations]] between such semifunctors as morphisms.

[[!redirects semipresheaves]]


semiprojective morphism

# Semiprojective morphisms
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{: toc}

## Definitions

Let $A,B$ be separable $C^\ast$-[[C-star-algebra|algebras]]. A morphism $f\colon A \to B$ is __semiprojective__ if for any separable $C^\ast$-algebra $C$, any increasing sequence $J_n \subset C$ of [[ideal of a ring|ideals]] with $J = \overline{\cup_{n=0}^\infty J_n}$ and any morphism $\sigma\colon B \to C/J$, there exist $n$ and an "$f$-relative lift" $\tilde{\sigma}\colon A \to C/J_n$ in the sense that the composition $A \stackrel{\tilde\sigma} \to C/J_n \to C/J$ equals the composition $A \stackrel{f}\to B \stackrel{\sigma}\to C/J$, where $C/J_n\to C/J$ is the [[epimorphism]] induced by the inclusion $J_n \subset J$ of ideals.

A separable $C^\ast$-algebra is __semiprojective__ if the identity $id_A\colon A \to A$ is a semiprojective morphism. In particular, every projective separable $C^\ast$-algebra is semiprojective. They are viewed as a generalization of (continuous function algebras) of [[ANR]]s for metric spaces.

This notion is used in the strong [[shape theory]] for separable $C^\ast$-algebras.

## References

* [[Bruce Blackadar]], _Shape theory for $C^\ast$-algebras_.
Math. Scand. 56 (1985), no. 2, 249–275, [MR87b:46074](http://www.ams.org/mathscinet-getitem?mr=813640), [pdf](http://www.mscand.dk/article.php?id=2767)

* [[Marius Dādārlat]], _Shape theory and [[asymptotic morphism]]s for $C^\ast$-algebras_, Duke Math. J. __73__ (3):687-711, 1994, [MR95c:46117](http://www.ams.org/mathscinet-getitem?mr=1262931), [pdf](http://www.math.purdue.edu/~mdd/Publications/shape.pdf)

[[!redirects semiprojective morphism]]
[[!redirects semiprojective morphisms]]


semiquantale

A __right semiquantale__ is a [[complete lattice]] $Q$ with an associative binary operation $\bullet$ satisfying

$$ a\bullet  \Vee_\alpha b_\alpha = \Vee_\alpha (a \bullet b_\alpha),\,\,\,\,\,a, b_\alpha\in Q.$$

Symmetrically, a left semiquantale is a complete lattice $P$ with an associative binary operation $\bullet$ satisfying

$$  \left(\Vee_\alpha b_\alpha\right)\bullet c= \Vee_\alpha (b_\alpha \bullet c),\,\,\,\,\,c, b_\alpha\in P.$$

The main example of a right semiquantale is the lattice of the [[topologizing filter]]s of right (or left) ideals.
These are the filters $\mathcal{F}\subset I_r R$ such that for all $I\in\mathcal{F}$
$$
   (I:r) := \{s\in R \,|\, r s\in I\}
$$
is also in $\mathcal{F}$ (i.e. $\mathcal{F}$ is a [[uniform filter]]) and if $J\in \mathcal{F}$ and $(I:r)\in I$ for all $r\in J$ then $I\in \mathcal{F}$.

The ordering is the reverse inclusion, thus the intersection is the supremum. The intersection of topologizing filters is topologizing, the lattice is complete and the product is the [[Gabriel multiplication]]

$$\mathcal{F}\bullet \mathcal{G} = \{ K\in I_r R\,|\,\exists L\in\mathcal{G}, K\subset L, \forall r\in K, (K:r)\in\mathcal{F}\} $$

The operation $\bullet$ preserves arbitrary intersections in the right variable.

[[!redirects semiquantales]]
[[!redirects right semiquantale]]
[[!redirects left semiquantale]]


semiring

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## Definitions

Similarily to the definition of a ring, mathematicians disagree on the definition of a semiring. There are 4 different definitions of a semiring used in mathematics:

1. A [[semigroup object]] in the [[category]] of [[commutative semigroups]]
2. A [[monoid object]] in the [[category]] of [[commutative semigroups]]
3. A [[semigroup object]] in the [[category]] of [[commutative monoids]]
4. A [[monoid object]] in the [[category]] of [[commutative monoids]]

The last algebraic structure is also called a [[rig]].

Historically, rings were defined without a multiplicative identity element, and the primary definition of semiring used was the third definition.

However, more recently, rings have been defined with multiplicative units, and thus nowadays the primary definition of semiring used is the fourth definition. In this context, semirings and rings lacking multiplicative identity elements are called [[nonunital ring|nonunital]].

Moreover, some mathematicians argue that semiring should be used in analogy to semigroup, where semirings do not have additive identity elements, and semirings with 0 are instead called [[rigs]], resulting in the first and second definitions of a semiring respectively.

The first definition is the most general of the four definitions of a semiring, and some authors use it precisely because of its generality.

If one adopts the second definition to define a semiring, and the fourth definition to define a [[rig]], then the first and third are then called *nonunital semirings* and *nonunital rigs* respectively.

## Examples

* [[tropical semiring]]

## Related concepts

* [[rig]]

* [[ring]]

* [[semigroup]]

## References

* Wikipedia, *[Semiring](http://en.wikipedia.org/wiki/Semiring)*

* Wolfram Mathworld, *[Semiring](https://mathworld.wolfram.com/Semiring.html)*

* Kazimierz Glazek, [A guide to the literature on semirings and their applications in mathematics and information sciences](https://zbmath.org/?format=complete&q=an:1072.16040)

[[!redirects semiring]]
[[!redirects semirings]]

[[!redirects commutative semiring]]

[[!redirects semi-ring]]
[[!redirects semi-rings]]


semiring scheme

#Contents#
* table of contents
{: toc}

## Idea

A **semiring scheme** is like a [[scheme]] but where the base [[commutative ring]] can be any [[commutative semiring]]. Note that we consider the simplest definition of semirings as defined under the name "rig" on the page [[rig]] ie. we have an additive unit 0, a multiplicative unit 1 and $0.x=0$.

## References

[[Oliver Lorscheid]], _The geometry of blueprints: Part I: Algebraic background and scheme theory_, Advances in Mathematics, **229** 3 (2012) 1804-1846 ([pdf](https://arxiv.org/pdf/1103.1745.pdf), [doi:10.1016/j.aim.2011.12.018](https://doi.org/10.1016/j.aim.2011.12.018))


semiseparated scheme

A __cover__ $\{U_i\}_{i\in I}$ of a scheme by affine open subsets is __semiseparated__ if $U_i\cap U_j$ is also affine for every pair $(i,j)\in I\times I$. An [[algebraic scheme]] is __semiseparated__ if it has an affine cover which is semiseparated. The notion is introduced in

* Thomason, Trobaugh, in Grothendieck Festschrift 1989

category: algebraic geometry


semisimple abelian category

An [[abelian category]] is **semisimple** if every object is a [[direct sum]] of [[simple object|simple objects]].  In other words, it is an abelian category that is [[semisimple category|semisimple]].


semisimple algebra

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## Definition

An [[associative algebra|associative]] [[unital algebra|unital]] algebra $A$ over a [[field]] $k$ is **semisimple** if its [[Jacobson radical]] is trivial.

## Properties

If $A$ is finite-dimensional, this is equivalent to saying that $A$ is a finite product of finite-dimensional [[simple algebra|simple algebras]].

By the [[Artin-Wedderburn theorem]], any finite-dimensional simple algebra over $k$ is a [[matrix algebra]] with entries lying in some [[division algebra]] whose [[center]] is $k$.  So, every finite-dimensional semisimple algebra is a finite product of such matrix algebras.

[[!redirects semisimple algebras]]
[[!redirects semi-simple algebra]]
[[!redirects semi-simple algebras]]


semisimple category

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## Idea
 {#Idea}

A _semisimple category_ is a [[category]] in which each [[object]] is a [[direct sum]] of [[finite number|finitely many]] [[simple objects]], and all such direct sums exist.

## Definition
 {#Definition}

\begin{definition}
**([[semisimple abelian category]])**
\linebreak
An [[abelian category]] is called **semisimple** if every [[object]] is a [[semisimple object]], hence a [[direct sum]] of [[finite number|finitely many]] [[simple objects]].
\end{definition}

Alternatively, at least over an [[algebraically closed field|algebraically closed]] [[ground field]] $k$:

\begin{definition}\label{SemisimpleLinearCategory}
**(semisimple linear category)**
\linebreak
A [[linear category]] (that is, a [[category]] [[enriched category|enriched over]] [[Vect]], cf. *[[tensor category]]*) is called **semisimple** if:

1. it has [[finite product|finite]] [[biproducts]]

   (usually called '[[direct sums]]'),

1. [[split idempotent|idempotents split]]

   (so we say that it 'has [[subobjects]]' or, perhaps better, 'has [[retracts]]'),

1. there exist [[objects]] $X_i$ labeled by an index [[set]] $I$ such that

   1. for any [[pair]] $i, j$ of indices we have an [[isomorphism]]

      \[
        \label{SimplicityIsomorphism}
        Hom(X_i, X_j)
          \;\cong\;
        \delta_{i j} k
        \,,
      \]

      where $\delta$ denotes the [[Kronecker delta]] and $k$ the [[ground field]]

      (such objects are called *[[simple object|simple]]*),

   1. for any [[pair]] of [[objects]] $V,\,W$, the natural [[composition]] map

      \[
        \label{CompositionIsomorphism}
        \bigoplus_{i \in I}
        \,
        Hom(V, X_i)
        \,\otimes\,
        Hom(X_i, W)
          \longrightarrow
        Hom(V, W)
      \]

      is an [[isomorphism]].

   (If the field $k$ is not [[algebraically closed field|algebraically closed]], a [[simple object]] $X$ can have a [[finite-dimensional vector space|finite-dimensional]] [[division algebra]] other than $k$ itself as its [[endomorphism algebra]].)

\end{definition}

(e.g. [Müger (2003, p. 6)](#Müger03))

\begin{proposition}\label{DirectSumDecompositionintoSimples}
**(direct sum decomposition into simple objects)**
\linebreak
Definition \ref{SemisimpleLinearCategory} implies that every object $V$ is a [[direct sum]] of [[simple objects]] $X_i$.
\end{proposition}
\begin{proof}
The third item of the definition is equivalent to stipulating that the [[vector space]] $Hom(X_i, V)$ is in canonical [[dual vector space|duality]] with the vector space $Hom(V, X_i)$. Indeed, we have:

1. a pairing

   \[
     \begin{array}{ccc}
        Hom(V, X_i)
         \,\otimes\,
        Hom(X_i, V)
        &
          \longrightarrow
        &
        Hom(X_i, X_i)
        &\simeq&
        k
        \\
        f \,\otimes\, g
        &\mapsto&
        f \,\circ\, g
        \mathrlap{\,,}
     \end{array}
   \]

   where on the right we used (eq:SimplicityIsomorphism),

1. a copairing

   \[
     k \cdot Id_B
     \;\hookrightarrow\;
     Hom(V,V)
        \longrightarrow
      Hom(X_i, V)
        \otimes
     Hom(V, X_i)
     \mathrlap{\,,}
   \]

   where on the right we used (eq:CompositionIsomorphism),

and one checks that these satisfy the [[triangle identities]] and as such exhibit [[dual vector spaces]].

Hence if we choose a [[linear basis]]

\[
   \big\{
     a_{i,p}
      \,\colon\,
     X_i
      \rightarrow V
   \big\}
\]

for each [[vector space]] $Hom(X_i, V)$, we get a corresponding dual basis

\[
  \big\{
    a_i^p
     \,\colon\,
    V \rightarrow X_i
  \big\}
\]

satisfying

\[
  a_i^p a_{j,q}
   \;=\;
  \delta_{i j} \delta_p^q
  \;\;\; \text{and} \;\;\;
  \sum_{i,p} a_{i,p} a_i^p
    \;=\; \id_V
  \mathrlap{\,.}
\]

This says precisely that $V$ has been expressed as a [[direct sum]] of the $X_i$.
\end{proof}

\begin{remark}
Def. \ref{SemisimpleLinearCategory} (of semisimple linear categories) does not use the concept of *[[abelian category]]*. This is because the concepts that one thinks about with abelian categories such as [[kernel]]s and [[cokernel]]s do not play an important conceptual role in semisimple categories, being replaced by the more important concepts of [[biproduct]] and [[retract]]. Hence it is best to give a streamlined definition from first principles without going through the language of abelian categories which would have muddied the waters.
\end{remark}

\begin{remark}
For a category to be semisimple, it needs to have a certain directional symmetry in its hom-sets, namely that $Hom(V, W)$ must at least have the same dimension as $Hom(W,V)$. This is the easiest way to check if a category will _fail_ to be semisimple. For instance, the category $Rep(A)$ of [[representations]] of an algebra $A$ will rarely be semisimple, precisely because there is no relation between $Hom(V, W)$ and $Hom(W,V)$ in general. Again, this can be traced back to the original algebra $A$ not having any 'symmetry' like the inverse operation in a group.
\end{remark}

\begin{remark}
As far as 'duality' on the hom-sets is concerned, one might have a $S \colon C \rightarrow C$ from the category to itself with the property that there are canonical isomorphisms

\[
  Hom(V, W) \cong Hom(W, S V)^\vee
\]

where "$\vee$" denotes the ordinary linear dual of a vector space. Such a functor is called a _Serre functor_ in algebraic geometry, and indeed there is precisely such a functor on the derived category of coherent sheaves on a complex manifold --- it is given by tensoring with the [[canonical line bundle]].
\end{remark}

\begin{remark}
For [[2-Hilbert space|2-Hilbert spaces]], there is an antilinear $*$-operation on the hom-sets $* : Hom(V, W) \rightarrow Hom(W,V)$. The presence of this duality in fact forces the category to be semisimple (this comes down to the fact that a finite-dimensional $*$-algebra, such as the hom's between a bunch of objects in the category, must be a full matrix algebra)
\end{remark}

## Examples

* The archetypical simple example is [[FinDimVect]] itself, the [[category]] of [[finite dimensional vector spaces]] over some [[ground field]] $k$. This has a single [[isomorphism]] class of simple objects: given by $k$ itself.

* The [[category of representations|category of finite-dimensional complex representations]] of a [[compact Lie group]] $G$ is semisimple, with the simple objects being precisely the [[irreducible representation]]s (this is the content of [[Schur's lemma]]). If $G$ is not a [[compact Lie group]], one needs to pass from the concept of 'direct sum' to '[[direct integral]]'.

* Every [[fusion category]] is a semisimple category.

## References

Various alternative definitions of semisimple category appear in the literature, and a number are compared here:

* James Bailie, Semisimple categories ([pdf](https://jameshbailie.github.io/Papers/papers/Semisimple.pdf).

For example, he compares a definition of "abelian semisimple category"(an abelian category where every object is a direct sum of a possibly infinite number of simple objects) with  "Müger semisimple" categories as discussed here:

* {#Müger03} [[Michael Müger]], p. 6 in: *From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence classes of tensor categories*, J. Pure Appl. Alg. **180** (2003) 81-157 [[arXiv:math/0111204](https://arxiv.org/abs/math/0111204), doi:10.1016/S0022-4049(02)00247-5]

There is a [related discussion on the nForum](http://nforum.ncatlab.org/discussion/1120/semisimple-category/?Focus=42783#Comment_42783) and a [discussion on MathOverflow](https://mathoverflow.net/questions/442953/what-is-the-correct-definition-of-semisimple-linear-category).

[[!redirects semisimple categories]]


semisimple Lie algebra

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## Definition

A [[Lie algebra]] is __semisimple__ if it is the [[direct sum]] of [[simple Lie algebra]]s.

(Notice that this is not quite the same as a [[semisimple object]] in the [[category]] of Lie algebras, because a simple Lie algebra is not quite the same as a [[simple object]] in the [[LieAlg]]. But this is the standard terminology convention.)

By [[Lie integration]] semisimple Lie algebras correspond to [[Lie group]]s that are [[semisimple Lie group]]s.

## Properties

* Every semisimple Lie algebra is a [[reductive Lie algebra]].

* A Lie algebra $\mathfrak{g}$ is semisimple precisely if the [[Killing form]] [[invariant polynomial]]

  $$
    \langle x,y \rangle \coloneqq tr (ad_x \circ ad_y)
  $$

  is non-degenerate as a [[bilinear form]] (hence making $\mathfrak{g}$ a [[metric Lie algebra]]).

  The corresponding cocycle $\langle -,[-,-]\rangle$ in [[Lie algebra cohomology]] is the one that classifies the [[string Lie 2-algebra]]-extension of $\mathfrak{g}$.

* [[Jacobson-Morozov theorem]]

## Classification

Since [[simple Lie algebras]] have a classification, so do semisimple Lie algebras: for each simple Lie algebra, we simply indicate how many times it appears in the direct-sum decomposition.  (There is a theorem to prove here: that the decomposition of a semisimple Lie algebra is unique.)

## Related concepts

* An infinite-dimensional generalization of semisimple Lie algebras are [[Kac-Moody Lie algebras]].

* [[exceptional Lie algebra]]

## References

* Robert Cahn, _Semisimple Lie algebras and their representation_ ([pdf](http://phyweb.lbl.gov/~rncahn/www/liealgebras/texall.pdf))

Basics of the representation theory of semisimple Lie algebras is surveyed in

* [[Joseph Bernstein]], _Lectures on Lie Algebras_, in:
Representation Theory, Complex Analysis and Integral Geometry, Birkhauser (2012), 97-133, [pdf](http://wwwmath.tau.ac.il/~bernstei/Publication_list/publication_texts/bernsteinLieNotes_book.pdf)

[[!redirects semisimple Lie algebra]]
[[!redirects semisimple Lie algebras]]


semisimple Lie group

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## Definition

A [[Lie group]] is __semisimple__ if its [[Lie algebra]] is a [[semisimple Lie algebra]].

## Related concepts

* [[Lie group]]

  * [[simple Lie group]]

* [[Lie algebra]]

  * [[semisimple Lie algebra]]

## References

* {#Knapp97} A. W. Knapp, _Structure theory of semisimple Lie groups_, 1997 ([pdf](http://www.math.sunysb.edu/~aknapp/pdf-files/1-27.pdf))

[[!redirects semisimple Lie groups]]


semisimple object

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## Definition

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###### Definition

An [[object]] $X$ in an [[abelian category]] is said to be __semisimple__ or __completely reducible__ if it is a [[coproduct]] ([[direct sum]]) of [[simple objects]].

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###### Remark

Sometimes this notion is considered in a bit more general setup than [[abelian category|abelian]] (where it is most often used); sometimes with subtle variants. For [[Lie algebras]], it is a bit different by convention: a Lie algebra is a __[[semisimple Lie algebra]]__ if it is a [[direct sum]] of *nonabelian* simple Lie algebras; though a 1-dimensional abelian Lie algebra is simple in the categorical (and in common) sense.

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###### Definition

A semisimple object is __isotypic__ if it is a [[direct sum]] of [[isomorphism|isomorphic]] simple objects (the isomorphism class of a simple object is called its _type_).

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## Properties

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###### Proposition

An object in an abelian category is semisimple if and only if it is the coproduct of all its simple subobjects.

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###### Proposition

If an object $X$ in an abelian category is semisimple then every [[short exact sequence]] $0 \to A \to X \to B \to 0$ [[split exact sequence|splits]].

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An abelian category in which all short exact sequences split is called a *spectral category*, and not every spectral category is semisimple (see [MathOverflow](https://mathoverflow.net/a/327944/2893)).

## Special classes

* semisimple left $R$-[[modules]] where $R$ is a [[ring]];

* completely reducible [[representations]] of a [[group]]

## Related concepts

* [[reductive group]]

* [[semisimple category]]

## References

* Wikipedia, [Semisimplicity](http://en.wikipedia.org/wiki/Semisimplicity).

* Wikipedia, [Semisimple module](http://en.wikipedia.org/wiki/Semisimple_module).

[[!redirects semisimple objects]]

[[!redirects semisimple]]

[[!redirects isotypic object]]
[[!redirects semisimple module]]
[[!redirects isotypic module]]

[[!redirects semisimple objects]]

[[!redirects isotypic objects]]
[[!redirects semisimple modules]]
[[!redirects isotypic modules]]

[[!redirects completely reducible]]
[[!redirects completely reducible object]]
[[!redirects completely reducible objects]]


semisimple ring

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## Definition

A **semisimple ring** $R$ is one obeying any of the following equivalent conditions:

* $R$ is an [[Artinian ring]] with vanishing [[Jacobson radical]].

* $R$ is a finite [[product]] of [[simple ring|simple]] Artinian rings.

* As a left [[module]] over itself, $R$ is a [[semisimple object]] in the category of left $R$-modules.

* As a right module over itself, $R$ is a semisimple object in the category of right $R$-modules.

* The category of [[finitely generated module|finitely generated]] left $R$-modules is a [[semisimple category]].

* The category of finitely generated right $R$-modules is a semisimple category.

* Every left $R$-module is [[projective module|projective]].

* Every right $R$-module is projective.

A [[commutative ring]] is semisimple if and only if it is both Artinian and a [[reduced ring]].

## Examples

By the [[Wedderburn-Artin theorem]], every semisimple ring is a finite product of matrix algebras over [[division rings]].

As a special case, every semisimple [[commutative ring]] is a finite product of [[fields]].  In particular, every semisimple [[integral domain]] is a [[field]].  Similarly, every semisimple [[local ring]] is a [[field]].

Beware: not every [[simple ring]] is a semisimple ring.  A simple ring is semisimple if and only if it is Artinian.  For example the [[Weyl algebra]] over any field is a simple ring that is not Artinian and not semisimple: intuitively speaking, it is "too infinite" to be Artinian, which is a kind of finiteness condition.

## See also

| [[commutative ring]] | [[reduced ring]] | [[integral domain]] |
---------------------|-----------------|-----------------|
| [[local ring]] | [[reduced local ring]] | [[local integral domain]] |
| [[Artinian ring]] | [[semisimple ring]] | [[field]] |
| [[Weil ring]] | [[field]] | [[field]] |

## References

* Wikipedia, [Semisimple ring](https://en.wikipedia.org/wiki/Semisimple_ring).

* Frank W. Anderson and Kent R. Fuller, _Rings and Categories of Modules, Springer Graduate Texts in Mathematics, Vol. 13, 2012.  Chapter 4, Section 13: Semisimple rings.

[[!redirects semisimple ring]]
[[!redirects semisimple rings]]


SemiSpin(16)

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## Idea

The [[semi-spin group]] in [[dimension]] 16.

## Properties

### As a subgroup of $E_8$

The [[subgroup]] of the [[exceptional Lie group]] [[E8]] which corresponds to the [[Lie algebra]]-inclusion $\mathfrak{so}(16) \hookrightarrow \mathfrak{e}_8$ is the [[semi-spin group]] in that dimension

$$
  SemiSpin(16)
  \;\subset\;
  E_8
$$

On the other hand, the [[special orthogonal group]] $SO(16)$ is _not_ a [[subgroup]] of $E_8$ (e.g. [McInnes 99a, p. 11](#McInnes99a)).

### In heterotic string theory

In [[heterotic string theory]] with [[gauge group]] the [[direct product group]] $E_8 \times E_8$ it is usually only this [[subgroup]] $Semispin(16) \times SemiSpin(16)$ which is considered (but typically denoted $Spin(16)/\mathbb{Z}_2$, see also [Distler-Sharpe 10, Sec. 1](parameterized+WZW+model#DistlerSharpe10)).

## Related entries

* [[spin group]]

* [[Spin(n1).Spin(n2)]]

* [[heterotic string theory]], [[Green-Schwarz anomaly cancellation]]

[[!include low dimensional rotation groups -- table]]

## References

* {#McInnes99a} [[Brett McInnes]], _The Semispin Groups in String Theory_, J. Math. Phys. 40:4699-4712, 1999 ([arXiv:hep-th/9906059](https://arxiv.org/abs/hep-th/9906059))

* {#McInnes99b} [[Brett McInnes]], _Gauge Spinors and String Duality_,  Nucl. Phys. B577:439-460, 2000 ([arXiv:hep-th/9910100](https://arxiv.org/abs/hep-th/9910100))

[[!redirects SemiSpin16]]


SemiSpin(32)

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#Contents#
* table of contents
{:toc}

## Idea

The [[semi-spin group]] in [[dimension]] 32.

## Properties

### In heterotic string theory

In [[heterotic string theory]] precisely two ([[isomorphism classes]] of) [[gauge groups]] are consistent (give [[quantum anomaly cancellation]]): one is the [[direct product group]] $E_8 \times E_8$ of the [[exceptional Lie group]] [[E8]] with itself, the other is in fact the [[semi-spin group]] $SemiSpin(32)$ (see [McInnes 99a, p. 5](#McInnes99a)).

Beware  that the [[string theory]] literature often writes this as [[Spin(32)]]$/\mathbb{Z}_2$, which is at best ambiguous and misleading, or even as [[SO(32)]], which is wrong. Of course this follows the general tradition in the [[physics]] literature to write identifications of [[Lie groups]] that are really only identifications of their [[Lie algebras]], see also ["SO(10)-GUT theory"](GUT#TheGUTKnownAsSO10).

## Related entries

* [[spin group]]

* [[Spin(n1).Spin(n2)]]

* [[heterotic string theory]], [[Green-Schwarz anomaly cancellation]]

[[!include low dimensional rotation groups -- table]]

## References

* {#McInnes99a} [[Brett McInnes]], _The Semispin Groups in String Theory_, J. Math. Phys. 40:4699-4712, 1999 ([arXiv:hep-th/9906059](https://arxiv.org/abs/hep-th/9906059))

* {#McInnes99b} [[Brett McInnes]], _Gauge Spinors and String Duality_,  Nucl. Phys. B577:439-460, 2000 ([arXiv:hep-th/9910100](https://arxiv.org/abs/hep-th/9910100))


semistable symmetric spectrum

[Schwede 12, part I, def. 3.14](#Schwede12)

## References

* {#Schwede12} [[Stefan Schwede]], _[[Symmetric spectra]]_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

[[!redirects semistable symmetric spectra]]