semi-spin group

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## Idea

What are called _half-spin groups_ or _semi-spin groups_ ([McInnes 99a](#McInnes99a), [McInnes 99b](#McInnes99b)) are [[quotient groups]] of [[spin groups]] $Spin(n)$ by a _non-standard_ [[Z/2]]-[[subgroup]]:

Generally, every [[spin group]] $Spin(n)$ is, essentially by definition, a $\mathbb{Z}/2$-[[group extension]] of the corresponding [[special orthogonal group]], so that the [[quotient group]] by the resulting canonical [[subgroup]] inclusion $\mathbb{Z}/2 \overset{\iota}{\hookrightarrow} Spin(n)$ recovers [[SO(n)]]

$$
  Spin(n)/_{\iota}(\mathbb{Z}/2) \simeq SO(n)
  \,.
$$

But in the special case that the [[dimension]] $n = 4k$ is a [[positive number|positive]] multiple of 4 distinct from 8 (i.e. $k \in \mathbb{N}_{\gt 0}, k \neq 2$), there is _another_ $\mathbb{Z}/2$-[[conjugacy class of subgroups]] $\mathbb{Z}/2 \overset{\iota_{s}}{\hookrightarrow} Spin(4k)$, which is distinct from the canonical $\iota$, and hence yields a [[quotient group]]

$$
  SemiSpin\big(4k \big)
  \;\coloneqq\;
  Spin\big(4k\big)/_{\iota_s} (\mathbb{Z}/2)
$$

which is distinct from (i.e. not [[isomorphism|isomorphic]] to) [[SO(n)]].

This is called the _semi-spin group_ or _half-spin_ group in that dimension.

## Examples

### SemiSpin(4)

The semi-spin group in [[dimension]] 4 is just the [[direct product group]] of [[SU(2)]] with [[SO(3)]]:

$$
  SemiSpin(4)
  \;\simeq\;
  SU(2) \times SO(3)
$$

### SemiSpin(8)
 {#SemiSpin8}

While also for [[Spin(8)]] it is the case that the [[center]] contains two copies of [[Z/2]], $Z\big( Spin(8)\big) \simeq  \mathbb{Z}_2 \times \mathbb{Z}_2$, in this case the existence of [[triality]] [[automorphisms]] actually makes these two copies behave identically, so that here the would-be semi-spin groups happens to coincide with [[SO(8)]] after all:

$$
  Spin(8)/_{\iota_s} \mathbb{Z}_2
  \;\simeq\;
  SO(8)
  \;\simeq\;
  Spin(8)/_{\iota} \mathbb{Z}_2
$$

(e.g. [McInnes 99a, p. 9](#McInnes99a))

### SemiSpin(16)

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]] [[SemiSpin(16)]]:

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

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]] with [[gauge group]] the [[direct product group]] $E_8 \times E_8$ it is typically 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)).

### SemiSpin(32)

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** (1999) 4699-4712 \[arXiv:hep-th/9906059, [doi:10.1063/1.532999](https://doi.org/10.1063/1.532999)\]

* {#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))

* Mboyo Esole, Monica Jinwoo Kang, _Flopping and Slicing: $SO(4)$ and $Spin(4)$-models_ ([arXiv:1802.04802](https://arxiv.org/abs/1802.04802))

[[!redirects semi-spin groups]]

[[!redirects semispin group]]
[[!redirects semispin groups]]

[[!redirects half-spin group]]
[[!redirects half-spin groups]]


semi-strict > history

< [[semi-strict]]
[[!redirects semi-strict -- history]]


semi-strict infinity-category

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## Idea

In [[higher category theory]] a notion of $\infty$-[[infinity-category|categories]] or $n$-[[n-category|categories]] is said to be **semi-strict**, if these higher categories are, somewhat vaguely, _as strict as possible while still being equivalent to general weak higher categories_ -- a kind of [[rectification]] statement.

For $n \leq 2$, even [[strict n-categories]] are semi-strict, but this does not hold for $n \gt 2$.

For $n \leq 3$ two alternative semi-strictifications are known:

1. **Gray-semistrictness**: horizontal composition is strict, but the [[exchange law]]s are nontrivial; see [[Gray-category]].

2. **Simpson-semistrictness**: everything except the [[unit law]]s hold strictly; see [[Simpson's conjecture]].

## Examples for strictification of horizontal composition

* Every [[quasi-category]] is equivalently modeled by a [[simplicially enriched category]], which is a model for an [[(∞,1)-category]] in which
all horizontal composition is strict. See [[relation between quasi-categories and simplicial categories]].

* A [[dg-category]] is an [[A-infinity-category]] in which horizontal composition is defined strictly. Every $A_\infty$-category is $A_\infty$-equivalent to a [[dg-category]]. This is at least roughly the [[stable (∞,1)-category]] analog of the above statement.

## Related concepts

* [[rectification]]

* [[rigidification of quasi-categories]]

* [[associative n-category]]

* _[[Globular]]_ proof assistant

## References

A review, some references and further discussion is at

* [_Semistrict $\infty$-categories and $\omega$-Semi-Categories_](http://golem.ph.utexas.edu/category/2008/10/semicategories.html)

[[!redirects semistrict]]
[[!redirects semi-strict]]
[[!redirects semistrict categories]]
[[!redirects semi-strict categories]]
[[!redirects semistrict n-categories]]
[[!redirects semi-strict n-categories]]
[[!redirects semistrict higher categories]]
[[!redirects semi-strict higher categories]]
[[!redirects semistrict ∞-categories]]
[[!redirects semi-strict ∞-categories]]
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[[!redirects semi-strict omega-categories]]
[[!redirects semistrict ∞-categories]]
[[!redirects semi-strict ∞-category]]
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semi-topological 4d Chern-Simons theory

[[!redirects semi-holomorphic 4d Chern-Simons theory]]

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## Idea

In [[mathematical physics|mathematical]]/[[theoretical physics]], _semi-topological_ or _semi-holomorphic 4d Chern-Simons theory_ is a variant of ordinary [[3d Chern-Simons theory]] where, roughly, one of the three [[coordinate functions]] is promoted from an ordinary real-valued coordinate to a [[holomorphic function|holomorphic coordinate]].

Hence semi-topological/holomorphic 4d Chern-Simons theory is a [[field theory]] defined on [[spacetimes]] of the form $\Sigma^2 \times C$ where $\Sigma^2$ is a real [[smooth manifold]], while $C$ is a [[complex curve]], and whose [[Lagrangian density]] is schematically the ordinary [[Chern-Simons form]], but with one of the wedge factors promoted from $d x^i$ to $d z$.

Accordingly, this [[field theory]] is a [[topological field theory]] (only) with respect to $\Sigma^2$, while it behaves like a [[conformal field theory]] with respect to $C$, whence "semi-topological/holomorphic" Chern-Simons theory: If one instead promotes all three coordinates of ordinary 3d Chern-Simons theory to holomorphic coordinates, one obtains full 6d [[holomorphic Chern-Simons theory]].

But beware that in the literature this is often referred to just as _[[4d Chern-Simons theory]]_.

## Properties

### Wilson lines and relation to integrable lattice models

Due to this particular property of the theory, [[Wilson line]] [[observables]] that form a network on $\Sigma^2$ but with each line located at a distinct point in the [[complex curve]] $C$ play a special role: Arranging them along the lines of a rectangular grid makes these [[observables]] behave much like those of a [[lattice model]] with their intersection points as [[vertices]], but the topological invariance of the theory with respect to $\Sigma^2$ implies certain symmetry operations on these observables, coming from invariance of the theory under moving Wilson lines across each other. These symmetries turn out to behave like [[R-matrices]] making the corresponding lattice model an [[integrable system]].

### Yangian symmetry

... [[Yangian]] ...

Consider the example when the spacetime is of the form $\mathbb{C}_w \times \mathbb{C}_z$ with subscripts to denote the respective coordinates. Pushing forward the factorization algebra for gives a locally constant factorization algebra on $\mathbb{C}_w$.

Applying Koszul duality gives a Hopf algebra because it is augmented as an $E_1$ algebra. This is a result of Tamarkin. In this case the Hopf algebra is the linear dual of the completed Yangian. This is seen by evaluating the classical limit and the first order bialgebra structure and then invoking a uniqueness result of Drinfeld.

## References

The idea of semi-topological 4d CS theory, and its relation to [[Yangians]] and [[integrable system|integrable]] [[lattice models]], is due to

* {#Costello13} [[Kevin Costello]], _Supersymmetric gauge theory and the Yangian_ ([arXiv:1303.2632](https://arxiv.org/abs/1303.2632))

* {#Witten16} [[Edward Witten]], _Integrable Lattice Models From Gauge Theory_, Adv.Theor.Math.Phys. 21 (2017) 1819-1843 ([arXiv:1611.00592](https://arxiv.org/abs/1611.00592))

* {#CostelloWittenYamazaki17} [[Kevin Costello]], [[Edward Witten]], [[Masahito Yamazaki]], _Gauge Theory and Integrability, I_, ICCM Not. 6, 46-119 (2018) ([arXiv:1709.09993](https://arxiv.org/abs/1709.09993))

* {#CostelloWittenYamazaki17} [[Kevin Costello]], [[Edward Witten]], [[Masahito Yamazaki]], _Gauge Theory and Integrability, II_, ICCM Not. 6, 120-146 (2018) ([arXiv:1802.01579](https://arxiv.org/abs/1802.01579))

* [[Kevin Costello]], [[Masahito Yamazaki]],  _Gauge Theory And Integrability, III_ ([arxiv:1908.02289](https://arxiv.org/abs/1908.02289))

* [[Meer Ashwinkumar]], _Integrable Lattice Models and Holography_, JHEP 02 (2021) 227 ([arXiv:2003.08931](https://arxiv.org/abs/2003.08931))

* [[Meer Ashwinkumar]], Jun-ichi Sakamoto, [[Masahito Yamazaki]], *Dualities and Discretizations of Integrable Quantum Field Theories from 4d Chern-Simons Theory* [[arXiv:2309.14412](https://arxiv.org/abs/2309.14412)]

Rigorous discussion using [[homotopy theory]] (see at [[homotopical AQFT]]):

* [[Marco Benini]], [[Alexander Schenkel]], Benoit Vicedo, _Homotopical analysis of 4d Chern-Simons theory and integrable field theories_ ([arXiv:2008.01829](https://arxiv.org/abs/2008.01829))

Lecture notes:

* Sylvain Lacroix, *4-dimensional Chern-Simons theory and integrable field theories* ([arXiv:2109.14278](https://arxiv.org/abs/2109.14278))

Discussion of realizations of semi-holomorphic 4d Chern-Simons theory as the [[worldvolume]] theory of [[intersecting brane|intersecting]] [[D-brane]]/[[NS5-brane]] models:

* [[Meer Ashwinkumar]], [[Meng-Chwan Tan]], Qin Zhao, _Branes and Categorifying Integrable Lattice Models_ ([arXiv:1806.02821](https://arxiv.org/abs/1806.02821))

and further relating to the [[quantum geometric Langlands correspondence]]:

* [[Meer Ashwinkumar]], [[Meng-Chwan Tan]], _Unifying Lattice Models, Links and Quantum Geometric Langlands via Branes in String Theory_ ([arXiv:1910.01134](https://arxiv.org/abs/1910.01134))

Relation to the [[Berkovits superstring]]:

* [[Nathan Berkovits]], Rodrigo S. Pitombo, *4D Chern-Simons and the pure spinor $AdS_5 \times S^5$ superstring* [[arXiv:2401.03976](https://arxiv.org/abs/2401.03976)]


semi-topological K-theory

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## Idea

_Semi-topological K-theory_ $K_*^{st}$ is a [[homology theory|homological invariant]] of complex [[noncommutative spaces]] that interpolates between [[algebraic K-theory]] and [[topological K-theory]]. It is defined by starting with the presheaf defined by [[nonconnective algebraic K-theory]], taking the associated [[etale topology|etale]] [[infinity-sheaf|sheaf]], making it $\mathbf{A}^1$-[[homotopy invariant]], and finally taking the topological realization of the resulting presheaf. One recovers [[topological K-theory]] from this by inverting the [[Bott element]].

Semi-topological K-theory of [[complex varieties]] is related to [[morphic cohomology]] in the same way that [[algebraic K-theory]] is related to [[motivic cohomology]] and [[topological K-theory]] is related to [[singular cohomology]]. In fact there is an analogue of the [[Atiyah-Hirzebruch spectral sequence]].

Some [examples](#Examples) suggest that semi-topological K-theory may be a better suited invariant for complex varieties than either algebraic or topological K-theory.

## Definition

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###### Definition
**(topological realization of presheaves)**.
Taking underlying [[topological spaces]] of complex points induces a functor of [[infinity-categories]]
  $$ |-| : Aff_C \longrightarrow Spc $$
from complex [[affine schemes]] to [[homotopy types]]. By [[left Kan extension]] it extends to a [[colimit]]-preserving functor
  $$ |-| : P(Aff_\mathbf{C}) \longrightarrow Spc $$
on the [[infinity-category of infinity-presheaves]].
This further extends to a [[colimit]]-preserving functor
  $$ |-| : P^{Spt}(Aff_\mathbf{C}) \longrightarrow Spt $$
on the [[infinity-category]] of [[infinity-presheaf|presheaves]] of [[spectra]], by taking the [[left Kan extension]] of the composite
  $$ P(Aff_\mathbf{C}) \stackrel{|-|}{\longrightarrow} Spc \stackrel{\Sigma^{\infty}}{\longrightarrow} Spt $$
along the canonical functor $P(Aff_\mathbf{C}) \to P^{Spt}(Aff_\mathbf{C})$ induced by taking [[suspension spectra]] objectwise.
=--

Let $T$ be a [[dg-category]] over $\mathbf{C}$. Let $K(T)$ denote the [[non-connective algebraic K-theory|nonconnective]] [[Waldhausen K-theory of a dg-category|Waldhausen K-theory]] of $T$. This defines a [[infinity-presheaf|presheaf]] of [[spectra]]
  $$ K(T \otimes_{\mathbf{C}} -) : Aff_C^{op} \longrightarrow Spt $$
which sends $Spec(A)$ to $K(T \otimes_{\mathbf{C}} A)$.

+-- {: .num_defn}
###### Definition
The **semi-topological K-theory** of $T$ is the [[spectrum]] defined by the formula
  $$ K^{st}(T) = |L^{A^1}(L^{et}(K(T \otimes_{\mathbf{C}} -)))| $$
where $L^{et}$ and $L^{A^1}$ are the [[etale topology|etale]] and $\mathbf{A}^1$-[[homotopy invariant]] [[localization of an (infinity,1)-category|localization]] endofunctors, respectively.
=--

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###### Definition
The **semi-topological K-theory** of a complex [[scheme]] $X$ is the [[spectrum]]
  $$ K^{st}(T) = K^{st}(Perf(X)) $$
where $Perf(X)$ is the [[dg-category]] of [[perfect complexes]] on $X$.
=--

## Properties

Let $T$ be a [[dg-category]] over $\mathbf{C}$ and $\mathcal{M}_T$ the [[derived moduli stack of objects in a dg-category|moduli space of pseudo-perfect dg-modules]] over $T$. This is a [[derived stack]], hence an [[infinity-stack]] on the [[infinity-category]] of [[simplicial commutative rings]]. Applying the topological realization (which extends to presheaves on [[simplicial commutative rings]] in the obvious way), one gets a spectrum $|\mathcal{M}_T|$.

+-- {: .num_prop}
###### Proposition
$|\mathcal{M}_T|$ is an [[infinite loop space]], and there is a canonical identification
  $$ \tilde{K}^{st}(T) \simeq |\mathcal{M}_T| $$
where the left hand side is a [[connective spectrum|connective]] version of semi-topological K-theory.
=--

See ([Blanc 13](#Blanc13), 4.3), ([Toen 10](#Toen10)), ([Kaledin 10](#Kaledin10), section 8).

## Conjectures

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###### Conjecture
**(lattice conjecture)**.
For every smooth proper [[dg-category]] $T$ over $\mathbf{C}$, the canonical morphism
  $$ ch^{top} \wedge_{\mathbf{S}} H \mathbf{C}
       : K^{top}(T) \wedge_{\mathbf{S}} H\mathbf{C}
       \longrightarrow HP(T) $$
induced by the noncommutative [[Chern character]] is an equivalence of [[spectra]].
=--

(move this to [[topological K-theory of a dg-category]]...)

## Examples
 {#Examples}

* On a [[point]], semi-topological K-theory coincides with [[topological K-theory]]:
  $$ K^{st}(\Spec(\mathbf{C})) \simeq K^{top}(\Spec(\mathbf{C})), $$
while the [[homotopy groups]] of [[algebraic K-theory]] are uncountable in degree $i \gt 0$.

* With finite [[coefficients]], semi-topological K-theory coincides with [[algebraic K-theory]]:
  $$ K_*^{st}(X, \mathbf{Z}/n) \simeq K_*^{alg}(X, \mathbf{Z}/n) $$
while the [[topological K-theory]] only sees the [[homotopy type]] of the variety and not the finer algebraic structure.

* Rationally, semi-topological K-theory contains information about the cycles on $X$, and conjecturally, the rational [[Hodge filtration]] on [[singular cohomology]].

* $K_0^{st}(X)$ coincides with the [[Grothendieck group]] of [[vector bundles]] up to [[algebraic equivalence]]. Rationally, an analogue of the [[Chern character]] map induces canonical isomorphisms
  $$ K_0^{st}(X, \mathbf{Q}) \stackrel{\sim}{\longrightarrow} CH^*_alg(X, \mathbf{Q}), $$
where on the right hand side is the group of [[algebraic cycles]] modulo [[algebraic equivalence]].

## References

A first definition of semi-topological K-theory for [[complex varieties]] was given in

* [[Eric M. Friedlander]] and [[Mark E. Walker]], _Semi-topological K-theory using function complexes_, Topology, 41(3):591–644, 2002.

An analogue of the [[Atiyah-Hirzebruch spectral sequence]], and computations of semi-topological K-theory for a large class of varieties, are in

* [[Eric M. Friedlander]], [[Christian Haesemeyer]], and [[Mark E. Walker]]. _Techniques, computations, and conjectures for semi-topological K-theory_, Math. Ann., 330(4):759–807, 2004, [web](http://www.math.uiuc.edu/K-theory/0621/).

A survey is

* [[Eric M. Friedlander]], [[Mark E. Walker]], _Semi-topological K-theory_, in [[Handbook of K-theory]], Springer, 2005, [web](http://www.math.illinois.edu/K-theory/handbook/).

Semi-topological K-theory for [[dg-categories]] was introduced by [[Bertrand Toen]] in lecture III of

* {#Toen10} [[Bertrand Toen]], _Saturated dg-categories_, lectures at Workshop on Homological Mirror Symmetry and Related Topics, January 2010, University of Miami, [notes](https://math.berkeley.edu/~auroux/frg/miami10-notes/).

Some discussion and interesting conjectures are in the last section of

* {#Kaledin10} [[D. Kaledin]], _Motivic structures in non-commutative geometry_, [arXiv:1003.3210](http://arxiv.org/abs/1003.3210).

A state of the art treatment via [[dg-categories]], from the point of view of [[derived noncommutative algebraic geometry]], is in

* {#Blanc13} [[Anthony Blanc]], _Topological K-theory of complex noncommutative spaces_, Ph.D. thesis, 2013, [arXiv:1211.7360](http://arxiv.org/abs/1211.7360).

[[!redirects semitopological K-theory]]

[[!redirects semi-topological K-theory of a dg-category]]
[[!redirects semitopological K-theory of a dg-category]]


semi-transitive coring

[[!redirects semitransitive coring]]
Let $k$ be a (commutative) [[field]] and $R$ a $k$-algebra.

An $R$-coring $C$ is semi-transitive if the category  $M^C_f$ of $C$-comodules which are finitely generated as modules over $R$

* $M^C_f$ is artinian and noetherian (has all objects of finite composition length), all its objects are projective as $R$-modules, and all its hom-spaces are finite dimensional over $k$
* the category of all $C$-comodules is Ind-generated by $M^C_f$ (Every $C$-comodule is a filtered colimit of $C$-comodules finitely generated as $R$-modules)

It appears that semi-transitive corings are those corings which can be reconstructed from fiber functors on categories which are artinian, noetherian and with finite dimensional hom-spaces.

* A. Bruguières, _Théorie tannakienne non commutative_, Comm. Algebra __22__, 5817–5860, 1994

* K. Szlachanyi, _Fiber functors, monoidal sites and Tannaka duality for bialgebroids_, [arxiv/0907.1578](http://arxiv.org/abs/0907.1578)


semiadditive (∞,1)-category

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## Idea

An [[(∞,1)-category]] is _$n$-semiadditive_ for $n \in \mathbb{N}$ if [[(∞,1)-colimits]] over certain [[diagrams]] coincide with their [[(∞,1)-limits]] in a canonical way (see [Hopkins-Lurie 14, def. 4.4.2](#HopkinsLurie14)). Here for $n=0$ the condition is that [[coproducts]] coincide with [[products]], hence that [[bilimits]] exist.

So an [[(∞,1)-category]] that happens to be an ordinary [[category]] is 0-semiadditive precisely if it is a [[semiadditive category]] in the traditional sense. Generally, an [[(∞,1)-category]] is 0-semiadditive precisely if its [[homotopy category of an (∞,1)-category|homotopy category]] is [[semiadditive category|semiadditive]] ([Hopkins-Lurie 14, remark 4.4.13](#HopkinsLurie14)).

## Examples

* For $R$ an [[A-∞ ring]] such that its connected component [[ring]] $\pi_0(A)$ is a [[vector space]] over the [[rational numbers]], then the [[(∞,1)-category of modules]] $A Mod$ is $n$-semiadditive for all $n \in \mathbb{N}$ ([Hopkins-Lurie 14, example 4.4.22](#HopkinsLurie14)).

## Related concepts

* [[additive (∞,1)-category]]

* [[stable (∞,1)-category]]

## References

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

[[!redirects semiadditive (∞,1)-categories]]

[[!redirects semiadditive (infinity,1)-category]]
[[!redirects semiadditive (infinity,1)-categories]]

[[!redirects 0-semiadditive (∞,1)-category]]
[[!redirects 0-semiadditive (∞,1)-categories]]
[[!redirects 0-semiadditive (infinity,1)-category]]
[[!redirects 0-semiadditive (infinity,1)-categories]]

[[!redirects 0-semiadditivity]]

[[!redirects 1-semiadditive (∞,1)-category]]
[[!redirects 1-semiadditive (∞,1)-categories]]
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[[!redirects 1-semiadditive (infinity,1)-categories]]


semiadditive dagger 2-poset

[[!redirects semiadditive dagger 2-posets]]

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## Definition ##

A **semiadditive dagger 2-poset** is a [[dagger 2-poset]] $C$ such that

* There exists an object $0:Ob(C)$ called the [[zero object]] such that for each object $A:Ob(C)$, there exists a morphism $Z_A:Hom(0, A)$ such that for each object $B:Ob(C)$ and morphism $R: Hom(A, B)$, $R \circ Z_A = Z_B$.

* For each object $A:Ob(C)$ and $B:Ob(C)$, there exists a object $A \oplus B:Ob(C)$ called the [[biproduct]] of $A$ and $B$, with morphisms $I_A: Hom(A,A \oplus B)$ and $I_B: Hom(B,A \oplus B)$, such that

  * For each object $A:Ob(C)$, $B:Ob(C)$, and $C:Ob(C)$ and morphism $R_A:Hom(A,C)$ and $R_B:Hom(B,C)$, there exist a morphism $R: Hom(A \oplus B ,C)$ such that $R \circ I_A = R_A$ and $R \circ I_B = R_B$

  * For each object $A:Ob(C)$, $B:Ob(C)$, $C:Ob(C)$, and $D:Ob(C)$, and morphism $R:Hom(A, B)$ and $S:Hom(C,D)$, there exists a morphism $R \oplus S:Hom(A \oplus B,C \oplus D)$ where $(R \oplus S)^\dagger = R^\dagger \oplus S^\dagger$.

## Examples ##

* The dagger 2-poset [[Rel]] of [[sets]] and [[relations]] is a semiadditive dagger 2-poset.

## See also ##

* [[dagger 2-poset]]
* [[semiadditive category]]
* [[semiadditive dagger category]]


semiadditive dagger category

[[!redirects semiadditive dagger categories]]

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## Definition ##
A **semiadditive dagger category** is a [[cocartesian monoidal dagger category]] $(C, \oplus, 0, i_A, i_B, 0_A)$ such that

* for all objects $A \in Ob(C)$, $i_A^\dagger \circ i_A = id_A$ and $i_A^\dagger \circ i_A = id_A$
* for all objects $A \in Ob(C)$ and $B \in Ob(C)$, $i_B^\dagger \circ i_A = 0_B \circ 0_A^\dagger$

In a semiadditive dagger category, the coproduct is called a **biproduct** and the initial object is called a **zero object**.

## Examples ##

* The dagger category [[Hilb]] of [[Hilbert spaces]] and [[continuous map|continuous]] [[linear maps]] is a semiadditive dagger category.

* The dagger category [[Rel]] of [[sets]] and [[relations]] is a semiadditive dagger category.

## See also ##

* [[dagger category]]
* [[monoidal dagger category]]
* [[cocartesian monoidal dagger category]]
* [[cartesian monoidal dagger category]]
* [[semiadditive dagger 2-poset]]

## References ##

* Martti Karvonen. Biproducts without pointedness ([abs:1801.06488](https://arxiv.org/abs/1801.06488))
* Chris Heunen and Martti Karvonen. Limits in dagger categories. Theory and Applications of Categories, 34(18):468–513, 2019.
* Chris Heunen, Andre Kornell. Axioms for the category of Hilbert spaces ([arXiv:2109.07418](https://arxiv.org/abs/2109.07418))


semialgebraic manifold

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## Idea

A _semi-algebraic set_ is much like a [[variety]] over the [[real numbers]], a [[subset]] of a [[Cartesian space]] $\mathbb{R}^n$ being the joint [[vanishing locus]] in  of a [[finite set]] of [[polynomials]] [[intersection|intersected]] with [[inequalities]].

A _semi-algebraic manifold_ is much like a corresponding [[scheme]], a space that is locally like a semi-algebraic set.

## Examples

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

The [[Fulton-MacPherson compactifications]] $FM_n\left( \mathbb{R}^d\right)$ of ordered [[configuration spaces of points]] are semi-algebraic manifolds ([Lambrechts-Volic 14, Prop. 5.2](#LambrechtsVolic14)).

Moreover, the canonical forgetful functions $FM_{n + k}\left( \mathbb{R}^d\right) \longrightarrow FM_{n}\left( \mathbb{R}^d\right)$ are semi-algebraic [[fiber bundles]] ([Lambrechts-Volic 14, Theorem 5.8](#LambrechtsVolic14)).

=--

## References

* J. Bochnak, M. Coste, and M.-F. Roy. Geometrie algebrique reelle, volume 12 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1987

A theory of [[differential forms]] on semi-algebraic manifolds was sketched in Appendix 8 of

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Deformations of algebras over operads and the Deligne conjecture. In Conference Moshe Flato 1999, Vol. I (Dijon) , volume 21 of Math. Phys. Stud., pages 255–307. Kluwer Acad. Publ., Dordrecht, 2000 ([arXiv:math/0001151](https://arxiv.org/abs/math/0001151))

Details are provided, the [[de Rham complex]] of semi-algebraic [[differential forms]] on a semi-algebraic manifold is constructed in and their [[rational homotopy theory|real homotopy theory]] is studied in

* Robert Hardt, [[Pascal Lambrechts]], Victor Tourtchine, [[Ismar Volić]], _Real Homotopy Theory of Semi-Algebraic Sets_, Algebr. Geom. Topol. 11 (2011) 2477-2545 ([arXiv:0806.0476](https://arxiv.org/abs/0806.0476))

Discussion of the example of [[Fulton-MacPherson compactifications]] of [[configuration spaces of points]] includes

* {#LambrechtsVolic14} [[Pascal Lambrechts]], Ismar Volic, section 5 of _Formality of the little N-disks operad_, Memoirs of the American Mathematical Society ; no. 1079, 2014  ([arXiv:0808.0457](https://arxiv.org/abs/0808.0457), [doi:10.1090/memo/1079](http://dx.doi.org/10.1090/memo/1079))

See also

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

[[!redirects semialgebraic manifolds]]

[[!redirects semi-algebraic manifold]]
[[!redirects semi-algebraic manifolds]]

[[!redirects semialgebraic set]]
[[!redirects semialgebraic sets]]

[[!redirects semi-algebraic set]]
[[!redirects semi-algebraic sets]]

[[!redirects semialgebraic differential form]]
[[!redirects semialgebraic differential forms]]

[[!redirects semi-algebraic differential form]]
[[!redirects semi-algebraic differential forms]]

[[!redirects semialgebraic de Rham complex]]
[[!redirects semialgebraic de Rham complexes]]

[[!redirects semi-algebraic de Rham complex]]
[[!redirects semi-algebraic de Rham complexes]]


semialgebraic set

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## Definition

Let $R$ be an [[order]]ed [[field]]. A **semialgebraic set** is a subset of a product space $R^n$ that is a Boolean combination of sets defined by polynomial inequalities:

$$\{(x_1, \ldots, x_n) \in R^n: p(x_1, \ldots, x_n) \geq 0\}$$

In other words, a semialgebraic set is a finite union of loci defined by systems of polynomial inequalities

$$p_1(x_1, \ldots, x_n) \geq 0, \ldots, p_k(x_1, \ldots, x_n) \geq 0, q_1(x_1, \ldots, x_n) = 0, \ldots, q_l(x_1, \ldots, x_n) = 0$$

## The Tarski-Seidenberg theorem
 {#TarskiSeidenbergTheorem}

The fundamental theorem for semialgebraic sets over [[real closed field]]s $R$ is as follows.

+-- {: .num_theorem}
######Theorem
**([[Tarski-Seidenberg theorem|Tarski-Seidenberg]])**
The image of a semialgebraic set in $R^{n+1}$ under the projection map
$$\pi: R^{n+1} \to R^n: (x_1, \ldots, x_{n+1}) \mapsto (x_1, \ldots, x_n)$$
is also a semi-algebraic set.
=--

This remarkable theorem has far-reaching consequences for the theory of ordered fields. For one, it is called a "quantifier elimination" result because it says that any first-order [[predicate]] in the theory of ordered fields (with [[signature]] $(0, 1, +, -, \cdot, \lt)$) is equivalent to a predicate which is [[quantifier]]-free. This follows from a straightforward induction coupled with the observation that the Tarski--Seidenberg theorem directly says that if a predicate $R(x_1, \ldots, x_n, y)$ is a Boolean combination of atomic formulas (thus defining a semi-algebraic set), then

$$\{(x_1, \ldots, x_n): \exists_y R(x_1, \ldots, x_n, y)\}$$

is also definable by a Boolean combination of atomic formulas; hence the existential quantifier can be eliminated.

From here, it may be shown that the theory of real closed fields is *decidable*: that each sentence in the theory is provably true or provably false. In fact, a real closed field is elementarily equivalent to the ordered field of real numbers, and so the theory of the real numbers (as ordered field) is decidable. As special cases, we have that

* Euclidean geometry is decidable.

* Differential calculus over the real numbers is decidable.

The collection of semialgebraic sets is the archetypal example of an [[o-minimal structure]].

## Semialgebraic relations

A **semialgebraic relation** of arity $m \to n$ is an ordered triple $(m, n, R)$ where $R$ is a semialgebraic subset of $R^{m+n}$.

The composite of two semialgebraic relations $R: m \to n$, $S: n \to p$ is the triple $(m, p, S \circ R)$ where

$$S \circ R = \{(x, z) \in R^m \times R^p: \exists_{y \in R^n} R(x, y) \wedge S(y, z)\};$$

here $S \circ R$ is semialgebraic by the Tarski-Seidenberg theorem. In brief, semialgebraic relations form a [[cartesian bicategory]] of relations.

[[!redirects semi-algebraic set]]


semicartesian monoidal category

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## Definition

A [[monoidal category]] is **semicartesian** if the [[unit object|unit]] for the [[tensor product]] is a [[terminal object]]. This is a weaker version of the concept of *[[cartesian monoidal category]]* (where moreover the teson product is required to be the [[Cartesian product]]).

[[formal duality|Dually]], a monoidal category is **semicocartesian** if the unit for the tensor product is an [[initial object]]. This is a weaker version of the concept of *[[cocartesian monoidal category]]* (where in addition the tensor product is required to be the [[coproduct]].)

Many semicartesian monoidal categories are also [[symmetric monoidal category|symmetric]], and sometimes that is included in the definition.

## Examples

Some examples of semicartesian monoidal categories that are not cartesian include the following.

* The category of [[Poisson manifold]]s with the usual product of Poisson manifolds as its tensor product.

* The opposite of the category of [[associative algebra]]s over a given [[base field]] $k$ with its usual tensor product $A \otimes B$.

* The category of [[measurable locales]], equivalently, the opposite category of [[commutative von Neumann algebras]], equipped with the [[spatial tensor product]], which corresponds to the measure-theoretic product of measurable spaces (but not the categorical product).

* The category [[Cat]] with the non-cartesian [[funny tensor product]].

* The category of [[strict 2-categories]] with the [[Gray tensor product]], and the category of [[strict omega-categories]] with the [[Crans-Gray tensor product]].

* The category of [[affine space|affine spaces]] made into a [[closed monoidal category]] where the [[internal hom]] has $hom(x,y)$ being the set of affine linear maps from $x$ to $y$, made into an affine space via pointwise operations.

* The category of [[convex space|convex spaces]], also known as 'barycentric algebras', made into a closed monoidal category where the internal hom has $hom(x,y)$ being the set of convex linear maps from $x$ to $y$, made into an barycentric algebra via pointwise operations.

* The thin category associated to the linear order $([0, \infty], \ge)$ of extended nonnegative real numbers with addition as the tensor product and [[internal hom]] as truncated subtraction.

* If $(M, \otimes, I)$ is any monoidal category, $I$ being the monoidal unit, the [[slice category]] $M/I$ inherits a monoidal product given by
$$(X \stackrel{f}{\to} I) \otimes (Y \stackrel{g}{\to} I) = (X \otimes Y \stackrel{f \otimes g}{\to} I \otimes I \cong I)$$
where the isomorphism displayed is the canonical one. This monoidal product is semicartesian. The forgetful functor $\Sigma: M/I \to M$ is strong monoidal, and is universal in the sense of exhibiting the fact that semicartesian monoidal functors and strong monoidal functors form a coreflective sub-bicategory of the bicategory of monoidal categories and strong monoidal functors. (Check this.)

* The category of [[nominal sets]] is cartesian closed but also semi-cartesian closed if taken with the separated product, see Chapter 3.4 of Pitts' monograph Nominal Sets.

## Internal logic

The [[internal logic]] of a (symmetric) semicartesian monoidal category is [[affine logic]], which is like [[linear logic]] but permits the [[weakening rule]] (and also the [[exchange rule]], if the monoidal structure is symmetric).

## Properties

### Semicartesian vs. cartesian

In a semicartesian monoidal category, any tensor product of objects $x \otimes y$ comes equipped with morphisms
$$ p_x : x \otimes y  \to x $$
$$ p_y : x \otimes y \to y$$
given by
$$ x \otimes y \stackrel{1 \otimes e_y}{\longrightarrow} x \otimes I \stackrel{r_x}{\longrightarrow} x $$
and
$$ x \otimes y \stackrel{e_x \otimes 1}{\longrightarrow} I \otimes y \stackrel{\ell_y}{\longrightarrow} y $$
respectively, where $e$ stands for the unique morphism to the terminal object and $r$, $\ell$ are the right and left unitors.  We can thus ask whether $p_x$ and $p_y$ make $x \otimes y$ into the [[product]] of $x$ and $y$.  If so, it is a theorem that $C$ is a cartesian monoidal category.  (This theorem has been observed by Eilenberg and Kelly ([1966](#EilenbergKelly66), p.551), but they may not have been the first to note it.)

Alternatively, suppose that $(C, \otimes, I)$ is a **symmetric** monoidal category equipped with [[monoidal natural transformations]] $e_x : x \to I$ and $\Delta_x: x \to x \otimes x$ such that
$$ x \stackrel{\Delta_x}{\longrightarrow} x \otimes x \stackrel{e_x \otimes 1}{\longrightarrow} I \otimes x \stackrel{\ell_x}{\longrightarrow} x $$
and
$$ x \stackrel{\Delta_x}{\longrightarrow} x \otimes x \stackrel{1 \otimes e_x}{\longrightarrow} x \otimes I \stackrel{r_x}{\longrightarrow} x $$
are identity morphisms.  Then $(C, \otimes, I)$ is a cartesian monoidal category (see [this MO question](#dremodaris) for discussion of one of the technical details).

So, suppose $(C, \otimes, 1)$ is a semicartesian symmetric monoidal category.  The unique map $e_x : x \to I$ is a [[monoidal natural transformation]].  Thus, if there exists a monoidal natural transformation $\Delta_x: x \to x \otimes x$ obeying the above two conditions, $(C, \otimes, 1)$ is cartesian.  The converse is also true.

The characterization of cartesian monoidal categories in terms of $e$ and $\Delta$ is mentioned on page 47 of the slides at:

* John Baez, _Universal algebra and diagrammatic reasoning_, 2006. [link](http://math.ucr.edu/home/baez/universal/)

However, the characterisation was certainly known earlier: for instance, it appears in the thesis of Kegelmann, in which it is called [[folklore]].

### Definition in terms of projections

For a monoidal category to be semicartesian it suffices that it admit a family of "projection morphisms".  Specifically, suppose $C$ is a monoidal category together with natural "projection" transformations $\pi^1_{X,Y}:X\otimes Y \to X$ such that $\pi^1_{I,I}:I\otimes I\to I$ is the unitor isomorphism.  Then the composites $Y \cong I\otimes Y \to I$ form a cone under the identity functor with vertex $I$ whose component at $I$ is the identity; hence $I$ is a [[terminal object]] and so $C$ is semicartesian.

However, it doesn't follow from this that the given projections $\pi^1_{X,Y}$ are the same as those derivable from semicartesianness!  For that one needs extra axioms; see [this cafe discussion](https://golem.ph.utexas.edu/category/2016/08/monoidal_categories_with_proje.html#c056710) for details.

### Colax functors

It is well-known that any functor between cartesian monoidal categories is automatically and uniquely [[colax monoidal functor|colax monoidal]]; the colax structure maps are the comparison maps $F(x\times y) \to F x \times F y$ for the cartesian product.  (This also follows from abstract nonsense given that the [[2-monad]] for cartesian monoidal categories is [[colax-idempotent 2-monad|colax-idempotent]].)  An inspection of the proof reveals that this property only requires the domain category to be semicartesian monoidal, although the codomain must still be cartesian.

### Semicartesian operads

The notion of [[semicartesian operad]] is a type of [[generalized multicategory]] which corresponds to semicartesian monoidal categories in the same way that [[operads]] correspond to (perhaps symmetric) monoidal categories and [[Lawvere theories]] correspond to cartesian monoidal categories.  Applications of semicartesian operads include:

* [Generalized operads in classical algebraic topology](http://golem.ph.utexas.edu/category/2009/10/generalized_operads_in_classic.html) (blog post) -- this also uses the above fact about colax functors

* [Characterizing finite measure spaces](http://golem.ph.utexas.edu/category/2011/05/an_operadic_introduction_to_en.html#c038131) (blog comment)

## Related pages

* [[cartesian monoidal category]]

* A [[relevance monoidal category]] is the "dual" of a semicartesian monoidal category, with diagonals but not projections.

## Reference

* {#EilenbergKelly66} [[Samuel Eilenberg]], [[G. Max Kelly]], *Closed Categories*, pp. 421-562 in:  [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

* [[Malte Gerhold]], [[Stephanie Lachs]], [[Michael Schürmann]], _Categorial Lévy Processes_, [arXiv:1612.05139](https://arxiv.org/abs/1612.05139).

* [[Tom Leinster]], [Monoidal Categories with Projections](https://golem.ph.utexas.edu/category/2016/08/monoidal_categories_with_proje.html) (blog discussion)

* {#dremodaris} MO user 'dremodaris', _A semicartesian monoidal category with diagonals is cartesian: proof?_ [MO348480](https://mathoverflow.net/q/348480/)

* Mathias Kegelmann, _Continuous domains in logical form_, Electronic Notes in Theoretical Computer Science 49 (2002): 1-166.

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semicategory

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## Idea

The notion of _semicategory_ or _non-unital category_ is like that of _[[category]]_ but omitting the requirement of [[identity]]-morphisms.

This generalizes the notions of [[semigroup]], [[semiring]], etc:

* a [[semigroup]] is (the [[hom-set]] of) a semicategory with a single object;

* a [[semiring]] is (the [[hom-set]] of) a semicategory [[enriched category|enriched]] in [[Ab]] with a single object.

Semicategories, like categories, appear as [[semipresheaf|semipresheaves]] on the category with two objects and two morphisms.

## Definition

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

A (small) **semicategory** or **non-unital category** $\mathcal{C}$ consists of

* a [[set]] $\mathcal{C}_0$ of _[[objects]]_;

* a set $\mathcal{C}_1$ of _[[morphisms]]_ (or _arrows_);

* two [[functions]] $s, t : \mathcal{C}_1 \to \mathcal{C}_0$ called _source_ (or _domain_) and _target_ (or _codomain_);

  * one writes $f : x \to y$ if $s(f) = x$ and $t(f) = y$;

* a [[function]] $\circ \colon \mathcal{C}_1 \times_{t,s} \mathcal{C}_1 \to \mathcal{C}_1$ ([[composition]]) from the set of pairs of morphisms such that the target of the first is the source of the second;

such that the following properties are satisfied:

* source and target are respected by composition: $s(g \circ f) = s(f)$ and $t(g\circ f) = t(g)$;

* composition is _[[associativity|associative]]_: $(h \circ g)\circ f = h\circ (g \circ f)$ whenever $t(f) = s(g)$ and $t(g) = s(h)$.

=--

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

If one added to this definition the existence of a function $i \colon C_0 \to C_1$ such that for all $c \in C_0$ the morphism $i(c)$ is an [[identity]] on $c$ under the given [[composition]], then one has the defintion of a [[category]].

However, having [[identities]] is just an extra [[property]] on a semi-category, not extra [[structure]]. For more on this see below at _[Relation to categories](#RelationToCategories)_.

=--

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

One often writes $hom(x,y)$, $hom_C(x,y)$, or $C(x,y)$ for the collection of morphisms $f : x \to y$; the latter two have the advantage of making clear which category is being discussed.  People also often write $x \in C$ instead of $x \in C_0$ as a short way to indicate that $x$ is an object of $C$.  Also, some people write $Ob(C)$ and $Mor(C)$ instead of $C_0$ and $C_1$.

=--

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

For $\mathcal{C}, \mathcal{D}$ two semicategories, a [[semi-functor]] $F \colon \mathcal{C} \to \mathcal{D}$ is a pair of [[functions]] $F_0 \colon \mathcal{C}_0 \to \mathcal{D}_0$, $F_1 \colon \mathcal{C}_1 \to \mathcal{D}_1$ that respects all the given [[structure]] in the obvious way.

Write $SemiCat$ for the ([[large category|large]]) [[category]] whose objects are semicategories, and whose morphisms are semifunctors.

=--

## Properties

### Relation to categories
 {#RelationToCategories}

We discuss the relation of semicategories to [[categories]]. (See for instance the beginning of ([Harpaz](#Harpaz)) for a quick review of basics, with an eye towards their generalization to the relation between [[complete Segal spaces]] and [[complete semi-Segal spaces]].)

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

There is an evident [[forgetful functor]]

$$
  U \colon Cat \to SemiCat
$$

from the [[category]] [[Cat]] of [[categories]] to that of semicategories, def. \ref{SemiFunctor}, given simply by forgetting the identity-assigning map $i \colon \mathcal{C}_0 \to \mathcal{C}_1$ in a category.

=--

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

For $\mathcal{C}$ a semi-category, def. \ref{SemiCategory}, write

$$
  Id(\mathcal{C}_1) \hookrightarrow \mathcal{C}_1
$$

for the [[subset]] on those morphisms which are [[endomorphisms]] on some object $x \in \mathcal{C}_0$ and such that they are neutral elements with respect to composition in  $\mathcal{C}$.
=--

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

A semicategory is the semicategory underlying a category, hence is in the image of the functor $U$ of def. \ref{ForgetfulFromCatToSemicat}, precisely if every object has a neutral [[endomorphism]], hence precisely if the composite diagonal function in

$$
  \array{
    Id(\mathcal{C}_1) &\hookrightarrow& \mathcal{C}_1
    \\
    & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{s}}
    \\
    && \mathcal{C}_0
  }
$$

is an [[isomorphism]], where the horizontal function is that of def. \ref{SetNeutralElementsInEndoSemiMonoids}.

Moreover, if a semicategory lifts to a category, it does so in a unique way: the functor $U \colon Cat \to SemiCat$ is an [[injection]] on [[isomorphism classes]].

=--

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

Equivalently one could use the target map instead of the source map in the formulation of prop. \ref{CategoriesAreSemicategoriesWithUnits}.

=--

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

The diagram appearing in prop. \ref{CategoriesAreSemicategoriesWithUnits} is a simple version of the [[univalence]] condition appearing in definition of a [[complete semi-Segal space]], a [[category object in an (infinity,1)-category|semi-category object in an (infinity,1)-category]]. See there for more on this.

=--

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

The functor $U$ of def. \ref{ForgetfulFromCatToSemicat} has a [[left adjoint]], which [[free functor|freely]] adjoins identity morphisms to a semicategory in the obvious way.  It also has a [[right adjoint]], which sends a semicategory $S$ to the category whose objects are the [[idempotents]] of $S$ and whose morphisms are the morphisms of $S$ that commute suitably with them, as described at [[Karoubi envelope]].  Indeed, the [[monad]] on [[Cat]] generated by this latter [[adjunction]] is exactly the monad for _[[idempotent completion]]_, also called [[Cauchy completion]].  (Note, however, that this is not a [[2-monad]], because the right adjoint of $U$ is not a [[2-functor]].)

=--

### Transitive relations

A [[transitive relation]] is a semicategory [[enriched category|enriched]] on [[truth values]], or a semicategory $C$ where there is at most one morphism from every object $a$ to object $b$ in $C$

### Nerves and semi-simplicial sets

The [[nerve]] of a semicategory is a [[semi-simplicial set]] which satisfies the [[Segal conditions]].

## Examples

Start with the category of metric spaces and short maps. An occasionally useful semicategory can be formed from it by considering the nonempty spaces and strictly contractive functions.

This is a semicategory, since:

* the composition of two strictly contractive functions is strictly contractive
* identity maps are not contractive (they are trivial isometries)

The interest in this semicategory arises from the fact that all morphisms $f : A \to A$ have unique fixed points, by Banach's fixed point theorem.

## In higher category theory

The concept of semicategory has more or less evident analogs and generalizations in [[higher category theory]].

For models of higher categories by [[simplicial set]]s, i.e. presheaves on the [[simplex category]] (such as [[Kan complex]]es, [[quasi-categories]], [[weak complicial set]]s)  the corresponding semi-category notion is obtained by discarding the degeneracy maps (which are the identity-assigning maps in the simplicial framework), i.e. by considering just presheaves on the subcategory $\Delta_+ \subset \Delta$ on injective morphisms (see the discuss of $\Delta_+$ at [[Reedy model structure]] for more details).

Accordingly, there is the semi-category analog of a [[Segal space]], called a _[[semi-Segal space]]_.

[[Simpson's conjecture]] says that every $\infty$-category has a model where all [[composition]] operations are strict and only the [[unit law]]s hold up to [[coherent]] homotopies. This would mean that the $\infty$-semicategory underlying any $\infty$-category can always be chosen to be strict.

## Related concepts

* [[dagger semicategory]]

* [[semipresheaf]]

* [[regular semicategory]]

* [[non-unital ring]]

* [[semi-simplicial set]]

* [[semi-Segal space]]

[[!include oidification - table]]

## References

Semicategories were introduced in

* [[Barry Mitchell]], *The dominion of Isbell*, Transactions of the American Mathematical Society **167** (1972) 319-331 [[doi:10.2307/1996142](https://doi.org/10.2307/1996142)]

Enriched semicategory theory is developed in

* {#MBB02}M.-A. Moens, U. Bernani-Canani, [[Francis Borceux|F. Borceux]], _On regular presheaves and regular semi-categories_ , Cah. Top. Géom. Diff. Cat. **XLIII** no.3 (2002) pp.163-190. ([numdam](http://www.numdam.org/item?id=CTGDC_2002__43_3_163_0))

This is turned one notch further in

* {#Stubbe05}[[Isar Stubbe]], _Categorical structures enriched in a quantaloid : regular presheaves, regular semicategories_ , Cah. Top. Géom. Diff. Cat. **XLVI** no.2 (2005) pp.99-121. ([numdam](http://www.numdam.org/item/CTGDC_2005__46_2_99_0))

Semicategories and semigroups are mentioned in section 2 in

* W. Dale Garraway, _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))

Semicategories with an eye towards their generalization to [[semi-Segal spaces]] are briefly discussed at the beginning of

* {#Harpaz} [[Yonatan Harpaz]], _Quasi-unital $\infty$-Categories_ ([arXiv:1210.0212](http://arxiv.org/abs/1210.0212))

Structures obtained by further relaxing also the [[associativity]] law are discussed in

* Salvatore Tringali, _Plots and Their Applications - Part I: Foundations_  ([arXiv:1311.3524](http://arxiv.org/abs/1311.3524))

Topologically enriched semicategories are used for studying some aspects of concurrency theory in computer science. It is necessary to work with semicategories to have functorial definitions of the branching and merging homologies of a concurrent system. A starting point for reading the theory can be the paper

* {#Gaucher} [[Philippe Gaucher]], _Flows revisited: the model category structure and its left determinedness_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, vol LXI-2 (2020) ([published](http://cahierstgdc.com/wp-content/uploads/2020/04/GAUCHER-LXI-2.pdf), [arXiv:1806.08197](https://arxiv.org/abs/1806.08197))

Semi-categories, semi-adjunctions and semi-cartesian closed categories have been used to study the [[lambda calculus]] since

* Susumu Hayashi, _Adjunction of semifunctors: Categorical structures in nonextensional lambda calculus_, Theoretical Computer Science
Volume 41, 1985, Pages 95-104 [doi:10.1016/0304-3975(85)90062-3](https://doi.org/10.1016/0304-3975%2885%2990062-3)

[[!redirects semicategories]]

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

[[!redirects nonunital category]]
[[!redirects nonunital categories]]
[[!redirects non-unital category]]
[[!redirects non-unital categories]]

[[!redirects category without units]]
[[!redirects categories without units]]


semiclassical approximation

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## Overview

To some extent, [[quantum mechanics]] and [[quantum field theory]] are a [[deformation theory|deformation]] of [[classical mechanics]] and [[classical field theory]], with the deformation parameterized by [[Planck's constant]] $\hbar$. The _semiclassical approximation_ or _quasiclassical approximation_ to [[quantization]]/[[quantum mechanics]] is the restriction of this deformation to just first order (or some finite order) in $\hbar$.

[[!include classical-to-quantum notions - table]]

Applied to [[path integral]] [[quantization]], the semiclassical approximation is meant to approximate the [[path integral]]
$\int_{\phi \in \mathbf{Fields}} D\phi\; F(\phi) e^{iS(\phi)/\hbar}$ by an expansion in $\hbar$ about the [[critical points]] of the [[action functional]] $S$ (hence the solutions of the [[Euler-Lagrange equations]], hence to the classical trajectories of the system). As usual for the [[path integral]] in [[physics]], this often requires work to make precise, but at a heuristic level the idea is famous as the _[[rotating wave approximation]]_: the idea is that in regions of [[field (physics)|field]]-space where $S$ varies fast as measured in units of [[Planck's constant]], the [[complex number|complex phases]] of the integrand $\exp(i S / \hbar )$ tend to cancel each other in the integral so that substantial contributions to the integral come only from the vicininity of critical points of $S$ (classical [[trajectories]]).

But semiclassical approximations can be applied to most other formulations of [[quantum physics]], where they often lead to precise and powerful mathematical tools.

Notably in the [[Schrödinger picture]] of quantum evolution, solutions to the [[Schrödinger equation]] $i \hbar \frac{d}{d t} \psi = \hat H \psi$ (which characterizes [[quantum states]] given by [[wave functions]] $\psi$ for [[Hamiltonian mechanics|Hamiltonian dynamics]] induced by a [[Hamilton operator]] $\hat H$) are usefully considered to first (or any finite) order in $\hbar$.
This method, known after (some of) its inventors as the **[[WKB method]]** or similar, amounts to expressing the [[wave function]] in the form $\psi = exp(S)$ where $S$ is a slowly varying function and solving the equation for $S$.
Globally consistent such solutions to first order lead to what are called [[Bohr-Sommerfeld leaf|Bohr-Sommerfeld quantization conditions]].
For the formalization of this method in [[symplectic geometry]]/[[geometric quantization]] see at _[[semiclassical state]]_.

This [[WKB method]] makes sense for a more general class of
[[wave equations]]. For instance in [[wave mechanics|wave]] [[optics]] this yields the short-[[wavelength]] limit of the [[geometrical optics]] approximation. Here $S$ is called the **[[eikonal]]**.

Multidimensional generalization of the [[WKB method]] appear to be rather nontrivial; they have been pioneered by [[Victor Maslov]] who introduced a topological invariant to remove ambiguities of the naive version of the method, called the _[[Maslov index]]_.

## Related phenomena

### Equivariant localization

In some special cases (most often in the presence of supersymmetry) the main contribution (the first term in expansion) amounts to the true result; the quantum correction sometimes leads however to an overall scalar factor. This is the case of so-called localization (related directly in some cases to the equivariant localization in cohomology and Lefshetz-type fixed point formulas). Most of well known examples of integrable systems and TQFTs lead to localization.

### Large $N$-limit in gauge theories

The [[large N limit]] of gauge theories, which is of importance in [[collective field theory]] and in the study of relation between gauge and string theories is formally very similar to semiclassical expansion, where the role of Planck constant is played by $1/N^2$.

### In radiation theory

In the theory of radiation there is a different meaning of semiclassical treatment: one considers particles in a sorrounding electromagnetic field and the particles are treated as in finite-dimensional quantum mechanics, with the electromagnetic field as an external classical field coupled to the particles via an interaction term.

## Literature

* M.V. Fedoryuk, _Semi-classical approximation_, Springer [Online](http://eom.springer.de/S/s083990.htm) Enc. of Math.

* Sean Bates, Alan Weinstein, _Lectures on the geometry of quantization_, [pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)

* [[Victor Maslov]], _Stationary-phase method for Feynman's continual integral_, Theoret. and Math. Phys., 2:1 (1970), 21–25; Russian original: ТМФ, 2:1 (1970), 30–35 [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=tmf&paperid=3986&what=fullt&option_lang=rus).

* [[Victor Maslov]], _Theory of perturbations and asymptotic methods_ (Russian), Izdat. Moskov. Gos. Univ. 1965.

* [[Vladimir Arnold]], _Characteristic class entering in quantization conditions_, Funct. Anal. its Appl. 1967, 1:1, 1–13, [doi](http://dx.doi.org/10.1007/BF01075861) (В. И. Арнольд, "О характеристическом классе, входящем в условия квантования", Функц. анализ и его прил., 1:1 (1967), 1–14, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=2802&what=fullt&option_lang=rus))

* [[Victor Guillemin]], [[Shlomo Sternberg]], _Geometric asymptotics_, AMS 1977, [online](http://www.ams.org/online_bks/surv14);  _Semi-classical analysis_, 499 pages, [pdf](http://www-math.mit.edu/~vwg/semistart.pdf)

* A. S. Mishchenko, B. Yu. Sternin, V. E. Shatalov, _Lagrangian manifolds
and the canonical operator method_, Nauka, Moscow, 1978.
(in Russian). English transl.: _Lagrangian manifolds and the Maslov operator_, Springer, Berlin, 1990.

* [[Richard Szabo]], _Equivariant cohomology and localization of path integrals_, Lecture Notes in Physics, N.S. Monographs __63__. Springer  2000. xii+315 pp. (early version: _Equivariant localization of path integrals_, [hep-th/9608068](http://arxiv.org/abs/hep-th/9608068))

* [[Michael Atiyah]], _Circular symmetry and stationary phase approximation_, Asterisque __131__ (1985) 43--59

* [[Nicole Berline]], [[Ezra Getzler]],  [[Michèle Vergne]], _Heat kernels and Dirac operators_, Grundlehren __298__, Springer 1992, "Text Edition" 2003.

* [[Albert Schwarz]], Oleg Zaboronsky, _Supersymmetry and localization_, Comm. Math. Phys. __183__, 2 (1997), 463-476, [euclid](http://projecteuclid.org/euclid.cmp/1158328185)

* [[Albert Schwarz]], _Semiclassical approximation in [[Batalin-Vilkovisky quantization|Batalin-Vilkovisky formalism]]_, Comm. Math. Phys.  __158__ (1993), no. 2, 373--396, [euclid](http://projecteuclid.org/euclid.cmp/1104254246).
* A. Laptev, I.M. Sigal, _Global Fourier integral operators and semiclassical asymptotics_, Review of Math. Physics, __12__:5 (2000) 749--766 [pdf](http://www.math.kth.se/~laptev/Research/Papers/LSig.pdf)
* Maurice A. de Gosson, _Symplectic geometry, Wigner-Weyl-Moyal calculus, and quantum mechanics in phase space_, 385 pp. [pdf](http://opus.kobv.de/ubp/volltexte/2009/3021/pdf/2006_06.pdf)
* Semyon Dyatlov, _Semiclassical Lagrangian distributions_, [pdf](http://math.mit.edu/~dyatlov/files/2012/hlagrangians.pdf); _Hoermander–Kashiwara and Maslov indices_, [pdf](http://math.berkeley.edu/~dyatlov/files/2009/maslov.pdf)
* Shanzhong Sun, _Gutzwiller's semiclassical trace formula and Maslov-type index theory for symplectic paths_, [arxiv/1608.08294](https://arxiv.org/abs/1608.08294)

Borel summability may make sense of the semiclassical expansion to all orders; this approach is sometimes called exact WKB method:

* A. Voros, _The return of the quartic oscillator. The complex WKB method_, Annales de l'institut Henri Poincaré A39:3, 211-338 (1983) [euclid](http://eudml.org/doc/76217)

* Alexander Getmanenko, Dmitry Tamarkin, _Microlocal properties of sheaves and complex WKB_, [arxiv/1111.6325](http://arxiv.org/abs/1111.6325)

* Kohei Iwaki, Tomoki Nakanishi, _Exact WKB analysis and cluster algebras_, J. Phys. A 47 (2014) 474009 [arxiv/1401.7094](http://arxiv.org/abs/1401.7094); _Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras_, [arXiv:1409.4641](http://arxiv.org/abs/1409.4641)

Relation to quantum [[integrable system]]s is in a series of works of Vũ Ngọc, e.g.

* San Vũ Ngọc, _Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type_, Preprint Institut Fourier 433, 1998 15 [pdf](http://perso.univ-rennes1.fr/san.vu-ngoc/articles/focus-tout.pdf); _Quantum monodromy in integrable systems_,  Comm. Math. Phys. 203 (1999), no. 2, 465–479 [doi](http://dx.doi.org/10.1007/s002200050621)

For large N-limit compared to semiclassical expansion see

*  L. G. Yaffe, _Large N limits as classical mechanics_, Rev. Mod. Phys. __54__, 407–435 (1982), [pdf](http://rmp.aps.org/pdf/RMP/v54/i2/p407_1)

For the semiclassical method in [[superstring theory]] see

* J. Maldacena, G. Moore, N. Seiberg, D. Shih, _Exact vs. semiclassical target space of the minimal string_, [hep-th/0408039](http://arxiv.org/abs/hep-th/0408039)

* K. Hori, A. Iqbal, C. Vafa, _D-Branes and mirror symmetry_, [hep-th/0005247](http://arxiv.org/abs/hep-th/0005247)

[[!redirects quasiclassical approximation]]
[[!redirects quasi-classical approximation]]
[[!redirects semiclassical expansion]]
[[!redirects semiclassical analysis]][[!redirects semi-classical analysis]]

[[!redirects semiclassical quantization]]

[[!redirects semiclassical limit]]
[[!redirects semiclassical limits]]


semiclassical state

> under construction

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## Idea

In [[physics]], a _semiclassical state_ is the approximation to a [[quantum state]] in [[semiclassical approximation]].

In the original sense of the [[WKB approximation]], in the [[Schrödinger picture]] a semiclassical state is a [[wave function]] which solves the [[Schrödinger equation]] to first order in [[Planck's constant]] $\hbar$.

In the broader formalization of quantum physics in [[symplectic geometry]]/[[geometric quantization]] one finds that such WKB semiclassical states are formalized as being [[Lagrangian submanifolds]] of the given [[phase space]] [[symplectic manifold]] equipped with with a [[half-density]].

## Definition

We first give the traditional definition of semiclassical states according to the [[WKB method]] for a non-relativistic [[particle]] propagating on the [[Euclidean space]] $\mathbb{R}^n$ with its standard [[kinetic action]] and some arbitrary [[force]] potential

* [Of a non-relativistic particle in a potential](#TraditionalInSchrödingerPicture)

Then we discuss the formalization of this in the broader context of [[symplectic geometry]]/[[geometric quantization]] in

* [In symplectic geometry](#InSymplecticGeometry)

### Semiclassical state of the non-relativistic particle in a potential
 {#TraditionalInSchrödingerPicture}

Consider the [[physical system]] given by a non-relativistic [[particle]] of [[mass]] $m$ propagating on the [[Cartesian space]] $\mathbb{R}^n$ with standard [[kinetic action]] and sunbject to a [[force]] induced by a given potential [[smooth function]] $V \colon \mathbb{R}^n \to \mathbb{R}$.

#### As a wave function

The [[Hamilton operator]] for this system is the standard

$$
  \hat H \coloneqq
  -
  \frac{\hbar^2}{2 m} \Delta  + V
  \,,
$$

where

$$
  \Delta = \mathbf{d}^\dagger \mathbf{d}  =   \sum_{i = 1}^n \frac{\partial}{\partial x^i} \frac{\partial}{\partial x^i}
$$

is the [[Laplace operator]] on $\mathbb{R}^n$ regarded as a [[Riemannian manifold]] with its canonical flat [[metric]] ($\mathbf{d}$ is the [[de Rham differential]]).

Then for

$$
  \psi \colon \mathbb{R}^n \times \mathbb{R} \to \mathbb{C}
$$

a smooth 1-parameter collection of [[smooth functions]] (of [[wave functions]]), the [[Schrödinger equation]] is

$$
  i \hbar \frac{d}{ dt} \psi
  =
  \hat H \psi
  \,,
$$

where $\frac{d}{d t}$ is the [[differentiation]] with respect to the additional parameter ([[time]]).

We say that $\psi$ is a _stationary solution_ to the Schrödinger equation if it is a solution of the form

$$
  \psi(x, t) = \phi(x)\exp(- i \omega t)
$$

for some $\omega \in \mathbb{R}$. For the following it is useful to decompose the remaining [[complex number|complex]]-valued [[smooth function]]

$$
  \phi \colon \mathbb{R}^n \to \mathbb{C}
$$

into its modulus and phase by writing it as

$$
  \phi(x)  = \exp(i S(x)/\hbar) a(x)
$$

for two smooth functions $S \colon \mathbb{R}^n \to \mathbb{R}$ and $a \colon \mathbb{R}^n \to \mathbb{R}_{\geq 0}$.

In fact it is often useful (such as in the symplecto-geometric interpretation that we turn to [below](#InSymplecticGeometry)) to restrict attention to non-vanishing solutions (or else to solutions restricted to their [[support]]) in which case we can regard $\phi$ as a function of the form

$$
  \phi \colon \mathbb{R}^n \to \mathbb{C}^\times \simeq U(1) \times \mathbb{R}_{\gt 0}
$$

and then this decomposition is unique up to a global global offset of $S$ by $2\pi i \cdot n$ for  $n \in \mathbb{Z}$.

In terms of this decomposition the [[Schrödinger equation]] becomes

$$
  \begin{aligned}
    0
    &=
    \left(i \hbar \frac{d}{dt} - \hat H\right) \psi
    \\
    & =
    \left(
      \left(
        \frac{{\vert \mathbf{d} S \vert}^2 }{2 m }
        +
        (V - \hbar \omega)
      \right)
      -
      \frac{i \hbar}{ 2m a}
      \mathbf{d}^\dagger \left(a^2 \mathbf{d} S\right)
    \right)
    \exp( i S / \hbar ) a
    +
    \mathcal{O}(\hbar^2 )
  \end{aligned}
  \,,
$$

where $\mathbf{d} S$ is the [[gradient]] [[covector field]] of $S$, where $\mathbf{d}^\dagger ( a^2 \mathbf{d}S)$ is the [[divergence]] of $a ^2 \mathbf{d}S$, and where $\mathcal{O}(\hbar^2)$ denotes all further terms that are non-linear in $\hbar$.

This means that $\psi(-,t) = \exp(i S / \hbar) a \exp(- i \omega )$ is a **semiclassical stationary state** with [[energy]]

$$
  E \coloneqq \hbar \omega
$$

if the phase $S$ and the modulus $a$ satisfy the following two conditions:

1. The phase function $S$ satisfies the [[Hamilton-Jacobi equation]] or [[eikonal]] equation

   $$
     H(x, \nabla S(x)) =
     \frac{\vert \mathbf{d} S\vert^2}{2 m } + V =
     E
     \,,
   $$

1. The modulus $a$ is such that $a^2 \mathbf{d} S$ satisfies the homogeneous [[transport equation]] in that it is a [[divergence]]-free [[vector field]].

#### As a Lagrangian submanifold of phase space equipped with a half-density

The above characterization of semiclassical [[wave functions]]
of the non-relativistic particle in a potential
has a natural equivalent reformulation in terms of
[[symplectic geometry]]/[[geometric quantization]].

The [[phase space]] is

$$
  T^* \mathbb{R}^n \simeq \mathbb{R}^{2 n}
  \,.
$$

Into this space is canonically embedded as the 0-section:

$$
  0
  =
 (x \mapsto (x, p = 0))
  \;
  \colon
  \;
  \mathbb{R}^n
   \hookrightarrow
  T^* \mathbb{R}^n
$$

which is a [[Lagrangian submanifold]].

Now every phase function $S \colon \mathbb{R}^n \to \mathbb{R}$ as above induces a deformation of this by regarding the [[de Rham differential]] $\mathbf{d}S$ as a [[section]] of the [[cotangent bundle]]

$$
  \mathbf{d}S \colon \mathbb{R}^n \hookrightarrow T^* X
$$

(This is what related [[phase and phase space in physics]].)

This is again a [[Lagrangian submanifold]]. We write

$$
  \pi \colon im(\mathbf{d}S) \to \mathbb{R}^n
$$

for the restriction of the [[cotangent bundle]] projection to this Lagrangian submanifold.

The fact that $S$ satisfies the [[Hamilton-Jacobi equation]] means equivalently that this Lagrangian submanifold is the level-set of the [[Hamiltonian]] $H \colon \mathbb{R}^n \to \mathbb{R}$ at energy $E = \hbar \omega$

$$
  im(\mathbf{d}S) = H^{-1}(E)
  \,.
$$

For the interpretation of the modulus function $a$ in this reformulation, first notice that for $vol$ the canonical [[volume form]] on $\mathbb{R}^n$, the homogeneous transport equation

$$
  div(  a^2 \mathbf{d}S) = 0
$$

is equivalent to

$$
  \mathcal{L}_{\nabla S} ( a^2 vol ) = 0
$$

where on the left we have the [[Lie derivative]] along the [[gradient]] of $S$. Next observe that

$$
  \nabla S = \pi_* (v_H)|_{im(\mathbf{d}S)}
$$

where $v_{H}$ is the [[Hamiltonian vector field]] corresponding to $H$.

This means that the transport equation is equivalently

$$
  \mathcal{L}_{(v_H)} \pi^* (a^2 vol) = 0
  \,.
$$

Hence this says that $\pi^* a^2 vol$ is a [[volume form]] on $im(\mathbf{d}S)$ which is invariant with respect to the Hamiltonian flow of time evolution.

Finally, if instead of a [[volume form]] we choose a [[half-density]] $\sqrt{vol}$, then $a \sqrt{vol}$ is another half-density and the condition is that this be invariant under the Hamiltonian flow.

In summary then, the semiclassical wave fuction is equivalently

1. a [[Lagrangian submanifold]]

   * which is a level-set of the [[Hamiltonian]] at the given [[energy]]

1. such that $\mathbf{a} \coloneqq a \sqrt{vol}$ is a [[half-density]] on the Lagranian submaifold

   * which in addition is invariant under the Hamiltonian flow.

This formulation now suggests a more general definition of semiclassical states in [[symplectic geometry]]/[[geometric quantization]].

### In symplectic geometry / geometric quantum theory
 {#InSymplecticGeometry}

(...)

abstracting the above we have that

(...)

## Related concepts

[[!include classical-to-quantum notions - table]]

## References

An introduction to the formulation of semiclassical states in [[symplectic geometry]] is in the first section of

* Sean Bates, [[Alan Weinstein]], _Lectures on the geometry of quantization_ ([pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf))

[[!redirects semiclassical states]]


semiconductor

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#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]], a *semiconductor* is a [[crystal|crystalline]] material for which the [[electron]] [[chemical potential]] lies in a *small* gap between a [[valence band]] and a [[conduction band]].

This means that a semi-conductor behaves like an [[insulator]] unless a little extra energy is supplied to lift the electrons out of the [[valence bundle]] into the [[conduction bundle]], in which case it starts behaving like a [[conductor]] (or [[metal]]).

This "external switching" of electron conductivity in semi-conductors is the basis of [[transistors]] and hence of [[digital computers]].

## Related concepts

* [[semi-metal]]

* [[solid state physics]]

## References

Textbook accounts:

* [[Karlheinz Seeger]], *Semiconductor Physics*, Advanced texts in physics, Springer (2004) $[$[doi:10.1007/978-3-662-09855-4](https://doi.org/10.1007/978-3-662-09855-4)$]$

* [[Sheng San Li]] (ed.), *Semiconductor Physical Electronics*, Springer (2006)  $[$[doi:10.1007/0-387-37766-2_4](https://doi.org/10.1007/0-387-37766-2)$]$

See also:

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

[[!redirects semiconductors]]

[[!redirects semi-conductor]]
[[!redirects semi-conductors]]


semicontinuous map

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# Semicontinuous maps
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## Idea

Recall that a (say [[real number|real]]-valued) [[function]] $f$ is [[continuous map|continuous]] at a [[point]] $x$ if, roughly speaking, $f(x) \approx f(y)$ (meaning that $f(x)$ is close to $f(y)$) whenever $x \approx y$.  For a _lower_ semicontinuous map, we require only $f(x) \lesssim f(y)$ (meaning that $f(x)$ is close to or less than $f(y)$); for an _upper_ semicontinuous map, we require only $f(x) \gtrsim f(y)$.

## Definitions

Let $X$ be a [[topological space]], let $R$ be a [[linearly ordered set]], and let $f$ be a [[function]] from $X$ to $R$.

In [[nonstandard analysis]], the vague idea above becomes a precise definition, so long as we use the appropriate [[quantification|quantifiers]] for $x$ and $y$.
+-- {: .num_defn #nonstandard}
###### Definition

The function $f$ is __lower semicontinuous__ if, for each [[standard point]] $x$ of $X$ and each [[hyperpoint]] $y$ in the [[infinitesimal neighbourhood]] of $x$, $f(y)$ is either greater than or in the infinitesimal neighbourhood of $f(x)$.  Similarly, $f$ is __upper semicontinuous__ if, for each standard point $x$ of $X$ and each hyperpoint $y$ in the [[infinitesimal neighbourhood]] of $x$, $f(y)$ is either less than or in the infinitesimal neighbourhood of $f(x)$.
=--

In [[classical analysis]], we must phrase this another way:
+-- {: .num_defn #standard}
###### Definition

The function $f$ is __lower semicontinuous__ if, for each [[point]] $x$ of $X$ and each $a \lt f(x)$ in $R$, there is some [[neighbourhood]] $U$ of $x$ such that, for each $y \in U$, $f(y) \gt a$.  Similarly, $f$ is __upper semicontinuous__ if, for each point $x$ of $X$ and each $b \gt f(x)$ in $L$, there is some neighbourhood $U$ of $x$ such that, for each $y \in U$, $f(y) \lt b$.
=--

We can also refer to an appropriate [[topological structure]] on $R$:
+-- {: .num_defn #continuous}
###### Definition

The function $f$ is __lower semicontinuous__ if it is [[continuous function|continuous]] from $X$ to $R$ with the [[semicontinuous topology|lower semicontinuous topology]].  Similarly, $f$ is __upper semicontinuous__ if it is continuous from $X$ to $R$ with the [[semicontinuous topology|upper semicontinuous topology]].
=--
Of course, this doesn\'t really say anything if you don\'t know what those topologies on $R$ are, and the easiest way to figure that out is to refer to Definition \ref{standard} (or to Definition \ref{nonstandard} if you know enough nonstandard analysis to interpret it).

## Examples

A function is [[continuous function|continuous]] (with respect to the usual [[order topology]] on $R$) iff it is both upper and lower semicontinuous.

The [[characteristic function]] of a [[subset]] $A$ (either valued in the [[poset]] of [[truth values]] with its usual order or valued in the [[real numbers]] with $1$ for true and $0$ for false) is lower semicontinuous iff $A$ is [[open subset|open]], and upper semicontinuous iff $A$ is [[closed subset|closed]] (hence continuous iff $A$ is [[clopen subset|clopen]]).

## Semicontinuity for multi-valued maps

We define the __small__ and __large preimages__ of a subset $V \subset Y$ under a [[multi-valued function]] $F\colon X \to Y$ by
$$\begin{aligned}
  F^{*s}(V) &\coloneqq \{ x \in X \mid F(x) \subset V \} \;\text{and} \\
  F^{*l}(V) &\coloneqq \{ x \in X \mid F(x) \cap V \neq \emptyset \}.
\end{aligned}$$
A [[multi-valued function]] $F\colon X \to Y $ is said to be __upper semicontinuous__ if the small preimage of all open sets are open and is said to be __lower semicontinuous__ if the large preimages of all open sets are open. Both properties have also a point-wise variant. The map $F$ is __upper semicontinuous at $x$__ for some $ x \in X$ if for every open neighborhood $V$ of $F(x)$ there is a neighborhood $U$ of $x$ such that for all $x'\in U$ the set $F(x')$ is contained in $V$. Likewise, $F$ is __lower semicontinuous at $x$__ for some $ x \in X$ if for every open neighborhood $V$ intersecting $F(x)$ (i.e. $ V \cap F(x) \neq \emptyset $) there is a neighborhood $U$ of $x$ such that for all $x'\in U$ the set $F(x')$ intersects $V$.

## Generalizations

We need to say something about when $R$ is a [[dcpo]] or something like that (involving the [[Scott topology]]), as well as semicontinuity in [[constructive mathematics]] involving [[locales]].  Compare also the [[one-sided real numbers]].

## Related concepts

* [[selection theorem]]

## References

To read later:

* {#LiWang} Li Yong-ming and Wang Guo-jun, Localic Katětov–Tong insertion theorem and localic Tietze extension theorem, [pdf](http://www.kurims.kyoto-u.ac.jp/EMIS/journals/CMUC/pdf/cmuc9704/yong.pdf).

* {#GarciaPicado} Gutiérrez García and Jorge Picado, On the algebraic representation of semicontinuity, [doi](http://dx.doi.org/10.1016/j.jpaa.2006.09.004).

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semicontinuous topology

# Semicontinuous topologies
* table of contents
{: toc}

## Idea

The (lower or upper) semicontinuous topology is a [[topological structure|topology]] on the [[real line]] (or a generalization thereof) such that a [[continuous function]] (from some [[topological space]] $X$) to the real line equipped with this semicontinuous topology is the same thing as a (lower or upper) [[semicontinuous map]] from $X$ to the real line.

Thus one replaces discussion of semicontinuous maps with continuous maps by using a different topological structure.

## Definition

Let $R$ be any [[linear order]]; think of the [[real line]] with its usual order.  For each [[element]] $a$ of $R$, consider the [[subsets]]

$$ L_a \coloneqq \{ x \in R \;|\; x \gt a \} ,$$
$$ R_a \coloneqq \{ x \in R \;|\; x \lt a \} .$$

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

The __lower semicontinuous topology__ on $R$ is generated by the [[topological base|base]] (of open sets) given by the sets $L_a$; the __upper semicontinuous topology__ on $R$ is generated by the [[topological base|base]] (of open sets) given by the sets $R_a$.
=--

(more to come)

## References

To read later:

* Li Yong-ming and Wang Guo-jun, Localic Katětov–Tong insertion theorem and localic Tietze extension theorem, [pdf](http://www.kurims.kyoto-u.ac.jp/EMIS/journals/CMUC/pdf/cmuc9704/yong.pdf).
  {#LiWang}

* Gutiérrez García and Jorge Picado, On the algebraic representation of semicontinuity, [doi](http://dx.doi.org/10.1016/j.jpaa.2006.09.004).
  {#GarciaPicado}

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semidefinite element

# (Semi)definiteness
* table of contents
{: toc}

## Idea

There are many contexts where terms such as 'positive definite', 'negative semidefinite', and 'indefinite' appear.  This is an attempt to describe a general framework in which these may be used.  Warning: this may be [[centipede mathematics]].

## Definitions

Let $A$ be a [[set]] equipped with both a [[partial order]] $\leq$ (whose [[opposite relation|opposite]] is denoted $\geq$), a [[symmetric relation]] $#$, and a [[basepoint]] $0$.  (The notation $#$ is to put one in mind of [[inequality]], but it is nontrivial even in [[classical mathematics]], and it is rarely [[tight relation|tight]].  It\'s not even always [[irreflexive|irreflexive]], although that is very common.)

We write $\lt$ for the conjunction of $\leq$ and $#$ (and $\gt$ for its opposite, the conjunction of $\geq$ and $#$).

An element $x$ of $A$ is:

* __positive__ (or _positive semidefinite_) if $x \geq 0$,
* __negative__ (or _negative semidefinite_) if $x \leq 0$,
* __semidefinite__ if $x$ is positive or negative,
* __indefinite__ if $x$ is not semidefinite,
* __nonsingular__ (or _nondegenerate_) if $x # 0$,
* __definite__ if $x$ is semidefinite and nonsingular,
* __positive definite__ if $x \gt 0$ (that is if $x$ is both positive and nonsingular),
* __negative definite__ if $x \lt 0$ (that is if $x$ is both negative and nonsingular).

Note that a positive or negative element must be semidefinite, so 'positive semidefinite' and 'negative semidefinite' are redundant; in many contexts, however, this redundancy is useful, since 'positive' and 'negative' are so widely used in other contexts.  Also, since a semidefinite element is definite iff it\'s nonsingular, 'positive definite' and 'negative definite' really mean what they say.

In [[constructive mathematics]], it is not the partial order $\leq$ that is most relevant but rather the relation $\nleq$, which classically is the [[negation]] of $\leq$ but which constructively is generally stronger.  This relation $\nleq$ is required to satisfy properties [[de Morgan duality|dual]] to those of a partial order ([[irreflexive relation|irreflexivity]], [[comparison]], and [[connected relation|connectedness]]), and $\leq$ is then defined as the negation of $\nleq$ (and can be proved to be a partial order).  Constructively, we say that $x$ is __indefinite__ if $x \nleq 0$ and $x \ngeq 0$, and that $x$ is __semidefinite__ if it is not indefinite; these are equivalent to the above in [[classical logic]], and all of the other definitions read the same.  In particular, ‹semidefinite› is not ‹positive or negative› but rather the [[double negation]] of that.  (All that said, if one is willing to give up ‹indefinite›, then there is no need for $\nleq$, although ‹semidefinite› should still be a double negation.)

We will also say that $A$ (or rather, the structure $(A,\leq,#,0)$) is __nontrivial__ if $#$ is [[irreflexive|irreflexive]].

Suppose that $A$ has the structure of a [[group]], which we will write additively (but without assuming commutativity).  We will say that the group structure is _right-compatible_ with the other structure ($\leq$, $#$, and $0$) if

* $0$ is the [[identity element]] of the group,
* $x \leq y$ iff $y - x \geq 0$, and
* $x # y$ iff $y - x # 0$.

Thus $\leq$ and $#$ are entirely recoverable from the positive and nonsingular elements, respectively.  (Constructively, we require that $x \nleq y$ iff $y - x \nleq 0$.)  Note that in this case, $A$ is nontrivial iff $0 # 0$ is false.

We can similarly say when $A$ is _left-compatible_.  (By symmetry of $#$, this half of the compatibility condition is the same.)  Of course, if $A$ is [[abelian group|commutative]], then left- and right-compatibility are the same entirely.

## Examples

If $A$ is the set of [[real numbers]], then give $\leq$, $#$, and $0$ their usual meanings.  (The usual meaning of $#$ is [[apartness]], which classically is the same as [[denial inequality|ordinary inequality]].)  Then $\lt$ and $\gt$ also have their usual meanings.  (We also have that $\nleq$ is the same as $\gt$, and $\ngeq$ is the same as $\lt$.)  The positive-semidefinite elements are the nonnegative ones, but the positive definite elements are the (strictly) [[positive numbers]]; every real number is semidefinite, and none is indefinite.  The (commutative) additive group structure is compatible, and $A$ is nontrivial.

The previous example (except for the final sentence) can be used with any [[pointed set|pointed]] [[linear order]].  (Constructively, define $\nleq$ to be $\gt$ and go from there.)  This is always nontrivial.

If $A$ is the set of [[upper real number|upper]], [[lower real number|lower]], or [[MacNeille real number|MacNeille]] real numbers, then there is really nothing new classically; $\infty$ is positive definite, $-\infty$ is negative definite, and that is all.  Constructively, however, neither $\leq$ and $\lt$ (with their usual meanings in the relevant context) is the negation of the other.  Nevertheless, if we define $x # y$ to mean that $x \lt y$ or $x \gt y$, then everything goes through.  If we restrict to the bounded numbers, then the additive group structure is compatible too.

If $A$ is the set of [[complex numbers]], then let $x \leq y$ mean that $y - x$ is a nonnegative real, and give $#$ its usual meaning (apartness, which classically is ordinary inequality).  Then the positive-semidefinite elements are the nonnegative real numbers, the positive-definite elements are the strictly positive real numbers, the semidefinite elements are the real numbers, and the indefinite elements are the (not necessarily purely) [[imaginary numbers]].  The additive group structure is compatible, and this $A$ is nontrivial.

The previous example generalizes to any algebra of [[hypercomplex numbers]].  (Even the [[trivial algebra]] is nontrivial, for once.)

If $A$ is the set of sets of real or complex numbers (so a [[power set]]), then let $x \leq y$ mean that every element of $x$ is $\leq$ every element of $y$, $x # y$ mean that $x$ and $y$ are [[disjoint sets|disjoint]], and $0$ be the [[singleton subset|singleton]] $\{0\}$.  (Constructively, let $x \nleq y$ mean that some element of $x$ is $\nleq$ some element of $y$.)  Then $x \lt y$ mean that every element of $x$ is $\lt$ every element of $y$.  The positive-semidefinite elements are those sets whose members are all nonnegative reals, the positive-definite ones are those whose members are all strictly positive reals, the indefinite ones are those that have at least one imaginary member *or* at least one positive member and at least one negative member, and the nonsingular elements are those whose members are all nonzero.  (In particular, the [[empty subset|empty set]] is nondegenerate, for once.)  There is no compatible group structure, and this example is *not* nontrivial (since $\emptyset # \emptyset$).

The power-set construction above can be applied to any example to produce a new example.  This new example never has a compatible group structure and is never nontrivial.

If $A$ is any real‑ or complex-valued [[function set]], then let $x \leq y$ mean that $x_n \leq y_n$ for every $n$ in the [[domain]] of the functions, let $x # y$ mean that $x_n # y_n$ for every $n$, and let $0$ be the [[constant function]] with value zero.  (Constructively, let $x \nleq y$ if $x_n \nleq y_n$ for *some* $n$.)  Then $x \lt y$ means that $x_n \lt y_n$ for every $n$.  The positive semidefinite elements are those functions that take only nonnegative real values, the positive definite elements are those that take only strictly positive real values, the indefinite elements are those that take at least one imaginary value *or* at least one positive value and at least one negative value, and the nonsingular elements are those that take only nonzero values.  The pointwise-defined additive group structure is compatible, and $A$ is nontrivial iff the domain of the functions is [[inhabited set|inhabited]].  Every function has a [[range]], and a function is positive definite, etc, if and only if its range is (in the sense of the power-set example above).

The function-set construction above can be applied to any example to produce a new example.  If the original example had a compatible group structure, so does the new example; if the original example was nontrivial, then so is the new example iff the domain is inhabited.

If $A$ is a real‑ or complex-valued set of [[pointed function|pointed functions]], then let $x \leq y$ mean that $x_n \leq y_n$ for every $n$ in the [[domain]] of the functions, and let $0$ be the [[constant function]] with value zero, as before; but now let $x # y$ mean that $x_n # y_n$ for every $n$ *other than* the basepoint of the domain (because of course $x_n = y_n = 0$ when $n$ is the basepoint).  Then $x \lt y$ means that $x_n \lt y_n$ for every $n$ other than the basepoint.  (In [[constructive mathematics]], we often a strong notion of pointed function with a strong notion of being distinct from the basepoint, and then we can use that here.)  The positive semidefinite elements are those functions that take only nonnegative real values, as before, but now the positive definite elements are those that take only strictly positive real values *except* at the basepoint.  The indefinite elements are still those that take at least one imaginary value *or* at least one positive value and at least one negative value, but now the nonsingular elements are those that take only nonzero values except at the basepoint.  The pointwise-defined additive group structure is compatible, and $A$ is nontrivial iff the set of non-basepoints in the domain of the functions is [[inhabited set|inhabited]].  Every pointed function gives rise to a function on the set of non-basepoints (and classically, this information determines the pointed function completely), and a pointed function is positive definite, etc, if and only if its corresponding function of the non-basepoints is, in the sense of the previous paragraph.

If $A$ is the set of symmetric [[bilinear forms]] on some real [[vector space]] $V$ or the set of conjugate-symmetric [[sesquilinear forms]] on some complex vector space $V$, then 'semidefinite' etc have the meanings given at [[inner product space]].  (Constructively, we require $V$ to have a compatible [[apartness relation]], or else we cannot define 'indefinite'.)  The obvious additive group structure is compatible (which explains what $\leq$, $#$, and $0$ mean), and $A$ is nontrivial iff $V$ is nonzero.  If we replace a symmetric bilinear form with its corresponding [[quadratic form]], then this becomes a special case of the previous paragraph.

If $A$ is the set of $n$-by-$n$ real or complex [[matrices]] for some [[natural number]] $n$, then 'positive definite' and the rest have all of their usual meanings.  The additive group structure is compatible, and $A$ is nontrivial iff $n \gt 0$.  In [[classical mathematics]] (and more generally, assuming [[weak countable choice]]), every matrix gives rise to a set of complex numbers, its set of complex [[eigenvalues]], and then a matrix is positive definite, etc, exactly when its set of complex eigenvalues is, in the sense of the power-set example above.  (Without WCC, we can constructively form approximate sets of eigenvalues, and these still determine the definiteness properties, although this is a little more complicated.)

Generalizing the previous example, if $A$ is a real or complex $*$-[[star-algebra|algebra]] (with apartness, constructively, if we wish to define 'indefinite'), then all of the terms have their usual meanings there too.  (Note that $x \leq y$ means that $y - x = u^* u$ for some $u$, and $x # y$ means that $y - x$ is invertible.  Constructively, $x \nleq y$ means that $y - x$ is apart from $u^* u$ for all $u$.)  The additive group structure is compatible, and $A$ is nontrivial in our sense iff it is [[trivial algebra|nontrivial as a real algebra]].

From any example, we may form a sub-example by restricting to a [[subset]] of $A$, as long as $0$ belongs.  If the original example has a compatible group structure, then so does the subset if it is a [[subgroup]].  If the original example is nontrivial, then so is the subset.

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semidirect product

* [[semidirect product group]]

* [[semidirect product Lie algebra]]

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