suplattice

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## Definition

A **suplattice** is a [[poset]] that has [[joins]] of arbitrary [[subsets]] (and in particular is a join-[[semilattice]]).  By the [[adjoint functor theorem]] for posets, a suplattice necessarily has all [[meet|meets]] as well and so is a [[complete lattice]].  However, a **suplattice homomorphism** preserves joins, but not necessarily meets.  Furthermore, a *[[proper class|large]]* semilattice which has all *small* joins need not have all meets, but might still be considered a large suplattice (even though it may not even be a lattice).

Dually, an __inflattice__ is a poset which has all [[meets]], and an __inflattice homomorphism__ is a monotone function that preserves all meets.

A __[[frame]]__ (dual to a [[locale]]) is a suplattice in which finitary meets distribute over arbitrary joins.  (Frame homomorphisms preserve all joins and finitary meets.)

The [[category]] [[SupLat]] of suplattices and suplattice homomorphisms admits a [[tensor product]] which represents "[[bilinear maps]]", i.e. functions which preserve joins separately in each variable.  Under this tensor product, the category of suplattices is a [[star-autonomous category]] in which the [[dualizing object]] is the suplattice dual to the object $TV$ of [[truth-values]].  A [[semigroup object|semigroup]] in this [[monoidal category]] is a __[[quantale]]__, including [[frames]] as a special case when the quantale is idempotent and unital. Modules over them are [[modules over quantales]] (quantic modules with special case of localic modules, used in the localic analogue of the Grothendieck's descent theory in [Joyal-Tierney 84](#JoyalTierney84)).

## The free suplattice on a poset

There is a [[forgetful functor]]

$$
U \colon SupLat \to Poset
$$

This has a [[left adjoint]]

$$
F \colon Poset \to SupLat
$$

where for any poset $P$, the suplattice $F(P)$ is the poset of 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 $\{s \in P : \; s \le p \}$.   (To give a downset is to give an [[antichain]], and so the free suplattice is sometimes described equivalently in terms of antichains.)

To understand this description of the free suplattice on a poset, some [[enriched category theory]] is useful.  [[preorder|Preorders]] are the same as $Bool$-[[enriched categories]], where $Bool$ is the [[monoidal category]] with two objects $F$, $T$ and one nontrivial morphism $F \implies T$, its monoidal structure being "and".   Using this idea, the downsets of a poset $P$ correspond in a one-to-one way with $Bool$-[[enriched functors]] $f \colon P^{op} \to Bool$, just as [[presheaves]] on a category $C$ are functors $f \colon C^{op} \to Set$.   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 is just the same as a suplattice.

## The category of suplattices

The category of suplattices is [[monadic adjunction|monadic]] over the category of posets, and each algebra structure $\xi: \hat{P} \to P$ is left adjoint to the [[Yoneda embedding]] $y: P \to \hat{P}$. This makes suplattices the same thing (up to equivalence) as [[total categories]] in the $Bool$-[[enriched category theory|enriched]] sense. Notice that algebra structure maps, being left adjoints, are cocontinuous and therefore suplattice morphisms. This makes the monad a [[commutative monad]], and therefore according to general theory, $SupLat$ is a [[symmetric monoidal closed category]] where the internal hom $Hom(P, Q)$ between two suplattices is the suplattice of cocontinuous maps $P \to Q$, which are the same as left adjoints $P \to Q$ according to the poset version of the [[adjoint functor theorem]].

$SupLat$ is also monadic over $Set$, where the monad $P: Set \to Set$ is the covariant [[power set]] functor. It therefore is a complete and cocomplete [[Barr-exact category]]. The [[Kleisli category]] of this monad is equivalent to [[Rel]], the category of sets and relations. Thus the objects of $Rel$ may be thought of as the free suplattices.

As stated above, the [[symmetric monoidal closed category]] $SupLat$ is a [[star-autonomous category]] where the star-[[involution]] takes a suplattice $P$ to the [[opposite category|opposite]] poset $P^{op}$. In part this says that a suplattice is also an [[inflattice]], a fact which holds internally in any [[topos]] (where we use an internal covariant power-object functor to form an appropriate monad). Thus the tensor product $P \otimes Q$ may be formed as the suplattice $Hom(P, Q^{op})^{op}$. The presence of the equivalence

$$\ast = (-)^{op}: SupLat^{op} \to SupLat$$

(which takes a morphism $f: P \to Q$ to $g^{op}: Q^{op} \to P^{op}$, where $g$ is right adjoint to $f$) also means that colimits may be formed as appropriate limits, which are in turn formed pointwise by monadicity over $Set$.

## References

* {#JoyalTierney84} [[André Joyal]], [[Miles Tierney]], _An extension of the Galois theory of Grothendieck_, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp. ([ISBN: 978-1-4704-0722-3](https://bookstore.ams.org/memo-51-309/))

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supply in a monoidal category

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## Idea

Consider a type $P$ of [[algebra|algebraic]] [[structure]] (such as [[monoid]]-structure, [[comonoid]]-structure, etc.)
that may be put on an [[object]] of a [[symmetric monoidal category]].  We say that a symmetric monoidal category $C$ *supplies* $P$ if every object of $C$ can be equipped with such $P$-structure, compatibly with the given [[tensor product]].  Importantly, the morphisms of $C$ are *not* all required to be $P$-homomorphisms.

## Definition

Let $P$ be a (single-sorted) [[prop]], and $C$ a symmetric monoidal category.  A **supply of $P$ in $C$** consists of

1. A $P$-algebra structure on every object of $C$, such that
2. The $P$-algebra structure on any tensor product $x\otimes y$ is that canonically induced from those on $x$ and $y$.

If there is a given supply of $P$ in $C$, we also say that $C$ **supplies** $P$-algebras.

## Remarks

The notion of "supply" does not satisfy the [[principle of equivalence]]: if $C\simeq D$ then a supply of $P$ in $C$ does not automatically transfer to a supply of $P$ in $D$.  Accordingly, a "symmetric monoidal category that supplies $P$" should be thought of as an atomic categorical structure, not as a [[category]] equipped with [[stuff, structure, property|structure]].

## Examples

* A [[hypergraph category]] is a symmetric monoidal category that supplies special commutative [[Frobenius algebras]].

* A [[Markov category]] is a [[semicartesian monoidal category|semicartesian]] symmetric monoidal category that supplies cocommutative comonoids.

* A [[gs-monoidal category]] is a symmetric monoidal category that supplies cocommutative comonoids.

## References

* [[Brendan Fong]], [[David Spivak]], _Supplying bells and whistles in symmetric monoidal categories_, [[arxiv:1908.02633](https://arxiv.org/abs/1908.02633)]

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support

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## Disambiguation

The term "support" means different things in different parts of mathematics:

### In set theory

* [[support of a function]]

* [[support of a set]]

### In category theory

* [[support object]]

### In functional analysis

* [[support of a distribution]]

### In measure theory

* [[tau-additive measure#null_sets_and_support|support of a measure]]
* [[valuation (measure theory)#null_sets_and_support|support of a valuation]]

### In field theory

* [[spacetime support]]

[[!redirects supports]]

category: disambiguation


support object

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## Idea

In [[set theory]], the [[support of a set]] turns the set into a [[subsingleton]]. In [[dependent type theory]], [[propositional truncation]] or the [[bracket type]] is an operation which takes a [[type]] and turns it into a [[h-propositions]], which are the type theoretic equivalent of subsingletons. By the [[relation between type theory and category theory]] and the relation between set theory and [[topos theory]], it should be possible to do the same process and turn an [[object]] of a [[category]] into a [[subterminal object]]. These are the **support objects**, **bracket objects**, or **propositional truncation objects** in category theory.

## Definition

The support object $[X]$ of an object $X$ in a [[category]] $C$ with a [[terminal object]] $1$ is the [[image]] of the unique morphism $X \to 1$.

$$X \to [X] \hookrightarrow 1$$

As a result, that all support objects exist is equivalent to the condition that every morphism into the terminal object has an [[image factorization]].

## Properties

In any [[well-pointed pretopos]] $\mathcal{E}$, the support is the [[coequalizer]] of the [[product]] projection morphisms $\pi_1:X \times X \to X$ and $\pi_2:X \times X \to X$

$$\array{
    X \times X
    &&
      \stackrel{\overset{\pi_1}{\longrightarrow}}{\underset{\pi_2}{\longrightarrow}}
    &&
    X
    \\
    & \searrow && \swarrow_{\mathrlap{p}}
    \\
    && [X]
  }.$$

## See also

* [[support]]
* [[subsingleton]]
* [[subterminal object]]
* [[complete loop graph]]
* [[product]]
* [[coequalizer]]
* [[propositional truncation]]

* [[inhabited object]]

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support of a distribution

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## Idea

The concept of _support of a distribution_ is an evident generalization of [[support of a function|support of a]] [[continuous function]] as functions are generalized to [[distributions]].

Of particular interest is the _singular support_ of distributions, which is the subset of those points inside their support around which they are singular when regarded as [[generalized functions]]. This "local analysis" of singularities of distributions may be further refined to their [[microlocal analysis]] by considering not such the singular points, but also the directions of the [[propagation of singularities theorem|propagation of the singularities]] at these points, which is the set of [[covectors]] over the singular support called the _[[wave front set]]_.

## Definition

+-- {: .num_defn #SupportOfADistribution}
###### Definition
**(support of a distribution)**

Let $X$ be a [[smooth manifold]] and let $\phi \colon C^\infty_c(X) \longrightarrow \mathbb{R}$ be a [[distribution]]. Then the _support_ of $\phi$ is the [[subset]] $supp(\phi) \subset X$ of all those points $x \in X$ such that for every [[open neighbourhood]] $U_x \subset X$ the [[restriction of distributions|restricted distribution]] $\phi\vert_{U_x}$ is not the zero-distribution

$$
  supp(\phi)
  \;\coloneqq\;
  \left\{
    x \in X \;\vert\; \underset{ \text{nbhd}\, U_x }{\forall}\left( \phi\vert_{U_x} \neq 0 \right)
  \right\}
  \,.
$$

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(e.g. [Hörmander 90, def. 2.2.2](#Hoermander90))

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###### Remark

By construction, the support of a distribution $\phi$ according to def. \ref{SupportOfADistribution} is a [[closed subset]].
If it happens to be a [[compact space|compact]] subset, then the
distribution is said to be a _[[compactly supported distribution]]_.

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(e.g. [Hörmander 90, section 2.3](#Hoermander90))

+-- {: .num_defn #SingularSupport}
###### Definition
**(singular support)**

Let $X$ be a [[smooth manifold]] and let $\phi \colon C^\infty_c(X) \longrightarrow \mathbb{R}$ be a [[distribution]]. Then the _singular support_  $supp_{sing}(\phi) \subset X$ is the [[subset]] of points such that for every [[open neighbourhood]] $U_x \subset X$ the [[restriction of distributions|restriction]] $\phi\vert_{U_x}$ is singular, hence not a [[non-singular distribution]].

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###### Remark
**([[wave front set]])**

A refinement of the information contained in the singular support of a distribution (def. \ref{SingularSupport}) is its _[[wave front set]]_, which over each point of the singular support records those [[covectors]] "along which the singularity propagates", as made precise by the [[propagation of singularities theorem]].

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## Properties

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###### Remark

If a [[bump function]] $f \in C^\infty_c(X)$ has its ([[compact support|compact]]) [[support]] $supp(f)$ [[disjoint subset|disjoint]] from the support of of a distribution $\phi$ (def. \ref{SupportOfADistribution}), then the [[evaluation]] vanishes:

$$
  supp(f) \cap supp(\phi) = \emptyset
  \;\Rightarrow\;
  \phi(f) = 0
  \,.
$$

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(e.g. [Hörmander 90, theorem 2.2.1 and (2.2.1)](#Hoermander90))

## Examples

## Related entries

* [[support of a function]]

* [[point-supported distribution]]

* [[order of a distribution]]

* [[scaling degree of a distribution]]

## References

* {#Hoermander90} [[Lars Hörmander]], _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

* [[Sergiu Klainerman]], chapter 3, section 3.5 of _Lecture notes in analysis_, 2011 ([pdf](https://web.math.princeton.edu/~seri/homepage/courses/Analysis2011.pdf))

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support of a function

> This article is about support of a [[function]]. For other notions of support, see [[support]].

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## Definition

### In set theory

Given a [[pointed set]] $A$ with specified [[element]] $0 \in A$, a [[set]] $X$, and a [[function]] $f \colon X \to A$, the _support_ of $f$ is the [[subset]] of $X$ on which $f$ is not equal to $0$.

#### In constructive mathematics

In [[constructive mathematics]], there are multiple notions of [[inequality]], due to the failure of the [[double negation law]]. As a result, there are multiple notion of support of a function. Thus, we define the following:

Given a [[pointed set]] $A$ with specified [[element]] $0 \in A$, a [[set]] $X$, and a [[function]] $f \colon X \to A$, the **support** of $f$ is the [[subset]] of $X$ on which $f$ is not equal to $0$.

Given a [[pointed set]] $A$ with an [[tight apartness relation]] $\#$ and specified [[element]] $0 \in A$, a [[set]] $X$, and a [[function]] $f \colon X \to A$, the **strong support** of $f$ is the [[subset]] of $X$ on which $f$ is apart from $0$.

### In topology

 {#InTopology}

In [[topology]] the support of a [[continuous function]] $f \colon X \to A$ as above is the [[topological closure]] of the set of points on which $f$ does not vanish:

$$
  Supp(f) = Cl(\{x \in X \vert f(x) \neq 0 \in A\})
  \,.
$$

If $Supp(f) \subset X$ is a [[compact subspace]], then one says that $f$ has _[[compact support]]_.

## See also

* [[compact support]]
* [[compactly supported cohomology]]
* [[support of a distribution]]

## References

* Wikipedia, _[Support (mathematics)](https://en.wikipedia.org/wiki/Support_%28mathematics%29)_


support of a set

> This article is about support of a [[set]]. For other notions of support, see [[support]].

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## Definition

The *support* $[X]$ of any set $X$ is the [[image]] of the unique function into any [[singleton]] $X \to 1$. By definition of [[image]] the support is thus a [[subsingleton]], and a [[singleton]] if $X$ is pointed. Note that this is different from the [[support of a function|support of the unique function]] $X \to 1$, which is always the [[empty set]] $\emptyset$.

## Generalizing to other categories

The above definition could be interpreted not just in [[Set]] but in any [[category]] with a [[terminal object]]. This leads to the notions of a [[support object]].

The [[support object]] of an [[object]] $A$ of a [[category]] is the [[image]] of its map to the [[terminal object]]. In the [[internal logic]] of a category, this corresponds to the [[propositional truncation]].

## Related concepts

* [[split support]]
* [[support object]]
* [[propositional truncation]]

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Suren Arakelov

* [Wikipedia entry](https://en.wikipedia.org/wiki/Suren_Arakelov)

## related $n$Lab entries

* [[Arakelov geometry]]

category: people


Suresh Govindarajan

* [official webpage](http://sgovindarajan.wikidot.com)

* [other webpage](http://sgovindarajan.wikidot.com)

## related $n$Lab entries

* [[torsion constraints in supergravity]]

category: people


surface

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## Idea

A **surface** is a [[space]] of [[dimension]] 2.

In [[differential geometry]] this means a 2-dimensional [[smooth manifold]] or something thereby [[parametrization|parametrized]].

In [[complex analytic geometry]] this usually means a [[complex manifold]] of _complex_ dimension 2.

Similarly and more generally, in [[algebraic geometry]] an _[[algebraic surface]]_ is a [[variety]] of algebraic dimension 2.

## Examples

* [[plane]]

* [[sphere]]

* [[torus]]

* [[Klein bottle]]

* [[3-manifold]], [[4-manifold]], [[8-manifold]]

## Related concepts

* [[differential geometry of curves and surfaces]]

* analog for dimension 1: [[curve]], [[algebraic curve]]

* [[genus of a surface]]

* [[area]]

* [[moduli space of curves]]

* [[complex surface]]

* [[surface knot]]

* [[hypersurface]]

* [[minimal surface]]

## References

* [[Manifold Atlas]], _[2-manifolds](www.map.mpim-bonn.mpg.de/2-manifolds)_

Discussion of [[de Rham cohomology]] of surfaces:

* [[William Fulton]], §18 of: *Algebraic Topology -- A First Course*, Graduate Texts in Mathematics **153**, Springer (1995) \[doi:10.1007/978-1-4612-4180-5\]

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surface diagram

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## Idea

Surface diagrams [[categorification|categorify]] [[string diagram|string diagrams]]: while string diagram calculus describes the composition of morphisms in 2-categories, surface diagram calculus describes composition of morphisms in (non-strict) [[3-category|3-categories]].

## Variants

There are several discussions and variation of surface diagrams available.

* [Trimble](#Trimble) discusses surface diagrams via stratified Morse theory,
* [Barret-Meusburger-Schaumann](#BMS) define a notion of surface diagrams for [[Gray-category|Gray categories]] with duals,
* [Hummon](#Hummon) defines a notion for [[Gray-category|Gray categories]],
* [Dorn-Douglas](#DornDouglas22) define a notion of [[manifold diagram]], which specializes to surface diagrams in dimension 3.

While all of the above notions are "in essence" equivalent, they differ in some of their technical details.

## Related concepts

* [[manifold diagram]]
* [[string diagram]]
* [[tangle]]

## References

* {#Trimble}  [[Todd Trimble]], _[[toddtrimble:Surface diagrams]]_

* {#BMS} John Barrett, Catherine Meusburger, Gregor Schaumann. _Gray categories with duals and their diagrams_. [arXiv](https://arxiv.org/abs/1211.0529)

* {#Hummon} Benjamin Hummon. _Surface diagrams for gray-categories_. Phd Thesis, UC San Diego. [pdf](https://escholarship.org/uc/item/5b24s9cc)

* {#DornDouglas22} Christoph Dorn and Christopher Douglas, _Manifold diagrams and tame tangles_, 2022 ([arXiv](https://arxiv.org/abs/2208.13758))

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surface knot

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## Idea

The notion of _surface knots_ is the next higher [[dimension|dimensional]] analog of that of _[[knots]]_: Where a [[knot]] is a 1-dimensional [[submanifold]] of the [[Euclidean space|3d Euclidean space]]/[[3-sphere]], so a _surface knot_ is a 2-dimensional submanifold (hence a [[surface]]) of the [[Euclidean space|4d Euclidean space]]/[[4-sphere]].

## Related concepts

* [[movie move]]

## References

Exposition:

* Lawrence Hook, _Knots in four dimensions_ ([pdf](http://people.virginia.edu/~lph5s/t/2.pdf))

Textbook accounts:

*  Seiichi Kamada, _Surface-Knots in 4-Space_, Springer 2017 ([doi:10.1007/978-981-10-4091-7](https://link.springer.com/book/10.1007/978-981-10-4091-7))

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surgery theory

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## Idea

(...)

## Related entries

* [[Dehn surgery]]

## References

* {#Wall69} [[C. T. C. Wall]] (ed. [[Andrew Ranicki]]), *Surgery on compact Manifolds*, Math. Surveys and Monographs **69** (1999) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/books/scm.pdf)]

* [[Rustam Sadykov]], _Elements of Surgery Theory_, 2013 ([pdf](https://www.math.ksu.edu/~sadykov/Lecture%20Notes/Surgery%20Theory.pdf), [[SadykovSurgeryTheory.pdf:file]])

In relation to [[Hopf invariants]]:

* [[Michael Crabb]], [[Andrew Ranicki]], *The Geometric Hopf Invariant and Surgery Theory*,  Springer Monographs in Mathematics, Springer (2017) $[$[doi:10.1007/978-3-319-71306-9](https://doi.org/10.1007/978-3-319-71306-9), [slides pdf](https://www.maths.ed.ac.uk/~v1ranick/slides/hopfbeam.pdf), [[CrabbRanicki-GeometricHopf.pdf:file]]$]$

See also:

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

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Surj

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## Idea
 {#Idea}

__$Surj$__ (or __$Set_{surj}$__) is the [[wide subcategory]] of the [[topos]] [[Set]] with its [[morphisms]] restricted to [[epimorphisms]] ([[surjective functions]]).

## Related concepts

* [[Inj]]


surjection

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## Definition

A [[function]] $f$ (of sets) from $A$ to $B$ is __surjective__ if, given any element $y$ of $B$, $y = f(x)$ for some $x$.  A surjective function is also called __onto__ or a __surjection__.

Surjections are the same as [[epimorphisms]] in [[Set|the category of sets]] and [[effective epimorphism in an (infinity,1)-category|effective epimorphisms]] in [[Set|the (infinity,1)-category of sets]].

A _[[bijection]]_ is a function that is both [[surjection|surjective]] and [[injection|injective]].

## The surjection preorder

In [[classical mathematics|classical]] [[set theory]], one writes $|B| \leq^* |A|$ to mean that either there is a surjection $A \to B$ *or* $B=\empty$. The relation $\leq^*$ is a [[preorder]] on the class of all sets, and its restriction to [[inhabited]] sets is the preorder reflection of the category $Surj_{inh}$ of inhabited sets and surjections.

To make this definition less piecemeal and more constructive, one can define $|B| \leq^* |A|$ to mean $B$ is a [[subquotient]] of $A$, in other words one has a surjection $B' \to B$ and an injection $B' \to A$. For $B$ inhabited and a subquotient of $A$, [[excluded middle]] implies there is a surjection from $A$ to $B$, so in [[classical mathematics]] this coincides with the piecemeal definition.

Contrast with the notation $|B| \leq |A|$ if there is an [[injection]] $B\to A$.  Since subobjects are subquotients (e.g. we can take $B'=B$ above), $|B| \leq |A|$ implies $|B| \leq^* |A|$.  (If we wanted to prove this using the piecemeal definition, we would require excluded middle.)

## Axioms of choice

The [[axiom of choice]] states precisely that every surjection in the category of sets has a [[section]]. Thus in this setting one has: $|B| \leq^* |A|$ implies $|B| \leq |A|$, and so $|B| \leq^* |A|$ iff $|B| \leq |A|$ assuming AC. Some authors who doubt the axiom of choice use the term 'onto' for a surjection as defined above and reserve 'surjective' for the stronger notion of a function with a section (a [[split epimorphism]]).

The axiom [[WISC]] has an equivalent statement (that works in any Boolean [[topos]]) due to François Dorais phrased almost entirely in terms of surjections (or epimorphisms):

>For every set $X$ there is a set $Y$ such that for every surjection $q\colon Z \to X$ there is a function $s\colon Y \to Z$ such that $q\circ s\colon Y\to X$ is a surjection.

One can view this as really a statement about the [[Grothendieck fibration]] over [[Set]] with fibre over $X$ the full subcategory of $Set/X$ on the surjections: every fibre has a [[weakly initial object]].

## In other categories
Since an element $a$ in a set $A$ in the [[category of sets]] is just a [[global element]] $a:1\rightarrow A$, one could define surjections in any category $\mathcal{C}$ with a [[terminal object]] $1$:

+-- {: .num_defn}
###### Definition
A morphism $f:A\rightarrow B$ in a category $\mathcal{C}$ with a [[terminal object]] $1$ is called a __surjection__. a __surjective morphism__, or an __onto morphism__ if, given any global element $y:1\rightarrow B$, there exists a global element $x:1\rightarrow A$ such that $y = f \circ x$.
=--

+-- {: .num_remark}
###### Remark
Some authors regard surjection as a synonym of [[split epimorphism]], and only use 'onto' for the definition above.
=--

+-- {: .num_prop}
###### Proposition
In a category $\mathcal{C}$ with a [[terminal object]] $1$, the unique morphism $!:A\rightarrow 1$ is a surjection for every object $A$.
=--

+-- {: .proof}
###### Proof
By definition of a terminal object, for every object $A$ there exists a unique morphism $!:A\rightarrow 1$, and the identity morphism is the unique global element $1_{1}:1\rightarrow 1$. The composite of a global element $x:1\rightarrow A$ with the function $!:A\rightarrow 1$ results in a function $! \circ x:1\rightarrow 1$, which by definition of a terminal object is the same as $1_{1}:1\rightarrow 1$. Since $! \circ x = 1_{1}$, for every object $A$, $!:A\rightarrow 1$ is a surjection.
=--

+-- {: .num_prop}
###### Proposition
In a category of [[pointed object|pointed objects]] $\mathcal{C}$, every morphism $f:A\rightarrow B$ is a surjection for every object $A$.
=--

+-- {: .proof}
###### Proof
A pointed object in a category with terminal objects is a object $A$ with a global element $a:1\rightarrow A$ to the object, and morphisms in the category preserve the global element, which means there is only a unique global element for each object $A$. Therefore, for every morphism $f:A\rightarrow B$ and global element $y:1\rightarrow B$, there exists a global element $x:1\rightarrow A$ such that $y = f \circ x$.
=--

Just because a morphism is a surjection in a concrete category does not mean that the morphism in the underlying category of sets is a surjection; equivalently, the [[forgetful functor]] from any category $\mathcal{C}$ to Set does not preserve surjections.

### Well-pointedness
The fact that surjections are epimorphisms in [[Set]] is a result of the fact that Set is well-pointed. This could be generalised to any category with a terminal separator $1$.

+-- {: .num_prop}
###### Proposition
In a category $\mathcal{C}$ with a [[terminal object]] $1$ such that $1$ is a [[separator]], every surjection is an [[epimorphism]].
=--

+-- {: .proof}
###### Proof
For any surjection $f:A\rightarrow B$, suppose there are parallel morphisms $g, h:B\rightarrow C$ such that there $g \circ f = h \circ f$ (a [[fork]]). Then for every global element $y:1\rightarrow B$ there exists a global element $x:1\rightarrow A$ such that $y = f \circ x$, and thus $g \circ y = g \circ f \circ x$ and $h \circ y = h \circ f \circ x$. But since $g \circ f = h \circ f$, $g \circ f \circ x = h \circ f \circ x$, which implies that $g \circ y = h \circ y$. Since $1$ is a [[separator]], then for every global element $y:1\rightarrow B$, if $g \circ y = h \circ y$, then $g = h$. Therefore, every surjection is an epimorphism.
=--

### Axiom of choice

The axiom of choice for surjections in [[Set]] is the following statement:

* Every [[surjection]] is a [[split epimorphism]].

This axiom could be defined in every category with a terminal object, and could be contrasted with the axiom of choice for epimorphisms, as not every epimorphism is an surjection, nor is every surjection an epimorphism.

+-- {: .num_prop}
###### Proposition
In a category $\mathcal{C}$ with a [[terminal object]] $1$ and binary [[equalisers]] such that every [[surjection]] is a [[split epimorphism]], the terminal object $1$ is a [[separator]].
=--

+-- {: .proof}
###### Proof
Suppose there are parallel morphisms $g, h:B\rightarrow C$ such that for every global element $y:1\rightarrow B$, $g \circ y = h \circ y$. One could construct an equaliser $f:eq(g,h)\rightarrow B$, which implies that $g \circ f = h \circ f$ and that for any global element $x:1\rightarrow eq(g,h)$, $g \circ f \circ x = h \circ f \circ x$. This implies that for every global element $y:1\rightarrow B$, there exists a global element $x:1\rightarrow eq(g,h)$ where $y = f \circ x$, and the equaliser $f:eq(g,h)\rightarrow B$ is a surjection. Since every surjection is a [[split epimorphism]], $f$ has a section $i:B\rightarrow eq(g,h)$ such that $f \circ i = 1_{B}$, the identity morphism on $B$, $g \circ f \circ i = h \circ f \circ i$, $g \circ 1_{B} = h \circ 1_{B}$, and $g = f$. Because for every global element $y:1\rightarrow B$, $g \circ y = h \circ y$ implies $g = h$, the terminal object $1$ is a [[separator]].
=--

## Related concepts

* [[injection]]

* [[epimorphism]]

* [[subquotient]]

* [[monotone function]]

* [[equivariant function]]

[[!redirects surjective function]]
[[!redirects surjective functions]]

[[!redirects surjections]]
[[!redirects surjective]]
[[!redirects surjective map]]
[[!redirects surjective maps]]

[[!redirects surjectivity]]


surjective geometric morphism

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

#Contents#
* table of contents
{:toc}

## Definition

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

A [[geometric morphism]] between [[toposes]] $(f^* \dashv f_*) : \mathcal{E} \to \mathcal{F}$ is **surjective** or a **geometric surjection** if it satisfies the following equivalent criteria:

* its [[inverse image functor]] $f^*$ is [[faithful functor|faithful]] (in contrast to the [[direct image]] being [[full and faithful functor|full and faithful]] as for a [[geometric embedding]]);

* its [[inverse image functor]] $f^*$ is [[conservative
functor|conservative]];

* the components $X \to f_* f^* X$ of the [[unit of an adjunction|adjunction unit]] are [[monomorphism]]s, for all $X \in \mathcal{F}$;

* $f^*$ induces a injective homomorphism of [[subobject]] [[lattice]]s

  $$
    Sub(X) \hookrightarrow Sub(f^* X)
  $$

  for all $X \in \mathcal{F}$;

* $f^*$ reflects the order on [[subobject]]s;

* $(f^* \dashv f_*)$ is a [[comonadic adjunction]].

=--

The equivalence of these condition appears for instance as [MacLaneMoerdijk, VII 4. lemma 3 and prop. 4](#MacLaneMoerdijk).

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

We discuss the equivalence of these conditions:

The equivalence $(f^* \; faithful) \Leftrightarrow (Id \to f^* f_* \;is \; mono)$ is a general property of [[adjoint functors]] (see there).

The implication $(f^* faithful) \Rightarrow (f^* induces\;injection\;on\;subobjects)$ works as follows:

first of all $f^*$ does indeed preserves subobjects: since it respects [[pullback]]s and since [[monomorphism]]s are characterized as those morphisms whose domain is stable under pullback along themselves.

To see that $f^*$ induces an injective function on subobjects let $U \hookrightarrow X$ be a subobject with characteristic morphism $char U : X \to \Omega$ and consider the image

$$
  \array{
    f^* U &\to& f^* * \simeq *
    \\
    \downarrow && \downarrow
    \\
    f^* X &\stackrel{f^* char U}{\to}& f^* \Omega
  }
  \,.
$$

of the pullback diagram that exhibits $U$ as a subobject. Since $f^*$ preserves pullbacks, this is still a pullback diagram.

If now $U \leq \tilde U$ but $f^* (U) = f^*(\tilde U)$ then both corresponding pullback diagrams are sent by $f^*$ to the same such diagram. By faithfulness this implies that also

$$
  \array{
    \tilde U &\to& *
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{char U}{\to}& \Omega
  }
$$

commutes, and hence that also $\tilde U \subset U$, so that in fact $\tilde U \simeq U$.

Next we consider the implication ($f^*$ induces injection on subobjects) $\Rightarrow$ ($f^*$ is conservative).

Assume $f^* (X \stackrel{\phi}{\to} X')$ is an isomorphism. We have to show that then $\phi$ is an isomorphism. Consider the [[image]] factorization $X \to im(\phi) \hookrightarrow X'$. Since $f^*$ preserves pushouts and pullbacks, it preserves epis and monos and so takes this to the image factorization

$$
  f^* X \to f^* (im \phi) \stackrel{\simeq}{\to} f^* X'
$$

of $f^* \phi$, where now the second morphism is an iso, because $f^* \phi$ is assumed to be an iso. By the assumption that $f^* $ is injective on subobjects it follows that also $im \phi \simeq X'$ and thus that $\phi$ is an epimorphism.

It remains to show that $\phi$ is also a monomorphism. For that it is sufficient to show that in the pullback square

$$
  \array{
    X \times_{X'} X &\to& X
    \\
    \downarrow && \downarrow^{\mathrlap{\phi}}
    \\
    X &\stackrel{\phi}{\to}& X'
  }
$$

we have $X \times_{X'} X \simeq X$. Write $\Delta : X \to X \times_{X'} X$ for the [[diagonal]] and let

$$
  X \to im \Delta_\phi \to X \times_{X'} X
$$

be its [[image]] factorization. Doing the same for $f^* \phi$, which we have seen is a monomorphism, and using that $f^*$ preserves the pullback,  we get

$$
  f^* im \Delta_\phi \simeq f^* (X \times_{X'} X)
  \,.
$$

Now using again the assumption that $f^*$ is injective on subobjects, this implies $im \Delta_\phi = X \times_{X'} X$ and hence that $\phi$ is a monomorphism.

(...)

The statement about the comonadic adjunction we discuss below as prop. \ref{Comonadicity}.

=--

## Properties

### Surjection/embedding factorization

+-- {: .num_prop #CofreeAlgebraFunctorIsSurjective}
###### Observation

For $T : \mathcal{E} \to \mathcal{E}$ a [[left exact functor|left exact]] [[comonad]] the cofree algebra functor

$$
  F : \mathcal{E} \to T CoAlg(\mathcal{E})
$$

to the [[topos of coalgebras]] is a geometric surjection.

=--

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

By the discussion at [[topos of coalgebras]] the [[inverse image]] is the [[forgetful functor]] to the underlying $\mathcal{E}$-objects. This is clearly a [[faithful functor]].

=--

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

Up to [[equivalence of categories|equivalence]], every geometric surjection is of this form.

=--

This appears for instance as ([MacLaneMoerdijk, VII 4., prop 4](#MacLaneMoerdijk)).

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

With observation \ref{CofreeAlgebraFunctorIsSurjective} we only need to show that if $f : \mathcal{E} \to \mathcal{F}$ is surjective, then there is $T$ such that

$$
  \array{
     \mathcal{E} &\stackrel{f}{\to}& \mathcal{F}
     \\
     & {}_{\mathllap{F}}\searrow & \downarrow^{\mathrlap{\simeq}}
     \\
     && T CoAlg(\mathcal{E})
  }
  \,.
$$

For this, take $T := f^* f_*$. This is a [[left exact functor]] by definition of [[geometric morphism]]. By assumption on $f$ and using the equivalent definition of def. \ref{EquivalentDefinitions} we have that $f^*$ is a [[conservative functor]]. This means that the conditions of the [[monadicity theorem]] are met, so so $f^*$ is a [[monadic functor|comonadic functor]].

=--

For more on this see _[[geometric surjection/embedding factorization]]_  . Also at _[[monadic descent]]_.

## Examples

Trivially, any [[connected geometric morphism]] is surjective.

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

For $f : X \to Y$ a [[continuous function]] between [[topological space|topological spaces]] and $(f^* \dashv f_*) : Sh(X) \to Sh(Y)$ the corresponding geometric morphism of [[sheaf topos|sheaf toposes]]: if $f$ is surjective then $(f^* \dashv f_*)$ is a surjective geometric morphism, conversely, if $(f^* \dashv f_*)$ is a surjective geometric morphism and $Y$ a $T_1$-space then $f$ is surjective.
=--

For a proof see e.g [MacLane-Moerdijk](#MacLaneMoerdijk), p.367. A similar result holds for injective functions and [[geometric embedding|geometric embeddings]] but there $T_0$ suffices as a [[separation axiom|separation requirement]] on $X$.

## Related concepts

* [[geometric morphism]]

  * **geometric surjection**

  * [[geometric embedding]]

  * [[connected geometric morphism]]

* [[monadic descent]]

## References

* {#Johnstone77} [[Peter Johnstone]], _Topos Theory _ , Academic Press New York 1977. (Available as Dover reprint 2014; Section 4.1)

* {#MacLaneMoerdijk} [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994. (Section VII.4)

[[!redirects surjective geometric morphisms]]
[[!redirects geometric surjection]]
[[!redirects geometric surjections]]


surjective on objects functor

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

#Contents#
* table of contents
{:toc}

## Idea

A [[functor]] $F:C\to D$ is **surjective on objects** if for each object $y\in D$ there is an object $x\in C$ such that $F(x)=y$.

This is of course [[principle of equivalence|not an invariant notion]]; the corresponding invariant notion is [[essentially surjective functor]].  However, surjective-on-objects functors are sometimes useful when doing [[strict 2-category]] theory.

## Properties

Surjective-on-objects functors are the left class in an [[orthogonal factorization system]] on [[Cat]] whose right class consists of the [[injective-on-objects functor|injective-on-objects]] and [[fully faithful functors]].

## Related pages

[[!include properties of functors -- contents]]

## References

* [[Ross Street]], *Two-dimensional sheaf theory* refers to surjective-on-objects functors as *acute* and their corresponding right class as *chronic*.

[[!redirects surjective on objects functor]]
[[!redirects surjective on objects functors]]
[[!redirects surjective on objects]]
[[!redirects surjective-on-objects functor]]
[[!redirects surjective-on-objects functors]]
[[!redirects surjective-on-objects]]
[[!redirects so functor]]
[[!redirects so functors]]
[[!redirects s.o. functor]]
[[!redirects s.o. functors]]
[[!redirects acute functor]]
[[!redirects acute functors]]
[[!redirects chronic functor]]
[[!redirects chronic functors]]


Surreal geometry

The previous content of this page was deemed inappropriate; please don't continue editing it.  If you insist on discussing your ideas here, you can start out on the [nForum](https://nforum.ncatlab.org), where regulars can give feedback and guidance.  But we suggest that you try to take this discussion to another, more suitable forum.

-- The [[nlabmeta:steering committee|Steering Committee]]


surreal number

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###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

**Surreal numbers** (or Conway numbers, after John H. Conway) are a kind of [[numbers]] that originally arose in the context of [[Conway game|combinatorial games]], as a kind of generalized quantity for measuring strength of positions. The class of surreal numbers forms an ordered field in which any small ordered field may be embedded, so that the field of surreal numbers forms a universal field for expressing any set-bounded degree of infinite and infinitesimal quantities. Any ordinal number may be construed as a surreal number, and new surreal numbers may be defined from old ones by generalizing the notion of Dedekind cut. Surreal numbers are a very large extension of real numbers, where one may make sense of fanciful quantities such as

$$(\omega^2 + 5\omega - 13)^{\frac1{3} - \frac{2}{\omega}} + \pi$$

The term "surreal number" was introduced by Donald Knuth, who wrote a short novel about a couple who check out from Western civilization and live on a remote beach in India and who, finding a mysterious engraved stone and with an abundance of time to spend on things besides food and sex, discover for themselves surreal numbers and their delightful properties.

## Conway-style definitions

The notions of Conway game and Conway number, as given by Conway in his book _On Numbers and Games_, are disarmingly simple-looking. Here is a Conway-style definition of game:

* A game $G$ consists of a set of games called **left options** of $G$, and a set of games called **right options** of $G$. All games are constructed this way.

The second sentence is a shorthand way of saying this is supposed to be a recursive or inductive definition: if a collection $G^L$ of left options and a collection $G^R$ is given, one can construct a game $G = \{G^L \vert G^R\}$. Games are built recursively, starting with $G = \{\emptyset \vert \emptyset\}$.

Here is a Conway-style definition of number:

* A number $x$ consists of a set $L_x$ of numbers and a set $R_x$ of numbers, under the condition that no member of $L_x$ is $\geq$ any member of $R_x$

followed immediately by a recursive definition of $\geq$:

* $x \geq y$ if $y \geq x'$ is false for any $x' \in R_x$ and $y' \geq x$ is false for any $y' \in L_y$.

Ultimately, each number $x$ is considered to be a (Dedekind) cut between the set of left options (which are 'below' $x$) and the set of right options (which are 'above' $x$).

Equality of numbers is a derived notion:

* $x = y$ is defined to mean $x \geq y$ and $y \geq x$.

Finally, field operations on surreal numbers are defined recursively:

* $x + y$ is defined by letting $L_{x+y}$ be the set $\{x' + y \vert x' \in L_x\} \cup \{x + y' \vert y' \in L_y\}$, and $R_{x+y}$ the set $\{x' + y \vert x' \in R_x\} \cup \{x + y' \vert y' \in R_y\}$;

* $-x$ is defined by letting $L_{-x}$ be the set of negatives of elements of $R_x$, and $R_{-x}$ the set of negatives of $L_x$;

* $x y$ is defined by letting $L_{x y}$ be the set
$$\{x' y + x y' - x'' y'' \vert x', x'' \in L_x, y', y'' \in L_y\} \cup \{x' y + x y' - x'' y'' \vert x', x'' \in R_x, y', y'' \in R_y\},$$
and $R_{x y}$ be the set
$$\{x' y + x y' - x'' y'' \vert x', x'' \in L_x, y', y'' \in R_y\} \cup \{x' y + x y' - x'' y'' \vert x', x'' \in R_x, y', y'' \in L_y\}.$$

The style of such definitions is familiar from [[material set theory]], where elements are conceived to be sets of elements, which themselves have elements, and so on, down to a bedrock established by a foundation axiom. In fact, games can be axiomatized as elements of a structure with two predicates $\in_L$, $\in_R$ satisfying some set-theoretic axioms.

The definition of multiplication is the most involved. We can summarize it by saying that the left set gets contributions from like sets, and the right set gets contributions from opposite kinds. For example, $0 \cdot 1$ is $\{|\} \cdot \{0 | \emptyset\}$, so the left-hand contributions to the left set are

$$\emptyset \cdot \{0|\} + 0 \cdot \{0\} - \emptyset \cdot \{0\} = \emptyset + 0 \cdot \{0\} - \emptyset \cdot \{0\} = \emptyset.$$

The other contributions work out similarly: everything ends up being the empty set, so the answer is $\{\emptyset | \emptyset\} = 0$.

Conway motivates this definition from the observation that, if $x' \in L_x$ and $y' \in L_y$, then $x - x' \gt 0$ and $y - y' \gt 0$. And so, if multiplication makes any sense at all, we should have $(x - x')(y - y') \gt 0$ and thus $x y \gt x' y + x y' - x' y'$. Therefore, elements of this form should appear in $L_{x y}$. Following this kind of logic for the other possible combinations leads to the definition of multiplication given above.

## In constructive mathematics

The definition of $\geq$ for numbers as formulated by Conway leans heavily upon properties of negation, and in particular the classical [[principle of excluded middle]].  However, it can be made more constructive by defining $\lt$ separately from $\le$:

* A number $x$ consists of a set $L_x$ of numbers and a set $R_x$ of numbers, under the condition that every member of $L_x$ is $\lt$ every member of $R_x$.
* $x \leq y$ if $x' \lt y$ is true for any $x' \in L_x$, and $x \lt y'$ is true for any $y' \in R_y$.
* $x \lt y$ if there exists an $x'\in R_x$ such that $x'\le y$, or there exists a $y'\in L_y$ such that $x\le y'$.
* $x = y$ is defined to mean $x \le y$ and $y \le x$.

This approach works up to a point.  However, it encounters a problem in proving that multiplication is well-defined, i.e. that it respects equality.  We can prove that if $x\ge 0$ then $y\le z \Rightarrow x y \le x z$ and hence $y = z \Rightarrow x y = x z$, and that if $x\le 0$ then $y\le z \Rightarrow x y \ge x z$ and hence $y = z \Rightarrow x y = x z$; but without excluded middle we can't say that every $x$ is either $\ge 0$ or $\le 0$, so how can we conclude that $y = z \Rightarrow x y = x z$?  It is unknown whether there is a solution to this problem.

## Structural approach

It is a routine matter to rewrite the definitions in a more structural fashion, say within [[ETCS]]. In outline, a game is a rooted [[tree]] with no infinite branches and equipped with a labeling of each edge by a symbol $L$ or $R$. The nodes are called **positions** of the game. The root is the initial position, and starting from any position, there are positions that a player named $Left$ may move to, and positions that the other player $Right$ may move to, according to the edge-labelings. Any sequence of moves proceeding from the initial position (and not necessarily alternating between $Left$ and $Right$) must eventually terminate, since there are no infinite branches.

Each position has a definite depth $n$, where $n$ is the number of edges one must back-track through along the unique path that leads back to the initial position. A tree is thus an internal presheaf $X$ on the ordered set $\omega$, whose value at the initial object $0 \in \omega$ is terminal (occupied by the initial position), and where $X(n)$ is the set of positions at depth $n$. A move is a morphism in the corresponding category of elements of the presheaf, and moves are labeled by $L$ or $R$.

In this set-up, the condition of no infinite branches plays the role of a foundation axiom, and inductive arguments tend to devolve upon the use of the principle of [[dependent choice]].

Numbers may be similarly elaborated within this context. The definition of $\geq$ on numbers is ultimately derived from a posetal reflection of a category of games (with strategies as morphisms), and the definition of addition of numbers is derived from a Day convolution product on the category of games. More information may be found at [[Conway game]].

## In homotopy type theory

In [[homotopy type theory]], the surreal numbers can be defined as a [[higher inductive type]], roughly following Conway's definition but imposing the "defined" notion of equality as the "actual" equality with an identification-constructor.  This can also be done constructively by separating $\lt$ from $\le$, as remarked above.  See chapter 11.6 of the [[HoTT Book]].

## In algebraic set theory

An axiomatic approach to surreal numbers analogous to the approach to sets and ordinals in [[algebraic set theory]] can be found in [Rangel-Mariano 19](#RM19).

## Properties of surreal numbers

The surreal numbers form a large (i.e., proper-class size) [[real closed field]] with exponentiation.

To some degree, and with appropriate caveats, one can do analysis and number theory on surreal numbers. We refer to Conway's book for more information.

## Related pages

* [[ordinal number]]
* [[real number]]
* [[Hahn series]]
* [[infinitesimal]]
* [[∞-number]]
* [[Surreal space]]
* [[Surreal geometry]]

## References

* John H. Conway, _On Numbers and Games_ ($2^{nd}$ edition), A.K. Peters, Ltd (2001).

* [[Donald Knuth|Donald E. Knuth]], _Surreal numbers_, Addison-Wesley (1974)

* The [[HoTT Book]], chapter 11.6.

* Dimi Rocha Rangel, Hugo Luiz Mariano, *An algebraic (set) theory of surreal numbers, I*, 2019, [arxiv](https://arxiv.org/abs/1911.12726)
 {#RM19}

[[!redirects surreal numbers]]


Surreal space


Susan Montgomery

__Susan Montgomery__ is an American algebraist, specialized in [[ring theory]] and [[Hopf algebra]]s.

* [wikipedia](https://en.wikipedia.org/wiki/Susan_Montgomery)

## Selected works

* Susan Montgomery, _Hopf algebras and their action on rings_, AMS 1994, 240p.
* [[Susan Montgomery]], _Hopf Galois theory: a survey_, Geometry and topology monographs 16 (2009) 367--400; [link](http://www.msp.warwick.ac.uk/gtm/2009/16/p012.xhtml), [doi](https://doi.org/10.2140/gtm.2009.16.367).

category: people


Susan Niefield

* [Homepage](http://www.math.union.edu/~niefiels/)

## Selected writings

On [[exponential objects]] among [[topological spaces]] and [[compactly generated topological spaces]]:

* [[Susan Niefield]], *Cartesianness*, PhD thesis, Rutgers 1978 ([proquest:302920643](https://www.proquest.com/docview/302920643))

* {#Niefield82} [[Susan Niefield]], _Cartesianness: topological spaces, uniform spaces, and affine schemes._, Journal of Pure and Applied Algebra, 23.2, 1982, pp. 147--167 (doi:10.1016/0022-4049(82)90004-4, [pdf](https://core.ac.uk/download/pdf/81936509.pdf))

* {#Niefield81} [[Susan Niefield]], _Cartesian inclusion: locales and toposes._, Communications in Algebra __9__:16 (1981) 1639--1671  ([doi:10.1080/00927878108822672](https://doi.org/10.1080/00927878108822672))

On [[exponential objects]] among [[internal groupoids]] in [[Top]] ([[topological groupoids]]):

* [[Susan B. Niefield]], [[Dorette A. Pronk]], *Internal groupoids and exponentiability*, Cahiers de topologie et géométrie différentielle catégoriques, [**LX** 4 (2019)](http://cahierstgdc.com/index.php/volume-lx/) ([pdf](http://cahierstgdc.com/wp-content/uploads/2019/10/Niefield-Pronk-LX-4.pdf))

Discrete [[Conduché functor]]s are considered in

* [[Marta Bunge]], [[Susan Niefield]], _Exponentiability and single universes_, JPAA **148** (2000) 217--250

On [[quantale]]s and [[quantic nuclei]]

* [[Susan B. Niefield]], Kimmo I. Rosenthal, _Constructing locales from quantales_, Mathematical Proceedings of the Cambridge Philosophical Society __104__:2 (1988) 215--234 [doi](https://doi.org/10.1017/S0305004100065403)

category: people

[[!redirects Susan B. Niefield]]

[[!redirects S. B. Niefield]]


Susan Tolman

* [institute page](https://math.illinois.edu/directory/profile/tolman)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Susan_Tolman)

## Selected writings

On [[geometric quantization]]:

*  [[Ana Cannas da Silva]], [[Yael Karshon]], [[Susan Tolman]], *Quantization of Presymplectic Manifolds and Circle Actions*, Transactions of the AMS **352** 2 (2000) 525-552  [[arXiv:dg-ga/9705008](http://arxiv.org/abs/dg-ga/9705008), [jstor:118052](https://www.jstor.org/stable/118052)]

## Related entries

* [[Guillemin-Sternberg geometric quantization conjecture]]

category: people


suspended category

#Contents#
* automatic table of contents goes here
{:toc}

## Definition

A **suspended category** is an [[additive category]] $C$ equipped with an additive functor $S:C\to C$ called __suspension__ and a class of $S$-sequences called __triangles__ satisfying axioms below. Here one calls an _$S$-sequence_ a sequence of morphisms of the form
$$
X\stackrel{f}\to Y\stackrel{g}\to Z\stackrel{h}\to S X,
$$
and morphisms are ladders of the type
$$\array{
X&\stackrel{f}\to &Y&\stackrel{g}\to &Z&\stackrel{h}\to& S X\\
a\downarrow&&b\downarrow&&c\downarrow&&\downarrow S a\\
 X'&\stackrel{f'}\to &Y'&\stackrel{g'}\to &Z'&\stackrel{h'}\to& S X',\\
}$$
where all the squares commute. Axioms:

(SP0) Each sequence [[isomorphism|isomorphic]] to a triangle is a triangle.

(SP1) Each sequence of the form $0\to X\stackrel{id}\to X\to S0$ is a triangle.

(SP2) If $X\stackrel{f}\to Y\stackrel{g}\to Z\stackrel{h}\to S X$ is a triangle, then $Y\stackrel{g}\to Z\stackrel{h}\to S X\stackrel{-S f}\to S Y$ is also a triangle.

(SP3) Every [[diagram]] of the form

$$\array{
X&\stackrel{f}\to &Y&\stackrel{g}\to &Z&\stackrel{h}\to& S X\\
a\downarrow&&b\downarrow&&&&\downarrow S a\\
 X'&\stackrel{f'}\to &Y'&\stackrel{g'}\to &Z'&\stackrel{h'}\to& S X'\\
}$$

can be completed to a morphism of $S$-sequences.

(SP4) For any two morphisms $X\stackrel{f}\to Y$ and
$Y\stackrel{g}\to Z$ there is a commuting diagram
$$\array{
X&\stackrel{f}\to &Y&\stackrel{i}\to &Z'&\to& S X\\
=\downarrow&&g\downarrow&&\downarrow&&=\downarrow \\
 X&\to &Z&\to &Y'&\to& S X\\
&&\downarrow&&\downarrow&&\downarrow S f\\
 & &X'&\stackrel{id}\to &X'&\stackrel{j}\to& S Y\\
&&j\downarrow&&\downarrow&&\\
 &&S Y&\stackrel{S i}\to &S Z'&&\\
}$$
where the first two rows and the middle two columns are triangles.

## Examples

* Every [[triangulated category]] is suspended.

* Every suspended category in which $S$ is an equivalence is triangulated.

* Under mild assumptions, the [[stable category]] of a [[Quillen exact category]] $(A,E)$ is a suspended category. If $A$ is a [[Frobenius category]], then $A$ is a [[triangulated category]].

## References

Suspended categories were introduced in

* [[Bernhard Keller]], Dieter Vossieck,
Sous les catégories dérivées. [Beneath the derived categories] C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 6, 225--228.

See also

* B. Keller, Chain complexes and stable categories, Manus. Math. 67 (1990), 379-417.


suspension

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea
 {#Idea}

* [In point-set topology](#InPointSetTopology)

* [In homotopy theory](#InHomotopyTheory)

### In point-set topology
 {#InPointSetTopology}

In [[point-set topology]], the (un-[[reduced suspension|reduced]]) *suspension* $\mathrm{S} X$ of an [[inhabited set|inhabited]] [[topological space]] $X$ is the [[quotient space]] of the [[cylinder]] space $X \times [-1,1]$ (the [[product space]] with a [[closed interval]]) by the relation which identifies all the points at either end:

\[
  \label{PointSetFormula}
  \mathrm{S}X
  \;\coloneqq\;
  \frac{
    X \,\times\, [-1,\, +1]
  }{
    \left(
    \begin{array}{c}
      X \times \{-1\}\mathrlap{,\,}
      \\
      X \times \{+1\}
    \end{array}
    \right)
  }  \,.
\]

{#Illustration} The following graphics gives an artistic impression of the suspension of a (non-empty) space $X$, indicated in gray:

> (adapted from [Muro 2010](#Muro10))

In particular, the suspension of an [[n-sphere|$n$-sphere]] is [[homeomorphism|homeomorphic]] to the $(n+1)$-sphere

$$
  S^{n+1}
    \;\underset{\mathrm{homeo}}{\simeq}\;
  \mathrm{S}(S^n)
  \,,
$$

namely the [[union]] of (in terms of the above graphics):

1. a north pole "$nth$"

1. a south pole "$sth$"

1. the [[meridians]] "$mer(x)$" through all points $x \in S^{n}$ on the [[equator]].

Hence, by [[induction]], the $(n+1)$-sphere is homeomorphic to the $n+1$-fold suspension of the [[0-sphere]]:

$$
  S^{n+1}
    \;\underset{\mathrm{homeo}}{\simeq}\;
  \mathrm{S}^{n+1}(S^0)
  \,.
$$

### In homotopy theory
 {#InHomotopyTheory}

Regarded in the [[classical homotopy category]], the suspension construction (eq:PointSetFormula) on a [[CW-complex]] $X$ is a canonical model for the *[[suspension object]]* of the [[homotopy type]] represented by $X$ --- namely the [[homotopy pushout]] of the [[terminal object|terminal]] [[map]] $X \to \ast$ along itself) --- and as such plays a key role in [[homotopy theory]] and [[stable homotopy theory]].

This also explains the usual definition of the suspension of the [[empty space]] as the [[0-sphere]]

$$
  \mathrm{S}(\varnothing)
  \;=\;
  S^0
  \,,
$$

since the latter does model the [[homotopy pushout]] of $\varnothing \to \ast$ along itself (which is just a [[1-category]][[category theory|theoretic]] [[pushout]] and hence the [[coproduct]] of the [[point space]] with itself).

For more on this see at *[[suspension object]]*.

More precisely, what appears in most of [[algebraic topology]] and [[stable homotopy theory]] is the *[[reduced suspension]]* of [[pointed topological spaces]] $(X,x_0)$ (and eventually their [[suspension spectra]]):

$$
  \Sigma X \,\coloneqq\, \mathrm{S}X / \{x_0\}
  \,,
$$

where in addition the copy of the interval over the given basepoint is identified with a single point. For [[CW-complexes]] the [[reduced suspension]] is [[weak homotopy equivalence|weakly homotopy equivalent]] to the ordinary suspension.

In this context of ([[stable homotopy theory|stable]]) [[homotopy theory]], the suspension construction is part of the canonical [[point-set topology|point-set]] model for [[cofiber sequences]] induced from any [[mapping cone]] construction:

> (from [Muro 2010](#Muro10))

## Definitions

Let $X$ be a [[space]] (such as a [[topological space]], or something more interesting like a [[generalized smooth space]]).  Let $I$ be the unit [[interval]] $[0,1]$ in the [[real line]]; let $2$ be the $2$-point [[discrete space]] $\{\bot,\top\}$.  Let $X \times I \times 2$ be the [[cartesian product]] of $X$, $2$, and $I$; let $X + (X \times 2 \times I) + 2$ be the [[disjoint union]] of $X$, $X \times I \times 2$ and $2$.  We will suppress reference to the inclusion maps into $X + (X \times I) + 2$; it will be clear from context how to parse an element of the latter.

### Generalisable definition

The __suspension__ $S X$ of $X$ is the [[quotient space]] of $X + (X \times I \times 2) + 2$ by the [[equivalence relation]] $\sim$ generated by:

*  $a \sim (a,0,\bot)$ for $a$ in $X$;
*  $a \sim (a,0,\top)$ for $a$ in $X$;
*  $(a,1,\bot) \sim \bot$ for $a$ in $X$;
*  $(a,1,\top) \sim \top$ for $a$ in $X$.

### Generalisation: the join

This generalises immediately to an operation called the _[[join of topological spaces|join]]_ $X \star Y$ of two spaces $X$ and $Y$; this is the quotient space of $X + (X \times I \times Y) + Y$ by the equivalence relation generated by:

*  $a \sim (a,0,b)$ for $a$ in $X$ and $b$ in $Y$;
*  $(a,1,b) \sim b$ for $a$ in $X$ and $b$ in $Y$.

(Compare [[join of simplicial sets]] and [[join of topological spaces]], the same operation in another guise.)  Then we have $S X = X \star 2$.

### Simplified definition

It is somewhat simpler to define $S X$ as the quotient space of $(X \times I) + 2$ by the equivalence relation generated by:

*  $(a,0) \sim \bot$ for $a$ in $X$;
*  $(a,1) \sim \top$ for $a$ in $X$.

This works to define a topological space, but it does not directly give the smooth structure that matches the picture above.

### Further simplified for pointed spaces

If $X$ has a point $p$, hence if it is [[inhabited set|inhabited]], then we can define $S X$ as the quotient space of $X \times I$ by the equivalence relation generated by:

*  $(a,0) \sim (b,0)$ for $a, b$ in $X$;
*  $(a,1) \sim (b,1)$ for $a, b$ in $X$.

This is probably the most common definition seen, but it only works for $X$ an inhabited space (and even then gives only the topological structure).

### Reduced suspension

To make the suspension of a [[pointed space]] $(X,p)$ again a [[pointed space]] one may further collapse in $S X$ the set $\{p\} \times I$ to the point. The result then is called the _[[reduced suspension]]_ of $(X,p)$ and is denoted

$$
  \Sigma X \coloneqq
  S X / \{p \} \times I
  \simeq
  X \times I / \left(
    X \times \{0,1\}
    \cup
    \{p\} \times I
  \right)
  \,.
$$

## Properties

### As a functor

It\'s easy to extend the suspension operation $S$ to a [[functor]] from [[Top]] to itself.

### Relation between suspension and reduced suspension

For [[CW-complexes]] suspension and [[reduced suspension]] agree, up to [[weak homotopy equivalence]].

### Cogroup structure

In the pointed case ([[reduced suspension]]): [[suspensions are H-cogroup objects]]

## Examples

### Suspension of cubes

The suspension of the $n$-[[cube]] is the $(n+1)$-cube, probably best visualised as a diamond.  This gives a [[recursion|recursive]] definition of cube, starting with the $0$-cube as the [[point]], which is not the suspension of anything.  Note that this not only gives us the [[topological structure]] of the cube, but also (by working in an appropriate [[category]] of [[smooth spaces]] throughout) the correct [[smooth structure]] on the cube as a [[manifold with corners]].  You can probably even get the correct [[metric space|metric]] on the cube (normalised to have diagonals of length $1$) automatically by using a more complicated quotienting process.

### Suspension of spheres

Up to [[homeomorphism]], the suspension of the $n$-[[sphere]] is the $(n+1)$-sphere, and the [[reduced suspension]] is

$$
  \Sigma S^n \simeq S^{n+1}
  \,.
$$

See at _[one-point compactification -- Examples -- Spheres](one-point+compactification#ExamplesSpheres)_ for details.

Notice that the $n$-sphere is (topologically) the [[boundary]] of the $(n+1)$-cube.  The coincidence that 'sphere' and 'suspension' both begin with 's' has not been ignored; we can write $S^n \cong S^n(2)$, where $S^n$ on the left is the $n$-sphere and $S^n$ on the right is the $n$-fold [[composite]] of the suspension functor.  (Actually, you should start with the $(-1)$-sphere as the [[empty space]], which is not the suspension of anything; then the $0$-sphere is $S \empty = 2$.)  However, this does not give the correct smooth structure on the sphere, unless perhaps there is some more sophisticated definition that fixes this (but then that would break the cube).  It might be more appropriate to say that the suspension of the $n$-[[globe]] is the $(n+1)$-globe.

### Suspension of simplices

Up to topological structure, the suspension of the $n$-[[simplex]] is the $(n+1)$-simplex, but now this is not very useful.  To study simplices, you should use the __cone__ functor instead, which is $\Lambda X = X \star 1$, where $1$ is the [[point]].

## Related concepts

* [[EHP spectral sequence]]

* [[loop space object]], [[free loop space object]],

  * [[delooping]]

  * [[loop space]], [[free loop space]], [[derived loop space]]

* [[suspension object]]

  * [[suspension type]], [[reduced suspension type]]

  * **suspension**, [[reduced suspension]]

    * [[Freudenthal suspension theorem]]

    * [[Spanier-Whitehead category]]

    * [[suspension isomorphism]]

    * [[Sullivan model of suspensions]]

  * [[suspension of a chain complex]]

## References
 {#References}

> For discussion of *[[reduced suspension]]* see [there](reduced+suspension#References). For more general discussion of [[homotopy pushouts]] see also [there](homotopy+pushout#References). Most introductions to [[homotopy theory]] discuss suspension, in one form or other, see [there](homotopy+theory#References).

Textbook accounts with the basics definition:

* [[Klaus Jänich]], p. 41 in: *Topology*, Undergraduate Texts in Mathematics, Springer (1984, 1999) [[ISBN:9780387908922](https://link.springer.com/book/9780387908922), [doi:10.1007/978-3-662-10574-0](https://doi.org/10.1007/978-3-662-10574-0), Chapters 1-2: [pdf](http://topologicalmedialab.net/xinwei/classes/readings/Janich/Janich_Topology_ch1-2.pdf)]

* [[Allen Hatcher]], p. 8 in: *Algebraic Topology*, Cambridge University Press (2002) [[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)]

A textbook which knows that the suspension of the [[empty set]] is the [[0-sphere]] (although even it regards this as an exception):

* Anatole Katok, Alexey Sossinsky, Chapter 1 of: *Introduction to Modern Topology and Geometry* (2010) [[toc pdf](http://akatok.s3-website-us-east-1.amazonaws.com/TOPOLOGY/Contents.pdf), [[KatokSossinsky-Topology-Ch1.pdf:file]]]

Discussion of the [[suspension type]] as a [[higher inductive type]] in [[homotopy type theory]]:

* [[Univalent Foundations Project]], §6.5 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

On the [[free loop space]] of a suspension:

* [[Ralph Cohen]], _A model for the free loop space of a suspension_ Lecture Notes in Mathematics, 1987, Volume 1286/1987, 193-207

On the question on what is the Eckmann-Hilton dual to $X\star Y$:

* D. B. Fuks, Eckmann--Hilton duality and the theory of functors in the category of topological spaces, 1966 Russ. Math. Surv. 21 1--33 [doi](http://dx.doi.org/10.1070/RM1966v021n02ABEH004149), [free Russian original pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=5847&what=fullt&option_lang=rus)

The above graphics is taken from

* {#Muro10} [[Fernando Muro]], _Representability of Cohomology Theories_, Joint Mathematical Conference CSASC 2010, 22–27 January 2010, Prague, Czech Republic [[slides](https://personal.us.es/fmuro/files/slides/praha.pdf), [[Muro-Representability.pdf:file]]]

[[!redirects suspension]]
[[!redirects suspensions]]
[[!redirects suspension functor]]

[[!redirects join of spaces]]
[[!redirects joins of spaces]]

[[!redirects suspensions]]


suspension isomorphism

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

#Contents#
* table of contents
{:toc}

## Idea

In [[cohomology]] of [[topological spaces]]/[[homotopy types]], the _suspension isomorphism_ identifies the degree-$n$ [[reduced cohomology]] of a [[pointed topological space|pointed]] space $X$ with the degree-$(n+k)$ cohomology of its $k$-fold [[suspension]], hence of its [[smash product]] with the $k$-[[sphere]]

$$
  \tilde H^n(X,E) \simeq \tilde H^{n+k}(S^k \wedge X, E)
  \,.
$$

In particular this serves to express cohomology in negative degree in terms of cohomology in non-negative degree of suspended spaces.

Requiring this to hold in [[equivariant cohomology theory]] not just for [[integer]] grading and spheres but also for [[RO(G)-grading]] and [[representation spheres]] leads to the concept of _genuine_ equivariant cohomology represented by [[genuine G-spectra]].

## Related concepts

* [[Freudenthal suspension theorem]]

* [[Steenrod square]]

* [[cohomology theory]]

## References

Textbook accounts include

* {#KonoTamaki02} Akira Kono, Dai Tamaki, p. 19 in _Generalized cohomology_, AMS 2002, esp. chapter 2 ([[GeneralizedCohomology.pdf:file]])

In [[equivariant homotopy theory]]:

* {#May96} [[Peter May]], chapter XI of _Equivariant homotopy and cohomology theory_, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. ([pdf](http://www.math.rochester.edu/u/faculty/doug/otherpapers/alaska1.pdf))

[[!redirects suspension isomorphisms]]


suspension object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Stable homotopy theory
+-- {: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

# Contents
* automatic table of contents goes here
{: toc}

## Definition

In a [[(∞,1)-category]] $C$ admitting a [[final object]] ${*}$, for any object $X$ its **suspension object** $\Sigma X$ is the [[homotopy pullback|homotopy pushout]]

$$
  \array{
     X &\to& {*}
     \\
     \downarrow && \downarrow
     \\
     {*} &\to& \Sigma X
  }
  \,,
$$

This is the [[mapping cone]] of the terminal map $X \to {*}$. See there for more details.

This concept is dual to that of [[loop space object]].

## Suspension functor

### As an (infinity,1)-functor

Let $C$ be a [[pointed (infinity,1)-category]].  Write $M^\Sigma$ for the [[(infinity,1)-category]] of [[pushout|cocartesian squares]] of the form

$$
  \array{
     X &\to& {*}
     \\
     \downarrow && \downarrow
     \\
     {*} &\to& Y
  }
  \,,
$$

where $X$ and $Y$ are [[objects]] of $C$.  Supposing that $C$ admits [[cofibres]] of all [[morphisms]], then one sees that the functor $M^\Sigma \to C$ given by evaluation at the initial vertex ($X$) is a [[trivial fibration]].  Hence it admits a [[section]] $s : C \to M^\Sigma$.  Then the **suspension functor** $\Sigma_C : C \to C$ is the [[composite]] of $s$ with the functor $M^\Sigma \to C$ given by evaluating at the final vertex ($Y$).

$\Sigma_C$ is [[left adjoint]] to the [[loop space functor]] $\Omega_C$.

For $X$ a pointed object of a [[Grothendieck (∞,1)-topos]] ${\mathcal{H}}$, the suspension object $\Sigma X$ is homotopy equivalent to $B{\mathbb{Z}}\wedge X$, the smash product by the classifing space of the discrete group of integers.

We outline a proof below. For $X$ a pointed object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, its _reduced free group_, denoted by $F[X]$, is the left adjoint to the functor $\Omega {\mathbf{B}}:Grp(\mathcal{H})\to \mathcal{H}_*$ which sends a [[groupoid object in an (∞,1)-category|group object internal]]  to ${\mathcal{H}}$ to the loop space of its delooping object.

+-- {: .num_prop #ConstructionOfFreeGroup}
###### Proposition
 For $X$ a pointed object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, there is a natural equivalence ${\mathbf{B}}F[X]\simeq \Sigma X$.
 =--

+-- {: .proof}
###### Proof
This is due to the adjunction $(\Sigma \vdash \Omega):\mathcal{H}_*\leftrightarrows\mathcal{H}_*$ between suspending and looping and the the adjunction $(\Omega \vdash {\mathbf{B}}):PathConn(\mathcal{H}_*)\leftrightarrows Grp(\mathcal{H})$ between looping and delooping. Indeed, for any group object $H$, the above-mentioned adjunctions imply the following natural equivalences:
$$
  \begin{aligned}
    Grp({\mathcal{H}})(\Omega \Sigma X, H)
    &  \simeq
    PathConn({\mathcal{H}}_*)(\Sigma X, {\mathbf{B}}H)
    \\
    & \simeq
    PathConn({\mathcal{H}}_*)(X, \Omega{\mathbf{B}}H)
    \,,
  \end{aligned}
$$
Hence $\Omega \Sigma X$ has the universal property of the reduced free group. Delooping gives the required result.
 =--

The [[(∞,1)-category]] $Grp(\mathcal{H})$ of group objects internal ${\mathcal{H}}$ is tensored over ${\mathcal{H}}_*$; in particular, for $G$ a group object and $X$ a pointed object, we can form the tensor product $X\otimes G$, which is a group object. Explicitly, this tensor product is required to satisfy a homotopy equivalence $Grp({\mathcal{H}})(\Omega (X\otimes G, H)\simeq PathConn({\mathcal{H}}_*)(X, Grp({\mathcal{H}})(G, H))$, natural in group objects $H$.

+-- {: .num_prop #ConstructionOfTensorProduct}
###### Proposition
 For $X$ a pointed object and $G$ a group object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, there is a natural equivalence ${\mathbf{B}}(X\otimes G)\simeq X\wedge {\mathbf{B}}G$.
 =--

+-- {: .proof}
###### Proof
This is due to the adjunction $(\Omega \vdash {\mathbf{B}}):PathConn(\mathcal{H}_*)\leftrightarrows Grp(\mathcal{H})$ between looping and delooping and the [[internal hom]] adjunction. Indeed, for any group object $H$, the above-mentioned adjunctions gives the following natural equivalences:
$$
  \begin{aligned}
    Grp({\mathcal{H}})(\Omega (X\wedge {\mathbf{B}}G), H)
    &  \simeq
    PathConn({\mathcal{H}}_*)(X\wedge {\mathbf{B}}G, {\mathbf{B}}H)
    \\
    & \simeq
    PathConn({\mathcal{H}}_*)(X, PathConn({\mathcal{H}}_*)({\mathbf{B}}G, {\mathbf{B}}H))
    \\
    & \simeq
    PathConn({\mathcal{H}}_*)(X, Grp({\mathcal{H}})(G, H))
    \,,
  \end{aligned}
$$
Hence $\Omega (X\wedge {\mathbf{B}}G)$ has the universal property of the tensor product. Delooping gives the required result.
=--

+-- {: .num_lemma #FreeGroupAsTensor}
###### Lemma
 For $X$ a pointed object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, there is a natural equivalence $F[X]\simeq X\otimes Z$, where $Z$ is the group object whose delooping object is $B {\mathbb{Z}}$, the classifying space of the discrete group of integers.
 =--

+-- {: .proof}
###### Proof
Since ${\mathcal{H}}$ is a Grothedieck $(\infty,1)$-topos, the $(\infty,1)$-functor $*\to {\mathbf{B}}-:Group(\mathcal{H})\to Func(\Delta^1,\mathcal{H})$ which sends a group object to the map from the terminal object to its delooping object is a $(\infty,1)$-categorial equivalence onto its image, which is the full subcategory of $Func(\Delta^1,\mathcal{H})$ spanned by the effective epimorphisms from the terminal object. Hence, for $H$ a group object, we have
$$
  \begin{aligned}
    Grp(\mathcal{H})(Z,H)
    &  \simeq
    Func(\Delta^1,{\mathcal{H}})(*\to B{\mathbb{Z}},*\to {\mathbf{B}}H)
    \\
    & \simeq
    {\mathcal{H}}_*(B{\mathbb{Z}},{\mathbf{B}}H)
    \,,
  \end{aligned}
$$
This latter based mapping object is equivalent to the based object of deloopable maps from ${\mathbb{Z}}$ to $\Omega{\mathbf{B}}H$, which is just $\Omega{\mathbf{B}}H$, since ${\mathbb{Z}}$ is the discrete free group on one generator.

Hence, there are the following natural equivalences:
$$
  \begin{aligned}
    Grp({\mathcal{H}})(F[X], H)
    &  \simeq
    PathConn({\mathcal{H}}_*)(X, \Omega{\mathbf{B}}H)
    \\
    & \simeq
    PathConn({\mathcal{H}}_*)(X, Grp(Z, H)
    \,,
  \end{aligned}
$$
Therefore $F[X]$ has the universal property of the tensor product $X\otimes Z$. The required natural equivalence follows by abstract nonsense.
=--

+-- {: .num_theorem #SuspendingAsSmashProd}
###### Theorem
 For $X$ a pointed object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, there is a natural equivalence $\Sigma X\simeq B{\mathbb{Z}}\wedge X$.
 =--

+-- {: .proof}
###### Proof
Deloop the natural equivalence in Lemma \ref{FreeGroupAsTensor} to obtain the natural equivalence ${\mathbf{B}}F[X]\simeq {\mathbf{B}}(X\otimes Z)$. By propositions \ref{ConstructionOfFreeGroup} and \ref{ConstructionOfTensorProduct}, this gives the required natural equivalence.
 =--

### As an ordinary functor

Let $C$ be a [[category]] admitting small [[colimits]].  Let $\Phi$ be a [[graded monoid]] in the [[category]] of [[groups]] and $F : C \to C$ a $\Phi$-symmetric [[endofunctor]] of $C$ that commutes with small [[colimits]].  Let $Spect_F^{\Phi}(C)$ denote the [[category]] of $\Phi$-symmetric $F$-[[spectrum objects]] in $C$.

Following [Ayoub](#Ayoub), the [[evaluation]] functor
  $$ Ev^n : Spect_F^{\Phi}(C) \to C, $$
which "evaluates" a symmetric spectrum at its $n$th component, admits under these assumptions a [[left adjoint]]
  $$ Sus^n : C \to \Spect_F^\Phi(C) $$
called the $n$th **suspension functor**, more commonly denoted $\Sigma_C^{\infty-n}$.

When $C$ is [[symmetric monoidal category|symmetric monoidal]], and in the case $\Phi = \Sigma$ and $F = T \otimes -$ for some object $T$, there is an induced [[symmetric monoidal structure]] on $Spect^\Sigma_T(C)$ as described at [[symmetric monoidal structure on spectrum objects]].

**Proposition.** One has
  $$ Sus^p_T(X) \otimes Sus^q_T(Y) \simeq Sus^{p+q}_T(X \otimes Y) $$
for all $X,Y \in C$.  In particular, $Sus = Sus^0 : C \to \Spect^\Sigma_T(C)$ is a [[symmetric monoidal functor]].

## Examples

* In [[Top]], this is the [[reduced suspension]] of a space.

* In a [[category of chain complexes]] the [[suspension of a chain complex]] is given by shifting the degrees of the chain complex up by one.

## Related concepts

* [[loop space object]], [[free loop space object]],

  * [[delooping]]

  * [[loop space]], [[free loop space]], [[derived loop space]]

* **suspension object**

  * [[suspension type]], [[reduced suspension type]]

  * [[suspension]], [[reduced suspension]]

## References

A detailed treatment of the 1-categorical case is in the last chapter of

* [[Joseph Ayoub]], _Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, I_. Astérisque, Vol. 314 (2008). Société Mathématique de France. ([pdf](http://user.math.uzh.ch/ayoub/PDF-Files/THESE.PDF))
{#Ayoub}

[[!redirects suspension object]]
[[!redirects suspension objects]]


suspension of a chain complex

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#### Homological algebra
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=--
#### Homotopy theory
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[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In a [[category of chain complexes]] $Ch_\bullet(\mathcal{A})$, the [[suspension object]] of a [[chain complex]] $C_\bullet$ is the complex

$$
  \Sigma C_\bullet = C[1]_\bullet
$$

(or sometimes denoted $C[-1]_\bullet$, depending on an unessential choice of sign convention) obtained by shifting the [[graded object|degrees]] up by one:

$$
  C[1]_n \coloneqq C_{n-1}
$$

with the [[differential]] the original one but equipped with a sign:

$$
  d^{C[1]}_n \coloneqq - d^C_{n-1}
  \,.
$$

Generally, for $p \in \mathbb{Z}$, $C[p]$ is the chain complex with

$$
  C[p]_n \coloneqq C_{n-p}
$$

$$
  d^{C[p]}_n \coloneqq (-1)^p d^C_{n-p}
  \,.
$$

## Related concepts

* [[mapping cone]]

* [[loop space object]], [[free loop space object]],

  * [[delooping]]

  * [[loop space]], [[free loop space]], [[derived loop space]]

* [[suspension object]]

  * [[suspension]], [[reduced suspension]]

  * [[Sullivan model of a suspension]]

[[!redirects suspension of chain complexes]]
[[!redirects suspensions of a chain complex]]
[[!redirects suspensions of chain complexes]]


suspension spectrum

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### Context
#### Stable Homotopy theory
+-- {: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

For $X$ a [[pointed object|pointed]] [[topological space]], its **suspension spectrum** $\Sigma^\infty X$ is the [[spectrum]] given by the pre-spectrum whose degree-$n$ space is the $n$-fold [[reduced suspension]] of $X$:

$$
  (\Sigma^\infty X)_n = \Sigma^n X
  \,.
$$

(e.g. [Elmendorf-Kriz-May, example 1.1](#ElmendorfKrizMay))

As a [[symmetric spectrum]]: ([Schwede 12, example I.2.6](#Schwede12))

## Properties

### Completion to an $\Omega$-spectrum

See at [Omega spectrum -- Completion of a suspension spectrum](Omega-spectrum#CompletionOfSuspensionSpectra).

### Relation to looping and stabilization

As an [[infinity-functor]] $\Sigma^\infty\colon Top_* \to Spec$ the suspension spectrum functor exhibits the [[stabilization]] of [[Top]].

$$
  (\Sigma^\infty \dashv \Omega^\infty)\colon
  Top_* \stackrel{\overset{\Omega^\infty}{\leftarrow}}{\underset{\Sigma^\infty}{\to}}
  Spec
$$

### Strong monoidalness
 {#StrongMonoidalness}

The suspension spectrum functor is [[strong monoidal functor|strong monoidal]].

On the one hand, this is the case for its incarnation as a [[1-functor]] with values in [[structured spectra]] ([this Prop.](Introduction+to+Stable+homotopy+theory+--+1-2#SmashProductOfFreeSpectra))
Via the corresponding [[symmetric monoidal smash product of spectra|symmetric monoidal]] [[Model categories of diagram spectra|model structure on structured spectra]] this exhibits strong monoidalness also as an [[(infinity,1)-functor]].

More abstractly this follows from general properties of [[stabilization]] when regarding [[stable homotopy theory]] as the result of inverting [[smash product]] with the [[circle]], via [Robalo 12, last clause of Prop. 4.1 with last clause of Prop. 4.10 (1) ](stabilization#Robalo12). For emphasis see also [Hoyois 15, section 6.1](stabilization#Hoyois15), specifically [Hoyois 15, Def. 6.1](stabilization#Hoyois15).

[[!include smash-monoidal diagonals -- section]]

## Related concepts

* [[free structured spectrum]]

* [[equivariant suspension spectrum]]

## References

* {#Adams74} [[Frank Adams]], part III, section 2 of _[[Stable homotopy and generalised homology]]_, 1974

* {#ElmendorfKrizMay} [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], example 1.1 of _[[Modern foundations for stable homotopy theory]]_, in [[Ioan Mackenzie James]], _[[Handbook of Algebraic Topology]]_, Amsterdam: North-Holland (1995) pp. 213–253,  ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

* {#Kuhn84} Nicholas J. Kuhn, _Suspension spectra and homology equivalences_, Trans. Amer.
Math. Soc. 283, 303–313 (1984) ([JSTOR](http://www.jstor.org/stable/2000005))

* {#Klein02} [[John Klein]], _Moduli of suspension spectra_ ([arXiv:math/0210258](http://arxiv.org/abs/math/0210258), [MO](http://mathoverflow.net/a/84686/381))

* {#Schwede12} [[Stefan Schwede]], Example I.2.6 in _Symmetric spectra_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec.pdf))

Suspension spectra of [[infinite loop spaces]] are discussed (in a context of [[Goodwillie calculus]] and [[chromatic homotopy theory]]) in

* Nicholas J. Kuhn, section 6.2 of _Goodwillie towers and chromatic homotopy: An overview_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/KuhnKinosaki.pdf))

Formalization of suspension spectra in [[dependent linear homotopy type theory]] (see also on the "[[exponential modality]]" [here](!-modality#RealizationInLinearHomotopyTypeTheory)):

* {#Riley22Thesis} [[Mitchell Riley]], Def. 2.1.29 & Rem. 2.4.13 in: *A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory*, PhD Thesis (2022) [[doi:10.14418/wes01.3.139](https://doi.org/10.14418/wes01.3.139)]

[[!redirects suspension spectrum]]
[[!redirects suspension spectra]]


suspension type

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[homotopy type theory]], a _suspension type_ of any [[type]] $X$ is the [[pushout type]] of the terminal function $X \to \ast$ (to the [[unit type]]) along itself, hence the pushout of the [[span]] $\ast \longleftarrow X \longrightarrow \ast$.

The [[categorical semantics]] is given by [[suspension objects]] in [[homotopy theory]] (and hence by [[suspension]] of [[topological spaces]] in, say, the [[classical model structure on topological spaces]]).

## Definition
 {#Definition}

As a [[higher inductive type]], the (unreduced) suspension $\mathrm{S} X$ of a type $X$ is generated by

1. A term $\mathrm{nth} \colon \mathrm{S} X$

1. a term $\mathrm{sth} \colon \mathrm{S} X$

1. A function $mer \colon X \to Id_{\mathrm{S}X}\big(nth,\, sth\big)$

   (to the [[identity type]] of $\mathrm{S}X$).

> (graphics adapted from [Muro 2010](suspension#Muro10))

The resulting [[inference rules]] are the specialization of those of [[pushout types]] (see [there](pushout+type#InferenceRules)).

In [[Coq]] [[pseudocode]] this becomes

    Inductive Suspension (X \colon Type) : Type
      | nth : Suspension X
      | sth : Suspension X
      | mer : X -> Id Suspension X nth sth

This says that the type is [[dependent type|dependent]] on the type $X$ and inductive constructed from two [[terms]] in the suspension, whose [[semantics|interpretation]] is as the north and south poles of the suspension, together with a term in the [[function type]] from $X$ to the [[identity type]] of paths between these two terms, representing the [[meridians]] from the north to the south pole.

## Examples

* The [[two-valued type]] $\mathbf{2}$ is the suspension type of the [[empty type]] $\mathbf{0}$.

* The [[interval type]] $I$ is the suspension type of the [[unit type]] $\mathbf{1}$.

* The [[circle type]] $S^1$ is the suspension type of $\mathbf{2}$.

* The homotopical disk type $G_2$ is the suspension type of $I$.

* Homotopical $n$-sphere types of dimension $n:\mathbb{N}$, $S^n$, are suspension types of $S^{n-1}$.

* Homotopical $n$-[[globe]] types of dimension $n:\mathbb{N}$, $G_n$, are suspension types of $G_{n-1}$.

## Related concepts

* [[suspension object]]

  * **suspension type**, [[reduced suspension type]]

  * [[suspension]], [[reduced suspension]]

* [[homotopy pushout]]

* [[acyclic type]]

## References

* [[Univalent Foundations Project]], §6.5 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

[[!redirects suspension types]]


suspensions are H-cogroup objects

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###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[suspension objects|Suspension objects]] are canonically [[cogroup objects]] up to [[homotopy]], via their "pinch map". In particular this is the case for positive dimensional [[n-spheres]].

## Statement

Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-colimits]] and with a [[zero object]]. Write $\Sigma \colon X \mapsto 0 \underset{X}{\coprod} 0$ for the [[reduced suspension]] functor.

Then the pinch map

$$
  \Sigma X
  \simeq
  0 \underset{X}{\sqcup} 0
  \simeq
  0 \underset{X}{\sqcup} X \underset{X}{\sqcup} 0
    \longrightarrow
  0 \underset{X}{\sqcup} 0 \underset{X}{\sqcup} 0
    \simeq
  \Sigma X \coprod \Sigma X
$$

exhibits a [[cogroup]] structure on the image of $\Sigma X$ in the [[homotopy category of an (∞,1)-category|homotopy category]] $Ho(\mathcal{C})$.

This is equivalently the [[group]]-structure of the first ([[fundamental group|fundamental]]) [[homotopy group]] of the values of the [[representable functor|functor co-represented]] by $\Sigma X$:

$$
  Ho(\mathcal{C})(\Sigma X, -)
   \;\colon\;
  Y
   \mapsto
  Ho(\mathcal{C})(\Sigma X, Y)
   \simeq
  Ho(\mathcal{C})(X, \Omega Y)
   \simeq
  \pi_1 Ho(\mathcal{C})(X, Y)
  \,.
$$

## Examples

### Positive-dimensional spheres in $\infty Grpd$

All [[n-spheres]], regarded as their [[homotopy types]] in $\mathcal{C} = $[[∞Grpd]] suspensions for $n \geq 1$: $S^{n+1}\simeq \Sigma S^n$, hence they carry [[cogroup]] structure in the classical [[homotopy category]]. Under the $\Sigma \dashv \Omega$-[[adjunction]], this cogroup structure turns into the [[group]] structure on all [[homotopy groups]] in positive degree.

[[!redirects positive dimension spheres are H-cogroup objects]]


Susumu Hayashi

* [PA/LCM page](https://shayashiyasugi.com/wwwshayashijp/PALCM/)

## Selected writings

On [[adjoint pairs]] of [[semifunctors]] in the context of [[categorical semantics]] for a version of the [[lambda calculus]], and introducing the terminology (if not the notion) of *[[Karoubi envelopes]]*:

* {#Hayashi1985} [[Susumu Hayashi]], Def. 1.2 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]

category: people


Suvrat Raju

* [personal page](http://www.suvratraju.net/)

## Selected writings

On breakdown of the [[locality]] due to ill-behaved [[perturbative quantum field theory|perturbation series]] of [[perturbative quantum gravity]] and [[perturbative string theory]] for [[n-point functions]] in the limit of large $n$:

* [[Sudip Ghosh]], [[Suvrat Raju]], *Loss of locality in gravitational correlators with a large number of insertions*, Phys. Rev. D 96, 066033 (2017) ([arXiv:1706.07424](https://arxiv.org/abs/1706.07424))

* [[Sudip Ghosh]], [[Suvrat Raju]], *The Breakdown of String Perturbation Theory for Many External Particles*, Phys. Rev. Lett. 118, 131602 (2017) ([arXiv:1611.08003](https://arxiv.org/abs/1611.08003))

Argument that the [[black hole information paradox]] is not resolved by "islands" unless for massive gravity:

* [[Hao Geng]], [[Andreas Karch]], [[Carlos Perez-Pardavila]], [[Suvrat Raju]], [[Lisa Randall]], [[Marcos Riojas]], [[Sanjit Shashi]], *Inconsistency of islands in theories with long-range gravity*, J. High Energ. Phys. **2022** 182 (2022) [doi:10.1007/JHEP01(2022)182]

* [[Suvrat Raju]], *Failure of the split property in gravity and the information paradox*,  Classical and Quantum Gravity, **39** 064002 [[doi:10.1088/1361-6382/ac482b](https://iopscience.iop.org/article/10.1088/1361-6382/ac482b), [arXiv:2110.05470](https://arxiv.org/abs/2110.05470)]

and on what to do about it:

* [[Hao Geng]], [[Andreas Karch]], [[Carlos Perez-Pardavila]], [[Suvrat Raju]], [[Lisa Randall]], [[Marcos Riojas]], [[Sanjit Shashi]], *Information Transfer with a Gravitating Bath*, SciPost Phys. **10** 103 (2021)  \[arXiv:2012.04671, [doi:10.21468/SciPostPhys.10.5.103](https://scipost.org/10.21468/SciPostPhys.10.5.103)\]

category: people


Suyay Ashok

* [webpage](https://www.perimeterinstitute.ca/people/sujay-ashok)

## related $n$Lab entries

* [[landscape of string theory vacua]]

category: people


SVect > history

> see at _[[super vector space]]_


svelte category

A [[category]] is __svelte__ if the [[class]] of objects of its [[skeleton]]
 is a [[set]]. In other words svelte category is a category equivalent to a small category. The expression is probably due to Pierre Gabriel.

This is the same as an [[essentially small category]].


Sven Bachmann

* [personal page](https://personal.math.ubc.ca/~sbach/)

## Selected writings

On the [[quantum adiabatic theorem]] in [[condensed matter theory]] with an eye towards [[topological phases of matter]]:

* [[Sven Bachmann]], Spyridon Michalakis, [[Bruno Nachtergaele]], Robert Sims, *Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems*, Commun. Math. Phys. **309** (2012) 835-871 ([arXiv:1102.0842](https://arxiv.org/abs/1102.0842), [doi:10.1007/s00220-011-1380-0](https://doi.org/10.1007/s00220-011-1380-0))

* [[Sven Bachmann]], [[Wojciech De Roeck]], [[Martin Fraas]], *Adiabatic Theorem for Quantum Spin Systems*, Phys. Rev. Lett. **119** (2017) 060201 ([doi:10.1103/PhysRevLett.119.060201](https://doi.org/10.1103/PhysRevLett.119.060201))

* [[Sven Bachmann]], [[Wojciech De Roeck]], [[Martin Fraas]], *The adiabatic theorem in a quantum many-body setting*, in: *Analytic Trends in Mathematical Physics*, Contemporary Mathematics **741** (2020) 43-58 ([arXiv:1808.09985](https://arxiv.org/abs/1808.09985), [ISBN:978-1-4704-5388-6](https://bookstore.ams.org/conm-741))

category: people


Sven Bjarke Gudnason

* [webpage](http://bjarke.gudnason.net/)

## Selected writings

On [[Skyrmions]] in view of the [[complex Hopf fibration]]:

* {#GudnasonNitta20} [[Sven Bjarke Gudnason]], [[Muneto Nitta]], _Linking number of vortices as baryon number_ ([arXiv:2002.01762](https://arxiv.org/abs/2002.01762), [spire:1778698/](http://inspirehep.net/record/1778698/))

On [[Skyrmions]]:

* {#GudnasonHalcrow18} [[Sven Bjarke Gudnason]], [[Chris Halcrow]], *Vibrational modes of Skyrmions*, Phys. Rev. D **98** 125010 (2018) [[arXiv:1811.00562](https://arxiv.org/abs/1811.00562), [doi:10.1103/PhysRevD.98.125010](https://doi.org/10.1103/PhysRevD.98.125010)]

  *Database of Skyrmion Vibrations* [[www1.maths.leeds.ac.uk/pure/geometry/SkyrmionVibrations/](http://www1.maths.leeds.ac.uk/pure/geometry/SkyrmionVibrations/)]

On the [[Skyrmion model]] with [[omega-meson]]-contribution:

* {#GudnasonSpeight20} [[Sven Bjarke Gudnason]], James Martin Speight, _Realistic classical binding energies in the $\omega$-Skyrme model_ ([arXiv:2004.12862](https://arxiv.org/abs/2004.12862))

On [[baryons]] in the [[Sakai-Sugimoto model]] of [[holographic QCD]]:

On [[baryons]] in the [[Sakai-Sugimoto model]] of [[holographic QCD]]:

* Salvatore Baldino, [[Lorenzo Bartolini]], [[Stefano Bolognesi]], [[Sven Bjarke Gudnason]], _Holographic Nuclear Physics with Massive Quarks_,  Phys. Rev. D **103** 126015 (2021) \[doi:10.1103/PhysRevD.103.126015, arXiv:2102.00680\]

Computer-survey of [[Skyrmion]]-solutions:

* {#GudnasonHalcrow22} [[Sven Bjarke Gudnason]], [[Chris Halcrow]], *A Smörgåsbord of Skyrmions* [[arXiv:2202.01792](https://arxiv.org/abs/2202.01792)]

On [[skyrmions]]/[[solitons]] in the [[Sakai-Sugimoto model]] ([[AdS/QCD correspondence]] via [[D=5 Maxwell theory]]):

* Lorenzo Bartolini, [[Stefano Bolognesi]], [[Sven Bjarke Gudnason]], Tommaso Rainaldi, *Mass and Isospin Breaking Effects in the Skyrme Model and in Holographic QCD* \[arXiv:2312.15404\]

category: people

[[!redirects Sven Gudnason]]


Sven Möller

* [personal page](https://www.math.uni-hamburg.de/home/moeller/home.html)

## Selected writings

On classification of strongly rational holomorphic [[vertex operator algebras]] of [[central charge]] $\leq 24$ (of relevance in [[heterotic string theory]] and [[monstrous moonshine]]):

* [[Gerald Höhn]], [[Sven Möller]], *Classification of Self-Dual Vertex Operator Superalgebras of Central Charge at Most 24* [[arXiv:2303.17190](https://arxiv.org/abs/2303.17190)]

category: people

[[!redirects Sven Moeller]]

[[!redirects Sven Moller]]


Sven van Nigtevecht

* [personal page](https://svann.science/)

* [institute page](https://www.uu.nl/medewerkers/svannigtevecht)

## Selected writings

On the [[K-theory classification of topological phases of matter]]:

* [[Sven van Nigtevecht]], *Topological phases and K-theory*, Amsterdam (2019) $[$[web](https://svann.science/publication/bachelors), [pdf](https://webspace.science.uu.nl/~0554804/publications/bachelor.pdf)$]$

category: people


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