finitary monad

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## Definition

A [[monad]] $(T,\mu,i)$, is __finitary__ if the underlying [[endofunctor]] $T: C \to C$ preserves [[filtered colimits]].

In other words, a finitary monad is a [[monoid]] in the category of finitary endofunctors on some category.

## Finitary monads and Lawvere theories

A finitary monad $(T,\mu,i)$ on [[Set]] is completely determined by its value on all finite [[ordinal]]s $n\in\mathbb{N}_0$ considered as standard [[finite set]]s.  $T(n)$ is then the set of $n$-ary operations.  The notion of algebraic monad is hence similar to the notion of a nonsymmetric [[operad]] in $\mathrm{Set}$, but it is not equivalent, because of the possibility of duplicating or discarding inputs.

More precisely, each finitary monad $T$ defines a [[Lawvere theory]] $Th_T$, namely $Th_T = Free_{fin}^{op}$ where $Free_{fin}$ is the category of [[free construction|free]] algebras $T(n)$ on finite sets (as a full subcategory of $Alg_T$). In fact, the two notions are equivalent: the assignment

$$T \mapsto Th_T$$

defines an equivalence between the category of finitary monads on $Set$ and the category of Lawvere theories. Moreover, the category of $T$-algebras is equivalent to the [[category of models]] of $Th_T$. However, a technical advantage of Lawvere theories is that they can be interpreted in categories other than [[Set]]: a model of a Lawvere theory $\mathcal{T}$ in a category with cartesian products $C$ is just a product-preserving functor $\mathcal{T} \to C$.

## Properties

- The category of finitary monads on a [[locally finitely presentable category|locally finitely presentable]] $\mathscr{A}$ category is also [[locally finitely presentable category|locally finitely presentable]], and is [[monadic]] over $[|\mathscr{A}|, \mathscr{A}]$ (see [Lack](#Lack)).

## Applications

There is an interesting commutativity condition singling out the subclass of commutative algebraic/finitary monads, cf. [[commutative algebraic theory]]; they are useful to establish a theory of [[generalized scheme after Durov|generalized commutative schemes]].

## Related concepts

* [[classifying topos for the theory of objects]]

## References

For a proof of the equivalence between finitary monads and Lawvere theories, see

* J. Adamek and J. Rosicky, _[[Locally presentable and accessible categories]]_ (chapter 3), LMS Lecture Notes 189, Cambridge U. Press, 1994.

For properties of the category of finitary monads, see

* [[Stephen Lack]]. _On the monadicity of finitary monads_. Journal of Pure and Applied Algebra 140.1 (1999): 65-73.

Durov's application of finitary monads to the "field with one element" may be found in:

* {#Durov07} [[Nikolai Durov]], _A new approach to Arakelov geometry_, [arxiv/0704.2030](http://arxiv.org/abs/0704.2030)

## Discussion

This discussion appeared when the page was at [[algebraic monad]].

+--{: .query}
[[Mike Shulman|Mike]]: Does anyone besides Durov use this terminology?  In category theory these already have two standard names: "finitary monads" and "(Lawvere) theories."

[[Zoran Škoda]] Lawvere theories are equivalent to algebraic monads, but not in standard exposition literally a type of monad, but one says algebraic monad when one really talks about a condition on monads. Algebraic monad is just a monoid in the category of algebraic endofunctors. Lawvere and others studied few variants of notion of algebraic functor, but in all of them algebraic functor commutes with filtered colimits, isn't it ? I heard the term certainly in a number of algebra talks before [[Nikolai Durov]]'s thesis (I can say some names but I may misremeber sources). Definitely the term is getting much more influential after the Durov's work, and reused by algebraic geometers now. He is quite familiar with Lawvere's work so I hope the related expression "algebraic endofunctor" is not chosen incompatibly. Also there is some parallel with terminology [[cartesian monad]]. But what do you think ?
In any case, one should also list term finitary monad, both synonyms should be listed.

_Toby_:  Whether these are 'algebraic' or 'finitary' (I\'d be more inclined to the latter, but not through any familiarity with the literature), they\'re not obviously the same as Lawvere theories.  I\'ve tried to indicate the connection, although (vague as my statement is) I\'m not even sure that it\'s correct!

[[Zoran Škoda]] added below redirect [[finitary monad]].

_Toby_:  I moved this to [[finitary monad]] now, after Zoran made a link to it using that name from [[generalized scheme after Durov|an article on Durov's work]].
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[[!redirects finitary monad]]
[[!redirects finitary monads]]
[[!redirects algebraic monad]]
[[!redirects algebraic monads]]


finite (infinity,1)-category

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## Idea

The concept of _finite [[(∞,1)-category]]_ is the generalization of [[finite homotopy type]] from [[∞-groupoids]]([[(∞,0)-categories]]) to [[(∞,1)-categories]].

If we model [[(∞,1)-categories]] by [[quasicategories]], then this can be made precise by saying it is equivalent to the fibrant replacement in the [[Joyal model structure]] of a [[simplicial set]] with finitely many nondegenerate [[simplices]] (in the [[∞-groupoid]] case also: [[finite CW-complex]]).

## Related concepts

* [[finite category]]

* [[finite (∞,1)-limit]]

* [[finite homotopy type]]

* [[finite (∞,1)-category]]

[[!redirects finite (infinity,1)-categories]]

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


finite (infinity,1)-limit

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# Finite $(\infty,1)$-limits
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## Definition

A **finite $(\infty,1)$-limit** is an [[(∞,1)-limit]] over a finitely presented [[(∞,1)-category]] -- a [[finite (∞,1)-category]].

If we model our (∞,1)-categories by [[quasicategories]], then this can be made precise by saying it is a limit over some [[simplicial set]] with finitely many nondegenerate [[simplices]].  Note that such a simplicial set is rarely itself a quasicategory; we regard it instead as a finite presentation of a quasicategory.

## Properties

### Preservation of finite $(\infty,1)$-limits
 {#Preservation}

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

An [[(∞,1)-functor]] $F : C \to D$ out of an [[(∞,1)-category]] $C$ that has all finite $(\infty,1)$-limits preserves these finite $(\infty,1)$-limits as soon as it preserves [[(∞,1)-pullbacks]] and the [[terminal object]].

=--

This appears as ([Lurie, cor. 4.4.2.5](#Lurie)).

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

Let $C$ be a [[small (∞,1)-category]] with finite $(\infty,1)$-limits, and $\mathbf{H}$ an [[(∞,1)-topos]]. Write $PSh(C)$ for the [[(∞,1)-category of (∞,1)-presheaves]] on $C$.

If a functor $F : PSh_\infty(C) \to \mathbf{H}$ preserves [[(∞,1)-colimits]] and finite $(\infty,1)$-limits of [[representable functor|representables]], then it preserves all finite $(\infty,1)$-limits.

=--

This appears as ([Lurie, prop. 6.1.5.2](#Lurie)).

## Examples

* Binary products, pullbacks, and terminal objects are all finite $(\infty,1)$-limits.

* Unlike the case in [[1-category]] theory, the [[split idempotent|splitting of idempotents]] is *not* a finite $(\infty,1)$-limit.

## Related pages

* [[finite limit]]
* [[(∞,1)-limit]]
* [[finitely complete (infinity,1)-category]]

## References

* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_

Relation to [[homotopy type theory]]:

* [[Chris Kapulkin]], [[Karol Szumiło]], _Internal Language of Finitely Complete $(\infty,1)$-categories_ ([arXiv:1709.09519](https://arxiv.org/abs/1709.09519))

[[!redirects finite (infinity,1)-limit]]
[[!redirects finite (infinity,1)-limits]]
[[!redirects finite (∞,1)-limit]]
[[!redirects finite (∞,1)-limits]]
[[!redirects finite (infinity,1)-colimit]]
[[!redirects finite (infinity,1)-colimits]]
[[!redirects finite (∞,1)-colimit]]
[[!redirects finite (∞,1)-colimits]]


finite abelian category

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# Finite abelian categories
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## Definition

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

Let $k$ be a [[field]], and let $\mathcal{C}$ be a $k$-linear [[abelian category]] (i.e. one whose [[Ab-enriched category|Ab-enrichment]] is lifted to a [[Vect]]-[[enriched category|enrichment]]). Then $\mathcal{C}$ is called [[Etingof & Ostrik 2003 p. 3](#EtingofOstrik03)] *finite* (over $k$) if

* For any two [[objects]] $a$, $b$ of $C$, the [[hom-object]] ($k$-[[vector space]]) $\hom(a, b)$ has [[finite]] [[dimension]];

* Each object $a$ is of [[object of finite length|finite length]];

* There are only [[finite number|finitely many]] [[simple objects]] in $C$, and each of them admits a [[projective presentation]].

=--

+-- {: .num_theorem #Deligne}
###### Theorem (Deligne)

For any finite abelian category $C$, there exists a finite-dimensional $k$-algebra $A$ and an $k$-linear equivalence between $C$ and $A$-$Mod_{fd}$, the [[category of modules]] over $A$ that are finite-dimensional as vector spaces over $k$.

=--

## Related concepts

* [[semisimple category]]

* [[finite-dimensional vector space]]

* [[finite-dimensional Hilbert space]]

## References

* {#EtingofOstrik03} [[Pavel Etingof]], [[Victor Ostrik]], *Finite Tensor Categories* [[arXiv:math/0301027](https://arxiv.org/abs/math/0301027)]

* {#EGNO15} [[nLab:Pavel Etingof]], Shlomo Gelaki, Dmitri Nikshych, [[nLab:Victor Ostrik]], chapter 6 of _Tensor categories_, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society (2015) [[pdf](http://www-math.mit.edu/~etingof/egnobookfinal.pdf)]

[[!redirects finite abelian category]]
[[!redirects finite abelian categories]]


finite abelian group

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## Definition

A **finite abelian group** is a [[group]] which is both [[finite group|finite]] and [[abelian group|abelian]].

## Properties

As any [[finite group]], a finite abelian group is [[torsion subgroup|pure torsion]].

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

If a finite abelian group $A$ has [[order of a group|order]] ${\vert A \vert} = p$ a [[prime number]], then it is the [[cyclic group]] $\mathbb{Z}_p$.

=--

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

If $A$ is a finite abelian group and $p \in \mathbb{N}$ is a [[prime number]] that divides the [[order of a group|order]] ${\vert A \vert}$, then equivalently

* $A$ has an element of [[order of an element|order]] $p$;

* $A$ has a [[subgroup]] of [[order of a group|order]] $p$.

=--

This is [[Cauchy's theorem]] restricted to abelian groups.

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

We prodeed by [[induction]] on the [[order of a group|order]] of $A$. For ${\vert A \vert} = 2$ we have that $A = \mathbb{Z}_2$ is the unique [[group of order 2]] and the statement holds for $p =2$.

Assume then that the statement has been show for groups of order $\lt n$ and let ${\vert A \vert} = n$.

If $A$ has no non-trivial proper [[subgroup]] then $n$ must be prime and $A = \mathbb{Z}_n$ a [[cyclic group]] and the statement follows.

If $A$ does have a non-trivial proper [[subgroup]] $H \hookrightarrow A$ then $p$ divides either ${\vert H \vert}$ or $\vert A/H\vert$.

In the first case by [[induction]] assumption $H$ has an element of order $p$ which is therefore also an element of $G$ of order $p$.

In the second case there is by induction assumption an element $a \in A$ such that $a + H \in A/H$ has order $p$. Since the order of $a + H \in A/H$ divides the order of $a \in A$ it follows that $a$ has order $k p$ for some $k \in \mathbb{N}$. Then $k a$ has order $p$.

=--

+-- {: .num_theorem #FiniteAbelianGroupIsDirectSumOfCyclics}
###### Theorem
**([[fundamental theorem of finite abelian groups]])**

Every finite abelian group is the [[direct sum of abelian groups|direct sum]] of [[cyclic groups]] of [[prime power]] [[order of a group|order]] (its [[p-primary groups]]).

=--

See for instance ([Sullivan](#Sullivan)).

## References

A new proof of the [[fundamental theorem of finite abelian groups]] was given in

* {#Navarro03} Gabriel Navarro, _On the fundamental theorem of finite abelian groups_, Amer. Math. Monthly, February 2003

reviewed in

* {#Sullivan} John Sullivan, _Classification of finite abelian groups_ ([pdf](http://www.isama.org/jms/m317/handouts/finabel.pdf))

[[!redirects finite abelian groups]]


finite category

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# Finite categories
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## Definition

A **finite category** $C$ is a [[category]] [[internal category|internal to]] the category [[FinSet]] of finite sets.

This means that a finite category consists of

* a finite set of [[object]]s;
* a finite set of [[morphism]]s.

(Notice that the latter implies the former, since for every object there is the [[identity morphism]] on that object). Finite categories form the [[2-category]], [[FinCat]].

Similarly, a **locally finite category** is a category [[enriched category|enriched]] over $Fin Set$, that is a category whose [[hom-set]]s are all finite.

(Locally) finite categories may also be called (locally) **$\omega$-small**; this generalises from $\omega$ (the set of [[natural number]]s) to (other) [[inaccessible cardinal]]s (or, equivalently, [[Grothendieck universe]]s).

## Limits and colimits

One is often interested in whether an arbitrary category $D$ has [[limit]]s and [[colimit]]s indexed by finite categories.  A category with all finite limits is called **[[finitely complete category|finitely complete]]** or **left exact** (or **lex** for short).  A category with all finite colimits is called **finitely cocomplete** or **right exact**.

## In dependent type theory

In dependent type theory, there are multiple notions of category. Because the type of objects in a category is required to be a [[finite set]], all finite categories are [[strict categories]], and the only finite [[univalent categories]] are the finite [[gaunt categories]].

## Related concepts

* [[finite set]]

* [[finite graph]]

* [[homotopy type with finite homotopy groups]]

* [[finite homotopy type]]

* [[finite (∞,1)-category]]

* [[locally finite type]]

* [[finitely complete category]], [[finitely cocomplete category]]

[[!redirects finite category]]
[[!redirects finite categories]]
[[!redirects locally finite category]]
[[!redirects locally finite categories]]


finite choice

\tableofcontents

## Overview

The **principle of finite choice** states that the [[dependent product]] of a [[finite set|finite]] [[indexed set|family]] of [[inhabited sets]] is itself inhabited.

Finite choice is provable in most set theories, including [[strongly predicative]] and [[constructive mathematics|constructive]] set theories, as well as set theories without [[quotient sets]].

In [[dependent type theory]], finite choice is definable and provable, provided one has [[identity types]], [[dependent sum types]], [[bracket types]], [[finite sets]], and [[dependent tuple types]] in the type theory:

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma, i:\mathrm{Fin}(n) \vdash A(i) \; \mathrm{type}}{\Gamma, a:(i:\mathrm{Fin}(n)) \to \left[A(i)\right] \vdash \mathrm{finitechoice}(a):\left[(i:\mathrm{Fin}(n)) \to A(i)\right]}$$

One could express the principle of finite choice as two separate cases representing the case with zero and two inhabited sets:

* Every [[singleton]] is an [[inhabited set]]

* The [[product]] of two [[inhabited sets]] is inhabited.

In [[dependent type theory]], the former follows from the definition of a singleton or [[contractible type]] as an inhabited [[mere proposition]], and the latter is given by the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, a:[A], b:[B] \vdash \mathrm{binarychoice}(a, b):\left[A \times B\right]}$$

## See also

* [[countable choice]]
* [[axiom of choice]]

## References

* *Which axioms of ZF are used for finite choice?*, MathOverflow, ([web](https://mathoverflow.net/questions/179361/which-axioms-of-zf-are-used-for-finite-choice))

* *Finite choice without AC*, Mathematics Stackexchange, ([web](https://math.stackexchange.com/questions/64237/finite-choice-without-ac))

* *Finite axiom of choice: how do you prove it from just ZF?*, MathOverflow, ([web](https://mathoverflow.net/questions/32538/finite-axiom-of-choice-how-do-you-prove-it-from-just-zf))

[[!redirects finite choice]]
[[!redirects axiom of finite choice]]
[[!redirects principle of finite choice]]


finite cover

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## Idea

A _finite cover_ is a [[cover]] by a [[finite set]] of patches.

A _finite open cover_ is an [[open cover]] with a [[finite set]] of patches.
Finite open covers appear in the definition of [[compact topological spaces]]

## Related concepts

* [[locally finite cover]]

* [[countable cover]]

* [[compact topological space]]

* [[quasi-topological space]]

[[!redirects finite covers]]

[[!redirects finite open cover]]
[[!redirects finite open covers]]


finite CW complex

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## Definition

A _finite CW-complex_ is a [[CW-complex]] with a [[finite number]] of [[attaching maps]].

The [[homotopy type]] of a finite CW-complex is called a _[[finite homotopy type]]_.

## Related concepts

* [[finite set]], [[finite number]]

* [[finite topological space]]

* [[finite spectrum]]

* [[finite type]]

[[!redirects finite CW complexes]]

[[!redirects finite CW-complex]]
[[!redirects finite CW-complexes]]


finite element method

(under construction)

Finite element methods are certain discretized approximation schemes for [[partial differential equation]]s.

* wikipedia: [finite element method](http://en.wikipedia.org/wiki/Finite_element_method), [extended finite element method](http://en.wikipedia.org/wiki/Extended_finite_element_method), [list of finite element software packages](http://en.wikipedia.org/wiki/List_of_finite_element_software_packages)

* $n$Lab entries: [[homological algebra in finite element method]].

[[!redirects finite element methods]]


finite field

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## Idea

A [[field]] with [[finite set|finitely]] many elements.

## Structure

Let $F$ be a finite field. As $\mathbb{Z}$ is the [[initial object]] in the category of [[rings]], there is a unique ring [[homomorphism]] $\mathbb{Z} \to F$, whose [[(epi, mono) factorization system|regular epi-mono factorization]] is

$$\mathbb{Z} \twoheadrightarrow \mathbb{Z}/(p) \hookrightarrow F;$$

here $p$ is [[prime number|prime]] (irreducible) since any subring of a field has no [[zero divisors]]. Here $\mathbb{Z}/(p)$ is a [[prime field]], usually denoted $\mathbb{F}_p$.

Thus we have an inclusion of fields $\mathbb{F}_p \to F$; in particular $F$ is an $\mathbb{F}_p$-[[module]] or [[vector space]], clearly of finite dimension $n$. It follows that $F$ has $q = p^n$ elements.

In addition, since the multiplicative group $F^\times$ is [[cyclic group|cyclic]] (as shown for example at [[root of unity]]), of order $q - 1$, it follows that $F$ is a [[splitting field]] for the polynomial $x^{q-1} - 1 \in \mathbb{F}_p[x]$. As splitting fields are unique up to isomorphism, it follows that up to isomorphism there is just one field of cardinality $q = p^n$; it is denoted $\mathbb{F}_q$.[^fine]

[^fine]: There is no 'canonical' choice of such a splitting field, just as there is no canonical choice of algebraic closure. So 'morally' there is something wrong with saying '[[the]]' finite field $\mathbb{F}_q$, although this word usage can be found in the literature. In the special case $n = 1$ there is no problem: Given fields containing $p$ elements are isomorphic _in a unique way_.

The [[Galois group]] $Gal(\mathbb{F}_{p^n}/\mathbb{F}_p)$ is a cyclic group of order $n$, generated by the [[automorphism]] $\sigma: \mathbb{F}_{p^n} \to \mathbb{F}_{p^n}$ sending $x \mapsto x^p$. (One way to see that $\sigma$ preserves addition is to write ([[binomial theorem]])

$$\array{
(x + y)^p & = & \sum_{i=0}^p \binom{p}{i} x^i y^{p-i} \\
 & = & x^p + y^p
}$$

where the second equation follows from the fact that the integer $p$ divides the numerator of $\frac{p!}{i!(p-i)!}$, but neither factor of the denominator, if $0 \lt i \lt p$. This is true for any [[commutative algebra]] over $\mathbb{F}_p$; see [[freshman's dream]].)

This $\sigma$ is called the Frobenius (auto)morphism or [[Frobenius map]]. More generally, if $m$ divides $n$, then $\mathbb{F}_{p^m}$ is the fixed field of the automorphism $\sigma^m: x \mapsto x^{p^m}$, and $Gal(\mathbb{F}_{p^n}/\mathbb{F}_{p^m})$ is a cyclic group of order $n/m$ that is generated by this automorphism, which is also called the Frobenius map (for the field extension $\mathbb{F}_{p^n}/\mathbb{F}_{p^m}$), or just "the Frobenius" for short.

If $K = \widebar{\mathbb{F}_p}$ is an [[algebraic closure]] of $\mathbb{F}_p$, then $K$ is the union (a [[filtered colimit]]) of the system of such finite field extensions $\mathbb{F}_q$ and inclusions between them. If $q = p^n$, then $\mathbb{F}_q$ may be defined to be the fixed field of the automorphism $\sigma^n: K \to K$.

The Galois group $Gal(K/\mathbb{F}_p)$ is the inverse limit of the system of finite cyclic groups and projections between them; it is isomorphic to the profinite completion

$$\widehat{\mathbb{Z}} = \hom(\mathbb{Q}/\mathbb{Z}, \mathbb{Q}/\mathbb{Z}) \cong \prod_{primes\; p} \mathbb{Z}_p$$

where $\mathbb{Z}_p$ is the group of $p$-[[p-adic integers|adic integers]].

## Related concepts

* [[modular arithmetic]], [[number theory]], [[field of one element]]

## References

* Wikipedia, _[Finite field](http://en.wikipedia.org/wiki/Finite_field)_
* James Ax, _The elementary theory of finite fields_, Annals of Math. Series 2. 88 (2): 239–271 (1968) [doi](https://doi.org/10.2307/1970573)

category: algebra
[[!redirects finite fields]]


finite graph

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## Idea

A _[[graph]]_ is _finite_ if both its [[set]] of [[vertices]] as well as its set of [[edges]] are [[finite sets]].

## Related concepts

* [[finite set]]

* [[finite category]]

## References

See also

* MathWorld, _[Finite graph](http://mathworld.wolfram.com/FiniteGraph.html)_

[[!redirects finite graphs]]


finite group

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# Contents
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## Definition

A **finite group** is a [[group]] whose [[underlying]] [[set]] is [[finite set|finite]].

This is equivalently a [[group object]] in [[FinSet]].

## Properties

### Cauchy's theorem

Let $G$ be a finite group with [[order of a group|order]] ${\vert G\vert} \in \mathbb{N}$.

+-- {: .num_theorem}
###### Theorem
**(Cauchy)**

If a [[prime number]] $p$ divides ${\vert G\vert}$, then equivalently

* $G$ has an element of [[order of an element|order]] $p$;

* $G$ has a [[subgroup]] of [[order of a group]] $p$.

=--

See at _[[Cauchy's theorem]]_ for more.

### Feit-Thompson theorem

+-- {: .num_theorem}
###### Theorem

Every finite group of odd [[order]] is a [[solvable group]].

=--

See at _[[Feit-Thompson theorem]]_.

### Classification

The structure of finite groups is a very hard problem; the classification of finite [[simple group]]s alone is one of the largest theorems ever proved (certainly if measured by number of journal pages needed for a complete proof).

All finite groups are built out of [[simple group]]s, but the ways to do this have not (yet?) been fully classified.

A point of view that can be useful in particular cases -- more useful than the [[Jordan-Hölder theorem]] -- is provided by the [[F*-theorem]], due to [[Hans Fitting]] in the [[solvable group|solvable]] case and [[Helmut Bender]] in the general case. It states that $C_G(F^*(G))=Z(F^*(G))$, where $F^*(G)$ is the [[generalized Fitting subgroup]] of $G$, defined below, $C_G(F^*(G))$ is the [[subgroup]] of $G$ consisting of all elements commuting with every element of $F^*(G)$, and $Z(H)$ for any group $H$ is the [[center]] of $H$, the subgroup of $H$ consisting of all elements commuting with every element of $H$. Thus $G$ is somehow assembled from $F^*(G)$, whose structure has some easy features, and $G/C_G(F^*(G))$, which is isomorphic to a subgroup of the [[automorphism group]] of $F^*(G)$ and which has a [[quotient]] group [[isomorphic]] to $G/F^*(G)$.

One definition of $F^*(G)$ is that it is the subgroup generated by all [[normal subgroup]]s $N$ of $G$ possessing subgroups $N_1,N_2,\dots, N_r$ for some integer $r$ such that $N=N_1N_2\cdots N_r$; $x_i x_j=x_j x_i$ for all $x_i\in N_i$, $x_j\in N_j$, and distinct subscripts $i$ and $j$; and each $N_i$ either has prime power order or is a [[quasisimple group]]. Bender proved that $F^*(G)$ itself enjoys these properties.

Finally a group $H$ is called quasisimple if and only if $H=[H,H]$ and $H/Z(H)$ is simple. The finite quasisimple groups have been classified, as a consequence of the classification of finite simple groups and the calculation of the [[Schur multiplier]] of each finite simple group.

For more on this see

* [[classification of finite simple groups]].

#### "Most finite groups are nilpotent"

The meaning of the title is this curious fact (based on empirical evidence, anyway): if we are counting [[isomorphism classes]] of finite groups up to a given [[order of a grouporder]], then most of them are $2$-[[p-primary group|primary]] groups (and therefore [[nilpotent group|nilpotent]]; see at *[[class equation]]*).

For example, it is reported that "out of the 49,910,529,484 groups of order at most 2000, a staggering 49,487,365,422 of them have order 1024".

(It has also been suggested that a more meaningful weighting would divide each isomorphism class representative by the order of its [[automorphism group]]; as of this writing the nLab authors don't know how much this would affect the strength of the assertion "most finite groups are nilpotent".)

Discussion can be found [here](http://mathoverflow.net/a/114666/2926) and [here](https://golem.ph.utexas.edu/category/2012/11/almost_all_of_the_first_50_bil.html).

## Examples

* For every [[natural number]] $n \in \mathbb{N}$, the [[cyclic group]]

  $$
    \mathbb{Z}_n := \mathbb{Z}/n \mathbb{Z}
  $$

  is finite.

* The largest finite group that is also a [[sporadic simple group]], i.e., does not belong(up to isomorphism) to the infinite family of the alternating groups or to the infinite family of finite groups of Lie type, is the [[Monster group]].

* [[finite subgroups of SO(3)]] and [[finite subgroups of SU(2)]] have an [[ADE classification]]

* the [[general linear group]] over [[prime fields]] are finit, such as [[GL(2,3)]]

* [[wallpaper group]]

* see also at _[[finite abelian group]]_.

[[!include ADE -- table]]

## Related concepts

* [[profinite group]]

* [[finitely generated group]]

* [[finite abelian group]]

* [[finite group scheme]]

* [[Burnside ring]], [[Segal conjecture]]

## References
 {#References}

Early textbook accounts:

* {#Burnside1897} [[William Burnside]], _Theory of Groups of Finite Order_, 1897 ([pdf](http://www.gutenberg.org/files/40395/40395-pdf.pdf)), reprinted by Cambridge University Press 2012 ([doi:10.1017/CBO9781139237253]( https://doi.org/10.1017/CBO9781139237253))

* {#MillerBlichfeldtDickson16} G. A. Miller, H. F. Blichfeldt, [[Leonard E. Dickson]], _Theory and applications of finite groups_, Dover, New York, 1916 ([ark:/13960/t0ht2kb4x](https://archive.org/details/theoryapplicatio00milluoft/))

Discussion in [[homotopy type theory]]/[[univalent foundations of mathematics]]:

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: Chapter 7 of: *[[Symmetry]]* (2021) $[$[pdf](https://unimath.github.io/SymmetryBook/book.pdf)$]$

On the [[representation theory]] ([[linear representations]]) of finite groups over [[algebraic number fields]]:

* [[Louis Solomon]], _The representation of finite groups in algebraic number fields_, J. Math. Soc. Japan Volume 13, Number 2 (1961), 144-164 ([euclid:euclid.jmsj/1261147137](https://projecteuclid.org/euclid.jmsj/1261147137))

Discussion of [[group characters]] and [[group cohomology]] of finite groups:

* [[Michael Atiyah]], _Characters and cohomology of finite groups_, Publications Mathématiques de l'IHÉS, 9 (1961), p. 23-64  ([numdam:PMIHES_1961__9__23_0](http://www.numdam.org/item?id=PMIHES_1961__9__23_0))

* [[Alejandro Adem]], [[R. James Milgram]], _Cohomology of Finite Groups_, Springer 2004

* {#Epa10} [[Narthana Epa]], _Platonic 2-groups_, 2010 ([pdf](http://www.ms.unimelb.edu.au/documents/thesis/Epa-Platonic2-Groups.pdf))

See also:

* _[Atlas of finite group representations](http://brauer.maths.qmul.ac.uk/Atlas/v3/)_

Discussion of [[free actions]] of finite groups on [[n-spheres]] (see also at _[[ADE classification]]_) includes

* {#Milnor57} [[John Milnor]], _Groups which act on $S^n$ without fixed points_, American Journal of Mathematics Vol. 79, No. 3 (Jul., 1957), pp. 623-630 ([JSTOR](http://www.jstor.org/stable/2372566))

* {#Keenan03} [[Adam Keenan]], _Which finite groups act freely on spheres?_, 2003 ([pdf](http://www.math.utah.edu/~keenan/actions.pdf))

With an eye towards application to (the [[standard model of particle physics|standard model]] of) [[particle physics]]:

* [[Howard Georgi]], §1 in :*Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

[[!redirects finite groups]]


finite group scheme

## Idea

A finite [[group scheme]]

(...)

## Related concepts

* [[finite group]]

* [[Cartier duality]]

## References

* Richard Pink, _Finite group schemes_, Lecture notes 2004/05 ([pdf](http://www.math.ethz.ch/~pink/ftp/FGS/CompleteNotes.pdf))

[[!redirects finite group schemes]]


finite homotopy limits of spectra -- section

### Finite homotopy (co)limits of spectra
  {#FiniteHomotopyLimitsOfSpectra}

\begin{proposition}
  \label{InSpectraHomotopyFiberSequencesAreHomotopyCofiberSequences}

A sequence of [[morphisms]] of [[spectra]] $E \longrightarrow F \longrightarrow G$ is a [[homotopy fiber sequence]] if and only if it is a [[homotopy cofiber sequence]]:

\begin{xymatrix@R=18pt}
  E
  \ar[rr]_>>>{\ }="s1"
  \ar[dd]
    |-{
      \mathclap{\phantom{\vert^{\vert}}}
      \mathrm{fib}
      \mathclap{\phantom{\vert_{\vert}}}
    }
    ^>>>{\ }="t1"
  &&
  \ast
  \ar[dd]
  &&
  E
  \ar[rr]_>>>{\ }="s2"
  \ar[dd]^>>>{\ }="t2"
  &&
  \ast
  \ar[dd]
  \\
  && & \Leftrightarrow
  \\
  F
  \ar[rr]
  &&
  G
  &&
  F
  \ar[rr]
   |-{
     \;\mathrm{cofib}\;
   }
  &&
  G
  %
  \ar@{=>}
    %^{ \mbox{\tiny\rm(po)} }
    "s1"; "t1"
  \ar@{=>}
    %^{ \mbox{\tiny\rm(pb)} }
    "s2"; "t2"
\end{xymatrix}

\end{proposition}
A **proof** is spelled out at _[[Introduction to Stable homotopy theory]]_ ([this Prop.](Introduction+to+Stable+homotopy+theory+--+1-1#HomotopyCofiberSequencesAreHomotopyFiberSequencesInSpectra), following [Lewis-May-Steinberger 86, chapter III, theorem 2.4](equivariant+stable+homotopy+theory#LewisMaySteinberger86)
)

In fact:

\begin{proposition}
  \label{HomotopyPullbacksOfSpectraAreHomotopyPushouts}

A [[homotopy coherent diagram|homotopy]]-[[commuting square]] in [[Spectra]] is a [[homotopy pullback]] if and only it is a [[homotopy pushout]].

\begin{xymatrix@R=18pt}
  E
  \ar[rr]_>>>{\ }="s1"
  \ar[dd]^>>>{\ }="t1"
  &&
  H
  \ar[dd]
  &&
  E
  \ar[rr]_>>>{\ }="s2"
  \ar[dd]^>>>{\ }="t2"
  &&
  H
  \ar[dd]
  \\
  && & \Leftrightarrow
  \\
  F
  \ar[rr]
  &&
  G
  &&
  F
  \ar[rr]
  &&
  G
  %
  \ar@{=>}
    ^{ \mbox{\tiny\rm(po)} } "s1"; "t1"
  \ar@{=>}
    ^{ \mbox{\tiny\rm(pb)} } "s2"; "t2"
\end{xymatrix}

\end{proposition}

This follows from Prop. \ref{InSpectraHomotopyFiberSequencesAreHomotopyCofiberSequences} by the fact that [[Spectra]] is [[additive category|additive]] ([this Prop.](Introduction+to+Stable+homotopy+theory+--+1-1#TheStableHomotopyCategoryIsAdditive)).

See also [arXiv:1906.04773, Prop. 6.2.11](https://arxiv.org/abs/1906.04773), [MO:q/132347](https://mathoverflow.net/q/132347/381).

\begin{remark}
  This property of Spectra (Prop. \ref{InSpectraHomotopyFiberSequencesAreHomotopyCofiberSequences}, Prop. \ref{HomotopyPullbacksOfSpectraAreHomotopyPushouts}) reflects one of the standard defining [[axioms]] on [[stable (∞,1)-categories]] (see [there](stable+infinity-category#Definition)) and on [[stable derivators]] (see [there](stable+derivator#Definition)).
\end{remark}


finite homotopy type

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

#Contents#
* table of contents
{:toc}

## Definition

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

A [[homotopy type]] is called _finite_ if it is presented (via the discussion at _[[homotopy hypothesis]]_) by either

* a [[simplicial set]] with a [[finite number]] of non-degenerate [[simplices]];

* a [[CW-complex]] with a [[finite number]] of [[cell complex|cells]].

=--

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

Similarly a [[spectrum]] ([[stable homotopy type]]) given by a sequence of finite homotopy types is called a _[[finite spectrum]]_.

=--

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

Beware that a finite homotopy type in general does not have finite and finitely many [[homotopy groups]] (see e.g. at [[homotopy groups of spheres]]). Homotopy types with finite and finitely many homotopy groups have alternatively been called _$\pi$-finite_, or _[[groupoid cardinality|tame]]_, or (adapted from [[homological algebra]]) "[[of finite type]]" (which needs to be carefully distinguishes, therefore, from "finite homotopy type").
See at _[[homotopy type with finite homotopy groups]]_.

=--

## Properties

### Relation to compact homotopy type

The [[compact object in an (infinity,1)-category|compact objects]] in [[∞Grpd]] are the [[retracts]] of finite homotopy types. Not every such retract is itself a finite homotopy type; the vanishing of [[Wall's finiteness obstruction]] is a necessary and sufficient condition for this to happen.

## Related concepts

* [[π-finite homotopy type]], [[finite type]]

* [[finite set]], [[hereditarily finite set]]

* [[finite category]], [[finite limit]]

* [[finite spectrum]]

* [[finite (infinity,1)-category]], [[finite (infinity,1)-limit]]

* [[p-local homotopy type]], [[p-complete homotopy type]], [[rational homotopy type]]

* [[(infinity,1)-presheaves]] on $\infty Grpd_{fin}^{op}$ form the [[classifying (infinity,1)-topos]] for objects.

## References

* [[Doug Ravenel]], _The stable homotopy theory of finite complexes_, in [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ (1995)

[[!redirects finite homotopy types]]

[[!redirects finite ∞-groupoid]]
[[!redirects finite ∞-groupoids]]

[[!redirects finite infinity-groupoid]]
[[!redirects finite infinity-groupoids]]


finite intersection property

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

#Contents#
* table of contents
{:toc}

## Idea

The _finite intersection property_ on a set of [[subsets]] says that every [[finite number]] of them has [[inhabited set|non-empty]] [[intersection]] (def. \ref{FiniteIntersectionProperty} below).

This property is typically used in order to re-formulate the condition on [[topological space]] to be [[compact topological space|compact]] in terms of [[closed subsets]] (prop. \ref{CompactnessInTermsOfFiniteIntersectionProperty} below).

The formulation of compactness in terms of the finite intersection property is useful for proving the [[Tychonoff theorem]] as well as for proving that [[compact spaces equivalently have converging subnet of every net|compact spaces are equivalently those for which every net has a converging subnet]].

## Statement

+-- {: .num_defn #FiniteIntersectionProperty}
###### Definition
**(finite intersection property)**

Let $X$ be a [[set]]. Then a set of [[subsets]] $\{S_i \subset X\}_{i \in I}$ is said to satify the _finite intersection property_ if for every [[finite set|finite subset]] $J \subset I$ the corresponding intersection is [[inhabited]]: $\underset{i \in J \subset I}{\cap} S_i \neq \emptyset$.

=--

## Relation to compact topological spaces

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

Let $(X,\tau)$ be a [[topological space]].

Assuming [[excluded middle]], then:

The following are equivalent.

1. $(X, \tau)$ is a [[compact topological space]] in the sense that for every [[open cover]] $\{U_i \subset X\}_{i \in I}$ there is a [[finite set|finite]] [[subset]] $J \subset I$ such that $\{U_i \subset X\}_{i \in J \subset I}$ is still a cover.

1. Every set of [[closed subsets]] $\{C_i \subset X\}_{i \in I}$ which satisfies the finite intersection property (def. \ref{FiniteIntersectionProperty}) has even [[inhabited set|non-empty]] total intersection $\underset{i \in I}{\cap} C_i \neq \emptyset$.

=--

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

In one direction, assume that $(X,\tau)$ is compact, and that $\{C_i \subset X\}_{i \in I}$ satisfies the finite intersection property. We need to show that then $\underset{i \in I}{\cap} C_i \neq \emptyset$.

Assume that this were not the case, hence assume that $\underset{i \in I}{\cap} C_i = \emptyset$. This would imply that the open [[complements]] $U_i \coloneqq X \backslash C_i$ were an [[open cover]] of $X$, because (using [[de Morgan's law]])

$$
  \begin{aligned}
    \underset{i \in I}{\cup} U_i
    & \coloneqq
    \underset{i \in I}{\cup} X \backslash C_i
    \\
    & = X \backslash \left( \underset{i \in I}{\cap}C_i \right)
    \\
    & X \backslash \emptyset
    \\
    & = X
  \end{aligned}
  \,.
$$

But then by compactness of $(X,\tau)$ there were a finite subset $J \subset I$ such that $\{ U_i \subset X\}_{i \in J \subset I}$ were still an open cover, hence that $\underset{i \in J \subset I}{\cup} U_i = X $. Translating this back through the [[de Morgan's law]] again this would mean that

$$
  \begin{aligned}
    \emptyset
       & =
    X \backslash
    \left(
      \underset{i \in J \subset I}{\cup} U_i
    \right)
    \\
    & \coloneqq
    X \backslash
    \left(
      \underset{i \in J \subset I}{\cup} X \backslash C_i
    \right)
    \\
    & = \underset{i \in J \subset I}{\cap} X \backslash \left( X \backslash C_i\right)
    \\
    & =
    \underset{i \in J \subset I}{\cap}  C_i
    \,.
  \end{aligned}
$$

This would be in contradiction with the finite intersection property of $\{C_i \subset X\}_{i \in I}$, and hence we have [[proof by contradiction]].

Conversely, assume that every set of closed subsets in $X$ with the finite intersection property has non-empty total intersection. We need to show that the every open cover $\{U_i \subset X\}_{i \in I}$ of $X$ has a finite subcover.

Write $C_i \coloneqq X \backslash U_i$ for the closed complements of these open subsets.

Assume that there were no finite subset $J \subset I$ such that $\underset{i \in J \subset I}{\cup} U_i = X$. By [[de Morgan's law]] this means equivalently that there were no finite subset $J$ such that $\underset{i \in J \subset I}{\cap} C_i = \emptyset$, hence it would mean that $\{C_i \subset X\}_{i \in I }$ satisfied the finite intersection property.

But by assumption this would then imply that $\underset{i \in I}{\cap} C_i \neq \emptyset$, which, again by de Morgan, would mean that $\underset{i \in I}{\cup} U_i \neq X$. But this contradicts the assumption that the $\{U_i \subset X\}_{i \in I}$ are a cover. Hence we have a [[proof by contradiction]].

=--


finite limit

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A **finite limit** is a [[limit]] over a finite [[diagram]] - that is, one whose shape is a [[finite category]].

More generally, in [[higher category theory]], a finite limit is a [[limit]] of a diagram that is a finite [[(n,r)-category]].

A [[category]] that has all finite limits is called a *[[finitely complete category]]* or a (finitary) *[[essentially algebraic theory]]*.

A [[functor]] that [[preserved limit|preserves]] all finite limits is called *[[left exact functor]]*, a *lex functor*, a *cartesian functor*, or a *finitely continuous functor*.

The [[2-category]] of [[finitely complete categories]], [[left exact functors]] and [[natural transformations]] is often denoted *[[Lex]]*.

## Examples
  {#Examples}

\begin{example}
  A [[product]] is a finite limit iff it is a [[finite product]], hence if its factors are [[indexed set|indexed]] by a [[finite set]].

In particular, [[binary products]] are finite limits and the [[terminal object]] (being the [[limit]] over the [[empty diagram]]) is a finite limit.
\end{example}

\begin{example}
  Other common finite limits are [[pullbacks]] and [[equalizers]].
\end{example}

\begin{example}
  The [[fixed locus]] of a [[group action]] is a finite limit if the group in question is a [[finite group]].
\end{example}

\begin{example}
  The [[functor]] of [[geometric realization]] of [[simplicial sets]]
  into [[compactly generated topological spaces]]
  [[preserved limit|preserves]] finite limits (is a [[left exact functor]]), see [there](geometric+realization#GeometricRealizationIsLeftExact).

More generally, the [[functor]] of  [[geometric realization of simplicial topological spaces]] (with respect to [[compactly generated topological spaces]]) preserves finite limits (by [this Prop.](geometric+realization+of+simplicial+topological+spaces#GeometricRealizationPreservesFiniteLimits)).
\end{example}

## Properties
 {#Properties}

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

For a [[category]] $\mathcal{C}$ the following are equivalent:

1. $\mathcal{C}$ has all [[finite limits]].

1. $\mathcal{C}$ has all [[equalizers]] and [[binary  products]].

1. $\mathcal{C}$ has all [[pullbacks]] and a [[terminal object]].

In fact, either class of limits may be expressed in terms of another, so that for a [[functor]] $F \;\colon\; \mathcal{C} \to \mathcal{D}$ the following are equivalent:

1. $F$ [[preserved limit|preserves]] [[finite limits]].

1. $F$ [[preserved limit|preserves]] [[equalizers]] and [[binary products]].

1. $F$ [[preserved limit|preserves]] [[pullbacks]] and the [[terminal object]].

=--

(The first statement may be found, e.g., in [Borceux 1994, Prop. 2.8.2](#Borceux94). From the proof there the second statement immediately follows.)

\begin{remark}
The equivalence of the first two items in Prop. \ref{FiniteLimitsFromPullbacksAndTerminalObject} is the finite analog of
how a category with equalizers and all small products has all [[small limits]].
\end{remark}

\begin{remark}\label{LFiniteLimits}
**([[saturation of a class of limits|saturation]] to [[L-finite limits]])**
\linebreak
Prop. \ref{FiniteLimitsFromPullbacksAndTerminalObject} implies that finite limits are contained in the [[saturation of a class of limits|saturation]] of the class containing only [[finite products]] and [[equalizers]], and also that of the class containing only [[pullbacks]] and [[terminal objects]].

But the existence of finite limits in fact implies that of the larger class of [[L-finite limits]], and this is the largest class of limits implied by the existence of finite limits ([Paré 1990, Prop 7](#Pare90), see [there](L-finite+category#CharacterizationsOfLFiniteLimits)).

Therefore Prop. \ref{FiniteLimitsFromPullbacksAndTerminalObject} implies that [[L-finite limits]] constitute also the full [[saturation of a class of limits|saturation class]] of both equalizers+finite products as well as pullbacks+terminal objects.
\end{remark}

## Related concepts

* [[finite (∞,1)-limit]]

* [[finite set]], [[finite group]], [[finite category]]

## References

Textbook account:

* {#Borceux94} [[Francis Borceux]], around Def. 2.8.2 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory*, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994)
([doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858))

On the saturation to [[L-finite limits]]:

* {#Pare90} [[Robert Paré]], Section 3 of: *Simply connected limits*.  Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746 ([doi:10.4153/CJM-1990-038-6](https://doi.org/10.4153/CJM-1990-038-6))

[[!redirects finite limits]]
[[!redirects finite colimit]]
[[!redirects finite colimits]]


finite mathematics

# Finite mathematics
* table of contents
{: toc}

## Idea

__Finite mathematics__ is the [[mathematics]] of [[finite sets]].  The term is sometimes used more broadly for *[[discrete mathematics]]*.

We may say that finite mathematics is mathematics done [[internalization|internal]] to the category [[FinSet]] of finite sets or the mathematics of $\Fin\Set$ itself. The latter (but not the former) includes the basic arithmetic of [[natural numbers]], since these are the [[cardinal numbers|cardinalities]] of finite sets; we can go as far as [[rational numbers]] this way, but not [[real numbers]].

Finite mathematics also includes a great deal of [[combinatorics]], basic [[algebra]], and elementary [[formal logic]], although not many advanced topics.

## Finitism

In the [[foundations]] of mathematics and in [[philosophy of mathematics]], __finitism__ is the [[philosophy|philosophical]] sentiment that one "should" do only finite mathematics.  In a weak sense, one should not assume the [[axiom of infinity]]; in a strong sense, one should even deny it by an axiom of finiteness.  This makes it impossible to do [[analysis]] as we normally understand it.

Finitism (in the weak sense of not accepting an axiom of infinity) is essentially the mathematics that can be done internal to an arbitrary [[boolean topos]] (at least if one is not also being [[predicative mathematics|predicative]] or [[constructive mathematics|constructive]]).  For [[constructive mathematics]] as usually practised, one goes beyond finitism by positing a [[natural numbers object]].

Although often considered an extreme form of constructivism, finitism in the strong sense (actually denying the axiom of infinity) can make [[excluded middle]] and even the [[axiom of choice]] constructively acceptable (and similarly make [[power sets]] predicatively acceptable).  This is because even constructivists agree that these are true in $\Fin\Set$; it\'s the extension of them to [[infinite set]]s that the first constructivists objected to.

__Ultrafinitism__ is an even more extreme form of finitism, in which one doubts the existence of certain very large natural numbers. The theory of ultrafinite mathematics is most well developed by [[Edward Nelson]] in [[Nelson arithmetic]]. Foundational systems such as [[soft linear logic]] can also be argued to have an ultrafinitist flavor.

This is also true of foundational systems using [[paraconsistent logic]], where [[paraconsistent arithmetics]] can have only a finite number of natural numbers. In particular, there is a paraconsistent version of the [[Löwenheim-Skolem theorem]], which states:

> Any mathematical theory presented in first order logic has a finite paraconsistent model.

For the opinionated espousal of finitism (and much else), one can hardly do better than the [Opinions](http://www.math.rutgers.edu/~zeilberg/OPINIONS.html) of [[Doron Zeilberger]].

### The mathematics of FinSet

An alternative definition of finite mathematics is as the mathematics of [[FinSet]]. This includes the basic arithmetic of [[natural numbers]], since these are the [[cardinal numbers|cardinalities]] of finite sets. This may seem contradictory to the entire conception of "finite mathematics", since the [[natural numbers]] form an infinite set. The way out is to state that the natural numbers don't form a set (or class) at all; instead, they are formally defined outside of the [[set theory]] via an auxiliary theory like [[Peano arithmetic]]. This phenomenon is similar to defining the [[universe hierarchy]] of [[Russell universes]] or [[Coquand universes]] in [[dependent type theory]] without a separate [[type]] [[judgment]], where the natural numbers used in the universe indices are not elements of a type, but rather formally defined outside of the type theory using some other theory.

### $\pi$-finite mathematics

In [[homotopy type theory]], another possibility is available: $\pi$-finite mathematics, which can be said to be the study of only the $n$-[[truncated]] [[pi-finite types]] for $n:\mathbb{N}$, which have finite homotopy groups, and type operations that preserve finiteness. Finite sets are just finite 0-truncated tame homotopy types in homotopy type theory. Certain type operations do not preserve tameness, such as [[suspensions]] and [[homotopy pushouts]], and so wouldn't be part of finite mathematics. This extends traditional finite mathematics based in set theoretic foundations to more modern fields such as [[higher category theory]] and [[homotopy theory]]. Just as the cardinalities of finite sets are [[natural numbers]] (finite set cardinalities), the [[groupoid cardinality|cardinalities]] of these $n$-truncated homotopy types are positive [[rational numbers]].

## See also

* [[predicative mathematics]]

* [[constructive mathematics]]

* [[paraconsistent mathematics]]

* [[Nelson arithmetic]]

## References

* [[Edward Nelson]], _Warning Signs of a Possible Collapse of Contemporary Mathematics_ (~2011) [[pdf](https://web.math.princeton.edu/~nelson/papers/warn.pdf), [[Nelson-WarningSigns.pdf:file]]]

* Manuel Bremer, *Inconsistent Mathematics*, ([slides](https://www.mbph.de/Logic/Para/InconsistentMathematics.pdf))

[[!redirects finite mathematics]]

[[!redirects finitist mathematics]]
[[!redirects finitism]]
[[!redirects ultrafinitism]]


finite number

The [[cardinality]] of a [[finite set]] is a _finite number_.

## Category of finite numbers

Let's identify finite numbers with elements of $\mathbb{N}$.

$\mathbb{N}$ can be made into a [[category]] $FNumb$ with [[hom-sets]] $FNumb[n,p]=\{\bullet\}$ if $n \le p$ and $FNumb[n,p]= \emptyset$ else.

## Cardinality functor

Write $MonoFSet$ for the [[wide subcategory]] of $FSet$ with objects finite sets and morphisms injections.

We then have the cardinality functor $|.|: MonoFSet \rightarrow FNumb$ which associates its number of elements to every finite set.

This functor verifies the following properties:

* $|A \times B| = |A|.|B|$
* $|f \times g| = |f|.|g|$
* $|A \sqcup B| = |A|+|B|$
* $|f \sqcup g| = |f|+|g|$
* $|\emptyset| = 0$

[[!redirects finite numbers]]
[[!redirects cardinality functor]]


finite object

> This entry is about finite objects in generalization of [[finite sets]]. For generalization of [[finite-dimensional vector spaces]] see instead at *[[dualizable object]]*.

***

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

# Finite objects
* table of contents
{: toc}

## Idea

The notion of _[[finite object]]_ in a [[category]] -- notably in a [[topos]] -- is a generalisation of the notion of [[finite set]] in [[Set|the category of sets]].

As there are already at least five distinct notions of [[finite set]] in [[constructive mathematics]], so there must be at least five distinct notions of finite object internal to a [[topos]].  Additionally, the definitions may also be interpreted in an 'external' sense, giving even further notions.  Only some are mentioned below.

Also beware that in [[category theory]] the term 'finite object' is also used in a much more general sense to mean a _[[compact object]]_. Similar finiteness meaning may also be attributed to [[dualizable objects]] in [[monoidal categories]] and to [[perfect complexes]] (of [[abelian sheaves]]) in [[geometry]].

## Definitions
 {#Definitions}

Consider an ambient [[topos]] $\mathcal{T}$.
Assume that $\mathcal{T}$ is equipped with a [[natural numbers object]] $N$. Write $N_{\lt} \hookrightarrow N\times N$ for its [[strict total order]] relation.

### External version
 {#ExternalDefinition}

A "finite set" in $\mathcal{T}$ in the strictest sense is usually called a **finite cardinal**.  This is an [[object]] $[n] \in \mathcal{C}$ which is the [[pullback]] of $N_{\lt}\to N$ along some [[global element]] $n:1\to N$.

   We can then consider [[subobjects]], [[quotient objects]], and [[subquotient objects]] of finite cardinals to obtain external versions of subfinite, finitely indexed, and subfinitely indexed sets.

### Internal version
 {#InternalDefinition}

 The _internal_ version of a "finite set" is an object $X$ such that "$X$ is a finite cardinal" is true in the [[internal logic]].  This is equivalent to the following

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

An object $X \in \mathcal{T}$ is _locally_ isomorphic to a finite cardinal, if there is an [[epimorphism]] $U\to 1$ and a [[generalized element]] $n:U\to N$ such that $U\times X \cong n^*(N_\lt)$ over $U$.  Equivalently, there is a $U\to 1$ such that $U\times X$ is a finite cardinal in the
[[slice topos]] $\mathcal{T}/U$.

An  **internally finitely indexed object** is an object $X$ which is locally a [[quotient]] of a finite cardinal, hence such that there is an [[epimorphism]] $U \to *$, a finite cardinal in the slice topos $n \in \mathcal{T}_{/U}$ and an epimorphism $n \to U \times X$.

=--

An "internally finitely indexed" object is generally called a **Kuratowski-finite object**  or **$K$-finite object** for short, and an "internally subfinitely indexed" one is called a **$\tilde{K}$-finite object**.

There is a more general definition of K-finite objects that does not need to assume the presence of a [[natural number object]]. See ([Johnstone, theorem D5.4.13](#Johnstone)).

Since it is still provable in the [[internal logic]] that any decidable finitely indexed set is finite, the "internally finite" objects (those that are locally isomorphic to a finite cardinal, as above) can be characterized as the __decidable $K$-finite objects__.

### Dedekind finiteness

An object is **Dedekind finite** or **co-Hopfian** (in analogy to [[Hopfian group]]) if every [[monic]] [[endomorphism]] is an [[automorphism]]. This definition makes sense in any [[category]].

## Properties

### Closure of finite objects

The following lists closure properties of K-finite objects, def. \ref{KFinite}.

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

1. The [[initial object]] and the [[terminal object]] are K-finite.

1. The [[image]] of a K-finite object under an [[epimorphism]] is K-finite.

1. The [[union]] of two K-finite [[subobjects]] is K-finite.

1. A [[coproduct]] is K-finite precisely if both summands are.

1. A [[subterminal object]] is K-finite precisely if it is a [[complemented subobject]].

1. A [[product]] of two K-finite objects is K-finite.

=--

This appears in ([Johnstone](#Johnstone)) as lemma D5.4.4, corollary D5.4.5, pro. 5.4.8.

### Subcategories of finite objects

The [[full subcategory]] of finite cardinals in any [[topos]] is again a topos, and it is [[Boolean topos|Boolean]].  Its [[subobject classifier]] is $2=1\sqcup 1$, which in the ambient topos is the classifier only of [[decidable object|decidable]] subobjects.  This means that [[classical logic|classically]] valid arguments, including all of finitary combinatorics, can generally be applied easily to finite cardinals, as long as we always interpret "subset" to mean "decidable subset."

+-- {: .num_theorem}
###### Theorem

The [[full subcategory]] $\mathcal{T}_{dKf} \hookrightarrow \mathcal{T}$ of decidable $K$-finite objects in a topos $\mathcal{T}$ is a [[Boolean topos]] whose subobject classifier is $2$.

The category of $K$-finite objects is a topos if and only if every $K$-finite object is decidable, and the category of $\tilde{K}$-finite objects is a topos if (but not only if) the subobject classifier is $K$-finite.

=--

The first statement appears as ([Johnstone, theorem 5.4.18](#Johnstone)).

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

The full subcategory $\mathcal{T}_{dKf} \hookrightarrow \mathcal{T}$ can be regarded as the "[[stack]] completion" of the topos of finite cardinals.

=--

### Relation to slice toposes

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

An object $X \in \mathcal{T}$ is K-finite precisely if the [[étale geometric morphism]]

$$
  \mathcal{T}_{/X} \to \mathcal{T}
$$

out of the [[slice topos]] is a [[proper geometric morphism]].

=--

([Moerdijk-Vermeulen, examples III 1.4](#MoerdijkVermeulen))

## Examples

* In any [[Boolean topos]], all four internal notions coincide.  In a [[well-pointed topos]], each internal notion coincides with its external notion.  Therefore, in a well-pointed Boolean topos, including the topos [[Set]] as usually conceived, all notions of finiteness coincide.

* In a [[presheaf]] topos $[C^{op},Set]$, the finite cardinals are the finite-set-valued functors which are constant on each connected component.  In particular, if $C$ is a [[group]], then the topos of finite cardinals is equivalent to [[FinSet]].

* Likewise, in the [[Grothendieck topos]] $Sh(X)$ of [[sheaf|sheaves]] on a space $X$, the finite cardinals are the locally constant functions $X\to N$.  So if $X$ is connected, the topos of finite cardinals in $Sh(X)$ is also equivalent to $FinSet$.

* Examples of such are [[tiny object]]s and [[infinitesimal object]]s in sheaf toposes.

* By contrast, the $K$-finite objects in $[C^{op},Set]$ are the finite-set-valued functors each of whose transition functions is surjective, and the _decidable_ K-finite objects are the finite-set-valued functors each of whose transition functions is bijective.

* In particular, if $C$ is a groupoid, the topos of decidable $K$-finite objects is equivalent to $[C^{op},FinSet]$.  Since the topos of presheaves on a groupoid is Boolean, this gives an example of a Boolean topos in which the finite cardinals ("externally finite objects") and the (decidable) $K$-finite objects ("internally finite objects") fail to coincide.

* In the [[category of sheaves]] $Sh(X)$ over a [[topological space]], the decidable K-finite objects are those that are "locally finite;" i.e. there is an [[open cover]] of $X$ such that over each open in the cover, the sheaf is a [[locally constant function]] to $N$.  These are essentially the same as [[covering space|covering spaces]] of $X$ with finite fibres.

## Related concepts

* [[finite set]], [[hereditarily finite set]]

* [[finite category]], [[finite limit]]

* [[finite graph]]

* [[decidable object]], [[theory of objects]]

* [[finite type]]

* [[finite homotopy type]], [[finite CW-complex]], [[finite spectrum]]

* [[π-finite homotopy type]]

* [[finite (infinity,1)-category]], [[finite (infinity,1)-limit]]

[[!include finite objects -- table]]

## References

In

* [[Peter Johnstone]], _[[Sketches of an elephant]]_
 {#Johnstone}

finite cardinal objects are discussed in section D5.2, Kuratowski finite objects in section D5.4

See also

* O. Acuña-Ortega, [[Fred Linton]], _Finiteness and decidability: I_ , Springer Lecture Notes in Mathematics, (1979), Volume **753**, pp.80-100, (DOI: 10.1007/BFb0061813)

* [[Peter Johnstone]], [[Fred Linton]], _Finiteness and decidability: II_ , Cambridge Philosophical Society Mathematical Proceedings of the Cambridge Philosophical Society (1978).

* B. P. Chisala, M.-M. Mawanda, _Counting Measure for Kuratowski Finite Parts and Decidability_ , Cah.Top.Géom.Diff.Cat. **XXXII** 4 (1991) pp.345-353. ([pdf](archive.numdam.org/article/CTGDC_1991__32_4_345_0.pdf))

* S. J. Henry, _Classifying Topoi and Preservation of Higher Order Logic by Geometric Morphisms_ , PhD University of Michigan (2013). ([arxiv](http://arxiv.org/abs/1305.3254))

* C. Kuratowski, _Sur la notion d'ensemble fini_ , Fund. Math. **1** (1920) pp.129-131. ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm1/fm1117.pdf))

* [[Ieke Moerdijk]], J. Vermeulen,  _Relative compactness conditions for toposes_ ([pdf](http://igitur-archive.library.uu.nl/math/2001-0702-142944/1039.pdf)) and _Proper maps of toposes_ , American Mathematical Society (2000)
  {#MoerdijkVermeulen}

* L. N. Stout, _Dedekind finiteness in topoi_ , JPAA **49** (1987) pp.219-225.

* T. Streicher, P. Freyd, F. Linton, P.Johnstone, W. Lawvere, _catlist discussion 'finiteness in toposes'_, January 1997. ([link](http://www.mta.ca/~cat-dist/catlist/1999/finite-topos))

* [[Alfred Tarski|A. Tarski]], _Sur les ensembles finis_ , Fund. Math. **3** (1924) pp.45-95. ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm619.pdf))

* H. Volger, _Ultrafilters, ultrapowers and finiteness in a topos_ , JPAA **6** (1975) pp.345-356.

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finite objects -- table

**[[finite objects]]:**

| [[geometry]] | [[monoidal category|monoidal category theory]]| [[category theory]] |
|--|--|--|
| [[perfect module]] | ([[fully dualizable object|fully]]-)[[dualizable object]] | [[compact object]] |


finite open cover > history

see _[[finite cover]]_


finite product

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A _finite product_ is a [[product]] ([[Cartesian product]]) of a [[finite number]] of factors.

Finite products are generated from the empty product (the [[terminal object]]) and binary products (those with two factors, often -- but not always -- understood by default under "product".)

Similarly a _finite coproduct_ is a [[coproduct]] of a [[finite number]] of summands. This is generated from the empty coproduct (the [[initial object]]) and binary coproducts.

## Properties

+-- {: .num_example #CategoriesWithFiniteProductsAreCosifted}
###### Example
**([[categories with finite products are cosifted]])**

Let $\mathcal{C}$ be a [[small category]] which has [[finite products]]. Then $\mathcal{C}$ is a _[[cosifted category]]_, equivalently its [[opposite category]] $\mathcal{C}^{op}$ is a _[[sifted category]]_, equivalently [[colimits]] over $\mathcal{C}^{op}$ with values in [[Set]] are _[[sifted colimits]]_, equivalently [[colimits]] over $\mathcal{C}^{op}$ with values in [[Set]] _[[limits commuting with colimits|commute]] with [[finite products]]_, as follows:

For $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ to [[functors]] on the [[opposite category]] of $\mathcal{C}$ (hence two [[presheaves]] on $\mathcal{C}$) we have a [[natural isomorphism]]

$$
  \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
  \left(
    \mathbf{X} \times \mathbf{Y}
  \right)
  \;\simeq\;
  \left(
    \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
    \mathbf{X}
  \right)
  \times
  \left(
    \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
    \mathbf{Y}
  \right)
  \,.
$$

=--

## Related concepts

* [[finite limit]], [[sifted colimit]]

* [[Lawvere theory]], [[essentially algebraic theory]]

## References

* [[Saunders MacLane]], §III.5 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

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+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

This page collects material related to the textbook

* [[Günter Scharf]]

  \linebreak

  **Finite Quantum Electrodynamics -- The Causal Approach**

  \linebreak

  Springer 1995

  [doi:10.1007/978-3-642-57750-5](https://doi.org/10.1007/978-3-642-57750-5)

on [[perturbative quantum field theory]] via [[causal perturbation theory]] and its application to the construction of perturbative [[quantum electrodynamics]].

The second book

* [[Günter Scharf]], _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

applies [[causal perturbation theory]] to perturbative [[Yang-Mills theory]] ([[quantum chromodynamics]]) and to perturbative [[quantum gravity]].

## Related textbooks and lecture notes

* {#Rejzner16} [[Katarzyna Rejzner]], _[[Perturbative Algebraic Quantum Field Theory]]_, Mathematical Physics Studies, Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

* _[[geometry of physics -- perturbative quantum field theory]]_

* [[Michael Dütsch]], _[[From classical field theory to perturbative quantum field theory]]_, 2018

category: reference


finite rotation group

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

#Contents#
* table of contents
{:toc}

## Idea

By a _finite rotation group_ one means a [[finite]] [[subgroup]] of a [[group]] of [[rotations]], hence of a [[special orthogonal group]] $SO(n)$ or [[spin group]] $Spin(n)$ or similar.

The finite subgroups of [[SO(3)]] and [[SU(2)]] follow an [[ADE classification]] (theorem \ref{ClassificationOfFiniteSubgroupsOfSO3} below).

## Classifications

### Finite subgroups of $O(3)$, $SO(3)$ and $Spin(3)$

+-- {: .num_theorem #ClassificationOfFiniteSubgroupsOfSO3}
###### Theorem
**([[ADE classification]] of [[finite group|finite]] [[subgroups]] of [[SO(3)]] and [[spin group|Spin(3)]]$\simeq$ [[SU(2)]])**

The [[finite group|finite]] [[subgroups]] of the [[special orthogonal group]] $SO(3)$ as well as the [[finite group|finite]] [[subgroups]] of the [[special unitary group]] [[SU(2)]] are, up to [[conjugation]], given by the following classification:

[[!include ADE -- table]]

Here under the [[double cover]] projection (using the [exceptional isomorphism](spin+group#ExceptionalIsomorphisms) $SU(2) \simeq Spin(3)$)

$$
  SU(2) \simeq Spin(3) \overset{\pi}{\longrightarrow} SO(3)
$$

all the finite subgroups of $SU(2)$ except the [[odd number|odd]]-[[order of a group|order]] [[cyclic groups]] are the [[preimages]] of the corresponding finite subgroups of $SO(3)$, in that we have [[pullback]] [[diagrams]]

$$
  \array{
    \left\langle
     \exp
     \left(
       \pi \mathrm{i} \tfrac{1}{n}
     \right)
    \right\rangle
    &
    =
    &
    \mathbb{Z}/(2n)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(3)
    \\
    &&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow^{ \mathrlap{\pi} }
    \\
    \left\langle
     Ad_{
     \exp
     \left(
       \pi \mathrm{i} \tfrac{1}{n}
     \right)
     }
    \right\rangle
    &
    =
    &
    \mathbb{Z}/n
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)
  }
$$

exhibiting the [[even number|even]] [[order of a group|order]] [[cyclic groups]] as subgroups of [[Spin(2)]], including the minimal case of the [[group of order 2]]

$$
  \array{
    \left\langle
     \exp
     \left(
       \pi \mathrm{i}
     \right)
     =
     -1
    \right\rangle
    &
    =
    &
    \mathbb{Z}/2
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(3)
    \\
    &&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow^{ \mathrlap{\pi} }
    \\
    \left\langle
     Ad_{
     \exp
     \left(
       \pi \mathrm{i}
     \right)
     }
     =
     e
    \right\rangle
    &
    =
    &
    1
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)
  }
$$

as well as

$$
  \array{
    \left\langle
      \exp\left( \pi \mathrm{i} \tfrac{1}{n} \right), \, \mathrm{j}
    \right\rangle
    &=&
    2 D_{2n}
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Pin_-(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(3)
    \\
    &&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow^{ \mathrlap{\pi} }
    \\
    \left\langle
      Ad_{\exp\left( \pi \mathrm{i} \tfrac{1}{n}  \right) },
      \,
      Ad_{\mathrm{j}}
    \right\rangle
    &&
    D_{2n}
    &\overset{\phantom{AA}}{\hookrightarrow}&
    O(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)
  }
$$

exhibiting the [[binary dihedral groups]] as sitting inside the [[Pin(2)]]-[[subgroup]] of [[Spin(3)]],

but only [[commuting diagrams]]

$$
  \array{
    \left\langle
     \exp
     \left(
       2 \pi \mathrm{i} \tfrac{1}{{2n+1}}
     \right)
    \right\rangle
    &
    =
    &
    \mathbb{Z}/(2n+1)
    &&\overset{\phantom{AA}}{\hookrightarrow}&&
    Spin(3)
    \\
    &&
    \big\downarrow
      &&
      &&
    \big\downarrow^{ \mathrlap{\pi} }
    \\
    \left\langle
     Ad_{
     \exp
     \left(
       2 \pi \mathrm{i} \tfrac{1}{2n+1}
     \right)
     }
    \right\rangle
    &
    =
    &
    \mathbb{Z}/(2n+1)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)
  }
$$

for the [[odd number|odd]] [[order of a group|order]] [[cyclic group|cyclic]] [[subgroups]].

=--

This goes back to ([Klein 1884, chapter I](#Klein1884)).
Full proof for $SO(3)$ is spelled out for instance in ([Rees 05, theorem 11](#Rees05), [De Visscher 11](#DeVisscher11)). The proof for the case of $SL(2,\mathbb{C})$ is spelled out in ([Miller-Blichfeldt-Dickson 16](#MillerBlichfeldtDickson16)) reviewed in ([Serrano 14, section 2](#Serrano14)). The proof of the case for $SU(2)$ given the result for $SO(3)$ is spelled out in [Keenan 03, theorem 4](#Keenan03).

### Finite subgroups of $O(4)$, [[SO(4)]] and [[Spin(4)]]
  {#FiniteSubgroupsOfO4}

For classification of the finite subgroups of $O(4)$ see ([duVal 65](#DuVal65), [Conway-Smith 03](#ConwaySmith03))

For finite subgroups of [[Spin(4)]]: [MFF 12, appendix B](#MFF12).

In this classification, the [[symmetry group]] of the [[120-cell]] and hence that of the [[600-cell]] is the [[quotient group]] $(2 I \times 2 I)/\mathbb{Z}_2$  by the [[cyclic group of order 2]] of the [[direct product group]] of two copies of the [[binary icosahedral group]] ([SadocMosseri 89, p. 172](#SadocMosseri89), see [MFF 12, table 16](#MFF12)).

### Finite subgroups of $O(5)$

For classification of the finite subgroups of $O(5)$ see [Mecchia-Zimmermann 10](#MecchiaZimmermann10)

## Properties

### Subgroup lattice
 {#SubgroupLattice}

The [[subgroup]] [[lattice]] of [[SU(2)]] under the three exceptional [[finite group|finite]] subgroups [[2T]], [[2O]], [[2I]] (from Theorem \ref{ClassificationOfFiniteSubgroupsOfSO3}) looks as follows:

This is obtained from the subgroup lattice as shown on [GroupNames](https://people.maths.bris.ac.uk/~matyd/GroupNames/) for
$2I \simeq$ SL(2,5) and  $2O \simeq $ CSU(2,3)

See also [Goncalves-Guaschi 11, appendix](#GoncalvesGuaschi11).

### Group cohomology
 {#GroupCohomology}

+-- {: .num_prop #GroupCohomologyOfFiniteSubgroupsOfSU2}
###### Proposition
**([[group cohomology]] of [[finite subgroups of SU(2)]])

Let $G_{ADE} \xhookrightarrow{i} SU(2)$ be a [[finite subgroup of SU(2)]]. Then its [[group cohomology]] with [[integer]] [[coefficients]] is as follows:

\[
  \label{IntegralGroupCohomologyOfFiniteSubgroupsOfSU2}
  H^n_{grp}(G_{ADE}, \mathbb{Z})
  \;\simeq\;
  \left\{
  \array{
    \mathbb{Z} &\vert& n = 0
    \\
    G_{ADE}^{ab} &\vert&  n = 2 \, mod \, 4
    \\
    \mathbb{Z}/{\vert G_{ADE}\vert}
     &\vert&
    n \, \text{positive multiple of} \, 4
    \\
    0 &\vert& \text{otherwise}
  }
  \right.
\]

Here $G_{ADE}^{ab}$ denotes the [[abelianization]]  of $G_{ADE}$ and $\vert G_{ADE}\vert$ its [[cardinality]], hence $\mathbb{Z}/{\vert G_{ADE}\vert}$ is the [[cyclic group]] whose [[order of a group|order]] is the cardinality of $G_{ADE}$.

The [[group homology]] with [[integer]] [[coefficients]] is

\[
  \label{IntegralGroupHomologyOfFiniteSubgroupsOfSU2}
  H_n^{grp}(G_{ADE}, \mathbb{Z})
  \;\simeq\;
  \left\{
  \array{
    \mathbb{Z} &\vert& n = 0
    \\
    G^{ab}_{ADE} &\vert&  n = 1 \, mod \, 4
    \\
    \mathbb{Z}/{\vert G_{ADE}\vert} &\vert& n = 3 \,mod\, 4
    \\
    0 &\vert& \text{otherwise}
  }
  \right.
\]

=--

This general statement is summarized in [Epa & Ganter 16, p. 12](#EpaGanter16).
A detailed proof for the case of cyclic groups is spelled out in [BSST 07, Lem. 4.51](Pontrjagin+dual#BSST07).
The three exceptional cases are worked out in explicit detail in [Tomoda & Zvengrowski 08, Section 4](#TomodaZvengrowski08). The vanishing of $H^3(G_{ADE},\mathbb{Z}) = H^2(G_{ADE}, \mathbb{C}^\times)$ is also [FHHP 01, Cor. 3.1](#FHHP01).

+-- {: .num_remark #DiscreteTorsion}
###### Remark
**([[C-field]] [[discrete torsion]])**

In discussion of [[11-dimensional supergravity]] on [[spacetimes]] with [[ADE-singularities]], the special case

$$
  \underset{
    \simeq H^3( \mathbb{C}^4 \sslash G_{ADE}, U(1))
  }{
  \underbrace{
    H^4_{grp}(G_{ADE}, \mathbb{Z})
  }}
   \;\simeq\;
  \mathbb{Z}/{\vert G_{ADE} \vert }
$$

of Prop. \ref{GroupCohomologyOfFiniteSubgroupsOfSU2}, regarded as expressing [[orbifold cohomology]] of an [[ADE singularity]], as shown under the brace, witnesses the possible [[torsion subgroup|torsion]] [[supergravity C-field]] [[flux]] of [[M5-branes]] [[wrapped brane|wrapped]] on torsion homology 3-cycles ("[[discrete torsion]]", see [Aharony-Bergman-Jafferis 08, p. 8](discrete+torsion#AharonyBergmanJafferis08) and [BDHKMMS 01, section 4.6.2](discrete+torsion#BDHKMMS01)).

See also at _[[Platonic 2-group]] -- [Relation to String 2-group](Platonic+2-group#RelationToTheString2Group)_.

=--

\begin{proposition}
\label{RestrictingTheSecondChernClassFromSU2ToFiniteSubgroup}
For $G_{ADE} \xhookrightarrow{i} Spin(3)$ a [[finite subgroup of SU(2)]], the [[pullback in cohomology]] along the induced map $B G_{ADE} \xrightarrow{B i} B SU(2)$ of [[classifying spaces]]  sends

* the generator $c_2 \,\in\,H^4(B SU(2), \mathbb{Z}) \simeq \mathbb{Z}$ (the [[universal characteristic class|universal]] [[second Chern class]])

to

* a generator of $H^4(B G_{ADE}, \mathbb{Z}) \simeq \mathbb{Z}/\left\vert G_{ADE}\right\vert$ (eq:IntegralGroupCohomologyOfFiniteSubgroupsOfSU2)

in that the [[pullback in cohomology]] is identified with the [[quotient]] [[coprojection]] from the [[integers]] to the [[cyclic group]] of [[order of a group|order]] that of $G_{ADE}$:

\[
  \label{RestrictionOf4CohomologyOfBSU2ToBGADE}
  \array{
    H^4_{grp}(SU(2), \mathbb{Z})
    &=&
    H^4(B SU(2), \mathbb{Z})
    &\simeq&
    \mathbb{Z}
    \\
    {}^{\mathllap{i^\ast}}\big\downarrow
    &&
    {}^{\mathllap{B i^\ast}}\big\downarrow
    &&
    \big\downarrow {}^{\mathrlap{ {quotient} \atop {coprojection}  }}
    \\
    H^4_{grp}(G_{ADE}, \mathbb{Z})
    &=&
    H^4(B G_{ADE}, \mathbb{Z})
    &\simeq&
    \mathbb{Z}/\left\vert G_{ADE}\right\vert
    \mathrlap{\,.}
  }
\]

\end{proposition}

This is equivalently the statement of [Epa & Ganter 16, Prop. 4.1](#EpaGanter16), whose proof is analogous to that of Lemma 3.1 there, where the analogue of (eq:RestrictionOf4CohomologyOfBSU2ToBGADE) is the top left square of the commuting diagram on [p. 11](https://arxiv.org/pdf/1605.09192.pdf#page=11).

## Related concepts

* [[regular polytope]], [[regular polyhedron]]

* [[ADE singularity]]

* [[McKay correspondence]]

* [[Platonic 2-group]]

* [[wallpaper group]]

## References
  {#References}

### Finite subgroups of $SO(3)$ and $Spin(3)$

The classification in Theorem \ref{ClassificationOfFiniteSubgroupsOfSO3} goes back to

* {#Klein1884} [[Felix Klein]], chapter I of _Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade_, 1884, translated as _Lectures on the Icosahedron and the Resolution of Equations of Degree Five_ by George Morrice 1888, [online version](https://archive.org/details/cu31924059413439)

Expository review of the argument in [Klein 1884](#Klein1884):

* {#Senechal90} Marjorie Senechal, *Finding the Finite Groups of Symmetries of the Sphere*, The American Mathematical Monthly **97** 4 (1990) 329-335 ([jstor:2324519](https://www.jstor.org/stable/2324519))

Textbook accounts:

* {#MillerBlichfeldtDickson16} G. A. Miller, H. F. Blichfeldt, [[Leonard E. Dickson]], _Theory and applications of finite groups_, Dover, New York, 1916 ([ark:/13960/t0ht2kb4x](https://archive.org/details/theoryapplicatio00milluoft/))

* {#Lamotke86} [[Klaus Lamotke]], _Regular Solids and Isolated Singularities_, Vieweg 1986 ([doi:10.1007/978-3-322-91767-6](https://link.springer.com/book/10.1007/978-3-322-91767-6))

* {#Rees05} [[Elmer Rees]], _Notes on Geometry_, Springer 2005

see also

* {#Serrano14} Javier Carrasco Serrano, _Finite subgroups of $SL(2,\mathbb{C})$ and $SL(3,\mathbb{C})$_, Warwick 2014 ([pdf](https://homepages.warwick.ac.uk/~masda/McKay/Carrasco_Project.pdf))

Complete proof of the classification of the finite subgroups of $SO(3)$ is also spelled out in

* {#DeVisscher11} Maud De Visscher, _[Groups and Symmetry](http://www.staff.city.ac.uk/maud.devisscher.1/GS/)_ lecture notes handout: _Classification of finite rotation groups_  ([pdf](http://www.staff.city.ac.uk/maud.devisscher.1/GS/classif_rotation.pdf))

Based on the classification of the finite subgroups of $SO(3)$, full proof of that of the finite subgroups of $SU(2)$ is spelled out in

* {#Keenan03} [[Adam Keenan]], _Which finite groups act freely on spheres?_, 2003 ([pdf](http://www.math.utah.edu/~keenan/actions.pdf))

See also

* [[Groupprops]], _Classification of finite subgroups of SO(3,R)

Discussion of the [[lattice]] of [[subgroups]] of the three exceptional subgroups is in

* {#GoncalvesGuaschi11} Daciberg Lima Gonçalves, John Guaschi, _The Subgroups of the Binary Polyhedral Groups_, Appendix of _The classification of the virtually cyclic subgroups of the sphere braid groups_, Springer (Ed.) (2013) 112 ([arXiv:1110.6628](https://arxiv.org/abs/1110.6628), [pdf](https://link.springer.com/content/pdf/bbm%3A978-3-319-00257-6%2F1.pdf))

Discussion of the [[group cohomology]]:

* {#TomodaZvengrowski08} [[Satoshi Tomoda]], [[Peter Zvengrowski]], *Remarks on the cohomology of finite fundamental groups of 3-manifolds*, Geom. Topol. Monogr. 14 (2008) 519-556 ([arXiv:0904.1876](https://arxiv.org/abs/0904.1876))

see also

* {#FHHP01} [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], Nikolaos Prezas, Cor. 3.1 in: *Discrete Torsion, Non-Abelian Orbifolds and the Schur Multiplier*, JHEP 0101:033, 2001 ([arXiv:hep-th/0010023](https://arxiv.org/abs/hep-th/0010023))

The [[universal higher central extension]] of finite subgroups of $SU(2)$ ("[[Platonic 2-groups]]") are discussed in

* {#EpaGanter16} [[Narthana Epa]], [[Nora Ganter]], _Platonic and alternating 2-groups_, Higher Structures 1(1):122-146, 2017 ([arXiv:1605.09192](http://arxiv.org/abs/1605.09192))

### Finite subgroups of $O(4)$

* {#DuVal65} [[Patrick du Val]], _Homographies, Quaternions and Rotations_, Oxford Mathematical Monographs, Clarendon Press (1964)

  also(?): Journal of the London Mathematical Society, Volume s1-40, Issue 1 (1965) ([doi:10.1112/jlms/s1-40.1.569b](https://doi.org/10.1112/jlms/s1-40.1.569b))

* {#ConwaySmith03} [[John Conway]], D. A. Smith, *On quaternions and octonions: their geometry, arithmetic and  symmetry*, A K Peters Ltd., Natick, MA, 2003

* {#MFF12} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], appendix B of _Half-BPS M2-brane orbifolds_, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761), [Euclid](https://projecteuclid.org/euclid.atmp/1408561553))

* {#SadocMosseri89} J. F. Sadoc, R. Mosseri, _Icosahedral order, curved space and quasicrystals_, in Jaric, Gratias (eds.) _Extended icosahedral structures_, 1989 ([GoogleBooks](https://books.google.co.uk/books?id=HWyIUglWeXsC&pg=PA172&lpg=PA172#v=onepage&q&f=false))

### Finite subgroups of $O(5)$

* {#MecchiaZimmermann10} Mattia Mecchia, Bruno Zimmermann, _On finite groups acting on homology 4-spheres and finite subgroups of $SO(5)$_, Topology and its Applications 158.6 (2011): 741-747 ([arXiv:1001.3976](https://arxiv.org/abs/1001.3976))

[[!redirects finite rotation groups]]

[[!redirects classification of the finite rotation groups]]

[[!redirects finite subgroup of SO(3)]]
[[!redirects finite subgroups of SO(3)]]

[[!redirects finite subgroup of SU(2)]]
[[!redirects finite subgroups of SU(2)]]

[[!redirects finite subgroup of Spin(3)]]
[[!redirects finite subgroups of Spin(3)]]

[[!redirects finite subgroup of O(4)]]
[[!redirects finite subgroups of O(4)]]

[[!redirects finite subgroup of SO(4)]]
[[!redirects finite subgroups of SO(4)]]

[[!redirects finite subgroup of O(5)]]
[[!redirects finite subgroups of O(5)]]

[[!redirects finite subgroup of SO(5)]]
[[!redirects finite subgroups of SO(5)]]

[[!redirects classification of finite rotation groups]]
[[!redirects classifications of finite rotation groups]]


finite set

\tableofcontents

\section{Introduction}

A _finite set_ is, roughly speaking, a [[set]] with only finitely many [[element|elements]]. There are a number of ways to make this precise.

Classically, the finite sets are the [[finitely presentable object|finitely presentable objects]] in [[Set]]. Constructively, the same is true if _finitely presented_ is properly interpreted, see there for details.

The category [[FinSet]] of finite sets and functions between them is a prime example of an [[elementary topos]] which is not a [[Grothendieck topos]]. It is essentially the subject matter of [[combinatorics]]; it is fundamental in the subject of [[structure types]].

\section{Definitions}

\subsection{Standard definition}

We can for example make the following definition.

\begin{defn} A **finite set** is a [[set]] $A$ for which there exists a [[bijection]] between $A$ and the set $[n] \coloneqq \{k\in \mathbb{N} | k\lt n\}$ for some $n\in \mathbb{N}$, where $\mathbb{N}$ is the [[natural number|natural numbers]]. \end{defn}

\subsection{Finiteness constructively and internally}
{#Constructivist}

\subsubsection{Variations}

In [[constructive mathematics]], and internally to a [[topos]], a number of classically equivalent notions of finiteness become distinguishable:

* A set is **[[finite]]** (for emphasis __Bishop-finite__ or __$B$-finite__) if (as above) it admits a [[bijection]] with $[n]$ for some [[natural number]] $n$.

* A set is **[[subfinite]]** (or __$\tilde{B}$-finite__) if it admits an [[injection]] into some finite set $[n]$; that is, it is a [[subset]] of a finite set.

* A set is **finitely indexed** (or **[[Kuratowski-finite]]**, **$K$-finite**, or even sometimes, confusingly, _subfinite_) if it admits a [[surjection]] from some finite set $[n]$; that is, it is a [[quotient set]] of a finite set.

* A set is **subfinitely indexed** (or __[[Kuratowski-subfinite]]__ or **$\tilde{K}$-finite**) if it admits a surjection from a subfinite set, or equivalently admits an injection to a finitely indexed set; that is, it is a [[subquotient set]] of a finite set.

* A set $X$ is **[[Dedekind-finite]]** if it satisfies one of the following:

  * any [[injection]] $X\hookrightarrow X$ must be a [[bijection]].
  * for any function $f\colon \mathbb{N} \to X$ from the [[natural numbers]], there exist $n,m$ with $n \ne m$ such that $f(n) = f(m)$.

  In contrast to the previous three notions, Dedekind-finite [[infinite sets]] can coexist with [[excluded middle]], although [[countable choice]] suffices to banish them.  The above two versions of Dedekind-finiteness are equivalent with excluded middle, but constructively they may differ.  In addition, there are other forms of Dedekind-finiteness that are strictly stronger even with excluded middle; see [this MO question](https://mathoverflow.net/q/410013/) for instance.

In constructive mathematics, one is usually interested in the finite sets, although the finitely indexed sets are also sometimes useful, as are the Dedekind-finite sets in the second sense.

\subsubsection{Properties and relationships}

Of course, we have

$$
\array{ & & finitely\;indexed\\
  & \neArrow & & \seArrow\\
  finite & & & & subfinitely\;indexed\\
  & \seArrow & & \neArrow\\
  & & subfinite }
$$

Moreover:

* Finite and subfinite sets have [[decidable equality]].  Conversely, any [[complement|complemented]] subset of a finite set is finite.

* Finite sets are closed under finite limits and colimits.

* A finitely indexed set with decidable equality must actually be finite.  For it is then the quotient of a decidable equivalence relation, hence a coequalizer of finite sets.  In particular, a set which is both finitely indexed and subfinite must be finite, i.e. the above "commutative square" of implications is also a "pullback".

* In particular, a set with a [[split surjection]] from $[n]$ is finite, since it has both a surjection from and an injection into $[n]$.

* Finite sets are always [[projective object|projective]]; that is, the "finite [[axiom of choice]]" always holds.

* However, if a finite set with $2$ elements (or any set, finite or not, with at least $2$ distinct elements) is [[choice object|choice]], or if every finitely-indexed set (or even any $2$-indexed set) is projective, then the logic must be classical (see [[excluded middle]] for a proof).

* Finite sets are also Dedekind-finite (in either sense).

* If _[[filtered category]]_ means _admitting cocones of every Bishop-finite diagram_, then a set is Bishop-finite iff it is a [[compact object|finitely presented object]] in Set and it is Kuratowski-finite iff it is a [[finitely generated object]] in Set.

\subsubsection{Finiteness without infinity}
{#Finitist}

All of the above definitions except for Dedekind-finiteness only make sense given the set of natural numbers. However, they can all be rephrased to make sense even without the [[axiom of infinity]] in [[set theory]], and thus in a topos without a [[natural numbers object]]. Basically, you define (for a given set $S$) the concept of 'collection of subsets of $S$ that includes all of the finite subsets' by requiring it to be closed under inductive operations appropriate for the sense of 'finite' that you want; then $S$ is finite if and only if it is an element of all such collections.  Namely, for any set $S$ we define the following subsets of the [[power set]] $P(S)$.

* $K(S)$ is the smallest subset of $P(S)$ containing the empty set and closed under the operation $A \mapsto A \cup B$ for $A$ a subset of $S$ and $B$ a [[singleton]] in $S$.  Then $S$ is finitely-indexed iff $S \in K(S)$.  Note that $K(S)$ is also the free [[semilattice]] generated by $S$.
* $\tilde{K}(S)$ is the smallest subset of $P(S)$ containing the empty set and closed under the operation $A \mapsto A \cup B$ for $A$ a subset of $S$ and $B$ a [[subsingleton]] in $S$.  Then $S$ is subfinitely-indexed iff $S \in \tilde{K}(S)$.
* $B(S)$ is the smallest subset of $P(S)$ containing the empty set and closed under the operation $A \mapsto A \cup B$ for $A$ a subset of $S$ and $B$ a singleton in $S$ [[disjoint sets|disjoint]] from $A$.  Then $S$ is finite iff $S \in B(S)$.
* $\tilde{B}(S)$ is the smallest subset of $P(S)$ containing the empty set and closed under the operation $A \mapsto A \cup B$ for $A$ a subset of $S$ and $B$ a subsingleton in $S$ disjoint from $A$.  Then $S$ is subfinite iff $S \in \tilde{B}(S)$.

In addition, finite sets are [[decidable subsets]] of themselves, so one could use the set of decidable subsets $2^S$ in the definition of finite set instead of [[power sets]]:

* Let $B(S)$ denote the smallest subset of $2^S$ containing the empty set and closed under the operation $A \mapsto A \cup B$ for $A$ a decidable subset of $S$ and $B$ a singleton in $S$ [[disjoint sets|disjoint]] from $A$. Then a set $S$ is finite iff $S \in B(S)$.

The definitions of subfinite, finitely indexed, and subfinitely indexed follow from above:

* A set is [[subfinite]] if it admits an [[injection]] into some finite set.

* A set is finitely indexed if it admits a [[surjection]] from some finite set.

* A set is subfinitely indexed if it admits a surjection from a subfinite set, or equivalently admits an injection to a finitely indexed set.

\subsection{In a topos}
{#External}

In a topos, there are both "external" and "internal" versions of all the above notions of finiteness, depending on whether we interpret their meaning "globally" or in the [[internal logic]] of the topos.  See [[finite object]].

\section{Properties of the category of finite sets}

The [[category]] [[FinSet]] of finite sets is [[equivalence of categories|equivalent]] to that of finite [[Boolean algebras]] by the [[power set]]-[[functor]]. See at _[FinSet -- Opposite category](FinSet#OppositeCategory)_ for details and see at _[[Stone duality]]_ for more.

\section{Viewing as schemes}

Every finite set can be viewed as an [[affine scheme]]. Indeed, since a _finite_ coproduct of affine schemes $Spec R_i$, $i=1,\ldots,n$, is again affine, $Spec (R_1 \times \cdots \times R_n)$, given a finite set $X$, the coproduct of $X$ many copies of the terminal scheme, $Spec \mathbb{Z}$, is the affine scheme $Spec (\mathbb{Z}^X)$.

Equipping $X$ with a [[total order]], we can view it (up to isomorphism, that is, [[bijection]]) as a set of [[integer|integers]] $\left\{ x_1, \ldots, x_n \right\}$. One can then view $X$ as the set of zeroes of the set of [[polynomial|polynomials]] in one variable $y$ given by $\left\{ y - x_1, \ldots, y - x_n \right\}$, or of the single polynomial given by the product of all these.

Thus, one can view $X$ as the affine scheme (over $Spec(\mathbb{Z})$) given as the [[spectrum of a commutative ring|commutative ring spectrum]] $Spec\left( \mathbb{Z}[y] / I \right)$, where $I$ is the ideal generated by the afore-mentioned polynomial(s). Since $\mathbb{Z}[y] / I \simeq \mathbb{Z}^n$, this agrees with the above description, but additionally lets us see $X$ as a [[closed subscheme]] of the affine line $Spec(\mathbb{Z}[y])$.

One can view $X$ as a 'constant scheme' over any other base scheme $Y$ by [[base change]], that is, by means of the canonical [[projection]] morphism $Y \times X \to Y$.

\section{Related concepts}

* [[infinite set]]

* [[hereditarily finite set]]

* [[FinSet]]

* [[space of finite subsets]]

* [[finite object]], [[finite type]]

* the [[cardinality]] of a finite set is a [[finite number]]

* [[cofinite subset]]

* [[countable ordinal]]

* [[finite-dimensional vector space]]

* [[finite homotopy type]],

  * [[finite CW-complex]], [[finite spectrum]]

* [[profinite set]]

* [[finite cover]]

* [[finite graph]]

* [[finite category]]

* [[finite limit]], [[L-finite limit]]

* [[finite homotopy type]], [[π-finite homotopy type]]

* [[locally finite type]]

\section{References}

Original articles:

* [[Kazimierz Kuratowski|C. Kuratowski]], _Sur la notion d'ensemble fini_ , Fund. Math. **I** (1920) pp.130-131. ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm1/fm1117.pdf))

See also:

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

Discussion of notions of finite sets in [[constructive mathematics]]:

* {#SpiwackCoquand10} [[Arnaud Spiwack]], [[Thierry Coquand]], *Constructively Finite?*, in: *Contribuciones cientifícas en honor de Mirian Andrés Gómez*, Universidad de La Rioja (2010) 217-230 [ISBN:978-84-96487-50-5, [inria-00503917](https://inria.hal.science/inria-00503917)]

Formalization in [[homotopy type theory]]/[[univalent foundations of mathematics]]:

* [[Dan Frumin]], [[Herman Geuvers]], [[Léon Gondelman]], [[Niels van der Weide]], _Finite Sets in Homotopy Type Theory_, in *CPP 2018: Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs* (2018) 201–214 [[doi:10.1145/3167085](https://doi.org/10.1145/3167085), [pdf](http://cs.ru.nl/~nweide/FiniteSetsInHoTT.pdf)]

[[!redirects finite set]]
[[!redirects finite sets]]
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[[!redirects Dedekind-finite set]]
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[[!redirects Dedekind finite set]]
[[!redirects Dedekind finite sets]]
[[!redirects D-finite set]]
[[!redirects D-finite sets]]


finite spectrum

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

#Contents#
* table of contents
{:toc}

## Idea

A _finite spectrum_ is a [[spectrum]] presented by sequences of [[finite homotopy types]] ([[finite CW-complexes]]). Precisely, they are the [[CW-spectra]] with a finite number of [cells of a spectrum](CW-spectrum#Cells).

## Properties

Finite spectra are the [[compact objects]] in the [[stable homotopy category]]. The full subcategory on finite spectra is equivalently the [[Spanier-Whitehead category]] on [[finite CW complexes]] (e.g. [Schwede 12, chapter II, section 7](#Schwede12)).

Many of the basic constructions and theorems in [[chromatic homotopy theory]] apply to finite [[p-local spectra]], such as

* [[telescopic localization]]

* [[periodicity theorem]]

* [[chromatic convergence theorem]]

## Related concepts

* [[Spanier-Whitehead category]]

* [[finite set]], [[finite number]]

* [[finite topological space]], [[finite CW-complex]]

* [[finite type]]

* [[p-local spectrum]]

## References

* [[Neil Strickland]], Section 2.2 of _An introduction to the category of spectra_ ([pdf](http://neil-strickland.staff.shef.ac.uk/research/stableintro.pdf))

* {#Schwede12} [[Stefan Schwede]], chapter II, section 7 of  _[[Symmetric spectra]]_, 2012  ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

[[!redirects finite spectra]]