finite topological space

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# Contents
* table of contents
{:toc}

## Definition

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

A **finite topological space** is a [[topological space]] whose underlying [[set]] is a [[finite set]].

=--

## Properties

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

Every [[finite topological space]] is an [[Alexandroff space]], i.e. finite topological spaces are [[equivalence of categories|equivalent]] to finite [[preordered sets]], by the [[specialisation order]].

=--

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

Finite topological spaces have the same _[[weak homotopy equivalence|weak]]_ [[homotopy type]]s as finite [[simplicial complexes]] / finite [[CW-complexes]].

=--

This is due to ([McCord 67](#McCord67)).

+-- {: .proof}
###### Proof (sketch)
If $\mathbf{2}$ is [[Sierpinski space]] (two points $0$, $1$ and three opens $\emptyset$, $\{1\}$, and $\{0, 1\}$), then the continuous map $I = [0, 1] \to \mathbf{2}$ taking $0$ to $0$ and $t \gt 0$ to $1$ is a weak homotopy equivalence[^fine].

The essential construction in the proof is as follows: for any finite topological space $X$ with specialization order $\mathcal{O}(X)$, the topological [[interval]] map $I \to \mathbf{2}$ induces a weak homotopy equivalence $B\mathcal{O}(X) \to X$:

$$B\mathcal{O}(X) = \int^{[n] \in \Delta} Cat([n], \mathcal{O}(X)) \cdot Int([n], I) \to \int^{[n] \in \Delta} Cat([n], \mathcal{O}(X)) \cdot Int([n], \mathbf{2}) \cong X$$

(where we implicitly identify $\Delta^{op}$ with the category $Int$ of finite intervals with distinct top and bottom, so that $[n] \mapsto Int([n], I)$ is a covariant functor on $\Delta$). A few remarks on this construction:

* The *interval* $[n]$ has $n+2$ elements, two of which are the distinct top and bottom. The space $Int([n], I)$ is the $n$-dimensional affine simplex. The space $Int([n], \mathbf{2})$ has $n+1$ points $0, 1, \ldots, n$, where $j$ is in the closure of $j+1$ for $0 \leq j \lt n$. The map $Int([n], I) \to Int([n], \mathbf{2})$ induced by $I \to \mathbf{2}$ takes every interior point of $Int([n], I)$ to $n \in Int([n], \mathbf{2})$.

* Informally, the isomorphism on the right says that any finite topological space $X$ can be constructed by gluing together copies of Sierpinski space $\mathbf{2}$, just as any preorder can be constructed by gluing together copies of the preorder $\{0 \leq 1\}$. More formally, the isomorphism is established for objects $X$ in the equivalent category $PreOrd_{fin}$, by restricting an isomorphism over objects $X$ of the larger category $PreOrd$, given by the counit of a [[nerve and realization]] adjunction
$$\int^{[n] \in \Delta} Cat([n], X) \cdot Int([n], \{0 \leq 1\}) \cong \int^{[n] \in \Delta} Cat([n], X) \cdot [n] \stackrel{counit}{\cong} X$$
where the counit is an isomorphism because the inclusions $PreOrd \hookrightarrow Cat \stackrel{nerve}{\hookrightarrow} Set^{\Delta^{op}}$ are fully faithful.

On the other hand, any finite simplicial complex $K$ is homotopy equivalent to its [[barycentric subdivision]]. This is $B P K$, the [[geometric realization]] of the [[nerve]] of the [[poset]] $P K$ whose elements are simplices ordered by inclusion. Thus finite posets model the weak homotopy types of finite simplicial complexes.

=--

## Examples

* [[pseudocircle]]

## Related concepts

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

* [[finite CW-complex]]

* [[finite spectrum]]

* [[finite type]]

## References

A survey which includes the McCord theorems as background material is in

* [[Jonathan Barmak]], _Topología Algebraica de Espacios Topológicos Finitos y Aplicaciones,_ PhD thesis 2009 ([pdf](http://www.math.kth.se/~jbarmak/tesisfinal2.pdf))

published as

* [[Jonathan Barmak]], _Algebraic Topology of Finite Topological Spaces and Applications_, Lecture Notes in Mathematics,2032. Springer, Heidelberg (2011). [doi:10.1007/978-3-642-22003-6](https://doi.org/10.1007/978-3-642-22003-6)

The original results by McCord are in

* {#McCord66} [[Michael C. McCord]], _Singular homology groups and homotopy groups of finite topological spaces_ , Duke Math. J. 33 (1966), 465-474. ([EUCLID](http://projecteuclid.org/euclid.dmj/1077376525))

* {#McCord67} [[Michael C. McCord]], _Homotopy type comparison of a space with complexes associated with its open covers_ . Proc. Amer. Math. Soc. 18 (1967), 705-708, [doi:10.1090/S0002-9939-1967-0216499-0](https://doi.org/10.1090/S0002-9939-1967-0216499-0), [copy](http://www.maths.ed.ac.uk/~aar/papers/mccord2.pdf)

[^fine]: Any topological meet-semilattice $L$ with a bottom element $\bot$, for which there exists a continuous path $\alpha \colon I \to L$ connecting $\bot$ to the top element $\top$, is in fact contractible. The contracting homotopy is given by the composite $I \times L \stackrel{\alpha \times 1}{\to} L \times L \stackrel{\wedge}{\to} L$.

Generalization to [[ringed space|ringed]] finite spaces is discussed in

* [[Fernando Sancho de Salas]], _Ringed Finite Spaces_ ([arXiv:1409.4574](https://arxiv.org/abs/1409.4574))

* [[Fernando Sancho de Salas]], _Finite Spaces and Schemes_ ([arXiv:1602.02393](http://arxiv.org/abs/1602.02393))

and aspects of their [[homotopy theory]] is discussed in

* [[Fernando Sancho de Salas]], _Homotopy of ringed finite spaces_ ([arXiv:1511.06284](http://arxiv.org/abs/1511.06284))

[[!redirects finite topological space]]
[[!redirects finite topological spaces]]


finite type

The term "finite type" may refer to:

* In [[algebra]], [[homological algebra]] and [[rational homotopy theory]]:

  *[[objects of finite type]]*.

* In [[homotopy theory]]:

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

* In [[rational homotopy theory]] also:

  with [[finite-dimensional vector spaces|finite dimensional]] [[rational cohomology]] in each degree ("rational finite type")

* In [[type theory]]:

  [[types]] of the kind of [[finite sets]] (see at *[[family of finite types]]* and *[[inductive family]]*)

  also [[pi-finite types]], [[type of finite types]]

Related concepts:

* [[finite set]], [[profinite set]]

* [[finite number]]

* [[finite object]]

* [[finitely generated object]], [[finitely presented object]]

* ([[fully dualizable object|fully]]-)[[dualizable object]]

  [[perfect module]]

* [[compact object]]

* [[finite category]], [[finite graph]]

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

[[!redirects finite types]]


finite étale morphism of anabelioids

\tableofcontents

\section{Introduction}

Observing that an [[anabelioid]] is in particular a [[topos]], a finite étale morphism $\mathcal{X} \rightarrow \mathcal{Y}$ of [[anabelioid|anabelioids]] is the same as an [[étale geometric morphism]]: it is a 'local isomorphism', in some sense.

\section{Definition}

Recall that a [[morphism of anabelioids]] $\mathcal{X} \rightarrow \mathcal{Y}$ is simply an [[exact functor]] $\mathcal{Y} \rightarrow \mathcal{X}$, that is to say, a functor $\mathcal{Y} \rightarrow \mathcal{X}$ preserving both finite limits and finite colimits. The following definition follows [Mochizuki2004](#Mochizuki2004).

\begin{defn} A [[morphism of anabelioids]] $f: \mathcal{X} \rightarrow \mathcal{Y}$ is _finite étale_ if there is an object $S$ of $\mathcal{Y}$ such that $f = i_S \circ i$ for some isomorphism of anabelioids $i: \mathcal{X} \rightarrow \mathcal{Y}_{/ S}$, where $\mathcal{Y}_{/ S}$ denotes the [[overcategory]] of objects of $\mathcal{Y}$ over $S$, and where $i_{S}: \mathcal{Y}_{/ S} \rightarrow \mathcal{Y}$ is the canonical functor.  \end{defn}

\section{References}

* {#Mochizuki2004} _The geometry of anabelioids_, [[Shinichi Mochizuki]], 2004, Publ. Res. Inst. Math. Sci., 40, No. 3, 819-881. [paper](http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Geometry%20of%20Anabelioids.pdf) [Zentralblatt review](https://zbmath.org/?q=an%3A1113.14021)


finite ∞-group

#Contents#
* table of contents
{:toc}

## Idea

A **finite ∞-group** is an [[∞-group]] with finitely many non-trivial [[homotopy groups]] which are all finite. It is therefore a pointed [[connected]] [[truncated]] [[homotopy type with finite homotopy groups]].

##References

* [[Matan Prasma]], [[Tomer Schlank]], _Sylow theorems for ∞-groups_, ([arXiv:1602.04494](https://arxiv.org/abs/1602.04494))


finite-dimensional complex orientation and Ravenel's spectra -- references

### Finite-dimensional complex orientation and Ravenel's spectra
 {#ReferencesFiniteStageOrientations}

Discussion of [[complex oriented cohomology theory|complex orientation]] (in [[Whitehead generalized cohomology]]) on (only) those [[complex vector bundles]] which are [[pullback bundle|pulled back]] from base spaces of _bounded [[cell complex|cell]]-[[dimension]] ([Hopkins 84, 1.2](#Hopkins84), [Ravenel 86, 6.5.2](#Ravenel86))_ -- or rather, for the most part, of  [[Ravenel's spectrum|Ravenel's]] [[Thom spectra]] $X(n)$ and $T(m)$ ([Ravenel 84, Sec. 3](#Ravenel84))
which [[representable functor|co-represent]] these:

* {#Ravenel84} [[Douglas Ravenel]], section 3 of: _Localization with Respect to Certain Periodic Homology Theories_, American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 351-414 ([doi:10.2307/2374308](https://doi.org/10.2307/2374308), [jstor:2374308](https://www.jstor.org/stable/2374308))

* {#Hopkins84} [[Michael Hopkins]], _Stable decompositions of certain loop spaces_, Northwestern 1984 ([[HopkinsStableDecompositions.pdf:file]])

* {#Ravenel86} [[Douglas Ravenel]], section 6.5 of: _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986

* {#DevinatzHopkinsSmith88} [[Ethan Devinatz]], [[Michael Hopkins]], [[Jeffrey Smith]], Theorem 3 of: _Nilpotence and Stable Homotopy Theory I_, Annals of Mathematics Second Series, Vol. 128, No. 2 (Sep., 1988), pp. 207-241 ([jstor:1971440](https://www.jstor.org/stable/1971440))

* [[Doug Ravenel]], _The first Adams-Novikov differential for the spectrum $T(m)$_, 2000 ([pdf](https://web.math.rochester.edu/people/faculty/doug/mypapers/first.pdf), [[RavenelAdamsNovikovForTm.pdf:file]])

* Ippei Ichigi, Katsumi Shimomura, _The Modulo Two Homotopy Groups of the $L_2$-Localization of the Ravenel Spectrum_, CUBO A Mathematical Journal,
Vol. 10, No 03, (43–55). October 2008 ([cubo:1498](http://revistas.ufro.cl/ojs/index.php/cubo/article/view/1498))

* Gabe Angelini-Knoll, J. D. Quigley, _The Segal Conjecture for topological Hochschild homology of the Ravenel spectra_, Journal of Topology 4.3 (2011): 591-622 ([arXiv:1705.03343](https://arxiv.org/abs/1705.03343), [doi:10.1112/jtopol/jtr015]( https://doi.org/10.1112/jtopol/jtr015))

* [[Jonathan Beardsley]], _A Theorem on Multiplicative Cell Attachments with an Application to Ravenel's $X(n)$ Spectra_, Journal of Homotopy and Related Structures volume 14, pages 611–624 (2019) ([arXiv:1708.03042](https://arxiv.org/abs/1708.03042), [doi:10.1007/s40062-018-0222-6](https://link.springer.com/article/10.1007/s40062-018-0222-6))

* [[Jonathan Beardsley]], _Topological Hochschild homology of $X(n)$_ ([arXiv:1708.09486](https://arxiv.org/abs/1708.09486))

* Xiangjun Wang, Zihong Yuan, _The homotopy groups of $L_2 T(m)/\big(p^{[\tfrac{m}{2}]+2}, v_1 \big)$ for $m \gt 1$_, New York J. Math.24 (2018) 1123–1146 ([pdf](https://nyjm.albany.edu/j/2018/24-55v.pdf))

[[!redirects finite-rank complex orientation and MΩSUn -- references]]

[[!redirects finite-rank complex orientation and Ravenel's spectra -- references]]
[[!redirects finite-rank complex orientation and Ravenel's spectra -- references]]


finite-dimensional Hilbert space

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* table of contents
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## Idea

A *finite dimensional Hilbert space* is a [[Hilbert space]] whose [[underlying]] [[vector space]] is a [[finite-dimensional vector space]].

Since the completeness-condition on a [[Hilbert space]] is automatically satisfied in finite dimensions, finite-dimensional Hilbert spaces are equivalently [[finite dimensional vector spaces]] (usually [[complex vector spaces]], sometimes [[real vector spaces]]) equipped with a positive-definite [[Hermitian form|Hermitian]] [[inner product]], aka *Hermtian inner product spaces*.

Finite-dimensional Hilbert spaces are the [[spaces of quantum states]] of principal interest in [[quantum information theory]] and [[quantum computation]]. In [[quantum physics]] more broadly they appear in discussion of internal degrees of freedom (such as [[spin]]) or generally as the [[quantization]] of [[compact topological space|compact]] [[phase spaces]] (cf. eg. the [[geometric quantization of the 2-sphere]] or generally the [[orbit method]]).

The fact that they form a [[dagger-compact category]] [[cf. [Abramsky & Coecke 2004 p 10](#AbramskyCoecke04); Selinger 2012](#Selinger12)] has led to the [[string diagram]]-formulation of [[quantum information theory via dagger-compact categories]].

## Related concepts

* [[quantum information theory via dagger-compact categories]]

* [[finite-dimensional vector space]]

* [[finite abelian category]]

## References

* [[Paul R. Halmos]], *Finite-Dimensional Hilbert Spaces*, The American Mathematical Monthly, **77** 5 (1970) 457-464 [[doi:10.2307/2317378](https://doi.org/10.2307/2317378), [jstor:2317378](https://www.jstor.org/stable/2317378)]

* {#AbramskyCoecke04} [[Samson Abramsky]], [[Bob Coecke]], p. 10 of *A categorical semantics of quantum protocols*, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04), IEEE Computer Science Press (2004) [[arXiv:quant-ph/0402130](http://arxiv.org/abs/quant-ph/0402130)]

* {#Selinger12} [[Peter Selinger]], *Finite dimensional Hilbert spaces are complete for dagger compact closed categories*, Logical Methods in Computer Science, **8** 3 (2012) lmcs:1086 [[arXiv:1207.6972](https://arxiv.org/abs/1207.6972), doi:10.2168/LMCS-8(3:6)2012]

* {#Landsman17} [[Klaas Landsman]], §A.1 in: *Foundations of quantum theory -- From classical concepts to Operator algebras*, Springer Open (2017) [[doi:10.1007/978-3-319-51777-3](https://link.springer.com/book/10.1007/978-3-319-51777-3), [pdf](https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf)]

Lecture notes:

* [[Bergfinnur Durhuus]], §4.1, §4.2 in: *Operators on Hilbert Space* [[pdf](https://web.math.ku.dk/~durhuus/MatFys/MatFys4.pdf), [[Durhuus-OperatorsOnHilbertSpace.pdf:file]]]

[[!redirects finite-dimensional Hilbert spaces]]


finite-dimensional representation of the Poincaré group

#Contents#
* table of contents
{:toc}

## Idea

[[representations]] of the [[Poincaré group]] of [[finite number|finite]] [[dimension]]

## Related concepts

* [[unitary representation of the Poincaré group]]

## References

* S. M. Paneitz, _All linear representations of the Poincaré group up to dimension 8, Ann. Inst. H. Poincar´e 40, 35–57 (1984).

* S.M. Paneitz, _Indecomposable Finite Dimensional Representations Of The Poincaré Group And Associated Fields_, in _Differential Geometric Methods in Mathematical Physics_ Volume 1139 of the series Lecture Notes in Mathematics pp 6-9 (1983)

* Hans Plesner Jakobsen, _Indecomposable Finite-Dimensional Representations of a Class of Lie Algebras and Lie Superalgebras_, in _Supersymmetry in Mathematics and Physics_ Volume 2027 of the series Lecture Notes in Mathematics pp 125-138, June 2011 ([arXiv:1210.1958](https://arxiv.org/abs/1210.1958))

[[!redirects finite-dimensional representations of the Poincaré group]]

[[!redirects finite-dimensional representation of the Poincare group]]
[[!redirects finite-dimensional representations of the Poincare group]]


finite-dimensional space

Given a notion of [[space]] and a notion of [[dimension]] assigning a [[cardinal number]] or [[ordinal number]] to each space (or at least to some of them), a space is __finite-dimensional__ if this dimension exists as a [[natural number]].  Whether one is talking about [[vector spaces]], [[topological spaces]], [[manifolds]], or whatever, finite-dimensional spaces tend to have nicer properties than arbitrary spaces (such as [[infinite-dimensional manifolds]]).

Spaces with [[fractal]] dimension may also be considered finite-dimensional, but that\'s not usually what people are talking about when they say 'finite-dimensional space'.

[[!redirects finite-dimensional space]]
[[!redirects finite-dimensional spaces]]
[[!redirects finite dimensional space]]
[[!redirects finite dimensional spaces]]

[[!redirects finite dimension]]
[[!redirects finite dimensions]]


Finite-Dimensional topological vector space


finite-dimensional vector space

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* table of contents
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## Definition

A [[vector space]] is _finite-dimensional_ if it admits a [[finite set|finite]] [[basis]]: if there exists a [[natural number]] $n$ such that the vector space admits a [[linear isomorphism]] to the [[direct sum]] of $n$ copies of the [[underlying]] vector space of the [[ground field]].

The category [[FinDimVect]] of finite-dimensional vector spaces is of course the [[full subcategory]] of [[Vect]] whose objects are finite-dimensional.

## Properties

### Compact closure
 {#CompactClosure}

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

A vector space $V$ is finite-dimensional precisely if the [[hom-functor]] $\hom(V, -) \colon Vect \to Set$ [[preserved colimit|preserves]] [[filtered colimits]].

=--

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

Notice that every vector space $W$ is the  [[filtered colimit]] of the [[diagram]] of finite-dimensional subspaces $W' \subseteq W$ and inclusions between them. Applying this to $W = V$, the condition that $\hom(V,-)$ preserves filtered colimits implies that the canonical comparison map from the following [[filtered colimit]] over the finite-dimensional subspaces is an [[isomorphism]]:

$$
  \underset{
    \underset
     {\mathclap{fd\, V' \subseteq V}}
     {\longrightarrow}
  }{\lim}
  \;
  \hom(V, V')
    \overset{\sim}{\longrightarrow}
  \hom(V, V)
  \mathrlap{\,.}
$$

Therfore some element $[f]$ in the colimit represented by $f \colon V \to V'$ gets mapped to the [[identity morphism|identity]] $id_V$, i.e., $i \circ f = id_V$ for some inclusion $i \colon V' \hookrightarrow V$. This implies $i$ is an isomorphism, so that $V$ is finite-dimensional.

In the converse direction, observe that $\hom(V, -)$ has a [[right adjoint]] (and [[left adjoints preserve colimits|hence preserves all colimits]]) if $V$ is finite-dimensional.

To see this, first notice that the [[dual vector space]] $V^\ast$ of [[functionals]] $f \colon V \to k$ to the [[ground field]] is a [[dual object]] to $V$ in the [[category with duals|sense of monoidal category theory]], so that there is a [[counit of an adjunction|counit]] ([[evaluation map]])

$$
  \array{
    \mathllap{
      ev
      \,\colon\,
    }
    V^\ast \otimes V
    &\longrightarrow&
    k
    \\
    f \otimes v
    &\mapsto&
    f(v)
    \mathrlap{\,.}
  }
$$
The [[unit of an adjunction|unit]] is uniquely determined from this counit and can be described using any [[linear basis]] $e_i$ of $V$ and dual basis $f_j$ as the map

$$
  \array{
    \mathllap{
      coev
      \,\colon\,
    }
    k
    &
    \longrightarrow
    &
    V \otimes V^\ast
    \\
    1 &\mapsto& \sum_i e_i \otimes f_i
    \mathrlap{\,.}
  }
$$

We thus have an [[adjoint functor|adjunction]]

$$
  (- \otimes_k V)
    \;\;\dashv\;\;
  (- \otimes V^\ast)
  \,,
$$
whose [[mate]] is, by the [[internal hom]]-adjunction $(-) \otimes V \;\dashv\; hom(C,-)$ of this form:

$$
  \hom(V, -) \;\dashv\; \hom(V^\ast, -)
$$

exhibiting the desired [[right adjoint]] to $hom(V,-)$.

=--

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

Finite-dimensional vector spaces are exactly the [[compact objects]] of [[Vect]] in the sense of [[locally presentable categories]], but also the compact = [[dualizable objects]] in the sense of [[monoidal category]] theory. In particular the category [[FinDimVect]] is a [[compact closed category]].

=--

## Related concepts

* [[finite-dimensional Hilbert space]]

* [[finite abelian category]]

* [[Schur functor]]

* [[finite quantum mechanics in terms of dagger-compact categories]]

## References

Discussion of finite-dimensional vector spaces as [[categorical semantics]] for [[linear logic]]/[[linear type theory]]:

* [[Benoît Valiron]], [[Steve Zdancewic]], *Finite Vector Spaces as Model of Simply-Typed Lambda-Calculi*, in: *Proc. of ICTAC'14*, Lecture Notes in Computer Science **8687**, Springer (2014) 442-459 [[arXiv:1406.1310](https://arxiv.org/abs/1406.1310), [doi:10.1007/978-3-319-10882-7_26](https://doi.org/10.1007/978-3-319-10882-7_26), slides:[pdf](https://www.cs.bham.ac.uk/~drg/bll/steve.pdf), [[Zdancewic-LinearLogicAlgebra.pdf:file]]]

  > (focus on [[finite field|finite]] [[ground fields]])

* {#Murfet14} [[Daniel Murfet]], *Logic and linear algebra: an introduction* [[arXiv:1407.2650](http://arxiv.org/abs/1407.2650)]

[[!redirects finite-dimensional vector space]]
[[!redirects finite-dimensional vector spaces]]
[[!redirects finite dimensional vector space]]
[[!redirects finite dimensional vector spaces]]


finite-product theory > history

< [[finite-product theory]]
[[!redirects finite-product theory -- history]]


finitely cocomplete > history

see at *[[finitely cocomplete category]]*


finitely cocomplete category

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## Idea

A *finitely cocomplete category* is a [[category]] $\mathcal{C}$ which admits all _finite_ [[colimits]], that is a colimit for any [[finite category|finite]] [[diagram]] $F \colon \mathcal{J} \to \mathcal{C}$.

(A category with all [[small category|small]] colimit is called a *[[cocomplete category]]*).

This is the notion [[formal dual|formally dual]] to that of *[[finitely complete categories]]*, see there for more.

Finitely cocomplete categories are also called **rex categories**, although this is rarer than 'lex' for [[finitely complete categories]].

## Related concepts

* [[finitely complete category]]

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

[[!redirects finitely cocomplete categories]]

[[!redirects finitely cocomplete]]


finitely complete (infinity,1)-category

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## Idea

_Finitely complete $(\infty,1)$-categories_ are the analog in [[(∞,1)-category theory]] of [[finitely complete categories]] in [[category theory]].

## Definition

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

Let $\mathcal{C}$ be an [[(∞,1)-category]]. This is called a _finitely complete $(\infty,1)$-category_ if  $\mathcal{C}$ has [[finite (∞,1)-limits]], i.e.

* if it has a [[terminal object in an (∞,1)-category|terminal object]] and [[(∞,1)-fiber products]];
* if it has a [[terminal object in a quasi-category|terminal object]] and for every [[object]] $x \in C$, the [[over-(∞,1)-category]] $C_{/x}$ has finite [[products]].

=--

## Related concepts

* [[finitely complete category]]

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

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

* [[exact (infinity,1)-category]]

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

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


finitely complete > history

see at *[[finitely complete category]]*


finitely complete category

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#Contents#
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## Definition

A **finitely complete category** is a [[category]] $C$ which admits all __[[finite limit]]s__, that is all [[limit|limits]] for any [[diagrams]] $F: J \to C$ with $J$ a [[finite category]].  Finitely complete categories are also called **lex categories**.  They are also called (at least by Johnstone in the [[Elephant]]) **[[cartesian categories]]**, although this term more often means a [[cartesian monoidal category]].

Small finitely complete categories form a [[2-category]], [[Lex]].

## Variants

There are several well known reductions of this concept to classes of special limits. For example, a category is finitely complete if and only if:

* It has a [[terminal object]] and admits all binary [[product|products]] and [[equalizer|equalizers]];
* It has a [[terminal object]] and admits all [[fibers]] and binary [[product|products]];
* It has a [[terminal object]] and admits all [[pullback|binary pullbacks]].

An appropriate notion of morphism between finitely complete categories $C$, $D$ is a left [[exact functor]], or a functor that preserves finite limits (also called a _lex_ functor, a _cartesian_ functor, or a _finitely continuous_ functor). A functor preserves finite limits if and only if:

* it preserves terminal objects, binary products, and equalizers;
* it preserves terminal objects, fibers, and binary products;
* it preserves terminal objects and binary pullbacks.

Since these conditions frequently come up individually, it may be worthwhile listing them separately:

* $F: C \to D$ **preserves terminal objects** if $F(t_C)$ is terminal in $D$ whenever $t_C$ is terminal in $C$;

* $F: C \to D$ **preserves binary products** if the pair of maps
$$F(c) \stackrel{F(\pi_1)}{\leftarrow} F(c \times d) \stackrel{F(\pi_2)}{\to} F(d)$$
exhibits $F(c \times d)$ as a product of $F(c)$ and $F(d)$, where $\pi_1: c \times d \to c$ and $\pi_2: c \times d \to d$ are the product projections in $C$;

* $F: C \to D$ **preserves equalizers** if the map
$$F(i): F(e) \to F(c)$$
is the equalizer of $F(f), F(g): F(c) \stackrel{\to}{\to} F(d)$, whenever $i: e \to c$ is the equalizer of $f, g: c \stackrel{\to}{\to} d$ in $C$.

## Properties

In any finitely complete category, the [[kernel pair]] of the [[identity morphism]] $id$ on an object $X$ is the [[diagonal morphism]] $(id,id)$ of $X$ and has a [[coequalizer]] [[isomorphic]] to $X$ itself.

## Related concepts

* [[complete category]]

* [[finitely cocomplete category]]

* [[locally cartesian category]]

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

[[!include categories and logic - table]]

## References

Section A1.2 in

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

[[!redirects lex category]]
[[!redirects left exact category]]
[[!redirects finitely complete categories]]
[[!redirects lex categories]]
[[!redirects left exact categories]]
[[!redirects category with finite limits]]
[[!redirects categories with finite limits]]

[[!redirects finitely complete]]


finitely constructible object

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

A [[coherent object]] in an [[(∞,1)-topos]] which is also [[n-truncated object of an (∞,1)-category|n-truncated]] for some $n$ is called a _[[finitely constructible object]]_.

=--

([[Rational and p-adic Homotopy Theory|Lurie pAdic, def. 2.3.1]])

[[!redirects finitely constructible objects]]


finitely generated > history

< [[finitely generated]]


finitely generated algebra

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## Definition

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

Given a [[commutative ring]] $R$ and an $R$-[[associative algebra|algebra]] $A$, this algebra is _finitely generated over $R$_ if it is a [[quotient]] of a [[polynomial ring]] $R[x_1, \cdots, x_n]$ on finitely many variables.

If moreover $A = R[x_1, \cdots, x_n]/(f_1, \cdots, f_k)$ for a finite number of [[polynomials]] $f_i$, then $A$ is called _finitely presented_.

=--

## Examples

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

A [[morphism of finite presentation]] between [[schemes]] is one which is dually locally given by finitely presented algebras.

=--

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

A [[ring]] is an [[associative algebra]] over the [[integers]], hence a $\mathbb{Z}$-ring. Accordingly  a _finitely generated ring_ is a finitely generated $\mathbb{Z}$-algebra, and similarly for _finitely presented ring_.

For rings every finitely generated ring is already also finitely presented.

=--

## Related concepts

* [[morphism of finite type]], [[morphism of finite presentation]]

* [[Zariski site]]

* [[finitely generated object]]

  * [[finitely generated module]]

  * [[finitely generated group]]

## References

Finite generation of algebras plays a role in the choice of [[geometry (for structured (infinity,1)-toposes)]] in

* [[Jacob Lurie]], section 2.5 of _[[Structured Spaces]]_

[[!redirects finitely generated algebras]]

[[!redirects finitely presented algebra]]
[[!redirects finitely presented algebras]]

[[!redirects finitely generated ring]]
[[!redirects finitely generated rings]]

[[!redirects finitely presented ring]]
[[!redirects finitely presented rings]]


finitely generated group

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#Contents#
* table of contents
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## Idea

A *finitely generated group* is a ([[discrete group|discrete]]) [[group]] (equipped) with a [[finite set]] of [[generators]], hence of [[elements]] such that any other element can be written as a product of these.

## Properties

### Relation to Cayley graphs

Associated with a set of generators on a group is the corresponding [[Cayley graph]], whose [[vertices]] are the group elements which are connected by an [[edge]] if they differ by (left, say) multiplication with one of the generators.

The corresponding [[graph distance]] equips the group with a [[metric]] (the *[[word metric]]*) and thus makes it an object of [[geometric group theory]].

## Examples

\begin{example}
**([[symmetric group]])**
\linebreak
The [[symmetric group]] $Sym(n)$ of [[permutations]] of $n \in \mathbb{N}$ [[elements]] may be [[finitely generated group|generated]] from:

* all [[transposition permutations]] -- the corresponding Cayley [[graph distance]] is the original *[[Cayley distance]]*;

* the *adjacent* [[transpositions]] -- the corresponding Cayley [[graph distance]] is known as the *[[Kendall tau distance]]*.

\end{example}

\begin{example}\label{BraidGroupExample}
**([[braid group]])**
\linebreak
The [[braid group]] $Br(n)$ on $n \in \mathrm{N}$ strands is finitely generated via the *[Artin presentation](braid+group#ArtinPresentation)*.
\end{example}

## Related concepts

* [[finite group]], [[profinite group]]

* [[generators and relations]]

* [[finitely generated object]]

  * [[finitely generated algebra]]

  * [[finitely generated group]]

  * [[finitely generated module]], [[finitely presented module]]

* [[Solovay-Kitaev theorem]]

## References

Textbooks:

* [[Cornelia Druţu]], [[Michael Kapovich]], Chapter 7 of: *Geometric group theory*, Colloquium Publications **63**, AMS 2018 ([ISBN:978-1-4704-1104-6](https://bookstore.ams.org/coll-63/), [pdf](https://courses.maths.ox.ac.uk/node/view_material/35649))

See also

* Wikipedia, *[Finitely generated group](https://en.wikipedia.org/wiki/Finitely_generated_group)*

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

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: Chapter 6 of: *[[Symmetry]]* (2021) $[$[pdf](https://unimath.github.io/SymmetryBook/book.pdf)$]$

[[!redirects finitely generated groups]]


finitely generated module

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## Definition

Given a (not necessarily unital) [[ring]] $R$, a left $R$-[[module]] $(N,\nu)$, is __finitely generated__ if there exists a [[finite set]] $S\subseteq N$ such that the canonical morphism from the [[free module]] $F(S) \longrightarrow N$ is a [[surjection]].

## Related concepts

* [[generators and relations]]

* [[finitely generated object]]

  * [[finitely generated algebra]]

  * [[finitely generated group]]

  * [[finitely presented module]]

* [[coherent module]]

* [[free module]]

* [[Smith normal form]]

* [[Eilenberg swindle]]

## References

See also:

* Wikipedia, *[Finitely generated module](https://en.wikipedia.org/wiki/Finitely_generated_module)*

[[!redirects finitely generated modules]]


finitely generated object

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#### Category theory
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# Finitely generated objects
* table of contents
{: toc}

## Definition in arbitrary categories

Let $C$ be a [[locally small category]] that admits [[filtered colimits]] of monomorphisms. Then an [[object]] $X \in C$ is __finitely generated__ if the [[corepresentable functor]]

$$
  Hom_C(X,-) : C \to Set
$$

[[preserved limit|preserves]] these [[filtered colimits]] of monomorphisms.  This means that for every [[filtered category]] $D$ and every functor $F : D \to C$ such that $F(f)$ is a monomorphism for each morphism $f$ of $D$, the canonical morphism

$$
  \underset{\to_d}{\lim} C(X,F(d)) \stackrel{\simeq}{\to}
  C(X, \underset{\to_d}{\lim} F(d))
$$

is an [[isomorphism]].

## Definition in concrete categories

An object $A$ of a [[concrete category]] $C$ is __finitely generated__ if it is a [[quotient object]] (in the sense of a [[regular epimorphism]]) of some [[free object]] $F$ in $C$, where $F$ is free on a [[finite set]].

The object $A$ is __finitely presented__ if it is the [[coequalizer]] of a [[parallel pair]] $R \rightrightarrows F$ such that $R$ is also free on a finite set.

## Examples

* A set $X$ is a finitely generated object in [[Sets]] iff it is [[finite set|(Kuratowski-)finite]]. For this to hold constructively, [[filtered categories]] (appearing in the definition of _[[filtered colimit]]_) have to be understood as categories admitting cocones of every _Bishop-finite_ diagram.

*  For $R$ a [[ring]], an $R$-[[module]] $N$ is a [[finitely generated module]] if it is a [[quotient]] of a [[free module]] with a finite [[basis of a free module|basis]].

* [[finitely generated group]]

## Related concepts

* [[finitely presentable object]]

* [[generators and relations]]

* [[finite object]]

* [[object of finite type]]

## References

The general definition is Def. 1.67 in:

* [[Jiří Adámek]], [[Jiří Rosický]], *[[Locally Presentable and Accessible Categories]]*, London Mathematical Society Lecture Note Series, number 189, Cambridge University Press 1994 ([doi:10.1017/CBO9780511600579](https://doi.org/10.1017/CBO9780511600579))

Further development and connections to locally generated categories in

* [[Jiří Adámek]], [[Jiří Rosický]], *[[What are locally generated categories,]]*, Proc. Categ. Conf. Como 1990, Lect. Notes in Math. 1488 (1991), 14-19.

* [[Ivan Di Liberti]], [[Jiří Rosický]], *[[Enriched Locally Generated Categories]]*,  ([arxiv:2009.10980](https://arxiv.org/abs/2009.10980))

[[!redirects finitely generated object]]
[[!redirects finitely generated objects]]
[[!redirects finitely generated]]

[[!redirects finitely presented object]]
[[!redirects finitely presented objects]]
[[!redirects finitely presented]]


finitely presentable group

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# Finitely presented groups
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## Idea

A group $G$ is *finitely presentable* if it has a *finite presentation*, i.e., there is a [[group presentation]], $\langle X: R\rangle$, for $G$ with both its set, $X$, of generators and its set, $R$, of relations being finite sets.

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

The term 'finitely presented' is often used rather than `finitely presentable', however 'finitely presented' would seem to imply that a given finite presentation was intended, whilst here only the existence of one is required.
=--

## Related concepts

* [[finitely generated group]]

## References

Textbook accounts on [[group presentations]]:

* [[David Lawrence Johnson]], *Topics in Theory of Group Presentations*, London Mathematical Society Lecture notes series **42**, Cambridge Univ. Press (1980) [[doi:10.1017/CBO9780511629303](https://doi.org/10.1017/CBO9780511629303)]

* [[David Lawrence Johnson]], *Presentations of Groups*, London Mathematical Society Student Texts **15**, Cambridge Univ. Press (1990) [[doi:10.1017/CBO9781139168410](https://doi.org/10.1017/CBO9781139168410)]

Lecture notes:

* Derek Holt (notes by Florian Bouyer), *Presentation of Groups* (2013) [[pdf](https://warwick.ac.uk/fac/sci/maths/people/staff/fbouyer/presentation_of_group.pdf)]

On (finitely) presented groups as [[fundamental groups]] of ([[finite cell complex|finite]]) [[simplicial complexes]]/[[CW-complexes]]:

* [[Joseph J. Rotman]], around Thm. 7.34 in: *An Introduction to Algebraic Topology*, Graduate Texts in Mathematics **119** (1988) $[$[doi:10.1007/978-1-4612-4576-6](https://doi.org/10.1007/978-1-4612-4576-6)$]$

* Behrooz Mashayekhy, Hanieh Mirebrahimi, *Some Properties of Finitely Presented Groups with Topological Viewpoints*, International Journal of Mathematics, Game Theory and Algebra **18** 6 (2010) 511-515 [[arXiv:1012.1744](https://arxiv.org/abs/1012.1744)]

[[!redirects finitely presented group]]
[[!redirects finitely presented groups]]
[[!redirects finitely presentable group]]
[[!redirects finitely presentable groups]]


finitely presentable object > history

< [[finitely presentable object]]


finitely presented module

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## Definition

Given a (not necessarily [[commutative ring|commutative]]) [[unital ring]] $R$, an $R$-[[module]] is _finitely presented_ (or of finite presentation) if there exists an [[exact sequence]] $R^q\to R^p\to M\to 0$ where $p,q$ are [[natural numbers]].

## Equivalent characterisations

\begin{proposition}
For a module $M$ over a ring $R$, the following are equivalent :

1. $M$ is finitely presented ;
1. $M$ is a [[compact object]] of the category of $R$-modules.

\end{proposition}

\begin{proposition}
For a module $M$ over a ring $R$, the following are equivalent :

1. $M$ is finitely presented ;
1. for every family $\{Q_\alpha\}_{\alpha \in A}$ of $R$-modules, the canonical map
$$
  M \otimes_R \prod_\alpha Q_\alpha \rightarrow \prod_\alpha M \otimes_R Q_\alpha
$$
is an isomophism ;
1. for every $R$-module $Q$ and every set $A$, the canonical map
$$
  M \otimes_R R^A \rightarrow M^A
$$
is an isomorphism.

\end{proposition}

\begin{proof}
$1 \Rightarrow 2$. Using a finite presentation $R^q \to R^p \to M$
of $M$ one can write the commutative diagram
\begin{centre}
    \begin{tikzcd}
        R^q \otimes \prod_\alpha Q_\alpha \ar[r] \dar["\simeq"]
       & R^p \otimes \prod_\alpha Q_\alpha \rar \dar["\simeq"]
       & M \otimes \prod_\alpha Q_\alpha \dar \rar & 0
       \\
        \prod_\alpha R^q \otimes Q_\alpha \ar[r]
       & \prod_\alpha R^p \otimes Q_\alpha \rar
       & \prod_\alpha M \otimes Q_\alpha \rar & 0
    \end{tikzcd}
\end{centre}
and deduce the isomorphism.

$2 \Rightarrow 3$. Obvious

$3 \Rightarrow 1$. The map $M \otimes R^M \to M^M$ is surjective, so there exists a finite number of elements $\{m_i\}_{i \in I}$ and a finite number of functions $\{f_i\}_{i \in I} : M \to R$ such that the sum $\sum_i m_i f^i$ be the identity of $M$. As a consequence $M$ must be finitely generated.

Consider a short exact sequence $0 \to K \to R^p \to M \to 0$.
Then using the commutative diagram
\begin{centre}
    \begin{tikzcd}
        {} &
        K \otimes R^K \ar[r] \dar
       & R^p \otimes R^K \rar \dar["\simeq"]
       & M \otimes R^K \dar["\simeq"] \rar & 0
       \\
        0 \rar &
        K^K \ar[r]
       & (R^p)^K \rar
       & M^K \rar & 0
    \end{tikzcd}
\end{centre}
one gets that $K \otimes R^K \to K^K$ is surjective and thus that $K$ is finitely generated.
\end{proof}

## Related concepts

* [[finitely presented object]]

* [[finitely generated module]]

* [[coherent module]]

* [[dualizable module]]

[[!redirects finitely presented modules]]


finiteness space

# Finiteness spaces

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{: toc}

## Idea

A **finiteness space** is a [[set]] together with a collection of its subsets that behave, in certain ways, like the collection of finite subsets.

## Definition

Given a set $X$ and a subset $\mathcal{U} \subseteq P(X)$ of its [[power set]], define

$$\mathcal{U}^\perp = \{ v \in P(X) \mid u \cap v \;\text{ is finite for all }\; u\in \mathcal{U} \}.$$

This defines a [[Galois connection]] on $P(X)$.

+--{: .un_defn}
###### Definition
A **finiteness space** is a set $X$ equipped with a $\mathcal{U} \subseteq P(X)$ such that $\mathcal{U} = \mathcal{U}^{\perp\perp}$.
=--

We refer to the elements of $\mathcal{U}$ as **finitary**, and the elements of $\mathcal{U}^\perp$ as **cofinitary**.

## Properties

* All finite subsets of $X$ belong to $\mathcal{U}^\perp$ for any $\mathcal{U}$.  Thus, in a finiteness space all finite subsets are both finitary and cofinitary.

* Any set of the form $\mathcal{U}^\perp$ is down-closed, i.e. contains all subsets of its elements.  Thus, any subset of a finitary subset is finitary, and any subset of a cofinitary subset is cofinitary.

* Any set of the form $\mathcal{U}^\perp$ is closed under finite unions.  Thus, any finite union of finitary subsets is finitary, and any finite union of cofinitary subsets is cofinitary.

## Examples

* Taking $\mathcal{U} =$ all finite subsets of $X$ gives a **minimal** finiteness structure on $X$.

* Dually, taking $\mathcal{U}=$ all subsets of $X$ gives a **maximal** finiteness structure.

* In fact, if $(X,\mathcal{U})$ is a finiteness space, so is $(X,\mathcal{U}^\perp)$, its **dual**.  The maximal and minimal finiteness structures are dual.

* We cannot obtain any other finiteness spaces purely by cardinality restriction.  For instance, if all countable subsets of $X$ are finitary, then all cofinitary subsets must be finite, hence $X$ is maximal.  Thus, to obtain finiteness spaces other than the minimal or maximal ones, we must invoke some other structure on $X$.

* Let $X$ be a [[poset]] and $\mathcal{U}$ the set of all subsets of $X$ that are *noetherian*, i.e. contain no infinite strictly increasing chains.  Then this is a finiteness space.  Its dual (at least, assuming the [[axiom of choice]]) consists of subsets that are *artinian* (contain no infinite strictly decreasing chains) and *narrow* (contain no infinite [[antichains]]).

* Let $X$ be a [[linearly ordered set]] and let $\mathcal{U}$ the set of all *left-finite* subsets, i.e. those $u$ such that $u\cap \{ x \mid x \le y\}$ is finite for all $y$.  Since this is of the form $\mathcal{V}^\perp$ for $\mathcal{V}$ the set of subsets $\{x \mid x\le y\}$, it is a finiteness structure.

## Categories of finiteness spaces

A **relation** or **morphism** of finiteness spaces is a [[relation]] $R$ from $X$ to $Y$ such that

1. If $u\subseteq X$ is finitary, then $R[u] = \{ y\in Y \mid \exists x\in u, x R y \}$ is finitary.
2. If $v\subseteq Y$ is cofinitary, then $R^{-1}[v] = \{ x\in X \mid \exists y\in v, x R y \}$ is cofinitary.

An equivalent way to say this is that

* If $u\subseteq X$ is finitary and $v\subseteq Y$ is cofinitary, then the set $\{ (x,y) \mid x\in u \wedge y\in v \wedge x R y \} \subseteq X \times Y$ has finite images under both projections $X\times Y\to X$ and $X\times Y\to Y$.

Additionally, in the presence of condition (1), condition (2) is equivalent to its special case when $v$ is a [[singleton]].

Let $FinSp_{rel}$ denote the category of finiteness spaces and relations, and $FinSp_{par}$ and $FinSp_{fun}$ its [[wide subcategories]] in which the morphisms are restricted to be [[partial functions]] or [[functions]], respectively.

Note that a partial function lies in $FinSp_{par}$ if and only if the preimage of any cofinitary subset is cofinitary.  For this is precisely the second condition above specialized to a partial function, while conversely if this holds then for any finitary $u\subseteq X$ and cofinitary $v\subseteq Y$, we have a surjection $u\cap f^{-1}(v) \to f(u) \cap v$ so that finiteness of the former implies finiteness of the latter.

### Monoidal structures

* All three categories are [[symmetric monoidal category|symmetric monoidal]] and the inclusions $FinSp_{fun} \hookrightarrow FinSp_{par} \hookrightarrow FinSp_{rel}$ are strict symmetric monoidal.
* $FinSp_{rel}$ is [[star-autonomous]] (see [Ehrhard](#Ehrhard05)).
* $FinSp_{par}$ is [[closed monoidal category|closed]] (see [BCJS](#BCJS18)).

### Initial and terminal structures

Let $Set_{par}$ denote the category of sets and [[partial functions]].  Note that $Set_{par}$ is equivalent to the category $Set_*$ of [[pointed sets]], and hence in particular is complete and cocomplete.

The following theorem says that $FinSp_{par}$ and $FinSp_{fun}$ are *almost* [[topological concrete categories]] over $Set_{par}$ and $Set$ respectively.

+-- {: .un_theorem}
###### Theorem
The ([[faithful functor|faithful]]) forgetful functors $U_{par}:FinSp_{par} \to Set_{par}$ and $U_{fun}:FinSp_{fun} \to Set$ have [[initial lifts]] for all (possibly large) [[sink|sources]] $\{ f_i : X \rightharpoonup U(S_i) \}$ having the property that for all $x\in X$ there exists an $i$ such that $x\in dom(f_i)$.
=--
+-- {: .proof}
###### Proof
The functor $U_{fun}$ is simply the pullback $U_{par}$ along the inclusion $Set \hookrightarrow Set_{par}$, so it suffices to consider the former, which we denote simply $U$.

Let $X$ be a set and $\{ f_i : X \rightharpoonup U(S_i) \}$ a $U$-structured [[sink|source]] satisfying the given condition, where each $(S_i,\mathcal{V}_i)$ is a finiteness space.  Let $\mathcal{V} = \big(\bigcup_i f_i^{-1}(\mathcal{V}_i^\perp)\big)^\perp$, making $X$ a finiteness space for which $u\subseteq X$ is finitary if and only if $u \cap f_i^{-1}(v)$ is finite for all cofinitary $v\subseteq S_i$.  Thus each $f_i$ becomes a morphism in $FinSp_{par}$.

For universality, suppose given a finiteness space $T$ and a partial function $g:U(T)\rightharpoonup X$ such that each composite $f_i\circ g$ is a morphism in $FinSp_{par}$.  The latter means that $(f_i\circ g)^{-1}(v)$ is cofinitary for all cofinitary $v\subseteq S_i$, which is to say $u \cap g^{-1}(f_i^{-1}(v))$ is finite for all finitary $u\subseteq T$.  It follows that $g(u) \cap f_i^{-1}(v)$ is finite (being a surjective image of $u \cap g^{-1}(f_i^{-1}(v))$), hence $g(u)$ is finitary, i.e. condition (1) in the definition of morphism holds.

Thus it remains to show that for any $x\in X$ the preimage $g^{-1}(x)$ is cofinitary in $T$.  By assumption, there exists an $i$ such that $x\in dom(f_i)$, which is to say $x\in f_i^{-1}(f_i(x))$.  But then $g^{-1}(x) \subseteq g^{-1}(f_i^{-1}(f_i(x)))$, hence is cofinitary.
=--

In the case of $U_{fun}$, the extra condition reduces to the statement that if $X$ is inhabited then so is the set of indices $i$.  This condition is necessary: empty $U_{fun}$-structured sources on inhabited sets do not have initial lifts.  The only candidate is the maximal finiteness structure in which all subsets are finitary and only finite sets are cofinitary, but not all functions into a maximal finiteness space are finiteness functions (only those with finite fibers).

In particular, $FinSp_{fun}$ does not have a terminal object, though it does have all inhabited limits, and the forgetful functor $FinSp_{fun} \to Set$ is a Grothendieck fibration.

A similar argument shows that $FinSp_{par}$ has all inhabited limits.  (The above is essentially the argument given for this in [BCJS](#BCJS18), unraveled into an explicit construction of limits in $Set_{par}$ in terms of the equivalence to $Set_*$.)  But $FinSp_{par}$ also has a terminal object, namely the empty set with its unique finiteness structure, so it is a [[complete category]].

In fact, $FinSp_{par}$ is also a [[cocomplete category]], as shown in [BCJS](#BCJS18).  But its coequalizers are not preserved by $U_{par}$, and in particular not constructed as final lifts of $U_{par}$-structured sinks.

## Matrices {#Matrices}

Let $R$ be any [[semiring]]; we can define a category $FMat(R)$ as follows:

* Its objects are finiteness spaces.
* Its morphisms $X\to Y$ are functions $M:X\times Y \to R$ whose support $\{ (x,y) \mid M(x,y) \neq 0 \}$ is a morphism $X\to Y$ in $FinSp_{rel}$.  We write this morphism in $FinSp_{rel}$ as $supp(M):X\to Y$.

The composite of $M:X\to Y$ and $N:Y\to Z$ is defined by

$$ (N M)(x,z) = \sum_{M(x,y)\neq 0 \neq N(y,z)} N(y,z) \cdot M(x,y). $$

To see that this is well-defined, first note that by assumption, for any $x$ the set $\{ y \mid M(x,y)\neq 0 \}$ is finitary, and for any $z$ the set $\{ y \mid N(y,z)\neq 0 \}$ is cofinitary.  Thus their intersection is finite, so the above sum is finite and thus makes sense.  Now to check that $N M$ is indeed a morphism $X\to Y$ in $FMat(R)$, note that if $(N M)(x,z)\neq 0$ it must be that there exists a $y$ such thath $M(x,y)\neq 0 \neq N(y,z)$, and therefore $(supp(N) \circ supp(M))(x,z)$ holds.  In other words, $supp(N M) \subseteq supp(N) \circ supp(M)$, and therefore $supp(N M)$ is also a morphism in $FinSp_{rel}$.

It is straightforward to check associativity and unitality, so we have a category $FMat(R)$.  We note the following properties:

* $FMat(R)$ is enriched over abelian monoids (abelian groups if $R$ is a ring), and indeed over $R$-modules.

* $FMat(R)$ has a [[zero object]] (the empty finiteness space) and finite [[biproducts]] (disjoint unions of finiteness spaces).

* Any [[distributive lattice]] is a semiring with [[join]] as addition and [[meet]] as multiplication.  In the case $R= \{\bot,\top\}$ the set of [[truth values]], we have $FMat(R) = FinSp_{rel}$.

* The proof above that $supp(N M) \subseteq supp(N) \circ supp(M)$, plus a simpler argument in the unary case, shows that for any $R$ we have an identity-on-objects [[colax functor]] $supp: FMat(R) \to FinSp_{rel}$.  (For general $R$, $FMat(R)$ is not a 2-category, but it is when $R$ is a distributive lattice, including the case of $FinSp_{rel}$; thus this makes sense.)

* For general $R$, the category $FMat(R)$ is [[star-autonomous]], and the functor $supp: FMat(R) \to FinSp_{rel}$ preserves this structure strictly.

* Any morphism $H : X\to Y$ in $FinSp_{rel}$ induces a morphism $\hat{H}:X\to Y$ in $Mat(R)$ by setting $\hat{H}(x,y) = 1$ if $H(x,y)$ and $0$ otherwise.  However, this does not define a functor $FinSp_{rel} \to FMat(R)$ since morphisms of the form $\hat{H}$ need not be closed under composition, as the composite of two such might involve a finite sum of copies of $1$.  There are two cases when this does work: when $1$ is an additive idempotent in $R$ (such as when $R$ is a distributive lattice), and when we restrict the domain to $FinSp_{par}$.  In particular, we have a strict symmetric monoidal functor $\chi : FinSp_{par} \to FMat(R)$ for any $R$ that is a partial section of $supp$.

## Linearization

Let $(X,\mathcal{U})$ be a finiteness space and $A$ an [[abelian group]] (or more generally an abelian [[monoid]]), and define

$$ A \langle X\rangle = \{ f : X\to A \mid supp(f) \in \mathcal{U} \}$$

where $supp(f) = \{ x\in X \mid f(x)\neq 0 \}$ is the [[support]] of $f$.  Then $A\langle X\rangle$ is an abelian group called the **linearization** of $X$ (with coefficients in $A$).

Note that if $A=\{\bot,\top\}$, then $A\langle X\rangle$ is just the set $\mathcal{U}$ of finitary subsets of $X$.

It is shown in [BCJS](#BCJS18) that if $X$ is a partial finiteness monoid (i.e. a monoid in $FinSp_{par}$) and $R$ a [[ring]], then $R\langle X\rangle$ is also a ring with

$$ (f\cdot g)(m) = \sum_{(m_1,m_2)\in X_m(f,g)} f(m_1)\cdot g(m_2) $$

where $X_m(f,g)$ is the set of pairs $(m_1,m_2)\in supp(f) \times supp(g)$ such that $m_1 \cdot m_2 = m$.  In fact $R$ need only be a semiring (in which case so is $R\langle X\rangle$).

If $X$ is merely a monoid in $FinSp_{rel}$, then the above multiplication is still defined, but may not be associative or unital.  This can be explained more abstractly as follows.  We have $R\langle X\rangle = FMat(R)(1,X)$, where $1$ is the one-element finiteness space (the monoidal unit, which is terminal in $FinSp_{fun}$ but not in any of the other categories of finitenes spaces).  Thus, $R\langle X\rangle$ inherits a multiplicative monoid structure as soon as $X$ is a monoid object in $FMat(R)$.  As noted above, we have a monoidal functor $\chi : FinSp_{par} \to FMat(R)$, which therefore preserves monoids; but no such functor whose domain is $FinSp_{rel}$.

### Examples

* If $X$ is a minimal finiteness space, then the linearization $A\langle X\rangle$ is then the [[copower]] of $A$ by the set $X$ in [[Ab]].  Any monoid (or partial monoid) $M$ is a finiteness monoid with the minimal finiteness structure, and when $R$ is a ring the resulting ring structure on $R\langle M\rangle$ is induced on the basis elements by the multiplication of $M$.  When $M$ is a group $G$, this is the [[group algebra]] $R[G]$.

* If $X$ is a poset whose finitary subsets are the artinian and narrow ones, and whose multiplication preserves the strict order on each side, then it is a finitenes monoid, and its linearization is the ring of [[Ribenboim power series]] from $X$ with coefficients in $R$ (see [BCJS](#BCJS18)).  Thus, this includes rings of [[formal power series]], [[formal Laurent series]], [[polynomials]], [[Laurent polynomials]], and [[Hahn series]].

* If $X$ is a linearly ordered abelian group and the finitary subsets are the left-finite ones, then its linearization is the corresponding [[Novikov field]].

* As shown in [BCJS](#BCJS18), other examples include [[Puiseux series]], formal power series over a free monoid, and polynomials of bounded degree.

## Related pages

* Finiteness spaces were defined by analogy to [[coherence spaces]] in which the property "is finite" is replaced by "is a [[subsingleton]]".  Thus, both are special cases of [[arity spaces]] for two different sets of arities.

## References

* T. Ehrhard, *Finiteness spaces*, Mathematical Structures in Computer Science 15, pp. 615--646 (2005)
 {#Ehrhard05}

* Richard Blute, Robin Cockett, Pierre-Alain Jacqmin, and Philip Scott, *Finiteness spaces and generalized power series*, Electronic Notes in Theoretical Computer Science, 341 (2018), 5-22, [doi](https://doi.org/10.1016/j.entcs.2018.11.002), [arXiv:1805.09836](https://arxiv.org/abs/1805.09836)
 {#BCJS18}

[[!redirects finiteness spaces]]


finitism > history

< [[finitism]]
[[!redirects finitism -- history]]


Finn Larsen

* [webpage](https://lsa.umich.edu/physics/people/faculty/larsenf.html)

## Selected writings

On [[AdS3-CFT2 and CS-WZW correspondence]]:

* [[Per Kraus]], [[Finn Larsen]], _Partition functions and elliptic genera from supergravity_, JHEP 0701:002, 2007 ([arXiv:hep-th/0607138](https://arxiv.org/abs/hep-th/0607138))

category: people


Finn Lawler

I'm a Ph.D. student in the Department of Computer Science at Trinity College, Dublin.  My website is
[here](https://www.cs.tcd.ie/~flawler/), though there's not much of interest on it.

category: people


Finnur Larusson

__Finnur Lárusson__ is mathematician at the University of Adelaide (formerly at the Univ. of Western Ontario). His current research is at the interface between [[several complex variables]] and [[Quillen model categories]], especially in the context of the [[Oka principle]].

* [webpage](http://www.maths.adelaide.edu.au/finnur.larusson)

## Selected writings

On [[Oka manifolds]] and the [[Oka principle]]:

* [[Finnur Lárusson]], *Affine simplices in Oka manifolds*, Documenta Mathematica (2009) Volume: 14, page 691-697 ([eudml:228723](https://eudml.org/doc/228723))

* [[Finnur Lárusson]], _What is an Oka manifold?_, Notices AMS Volume 57, Number 1, 2010 ([pdf](http://www.ams.org/notices/201001/rtx100100050p.pdf), [[Larussen_OkaManifolds.pdf:file]])

* {#Larusson03} [[Finnur Lárusson]], _Model structures and the Oka principle_, Journal of Pure and Applied Algebra Volume 192, Issues 1–3, 1 September 2004, Pages 203-223 ([math.CV/0303355](http://arxiv.org/abs/math/0303355), [doi:10.1016/j.jpaa.2004.02.005](https://doi.org/10.1016/j.jpaa.2004.02.005))

* _Affine simplices in Oka manifolds_, Documenta Mathematica __14__ (2009) 691-697.

* {#ForstnericLarusson11} [[Franc Forstnerič]], [[Finnur Lárusson]], *Survey of Oka theory*, New York J. Math., 17a (2011), 1-28 ([arXiv:1009.1934](https://arxiv.org/abs/1009.1934), [eudml:232963](https://eudml.org/doc/232963))

* {#Forstneric13} [[Franc Forstnerič]] (appendix by [[Finnur Lárusson]]), *Oka manifolds: From Oka to Stein and back*, Annales de la Faculté des sciences de Toulouse, Mathématiques, Série 6, Tome 22 (2013) no. 4, pp. 747-809 ([numdam:AFST_2013_6_22_4_747_0](https://afst.centre-mersenne.org/item/?id=AFST_2013_6_22_4_747_0))

and via [[model structures on simplicial presheaves]] ([[higher complex analytic geometry]] and [[complex analytic ∞-groupoids]]):

* {#Larusson01} [[Finnur Lárusson]], _Excision for simplicial sheaves on the Stein site and Gromov's Oka principle_,  International Journal of Mathematics Vol. 14, No. 02, pp. 191-209 (2003)  ([arXiv:math/0101103](http://arxiv.org/abs/math/0101103), [doi:10.1142/S0129167X03001727](https://doi.org/10.1142/S0129167X03001727))

* {#Larusson03} [[Finnur Lárusson]], _Model structures and the Oka principle_, Journal of Pure and Applied Algebra Volume 192, Issues 1–3, 1 September 2004, Pages 203-223 ([math.CV/0303355](http://arxiv.org/abs/math/0303355), [doi:10.1016/j.jpaa.2004.02.005](https://doi.org/10.1016/j.jpaa.2004.02.005))

## Related $n$Lab entries

* [[Oka principle]], [[Oka manifold]]

* [[model structure on simplicial presheaves]]

* [[higher complex analytic geometry]]

* [[complex analytic ∞-groupoids]]

category: people

[[!redirects Finnur Lárusson]]


FinRel

+-- {: .rightHandSide}
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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

$FinRel$ is the [[category]] of [[finite sets]] and all [[relations]] between them: the [[full subcategory]] of [[Rel]] on finite sets.  Like [[Rel]], [[FinRel]] can also be seen as a [[2-poset]], and a [[cartesian bicategory]].

## Properties

### As a category of matrices

$FinRel$ is [[equivalence of categories|equivalent]] to $Mat(Bool)$, the category whose [[objects]] are [[natural numbers]] and where a [[morphism]] $f \colon m \to n$ is an $n \times m$ [[matrix]] with entries in the [[semiring]] of [[booleans]], $Bool = \{F,T\}$, where [[addition]] is the logical operation "[[logical disjunction|or]]" and [[multiplication]] is "[[logical conjunction|and]]".

For any semiring $R$, the category $Mat(R)$ has [[biproducts]], corresponding to addition of natural numbers, and the [[initial object]] $0$ is also [[terminal object|terminal]] (hence is a [[zero object]]).  For any [[commutative magma|commutative]] semiring $R$, $Mat(R)$ also has another [[symmetric monoidal category|symmetric monoidal structure]]: a [[tensor product]] with $m \otimes n = m n$, which [[distributivity|distributes]] over biproducts.  Thus $FinRel$ has all these properties.

### As a PROP

We can define [[bimonoids]] in any [[symmetric monoidal category]]: roughly, a bimonoid is a [[monoid]] and [[comonoid]] in a compatible way.  A bimonoid is said to be **special** if comultiplication followed by multiplication is the [[identity]], and **bicommutative** if it is both a [[commutative monoid]] and a [[cocommutative comonoid]].

In a category with [[biproducts]] and a [[zero object]], every object $c$ is a bicommutative bimonoid where the multiplication is the [[fold map]] $\nabla \colon c \oplus c \to c$ and the comultiplication is the [[diagonal]] $\Delta \colon c \to c \oplus c$.  Thus, every object of $FinRel$ is a bicommutative bimonoid.  One can easily check that it is also special.

But something stronger is true.  $FinRel$ made symmetric monoidal with biproducts is the *[[free construction|free]]* symmetric monoidal category on a special bicommutative bimonoid.  That is, given a symmetric monoidal category $C$ and a special bicommutative bimonoid object $c \in C$, there is symmetric [[monoidal functor]] $F \colon FinRel \to C$, unique up to monoidal [[natural isomorphism]], mapping $1 \in FinRel$ together with its bimonoid operations to $c$ and its bimonoid operations.

This is conveniently expressed in the language of [[PROPs]]: $Mat(Bool)$ is the PROP for special bicommutative bimonoids.   This is Theorem 7.2 in [Coya-Fong 17](#CoyaFong17), based on techniques from [Lack 04](#Lack04).

## Related entries

* [[Rel]]

* [[FinSet]]

## References

* {#Lack04} [[Steve Lack]], _Composing PROPs_,  Theory and Applications of Categories 13(9):147–163, 2004. ([pdf](http://www.tac.mta.ca/tac/volumes/13/9/13-09.pdf))

* {#CoyaFong17} Brandon Coya and [[Brendan Fong]], _Corelations are the prop for extraspecial commutative Frobenius monoids_, Theory and Applications of Categories, Vol. 32, 2017, No. 11, pp. 380-395.   ([arxiv](https://arxiv.org/abs/1601.02307))

category: category


FinSet

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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Definition

$\Fin\Set$ is the [[category]] of [[finite set]]s and all [[function]]s between them: the [[full subcategory]] of [[Set]] on finite sets.

(For [[constructive mathematics|constructive]] purposes, take the strictest sense of 'finite'.)

It is easy (and thus common) to make $\Fin\Set$ [[skeleton|skeletal]]; there is one [[object]] for each [[natural number]] $n$ (including $n = 0$), and a [[morphism]] from $m$ to $n$ is an $m$-tuple $(f_0, \ldots, f_{m-1})$ of numbers satisfying $0 \leq f_i \lt n$.  This amounts to identifying $n$ with the set $\{0, \ldots, n - 1\}$.  (Sometimes $\{1, \ldots, n\}$ is used instead.)

## Properties

### Opposite category
 {#OppositeCategory}

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

The [[opposite category]] of $FinSet$ is [[equivalence of categories|equivalent]] to that of finite [[Boolean algebras]]

$$
  FinSet^{op} \simeq FinBool
  \,.
$$

This equivalence is induced by the [[power set]]-[[functor]]

$$
  \mathcal{P} \;\colon\; FinSet^{op} \stackrel{\simeq}{\to} FinBool
  \,.
$$

=--

This is discussed for instance as ([Awodey, prop. 7.31](#Awodey)). For the generalization to all sets see at _[Set -- Properties -- Opposite category and Boolean algebras](Set#OppositeCategory)_. See at _[[Stone duality]]_ for more on this.

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

In constructive mathematics, for any flavor of _[[finite set|finite]]_, $\mathcal{P}$ defines an equivalence of $FinSet$ with the opposite category of that of those [[complete lattice|complete]] [[atom|atomic]] [[Heyting algebras]] whose set of atomic elements is finite (in the same sense as in the definition of $FinSet$). However, all finite [[complete lattice|complete]] [[atom|atomic]] [[Heyting algebras]] are Boolean.

=--

### Universal properties

$FinSet$ is the free category with finite coproducts on one object: that is, for any category $C$ with finite coproducts and any object $c \in C$ there is a functor $F : FinSet \to C$ preserving finite coproducts with $F(1) = c$, and $F$ is unique up to natural isomorphism.

The first fact is closely connected to the fact that $FinSet$ is the [[vertical categorification]] of the set of natural numbers $\mathbb{N}$, and $\mathbb{N}$ is the [[free monoid]] on one generator.

$FinSet$ is also the free category with finite colimits on one object: that is, for any category $C$ with finite colimits and any object $c \in C$ there is a functor $F : FinSet \to C$ preserving finite colimits with $F(1) = c$, and $F$ is unique up to natural isomorphism.

$FinSet$ is also the free symmetric monoidal category on a commutative monoid object: that is, for any symmetric monoidal category $(C, \otimes)$ and any commutative monoid object $c \in C$ there is a symmetric monoidal functor $F : (FinSet,+) \to (C, \otimes)$ with $F(1) = c$, and $F$ is unique up to monoidal natural isomorphism.

The last fact is closely connected to this: $FinSet$, made symmetric monoidal using $+$, is equivalent to the [[PROP]] for commutative monoids. A proof is given in Lafont's paper below.  Moreover, the sub-prop generated by the unit of the monoid consists of the monics, while the free prop generated by the multiplication of the monoid are the epics.

All these universal properties have useful duals.  $FinSet^{op}$ is the free category with finite products on one object and also the free category with finite limits on one object; the symmetric monoidal category $(FinSet, +)^{op}$ is equivalent to the PROP for cocommutative comonoids.

### Equivalence with the category of finite $T_1$-topological spaces

Given a [[continuous map]] $f \colon X \to Y$ between [[topological spaces]], it is order-preserving relative to the [[specialisation order]]. Thus, we have a [[faithful functor]] $Spec$ from the category of [[Top|$\mathrm{Top}$]] of topological spaces to the category $\mathrm{PreOrd}$ of [[preordered sets]].

In the other direction, to each [[preordered set]] $(X, \leq)$ we may associate a topological space whose [[elements]] are those of $X$, and whose [[open subsets]] are precisely the upward-closed sets with respect to the preordered set. This topology is called the [[specialization topology]]. This defines a [[functor]]

$$i \colon  PreOrd \to Top$$

which is a [[full subcategory|full embedding]]; the [[essential image]] of this functor is the category of [[Alexandroff spaces]] (spaces in which an arbitrary [[intersection]] of open sets is open). Hence the category of preorders is equivalent to the category of Alexandroff spaces.

There is an an [[adjoint functor|adjunction]] $i \dashv Spec$ between the categories [[PreOrd]] and [[Top]] making $PreOrd$ a [[coreflective subcategory]] of $Top$. In particular, the [[adjunction counit]] evaluated at a space $X$,

$$i(Spec(X)) \to X,$$

is the [[identity function]] at the level of sets, and is [[continuous function|continuous]] because any open $U$ of $X$ is upward-closed with respect to $\leq$, according to the second equivalent condition of the definition of the specialization order.

This [[adjunction]] restricts to an [[adjoint equivalence]] between the categories $\mathrm{FinPreOrd}$ and $\mathrm{FinTop}$ of finite preordered sets and finite topological spaces. The [[adjunction unit|unit]] and [[adjunction counit|counit]] are both [[identity functions]] on [[underlying sets]], so that in fact we have an equivalence between these categories as [[concrete categories]].

Since the [[specialization order]] of [[T_1-topological space|$T_1$-topological spaces]] is [[equality]] in $T_1$-spaces, and every subset is upward-closed with respect to the [[equality]] relation used to define the [[specialization topology]] of a [[set]], the above adjunction further restricts to an [[adjoint equivalence]] between the categories $\mathrm{FinSet}$ of [[finite sets]] and $\mathrm{Fin}T_1\mathrm{Sp}$ of finite $T_1$-topological spaces.

### In Grpd

$FinSet$ is a [[natural numbers object]] in the [[(2,1)-topos]] [[Grpd]] of [[groupoids]] and [[functors]], as it is an initial algebra of the 2-endofunctor $F(X) \cong 1 + X$. It is also a [[free object|free]] [[monoid object]] on one generator in [[Grpd]], an [[initial object|initial]] [[rig object]] in [[Grpd]], and a [[category]]. This all follows from the fact that the category $FinSet$ is a [[vertical categorification]] of the [[poset]] $\mathbb{N}$ of natural numbers.

As a groupoid itself, the [[core]] of $Fin Set$ is (with the operation of [[disjoint union]]) the free [[symmetric monoidal category]] on one object.  (There is no way to generate non-invertible morphisms from this data.)

### In topos theory

The category $FinSet$ is an [[elementary topos]] and the inclusion $FinSet \hookrightarrow Set$ is a [[logical morphism]] of toposes. ([[Elephant|Elephant, example 2.1.2]]).

Mathematics done within or about $FinSet$ is [[finite mathematics]].

A [[presheaf]] of sets on $\Fin\Set$ is a [[symmetric set]]; one generally uses the skeletal version of $\Fin\Set$ for this.

The [[copresheaf]] category $[FinSet,Set]$ is the [[classifying topos]] for the _[[theory of objects]]_ (the empty theory over the signature with one sort and no primitive symbols except equality). ([[Elephant|Elephant, D3.2]]).

### Subcategories of $FinSet$

The [[simplex category]] $\Delta$ embeds into $\Fin\Set$ as a category with the same objects but fewer morphisms. The category of [[cyclic set]]s introduced by Connes lies in between. All the three are special cases of extensions of $\Delta$ by a group in a particularly nice way. Full classification of allowed [[skew-simplicial sets]] has been given by Krasauskas and independently by Loday and Fiedorowicz.

### As a Lawvere theory

The [[cartesian monoidal category]] $FinSet_+$ of nonempty finite sets is the multi-sorted [[Lawvere theory]] of unbiased [[boolean algebra]]s. As a [[Lawvere theory]], $FinSet$ has one more sort, corresponding to $\emptyset$, and one more model, in which every sort has exactly one element (in all the other models, the sort corresponding to $\emptyset$ is empty).

## Related concepts

* [[permutation category]]

* [[braid category]]

* [[finite homotopy type]]

* [[FinRel]]

* [[FinInj]]

* [[axiom of finiteness]]

## References

* [[Steve Awodey]], _[Category Theory](http://www.andrew.cmu.edu/course/80-413-713/) -- lecture 7_ ([pdf](http://www.andrew.cmu.edu/course/80-413-713/notes/chap07.pdf))
 {#Awodey}

* [[Yves Lafont]], Towards an algebraic theory of Boolean circuits, February 12, 2013.  ([pdf](http://iml.univ-mrs.fr/~lafont/pub/circuits.pdf))

category: category

[[!redirects category of finite sets]]
[[!redirects FinSet]]
[[!redirects Fin Set]]

[[!redirects FiniteSet]]
[[!redirects FiniteSets]]


FinStoch

+-- {: .rightHandSide}
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### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[category]] whose [[objects]] are [[finite sets]] and whose [[morphisms]] are [[stochastic maps]] (or [[stochastic matrices]]) is often denotes _FinStoch_, or similar; a [[full subcategory]] of [[Stoch]].

This is an elementary but nontrivial example of a [[Markov category]], and together with its subcategory [[Stoch#borelstoch|BorelStoch]], one of the most important categories of [[category-theoretic probability]].

## Definition

_FinStoch_ is the category whose

* [[Objects]] are [[finite sets]];
* [[Morphisms]] are [[stochastic matrices]].

See at [[stochastic matrix]] for more details.

## Properties

* _FinStoch_ is closely related to the Kleisli category of the [[distribution monad]]: it is its [[full subcategory]] whose objects are finite sets.

* _FinStoch_ is a [[Cauchy-complete category]].

* As a [[Markov category]], it has [[Markov category#conditionals|conditionals]], and is hence [[Markov category#positivity_and_causality|positive and causal]].

## Related concepts

* [[Markov category]]
* [[Markov kernel]]
* [[category of couplings]]
* [[probability monad]]
* [[Giry monad]]
* [[Kleisli category]]
* [[categorical probability]]

## References:

* {#markov_support} [[Tobias Fritz]], Tomáš Gonda, Antonio Lorenzin, [[Paolo Perrone]], Dario Stein, _Absolute continuity, supports and idempotent splitting in categorical probability_ [[arXiv:2308.00651](https://arxiv.org/abs/2308.00651)]

category: category, probability

[[!redirects Finstoch]]


Fiona Torzewska

* [personal page](https://fionatorzewska.github.io/)

## Selected writings

On [[motion groupoids]] and [[mapping class groups]]:

* [[Fiona Torzewska]], [[João Faria Martins]], [[Paul Purdon Martin]], *Motion groupoids and mapping class groupoids*, Comm. Math. Phys. **402** (2023) 1621-1705 [[arXiv:2103.10377](https://arxiv.org/abs/2103.10377), [doi:10.1007/s00220-023-04755-0](https://doi.org/10.1007/s00220-023-04755-0)]

On [[topological quantum field theory]]:

* {#Torzewska22} [[Fiona Torzewska]], *Topological quantum field theories and homotopy cobordisms* [[arXiv:2208.14504](https://arxiv.org/abs/2208.14504)]

* [[Fiona Torzewska]], *Topological Quantum Field Theories and Homotopy Cobordisms*, [talk at](CQTS##TorzewskaDec2023) [[CQTS]] (Dec 2023) [slides:[[Torzewska-TQFTandHomCob.pdf:file]], video:[YT](https://youtu.be/7rtJ61EPL-M)]

category: people


first Chern class

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###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The first of the [[Chern classes]]. The unique [[characteristic class]] of [[circle bundles]] / complex [[line bundles]].

## Definition

### In bare homotopy-type theory

As a [[universal characteristic class]], the first Chern class is the
[[weak homotopy equivalence]]

$$
  c_1 : B U(1) \stackrel{\simeq}{\to} K(\mathbb{Z},2)
  \,.
$$

### In complex analytic geometry
 {#InComplexAnalyticGeometry}

In [[complex analytic geometry]] consider the [[exponential exact sequence]]

$$
  \mathbb{Z}\to \mathbb{G}\to \mathbb{G}^\times
  \,.
$$

For any [[complex analytic space]] $X$ this induces the [[long exact sequence in cohomology]] with [[connecting homomorphism]]

$$
  c_1\;\colon\;H^1(X,\mathbb{G}^\times ) \longrightarrow H^2(X,\mathbb{Z})
  \,.
$$

This is the first Chern-class map. It sends a [[holomorphic line bundle]] ($H^1(X,\mathbb{G}^\times)$ is the [[Picard group]] of $X$) to an integral cohomology class.

If $D$ is a [[divisor]] in $X$, then $c_1(\mathcal{O}_X(D))$ is the [[Poincaré duality|Poincaré dual]] of the [[fundamental class]] of $D$ (e.g. [Huybrechts 04, prop. 4.4.13](#Huybrechts04)).

Over a [[Riemann surface]] $\Sigma$ the evaluation of the Chern class $c_1(L)$ of a [[holomorphic line bundle]] $L$ on a [[fundamental class]] is the [[degree of a coherent sheaf|degree]] of $L$:

$$
  deg(L) = \langle c_1(L), X\rangle \in H^2(\Sigma, \mathbb{Z}) \simeq \mathbb{Z}
  \,.
$$

## Related concepts

* [[Chern class]], [[top Chern class]]

* [[Dixmier-Douady class]]

* [[complex oriented cohomology]]

* [[Chern insulator]]

## References

See the references at _[[Chern class]]_ and _[[characteristic class]]_.

In [[complex geometry]]

* {#Huybrechts04} [[Daniel Huybrechts]], _Complex geometry - an introduction_. Springer (2004). Universitext. 309 pages. ([pdf](http://www.math.uh.edu/~shanyuji/2012/Geometry/Huybrechts.pdf))

[[!redirects first Chern classes]]


first class constraint

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Examples

* [[Gauss law]]

## Related concepts

* [[constrained mechanics]]

* [[moment map]], [[symplectic reduction]]

## References

* Wikipedia, _[First class constraints](http://en.wikipedia.org/wiki/First_class_constraint)_

[[!redirects first class constraints]]

[[!redirects first-class constraint]]
[[!redirects first-class constraints]]


first law of thought

+-- {: .rightHandSide}
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### Context
#### Equality and Equivalencefreflectio
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[[!include equality and equivalence - contents]]
=--
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[philosophy]], the statement that given any $A$, then $A$ [[equality|equals]] $A$

$$
  A = A
$$

(i.e. [[reflexive relation|reflexivity]]) has been called (see the [references below](#References)) the _first law of thought_.

In terms of [[type theory]] this is the [[judgement]] that exhibits the [[term introduction]] for the [[reflexive relation|reflector]] in an [[identity type]]

$$
  A \colon Type \;\vdash\; refl_A \;\colon A = A\;
  \,.
$$

In fact, [[identity types]] may be regarded as [[inductive types]] generated by just this [[reflexive relation|reflector]] ([Licata 11](#Licata11), [Shulman 12,  slide 35](#Shulman12), see [there](identity+type#InTermsOfInductiveTypes)).

## Related concepts

* [[ontology]]

## References
 {#References}

In [[Gottfried Leibniz]]'s unpublished but famous manuscript on logic (from some time in 1683-1716), reproduced in the Latin original in

* {#Gerhard1890} K. Gerhard (ed.), Section XIX, [p. 228](https://archive.org/details/diephilosophisc00gerhgoog/page/228/mode/2up) in: *Die philosophischen Schriften von Gottfried Wilhelm Leibniz*, Siebenter Band, Weidmannsche Buchhandlung (1890) [[archive.org](https://archive.org/details/diephilosophisc00gerhgoog/page/n7/mode/2up)]

and English translation in

* {#Lewis18} [[Clarence I. Lewis]], Appendix (p. 373) of: *A Survey of Symbolic Logic*, University of California (1918) [[[LewisAppendix-LeibnizIndiscernibles.pdf:file]]]

it says, after statement of the [[identity of indiscernibles]] and then the [[indiscernibility of identicals]], that

> $A$ and $A$ are, of course, said to be the same

Later,

* [[Johann Fichte]],  _[[Grundlage der gesamten Wissenschaftslehre]]_ (1794)

calls "$A = A$"   "[den ersten, schlechthin unbedingent Grundsatz](Grundlage+der+gesamten+Wissenschaftslehre#ErsterGrundsatz)" hence the "absolutely imperative principle". (In fact Fichte there highlights the importance of the [[antecedent]] $A \colon Type$ to arrive at the [[consequent]] $A = A$.)

* [[Georg Hegel]], [[Science of Logic]] 1812,

in its [§875](https://ncatlab.org/nlab/show/Science+of+Logic#875) calls it the _[first original law of thought](https://ncatlab.org/nlab/show/Science+of+Logic#FirstOriginalLawOfThought)_:

> The [[first original law of thought]] ([WdL §875](Science+of+Logic#875)): everything is identical with itself ([WdL §863](Science+of+Logic#863)), no two things are like each other ([WdL §903](Science+of+Logic#903)).

* Thomas Hughes, _The Ideal Theory of Berkeley and the Real World_, (1865) Part II, Section XV,

in [Footnote, p. 38](https://archive.org/stream/idealtheoryberk01hughgoog#page/n48/mode/2up) calls it the _first_ of the four _primary laws of thought_.

See also

* Wikipedia, _[Law of thought -- Four laws](https://en.wikipedia.org/wiki/Law_of_thought#Four_laws)_,

which gives quotes by [[Arthur Schopenhauer]] to the same extent.

For the understanding of [[identity types]] as [[inductive types]] genrated just by the reflector $x = x$ see

* {#Licata11} [[Dan Licata]], _[Understanding Identity Elimination in Homotopy Type Theory](https://homotopytypetheory.org/2011/04/10/just-kidding-understanding-identity-elimination-in-homotopy-type-theory/)_, 2011

* {#Shulman12} [[Mike Shulman]], around slide 35 of _Inductive types and identity types_, 2012 ([pdf](https://home.sandiego.edu/~shulman/hottseminar2012/03induction-handout1up.pdf))

[[!redirects first original law of thought]]


first stable homotopy group of spheres

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

#Contents#
* table of contents
{:toc}

## Idea

The first [[stable homotopy group of spheres]] (the first [[stable stem]]) is the [[cyclic group of order 2]]:

\[
  \label{ComplexHopfFibrationGeneratingpi1}
  \array{
    \pi_1^s &\simeq& \mathbb{Z}/2
    \\
    [h_{\mathbb{C}}] &\leftrightarrow& [1]
  }
\]

where the generator $[1] \in \mathbb{Z}/2$ is represented by the [[complex Hopf fibration]] $S^3 \overset{h_{\mathbb{C}}}{\longrightarrow} S^2$.

\begin{imagefromfile}
        "file_name": "PuncturedSphereBordismBetweenTwoCircles.jpg",
        "web": "nlab",
        "width": 200,
        "unit": "px",
        "float": "right",
        "margin": {
            "top": -40,
            "right": 0,
            "bottom": 20,
            "left": 20,
            "unit": "px"
        },
        "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

## Properties

### As the first framed bordism group

Under the [[Pontrjagin-Thom isomorphism]], identifying the [[stable homotopy groups of spheres]] with the [[bordism ring]] $\Omega^{fr}_\bullet$ of [[stable framing|stably framed]] manifolds (see at _[[MFr]]_), the generator (eq:ComplexHopfFibrationGeneratingpi1) is represented by the [[1-sphere]] (with its left-invariant framing induced from the identification with the [[Lie group]] [[U(1)]])

$$
  \array{
    \pi_1^s & \simeq & \Omega_1^{fr}
    \\
    [h_{\mathbb{C}}] & \leftrightarrow & [S^1_{fr=1}]
    \,.
  }
$$

Moreover, the relation $2 \cdot [S^1_{Lie}] \,\simeq\, 0$ is represented by the [[bordism]] which is the [[complement]] of 2 [[open balls]] inside [[generalized the|the]] [[2-sphere]].

## Related concepts

* [[complex Hopf fibration]]

* [[second stable homotopy group of spheres]]

* [[third stable homotopy group of spheres]]

* [[homotopy groups of spheres in HoTT]]

## References

The original computation via [[Pontryagin's theorem]] in [[cobordism theory]]:

* [[Lev Pontrjagin]], _[[Classification of continuous maps of a complex into a sphere]]_, _Communication I_, Doklady Akademii Nauk SSSR 19(3) (1938), 147-149

with a more comprehensive account in:

* {#Pontrjagin55} [[Lev Pontrjagin]], Section 14 of: _[[Smooth manifolds and their applications in homotopy theory]]_, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) ([doi:10.1142/9789812772107_0001](https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001))

See also:

* Mehmet Kirdar, *On the First, the Second and the Third Stems of the Stable Homotopy Groups of Spheres* [[arXiv:2107.06103](https://arxiv.org/abs/2107.06103)]

Review:

* [[Daniel Freed]], [[Karen Uhlenbeck]], Appendix B of: _Instantons and Four-Manifolds_, Mathematical Sciences Research Institute Publications, Springer 1991 ([doi:10.1007/978-1-4613-9703-8](https://link.springer.com/book/10.1007/978-1-4613-9703-8))

* {#WangXu10} [[Guozhen Wang]], [[Zhouli Xu]], Section 2.3 of: _A survey of computations of homotopy groups of Spheres and Cobordisms_, 2010 ([[WangXuHomotopyGroupsOfSpheres2010.pdf:file]])

* {#Putnam} [[Andrew Putman]], Section 5 of: _Homotopy groups of spheres and low-dimensional topology_ ([pdf](http://www.math.rice.edu/~andyp/notes/HomotopySpheresLowDimTop.pdf), [[PutmanHomotopyGroupsOfSpheres.pdf:file]])

Discussion [[homotopy groups of spheres in homotopy type theory|in homotopy type theory]]:

* [[Guillaume Brunerie]]: *[[On the homotopy groups of spheres in homotopy type theory]]* $[$[arxiv:1606.05916](https://arxiv.org/abs/1606.05916)$]$

* {#Ljungstroem22} [[Axel Ljungström]], *The Brunerie Number Is -2* (June 2022) $[$[blog entry](https://homotopytypetheory.org/2022/06/09/the-brunerie-number-is-2/)$]$

[[!redirects 1st stable homotopy group of spheres]]

[[!redirects first stable stem]]
[[!redirects 1st stable stem]]


first uncountable ordinal

#Contents#
* table of contents
{:toc}

## Idea

The [[category]] of [[countable ordinals]] and [[simulations]], ordered by inclusion, is a [[preorder]] and in fact equivalent to a [[woset]]. In [[material set theory]], this woset is often identified with the first [[uncountable ordinal]] (well-ordered set under the membership relation), often denoted $\omega_1$. [[structural set theory|Structurally]], we can define $\omega_1$ up to [[isomorphism]] as a [[skeleton]] of the category of countable ordinals, or the [[Hartogs number]] of a countably infinite set, and so the study of countable ordinals becomes a study of the order structure of $\omega_1$.

This $\omega_1$ has no top element (of course not: otherwise $\omega_1$ would be a countable ordinal), and much of the interest and fun of the subject lies in playing the childhood game of seeing who can name the bigger number, or in this case the bigger countable ordinal. A key fact is that $\omega_1$ is countably cocomplete, and therefore behaves like a self-contained [[universe]] with respect to countably cocontinuous operations. There is quite a lot of scope in what one can build up to using such operations. After playing the larger ordinal game for a while, and becoming impressed by the sizes of ordinals one can define -- while recognizing at the same time that one is "never getting anywhere" relative to $\omega_1$ itself -- it is hard not to become awestruck by the staggering immensity of $\omega_1$.

On the other hand, the [[cardinality]] of $\omega_1$ is of course no bigger than that of, say, the real numbers $\mathbb{R}$ (and the same as it if the [[continuum hypothesis]] holds).

## Related concepts

* [[long line]]

## References

See also

* Wikipedia, _[First uncountable ordinal](https://en.wikipedia.org/wiki/First_uncountable_ordinal)_ [Spanish Version](https://chicksx.com/blog/primer-ordinal-incontable)by[Chicksx](https://chicksx.com).


first-countable space

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### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# First-countable spaces
* table of contents
{: toc}

## Idea

A [[space]] (such as a [[topological space]]) is _first-countable_ if, in a certain sense, there is only a [[countable set|countable]] amount of information locally in its topology.  (Change 'locally' to 'globally' to get a [[second-countable space]].)

## Definitions

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

A [[topological space]] is __first-countable__ if every [[point]] $x$ has a [[countable set|countable]] [[local basis]] $B_x$.
=--

## Generalizations

The __character__ of a space at a point $x$ is the [[minimum]] of the [[cardinalities]] of the possible bases $B_x$.  We are implicitly using the [[axiom of choice]] here, to suppose that this set of cardinalities (which really is a [[small set]] because bounded above by the number of [[neighbourhoods]] of $x$, and [[inhabited set|inhabited]] by this number as well) has a minimum.  But without Choice, we can still consider this collection of cardinalities.

Then a first-countable space is simply one whose characters are all countable.

The __character__, tout court, of a space is the [[supremum]] of the characters of its points; then a first-countable space is simply one with a countable character.

## Related countability properties

[[!include topology - countability axioms]]

## Related concepts

* [[sigma-topological space]]

* [[sigma-frame]]

[[!redirects first-countable space]]
[[!redirects first-countable spaces]]
[[!redirects first countable space]]
[[!redirects first countable spaces]]

[[!redirects first-countable topological space]]
[[!redirects first-countable topological spaces]]
[[!redirects first countable topological space]]
[[!redirects first countable topological spaces]]


first-order formulation of gravity

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### Context
#### Gravity
+-- {: .hide}
[[!include gravity contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The [[action functional]] of [[gravity]] was originally conceived as a [[nonlinear functional|functional]] on the space of [[pseudo-Riemannian metrics]] of a [[manifold]] $X$. Later on it was realized that it may alternatively naturally be thought of as a functional on the space of [[connection on a bundle|connections]] with values in the [[Poincaré Lie algebra]] -- subject to the constraint that the component in the [[translation Lie algebra]] defines a [[vielbein field]].

Mathematically this means that the [[field (physics)|field]] of [[gravity]] is modeled as a _[[Cartan connection]]_ for the [[Lorentz group]] inside the [[Poincaré group]]. In [[physics]] this is known as the _first order formalism_ or the _Cartan moving frame method_. The translation part of the [[Poincaré Lie algebra]]-connection is called the _[[vielbein]]_ and the remainder the _[[spin connection]]_. The [[field strength]] of gravity -- the [[Riemann tensor]] -- is the [[curvature]].

The reformulation of [[pseudo-Riemannian geometry]] in terms of [[Cartan geometry]] is suggestive of also re-rewriting the form of the [[Lagrangian density]]/[[action functional]] of the theory of [[gravity]], even though this is logically an independent issue. In [[spacetime]] of [[dimension]] 3+1 one such alternative is known as the _Palatini-Cartan-Holst action_. That its [[phase space]] coincides with that induced by the [[Einstein-Hilbert action]] is due to [Cattaneo-Schiavina 17a](#CattaneoSchiavina17a)

Promoting the first-order formulation of gravity from the [[Poincaré group]] to the [[super Poincaré group]] yields _[[supergravity]]_ formulated in [[super Cartan geometry]]. Promoting it further to the [[L-infinity algebra cohomology|Lie n-algebra]] extensions of the [[super Poincaré group]] (from the [[brane scan]]/[[schreiber:The brane bouquet|brane bouquet]]) yields [[type II supergravity]], [[heterotic supergravity]] and [[11-dimensional supergravity]] in [[higher Cartan geometry]]-formulation ([[D'Auria-Fré formulation of supergravity]]).

[[!include local and global geometry - table]]

## Properties

### Local and global framing

Discussion in the physics literature traditionally tends to ignore the global structure of [[spacetime]] manifolds and pretends that a [[vielbein]] field may be chosen globally, hence that spacetime admits a [[framed manifold|framing]].

In general that is only valid locally, but it so happens that in the archetypical case of interest, namely for 4-dimensional [[globally hyperbolic spacetimes]] with [[orientation|orientable]] spatial slices, it is valid globally. See _[this remark](framed+manifold#In4dFirstOrderGravity)_ at _[[framed manifold]]_ for more.

See also at _[[teleparallel gravity]]_.

## Related concepts

* [[Lorentzian geometry]]

* [[supergravity]]

* [[D'Auria-Fre formulation of supergravity]]

* [[gravity as a BF-theory]].

* [[teleparallel gravity]]

## References

A decent introduction is in

* [[Jorge Zanelli]], sections 4 and 5 of _Lecture notes on Chern-Simons (super-)gravities. Second edition (February 2008)_ ([arXiv:hep-th/0502193](http://arxiv.org/abs/hep-th/0502193))

A detailed account is in

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], section I.4.1 of _[[Supergravity and Superstrings - A Geometric Perspective]]_

which is aimed at [[supergravity]]. For more on this case see also

* [[Bernard Julia]], S. Silva, *On first order formulations of supergravities*, JHEP 0001 (2000) 026 [[arXiv:hep-th/9911035](https://arxiv.org/abs/hep-th/9911035), [doi:10.1088/1126-6708/2000/01/026](https://doi.org/10.1088/1126-6708/2000/01/026)]

The equivalence of the [[phase space]] of Palatini-Cartan-Holst [[Lagrangian  field theory]] with the [[Einstein-Hilbert action|Einstein-Hilbert version]] is established in

* {#CattaneoSchiavina17a} [[Alberto Cattaneo]], [[Michele Schiavina]], _The reduced phase space of Palatini-Cartan-Holst theory_, Ann. Henri Poincaré (2019) 20: 445 ([arXiv:1707.05351](https://arxiv.org/abs/1707.05351), [doi:10.1007/s00023-018-0733-z](https://doi.org/10.1007/s00023-018-0733-z))

Subtleties in extending to this to a [[local field theory]] (gluing pieces of spacetimes along their boundaries) are discussed in

* {#CattaneoSchiavina17b} [[Alberto Cattaneo]], [[Michele Schiavina]], _BV-BFV approach to General Relativity: Palatini-Cartan-Holst action_ ([arXiv:1707.06328](https://arxiv.org/abs/1707.06328))

following

* {#CattaneoSchiavina15} [[Alberto Cattaneo]], [[Michele Schiavina]], _BV-BFV approach to General Relativity, Einstein-Hilbert action_, J. Math. Phys. 57, 023515 (2016) ([arXiv:1509.05762](https://arxiv.org/abs/1509.05762))

Reviewed in:

* [[Alberto S. Cattaneo]], *Phase space for gravity with boundaries*,  Encyclopedia of Mathematical Physics (2023) [[arXiv:2307.04666](https://arxiv.org/abs/2307.04666)]

  > (following [Kijowski & Tulczyjew (2005)](phase+space#KijowskiTulczyjew05))

Relation to [[BF-theory]]:

* {#CattaneoMengerSchiavina23} [[Alberto S. Cattaneo]], [[Leon Menger]], [[Michele Schiavina]], *Gravity with torsion as deformed BF theory* [[arXiv:2310.01877](https://arxiv.org/abs/2310.01877)]

See also:

* J. Fernando Barbero G., Juan Margalef-Bentabol, Valle Varo, Eduardo J.S. Villaseñor, *Covariant phase space for gravity with boundaries: metric vs tetrad formulations* ([arXiv:2103.06362](https://arxiv.org/abs/2103.06362))

Discussion via [[L-infinity algebras|$L_\infty$-algebras]]:

* Marija Dimitrijević Ćirić, [[Grigorios Giotopoulos]], Voja Radovanović, [[Richard J. Szabo]], *$L_\infty$-Algebras of Einstein-Cartan-Palatini Gravity*, J. Math. Phys., **61** 112502 (2020) [[arXiv:2003.06173](https://arxiv.org/abs/2003.06173), [doi:10.1063/5.0011344](https://doi.org/10.1063/5.0011344)]

See also:

* Fernando T. Brandt, J. Frenkel, S. Martins-Filho, D. G. C. McKeon, *Quantization of Einstein-Cartan theory in the first order form* [[arXiv:2401.16343](https://arxiv.org/abs/2401.16343)]

[[!redirects first-order formulations of gravity]]

[[!redirects first order formulation of gravity]]
[[!redirects first order formulations of gravity]]

[[!redirects Palatini action]]
[[!redirects Palatini actions]]

[[!redirects Palatini formulation]]
[[!redirects Palatini formulations]]

[[!redirects Palatini formalism]]
[[!redirects Palatini formalisms]]