Lipschitz map

# Lipschitz maps
* table of contents
{: toc}

## Idea

Lipschitz maps are [[functions]] between [[metric spaces]] that lie intermediate between the [[uniformly continuous maps]] and the [[short maps]].  (That is, every short map is Lipschitz, and every Lipschitz map is uniform.)

One can also consider Lipschitz maps between [[gauge spaces]] with appropriate gauges, or between [[manifolds]] with appropriate atlases.

## Definitions

Let $X$ and $Y$ be [[metric spaces]] (with the metric denoted $d$ for both).  Let $f\colon X \to Y$ be a [[function]].

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

The __Lipschitz norm__ or __Lipschitz modulus__ ${\|f\|_{Lip}}$ of $f\colon X \to Y$ is the [[supremum]] of the absolute [[difference quotient]]s
$$ \frac{d(f(a), f(b))} {d(a, b)} $$
for $a \neq b$ in $X$.
=--

In general, this supremum is a [[lower real number]]; in particular, it may be infinite.  We work in the space $[0, \infty]$ of nonnegative lower reals, so that the Lipschitz norm is $0$ if $X$ is a [[subsingleton]] (so that $a \neq b$ never occurs).

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

The map $f\colon X \to Y$ is __Lipschitz continuous__, or simply __Lipschitz__, if its Lipschitz norm is finite.
=--

We can be more specific:

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

The map $f\colon X \to Y$ is __[[short map|short]]__ if its Lipschitz norm is at most $1$.  It is a __[[contraction mapping]]__ if its Lipschitz norm is strictly less than $1$.
=--

## Properties

In [[classical mathematics]], one calls $K \in [0, \infty]$ a __Lipschitz constant__ of $f\colon X \to Y$ if
$$ d(f(a), f(b)) \leq K d(a, b) $$
for all $a, b\colon X$.  Then the Lipschitz norm is the [[minimum]] of the Lipschitz constants (which is always attained, if there is any finite Lipschitz constant at all).  In [[constructive mathematics]], this procedure yields an extended [[upper real]] that is (in general) too large.  (For example, let $f$ be the sequence of [[partial sum]]s of a [[Specker sequence]], thought of as a map from $\mathbb{N} \subseteq \mathbb{R}$ to $\mathbb{R}$.)

If you want a definition that is constructively acceptable and yet involves no division, then take the Lipschitz norm of $f$ to be the supremum over those $k$ such that
$$ d(f(a), f(b)) \gt k d(a, b) $$
for some $a, b\colon X$.  (This is perhaps the best definition, as it will automatically take care of subsingletons, as long as you restrict attention to $[0, \infty]$ a priori, and pseudometric spaces, even constructively.)  You can even restrict attention to (say) [[rational number|rational]] $k$, or to $a, b$ in some [[dense subspace]] of $X$ (say if $X$ is [[separable space|separable]]).

Every Lipschitz map is [[uniformly continuous map|uniformly continuous]].  Every Lipschitz map from the [[real line]] $\mathbb{R}$ to itself is [[absolutely continuous map|absolutely continuous]].

If a map $f$ between the real line and itself is [[differentiable map|differentiable]], then its Lipschitz norm is the [[supremum norm]] ${\|f'\|_\infty}$ of its [[derivative]] $f'$.  (This result can actually be generalised quite a bit, using [[Rademacher's theorem]] and its generalisations.)

## Generalisations

The definitions above generalise immediately to [[quasimetric spaces]]; there is no asymmetry between left and right, and the Lipschitz norm of $f\colon X \to Y$ is the same as that of $f^op\colon X^op \to Y^op$ (although the map from $X^op$ to $Y$, equivalently the map from $X$ to $Y^op$, may be quite different).

The definitions also generalise to [[pseudometric spaces]] if we take the supremum over those $a \neq b$ such that $d(a, b) \gt 0$.  However, we must also require that a Lipschitz map preserve metric equivalence: $d(f(a), f(b)) = 0$ if $d(a, b) = 0$.  Then we may pass to the quotient metric spaces and preserve the Lipschitz norm.

Let $X$ be a [[gauge space]], that is a set equipped with a collection of pseudometrics (the _gauge_ of $X$).  Then $X$\'s gauge is a __Lipschitz gauge__ if the [[identity function]] of $X$ is Lipschitz continuous as a map between $X$ equipped with any two of its pseudometrics.  Then we may extend $X$\'s gauge to a unique maximal Lipschitz gauge and say that $X$ (so equipped) is a __Lipschitz gauge space__.  We may then unambiguously define Lipschitz continuous maps (but not their Lipschitz norms) between any two Lipschitz gauge spaces.

Let $X$ be a [[topological manifold]], and equip $X$ with a compatible [[atlas]] whose transition maps are all Lipschitz (a __Lipschitz atlas__).  Then we may extend $X$\'s atlas to a unique maximal compatible Lipschitz atlas and say that $X$ (so equipped) is a __Lipschitz manifold__.  We may then unambiguously define Lipschitz continuous maps (but not their Lipschitz norms) between any two Lipschitz manifolds.  Assuming that our Lipschitz manifolds are [[paracompact space|paracompact]], then we may think of them as Lipschitz gauge spaces (with gauges consisting of all Lipschitz-continuous pseudometrics), recovering the same notion of Lipschitz map.

A map $f\colon X \to Y$ is __Lipschitz of order $\alpha$__, or __Hölder of order $\alpha$__, if the supremum of
$$ \frac{d(f(a), f(b))} {d(a, b)^\alpha} $$
is finite.  Here, $\alpha$ may be any [[positive number]] (but is typically taken to be less than $1$).  Everything on this page generalises fairly well to these functions (changing also the definition of derivative to match).  Of course, a Lipschitz map of order $\alpha$ from $X$ to $Y$ is simply a Lipschitz map from $X^\alpha$ to $Y$, where $X^\alpha$ is $X$ with all distances raised to the power of $\alpha$.

Lipschitz (and Hölder) maps also have non-uniform versions (fix $a$, but let the norm vary with $b$, which introduces an asymmetry for quasimetric spaces), as well as local versions (every point has a [[neighbourhood]] on which $f$ is Lipschitz).  I kind of get lost in all of the possibilities.

## Related pages

* [[Lipschitz domain]]
* [[corkscrew domain]]
* [[convex function]]

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Lipót Fejér

Lipót Fejér was a Hungarian mathematician.
He defended his PhD thesis in 1902 at the University of Budapest, advised by [[Hermann Schwarz]], where he proved his theorem on summability of [[Fourier series]].

His PhD students include [[John von Neumann]].

* [Wikipedia entry](https://en.wikipedia.org/wiki/Lip%C3%B3t_Fej%C3%A9r)

category: people

[[!redirects Lipot Fejer]]


liquid vector space

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

\tableofcontents

## Idea

An alternative to [[complete topological vector spaces]] in the framework of [[condensed mathematics]].

Roughly, completeness is expressed as ability to integrate with respect to [[Radon measures]].

This doesn't quite work as stated, and to make this rigorous
one has to bring [[L^p-spaces]] for $0\lt p\le 1$ (i.e., the non-convex case)
into the picture.

## Definition

A [[condensed abelian group]] $V$ is __$p$-liquid__ ($0\lt p\le 1$)
if for every [[compact Hausdorff topological space]] $S$
and every morphism of [[condensed sets]] $f\colon S\to V$
there is a unique morphism of [[condensed abelian groups]]
$M_{\lt p}(S)\to V$ that extends $f$ along the inclusion $S\to M_{\lt p}(S)$.

Here for a [[compact Hausdorff topological space]] $S$
and for any $p$ such that $0\lt p\le 1$ we have
$$M_{\lt p}(S)=\bigcup_{q\lt p}M_q(S),$$
where
$$M_p(S)=\bigcup_{C\gt 0}M(S)_{\ell^p\le C},$$
where
$$M(S)_{\ell^p\le C}=\lim_i M(S_i)_{\ell^p\le C},$$
where $S_i$ are finite sets such that
$$S = \lim_i S_i$$
and
$$M(F)_{\ell^p\le C}$$
for a [[finite set]] $F$ denotes the subset of $\mathbf{R}^F$
consisting of [[sequence]] with [[p-norm|$\ell^p$-norm]] at most $C$.

## Related concepts

* [[solid module]]

## References

See at *[[condensed mathematics]]*.

[[!redirects liquid vector spaces]]


Liran Shaul

* [webpage](http://liranshaul.wordpress.com)

## related $n$Lab entries

* [[fracture theorem]]

category: people


Lisa Jeffrey

* [webpage](http://www.math.toronto.edu/jeffrey/)

## related $n$Lab entries

* [[eta invariant]]

## Writing

One of the editors and organizers of

* [[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], [[Lisa Jeffrey]], [[David Kazhdan]], [[John Morgan]], [[David R. Morrison]], [[Edward Witten]], (eds.), _[[Quantum Fields and Strings]]_

category: people


Lisa Randall

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

## Selected writings

Introducing the [[Randall-Sundrum model]]:

* {#RandallSundrum99a} [[Lisa Randall]], [[Raman Sundrum]], _A Large Mass Hierarchy from a Small Extra Dimension_, Phys.Rev.Lett.83:3370-3373, 1999 ([arXiv:hep-ph/9905221](https://arxiv.org/abs/hep-ph/9905221))

* {#RandallSundrum99b} [[Lisa Randall]], [[Raman Sundrum]], _An Alternative to Compactification_, Phys.Rev.Lett.83:4690-4693,1999 ([arXiv:hep-th/9906064](https://arxiv.org/abs/hep-th/9906064))

More on [[gravity]] in the RS model:

* [[Andreas Karch]], [[Lisa Randall]], *Locally Localized Gravity*, JHEP 0105:008 (2001) [[arXiv:hep-th/0011156](https://arxiv.org/abs/hep-th/0011156), [doi:10.1088/1126-6708/2001/05/008](https://doi.org/10.1088/1126-6708/2001/05/008)]

On [[AdS/CFT]] as [[open/closed string duality]] and early discussion of the concept of [[probe branes]] in [[AdS/CFT]]:

* [[Andreas Karch]], [[Lisa Randall]], _Open and Closed String Interpretation of SUSY CFT's on Branes with Boundaries_, JHEP 0106:063, 2001 ([arXiv:hep-th/0105132](https://arxiv.org/abs/hep-th/0105132))

On [[extremal black holes]] and [[black hole entropy]]:

* [[Sean Carroll]], Matthew C. Johnson, [[Lisa Randall]], _Extremal limits and black hole entropy_, JHEP 0911:109,2009 ([arXiv:0901.0931](https://arxiv.org/abs/0901.0931))

Argument that "islands" can resolve the [[black hole information paradox]] only for massive gravity:

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

and on what to do about it:

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

category: people


Lisbeth Fajstrup

Lisbeth Fajstrup is a mathematician at Aalborg University, Department of Mathematical Sciences. Her research these days is in the new field: Geometry and topology in Computer Science.

Her webpage is [here](http://people.math.aau.dk/~fajstrup/).

## Selected writings

On [[convenient categories of topological spaces]] and [[directed homotopy theory]]:

* {#FajstrupRosicky08} [[Lisbeth Fajstrup]], [[Jiří Rosický]], _A convenient category for directed homotopy_, Theory and Applications of Categories, Vol. 21, 2008, No. 1, pp 7-20. ([arXiv:0708.3937](https://arxiv.org/abs/0708.3937), [tac:21-01](http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html))

category:people

[[!redirects L. Fajstrup]]


lisse-étale site

#Contents#
* table of contents
{:toc}

## Idea

The lisse-étale site of a [[scheme]] $X$ is the category of $U \to X$ which are [[smooth morphisms]] with [[coverings]] defined as in the [[étale site]].

The lisse-étale site is not so well-behaved, but it is used in the study of [[algebraic stacks]] in order to have a site with more affine opens, hence better suited for the discussion of [[quasicoherent sheaves]].

## References

* [[The Stacks Project]], _[79.11. The lisse-étale and the flat-fppf sites](http://stacks.math.columbia.edu/tag/0786)_

[[!redirects lisse-étale sites]]

[[!redirects lisse-etale site]]
[[!redirects lisse-etale sites]]


list

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### Context
#### Type theory
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[[!include type theory - contents]]
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#### Deduction and Induction
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[[!include deduction and induction - contents]]
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#### Foundations
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[[!include foundations - contents]]
=--
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# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

In [[mathematics]] and specifically in [[formal logic]], by a "list" one means the fairly evident formalization of the colloquial notion of a "list of elements":

If a [[set]] $E$ of admissible elements is given --- also called an  "[[alphabet]]", in this context --- then a list of $E$-elements is a [[tuple]] $(e_1, e_2, \cdots, e_\ell)$ of elements $e_i \,\in\, E$, of any length $\ell \,\in\, \mathbb{N}$ --- also called a *[[word]]* in the given alphabet.

Typically one will write "$List(E)$" or similar for the [[set]] of all lists with entries in $E$.

For instance, for $E$ the ([[underlying]] set of) a [[field]], the component-expressions of [[vectors]] in the [[vector space]] $k^n$ may be thought of as lists of length $n$ with elements in $k$. However, doing so means to disregard the [[vector space]]-[[structure]] on the collection of all such lists.

Instead, the notion of *list* is a [[concept with an attitude]]: While lists are just [[tuples]] of any length, calling them lists indicates that one intends to consider the operation of *[[concatenation]]* of lists to larger lists, in particular the operations of appending an element to a list.

It is evident that the operation of concatenation makes $List(E)$ a *[[monoid]]* (the [[neutral element]] under concatenation is the [[empty list]], i.e. the unique list of length [[zero]]). In fact, a moment of reflection shows that, as such, $\big(List(E), conc\big)$ is the *[[free monoid]]* on $E$.

Therefore, in much of the mathematical literature, lists are understood as [[free monoids]].

Specifically in [[computer science]] one commonly deals with the corresponding notion of the [[data type]] of lists (with entries in a prescribed data type), which serve as a basic and common kind of [[data structure]]. In [[programming languages]] supporting something like a [[calculus of constructions]], this [[data type]], essentially with the [[free monoid]]-[[structure]] as above, may be defined as an  *[[inductive type]]* in an evident way (made explicit [below](#Definition)).

On the other hand, in [[dependent type theories]] which have a [[homotopy type theory]]-interpretation --- in that they do not verify [[uniqueness of identity proofs]] (or "[[axiom K (type theory)|axiom K]]") --- one may want to distinguish between lists and [[free monoids]]:

Because, in such theories it may happen that the given [[alphabet]] [[type]] $E$ is not actually a [[set]] but a higher [[homotopy type]], in that it is not [[0-truncated]]. In this case it still makes good sense to speak of *lists* of elements ([[terms]]) of $E$, only that now the corresponding [[type]] $List(E)$ is itself not [[0-truncated]].  But since the term "[[monoid]]" carries with it a connotation of being [[0-truncated]], one may no longer want to call this the *[[free monoid]]* on $E$. It is, of course, still a free monoid in the proper [[higher algebra|higher algebraic]] sense (cf. *[[monoid in a monoidal (infinity,1)-category|monoid in a monoidal $\infty$-category]]*).

## Definition
 {#Definition}

The type of lists on a type $A$ is defined as the [[dependent sum type]]

$$\mathrm{List}(A) \coloneqq \sum_{n:\mathbb{N}} \mathrm{Fin}(n) \to A$$

where $\mathrm{Fin}(n)$ is the [[finite set]] with $n$ elements. This shows that lists are just [[tuples]].

### Rules for types of lists

Assuming that [[identification types]], [[function types]] and [[dependent product types]] exist in the type theory, the type of lists on a type $A$ of [[generators]] is the [[inductive type]] $\mathrm{List}(A)$ generated by an element $\mathrm{nil}:\mathrm{List}(A)$ and a function $\mathrm{cons}:A \to \mathrm{List}(A) \to \mathrm{List}(A)$:

Formation rules for the type of lists:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{List}(A) \; \mathrm{type}}$$

Introduction rules for the type of lists:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{nil}:\mathrm{List}(A)} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{cons}:A \to (\mathrm{List}(A) \to \mathrm{List}(A))}$$

Elimination rules for the type of lists:
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\
      \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x)) \\
      \Gamma \vdash g:\mathrm{List}(A))
    \end{array}
  }{\Gamma \vdash \mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, g):C(g)}$$

Computation rules for the type of lists:

* Judgmental computation rules

$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\
      \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x))
    \end{array}
  }{\Gamma \vdash \mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, \mathrm{nil}) \equiv c_\mathrm{nil}:C(\mathrm{nil})}$$

\linebreak

$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\
      \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x)) \\
      \Gamma \vdash b:A \quad \Gamma \vdash g:\mathrm{List}(A))
    \end{array}
  }{\Gamma \vdash \mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, \mathrm{cons}(b)(g)) \equiv c_\mathrm{cons}(b)(g)(\mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, g)):C(\mathrm{cons}(b)(g))}$$

\linebreak

* Typal computation rules

$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\
      \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x))
    \end{array}
  }{\Gamma \vdash \beta_{\mathrm{List}(A)}^\mathrm{nil}(c_\mathrm{nil}, c_\mathrm{cons}):\mathrm{Id}_{C(\mathrm{nil})}(\mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, \mathrm{nil}), c_\mathrm{nil})}$$

\linebreak

$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \\
      \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x)) \\
      \Gamma \vdash b:A \quad \Gamma \vdash g:\mathrm{List}(A))
    \end{array}
  }{\Gamma \vdash \beta_{\mathrm{List}(A)}^\mathrm{cons}(c_\mathrm{nil}, c_\mathrm{cons}, b, g):\mathrm{Id}_{C(\mathrm{cons}(b)(g))}(\mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, \mathrm{cons}(b)(g)), c_\mathrm{cons}(b)(g)(\mathrm{ind}_{\mathrm{List}(A)}^C(c_\mathrm{nil}, c_\mathrm{cons}, g)))}$$

\linebreak

Uniqueness rules for the type of lists:

* Judgmental uniqueness rules

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:\mathrm{List}(A)} C(x) \quad \Gamma \vdash g:\mathrm{List}(A)}{\Gamma \vdash \mathrm{ind}_{\mathrm{List}(A)}^C(c(\mathrm{nil}), \lambda a:A.\lambda x:\mathrm{List}(A).c(\mathrm{cons}(a)(x)), g) \equiv c(g):C(g)}$$

* Typal uniqueness rules

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:\mathrm{List}(A)} C(x) \quad \Gamma \vdash g:\mathrm{List}(A)}{\Gamma \vdash \eta_{\mathrm{List}(A)}(c, n):\mathrm{Id}_{C(g)}(\mathrm{ind}_{\mathrm{List}(A)}^C(c(\mathrm{nil}), \lambda a:A.\lambda x:\mathrm{List}(A).c(\mathrm{cons}(a)(x)), g), c(g))}$$

The elimination, typal computation, and typal uniqueness rules for the type of lists type state that the type of lists satisfy the **dependent universal property of the type of lists**. If the dependent type theory also has [[dependent sum types]] and [[product types]], allowing one to define the [[uniqueness quantifier]], the dependent universal property of the type of lists could be simplified to the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{List}(A) \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{nil}:C(\mathrm{nil}) \quad \Gamma \vdash c_\mathrm{cons}:\prod_{a:A} \prod_{x:\mathrm{List}(A))} C(x) \to C(\mathrm{cons}(a)(x))}{\Gamma \vdash \mathrm{up}_\mathbb{Z}^C(c_\mathrm{nil}, c_\mathrm{cons}):\exists!c:\prod_{x:\mathrm{List}(A))} C(x).\mathrm{Id}_{C(\mathrm{nil})}(c(\mathrm{nil}), c_\mathrm{nil}) \times \prod_{a:A} \prod_{x:\mathrm{List}(A)} \mathrm{Id}_{C(\mathrm{cons}(a)(x))}(c(\mathrm{cons}(a)(x)), c_\mathrm{cons}(a)(c(x)))}$$

## See also

* [[natural numbers type]] (the list on the unit type $\mathrm{List}(\mathbb{1})$)

* [[underlying type of free infinity-group]] (the invertible version of the type of lists)

* [[James construction type]], [[free monoid]]

## References

Types of lists are defined in section 5.1 in:

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

[[!redirects list]]
[[!redirects lists]]

[[!redirects type of lists]]
[[!redirects types of lists]]

[[!redirects universal property of the type of lists]]
[[!redirects dependent universal property of the type of lists]]

[[!redirects cons]]


list monad

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### Context
#### Categorical algebra
+-- {: .hide}
[[!include categorical algebra -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The *list monad* or *free monoid monad* is the [[monad]] on [[Sets]] (or generally on a [[distributive monoidal category]])

$$
  \array{
    \mathllap{List \;\colon\;}
    Set &\longrightarrow& Set
    \\
    S
      &\mapsto&
    \displaystyle{
     \underset{n \colon \mathbb{N}}{\textstyle{\coprod}}
    }
    S^{\times n}
  }
$$

which sends a [[set]] $S$ to the [[underlying set]] $U\big(F(S)\big)$ of the [[free monoid]] $F(S) \equiv S^\ast$ on $S$, hence the set of  [[lists]] ([[string (computer science)|strings]]) of [[elements]] of $S$, hence of $n$-[[tuples]] $(s_1, s_2, \cdots, s_n) \colon S^{\times n}$ for any [[natural number]] $n$.

Its join operation is by [[concatenation|concatenating]] a list-of-lists to a single list,

\begin{tikzcd}[sep=0pt]
    \mathrm{List}
    \big(
      \mathrm{List}(D)
    \big)
    \ar[
      rr,
      "{
        \mathrm{join}^{\mathrm{List}}_D
      }"
    ]
    &&
    \mathrm{List}(D)
    \\
    \scalebox{.8}{$
    \left(
      \def\arraystretch{1}
      \begin{array}{l}
      (d_{11}, \cdots, d_{1 n_1}),
      \\
      \;\;\;\;\vdots
      \\
      (d_{k 1},  \cdots, d_{1 n_k})
      \end{array}
    \right)
    $}
    &\scalebox{.8}{$\mapsto$}&
    \scalebox{.8}{$
    \big(
      d_{11}, \cdots, d_{1 n_1}
      ,\,
      \cdots
      ,\,
      d_{k 1}, \cdots, d_{k n_k}
    \big)
    $}
\end{tikzcd}

and its [[unit of a monad|unit]] assigns to an element of $S$ the list consisting of that single element:

\begin{tikzcd}[sep=0pt]
    D
    \ar[
      rr,
      "{
        \mathrm{ret}
          ^{ \mathrm{List} }
          _{ D }
      }"
    ]
    &&
    \mathrm{List}(D)
    \\
    d &\mapsto& (d)
\end{tikzcd}

The [[Eilenberg-Moore category]] of [[module over a monad|modules]] over the list monad is the [[category of monoids]] $Mon$:

\begin{tikzcd}
  \mathrm{Mon}
  \ar[
    rr,
    shift right=8pt,
    "{ U^{\mathrm{List}} }"{swap}
  ]
  \ar[
    from=rr,
    shift right=8pt,
    "{ F^{\mathrm{List}} }"{swap}
  ]
  \ar[
    rr,
    phantom,
    "{ \scalebox{.7}{$\bot$} }"
  ]
  &&
  \mathrm{Set}
\end{tikzcd}

## Related pages

* [[monoid]], [[free monoid]]

* [[list]], [[string (computer science)]]

* [[writer monad]]

* [[nondeterministic computation]]

## References

Discussion of the list monad as a [[monad in computer science]] (in [[Haskell]]):

* [[Philip Wadler]], §2.1 in: *Comprehending Monads*, in: _Conference on Lisp and functional programming_, ACM Press (1990) [[[WadlerMonads.pdf:file]], [doi:10.1145/91556.91592](https://doi.org/10.1145/91556.91592)]

* [[Philip Wadler]], §5.1 in: *Monads for functional programming*, in M. Broy (eds.) *Program Design Calculi* NATO ASI Series, **118** Springer (1992) [[doi;10.1007/978-3-662-02880-3_8](https://doi.org/10.1007/978-3-662-02880-3_8), [pdf](https://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf)]

* [[Philip Wadler]], §2.7 in: *The essence of functional programming*, *POPL '92: Principles of programming languages* (1992) 1-14 [[doi:10.1145/143165.143169](https://doi.org/10.1145/143165.143169), [pdf](https://jgbm.github.io/eecs762f19/papers/wadler-monads.pdf)]

* [[Bartosz Milewski]], *The List Monad*, School of Haskell (2014) [[web](https://www.schoolofhaskell.com/school/starting-with-haskell/basics-of-haskell/13-the-list-monad)]

* {#Milewski19} [[Bartosz Milewski]], pp. 304 in: *Category Theory for Programmers*, Blurb (2019) [[pdf](https://github.com/hmemcpy/milewski-ctfp-pdf/releases/download/v1.3.0/category-theory-for-programmers.pdf), [github](https://github.com/hmemcpy/milewski-ctfp-pdf), [webpage](https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/), [ISBN:9780464243878](https://www.blurb.com/b/9621951-category-theory-for-programmers-new-edition-hardco)]

[[!redirects list monads]]

[[!redirects free monoid monad]]
[[!redirects free monoid monads]]

[[!redirects free monoid functor]]
[[!redirects free monoid functors]]


list of category theory papers that never appeared

This page is intended to list references to [[category theory]] papers cited as "to appear", "in preparation", or "forthcoming" by the authors, but never appeared (at least with the cited title). The intention is to prevent lost time searching for papers that do not exist. Sometimes a corresponding paper was published, but under a different name, in which case we record the name it was eventually published under.

This article is complementary to the [[list of lost manuscripts in category theory]], which records papers that appear to have been circulated, but not widely, or for which there is evidence they were complete but not circulated.

Do not include recent references that are likely still to appear. If the author is no longer alive, or the reference is not recent (e.g. 20 years or further), it is reasonable to include in this list.

|Title|Author|Cited in|Likely appeared as|URL/Comments|
|-----|------|--------|----------------------|------------|
|*Produits tensoriels d'esquisses*|[[Christian Lair]]|*Categories of sketched structures*, [[Andrée Bastiani]] and [[Charles Ehresmann]] (1972)| -- | On the tensor product of [[sketches]]|
|*Relative functorial semantics, II: Beck's Theorem*|[[Fred Linton]]|*Relative Functorial Semantics, III: Triples vs. Theories*, [[Fred Linton]] (1974)| -- | |
|*Extracts from the Archives of Categorical Folklore, I: Pro-categories*|[[Fred Linton]]|*Relative Functorial Semantics, III: Triples vs. Theories*, [[Fred Linton]] (1974)| -- | |
| |[[Max Kelly]] and [[H. Wolff]]|*Some existence theorems in the theory of doctrines*, [[Robert Blackwell]] (1976)| *Free monads and the orthogonal subcategory problem* by [[Harvey Wolff]] (1978) and *A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on* by [[Max Kelly]] (1980) | On [[free monads]] |
|*Essentially algebraic structures in the enriched context, II*|[[Max Kelly]]|*A Presentation of Topoi as Algebraic relative to Categories of Graphs*, [[Eduardo Dubuc]] and [[Max Kelly]] (1983)| -- | A sequel to *Structures defined by finite limits in the enriched context, I* by [[Max Kelly]] (1982)|
|*Categories with structure-Biadjoints for algebraic functors*|[[Max Kelly]]|*A Presentation of Topoi as Algebraic relative to Categories of Graphs*, [[Eduardo Dubuc]] and [[Max Kelly]] (1983)| -- | |
|*On finitary enriched monads and their presentations*|[[Max Kelly]] and [[John Power]]|*Two-dimensional monad theory*, Blackwell, [[Max Kelly]], and [[John Power]] (1989)|*Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads* (1993)| |
|*Equivalences in 2-categories, birepresentations, and biadjoints*|[[Max Kelly]]|*Two-dimensional monad theory*, Blackwell, [[Max Kelly]], and [[John Power]] (1989)| -- | |
|*On the abstract notion of club*|[[Max Kelly]]|*Two-dimensional monad theory*, Blackwell, [[Max Kelly]], and [[John Power]] (1989)|*On clubs and data-type constructors*  (1992)| |
|*Two-dimensional structures defined by limit-theories*|[[Max Kelly]]|*Flexible limits for 2-categories*, Bird, [[Max Kelly]], and [[John Power]] (1989)| -- | |
|*A criterion for flexibility of a 2-monad*|[[Max Kelly]], [[Steve Lack]], and [[John Power]]|*Monoidal functors generated by adjunctions, with applications to transport of structure*, [[Max Kelly]] and [[Steve Lack]] (2004)| -- | |
|*KZ-Adjunctions*|[[Francisco Marmolejo]] and [[Richard Wood]]|*Duality for CCD Lattices*, [[Francisco Marmolejo]], [[Robert Rosebrugh]], and [[Richard Wood]] (2009)| -- | |
|*Kleisli machines*|[[Michael Arbib]] and [[Ernest Manes]]|*Categorical theory of tree processing* (1974), Saud Alagić|*Fuzzy machines in a category*| |

## Related pages

* [[list of lost manuscripts in category theory]]
* [[list of theses in category theory]]

[[!redirects to appear]]


list of functorial field theories

This article is meant to give an exhaustive list of explicitly constructed nontopological [[functorial field theories]] in dimension 2 and higher.
All currently known explicit constructions are nonextended, and with the exception of the Kandel construction, have dimension 2.

## Free field theories

### Massless boson

* [[Hessel Posthuma]], _The Heisenberg group and conformal field theory_, [arXiv](https://arxiv.org/abs/0706.4262).

### Massive boson

* [[Santosh Kandel]], _Functorial Quantum Field Theory in the Riemannian setting_, [arXiv](https://arxiv.org/abs/1502.07219v3).

### Fermion

* [[James E. Tener]], _Construction of the unitary free fermion Segal CFT_, [arXiv](https://arxiv.org/abs/1608.02095).

## Field theories with interaction

### Pickrell

* [[Doug Pickrell]], _P(phi)_2 quantum field theories and Segal's axioms_, [arXiv](https://arxiv.org/abs/math-ph/0702077).

$$S(\Phi)=\int_\Sigma 2^{-1}(\|d\Phi\|^2+m^2\Phi^2)+P(\Phi)$$

### Liouville field theory

* Colin Guillarmou, Antti Kupiainen, Rémi Rhodes, Vincent Vargas, _Segal's axioms and bootstrap for Liouville Theory_, [arXiv](https://arxiv.org/abs/2112.14859).


list of journals publishing homotopy theory and category theory

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

#Contents#
* table of contents
{:toc}

The following journals have, at least at some time in the past, published homotopy theory, category theory, and related material.  More need to be added.  No guarantee is made that the mentioned editors are current; check the journal web sites for current editorial boards.

## Journals currently publishing homotopy theory and/or category theory
 {#List}

The following list is meant to be roughly in decreasing order of the number of articles they publish in the area of homotopy theory and category theory, as witnessed by the [[MathSciNet]] statistics. Unfortunately this webpage is in continuous update and it is not easy to keep track of this information consistently.

|Journal                                    |Publisher         |Relevant editor(s)|Access|Specialization|
|-------------------------------------------|------------------|-------------------|------|-------------|
|[Theory and Application of Categories](http://www.tac.mta.ca/tac/)|self|all|open|categories|
|[Journal of Pure and Applied Algebra](https://www.journals.elsevier.com/journal-of-pure-and-applied-algebra)|Elsevier|[[Jiri Adamek|Adamek]], [[John Bourke|Bourke]], [[Giuseppe Rosolini|Rosolini]]|subscription|algebra|
|[Applied Categorical Structures](https://link.springer.com/journal/10485)|Springer|all|subscription|categories|
|[Advances in Mathematics](https://www.journals.elsevier.com/advances-in-mathematics/)|Elsevier|[[Andrew Blumberg|Blumberg]]; [[Henning Krause|Krause]]; [[Ross Street|Street]]|subscription|general|
|[Journal of Algebra](https://www.journals.elsevier.com/journal-of-algebra)|Elsevier|[[Volodymyr Mazorchuk|Mazorchuk]]|subscription|algebra|
|[Journal of Homotopy and Related Structures](https://www.springer.com/mathematics/geometry/journal/40062)|[TCMS](http://tcms.org.ge/) via Springer|[[Ronnie Brown|Brown]]; [[Tim Porter|Porter]]; [[Ross Street|Street]]; [[Simon Willerton|Willerton]]|subscription|homotopy|
|[Homology, Homotopy, and Applications](https://intlpress.com/site/pub/pages/journals/items/hha/_home/_main/)|International Press|[[Ronnie Brown|Brown]]; [[George Janelidze|Janelidze]]; [[Emily Riehl|Riehl]]; [[Brooke Shipley|Shipley]]|subscription|homotopy|
|[Algebraic & Geometric Topology](https://msp.org/agt/)|MSP|[[Julie Bergner|Bergner]]|subscription|homotopy, algebraic topology, geometric topology|
|[Topology and its Applications](https://www.journals.elsevier.com/topology-and-its-applications)|Elsevier|[[Bill Dwyer|Dwyer]]|subscription|general topology, homotopy|
|[Cahiers de topologie et géométrie différentielle catégoriques](http://cahierstgdc.com/)|self|all|open|categories, [[synthetic differential geometry]]
|[Journal of Topology](https://londmathsoc.onlinelibrary.wiley.com/journal/17538424)|LMS via Wiley|[[Andrew Blumberg|Blumberg]]|subscription|topology|
|[Proceedings of the American Mathematical Society](https://www.ams.org/proc/)|AMS|[[Mark Behrens|Behrens]]|subscription|general|
|[Transactions of the American Mathematical Society](https://www.ams.org/tran/)|AMS|[[Craig Westerland|Westerland]]|subscription|general|
|[International Mathematics Research Notices](https://academic.oup.com/imrn/)|OUP|[[Andrew Blumberg|Blumberg]], [[Lars Hesselholt|Hesselholt]], [[Dmitry Kaledin|Kaledin]]|subscription|general|
|[Documenta Mathematica](https://www.elibm.org/series-details?q=se:2204)|DMV|[[Henning Krause|Krause]], [[Mike Hill|Hill]], [[Max Karoubi|Karoubi]]|open|general|
|[Communications in Algebra](https://www.tandfonline.com/toc/lagb20/current)|Taylor and Francis|varies|subscription|algebra|
|[Mathematische Zeitschrift](https://www.springer.com/journal/209)|Springer|[[Michael Hill|Hill]]|subscription|general|
|[Annals of K-theory](http://www.ktheoryfoundation.org/journal.html)|MSP|[[Paul Balmer|Balmer]]; [[Amnon Neeman|Neeman]]|subscription|K-theory|
|[Mathematical Structures in Computer Science](https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science)|Cambridge UP|[[Martin Hyland|Hyland]]; [[Steve Lack|Lack]]; [[Robert Seely|Seely]]|subscription|TCS|
|[Theoretical Computer Science](https://www.journals.elsevier.com/theoretical-computer-science)|Elsevier|varies|subscription|TCS|
|[Fundamenta Mathematicae](https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae)|IMPAN|[[Stefan Jackowski]], [[William Dwyer]], [[Jean Lannes]]|subscription|general|
|[Compositionality](http://www.compositionality-journal.org/)|self|all|open|categories|
|[Higher Structures](https://higher-structures.math.cas.cz/)|Institute of Mathematics, Czech Academy of Sciences|all|open|homotopy|
|[Categories and General Algebraic Structures with Applications](http://cgasa.sbu.ac.ir/)|Shahid Beheshti University|many|open|categories|
|[Logical Methods in Computer Science](https://lmcs.episciences.org/)|self|[[Valeria de Paiva|de Paiva]]; Jagadeesan; Schröder|open|logic and TCS|
|[New York Journal of Mathematics](http://nyjm.albany.edu/)|self|[[Michael Hopkins|Hopkins]]|open|general|
|[Extracta Mathematicae](https://publicaciones.unex.es/index.php/EM)|IMUEX|[[Philippe Gaucher]], [[Antonio M. Cegarra]], [[Tim Porter]]|open|general|
|[Quaestiones Mathematicae](https://www.tandfonline.com/toc/tqma20/current)|SAMS via Taylor and Francis|[[George Janelidze|Janelidze]]|subscription|general|
|[Journal of Algebra and Its Applications](https://www.worldscientific.com/worldscinet/jaa)|World Scientific|[[Alberto Facchini|Facchini]]|subscription|algebra|
|[Journal of the Australian Mathematical Society](https://www.austms.org.au/Journal+of+the+Australian+Mathematical+Society)|Cambridge University Press|[[Nora Ganter|Ganter]]|subscription|general|
|[Annals of Pure and Applied Logic](https://www.journals.elsevier.com/annals-of-pure-and-applied-logic)|Elsevier|[[Martin Hyland|Hyland]]|subscription|logic|
|[Logica Universalis](https://www.springer.com/journal/11787)|Springer|[[Olivia Caramello|Caramello]], [[Valeria de Paiva|de Paiva]]|subscription|logic|
|[The Journal of Symbolic Logic](https://www.cambridge.org/core/journals/journal-of-symbolic-logic)|Cambridge UP|[[Colin McLarty|McLarty]], [[Steve Awodey|Awodey]]|subscription|logic|

## Journals previously publishing homotopy theory and/or category theory

|Journal|Publisher|
|-------|---------|
|[[Comptes rendus hebdomadaires des séances de l'Académie des sciences]]|French Academy of Sciences|

## Related pages

* [[list of theses in category theory]]
* [[list of lost manuscripts in category theory]]

[[!redirects list of journals publishing category theory]]
[[!redirects list of journals publishing homotopy theory]]


list of lost manuscripts in category theory

This page is intended to list references to manuscripts in category theory (e.g. papers or books) that are cited but never became widely available. The hope is that some manuscripts may be recovered (in which case they will be uploaded here for posterity).

**If you have a copy of one of the manuscripts here that is not yet uploaded, please [upload it](https://ncatlab.org/nlab/show/HowTo#HowToUploadFiles).**

Do not include recent references that are likely still to appear. If the author is no longer alive, or the reference is not recent (e.g. 20 years or further), it is reasonable to include in this list. Prefer also to avoid self-references, in which authors cite their own forthcoming work that never appears, since it is likely no manuscript ever existed in this case: see [[list of category theory papers that never appeared]].

|Title|Author|Cited in|Contents|URL/Comments|
|-----|------|--------|--------|---|
| | [[Jon Beck]] | *Many sources* | The [[monadicity theorem]] | [PDF](https://ncatlab.org/nlab/files/Beck_MonadicityTheorem.pdf), provided by [[John Kennison]] |
| *Formal Category Theory II* | [[John Gray]] | [[Topos Theory]] [168] | A "detailed account of [[internal categories]] from a [[2-category|2-categorical]] point of view" | According to [[Peter Johnstone]], a manuscript was never circulated |
| | [[Claude Chevalley]] | [[Categories for the Working Mathematician]] | "A systematic treatment of all possible properties of [[limits]]" | According to [[Saunders Mac Lane]] in [[CWM]], the manuscript was lost by a shipping company |
| *A note on a construction of Kleisli* (1965) | [[Michael Barr]] | *An outline of functorial semantics*, [[Fred Linton]] (1969) | Universal property of the Kleisli construction |Michael Barr no longer has a copy|
| *Relative Functorial Semantics, III: Triples vs. Theories* (1974) | [[Fred Linton]] | *Functorial semantics and HSP type theorems*, [[Michael Barr]] (1994); *Monads of sets*, [[Ernest Manes]] (2003) | Characterisation of the [[Eilenberg--Moore category]] for an (enriched) monad as a [[pullback]] over [[presheaves]] on the [[Kleisli category]] | [[Linton_RelativeFunctorialSemanticsIII.pdf:file]], provided by [[Ernie Manes]] (also [here](https://oda.mfo.de/bitstream/handle/mfo/1474/full-text.pdf?sequence=1&isAllowed=y#page=33)) |
| *J-adjonctions et J-monades* (1970) | [[Yves Diers]] | *Monades involutives complémentées* [21], [[René Guitart]] (1975) | [[relative monads|Relative monads]] and [[relative adjunctions]] | |
|*Extension of functors to categories of algebras* (c. 1971)|H. Wolff|*Monad compositions I: general constructions and recursive distributive laws* [32], [[Ernie Manes]] and [[Philip Mulry]]| | |

## Related pages

* [[list of category theory papers that never appeared]]
* [[list of theses in category theory]]


list of mathematics software

# Mathematics software
* table of contents
{: toc}

Here is a very incomplete list of software available for mathematics, especially the sort of mathematics that nLabbers seem to be interested in.

##General

 * [swMATH](http://www.swmath.org/)
 * [Oberwolfach References on Mathematical Software](http://orms.mfo.de)

## Proof assistants

* [[Coq]]
* [[Agda]]
* [[Lean]]
* [[Nuprl]]
* [[HOL Light]]

### Logical frameworks

* [[Twelf]]
* [[Elf]]
* [[Automath]]
* [[Isabelle]]

## Quantum computation

* [[Quipper]]
* [PyZX](https://github.com/Quantomatic/pyzx)

## Computer algebra systems

###General

* [[Mathematica]]
* [Maple](http://www.maplesoft.com)
* [[Sage]]
* [[geogebra]] (dominantly used for educational purposes)
* [MathCAD](https://www.ptc.com/en/products/mathcad)
* [Calcpad] (http://calcpad.net)

###Computational Group Theory

* [[GAP]]

###Computational Homological Algebra (CHA)

* [CHA](http://hamilton.nuigalway.ie/CHA/)
* [[HAP]]

### Computational Algebraic Topology

* [[Kenzo]] for [[constructive algebraic topology]]

###Computational Higher-dimensional Discrete Algebra

* [CHDA Homepage](http://www.maths.bangor.ac.uk/chda/intro.html)

### Category theory

* [[TwoVect]]
* The [Fusion Atlas](http://tqft.net/wiki/Atlas_of_subfactors)
* [[Catlab.jl]]

### Globular n-categories

* [[Globular]], up to 4-categories
* [[homotopy.io]]

### Monoidal categories

* [Quantomatic](http://quantomatic.github.io/)
* [[DisCoPy]]

### Programming Environments

* [Maple IDE](http://digi-area.com/CasStudio/Maple/)
* [Wolfram Workbench](http://www.wolfram.com/products/workbench)
* [Wolphramalpha](http://www.wolframalpha.com) - online computational engine
* [[Sage]]

## Related pages

* [[programming language]]

category: software
[[!redirects list of mathematical software]]


list of notable initial algebras and terminal coalgebras

\begin{example}
A list of notable [[endofunctors]], and their [[initial algebras of an endofunctor|initial algebras]]/[[terminal coalgebra of an endofunctor|terminal coalgebras]].

> Nonexistent (co)algebras are labelled with '/', and unknown ones with '?'.

Base category | Endofunctor                | Initial Algebra        | Final Coalgebra               | Note, reference |
:-----: | :------------:                   | :-------------:         |:----------------:            |  :--------: |
[[Set]] | [[constant functor|Const]] $A$                        | $A$                    | $A$                           | |
[[Set]] | $X \mapsto X$                    | $\varnothing$            | $1$                           | |
[[Set]] | $X \mapsto 1 + X$                | $\mathbb{N}$           | [[conatural number|Conatural numbers]] $\mathbb{N}^\infty$ | [[extended natural number]] |
[[Set]] | $X \mapsto A + X$                | $A \times \mathbb{N}$  | $A \times \mathbb{N} + 1$, ie [[conatural numbers]] "terminated" (when they aren't $\infty$) with $A$ | [[partial map classifier]] |
[[Set]] | $X \mapsto X + X$ or $X \mapsto 2 \times X$                | $\emptyset$            | $2^\mathbb{N}$ (i.e. one definition of [[stream|Stream]] $2$)               | |
[[Set]] | $X \mapsto A \times X$                | $\emptyset$            | $A^\mathbb{N}$ (i.e. one definition of [[stream|Stream]] $A$)               | [[sequence]], [[writer monad]], [[stream]] |
[[Set]] | $X \mapsto X \times X$ or $X \mapsto [2, X]$          | $\emptyset \simeq [2, \emptyset]$            | 1 (the unique infinite unlabelled binary tree)     | |
[[Set]] | $X \mapsto [A, X]$               | $[A, \emptyset]$       | 1                             | [[reader monad]] |
[[Set]] | $X \mapsto 1 + A \times X$       | [[list|List]] $A$               | another definition of [[stream|Stream]] $A$; i.e. potentially infinite [[list|List]] $A$ | [[list]], [[stream]] |
[[Set]] | $X \mapsto 1 + A \times X^2$     | Finite [[binary tree]] with $A$-labelled nodes  | Potentially infinite [[binary tree]] with $A$-labelled [[nodes]] | [[tree]] |
[[Set]] | $X \mapsto B + A \times X^n$     | Finite $n$-ary tree with $A$-labelled nodes and $B$-labelled [[leaves]] | Potentially infinite $n$-ary tree with $A$-labelled nodes with and $B$-labelled leaves| |
[[Set]] | $X \mapsto O \times [I, X]$      | $O \times [I, \emptyset]$ | Potentially infinite [[Moore machine]] | |
[[Set]] | $X \mapsto [I, O \times X]$      | $[I, \emptyset]$       | Potentially infinite [[Mealy machine]]             | |
[[Set]] | $X \mapsto \mathcal{P}(X)$       | /                      | /                             | |
[[Set]] | $X \mapsto \mathcal{P}_{\text{fin}}(X)$ | Finite [[rooted forests]] | Potentially infinite finitely-branching [[rooted forests]] | |
[[Set]] | [[polynomial endofunctor]] | [[W-type]] | [[M-type]] | |
[[bi-pointed object|Bipointed]] [[set|Sets]] | $X \mapsto X \vee X$ | [[dyadic rational numbers]] in the [[interval]] $[0,1]$ | The [[closed interval]] $[0,1] \subseteq \mathbb{R}$ | [[coalgebra of the real interval]] | |
[[linearly ordered sets]] | $X \mapsto \omega \cdot X$, where $\omega \cdot X$ is the cartesian product of the natural numbers with the underlying set of $X$, equipped with the [[lexicographic order]]. | $\emptyset$ | The non-negative [[real numbers]] $\mathbb{R}_{\geq 0}$ | [[real number]] | |
[[Archimedean ordered fields]] | $X \mapsto X$ the [[identity functor]] | The [[rational numbers]] $\mathbb{Q}$ | The [[real numbers]] $\mathbb{R}$ | | |
[[Archimedean ordered fields]] | $X \mapsto \mathcal{D}(X)$ where $\mathcal{D}(X)$ is the Archimedean ordered field of two-sided [[Dedekind cuts]] of $X$ | The [[real numbers]] $\mathbb{R}$ | The [[real numbers]] $\mathbb{R}$ | | |
[[Archimedean ordered fields]] | $X \mapsto \mathcal{C}(X)$ where $\mathcal{C}(X)$ is the quotient of [[Cauchy sequences]] in the Archimedean ordered field $X$ | The [[HoTT book real numbers]] $\mathbb{R}_H$ | The [[Dedekind real numbers]] $\mathbb{R}$ | These are the same objects in the presence of [[excluded middle]] or [[countable choice]].| |
Any [[category]] | The [[constant functor]] $X \mapsto A$ given object $A$ | $A$ | $A$ | |
Any [[category]] | The [[identity functor]] $X \mapsto X$ | [[initial object]] | [[terminal object]] | |
Any [[extensive category]] | $X \mapsto 1 + X$ given [[terminal object]] $1$ and [[coproduct]] $+$ | [[natural numbers object]] | ? | |
Any [[closed monoidal category|closed]] [[abelian category]] | $X \mapsto I \sqcup (A \otimes X)$ given [[tensor unit]] $I$, [[tensor product]] $\otimes$, [[coproduct]] $\sqcup$, and object $A$ | [[tensor algebra]] of $A$ | [[cofree coalgebra]] over $A$ | [[tensor algebra]], [[cofree coalgebra]] |
[[Infinity-Grpd|$\infty$-Grpd]] | $X \mapsto \Sigma X$ | [[sphere spectrum]] $\mathbb{S}$ | ? | [[suspension]] |

\end{example}


list of theses in category theory

This page is intended to record doctoral theses whose focus is primarily [[category theory|category theoretic]], particularly those from earlier years that are harder to find online.

Note: the list is ordered first by year, then alphabetically by last name.

|Title|Author|Year|URL|Summary|
|-----|------|----|---|------|
|[[Des Catégories Abéliennes]]|[[Pierre Gabriel]]|1960|[numdam](http://www.numdam.org/item/?id=BSMF_1962__90__323_0)| |
|Functor theory|[[Peter Freyd]]|1960|[Category Theory Archive](https://github.com/CategoryTheoryArchive/archive/blob/main/resources/1960_freyd_functor-theory.pdf) | |
|[[Functorial Semantics of Algebraic Theories]]|[[William Lawvere]]|1963|[TAC reprint](http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html)| on [[algebraic theories]] |
|The Functorial Foundations of Measure Theory|[[Fred Linton]]|1963|[Scan](http://tlvp.net/~fej.math.wes/FredLintonPhDThesis_compressed.pdf)| |
|Categories of Set Valued Functors|[[Marta Bunge]]|1966|[ProQuest](https://www.proquest.com/docview/302207560/48E9615B59B04E17PQ/1?accountid=10474&sourcetype=Dissertations%20&%20Theses) | on [[categories of presheaves]] |
|Fibred Categories and the Theory of Structures (Part I)|[[John Duskin]]|1966|[McMaster repository](https://macsphere.mcmaster.ca/handle/11375/17610)| |
|Triples, algebras and cohomology|[[Jon Beck]]|1967|[TAC reprint](http://www.tac.mta.ca/tac/reprints/articles/2/tr2abs.html)| |
|A triple miscellany: some aspects of the theory of algebras over a triple|[[Ernest Manes]]|1967|[proquest](https://www.proquest.com/openview/85791ad4b7917802ca6692378e453d60/)| |
|Limit monads in categories|[[Anders Kock]]|1967|[proquest](https://www.proquest.com/openview/61883e900f1fd8585fe4dfc4c3f271dc)| on [[lax idempotent 2-monads]] |
|Homotopy Classification of Filtered Complexes|[[Ross Street]]|1968|[Scan](http://science.mq.edu.au/~street/Thesis.pdf)| |
|Absoluteness properties in category theory|[[Robert Paré]]|1969|[Scan](https://www.mathstat.dal.ca/~pare/PhDThesis.pdf)| on [[absolute colimits]] |
|Construction of Biclosed Categories|[[Brian Day]]|1970|[Scan](http://web.science.mq.edu.au/~street/DayPhD.pdf)| |
|A categorical approach to universal algebra|[[Bob Walters]]|1970|[ANU repository](https://openresearch-repository.anu.edu.au/handle/1885/133321)| on [[universal algebra]] |
|Self-Dual Structure-Semantics and Algebraic Categories|Michel Thiébaud|1971| | |
|Tensor and cotensor products of bifunctors|George D. Reynolds|1973| | |
|Some Aspects of Internal Category Theory in an Elementary Topos|[[Peter Johnstone]]|1974| | |
|Relative Bildzerlegungen und algebraische Kategorien|[[Walter Tholen]]|1974|[Scan](https://tholen.mathstats.yorku.ca/files/2021/01/PhD_Thesis.pdf)|English translation of the title is *Relative image decompositions and algebraic categories*|
|Coherence for a closed functor|Geoffrey Lewis|1974|[UNSW repository](https://unsworks.unsw.edu.au/entities/publication/3c1ee9e9-9d61-4fc9-9766-5fb9bf6415c2)| |
|Catégories internes et fibrations ; Cohomologie de Gel'fand-Fuks|[[Jean Celeyrette]]|1975|[[celeyrette-thesis-1975.pdf:file]]|[[internal categories]], [[fibrations]], [[Gelfand-Fuks cohomology]]|
|Théorie formelle des bicatégories|[[Syméon Bozapalides]]|1976| Scan | on 2-dimensional [[coends]], [[enriched bicategories]], etc. |
|Universal coalgebras|[[Thomas Fox]]|1976|[McGill repository](https://escholarship.mcgill.ca/concern/theses/xs55mc798)| |
|Indicial methods for relative categories|[[Richard Wood]]|1976|[Dalhousie repository](https://dalspace.library.dal.ca//handle/10222/55465)| |
|Some existence theorems in the theory of doctrines|Robert Blackwell|1976|[UNSW repository](https://unsworks.unsw.edu.au/server/api/core/bitstreams/820c3175-88e9-4201-8f4a-2fecd46f2b24/content)| |
|Abstract Families of Algebras|[[Robert Rosebrugh]]|1977| | |
|Catégories localisables & Aspects geometriques de la propriete de Radon-Nikodym dans les espaces de Banach|[[Yves Diers]]|1977|[Category Theory Archive](https://github.com/CategoryTheoryArchive/archive/blob/main/resources/1977_diers_cat%C3%A9gories-localisables.pdf)| on [[locally multipresentable categories]], among other things |
|Cartesianess|[[Susan Niefield]]|1978|[proquest](https://www.proquest.com/docview/302920643)| on [[exponentiable objects]] (there called "cartesian objects") |
|Algebraic and locally algebraic functors|James J. Dukarm|1980|[Scan](https://summit.sfu.ca/_flysystem/fedora/sfu_migrate/3281/b12244193a.pdf)| on locally equational categories |
|The Theory of Triposes|[[Andrew Pitts]]|1981|[Scan](https://www.cl.cam.ac.uk/~amp12/papers/thet/thet.pdf)| on [[triposes]] |
|Limits in 2-categories of locally presentable categories|Greg Bird|1984|[Scan](http://science.mq.edu.au/~street/BirdPhD.pdf)| |
|Operational Categories|Colin Barry Jay|1984|[Category Theory Archive](https://github.com/CategoryTheoryArchive/archive/blob/main/resources/1984_jay_generalising-structure-semantics.pdf) | |
|Butler's theorems and adjoint squares|[[John Power]]|1984|[Scan](https://central.bac-lac.gc.ca/.item?id=TC-QMM-121093&op=pdf&app=Library&is_thesis=1&oclc_number=1032920922)| on the [[adjoint lifting theorem]] and [[adjoint triangle theorem]] |
|Pasting diagrams in n-categories with applications to coherence theorems and categories of paths|[[Michael Johnson]]|1988|[postscript](http://web.science.mq.edu.au/~mike/papers/0.ps)| |
|Predicates and Fibrations|[[Duško Pavlović]]|1990|[Scan](https://dusko.org/wp-content/uploads/2020/03/1990-proefschrift-dusko.pdf)| |
|Dialectica Categories|[[Valeria de Paiva]]|1991|[Computer Lab Technical Report](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-213.html)| |
|Enriched categories, internal categories and change of base|[[Dominic Verity]]|1992|[TAC reprint](http://www.tac.mta.ca/tac/reprints/articles/20/tr20abs.html)| |
|Cauchy completion in enriched categories|Scott Russell Johnson|1993|[ANU repository](https://doi.org/10.25911/5d6513e948cab)| |
|Linear Logic, Bimodules, and Full Coherence for Autonomous Categories|[[Todd Trimble]]|1994| | |
|Cobordism Categories|Sean Carmody|1995| | |
|The algebra of distributive and extensive categories|[[Stephen Lack]]|1995| | |
|Groupoid Representations for Sheaves on Orbifolds|[[Dorette Pronk]]|1995| | |
|Internal categories and quantum groups|[[Marcelo Aguiar]]|1997|[pdf](https://ncatlab.org/nlab/files/Aguiar_InternalCategoriesAndQuantumGroups.pdf)| |
|Monadicity, Purity, and Descent Equivalence|Xiuzhan Guo|2000|[pdf](https://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/ftp02/NQ59136.pdf)| on [[monadicity]], [[descent]], and [[locally presentable categories]] |
|Higher-Dimensional Category Theory: Opetopic Foundations|[[Eugenia Cheng]]|2002|[pdf](https://eugeniacheng.com/wp-content/uploads/2017/02/cheng-thesis.pdf)| |
|Polycategories|[[Richard Garner]]|2006|[PDF](http://web.science.mq.edu.au/~rgarner/Thesis/Thesis.pdf)| on [[polycategories]] via Kleisli bicategories and [[mixed distributive laws]] |
| An algebraic theory of tricategories | [[Nick Gurski]] | 2007 | [[Gurski-AlgebraicTricategories.pdf:file]] | |
| Bigroupoid 2-torsors | [[Igor Baković]] | 2008 | [doi](https://doi.org/10.5282/edoc.9209)  | on [[principal 2-bundles]] |
| Coherence for Categorified Operadic Theories | Miles Gould | 2008 | [Glasgow repository](https://theses.gla.ac.uk/689/1/2009gouldphd.pdf) | on [[categorification]], [[coherence]] and [[operads]] |
| Double Categories and Base Change in Homotopy Theory | [[Michael Shulman]] | 2009 |[ProQuest](https://www.proquest.com/docview/305052318?pq-origsite=gscholar&fromopenview=true&sourcetype=Dissertations%20&%20Theses) | |
| Codescent objects in 2-dimensional universal algebra | [[John Bourke]] | 2010 | [pdf](http://www.math.muni.cz/~bourkej/papers/JohnBThesis.pdf) | on [[codescent objects]] and 2-dimensional generalisations of [[internal categories]] |
|Homology operations in the spectral sequence of a cosimplicial space|Philip Hackney|2010|[pdf](http://phck.net/thesis.pdf)| |
|Fundamental bigroupoids and 2-covering spaces|[[David Roberts]]|2010|[Adelaide repository](https://hdl.handle.net/2440/62680)| |
|Algebraic Model Structures|[[Emily Riehl]]|2011|[pdf](https://emilyriehl.github.io/files/phdthesis.pdf)| |
|Grothendieck categories of enriched functors|Hassan Al Hwaeer| 2014 |[core](https://core.ac.uk/outputs/161880844)| |
| Analysis and construction of logical systems: a category-theoretic approach | Michał R. Przybyłek | 2014 | [pdf](https://www.mimuw.edu.pl/~mrp/phd_mrp.pdf) | |
|Generalization of Algebraic Operations via Enrichment|[[Christina Vasilakopoulou]]|2014|[arXiv](https://arxiv.org/abs/1411.3038)| |
|Three studies in higher category theory: fibrations, skew monoidal structures and excision of extremals|[[Mitchell Buckley]]|2015|[Macquarie repository](http://hdl.handle.net/1959.14/1068202)| |
|Synthetic Lie theory|Matthew Burke|2015|[Macquarie repository](http://hdl.handle.net/1959.14/1068205)| |
|Fibrations of predicates and bicategories of relations|[[Finn Lawler]]|2015|[arXiv](https://arxiv.org/abs/1502.08017)| |
|Sketches in higher category theory|[[Remy Tuyeras|Remy Vincent Tuyeras]]|2015|[Macquarie repository](http://hdl.handle.net/1959.14/1061373)| |
|A higher categorical approach to Giraud's non-abelian cohomology|[[Alexander Campbell]]|2016|[Macquarie repository](http://hdl.handle.net/1959.14/1261186)| |
|The Algebra of Open and Interconnected Systems|Brendan Fong|2016|[arXiv](https://arxiv.org/abs/1609.05382)| |
|Oplax actions and enriched icons with applications to coalgebroids and quantum categories|Ramón Abud Alcalá|2017|[Macquarie repository](http://hdl.handle.net/1959.14/1267344)| |
|Tangent bundles, monoidal theories, and Weil algebras|Poon Leung|2017|[Macquarie repository](http://hdl.handle.net/1959.14/1260669)| |
|The Way of the Dagger|[[Martti Karvonen]]|2018|[arXiv](https://arxiv.org/abs/1904.10805)| |
|Morphisms of 2-dimensional structures with applications|Branko Nikolić|2018|[Macquarie repository](http://hdl.handle.net/1959.14/1266304)| |
|Compact symmetric multicategories and the problem of loops|Sophie Raynor|2018|[Aberdeen repository](https://abdn.alma.exlibrisgroup.com/discovery/delivery/44ABE_INST:44ABE_VU1/12153018780005941)| |
|Homotopy theory of Grothendieck ∞-groupoids and ∞-categories|Edoardo Lanari|2019|[Macquarie repository](http://hdl.handle.net/1959.14/1269609)| |
|Restriction presheaves and restriction colimits|Daniel Lin|2019|[Macquarie repository](http://hdl.handle.net/1959.14/1269712)| |
|Structures in two dimensional category theory and applications to polynomial functors|Charles Walker|2019|[Macquarie repository](http://hdl.handle.net/1959.14/1270211)| |
|The Scott adjunction|[[Ivan Di Liberti]]|2020|[arXiv](https://arxiv.org/abs/2009.07320)| |
|The Gray tensor product for 2-quasi-categories|[[Yuki Maehara]]|2020|[Macquarie repository](http://hdl.handle.net/1959.14/1284564)| |
|Toposes of Monoid Actions|[[Morgan Rogers]]|2021|[arXiv](https://arxiv.org/abs/2112.10198)| |
|Monadic and Higher-Order Structure|[[Nathanael Arkor]]|2022|[Cambridge repository](https://www.repository.cam.ac.uk/handle/1810/338940)| on [[relative monads]], [[second-order algebraic theories]], and monad–theory correspondences in a [[proarrow equipment]] |
|Pseudomonads, Relative Monads and Strongly Finitary Notions of Multicategory|[[Gabriele Lobbia]]|2022|[Leeds repository](https://etheses.whiterose.ac.uk/30578/)| |
|Topics in the theory of enriched accessible categories|[[Giacomo Tendas]]|2022|[Macquarie repository](https://figshare.mq.edu.au/articles/thesis/Topics_in_the_theory_of_enriched_accessible_categories/21638081/1)| |
|Introspective Theories and Geminal Categories|[[Sridhar Ramesh]]|2023|[eScholarship repository](https://escholarship.org/uc/item/3mn0c475)| on [[Löb's theorem]] and the [[Gödel incompleteness theorem]] |
|Scalar Enrichment and Cotraces in Bicategories|Callum William Reader|2023|[White Rose repository](https://etheses.whiterose.ac.uk/33327/)| on [[traces]] and [[cotraces]] in [[monoidal bicategories]] |
|Doctrinal and Groupoidal Representations of Classifying Topoi|Joshua L. Wrigley|2024|[PDF](https://jlwrigley.github.io/thesis/thesis_wrigley.pdf)| |

## Related pages

* [[list of journals publishing category theory]]
* [[list of lost manuscripts in category theory]]
* [[list of category theory papers that never appeared]]


lists of journals

# Lists of journals

Here are some lists of journals, generally collected by subject or purpose.

* [Dan Isaksen's list of algebraic topology journals](http://s.wayne.edu/isaksen/algebraic-topology-journals/).  Comprises journals whose editorial boards include at least one algebraic topologist.

* [[list of journals publishing category theory]]

* [AMS Digital Mathematics Registry](https://mathscinet.ams.org/dmr/JournalList.html), a fairly comprehensive list of 2915 journals in mathematics and related areas.


little cubes operad

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

#Contents#
* table of contents
{:toc}

## Idea

The **little $k$-disk operad** or **little $k$-cubes operad** (to distinguish from the [[framed little n-disk operad]]) is the [[topological operad]]/[[(∞,1)-operad]] $E_k$ whose $n$-ary operations are parameterized by rectilinear disjoint embeddings of $n$ $k$-dimensional cubes into another $k$-dimensional cube.

When regarded as a [[topological operad]], the topology on the space of all such embedding is such that a continuous path is given by continuously moving the images of these little cubes in the big cube around.

Therefore the [[algebra over an operad|algebras over]] the $E_k$ operad are "$k$-fold monoidal" objects. For instance [[k-tuply monoidal (n,r)-category|k-tuply monoidal (n,r)-categories]].

The limiting [[E-∞ operad]] is a [[resolution]] of the ordinary commutative monoid operad [[Comm]]. Its algebras are [[commutative algebra in an (infinity,1)-category|homotopy commutative monoid objects]] such as $E_\infty$-[[E-infinity-ring|rings]].

## Definition

(...)

An [[algebra over an operad]] over $E_k$ is an [[Ek-algebra]].

### Presentation by enriched operads

(...)

**Remark**

Many models for $E_\infty$-operads in the literature are not in fact cofibrant in the [[model structure on operads]], but are $\Sigma$-cofibrant. By the therem at [[model structure on algebras over an operad]], this is sufficient for their categories of algebras to present the correct $\infty$-categories of [[E-∞ algebras]].

(...)

### As $\infty$-operads

\begin{definition}
\label{DefinitionViaRectilinearEmbeddings}

Given a [[natural number]] $k \geq 0$, write

$$
  \square^k
  \;\coloneqq\;
  (-1, 1)^k
$$

for the open [[cube]] of dimension $k$ (the $k$-fold [[product topological space]] of the [[open interval]] with itself). We will say that a [[continuous map]] $f \colon \square^k \to \square^k $
is a _rectilinear embedding_ if, with respect to the canonical [[coordinate functions]] on $(-1,1) \,\subset\, \mathbb{R}$, it is given by an [[affine function]], hence by a formula of the form

$$
  f (x_1 , \cdots , x_k )
  \;=\;
  (a_1 x_1 + b_1 , \cdots , a_k x_k + b_k )
$$

for some [[real numbers]] $a_i$ and $b_i$ , with $a_i \gt 0$.

More generally, if $S$ is a [[finite set]], call a map $\square^k  \times S \to \square^k$ is a *rectilinear embedding* if it is an [[open embedding]] whose restriction to each connected component of $\square^k\times S$ is rectilinear in the above sense.

Let $Rect(\square^k \times S, \square^k )$ denote the collection of all rectitlinear embeddings from $\square^k \times S$ into $\square^k$ . We will regard $Rect(\square^2\times S, \square^k )$ as a [[topological space]], topologized as a [[subspace]] of the [[topological product space]] $\big(\mathbf{R}^{2k} \big)^S$.

The spaces $Rect(\square^k \times \{1, . . . , n\}, \square^k)$ constitute the $n$-ary operations of a [[topological operad]], which we
will denote by ${}^{t} \mathbb{E}_k$ and refer to as the _little k-cubes operad_.

\end{definition}

This is Definition 5.1.0.1 in [[Higher Algebra]].

+-- {: .num_defn}
###### Definition
 We define a topological category ${}^t \mathbb{E}^\otimes_k$ as follows:

* The objects of ${}^t \mathbb{E}^\otimes_k$ are the objects $[n] \in Fin_*$.

* Given a pair of objects $[m], [n] \in {}^t \mathbb{E}^\otimes_k$ , a morphism from $[m]$ to $[n]$ in $t E^\otimes_k$ consists of the following data:

  * A morphism $\alpha : [m] \to [n]$ in $Fin_*$ .

  * For each $j \in [n]^\circ$ a rectilinear embedding $\square^k \times \alpha^{-1} \{j\} \to \square^k$.

* For every pair of objects $[m], [n] \in tE^\otimes_k$ , we regard $Hom_{tE^\otimes_k} ([m], [n])$ as endowed with the [[topological space|topology]] induced by the presentation

  $$
    Hom_{{}^t \mathbb{E}^\otimes_k} ([m], [n])
     =
    \coprod_{f \colon [m]\to [n]}
    \prod_{1\le j\le n} Rect(\times \alpha^{-1} \{j\},\square^k)
    \,.
  $$

* Composition of morphisms in ${}^t \mathbb{E}^\otimes_k$ is defined in the obvious way. We let $\mathbb{E}^\otimes_k$ denote the nerve of the topological category ${}^t \mathbb{E}^\otimes_k$.

  Corollary T.1.1.5.12 implies that $\mathbb{E}^\otimes_k$ is an
[[(infinity,1)-category|$\infty$-category]]. There is an evident [[forgetful functor]] from ${}^t \mathbb{E}^\otimes_k$ to the (discrete) category $Fin_*$ , which induces a functor $\mathbb{E}^\otimes_k \to N(Fin_* )$.

This is [[Higher Algebra]] Definition 5.1.0.2.

=--

## Properties
  {#Propeties}

### Grouplike monoid objects
  {#GrouplikeMonoid}

Let $\mathcal{X}$ be an [[(∞,1)-sheaf]]  [[(∞,1)-topos]]

and $X \colon Assoc \to \mathcal{X}$ be a monoid object in $\mathcal{X}$.  Say that $X$ is _grouplike_ if the composite

$$
  \Delta^{op} \to Ass \to \mathcal{X}
$$

(see 1.1.13 of [[Commutative Algebra]])

is a [[groupoid object in an (infinity,1)-category|groupoid object]] in $\mathcal{X}$.

Say an $\mathbb{E}[1]$-algebra object is grouplike if it is grouplike as an $Ass$-monoid. Say that an $\mathbb{E}[k]$-algebra object in $\mathcal{X}$ is grouplike if the restriction along $\mathbb{E}[1] \hookrightarrow \mathbb{E}[k]$ is.
Write

$$
  Mon^{gp}_{\mathbb{E}[k]}(\mathcal{X})
  \subset
  Mon_{\mathbb{E}[k]}(\mathcal{X})
$$

for the [[(∞,1)-category]] of grouplike $\mathbb{E}[k]$-monoid objects.

### $k$-fold delooping, monoidalness and $\mathbb{E}[k]$-action###
{: #MainResult}

The following result of ([Lurie](#Lurie)) makes precise for _parameterized [[∞-groupoid]]s_  -- for [[∞-stack]]s -- the general statement that $k$-fold [[delooping]] provides a correspondence between [[n-category|n-categories]] that have trivial [[k-morphism|r-morphism]]s for $r \lt k$ and  [[k-tuply monoidal n-category|k-tuply monoidal n-categories]].

+-- {: .un_theorem}
###### Theorem (k-tuply monoidal ∞-stacks)

Let $k \gt 0$, let $\mathcal{X}$ be an [[∞-stack]] [[(∞,1)-topos]] and let $\mathcal{X}_*^{\geq k}$ denote the [[full subcategory]] of the category $\mathcal{X}_{*}$ of pointed objects, spanned by those pointed objects that are $k-1$-[[connected]] (i.e. their first $k$ [[homotopy groups in an (∞,1)-topos|∞-stack homotopy groups]]) vanish. Then there is a canonical equivalence of [[(∞,1)-category|(∞,1)-categories]]

$$
  \mathcal{X}_*^{\geq k} \simeq Mon^{gp}_{\mathbb{E}[k]}(\mathcal{X}) \,.
$$

=--

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

This is [[Ek-Algebras|EKAlg, theorem 1.3.6.]].

=--

Specifically for $\mathcal{X} = Top$, this refines to the classical theorem by ([May](#May)).

+-- {: .un_theorem}
###### Theorem (May recognition theorem)

Let $Y$ be a [[topological space]] equipped with an action of the [[little cubes operad]] $\mathcal{C}_k$ and suppose that $Y$ is grouplike. Then $Y$ is homotopy equivalent to a $k$-fold loop space $\Omega^k X$ for some pointed topological space $X$.

=--

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

This is [[Ek-Algebras|EkAlg, theorem 1.3.16.]]

=--

Proofs independent of higher order categories can be extracted from the literature. See this [MO answer](http://mathoverflow.net/a/202345/2926) by Tyler Lawson for details.

### Stabilization hypothesis {#StabilizationHypothesis}

A proof of the [[stabilization hypothesis]] for [[k-tuply monoidal n-category|k-tuply monoidal n-categories]] is a byproduct of corollary 1.1.10 of ([Lurie](#Lurie)), stated as example 1.2.3.

### Additivity theorem {#AdditivityTheorem}

It has been long conjectured that it should be true that when suitably defined, there is a tensor product of $\infty$-operads such that

$$
  \mathbb{E}_k \otimes \mathbb{E}_{k'} \simeq \mathbb{E}_{k + k'}
  \,.
$$

This is discussed and realized in section 1.2. of ([Lurie](#Lurie)). The tensor product is defined in appendix B.7.

### Relation to Fulton-MacPherson operad
  {#RelationToFultonMacPhersonOperad}

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

The [[Fulton-MacPherson operad]] is [[weak equivalence|weakly equivalent]] in the [[model structure on operads]] with respect to the [[classical model structure on topological spaces]], to the [[little n-disk operad]]

=--

([Salvatore 01, Prop. 4.9](#Salvatore01), summarized as [Lambrechts-Volic 14, Prop. 5.6](#LambrechtsVolic14))

### Cohomology: Formality

[[the little n-disk operad is formal]]

### Homology: Poisson operads

For an $E_k$-operad in a [[category of chain complexes]], its [[homology]] is the [[Poisson operad]] $P_{k}$.

See for instance ([Costello](#Costello)) and see at _[[Poisson n-algebra]]_.

## Examples

Explicit models of $E_\infty$-operads include

* [[Barratt-Eccles operad]]

(...)

## Related concepts

* [[configuration space of points]], [[Fulton-MacPherson operad]]

* [[factorization algebra]]

* [[A-∞ operad]]

* [[E-∞ operad]]

* [[L-∞ operad]]

## References

A comprehensive reference for many known models is

* [[André Beuckelmann]], [[Ieke Moerdijk]], _A small catalogue of $E_n$-operads_, [arXiv:2212.03842v1](https://arxiv.org/abs/2212.03842v1).

Review includes

* {#Fresse18} [[Benoit Fresse]], _Little discs operads, graph complexes and Grothendieck--Teichmüller groups_, in [[Haynes Miller]] (ed.) _[[Handbook of Homotopy Theory]]_ ([arXiv:1811.12536](https://arxiv.org/abs/1811.12536))

A standard textbook reference is chapter 4 of

* {#May} [[Peter May]], _The geometry of iterated loop spaces_ ([pdf](http://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf))

The equivalence to the [[Fulton-MacPherson operad]] is due to

* {#Salvatore01} [[Paolo Salvatore]], _Configuration spaces with summable labels_, Cohomological methods in homotopy theory. Birkhäuser, Basel, 2001. 375-395.

Proof that [[the little n-disk operad is formal]] was sketched by [[Maxim Kontsevich]] and spelled out in

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

[[John Francis]]' work on $E_n$-actions on $(\infty,1)$-categories is in

* [[John Francis]] PhD thesis ([web](http://dspace.mit.edu/handle/1721.1/43792))

This influenced the revised version of

* [[Jacob Lurie]], _[[Commutative Algebra]]_ ([arXiv](http://arxiv.org/abs/math/0703204))

and is extended to include a discussion of traces and centers in

* [[David Ben-Zvi]], [[John Francis]], [[David Nadler]], _Integral transforms and Drinfeld Centers in Derived Geometry_ ([arXiv](http://arxiv.org/abs/0805.0157))

  (see also [[geometric ∞-function theory]])

A detailed discussion of $E_k$ in the context of [[(∞,1)-operads]] is in

* [[Jacob Lurie]], _$\mathbb{E}[k]$-[[Ek-Algebras|Algebras]]_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-VI.pdf))
{#Lurie}

An elementary computation of the [[homology]] of the little $n$-disk operad in terms of _solar system calculus_ is in

* [[Dev Sinha]], _The homology of the little disks operad_ ([arXiv:math/0610236](http://arxiv.org/abs/math/0610236))

For the relation to Poisson Operads see

* [[Kevin Costello]], [[Owen Gwilliam]], _Factorization algebras in perturbative quantum field theory : $P_0$-operad_

[[!redirects little cube operad]]
[[!redirects little disk operad]]
[[!redirects little 2-disk operad]]

[[!redirects little k-cube operad]]
[[!redirects little k-cubes operad]]
[[!redirects little n-cube operad]]
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[[!redirects little n-disc operad]]
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[[!redirects E-k operads]]
[[!redirects E-n operads]]
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[[!redirects En operads]]
[[!redirects En-operad]]
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[[!redirects Ek operads]]
[[!redirects Ek-operad]]
[[!redirects Ek-operads]]

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[[!redirects E_n operads]]
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little etale topos

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#Contents#
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## Definition

For $X$ a [[scheme]] or more generally an [[algebraic stack]], let $X_{et}$ be its [[small étale site]]: the full [[subcategory]] of the [[slice category]] $Sch/X$ on the [[étale morphisms]] equipped with the induced [[étale topology]].

Then the [[category of sheaves]] $Sh(X_{et})$ is called the **[[étale topos]]** of $X$. (See there for more) This is the [[little topos]]-incarnation of $X$.

## Related concepts

* [[étale site]]

* [[étale (infinity,1)-site]]

[[!redirects little étale toposes]]
[[!redirects little étale topos]]


little site

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# Little sites
* table of contents
{: toc}

## Idea

Let $C$ be a category with a [[pretopology]] $J$ (i.e. a [[site]]) and $a$ an object of $C$. As an analogy with sheaves on a topological space $X$, which are defined on the site $Op(X)$ of open sets of $X$, we can try to define sheaves on $a$, using the elements of covering families of $a$ from $J$. This is called the **little site** of $a$, in contrast to the [[big site]] of $a$ which is the slice category $C/a$ with its induced topology.

The [[topos]] of [[sheaves]] on the little site is the [[petit topos]] of $a$.

A little site may sometimes be called a _[[small site]]_, but it\'s probably best to save that name for a site which is a [[small category]].

## Definition

>[[David Roberts]]: The following is experimental, use at own risk, although I'm sure it has been thought about before.

Consider the subcategory $J/a$ of $C/a$ with objects $u_0 \to a$ such that $u_0$ is a member of some covering family $U = \{u_i \to a\}$. Given two such objects $u_0 \to a$, $v_0 \to a$, and  covering families $U$, $V$ that contain them, there is a covering family $W = UV$ which is the pullback (or at least a weak pullback) of $U$ and $V$ in $C$. There is then some element $w$ of $W$ such that there is a square
$$\array{
w & \to & v_0 \\
\downarrow & & \downarrow \\
u_0  &\to  & a
}
$$
so $J/a$ is 'a bit like' the category of opens of a space (it's probably [[filtered category|cofiltered]], but I haven't checked that there are weak equalisers).

Now the morphisms of $J/a$ are those triangles
$$\array{
v_0 & \to & u_0 \\
& \searrow& \downarrow \\
  &  & a
}\, .
$$
such that $v_0 \to u_0$ is an element of a covering family of $u_0$, so the arrows $w \to u_0$ and $w \to v_0$ really are morphisms of $J/a$. Then we say a covering family of $u_0\to a$ is a collection of triangles that, when we forget the maps to $a$, form a covering family of $u_0$ in $C$. This is at the very least a [[coverage]], and so we have a site.

>To be continued...

## Related concept

* **little site**

* [[big site]]

[[!redirects little site]]
[[!redirects little sites]]

[[!redirects petit site]]
[[!redirects petit sites]]


little string

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#Contents#
* table of contents
{:toc}

## Idea

There is, by the [[brane scan]], a [[superstring]] propagating in the 6-dimensional [[worldvolume]] of an [[NS5-brane]]. This is often called the _little string_ of [[little string theory]], with respect to the [[superstrings]] in the ambient 10-dimensional [[heterotic supergravity]] and [[type II supergravity]] that in turn the [[NS5-brane]] is propagating in.

## Related concepts

* [[little string theory]]

[[!include brane scan]]

## References

See also

* Wikipedia, _[Little string theory](http://en.wikipedia.org/wiki/Little_string_theory)_

On [[D=3 N=4 super Yang-Mills theories]] with [[compact hyperkähler manifold]] [[Coulomb branches]] obtained by [[KK-compactification]] of [[little string theories]]:

* {#Intriligator99} [[Kenneth Intriligator]], _Compactified Little String Theories and Compact Moduli Spaces of Vacua_, Phys. Rev. D61:106005, 2000 ([arXiv:hep-th/9909219](https://arxiv.org/abs/hep-th/9909219))

[[!redirects little strings]]


little string theory

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#Contents#
* table of contents
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## Idea

A limit of the [[worldvolume]] theory of the [[NS5-brane]], which is a [[string theory]] of "[[little strings]]" in six dimensions.

## Related concepts

* [[D=6 N=(2,0) SCFT]]

[[!include table of branes]]

## References

### General

The original reference is

* [[Nathan Seiberg]], _New theories in six dimensions and matrix description of M-theory on $T^5$ and $T^5/\mathbb{Z}_2$_ Phys. Lett. B408 (1997) 98–104 ([arXiv:hep-th/9705221](https://arxiv.org/abs/hep-th/9705221))

Review is in

* [[Ofer Aharony]], _A brief review of "little string theories"_ , Class. Quant. Grav. 17 (2000) 929–938,
([arXiv:hep-th/9911147](http://arxiv.org/abs/hep-th/9911147)).

* [[David Kutasov]], _Introduction to Little String Theory_, Lectures 2001 ([pdf](http://users.ictp.it/~pub_off/lectures/lns007/Kutasov/Kutasov.pdf))

Little string theories on [[heterotic string theory]] [[NS5-branes]] with $N = (1,0)$ [[supersymmetry]]:

* E. Gava, K.S. Narain, M.H. Sarmadi, _Little String Theories in Heterotic Backgrounds_, Nucl.Phys. B626 (2002) 3-25 ([arXiv:hep-th/0112200](http://arxiv.org/abs/hep-th/0112200))

and via [[F-theory]] methods:

* [[Michele Del Zotto]], Muyang Liu, Paul-Konstantin Oehlmann, *6D Heterotic Little String Theories and F-theory Geometry: An Introduction* [[arXiv:2303.13502](https://arxiv.org/abs/2303.13502)]

The [[T-duality]] between type IIA and type IIB little strings is discussed in

* Jungmin Kim, Seok Kim, Kimyeong Lee, _Little strings and T-duality_, JHEP  2016:170 ([arXiv:1503.07277](https://arxiv.org/abs/1503.07277))

Construction within [[F-theory]] is discussed in

* Lakshya Bhardwaj, [[Michele Del Zotto]], [[Jonathan Heckman]], [[David Morrison]], [[Tom Rudelius]], [[Cumrun Vafa]], _F-theory and the Classification of Little Strings_ ([arXiv:1511.05565](http://arxiv.org/abs/1511.05565))

See also

* [[Mina Aganagic]], _Some mathematical applications of little string theory_, talk at [Strings 2016](http://ymsc.tsinghua.edu.cn:8090/strings/) ([pdf](http://ymsc.tsinghua.edu.cn:8090/strings/slides/8.1/Aganagic%20v2.pdf))

* Joonho Kim, Kimyeong Lee, _Little strings on $D_n$ orbifolds_ ([arXiv:1702.03116](https://arxiv.org/abs/1702.03116))

On [[D=3 N=4 super Yang-Mills theories]] with [[compact hyperkähler manifold]] [[Coulomb branches]] obtained by [[KK-compactification]] of [[little string theories]]:

* {#Intriligator99} [[Kenneth Intriligator]], _Compactified Little String Theories and Compact Moduli Spaces of Vacua_, Phys. Rev. D61:106005, 2000 ([arXiv:hep-th/9909219](https://arxiv.org/abs/hep-th/9909219))

Relation of little string theory to the [[quantum geometric Langlands correspondence]] is discussed in

* {#AganagicFrenkelOkounkov17} [[Mina Aganagic]], [[Edward Frenkel]], [[Andrei Okounkov]], _Quantum $q$-Langlands Correspondence_ ([arXiv:1701.03146](https://arxiv.org/abs/1701.03146))

also

* Babak Haghighat, Rui Sun, _M5 branes and Theta Functions_ ([arXiv:1811.04938](https://arxiv.org/abs/1811.04938))

Discussion with emphasis on terminology from [[higher gauge theory]]:

* [[Michele Del Zotto]], [[Kantaro Ohmori]], *2-Group Symmetries of 6D Little String Theories and T-Duality*, Ann. Henri Poincaré 22, 2451–2474 (2021) ([arXiv:2009.03489](https://arxiv.org/abs/2009.03489),
[doi:10.1007/s00023-021-01018-3](https://doi.org/10.1007/s00023-021-01018-3))

### Open D$p$-brane theories

For light [[D-branes]] ending on NS5-branes (see [there](NS5-brane#DBranesEndingOnNS5Branes))

* {#GopakumarMinwallaSeibergStrominger00} [[Rajesh Gopakumar]], [[Shiraz Minwalla]], [[Nathan Seiberg]], [[Andrew Strominger]], _OM Theory in Diverse Dimensions_, JHEP 0008:008, 2000 ([arXiv:hep-th/0006062](https://arxiv.org/abs/hep-th/0006062))

* {#Harmark00} Troels Harmark, _Open Branes in Space-Time Non-Commutative Little String Theory_,  Nucl.Phys. B593 (2001) 76-98 ([arXiv:hep-th/0007147](https://arxiv.org/abs/hep-th/0007147))

* {#Lu06} [[Jian Xin Lu]], *$(1 + p)$-Dimensional Open $D(p - 2)$ Brane Theories*, JHEP 0108:049 (2001) [[arXiv:hep-th/0102056](https://arxiv.org/abs/hep-th/0102056), [doi:10.1088/1126-6708/2001/08/049](https://doi.org/10.1088/1126-6708/2001/08/049)]

See also

* Dan Israel, Ari Pakman, Jan Troost, _D-branes in Little String Theory_, Nucl.Phys. B722 (2005) 3-64 ([arXiv:hep-th/0502073](https://arxiv.org/abs/hep-th/0502073))

### Via the plane wave matrix model

On a double-scaling limit of the [[BMN matrix model]] in which aspects of little string theory become visible:

* [[Hai Lin]], [[Juan Maldacena]], _Fivebranes from gauge theory_, Phys. Rev. D **74** 084014 (2006) [[arXiv:hep-th/0509235](https://arxiv.org/abs/hep-th/0509235), [doi:10.1103/PhysRevD.74.084014](https://doi.org/10.1103/PhysRevD.74.084014)]

* Henry Ling, Ali Reza Mohazab, Hsien-Hang Shieh, Greg van Anders, [[Mark Van Raamsdonk]], *Little String Theory from a Double-Scaled Matrix Model*, JHEP 0610:018 (2006) [[arXiv:hep-th/0606014](https://arxiv.org/abs/hep-th/0606014), [doi:10.1088/1126-6708/2006/10/018](https://doi.org/10.1088/1126-6708/2006/10/018)]

* [[Yuhma Asano]], [[Goro Ishiki]], [[Takaki Matsumoto]], [[Shinji Shimasaki]], [[Hiromasa Watanabe]], *On the existence of the NS5-brane limit of the plane wave matrix model*, PTEP 2023 (2023) 4, 043B01 [[arXiv:2211.13716](https://arxiv.org/abs/2211.13716), [doi:10.1093/ptep/ptad042](https://doi.org/10.1093/ptep/ptad042)]

[[!redirects little string theories]]


Littlewood-Richardson coefficient

When multiplying [[Schur functions]]:

$$s_\lambda\cdot s_\mu= \sum_\nu c^\nu_{\lambda,\mu}s_\nu$$

The coefficients, $c^\nu_{\lambda,\mu}$, are called  the _Littlewood–Richardson coefficients_.

There are various combinatorial models for the [[Littlewood]]–[[Richardson]] coefficients.

##References

* [Littlewood–Richardson rule entry in the symmetric functions catalog](https://www2.math.upenn.edu/~peal/polynomials/schur.htm#schurLittlewoodRichardson)

* [Littlewood-Richardson entry in the symmetric functions catalog](https://www2.math.upenn.edu/~peal/polynomials/littlewoodRichardson.htm)

[[!redirects Littlewood-Richardson coefficients]]
[[!redirects Littlewood-Richardson rule]]


Liviu Nicolaescu

* [webpage](https://www3.nd.edu/~lnicolae/)

## Selected writings

* [[Liviu Nicolaescu]], _The many faces of the Gauss-Bonnet theorem_ ([pdf](http://www.nd.edu/~lnicolae/GradStudSemFall2003.pdf))

  (on the [[Gauss-Bonnet theorem]])

* [[Liviu Nicolaescu]], _Lectures on the Geometry of Manifolds_, 2018 ([pdf](https://www3.nd.edu/~lnicolae/Lectures.pdf))

  (on [[differential geometry]] of [[smooth manifolds]])

## Related $n$Lab entries

* [[differential geometry]]

* [[smooth manifold]]

* [[Gauss-Bonnet theorem]]

* [[Euler form]]


Lluís Bacardit

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=136551)

## Selected writings

 On [[group action|actions]] of [[braid groups]] (such as via [[automorphisms]] of  [[free groups]]):

* [[Lluís Bacardit]], [[Warren Dicks]], *Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue*, Groups Complexity Cryptology **1** (2009) 77-129 [[arXiv:0705.0587](https://arxiv.org/abs/0705.0587), [doi;10.1515/GCC.2009.77](https://doi.org/10.1515/GCC.2009.77)]

[[!redirects Lluis Bacardit]]

category: people


LNL polycategory

[[!redirects LNL Polycategory]]
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# Idea

A _linear-non-linear polycategory_ or simply _LNL polycategory_ is a generalization of a symmetric [[polycategory]] to include "non-linear" objects in the same way that [[linear-non-linear logic]] generalizes [[symmetric multicategories]]. Just as linear-non-linear logic provides a setting in which the exponential [[comonad]] ! in intuitionistic linear logic can be decomposed into an [[adjunction]], LNL polycategories provide a setting in which the exponential comonad ! and the monad ? in classical linear logic can each be decomposed into adjunctions.

In addition to the study of classical linear-logic, LNL polycategories can be used as a unified setting in which to study [[multicategory]] and [[polycategory]]-like structures as "LNL [[doctrines]]".

# Doctrines

See the original paper for a definition of LNL doctrine.

The following is an incomplete list of structures that can be defined as LNL doctrines:

* [[cartesian multicategories]]
* [[symmetric multicategories]]
* [[symmetric polycategories]]
* [[linear-non-linear logic]]
* [[skew-multicategories]]
* [[call-by-push-value]]
* [[Freyd multicategories]]
* [[adjunctions]]
* models of [[classical linear logic]]

# References

Introduced in

* [[Michael Shulman]], _LNL polycategories and doctrines of linear logic_, [arxiv](https://arxiv.org/abs/2106.15042)


Loc

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# Contents
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{: toc}

## Definition

__$Loc$__ or **$Locale$** is the [[category]] whose [[objects]] are [[locales]] and whose [[morphisms]] are [[continuous maps]] between locales.  By definition, this means that $Loc$ is the [[opposite category]] of [[Frm]], the category of [[frames]].

$Loc$ is used as a substitute for [[Top]] if one wishes to do [[topology]] with locales instead of standard [[topological spaces]].

$Loc$ is naturally a [[(1,2)-category]]; its [[2-morphisms]] are the pointwise ordering of [[frame]] homomorphisms.

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

The [[2-category]] [[Locale]] has

* as [[object|objects]] $X$ [[frame|frames]] $Op(X)$;

* as [[morphisms]] $f : X \to Y$ frame homomorphisms $f^* : Op(Y) \to Op(X)$;

* a unique [[2-morphism]] $f \Rightarrow g$ whenever for all $U \in Op(Y)$ we have a (then necessarily unique) morphism $f^* U \to g^* U$.

=--

(For instance [Johnstone, C1.4, p. 514](#Johnstone).)

## Properties

For any [[base topos]] $S$ the [[2-category]] $Loc(S)$ of [[internal locales]] in $S$ is [[equivalence of categories|equivalent]] to the [[subcategory]] of the [[over topos|slice]] of [[Topos]] over $S$ on the [[localic geometric morphism|localic geometric morphisms]]. See there for more details.

See [[locale]] for more properties.

The [[2-functor]] that forms [[category of sheaves|categories of sheaves]]

$$
  Sh : Locale \to Topos
$$

exhibits $Locale$ as a [[full sub-2-category]] of [[Topos]]. See _[[localic reflection]]_ for more on this.

## References

For instance Section C1 of

*  [[Peter Johnstone]], _[[Elephant|Sketches of an elephant: a topos theory compendium]]_.
 {#Johnstone}

[[!redirects Loc]]
[[!redirects Locale]]
[[!redirects category of locales]]
[[!redirects the category of locales]]

category: category


local abelian category

An [[abelian category]] is local if it has the smallest topologizing subcategory.

An alternative description is the following.

Recall the [[preorder]] $\succ$ on the collection of nonzero objects of an abelian category $A$. Namely, $M\succ N$ if $N$ is a [[subquotient]] of finitely many copies of $M$. In other words, $M\succ N$ if there exists a positive number $k$ and a subobject $U$ of a direct sum $\oplus_{i = 1}^k M$ of $k$ copies of $M$ and an epimorphism from $U$ to $N$.

A nonzero object $M$ of an [[abelian category]] $A$ is said to be __quasifinal__ if $M\succ N$ for any nonzero object $N$ of $A$. The quasifinal object clearly belongs to the [[spectrum of an abelian category|spectrum of the abelian category]] $A$. An abelian category is called __local__ if it has a quasifinal object.

Local abelian categories in the theory of noncommutative spectra (in the sense of [[noncommutative algebraic geometry]]) have a role similar to local rings in the theory of usual spectra of rings and in the theory of locally ringed spaces. Moreover there is a "center" functor from the category of topological spaces equipped with stacks of local categories to the locally (commutatively) ringed spaces. It is basically induced by the construction of the [[center of abelian category]] which to an essentially small abelian category attaches the ring of endomorphisms of the identical functor, thanks to the functoriality of that construction with respect to the localizations.

* A. L. Rosenberg, _Topics in noncommutative algebraic geometry, homological algebra and K-theory_, preprint MPIM Bonn 2008-57 [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3589) (mainly Lec. 1.6 around page 30)

* [[A. L. Rosenberg]], _The spectrum of abelian categories and reconstructions of schemes_, in Rings, Hopf Algebras, and Brauer groups, Lectures Notes in Pure and Appl. Math. __197__, Marcel Dekker, New York, 257--274, 1998; MR99d:18011; and Max Planck Bonn preprint _Reconstruction of Schemes_, MPIM1996-108, [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3948) (1996).

* [[A. L. Rosenberg]], _Noncommutative local algebra_,  Geom. Funct. Anal. __4__  (1994),  no. 5, 545--585.