localized coinvariant

Let $B$ be a $k$-bialgebra and $E$ a right $B$-comodule algebra with coaction $\rho:E\to E\otimes B$. Let $j : E\to E_\mu$ be a biflat affine localization in the sense that it is a homomorphism of rings (affiness), and that the extension of scalars along $j$ is an exact localization functor whose right adjoint is also exact. If $B$ is a Hopf algebra of coordinate functions on an alegbraic $k$-group where $k$-is a field, and $E$ is the coordinate algebra of an affine $k$-variety with regular $G$-action, then we can not restrict the action to an arbitrary open sets, but only to the $G$-invariant open sets. In our case, this would mean that the  coaction does not necessarily extend to the localization $E_\mu$ respecting the algebra structure.

For this reason, one introduces the class of $\rho$-compatible localizations: a localization of a $B$-comodule algebra $(E,\rho)$ is **coaction compatible localization** if the coaction extends to an algebra morphism $\rho_\mu:E_\mu\to E_\mu\otimes B$ such that the diagram

$$\array{
E&\stackrel{\rho}\to &E\otimes B\\
j \downarrow &&\downarrow j\otimes B\\
E_\mu&\stackrel{\rho_\mu}\to &E_\mu\otimes B
}$$

commmutes. If such an extension exists it is unique and $E_\mu$ is automatically a coaction.

The algebra of **localized coinvariants** is the subalgebra $E_\mu^{co B}\subset E_\mu$ of $\rho_\mu$-[[coinvariants]] in $E_\mu$. It is important to notice that taking coinvariants and localization do not commute: there is no localization
$E^{co B}\to E^{co B}_\mu$ in general. A $B$-comodule algebra which is $\rho$-compatible may have many new $B$-coinvariants which do not come in any sense as images under some sort of a localization from the $B$-coinvariants of $E$. This localization can be used to get more local information on the noncommutative quotient space for coactions of Hopf algebra. In some cases one can construct a cover of the quotient spaces by affine charts whose coordinate algebras are the algebras of localized coinvariant for varying compatible localizations of the original comodule algebra; this can be further generalized by replacing the comodule algebra by a more general object which is also glued from the charts, but this requires consideration of the compatibility at the level of the localization functors and the actions are then globally expressed via an action of a monoidal category (which is still locally induced by a coaction).

The notion of compatibility of coactions and Ore localizations and the localized coinvariants have been introduced in 1997 when starting working on

* Z. Škoda, _Coset spaces for quantum groups_, thesis, Univ. of Wisconsin 2002.

This work is outlined in

* Z. Škoda, _Localizations for construction of quantum coset spaces_, in "Noncommutative geometry and Quantum groups", W.Pusz,  P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265--298, Warszawa 2003, [math.QA/0301090](http://arxiv.org/abs/math.QA/0301090).

and some aspects of it are generalized in

* Z. Škoda, _Some equivariant constructions in noncommutative algebraic geometry_, Georgian Mathematical Journal 16 (2009), No. 1, 183--202, [arXiv:0811.4770](http://arxiv.org/abs/0811.4770).

* Z. Škoda, _Compatibility of (co)actions and localization_,  [arXiv:0902.1398](http://arxiv.org/abs/0902.1398).


localized endomorphism

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#Contents#
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## Idea

Given a [[local net of observables]]

$$
  \mathcal{A} : Open(X) \to Algebras
$$

a [[natural transformation|natural]] [[associative algebra|algebra]] [[endomorphism]]

$$
  \rho : \mathcal{A} \to \mathcal{A}
$$

is called _local_ or _localized_ if outside of a bounded region of [[spacetime]] $X$ it is the [[identity]].

Localized endomorphisms play a central role in [[DHR superselection theory]].

## Definition

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

An endomorphim $\rho$ is **[[localized endomorphism|localized]]** or **localizable** if there is a bounded open set $\mathcal{O} \in \mathcal{J}$ such that $\rho$ is the identity on the algebra of the causal complement $\mathcal{A}(\mathcal{O}^{\perp})$. Such an endomorphism is **localized in $\mathcal{O}$**.

=--

[[!redirects local endomorphism]]


localizer

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#### Homotopy theory
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#### Locality and descent
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#Contents#
* table of contents
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## Idea

Given a [[category]] $\mathcal{C}$, by a _localizer_ in $\mathcal{C}$ (or on $\mathcal{C}$) is often meant to be a [[class]] $W \subset Mor(\mathcal{C})$ of [[morphisms]] in $\mathcal{C}$ such that $(\mathcal{C},W)$ is a [[category with weak equivalences]], intended as the input datum for the ([[simplicial localization|simplicial]]) [[localization]] of $\mathcal{C}$ at $W$. Typically there are some extra conditions imposed on what constitutes a localizer in a given context.

## Examples

The term appears for instance in the discussion of [[Cisinski model structures]], see that _[Cisinski model structure -- Localizers](http://ncatlab.org/nlab/show/Cisinski+model+structure#ALocalizers)_.

It also appears as the notion of _[[basic localizers]]_ on [[Cat]].

## Related concepts

* [[localizing subcategory]]

* [[modelizer]]

[[!redirects localizers]]


localizing subcategory

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#### Notions of subcategory
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#Contents#
* table of contents
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## Definition

A [[subcategory]] $T$ of an [[abelian category]] $A$ is a __localizing subcategory__ (French: *sous-catégorie localisante*) if there exists an [[exact functor|exact]] [[localization]] functor $Q:A\to B$ having a [[right adjoint]] $B\hookrightarrow A$ (which is automatically then [[full and faithful functor|fully faithful]]) and for which $T = Ker Q$ i.e. the [[full subcategory]] of $A$ generated by objects $a\in Ob(A)$ such that $Q(a) = 0$.

One sometimes says that $T$ is the localizing subcategory associated with quotient (or localized) category $B$ (which is then equivalent to the [[Serre quotient category]] $A/T$).

## Properties

### In general abelian categories

A localizing subcategory $Ker Q$ determines $Q:A\to B$ up to an [[equivalence of categories]] commuting with the localization functors; it is the quotient functor $Q_T : A\to A/T$ to the Serre [[quotient category]] $A/T$. The right adjoint $S_T : A/T\to A$ to $Q_T$ is usually called the __section functor__. Denote the unit of the adjunction $\eta : Id_A\to S_T Q_T$. Then for $X\in Ob A$, $Ker \eta_X\subset X$ is the maximal subobject of $X$ contained in $X$, called the $T$-torsion part of $X$. An object $X$ is $T$-torsionfree if the $T$-torsion part of $X$ is $0$, i.e. $\eta_X$ is a monomorphism, and $X$ is __$T$-closed__ ([[local object]] with respect to morphisms inverting under $Q$) if $\eta_X$ is an isomorphism.
The section functor
$S_T$ realizes the equivalence of categories between $A/T$ and the full subcategory of $A$ generated by $T$-closed objects.

A [[thick subcategory]] $T\subset A$ (in strong sense) is localizing iff every object $M$ in $A$ has the largest subobject among the subobjects from $T$ *and* if the only subobject from $T$ is a [[zero object]] then there is a monomorphism from $M$ to a $T$-closed object.

Localizing subcategories are precisely those which are [[topologizing subcategory|topologizing]], closed under extensions and closed under all colimits which exist in $A$. In other words, $A$ and $A''$ are in $T$ iff
any given extension $A'$ of $A$ by $A''$ is in $T$; and it is closed under colimits existing in $A$.

A strictly [[full subcategory]] $T\subset A$ is localizing iff the class $\Sigma_T$ of all $f\in Mor A$ for which $Ker f\in Ob T$ and $Coker f\in Ob T$ is *precisely* the class of all morphisms inverted by some left exact localization admiting right adjoint.

A [[reflective subcategory|reflective]] (strongly) [[thick subcategory]] $T$ is always localizing and the converse holds if $A$ has injective envelopes.

If $A$ admit [[colimits]] and has a set of generators, then any localizing subcategory $T\subset A$,and the [[quotient category|Serre quotient]] $A/T$, admit colimits and has a set of generators (Gabriel, Prop. 9) and the quotient functor $Q_T : A\to A/T$ preserves colimits (in the same [[Grothendieck universe]] if we work with universes). The generators of $A/T$ are the images of the generators in $A$ under the quotient functor $Q_T$. If $A$ is [[locally noetherian category|locally noetherian]] abelian category then any localizing subcategory $T\subset A$ and the quotient category $A/T$ are locally noetherian (Gabriel, Cor. 1). (If $A$ is locally finitely presented, $A$ and $A/T$ are locally finitely presented.?) If $A$ is locally noetherian and $A_{Noether}\subset A$ is the full subcategory of noetherian objects in $A$, then the assignment which to any localizing subcategory $T\subset A$ assigns the full subcategory $T_{Noether}\subset T$ of noetherian objects in $T$ is the bijection between the localizing subcategories in $A$ and (strongly) thick subcategories in $A_{Noether}$ (Gabriel Prop. 10).

### In locally finitely presentable abelian categories

In this setup, there is a bijective correspondence between hereditary torsion theories, localizing subcategories and exact localizations having right adjoint.

### In Grothendieck categories

For a strongly [[thick subcategory]] (i.e. weakly [[Serre subcategory]]) $T$ in a [[Grothendieck category]] $A$ the following are equivalent:

(i) $T$ is localizing

(ii) $T$ is closed under coproducts

(iii) $T$ is cocomplete (closed under arbitrary colimits)

(iv) any colimit of objects in $T$ in $A$ belongs to $T$

(v) the corresponding localizing functor $F: A\to A/T$ preserves colimits

### In $_R Mod$.

There is a canonical correspondence between [[topologizing filter]]s of a unital ring and localizing subcategories in the category $R$[[Mod]] of (say left) unital [[modules]] of the ring.

## Related concepts

* [[anti-modal type]]

## Literature

The notion is introduced by Gabriel:

* [[Pierre Gabriel]], [[Des catégories abéliennes]], Bulletin de la Société Mathématique de France __90__ (1962), 323-448 ([numdam](http://www.numdam.org/item?id=BSMF_1962__90__323_0))

* [[Francis Borceux]], Section II.1 in: *[[Handbook of Categorical Algebra]]*

* [[Henning Krause]], _The spectrum of a locally coherent category_, J. Pure Appl. Algebra __114__ (1997), 259-271, [pdf](http://www2.math.uni-paderborn.de/fileadmin/Mathematik/AG-Krause/publications_krause/publications_krause_mathscinet/MR1426488.pdf)

* Ryo Takahashi, _On localizing subcategories of derived categories_ (2000) ([pdf](http://math.shinshu-u.ac.jp/~takahasi/pdf/hcls6.pdf))

A comprehensive (and very reliable) source is

* N. Popescu, _Abelian categories with applications to rings and modules_, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. [MR0340375](http://www.ams.org/mathscinet-getitem?mr=0340375)

[[!redirects localizing subcategories]]


locally

> This page is concerned with notions of locality in [[category theory]]. For notions in [[physics]] see instead e.g. *[[causal locality]]* or *[[local field theory]]* (which overlaps with a category-theoretic meaning via the notion of *[[extended functorial field theories]]*).

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#### Category theory
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#### Mathematics
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[[!include mathematicscontents]]
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# locally
* table of contents
{: toc}

## Meanings

In [[category theory]],
the term "locally" is used in several ways with different (and not always compatible) meanings.

### Slice-wise

A [[category]] $C$ may be called "locally $P$" (for some property $P$) if each [[slice category]] $C/X$ is $P$.

Examples include the notion of *[[locally cartesian closed categories]]*, i.e. those categories all whose [[slice categories]] are [[cartesian closed categories]].

### Hom-wise

A [[2-category]] $K$ (or more generally an [[enriched category]]) may be called "locally $P$" (for some property $P$) if each [[hom-category]] (or more generally [[hom-object]]) $K[X, Y]$ is $P$. Examples include [[locally-small categories]], [[locally discrete 2-categories]], and locally graded and locally indexed categories. Corresponding, we have terminology like [[local colimits]].

### Stalk-wise

In [[topos theory]] the term "local" typically refers to properties that hold after, possibly, passage to a [[cover]] or (if there are [[enough points]]) [[stalk]]-wise.

Compare the notion of *[[local isomorphism]]* or the term *Local Homotopy Theory* for [[(infinity,1)-sheaf|$\infty$-sheaf]]-[[(infinity,1)-topos theory|theory]] [[Jardine 2015](infinity-stack#Jardine15)]

Via the case of [[2-sheaves]], i.e. [[category]]-valued [[stacks]] or [[internal categories]] in [[toposes]], this notion of "local" also applies to categories.

### Local presentability

Categories are called *[[locally presentable]]* if they are [[accessible category|accessible]] and [[cocomplete]]. The "locally" in the name is intended to disambiguate between such categories and the [[finitely presentable objects]] in the 2-category [[Cat]].

## Ambiguity

Unfortunately, these meanings can overlap. For instance, the term "locally cartesian closed 2-category" could in principle refer either to

* a 2-category whose [[slice 2-categories]] are [[locally cartesian closed]]

or

* a 2-category each of whose [[hom-categories]] are ([[locally cartesian closed category|locally]]) [[cartesian closed category|cartesian closed]].

For this reason, it is often worth considering whether a different term than "locally" may be used when naming such a concept.

(Note that for the [[bicategory of spans]] in some [[finitely complete category]] $E$ the [[hom-category]] $Span(E)(A, B)$ is the [[slice category]] $E/(A \times B)$.  Thus, hom-wise properties of $Span(E)$ coincide with slice-wise properties of $E$.)

## Related pages

- [[bi-]]

category: disambiguation


locally additive space

# Locally Kinematic Spaces # {: #title}

##### Andrew Stacey ##### {: #author}

* tic
{: toc}

+-- {: .num_section #sectiona }
=--
## Introduction ## {: .num_section}

The various categories of [[generalised smooth spaces]] were introduced to correct defects with the category of [[smooth manifolds]]. The category of manifolds, it is felt, is not big enough. There are things that behave like manifolds but which, for seemingly trivial reasons, are not considered as such: things like [[manifolds with corners]] or [[infinite dimensional manifolds]]. It is also relatively easy to break the structure of a manifold by doing things that most category theorists would regard as fairly trivial: taking subobjects, equalisers, and quotients being a few. So with the oft-heard cry of:   It is better to work in a nice category with some nasty objects than a nasty category with only nice objects;   people created various extensions of the category of smooth spaces to correct these limitations.

However, having built a suitable nice category, the "nice objects" (that is to say, the manifolds) are still inside it and it is a natural question as to how to identify these nice objects in amongst the nasty ones. The obvious answer is tautological: the manifolds are those objects that I first thought of. The real problem with this is that it is an external characterisation. A better characterisation would be one that could take an object from one of these categories and examine it purely within that category and conclude whether or not it was a manifold. This might lead to more things being called a manifold than were originally thought of, but should give a characterisation better knit to what a manifold actually is than just what happens to be our current best description thereof.

An example of this sort of result is [[Sta11](#citeStacey3), Proposition 8.3] where it was shown that in the category of [[Froelicher space|Frölicher spaces]], it is possible to *categorically* pick out $\mathbb{R}$.

The purpose of this page is to study a possible description of manifolds. The description is motivated by the construction of [[manifolds of smooth maps]]. A key component in constructing those spaces is that of a [[local addition]] on the target space. Therefore our first approximation is that of *locally additive* spaces, namely spaces that are equipped with a local addition.

The rough idea behind a local addition is simple. In any of the categories of [[generalised smooth spaces]], it is straightforward to define the notion of the tangent space at a point in a smooth space. Part of the data in a local addition says that near a point, the space is modelled on its tangent space. This would appear to instantly show that the space was locally Euclidean (or, at least, locally modelled on some vector space) except for the fact that tangent spaces are not automatically vector spaces.

The rest of the data in a local addition says that this modelling must vary smoothly over the space. This translates into the transition functions being smooth.

Thus a locally additive space is locally smoothly modelled in its tangent spaces. The key difference between this and the notion of a smooth manifold is that a locally additive space is somewhat tortoise-like in that it carries its model spaces around with it, rather than having them imposed from outside.

+-- {: .num_section #sectionb }
=--
## Definition ## {: .num_section}

We assume throughout that we have fixed upon one of the categories of [[generalised smooth space]].

The page [[local addition]] has the definition of a local addition for a smooth manifold. This is almost the same definition for us here, except that our underlying space is a generalised smooth space and we don't assume that the source is a vector bundle. We shall also work with the most precise version. Putting all of this together, we shall use the following definition.

+-- {: .num_defn #defna }
###### Definition ######
Let $X$ be a smooth space. A *local addition* on $X$ is a smooth map $\eta \colon T X \to X$ with the following properties:

1.

   +-- {: .num_enuma #enumaZa }
   =--
   The composition of $\eta $ with the zero section is the identity on $X$,

1.

   +-- {: .num_enuma #enumaZb }
   =--
   There is an open neighbourhood, say $V$, of the diagonal in $X \times X$ such that $\pi \times \eta \colon T X \to X \times X$ induces a diffeomorphism with image $V$, and

1.

   +-- {: .num_enuma #enumaZc }
   =--
   For $x \in X$, the composition
   $$
   T_{x} X \xrightarrow[\cong ]{} T_{0} T_{x} X \xrightarrow[\cong ]{d\eta } T_{x} X
   $$
   is the identity.
=--

From this definition, that of a locally additive space is obvious.

+-- {: .num_defn #defnb }
###### Definition ######
A *locally additive space* is a pair $(X, \eta )$ where $X$ is a smooth space and $\eta $ is a local addition on $X$.
=--

Although the obvious definition, as it stands it is not quite precise. This is because the various notions of [[tangent space]] diverge once one goes outside the realm of smooth manifolds. In this article, we shall use the [[kinematic tangent space]]. To allow other notions based on other definitions of tangent space, we use the notion *locally kinematic space* for a locally additive space where the tangent space used is the kinematic version.

The other part of the definition that needs pinning down is the notion of an *open neighbourhood*. This requires that we supply our smooth spaces with a topology. Some definitions have a topology built in, and those that do not can be supplied with at least one suitable topology. We shall not prescribe the exact topology but rather assume that a method of supplying a smooth space with a topology has already been decided upon. The only properties that we require are the following:

1.

   +-- {: .num_enumb #enumbZa }
   =--
   The assignment of a topology to a smooth space is functorial, and

1.

   +-- {: .num_enumb #enumbZb }
   =--
   The topology on $\mathbb{R}$ is the standard one.     These both seem quite reasonable! In fact, these two conditions define a particular topology which we refer to as the [[curvaceous topology]]: it is the strongest topology for which these two statements are true.

The really important property of the local addition is that it allows us to lift tangent vectors to curves. Of course, we can always do this for a single tangent vector (as we are using the kinematic tangent space) but a local addition allows us to do it for a family. As a direct consequence of this lifting, we get the following simple result. Recall that for the [[kinematic tangent space]] at a point there are potentially two smooth structures: the quotient structure and the fibre structure. The lifting property of a locally additive space implies that these two are the same.

+-- {: .num_lemma #lemmaa }
###### Lemma ######
Let $(X,\eta )$ be a locally kinematic space. For $x \in X$, the two smooth structures on $T_{x} X$ agree. That is, $T_{x} X \to T X$ is an embedding.
=--

+-- {: .proof }
Let us write $T_{x} X$ for the tangent space at $x$ and $(T X)_{x}$ for the fibre of $T X$ at $x$.

We define a map $T X \to C^{\infty } (\mathbb{R},X)$ by sending $v \in T X$ to the curve $t \mapsto \eta (t v)$. This is smooth and splits the quotient $C^{\infty } (\mathbb{R},X) \to T X$. It is also fibre-preserving, and so maps $(T X)_{x}$ in to $C^{\infty } ((\mathbb{R},0), (X,x))$. Composing with the quotient, we obtain a map $(T X)_{x} \to T_{x} X$ inverse to the given map $T_{x} X \to (T X)_{x}$.
=--

+-- {: .num_section #sectionc }
=--
## Mapping Spaces ## {: .num_section}

The motivation for locally kinematic spaces comes from considering mapping spaces. All of the categories of [[generalised smooth space]] are cartesian closed. This means that for $S$ and $X$ smooth spaces, the set of smooth maps $C^{\infty } (S,X)$ is again a smooth space. This construction is functorial and satisfies lots of nice properties.

Let $X$ be a locally kinematic space, with local addition $\eta $, and let $V \subseteq X \times X$ be the corresponding neighbourhood of the diagonal, so that $\pi \times \eta $ is a diffeomorphism on to $V$. Then simply by functorality, we see that $C^{\infty } (S,\eta )$ is a diffeomorphism from $C^{\infty } (S, T X)$ to $C^{\infty } (S, V)$. Moreover, $C^{\infty } (S, V)$ is a subset of $C^{\infty } (S, X \times X)$ which is diffeomorphic to $C^{\infty } (S,X) \times C^{\infty } (S,X)$ and $C^{\infty } (S,V)$ contains the diagonal. That $C^{\infty } (S,\pi \times \eta ) = C^{\infty } (S,\pi ) \times C^{\infty } (S,\eta )$ takes the zero section of $C^{\infty } (S, T X)$ to the diagonal is obvious. Hence $C^{\infty } (S,\eta )$ is almost a local addition on $C^{\infty } (S,X)$. There are two pieces missing:

1.

   +-- {: .num_enumc #enumcZa }
   =--
   A natural isomorphism $C^{\infty } (S,T X) \cong T C^{\infty } (S, X)$.

1.

   +-- {: .num_enumc #enumcZb }
   =--
   That $C^{\infty } (S,V)$ is an open neighbourhood of the diagonal in $C^{\infty } (S, X \times X)$.

As we are using the [[kinematic tangent space]], there is always a natural *transformation* relating to the first piece. The locally additive structure allows us to construct the inverse.

+-- {: .num_lemma #lemmab }
###### Lemma ######
For $X$ a locally additive space, the natural transformation
$$
TC^{\infty } (S, X) \cong C^{\infty } (S, T X)
$$
is a natural isomorphism
=--

+-- {: .proof }
We need to define an inverse for this map. This starts by defining a lift from $C^{\infty } (S, T X)$ to $C^{\infty } (S, C^{\infty } (\mathbb{R},X))$. This map is defined as follows. Let $\alpha \in C^{\infty } (S, T X)$. Define $\mathbb{R} \times S \to X$ by:
$$
(t,s) \mapsto \eta (t \alpha (s))
$$
where $t \alpha (s)$ uses the $\mathbb{R}$-action on $T X$. Using the adjunction between mappings and products, this defines a curve in $C^{\infty } (S,X)$ and thus, by quotienting, a tangent vector in $C^{\infty } (S,X)$ based at $s \mapsto \eta (0 \alpha (s)) = \alpha (s)$.

To show that this is the required inverse, we need to evaluate the resulting tangent vector at some $s \in S$. This leads to the curve $t \mapsto \eta (t \alpha (s))$ in $X$. Under the natural isomorphism $T_{0} T_{x} X \cong T_{x} X$, the curve $t \mapsto t \alpha (s)$ has derivative (at $0$) $\alpha (s)$ again. Thus we have the image of this under $d \eta \colon T_{0} T_{x} X \to T_{x} X$ which is just $\alpha (s)$ again.

Thus we have the required inverse.
=--

This result is the mathematics behind the statement above that a local addition allows us to lift families of tangent vectors to curves.

The second missing piece is a little harder to pin down because without knowing precisely which category of smooth spaces we are working in, we will find it hard to know exactly how to define the topology on a smooth space, and thus to know how to work with it. Thus rather than a detailed study of when this occurs, we shall adopt the mathematician's defence.

+-- {: .num_defn #defnc }
###### Definition ######
A smooth space $S$ is said to be *smoothly compact* if whenever $U \subseteq X$ is open then $C^{\infty } (S,U)$ is open in $C^{\infty } (S,X)$.
=--

With that in place, we can now fill in the missing pieces.

+-- {: .num_theorem #theorema }
###### Theorem ######
Let $X$ be a locally kinematic space and $S$ a smoothly compact space. Then $C^{\infty } (S,X)$ is again a locally kinematic space.

The tangent space of $C^{\infty } (S,X)$ at $\alpha \colon S \to X$ consists of all maps $\beta \colon S \to T X$ such that $\pi \beta = \alpha $.
=--

+-- {: .num_section #sectiond }
=--
## Local Structure ## {: .num_section}

Having a local addition means that the smooth space is locally modelled on its tangent spaces. It turns out that this also puts some restrictions on what those tangent spaces look like.

+-- {: .num_theorem #theoremb }
###### Theorem ######
Let $X$ be a locally kinematic space. Then for $x \in X$, the tangent space $T_{x} X$ is a vector space.
=--

+-- {: .proof }
As shown at [[local addition]], the only thing missing on $T_{x} X$ of the vector space structure is the addition. We have the zero vector, scalar multiplication, and that the identities of a vector space are satisfied whenever everything is defined. So we need to define the sum, $u + v$, of two vectors $u, v \in T_{x} X$. To define this sum we define a map $W \to X$ where $W \subseteq \mathbb{R} ^{2}$ is a neighbourhood of the origin, such that $\partial _{t}$ maps to $v$ and $\partial _{s}$ to $u$. Then the image of $\partial _{s} + \partial _{t}$ will be the required sum.

Let $\eta \colon T X \to X$ be the local addition and let $V \subseteq X \times X$ be the corresponding neighbourhood of the diagonal. Consider the curve $t \mapsto (\eta (t v), \eta (u))$. At time $t = 0$, this is at $(x,\eta (u))$ which lies in $V$. There is, therefore, some interval $(-\epsilon ,\epsilon )$ for which $(\eta (t v), \eta (u)) \in V$. Since $(\pi \times \eta ) \colon T X \to V$ is a diffeomorphism, we can apply the inverse of this to that path, resulting in:
$$
(\pi \times \eta )^{-1} (\eta (t v), \eta (u)) \in T_{\eta (t v)} X.
$$
This is a tangent vector moving along the curve $\eta (t v)$ starting at $u$ above $x$. As this is, in totality, a curve of tangent vectors we can scale them. Thus we define a map:
$$
\mathbb{R} \times (-\epsilon ,\epsilon ) \to T X, \quad (s,t) \mapsto s \cdot (\pi \times \eta )^{-1} (\eta (t v), \eta (u)).
$$
The final step is to map this back down to $X$ via $\eta $:
$$
(s,t) \mapsto \eta (s \cdot (\pi \times \eta )^{-1} (\eta (t v), \eta (u))).
$$
Let us write this as $\tau \colon \mathbb{R} \times (-\epsilon ,\epsilon ) \to X$ for short. We note that it is smooth, as it is a composition of smooth maps.

It may aid the   comprehension of this map if we describe it in the case of $\mathbb{R} ^{2}$ with $u = \partial _{x}$ and $v = \partial _{y}$. In this case, $\tau $ is the map $(s,(1 - s)t)$. Thus
$$
D_{(0,0)} \tau   = \left . \begin{bmatrix}   (1 - s) & -t \\   0 & 1 \end{bmatrix} \right |_{(0,0)} = \begin{bmatrix}   1 & 0 \\   0 & 1 \end{bmatrix}.
$$
At $s = 0$, we   have the curve:
$$
t \mapsto \eta   (0 \cdot (\pi \times \eta )^{-1} (\eta (t v), \eta (u))).
$$
Under $\eta $,   the zero section maps to the "foot" of the tangent space, which is also $\pi $ of the vector. Thus the path is $t \mapsto \eta (t v)$. Hence the image of $\partial _{t}$ is $v$.

At $t = 0$, we have the curve:
$$
s \mapsto \eta (s \cdot (\pi \times \eta )^{-1}(x, \eta (u)))
$$
since $\eta (0 v) = x$. Unravelling this, we get $s \mapsto \eta (s u)$. Hence the image of $\partial _{s}$ is $u$.

Then the claim is that the image of $\partial _{s} + \partial _{t}$ will do for $u + v$. To see this, consider a smooth function $f \colon X \to \mathbb{R} $. The composition $f \circ \tau $ is smooth, and so we have that:
$$
f(u) + f(v) = (f \circ \tau )(\partial _{s}) + (f \circ \tau )(\partial _{t}) = (f \circ \tau )(\partial _{s} + \partial _{t}) = f(\tau _{*}(\partial _{s} + \partial _{t})).
$$
Hence $\tau _{*}(\partial _{s} + \partial _{t})$ is the sum $u + v$.
=--

Thus a locally additive space is modelled on vector spaces. This is not quite as strong as saying that it is a smooth manifold (even ignoring paracompactness and Hausdorff) since we do not know that the smooth structure on a tangent space is "the" smooth structure on a vector space.

+-- {: .num_section #sectione }
=--
## Related Concepts ## {: .num_section}

A similar concept to this was devised and studied by Kriegl and Michor in [Mic84a](#citeMR764972), [Mic84b](#citeMR767683). The key difference between these two approaches is that of the starting point. The goal of Kriegl and Michor's construction was to devise a cartesian closed category of manifolds, starting from almost nothing. Thus they had to build the notions of "smooth map" and of "tangent bundle" in to the foundations. Our goal is to identify manifold-like objects inside a category of smooth spaces. Thus we already know what is a "smooth map" and how to construct the "tangent bundle". Nonetheless, there are similarities in the approaches: both are particularly suited to mapping spaces, and both have "path lifting" at their core.

Another concept that should be kept in mind with regard to the above is that of the [[tangent microbundle]] of a topological manifold. This is an equivalence class of neighbourhoods of the diagonal. Thus one can say that a locally additive space is one for which the topological and smooth tangent bundles agree.
## References ##

* {: #citeMR764972} **Mic84a** Peter Michor. A convenient setting for differential geometry and global analysis. Cahiers Topologie Géom. Différentielle, 25:63A109, 1984.

* {: #citeMR767683} **Mic84b** Peter Michor. A convenient setting for differential geometry and global analysis. II. Cahiers Topologie Géom. Différentielle, 25:113A178, 1984.

* {: #citeStacey3} **Sta11** A. Stacey. Comparative smootheology. TAC, 25:64-117, 2011, http://www.tac.mta.ca/tac/volumes/25/4/25-04abs.html. arXiv:0802.2225v2.


locally affine space

A __$\tau$-locally affine space__ is a [[sheaf]] of sets on the category $Aff$ ([[affine scheme]]s) opposite to the category of commutative unital rings in some subcanonical [[Grothendieck topology]] $\tau$. If $\tau$ is the [[Zariski topology]], then $\tau$-locally affine spaces are [[schemes]]. Other examples are sheaves in fppf, fpqc, étale or lisse (=smooth) topology. The relative version uses the category of relative affine schemes $Aff/X$ which is, in the case of $X=Spec(A)$ where $A$ is a commutative ring, simply the opposite of the category of $A$-algebras.

* [[Donald Knutson]], *Algebraic spaces*, Lecture Notes in Mathematics, **203**, Springer (1971) [[doi:10.1007/BFb0059750](https://link.springer.com/book/10.1007/BFb0059750)]

In [[noncommutative algebraic geometry]], one can consider $NAff$, that is the opposite to the category of all unital associative rings instead of $Aff$. There is a noncommutative smooth topology on $NAff$ (M. Konstevich, A. L. Rosenberg, _Noncommutative smooth spaces_, [arXiv:math/9812158](http://arxiv.org/abs/math/9812158)), which is a genuine Grothendieck topology. On the other hand, one can not obtain a Grothendieck topology using exact affine lcoalizations: the stability axiom of Grothendieck topologies fails. However some other generalizations of topologies on $NAff$ are available for the noncommutative case (e.g. [[Q-categories]] and "quasi-topologies" of Rosenberg) or one can instead of working with sheaves on the big site $NAff$ work with noncommutative generalization of small cotopologies on rings (cf. work of van Oystaeyen on "noncommutative topologies").

[[!redirects locally affine spaces]]


locally algebra-ed topos

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#### Higher geometry
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#### Higher algebra
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[[!include higher algebra - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The notion of a [[topos]] $X$ that is equipped with a [[local algebra]]-object $\mathcal{O}_X$ is a generalization of the notion of a [[locally ringed topos]]. The algebra object $\mathcal{O}_X$ is then also called the [[structure sheaf]].

For that reason in ([Lurie](#Lurie)) such pairs $(X, \mathcal{O}_X)$ are called **structured toposes**. But since the notion of _[[locally ringed topos]]_ is a special case, maybe a more systematic and descriptive term is **locally algebra-ed topos**. Elsewhere this is called a locally _$T$-modelled topos_, where $T$ the given [[algebraic theory]].

## Definition

Let $\mathcal{C}_{\mathbb{T}}$ be the [[syntactic category]] of an [[essentially algebraic theory]] $\mathbb{T}$, hence any category with [[finite limits]]. Let $J$ be a [[subcanonical coverage]] on $\mathcal{C}_{\mathbb{T}}$. Notice that this makes $(\mathcal{C}_{\mathbb{T}}, J)$ be a [[standard site]] and every standard site will do.

Then the [[sheaf topos]] $Sh(\mathcal{C}_{\mathbb{T}}, J)$ is the [[classifying topos]] for the [[geometric theory]] of $\mathbb{T}$-[[local algebra]]s.

For $\mathcal{E}$ any [[topos]], a local $\mathbb{T}$-algebra object in $\mathcal{E}$ is a [[geometric morphism]]

$$
  (\mathcal{O}_X \dashv A_*) : \mathcal{E}
   \stackrel{\overset{\mathcal{O}_X}{\leftarrow}}{\underset{A_*}{\to}}
  Sh(\mathcal{C}_{\mathbb{T}}, J)
  \,.
$$

By the discussion at [[classifying topos]] this is equivalently a functor

$$
  \mathcal{O}_X : \mathcal{C}_{\mathbb{T}} \to \mathcal{E}
$$

such that

1. it preserves [[finite limit]]s (and hence produces a $\mathbb{T}$-[[algebra over an algebraic theory|algebra]] in $\mathcal{E}$);

1. it sends $J$-[[covering]]s to [[epimorphism]]s; which makes it a _local_ $\mathbb{T}$-algebra.

The pair $(\mathcal{E}, \mathcal{O}_X)$ is called a **locally $\mathbb{T}$-algebra-ed topos**.

## Examples

All of the following notions are special cases of locally algebra-ed toposes:

* [[ringed space]], [[locally ringed space]];

* [[ringed site]], [[locally ringed site]]

* [[ringed topos]], [[locally ringed topos]].

The [[(∞,1)-category theory]]-version is that of

* [[locally algebra-ed (∞,1)-topos]].

## Related concepts

* [[Cole's theory of spectrum]]

## References

* [[Julian Cole]], _The Bicategory of Topoi, and Spectra_ , ms. ([pdf](http://www.oliviacaramello.com/Unification/ColeBicategoryTopoiSpectra.pdf))

* {#Lurie} [[Jacob Lurie]], _[[Structured Spaces]]_

[[!redirects locally algebra-ed toposes]]

[[!redirects modelled topos]]
[[!redirects T-modelled topos]]
[[!redirects modelled toposes]]
[[!redirects T-modelled toposes]]
[[!redirects modelled topoi]]
[[!redirects T-modelled topoi]]


locally algebraic scheme

A [[scheme]]  over a field $k$ is called *locally algebraic scheme* if it has a covering $\{X_i\}$ of (open affine sub functors which are) [[affine scheme|affine schemes]] and all $O(X_i)$ are finitely generated $k$-algebras, i.e. if it is [[morphism of finite type|locally of finite type]] over $Spec k$.

A scheme is called *algebraic scheme* if moreover there exist a finite such covering, i.e. if it is of finite type over $Spec k$.

category: algebraic geometry


locally bounded category

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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
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# Contents
* table of contents
{:toc}

## Idea

Local boundedness of a [[category]] is a generalization of the notion of [[locally presentable category|local presentability]] that includes the category of [[topological spaces]].

## Context

Let $C$ be a [[small category|small]] [[cocomplete category]] with a proper [[factorization system]], i.e., an [[orthogonal factorization system]] $(E,M)$ where every [[morphism|map]] in $E$ is an [[epimorphism]] and every map in $M$ is a [[monomorphism]].

The $M$-**union** of a small family of $M$-[[subobjects]] $(A_j \to B)_{j \in J}$ is the unique $M$-subobject $A \to B$ containing the $A_j$ and so that the induced map $\sum_j A_j \to A$ is in $E$. The union is calculated by applying the $(E,M)$ factorization to the canonical map $\sum_j A_j \to B$.

If the map $\sum_j A_j \to B$ is in $E$, we say $(A_j \to B)_{j \in J}$ is an $M$-**union**. The set $J$ is a preorder under the relation $j \leq k$ if $A_j \leq A_k$ as $M$-subobjects of $B$. Regarding the $A_j$ as a diagram of shape $J$, the family $(A_j \to B)_{j \in J}$ is an $M$-union if and only if the map colim$A_j \to B$ is in $E$. We say $(A_j \to B)_{j \in J}$ is a **filtered union of** $M$-**subobjects** if it is a union of $M$-subobjects and if the category $J$ is [[filtered category|filtered]].

A [[representable functor]] $C(X,-)$ **preserves** the $M$-union of $(A_j \to B)_{j \in J}$ if the functions $C(X,A_j) \to C(X,A)$ are jointly surjective, i.e., if each $X \to A$ factors through some $X \to A_j$.

## Definition

Let $\lambda$ be a [[regular cardinal]], and let $C$ be a cocomplete category with a proper factorization system $(E,M)$.

+-- {: .num_defn}
###### Definition (locally bounded)

An object $X$ in $C$ is $\lambda$-**bounded** if $C(X,-)$ preserves $\lambda$-[[filtered colimit|filtered]] [[unions]] of $M$-subobjects.

=--

+-- {: .num_defn}
###### Definition ($(E,M)$-generator)

A small set $G$ of objects of $C$ is an $(E,M)$-**generator**  if $f \colon A \to B$ in $M$ is invertible whenever $f_* \colon C(X,A) \to  C(X,B)$ is [[bijection|bijective]] for all $X \in G$. Equivalently, $G$ is an $(E,M)$-**generator** if for each $A \in C$ the family of maps $X \to A$ is jointly in $E$, i.e., if the map $\sum_{X \in G} \sum_{C(X,A)} X \to A$ is in $E$.

=--

+-- {: .num_defn}
###### Definition (locally bounded category)

A [[category]] $C$ is called **locally** $\lambda$-**bounded** with respect to a proper factorization system $(E,M)$ if

* it has an $(E,M)$-generator $G$ each of whose objects is $\lambda$-bounded

* it has arbitrary cointersections (even large ones) of maps in $E$ --- that is, it is [[E-cocomplete category|E-cocomplete]].

=--

## Features

+-- {: .num_prop}
###### Proposition (relation to local presentability)
In a locally $\lambda$-presentable category, every $\lambda$-presentable object is $\lambda$-bounded. Hence a $\lambda$-presentable category is $\lambda$-bounded.
=--

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

This appears as Lemma 2.3.1 of [Freyd-Kelly](#FreydKelly)
=--

+-- {: .num_prop}
###### Proposition (completeness)
Locally bounded categories are necessarily complete
=--

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

This appears as Corollary 2.2 of [Kelly-Lack](#KellyLack).

The essential point is an $(E,M)$-variant of the special adjoint functor theorem: if $C$ is cocomplete, has a proper factorization system $(E,M)$, admits arbitrary $E$-cointersections, and has an $(E,M)$-generator, then every cocontinuous functor $C \to D$ has a right adjoint.
=--

##Examples

The following examples are discussed in Section 6.1 of Kelly's _Basic concepts of enriched category theory_.

* [[locally presentable category|Locally presentable categories]] such as the categories of [[simplicial sets]], [[categories]], [[abelian groups]], [[Set|sets]].

* [[compactly generated topological space|Compactly generated spaces]], and likewise based compactly generated spaces, with $E$ the surjections and $M$ the subspace inclusions. The point is an $(E,M)$-generator.

* [[quasi-topological space|Quasi-topological spaces]].  Note that this category is not $E$-well-copowered.

* [[Banach spaces]] with $E$ the epimorphisms, equivalently the dense maps, and $M$ the [[extremal monomorphisms]], equivalently the inclusions of closed subspaces with the induced norm. The base field is an $(E,M)$-generator.

## Related notions
The term **locally ranked** is sometimes used to refer to a locally bounded category which in addition is co-wellpowered. For example, this terminology is used in [Adámek et. al.](#AHRT).

## References

The contents of this page are taken from:

* [[Max Kelly]], [[Steve Lack]], _$V$-cat is locally presentable or locally bounded if $V$ is so_ TAC (2001)
 {#KellyLack}

See also:

* [[Max Kelly]], _Basic concepts of enriched category theory_.

* [[Peter Freyd]], [[Max Kelly]], _Categories of continuous functors_ J. Pure. Appl. Algebra 2 (1972) 169-191.
 {#FreyKelly}
* [[Jírí Adámek]], [[Horst Herrlich]], [[Jírí Rosickỳ]], [[Walter Tholen]], _On a generalized small-object argument for the injective subcategory problem_. Cah. Topol. Géom. Différ. Catég 43 (2002) 83–106.
{#AHRT}

[[!redirects locally bounded categories]]


locally cartesian category

# Locally cartesian categories
* table of contents
{: toc}

## Definitions

A [[category]] $C$ is __locally cartesian__ if each of its [[slice categories]] $C/x$ is a [[cartesian monoidal category]], meaning that $C/x$ has all finite [[products]].  Another way to say this is that $C$ has all finite [[fibred products]] or equivalently that $C$ has all [[pullbacks]].

A __[[finitely complete category]]__ is precisely a locally cartesian category that has a [[terminal object]].

## Related entries

* [[locally cartesian closed category]]

[[!redirects locally cartesian category]]
[[!redirects locally Cartesian category]]


locally cartesian closed

See:

* [[locally cartesian closed (infinity,1)-category]],
* [[locally cartesian closed category]],
* [[locally cartesian closed functor]],
* [[locally cartesian closed model category]].

category: adjective

[[!redirects locally cartesian closed]]


locally cartesian closed (infinity,1)-category

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#Contents#
* table of contents
{:toc}

## Definition

An [[(∞,1)-category]] $C$ has [[finite limit|finite]] [[(∞,1)-limits]] 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]].

We say that such a $C$ is **locally cartesian closed** if moreover $C_{/x}$ is a [[cartesian closed (∞,1)-category]] for every object $x$.  This is equivalent to asking that the pullback functor $f^*\colon C_{\y} \to C_{\x}$, for any $f\colon x\to y$ in $C$, has a right adjoint $\Pi_f$.

## Examples

* Every (Grothendieck) [[(∞,1)-topos]] is locally cartesian closed.  This follows from the [[universal colimit|universality of colimits]] and the [[adjoint functor theorem]].

## Properties
 {#Properties}

### Presentations
 {#Presentations}

The following theorem should be compared with the fact that every [[locally presentable (∞,1)-category]] admits a presentation by a [[Cisinski model category]], indeed by a [[left Bousfield localization]] of a [[global model structure on simplicial presheaves]].

+-- {: .num_theorem #PresentationTheorem}
###### Theorem
For a locally presentable $(\infty,1)$-category $C$, the following are equivalent.

1. $C$ is locally cartesian closed.

1. [[(∞,1)-colimit|(∞,1)-Colimits]] in $C$ are [[universal colimits|stable under pullback]].

1. $C$ admits a presentation by a [[combinatorial model category|combinatorial]] [[locally cartesian closed model category]].

1. $C$ admits a presentation by a [[right proper model category|right proper]] [[Cisinski model category]].

1. $C$ admits a [[presentable (infinity,1)-category|presentation]] by a [[right proper model category|right proper]] [[Bousfield localization of model categories|left Bousfield localization]] of an [[model structure on simplicial presheaves|injective]] [[model structure on sSet-enriched presheaves]] over some small [[sSet-site]] (see [[model structure on simplicial presheaves#Properness|here]] for sufficient conditions).

=--

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

Since [[left adjoints]] preserve colimits, the first condition implies the second.  The converse holds by the [[adjoint functor theorem]] since each slice of $C$ is locally presentable.

Suppose $M$ is a right proper Cisinski model category.  Then pullback along a fibration preserves cofibrations (since they are the monomorphisms) and weak equivalences (since $M$ is right proper).  Since $M$ is a locally cartesian closed 1-category, pullback also has a right adjoint, so it is a [[Quillen adjunction|left Quillen functor]]; thus the fourth condition implies the third.  Since left Quillen functors preserve homotopy colimits, the third condition implies the second.

The fifth condition implies the fourth, since the model structure therein is Cisinski.  The only nonobvious part of this is that its underlying category is a topos, which follows from the fact that $sSet$-enriched presheaves on a $sSet$-enriched category $C$ can be identified with [[internal diagrams]] on $C$ regarded as an [[internal category]] in $sSet$, and the category of internal diagrams on any internal category in a topos (such as $sSet$) is again a topos.

It remains to show that the second condition implies the fifth.  For that, see [this blog comment](http://golem.ph.utexas.edu/category/2012/05/the_mysterious_nature_of_right.html#c041306) by [[Denis-Charles Cisinski]].  An alternative proof can be found in [(Gepner-Kock 12, Thm 7.10)](#GepnerKock12).

=--

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

Further equivalent characterizations of locally cartesian closed $(\infty,1)$-categories are in
([Lurie, prop. 6.1.1.4, lemma 6.1.3.3](#Lurie))

=--

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

Comparing the third and the fifth item in theorem \ref{Presentations}
notice that the projective and the injective [[model structure on simplicial presheaves]] are [[Quillen equivalence|Quillen equivalent]] (as discussed at _[[model structure on functors]]_.)

=--

### Internal logic and homotopy type theory
 {#InternalLogic}

It is expected that the [[internal logic]] of locally cartesian closed $(\infty,1)$-categories should be a sort of [[homotopy type theory]] (specifically, that with intensional [[identity types]] and [[dependent products]]), in higher analogy with the [[relation between type theory and category theory]].  More specifically, the following are hoped to be true:

* From any type theory, its [[syntactic category]] is (or gives rise to) an $(\infty,1)$-category, which is the [[initial object]] in an appropriate $(\infty,1)$-category (or perhaps $(\infty,2)$-category) of locally cartesian closed $(\infty,1)$-categories (perhaps with additional structure corresponding to additional axioms or rules in the type theory).  Thus, type theory can be "interpreted" in any lcc $(\infty,1)$-category by way of the universal morphism from this initial object.

* From any locally cartesian closed $(\infty,1)$-category $C$, one can construct a type theory with rules or axioms corresponding to its objects and morphisms, which has a canonical interpretation in $C$ itself.  This would be the "internal language" of $C$.

* These construction should set up some sort of adjunction, or perhaps equivalence, between some $(\infty,1)$- or $(\infty,2)$-category of type theories and that of lcc $(\infty,1)$-categories.

Some partial results in these directions are known.  For instance, Theorem \ref{Presentations} says in particular that every *locally presentable* and locally cartesian closed $(\infty,1)$-category has a presentation by a [[type-theoretic model category]].  As discussed there, this provides [[categorical semantics]] for [[homotopy type theory]] (without in general the [[univalence]] [[axiom]]), at least if we assume the initiality of the syntactic 1-category (which is known in a special case and expected to be true in all cases).  Together, this yields:

+-- {: .num_theorem #InterpretationTheoemForHoTT}
###### Theorem
**(Cisinski, Shulman)** Assuming the initiality of the syntactic 1-category, every locally presentable locally cartesian closed $(\infty,1)$-category admits an interpretation of [[Martin-Löf type theory]] with [[dependent sum types]], [[dependent product types]], and [[identity types]] ("[[homotopy type theory]]").

Moreover it also interprets [[function extensionality]] ([Shulman 12, lemma 5.9](#Shulman12), see this [this discussion](function+extensionality#EveryPresentableLocallyCartesinClosedInfinityCatInterpretsHoTTPlusFunExt).

Hence every presentable locally cartesian closed $\infty$-category interprets [[HoTT]]+[[FunExt]].

=--

{#UnivalentUniverses} This includes in particular all ([[∞-stack]]-) [[(∞,1)-toposes]], which in addition interpret [[univalence axiom|univalent]] [[type universes]] ([[(infinity,1)-topos#Shulman19|Shulman 19]]).

This statement is not fully satisfactory for several reasons.  Firstly, it assumes local presentability.  Secondly, rather than staying in the world of $(\infty,1)$-categories, it goes by way of a strict presentation.  Thus, the existence and behavior of a "universal model" is unclear.

A step in the latter direction was conjectured in ([Joyal 2011](#Joyal11)), and established in [[Chris Kapulkin]]'s PhD thesis (see [Kapulkin 14](#Kapulkin14), [Kapulkin 15](#Kapulkin15)):

+-- {: .num_theorem }
###### Theorem
**(Kapulkin)** If **T** is any [[dependent type theory]] with (at least) $\Sigma$-types, $\Pi$-types, and $\mathrm{Id}$-types, then the [[simplicial localisation]] of its [[syntactic category|classifying category]] $\mathrm{Cl}(\mathbf{T})$ is a locally cartesian closed $(\infty,1)$-category.
=--

However, this "syntactic $(\infty,1)$-category" will not in general be locally presentable, lacking appropriate colimits.  There is thus a mismatch between the two best statements we have at present, for the two directions of the internal language correspondence.

{#Initiality} Furthermore, the *initiality* of this syntactic $(\infty,1)$-category is subtle (see at *[[initiality conjecture]]*), but such initiality is a central aspect of what it means to have an internal language.  Finally, no one has yet explicitly constructed an "internal language" type theory from an $(\infty,1)$-category, and so of course the desired adjunction/equivalence cannot yet even be stated as a precise conjecture.

## Related concepts

* [[cartesian closed category]], [[locally cartesian closed category]]

* [[cartesian closed functor]], [[locally cartesian closed functor]]

* [[cartesian closed model category]], [[locally cartesian closed model category]]

* [[cartesian closed (∞,1)-category]], **locally cartesian closed (∞,1)-category**

## References

Characterizations of locally presentable locally cartesian closed $(\infty,1)$-categories (as [[locally presentable (∞,1)-categories]] with [[universal colimits]]) are discussed in section 6.1 of

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

Early discussion in the context of [[homotopy type theory]] is

* {#Joyal11} [[André Joyal]], _Remarks on homotopical logic_, Oberwolfach (2011) ([pdf](http://hottheory.files.wordpress.com/2011/06/report-11_2011.pdf#page=19))

proposing [[homotopy type theory]] as an internal language for locally cartesian closed $\infty$-categories ([[Awodey's conjecture]]).

[[Denis-Charles Cisinski]]'s argument in Theorem \ref{Presentations} above, that every locally presentable locally cartesian closed $(\infinity,1)$-category admits a presentation by a [[type-theoretic model category]], originally appeared on the Café ([this blog comment](http://golem.ph.utexas.edu/category/2012/05/the_mysterious_nature_of_right.html#c041306)) and is mentioned in print in Examples 2.16 of

* {#Shulman12} [[Michael Shulman]], _Univalence for inverse diagrams and homotopy canonicity_, Mathematical Structures in Computer Science, Volume 25, Issue 5 ( _From type theory and homotopy theory to Univalent Foundations of Mathematics_ ) June 2015 ([arXiv:1203.3253](https://arxiv.org/abs/1203.3253), [doi:/10.1017/S0960129514000565](https://doi.org/10.1017/S0960129514000565))

which gives a detailed account of the categorical semantics of homotopy type theory in [[type-theoretic model categories]] such as those presenting locally cartesian closed $(\infty,1)$-categories.

For more on this see also the relevant sections at _[[relation between type theory and category theory]]_.

Discussion of the converse direction, obtaining locally cartesian closed $(\infty,1)$-categories as [[syntactic categories]] of [[homotopy type theory|homotopy type theories]] is in

* {#Kapulkin14} [[Chris Kapulkin]], _Type theory and locally cartesian closed quasicategories_, Oxford 2014 ([video](https://www.youtube.com/watch?v=g87bZJ2bvYk))

* {#Kapulkin15} [[Chris Kapulkin]], _Locally Cartesian Closed Quasicategories from Type Theory_ ([arXiv:1507.02648](http://arxiv.org/abs/1507.02648))

A discussion of object classifiers, univalent families, and model category presentations is the context of $(\infty,1)$-categories (and hence in categorical semantics for what should be homotopy type theory with univalent universes "weakly a la Tarski") appeared also in

* {#GepnerKock12} [[David Gepner]], [[Joachim Kock]], "Univalence in locally cartesian closed ∞-categories", [arXiv](http://arxiv.org/abs/1208.1749)

[[!redirects locally cartesian closed (∞,1)-category]]
[[!redirects locally cartesian closed (infinity,1)-categories]]
[[!redirects locally cartesian closed (∞,1)-categories]]
[[!redirects locally cartesian closed infinity,1-category]]

[[!redirects locally Cartesian closed (∞,1)-category]]
[[!redirects locally Cartesian closed (∞,1)-categories]]
[[!redirects locally Cartesian closed (infinity,1)-category]]
[[!redirects locally Cartesian closed (infinity,1)-categories]]

[[!redirects locally Cartesian closed infinity,1-category]]
[[!redirects locally Cartesian closed infinity,1-categories]]


locally cartesian closed category

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#### Monoidal categories
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#Contents#
* table of contents
{:toc}

## Definition

A **locally cartesian closed category** is a [[category]] $C$ whose [[slice categories]] $C/x$ are all [[cartesian closed category|cartesian closed]].

If a locally cartesian closed category $C$ has a [[terminal object]], then $C$ is itself cartesian closed and in fact [[finitely complete category|has all finite limit]]s (because, cartesian products in $C/x$ are pullbacks in $C$); often this requirement is included in the definition.

Equivalently, a locally cartesian closed category $C$ is a category with [[pullbacks]] (and a [[terminal object]], if required) such that each [[base change]] functor $f^*: C/y \to C/x$ has a [[right adjoint]] $\Pi_f$, called the _[[dependent product]]_. (This equivalence is discussed in detail [below](#EquivalentCharacterizations).)

In particular, such pullbacks preserve all [[colimits]]. Therefore, if a locally cartesian closed category [[finitely cocomplete category|has finite colimits]], it is automatically a [[coherent category]] and in fact a [[Heyting category]].

## Properties

### Cartesian closure in terms of base change and dependent product
 {#EquivalentCharacterizations}

We show how the [[dependent product]] and the [[internal hom]] in the [[slice categories]] may be used to express each other.

#### In category theory

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

Let $\mathcal{C}$ be a [[category]] with [[pullbacks]] that has all [[dependent products]] $\prod_\bullet$, equivalently: such that every morphism $f \colon E \to X$ in $\mathcal{C}$ induces an [[base change]] [[adjoint triple]]

$$
  \mathcal{C}_{/E}
    \stackrel{\overset{\sum_f}{\longrightarrow}}{\stackrel{\overset{f^*}{\longleftarrow}}{\underset{\prod_f}{\longrightarrow}}}
  \mathcal{C}_{/X}
  \,.
$$

Then the [[internal hom]] in any [[slice category|slice]] $\mathcal{C}_{/X}$ exists and is given by

$$
  \big[
     \langle
       E \stackrel{f}{\longrightarrow} X
     \rangle
     \; ,\;
     \text{-}
  \big]_{\mathcal{C}_{/X}}
  \;\simeq\;
  \textstyle{\prod}_f \circ f^*(\text{-})
  \,.
$$

=--

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

By the discussion at *[[overcategory]] -- [limits and colimits](overcategory#LimitsAndColimits)*, the product in the slice $\mathcal{C}_{/X}$ of two objects $\langle E_1 \stackrel{f_1}{\longrightarrow} X\rangle$ and $\langle E_2 \stackrel{f_2}{\longrightarrow} X\rangle$ is given by the [[pullback]] $f_1^* E_2 = f_2^* E_1$ in $\mathcal{C}$, regarded again as a morphism over $X$. More formally this means that the product with $\langle E \stackrel{f}{\to} X\rangle $ is given by the composite

$$
  (-)
    \times_{\mathcal{C}_{/X}}
  \langle E \stackrel{f}{\to} X\rangle
  \;\;\;:\;\;\;
  \mathcal{C}_{/X}
    \stackrel{f^*}{\to}
  \mathcal{C}_{/E}
    \stackrel{\sum_f}{\to}
  \mathcal{C}_{/X}
$$

of the pullback along $f$ with the [[dependent sum]] along $f$. By the above [[adjoint triple]] both these morphisms have [[right adjoints]] and so the composite of the right adjoints is the right adjoint of the composite, hence is the [[internal hom]]:

$$
  \mathcal{C}_{/X}
    \stackrel{\prod_f}{\leftarrow}
  \mathcal{C}_{/E}
    \stackrel{f^* }{\leftarrow} \mathcal{C}_{/X}
  \;\;\;:\;\;\;
  \big[
    \langle E\stackrel{f}{\to} X \rangle
    \,,\,
    \text{-}
  \big]_{\mathcal{C}_{/X}}
  \,.
$$

=--

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

In a slice category $Set_{/X}$ of [[Sets]], whose [[objects]] are $X$-[[indexed sets]], the inner hom is explicitly given by the [[indexed set]] of [[function sets]] between [[preimages]]:
$$
  \big[
    \langle E \stackrel{f}{\to} X \rangle
    \,,\,
    \langle F \stackrel{g}{\to} X \rangle
  \big]_{Set_{/X}}
    \;=\;
  \big\{
    (x,h)
    \,\vert\,
    x \colon X,
    \;
    h \colon f^{-1}(x) \to g^{-1}(x)
  \big\}.
$$

=--

And conversely:

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

If for a [[category]] $\mathcal{C}$ every [[slice category]] is a [[cartesian closed category]], then for every morphism $f \colon X \to Y$ in $\mathcal{C}$ the [[dependent product]] $\prod_f$ exists and is given on an object $E \stackrel{p}{\to} X$ by the [[pullback]]

$$
  \array{
    \prod_f \langle E \stackrel{p}{\to} X\rangle &\to& [\langle X \stackrel{f}{\to}Y\rangle, \langle E \stackrel{p}{\to}X \stackrel{f}{\to} Y\rangle]_{\mathcal{C}_{/Y}}
    \\
    \downarrow && \downarrow
    \\
    Y &\stackrel{\bar id}{\to}& [\langle X \stackrel{f}{\to}Y \rangle,\langle X \stackrel{f}{\to}Y \rangle]_{\mathcal{C}_{/Y}}
  }
$$

in $\mathcal{C}_{/Y}$, where the bottom morphism is the [[adjunct]] of

$$
  Y \times_{\mathcal{C}_{/Y}} \langle X \stackrel{f}{\to} Y\rangle \simeq \langle X \stackrel{f}{\to}Y\rangle \stackrel{id}{\to} \langle X \stackrel{f}{\to}Y\rangle
  \,.
$$

=--

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

It is sufficient to check the $(f^* \dashv \prod_f)$-[[adjunction]] [[hom set]]-[[natural isomorphism]]

$$
  \mathcal{C}_{/Y}( \langle F \stackrel{}{\to} Y\rangle , \prod_f \langle E \stackrel{p}{\to} X\rangle)
  \simeq
  \mathcal{C}_{/X}(f^* \langle F \stackrel{}{\to} Y\rangle, \langle E \stackrel{p}{\to} X\rangle)
  \,,
$$

natural in $\langle F \stackrel{}{\to}Y \rangle$.

Since the [[hom functor]] $\mathcal{C}_{/Y}(\langle F \stackrel{}{\to} Y\rangle, -)$ preserves [[limits]] and hence [[pullbacks]], the expression on the left is exhibited as the pullback

$$
  \array{
    \mathcal{C}_{/Y}(\langle F \stackrel{}{\to} Y\rangle, \prod_f \langle E \stackrel{p}{\to} X\rangle) &\to&
  \mathcal{C}_{/Y}(\langle F \stackrel{}{\to} Y\rangle, [\langle X \stackrel{f}{\to}Y \rangle,\langle E \stackrel{p}{\to}X \stackrel{f}{\to} Y\rangle]_{\mathcal{C}_{/Y}})
    \\
    \downarrow && \downarrow
    \\
    * &\stackrel{}{\to}& \mathcal{C}_{/Y}(\langle F \stackrel{}{\to} Y\rangle,[\langle X \stackrel{f}{\to}Y \rangle,\langle X \stackrel{f}{\to}Y \rangle]_{\mathcal{C}_{/Y}})
  }
$$

in [[Set]]. Using the $((-) \times_{\mathcal{C}_{/Y}} \langle X \stackrel{f}{\to}Y\rangle \dashv [\langle X \stackrel{f}{\to}Y\rangle, -]_{\mathcal{C}_{/Y}})$-adjunction  this is equivalently

$$
  \array{
    \mathcal{C}_{/Y}(\langle F \stackrel{}{\to} Y\rangle, \prod_f \langle E \stackrel{p}{\to} X\rangle)
    &\to&
  \mathcal{C}_{/Y}(\langle F \stackrel{}{\to} Y\rangle \times_{\mathcal{C}_{/Y}} \langle X \stackrel{f}{\to}Y \rangle, \langle E \stackrel{p}{\to}X \stackrel{f}{\to} Y\rangle)
    \\
    \downarrow && \downarrow
    \\
    * &\stackrel{}{\to}&
    \mathcal{C}_{/Y}(\langle F \stackrel{}{\to} Y\rangle \times_{\mathcal{C}_{/Y} }\langle X \stackrel{f}{\to}Y \rangle ,
   \langle X \stackrel{f}{\to}Y \rangle)
  }
  \,.
$$

This pullback now manifestly computes
$\mathcal{C}_{/X}(f^* \langle F \to Y\rangle, \langle E \stackrel{p}{\to} X\rangle)$.

=--

+-- {: .num_prop #slicelcc}
###### Proposition
If $C$ is locally cartesian closed (i.e., if every slice $C/X$ is cartesian closed), then every slice $C/X$ is also locally cartesian closed.
=--

+-- {: .proof}
###### Proof
The slice of a slice is a slice, i.e., for every $f \colon X \to Y$ there is an equivalence

$$(C/Y)/(f \colon X \to Y) \simeq C/X$$

whence the statement immediately follows.
=--

+-- {: .num_prop #pres}
###### Proposition
If $C$ is locally cartesian closed (and has a terminal object), then the pullback functor $X \times - \colon C \to C/X$ preserves both finite products and exponentials up to isomorphism.
=--

+-- {: .proof}
###### Proof
Clearly $X \times - \colon C \to C/x$, being right adjoint to the forgetful functor $\sum_X \colon C/X \to C$, preserves limits, hence it preserves finite products in particular.

Let $\phi \colon A \to X$ be any morphism. From the pullback diagram

$$\array{
A \times Z & \stackrel{\phi \times 1}{\to} & X \times Z \\
 _\mathllap{\pi_A} \downarrow & & \downarrow _\mathrlap{\pi_X} \\
A & \underset{\phi}{\to} & X
}$$

we conclude $A \times_X (X \times Z) \cong A \times Z$, seen as an object over $X$ via $\phi \circ \pi_A \colon A \times Z \to X$. Thus arrows in the slice $C/X$ of the form

$$A \times_X (X \times Z) \to X \times Y$$

are in natural bijection with arrows in $C$ of the form

$$A \times Z \stackrel{\langle \phi \circ \pi_A, g\rangle}{\to} X \times Y$$

which in turn are in natural bijection with arrows in the slice $C/X$ of the form

$$A \stackrel{\langle \phi, \tilde{g} \rangle}{\to} X \times Y^Z$$

(where $\tilde{g} \colon A \to Y^Z$ is obtained by [[currying]] $g \colon A \times Z \to Y$ in $C$). This proves that $X \times - \colon C \to C/x$ preserves exponentials.
=--

+-- {: .num_cor}
###### Corollary
For any $f \colon X \to Y$ in $C$, the base change $f^\ast \colon C/Y \to C/X$ preserves exponentials. In other words, the dependent sum functor $\sum_f$ and the dependent product functor $\prod_f$ satisfy [[Frobenius reciprocity]].
=--

+-- {: .proof}
###### Proof
This is by combining proposition \ref{slicelcc} and proposition \ref{pres}, and recalling that the pullback functor

$$C/Y \to (C/Y)/(f \colon X \to Y) \simeq C/X$$

is identified with the pullback functor $f^\ast \colon C/Y \to C/X$.
=--

This state of affairs may be summarized in terms of the notion of _[[hyperdoctrine]]_:

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

If $C$ is a category with [[finite limits]], then it is locally cartesian closed precisely if regarded as an $C$-[[indexed category]] (by forming [[slice categories]]) it is a [[hyperdoctrine]].

=--

For a proof of the statement in this form, see for instance ([Freyd](#Freyd)).

#### In type theory
 {#RelationCartesianClosureBaseChangeInTypeTheory}

We formulate some of the above considerations in the [[syntax]] of
[[dependent type theory]].

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

Let

$$
  \array{
     X &&&& A
     \\
     & {}_{\mathllap{\phi}}\searrow && \swarrow_{\mathrlap{c}}
     \\
     &&
     B
  }
$$

be two [[display maps]]. Then the
[[category theory|category theoretic]] identification

$$
  [X,A]_{B} = \prod_\phi \phi^* \langle A \stackrel{c}{\to} B\rangle
$$

from prop. \ref{DependentProductImpliesLocalCartesinClosure} is the [[categorical semantics]] of the [[dependent type theory]] [[syntax]]

$$
  b : B \vdash X(b) \to A(b) : Type
  \;\;\;
  =_{def}
  \;\;\;
  b : B \vdash \prod_{x : X(b)} A(\phi(x))
  \,.
$$

=--

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

While equivalent under the [[relation between type theory and category theory]], the latter expression almost verbatim expresses the evident idea that:

 (collection of internal homs parameterized over $B$)
  =
 ( collection of sections of the pullback of $A$ to $X$ )

=--

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

By definition, the [[display map]] on the right is expressed as the [[dependent type]]

$$
  b : B \vdash A(b) : Type
  \,,
$$

the pullback $\phi^* \langle A \to B\rangle$ is expressed by [[substitution]]

$$
   x : X \vdash A(\phi(x)) : Type
$$

and next the [[dependent product]] $\prod_\phi \phi^* \langle A \to B\rangle$ by

$$
   b : B  \vdash \prod_{x : X(b)} A(\phi(x)) : Type
  \,.
$$

Now on the right $\phi(x) =_{def} b$, formally because $\phi$ is equivalently the projection $pr_1$ out of $X$ expressed as the [[direct sum]]

$$
  \frac{(b,x) :  \sum_{b : B} X(b)}
   { pr_1(b,x) =_{def} b =_{def} \phi(x)  : B}
  \,.
$$

Inserting this in the above expression makes it [[definitional equality|definitionally equal]] to

$$
   b : B  \vdash \prod_{x : X(b)} A(b) : Type
  \,.
$$

This is now a [[dependent product]] over a type that does not actually depend
on the context $B$, and hence by definition this is the [[dependent type|dependent]] [[function type]]

$$
  b : B \vdash X(b) \to A(b) : Type
  \,.
$$

which expresses the internal hom in the slice over $B$.

=--

### Internal logic

The [[internal logic]] of locally cartesian closed categories is an [[extensional type theory|extensional]] form of [[dependent type theory]].  In particular, the [[dependent product]] $\Pi_f$ represents an extensional [[dependent product type]] in the internal logic.

### Almost local cartesian closure

There are categories which are cartesian closed and not locally cartesian closed, but in which for some $f$ the  base change functor $f^*: C/y \to C/x$ has a right adjoint. This includes $Cat$, $Gpd$, and the category of [[crossed complex]]es; in the latter two cases, it is necessary and sufficient for $f$ to be a [[fibration]], while in $Cat$ it is sufficient for $f$ to be a fibration or an opfibration.

## Examples

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

Every [[sheaf topos]] is locally cartesian closed.

=--

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

By [[Giraud's theorem]], in a sheaf topos [[pullbacks]] preserve [[colimits]].
With the [[adjoint functor theorem]] this implies that for every morphism $f : X \to Y$, the pullback functor $f^* : \mathcal{C}_{/Y} \to \mathcal{C}_{/X}$ has a [[right adjoint]] $\prod_f$. By prop. \ref{DependentProductImpliesLocalCartesinClosure} this yields the local cartesian closure.

=--

More generally, every [[quasitopos]] is locally cartesian closed.

## Related concepts

* [[cartesian closed category]]

* [[cartesian closed functor]], [[locally cartesian closed functor]]

* [[locally cartesian closed enriched category]]

* [[cartesian closed model category]], [[locally cartesian closed model category]]

* [[cartesian closed (∞,1)-category]], [[locally cartesian closed (∞,1)-category]]

## References

A standard textbook account is around corollary A1.5.3 in

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

The relation between local cartesian closure and base change/[[hyperdoctrine]] structure is sometimes attributed to

* [[Peter Freyd]], _Aspects of topoi_, Bull. Australian Math. Soc. 7 (1972), 1-76.
 {#Freyd}

A discussion of [[dependent type theory]] as the [[internal language]] of locally cartesian closed categories is in

* [[R. A. G. Seely]], _Locally cartesian closed categories and type theory_, Math. Proc. Camb. Phil. Soc. (1984) 95 ([pdf](http://www.math.mcgill.ca/rags/LCCC/LCCC.pdf))

exposition:

* Xu Huang, *Locally Cartesian Closed Categories* [[arXiv:2202.04543](https://arxiv.org/abs/2202.04543)]

See also:

* [[Marta Bunge]],  and [[Susan Niefield]],   _Exponentiability and single universes_ J. Pure Appl. Algebra 148 (2000) 217--250.

* [[François Conduché]],   _Au sujet de l'existence d'adjoints à droite aux foncteurs "image réciproque" dans la catégorie des catégories_ (French)  C. R. Acad. Sci. Paris Sér. A-B  275  (1972), A891--A894.

* J. Howie, _Pullback functors and crossed complexes_ , Cahiers Topologie Géom. Différentielle, 20 (1979) 281--296.

[[!redirects locally cartesian closed categories]]
[[!redirects LCCC]]
[[!redirects lccc]]

[[!redirects locally Cartesian closed category]]
[[!redirects locally Cartesian closed categories]]

[[!redirects local cartesian closure]]


# Locally cartesian closed enriched categories

* table of contents
{: toc}

## Definition

Let $V$ be a [[monoidal category]] and $C$ a $V$-[[enriched category]] with [[pullbacks]] in the enriched sense.  Then for any morphism $f:x\to y$ in (the [[underlying ordinary category]] of) $C$, there is a pullback $V$-functor $f^*:C/y \to C/x$ between the [[enriched slice categories]], and each enriched slice category $C/x$ has $V$-enriched [[products]].  We say $C$ is **$V$-locally-cartesian-closed** if the following equivalent conditions hold:

* Each $V$-functor $f^*$ has a right $V$-adjoint $f_*$.
* Each $V$-category $C/x$ is $V$-[[cartesian closed enriched category|cartesian closed]].

## Relation to ordinary local cartesian closure

If $C$ is $V$-locally-cartesian-closed, then its [[underlying ordinary category]] $C_0$ is [[locally cartesian closed category|locally cartesian closed]] in the usual sense, since $V$-enriched right adjoints have underlying ordinary right adjoints.

The converse is true in some cases, such as the following:

* When $V=Set$, trivially.

* More generally, whenever the underlying-set functor $V(I,-) : V\to Set$ is [[conservative functor|conservative]].

* When $V$ is locally cartesian closed and cartesian monoidal and $C=V$.

However, the converse is false in general.  Counterexamples can be found in [[locally cartesian closed enriched category#MO|this mathoverflow discussion]] (the discussion is only about cartesian closed enriched categories, but the counterexamples given are in fact locally cartesian closed, being indeed [[presheaf categories]]).

## Examples

* Some kinds of [[type-theoretic model categories]] are required to be simplicially locally cartesian closed.

## Related entries

* [[locally cartesian closed category]]

* [[cartesian closed enriched category]]

* [[monoidal enriched category]]

* [[locally cartesian closed model category]]

## References

* {#MO} [Enriched cartesian closed categories on MO](https://mathoverflow.net/q/322085)

* {#MO2} [Simplicial cartesian closed categories on MO](https://mathoverflow.net/q/322917)

[[!redirects locally cartesian closed enriched category]]
[[!redirects locally cartesian closed enriched categories]]
[[!redirects enriched locally cartesian closed category]]
[[!redirects enriched locally cartesian closed categories]]
[[!redirects simplicially locally cartesian closed category]]
[[!redirects simplicially locally cartesian closed categories]]


locally cartesian closed functor

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

#Contents#
* table of contents
{:toc}

## Definition

A **locally cartesian closed functor** is a [[functor]] between [[locally cartesian closed categories]] that preserves the locally cartesian closed structure up to coherent [[isomorphism]].

## Examples

* Every [[logical functor]] between [[toposes]] is locally cartesian closed.
* Every [[locally connected geometric morphism]] is locally cartesian closed. ([[Elephant]] C3.3.1)

## Related concepts

* [[cartesian closed category]], [[locally cartesian closed category]]

* [[cartesian closed functor]], **locally cartesian closed functor**

* [[cartesian closed model category]], [[locally cartesian closed model category]]

* [[cartesian closed (∞,1)-category]] [[locally cartesian closed (∞,1)-category]]

## References

[[Elephant]] C3.3.1
[[!redirects locally cartesian closed functors]]


locally cartesian closed model category

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

#Contents#
* table of contents
{:toc}

## Idea

A _locally cartesian closed model category_ is a [[locally cartesian closed category]] which is equipped with the structure of a [[model category]] in a compatible way, namely such that all right [[base change]]-[[adjunctions]] along [[fibrations]] are [[Quillen adjunctions]], hence such that all their [[dependent product]]-functors are [[right Quillen functors]].

\begin{remark}\label{RelationToCartesianClosedModelCategories}
**(relation to [[cartesian closed model categories]])** \linebreak
Beware that, despite the terminology, the axioms on a locally cartesian closed model category (Def. \ref{LocallyCartesianClosedModelCategory}) do *not* imply that the underlying [[model category]] (or any of its [[slice model categories]]) is a [[cartesian closed model category]] -- and in most examples this is not the case. Namely, the axioms here (eq:RightBaseChangeQuillenAdjunction) only require [[Quillen functors]] in one variable (the second variable for [[internal homs]], with the other variable a fixed [[fibrant object]]) where those of a [[cartesian closed model category]] require [[Quillen bifunctors]].
\end{remark}

## Definition

\begin{definition}\label{LocallyCartesianClosedModelCategory}
**(locally cartesian closed model category)** \linebreak
A **locally cartesian closed model category** is

* a [[model category]] $\mathcal{C}$,

* whose underlying category is a [[locally cartesian closed category]]

* such that for every [[fibration]] between [[fibrant objects]]

  \[
    \label{FibrationBetweenFibrantObjects}
    \array{
      A &\underoverset{\;\;\;\;\;\in Fib\;\;\;\;\;}{g}{\longrightarrow}& B
      \\
      && \big\downarrow {}^{\mathrlap{\in Fib}}
      \\
      && \ast
    }
  \]

  the [[dependent product]]/[[base change]] [[adjunction]]

  $$
    g^*
    \;\colon\;
    \mathcal{C}/B
       \rightleftarrows
    \mathcal{C}/A
    \;\colon\;
    \Pi_g
  $$

  is a [[Quillen adjunction]] between the corresponding [[slice model structures]].

Concretely, this means that both [[cofibrations]] and [[trivial cofibrations]] are [[pullback-stability|stable]] under [[pullback]] along fibrations between [[fibrant objects]].

Equivalently this means that for all [[fibrations]] $A \to B$ the [[internal hom]] [[adjunction]] in the [[slice category]] over $B$

\[
  \label{RightBaseChangeQuillenAdjunction}
  (-) \times_{\mathcal{C}/_B} A
  \;:\;
  \mathcal{C}/_B
  \rightleftarrows
  \mathcal{C}/_B
  \;:\;
  [A, -]_{\mathcal{C}/_B}
\]

is a [[Quillen adjunction]].
\end{definition}

## Examples
 {#Examples}

\begin{example}\label{RightProperModelCategoriesWithCofibsTheMonosAreLcc}
Any [[right proper model category]] whose underlying category is [[locally cartesian closed category|locally cartesian closed]] and in which the [[cofibrations]] are the [[monomorphisms]] is a locally cartesian closed model category.
\end{example}
\begin{proof}
The [[fiber product]]/[[pullback]] functor $g^\ast$

  * is a left adjoint by local cartesian closure of the underlying category,

  * preserves cofibrations because these are the monomorphisms and hence are preserved by pullback (by [this prop.](monomorphism#MonomorphismsArePreservedByPullback)),

  * {#PullbackPreservesWeakEquivalences} preserves weak equivalences, and hence acyclic cofibrations by the previous item, due to right properness -- using here the assumption (eq:FibrationBetweenFibrantObjects) that $g$ is a fibration.

In summary this means that $g^\ast$ is a [[left Quillen functor]].
\end{proof}

\begin{example}
Example \ref{RightProperModelCategoriesWithCofibsTheMonosAreLcc} subsumes the following classes of examples, in increasing generality:

* the [[classical model structure on simplicial sets]],

* any [[injective global model structure on simplicial presheaves]],

* any [[injective local model structure on simplicial presheaves]] whose weak equivalences are detected [[stalk]]-wise (by the discussion [there](model+structure+on+simplicial+presheaves#StalkwiseWeakEquivalencesImpliesRightProperness)),

* any right proper [[Cisinski model structure]].

\end{example}

## Versus locally cartesian closed $(\infty,1)$-categories
 {#VersusLCCInfinityCategories}

{#LCCInfinityCategoriesFromLCCModelCategories}
It is easy to see that the $(\infty,1)$-category [[simplicial localization|presented]] by a locally cartesian closed model category is itself [[locally cartesian closed (infinity,1)-category|locally cartesian closed]]: With the assumption (eq:FibrationBetweenFibrantObjects) that $g$ is a fibration between fibrant objects, it follows (by [this Prop](homotopy+pullback#HomotopyPullbackByOrdinaryPullback)) that pullback along $g$ models the correct [[homotopy pullback]].

{#LCCModelCategoriesFromLCCInfinityCategories}
Conversely, any [[locally presentable (infinity,1)-category|locally presentable]] locally cartesian closed $(\infty,1)$-category can be presented by some right proper Cisinski model category, which is therefore a locally cartesian closed model category; see [[locally cartesian closed (infinity,1)-category#Presentations|there]] for the proof.

### Versus homotopy type theories
 {#VersusHomotopyTypesTheories}

* Modulo questions of strictness and [[coherence]] (see at *[[initiality conjecture]]* for details), locally cartesian closed model categories provide [[categorical semantics]] for [[homotopy type theory]] with [[function extensionality]].

## Related concepts

* [[cartesian closed category]], [[locally cartesian closed category]]

* [[cartesian closed functor]], [[locally cartesian closed functor]]

* [[cartesian closed model category]]

* [[locally cartesian closed enriched category]]

* [[cartesian closed (∞,1)-category]] [[locally cartesian closed (∞,1)-category]]

[[!redirects locally cartesian closed model categories]]


locally closed set

A [[subset]] $A$ of a [[topological space]] $X$ is **locally closed** if it is a [[closed subset]] of an [[open subspace]] of $X$. Equivalently, every point in $A$ has a neighborhood $U\subset X$ such that $A\cap U$ is closed in $U$.

A locally closed subset for the [[Zariski topology]] on an [affine space](http://ncatlab.org/nlab/show/affine%20space#further_remarks_11)
over an [[algebraically closed field]] is called an (embedded) _quasiaffine variety_, and a locally closed subset for the Zariski topology on a [[projective space]] over an algebraically closed field is called an (embedded) _quasiprojective variety_.

## Related entries

* [[locally closed subtopos]]

category: topology
[[!redirects locally closed subset]]


locally closed subtopos

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

#Contents#
* table of contents
{:toc}

## Idea

A **locally closed subtopos** generalizes the concept of a [[locally closed set|locally closed subspace]] from topology to toposes.

## Definition

A [[subtopos]] $\mathcal{E}_j\hookrightarrow\mathcal{E}$ is called _locally closed_ if it is the meet $\mathcal{E}_j=\mathcal{E}_o\cap\mathcal{E}_c$ of an [[open subtopos]] $\mathcal{E}_o$ and a [[closed subtopos]] $\mathcal{E}_c$ in the lattice of subtoposes of $\mathcal{E}$.

## Properties

* Since $\mathcal{E}$ is closed as well as open in itself, its closed and open subtoposes are trivially locally closed.

* Since open and closed subtoposes are [[complement|complemented]] in the [[lattice of subtoposes]], the join of the complements $\bar\mathcal{E}_{o}\cup\bar\mathcal{E}_{c}$ provides a **complement** for a locally closed subtopos $\mathcal{E}_j=\mathcal{E}_o\cap\mathcal{E}_c$.

* $\mathcal{E}_j$ is locally closed iff $\mathcal{E}_j\hookrightarrow\mathcal{E}$ can be factored into an open followed by a closed inclusion.

* A locally closed subtopos of an [[exponentiable topos]] is itself exponentiable (cf. Johnstone ([2002](#J02), p.749)).

## Examples

* A way to realize a topos $\mathcal{E}_j$ as a locally closed subtoposes is by repeated [[Artin gluing]] to appropriate left exact functors $f_1:\mathcal{E}_j\to \mathcal {F}_1$ and $f_2:\mathcal{F}_2\to Gl(f_1)$. Then $\mathcal{E}_j$ is open in $Gl(f_1)$ which itself is closed in $Gl(f_2)$ whence by the above remark on factorisations $\mathcal{E}_j\hookrightarrow Gl(f_2)$ is locally closed.

* Locally closed subspaces $Y$ of topological spaces $X$ yield locally closed subtoposes of the corresponding sheaf topos $Sh(Y)\hookrightarrow Sh(X)$ e.g. let $X$ be the space on $\{a,b,c\}$ with non-trivial open subsets $\{a\}$, $\{a,b\}$. Then $\{b\}$ as the intersection of $\{a,b\}$ with the closed subset $\{b,c\}$ is a locally closed subset to which a neither closed nor open but locally closed copy of $Set$ corresponds in the lattice of subtoposes of $Sh(X)$.

## Remark

Bunge and Niefield ([2000](#BN00)) consider general notions of _locally closed subcategory_ and _locally closed geometric morphism_.

## Related entries

* [[open subtopos]]
* [[closed subtopos]]
* [[dense subtopos]]
* [[locally closed set]]
* [[co-Heyting boundary]]

## References

* [[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, LNM **269** Springer Heidelberg 1972. (Exposé IV, ex.9.4.9., pp.462-463)

* {#BN00}[[Marta Bunge|M. Bunge]], [[Susan Niefield|S. Niefield]], _Exponentiability and single universes_ , JPAA **148** (2000) pp.217-250.

* [[Peter Johnstone]], _Conditions Related to de Morgan's Law_ , pp.479-491 in LNM **753** Springer Heidelberg 1979.

* {#J02}[[Peter Johnstone]], _Sketches of an Elephant vol. 2_ , Cambridge UP 2002.

* [[Anders Kock|A. Kock]], T. Plewe, _Glueing analysis for complemented subtoposes_ , TAC **2** (1996) pp.100-112. ([pdf](http://www.tac.mta.ca/tac/volumes/1996/n9/n9.pdf))


locally cocartesian coclosed category

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## Idea

The [[dual]] of a [[locally cartesian closed category]].

## Definition

A **locally cocartesian coclosed category** is a [[category]] $C$ whose [[coslice categories]] $x/C$ are all [[cocartesian coclosed categories]]. Equivalently, it is a category $C$ with all [[coexponentiable morphisms]].

## Properties

A locally cocartesian coclosed category should be the [[categorical semantics]] of [[codependent type theory]].

## See also

* [[coexponential object]]

* [[coslice category]]

* [[cocartesian coclosed category]]

* [[codependent type theory]]

* [[coinductive type]]

* [[cotopos]]

* [[locally cartesian closed category]]


locally compact and second-countable spaces are sigma-compact

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## Statement

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###### Proposition

Let $X$ be a [[topological space]] which is

1. [[locally compact topological space|locally compact]] (in the sense that every point has an [[open neighbourhood]] whose [[topological closure]] is [[compact topological space|compact]]),

1. [[second-countable topological space|second-countable]],

Then $X$ is [[sigma-compact topological space|sigma-compact]].

In particular then $X$ is also [[paracompact topological space|paracompact]] since [[locally compact and sigma-compact spaces are paracompact]].

=--

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

We need to produce a [[countable cover]] of $X$ by [[compact subspaces]].

By [[second-countable topological space|second-countability]] there exists a [[countable set|countable]] [[base for a topology|base]] of open subsets

$$
  \beta
    =
  \left\{
    B_i \subset X
  \right\}_{i \in I}
  \,.
$$

By [[locally compact topological space|local compactness]], every point $x \in X$ has an open neighbourhood $V_x$ whose [[topological closure]] $Cl(V_x)$ is [[compact topological space|compact]].

By definition of [[base of a topology]], there exists $B_x \in \beta$ such that ${x} \subset B_x \subset V_x$, hence $Cl(B_x) \subset Cl(V_x)$. Since $Cl(V_x)$ is compact by assumption, and since [[closed subspaces of compact spaces are compact]] it follows that $B_x$ is compact.

Applying this for each point yields that

$$
  X = \underset{x \in X}{\cup} Cl(B_x)
  \,.
$$

But since there is only a [[countable set]] of base elements $B$ to begin with, there is a countable subset $J \subset X$ such that

$$
  X = \underset{x \in J}{\cup} Cl(B_x)
  \,.
$$

Hence

$$
  \{Cl(B_x) \subset X\}_{x \in J}
$$

is a countable cover of $X$ by compact subspaces.

=--

## Related statements

* [[locally compact and sigma-compact spaces are paracompact]]

* [[second-countable regular spaces are paracompact]]

[[!redirects locally compact and second-countable spaces are paracompact]]


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## Statement

+-- {: .num_lemma #LocallyCompactAndSigmaCompactImpliesGoodNestedCover}
###### Lemma

Let $X$ be a [[topological space]] which is

1. [[locally compact topological space|locally compact]]

   in the sense that for every open neighbourhood $U$ there exists a smaller open neighbourhood whose topological closure is compact and still contained in $U$;

1. [[sigma-compact topological space|sigma-compact]].

Then there exists a [[countable open cover]] $\{U_i \subset X\}_{i \in \mathbb{N}}$ of $X$ such that for each $i \in I$

1. the [[topological closure]] $Cl(U_i)$ is a [[compact topological space|compact]] [[subspace]]

1. $Cl(U_i) \subset U_{i +1}$.

=--

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

By [[sigma-compact topological space|sigma-compactness]] of $X$ there exists a [[countable cover]] $\{K_i \subset X\}_{i \in \mathbb{N}}$ of [[compact topological space|compact]] [[subspaces]]. We use these to construct the required cover by [[induction]].

For $i = 0$ set

$$
  U_0 \coloneqq \emptyset
  \,.
$$

Then assume that for $n \in \mathbb{N}$ we have constructed a set $\{U_i \subset X\}_{i \in \{1, \cdots, n\}}$ with the required properties.

In particular this implies that

$$
  Q_n \coloneqq Cl(U_n) \cup K_{n-1} \;\subset X
$$

is a compact subspace. We now construct an open neighbourhood $U_{n+1}$ of this union as follows:

Let $\{U_x \subset X\}_{x \in Q_n}$ be a set of open neighbourhood around each of the points in $Q_n$. By [[locally compact topological space|local compactness]] of $X$, for each $x$ there is a smaller open neighbourhood $V_x$ with

$$
  \{x\} \subset V_x \subset \underset{\text{compact}}{\Cl(V_x)} \subset U_x
  \,.
$$

So $\{V_x \subset X\}_{x \in Q_n}$ is still an open cover of $Q_n$. By compactness of $Q_n$,  there exists a [[finite set]] $J_n \subset Q_n$ such that $\{V_x \subset X\}_{x \in J_n}$ is a [[finite open cover]]. The union

$$
  U_{n + 1} \coloneqq \underset{x \in J}{\cup} V_x
$$

is an open neighbourhood of $Q_n$, hence in particular of $Cl(U_n)$. Moreover,  since finite unions of compact spaces are compact ([this prop.](compact+space#UnionsAndIntersectionOfCompactSubspaces)) and since the closure of a finite union is the union of the closures ([this prop.](closed+subspace#ClosureOfAFiniteUnionIsUnionOfTheClosures)), the closure of $U_{n+1}$ is compact:

$$
  \begin{aligned}
    Cl(U_{n+1})
    &=
    Cl\left(  \underset{x\in J_n}{\cup} V_x \right)
    \\
    & =
    \underset{x \in J_n}{\cup} \underset{\text{compact}}{Cl( V_x )}
  \end{aligned}
  \,.
$$

This produces by [[induction]] a set $\{U_i \subset X\}_{i \in \mathbb{N}}$ with $Cl(U_i)$ compact and $Cl(U_i) \subset U_{i+1}$ for all $i \in \mathbb{N}$. It remains to see that this is a cover. This follows since by construction each $U_{i+1}$ is an open neighbourhood not just of $Cl(U_{i})$ but in fact of $Q_i$, hence in particular of $K_i$, and since the $K_i$ form a cover:

$$
  \underset{i \in \mathbb{N}}{\cup} U_i
  \supset
  \underset{i \in \mathbb{N}}{\cup} K_i
   =
  X
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition
**(locally compact and sigma-compact spaces are paracompact)**

Let $X$ be a [[topological space]] which is

1. [[locally compact topological space|locally compact]];

1. [[sigma-compact topological space|sigma-compact]].

Then $X$ is also [[paracompact topological space|paracompact]].

=--

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

Let $\{U_i \subset X\}_{i \in I}$ be an open cover of $X$. We need to show that this has a refinement by a [[locally finite cover]].

By lemma \ref{LocallyCompactAndSigmaCompactImpliesGoodNestedCover} there exists a countable open cover $\{V_n \subset X\}_{n \in \mathbb{N}}$ of $X$ such that for all $n \in \mathbb{N}$

1. $Cl(V_n)$ is compact;

1. $Cl(V_n) \subset V_{n+1}$.

Notice that the [[complement]] $Cl(V_{n+1}) \setminus V_n$ is compact, since $Cl(V_{n+1})$ is compact and $V_n$ is open (by [this lemma](compact+space#IntersectionCompactWithOpen)).

By this compactness, the cover $\{U_i \subset X\}_{i \in I}$ regarded as a cover of the [[subspace]] $Cl(V_{n+1})\setminus V_n$ has a finite subcover $\{U_i \subset X\}_{i \in J_n}$ indexed by a finite set $J_n \subset I$, for each $n \in \mathbb{N}$.

We consider the sets of intersections

$$
  \mathcal{U}_n
    \coloneqq
  \{ U_i \cap ( V_{n+2} \setminus Cl(V_{n-1}) )  \}
  \,.
$$

Since $V_{n+2} \setminus Cl(V_{n-1})$ is open, and since $Cl(V_{n+1}) \subset V_{n+2}$ by construction, this is still an open cover of $Cl(V_{n+1})\setminus V_n$. We claim now that

$$
  \mathcal{U} \coloneqq \underset{n\in \mathbb{N}}{\cup} \mathcal{U}_n
$$

is a locally finite refinement of the original cover, as required:

1. $\mathcal{U}$ is a refinement, since by construction each element in $\mathcal{U}_n$ is contained in one of the $U_i$;

1. $\mathcal{U}$ is still a covering because by construction it covers $Cl(V_{n+1}) \setminus V_n$ for all $n \in \mathbb{N}$, and since by the nested nature of the cover $\{V_n \subset X\}_{n \in \mathbb{N}}$ also  $\{Cl(V_{n+1}) \setminus V_n\}_{n \in \mathbb{N}}$ is a cover of $X$.

1. $\mathcal{U}$ is locally finite because each point $x \in X$ has an open neighbourhood of the form $V_{n+2} \setminus Cl(V_{n-1})$ (since these also form an open cover, by the nestedness) and since by construction this has trivial intersection with $\mathcal{U}_{\geq n+3}$ and since all $\mathcal{U}_n$ are finite, so that also $\underset{k \lt n+3}{\cup} \mathcal{U}_k$ is finite.

=--

## Related concepts

* [[second-countable regular spaces are paracompact]]

## References

* [[Akhil Mathew]], _[Paracompactness](https://amathew.wordpress.com/2010/08/17/paracompactness/)_


locally compact groupoid

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## Idea

There are a number of slightly different definitions in the literature, this page will try to give a general overarching definition.

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

A [[topological groupoid]] $X_1 \stackrel{\overset{s}{\to}}{\underset{t}{\to}} X_0$ is called **locally compact** if the maps $s$ and $t$ are [[locally proper]] [[open maps]].

=--

This is far more general than is usually assumed. One can additionally assume that either $X_1$ is locally compact or $X_0$ is locally compact, or that $X_0$ is Hausdorff or $X_1$ is locally Hausdorff, or that $X_1$ is [[sigma-compact|∞-compact]]

One usually wants to place a [[Haar system on a locally compact groupoid|Haar system]] on a locally compact groupoid. The obvious topological requirement of local compactness is further refined in order to make such a system of measures well-behaved.

## Examples

* Any finite-dimensional [[Lie groupoid]] is locally compact
* The action groupoid of a [[locally compact group]] on a [[locally compact topological space]] is (?should be) a locally compact groupoid.

## References

An early reference is

* Jean Renault, _A Groupoid Approach to C*-Algebras_, Lecture Notes in Mathematics **793** (1980) doi:[10.1007/BFb0091072](https://doi.org/10.1007/BFb0091072)

A version where Hausdorffness is weakened so that only the space of objects is Hausdorff is

* Jean-Louis Tu, _Non-Hausdorff groupoids, proper actions and K-theory_, Documenta Mathematica, Vol. 9 (2004) pp565-597, [journal page](https://www.math.uni-bielefeld.de/documenta/vol-09/26.html), arXiv:[math/0403071](https://arxiv.org/abs/math/0403071)


locally compact locale

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## Definition

A [[locale]] is **locally compact** if its [[frame]] of opens is a [[continuous poset]].

## Properties
 {#Propertis}

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

In [[classical mathematics]], every locally compact locale is [[spatial locale|spatial]], hence may be regarded as a [[locally compact topological space]].

=--

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

Prop. \ref{UsingClassicalLogicLocallyCompactLocaleIsSpatial} may fail in [[constructive mathematics]]. For instance any [[spectrum of a commutative ring]], considered as a locale, is locally compact, but may fail to be spatial (because of the lack of [[ideal|filters]]).

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

Locally compact locales are also exactly the [[exponentiable objects]] in the [[category of locales]].

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## Related concepts

* [[compact locale]]

[[!redirects locally compact locales]]


locally compact topological group

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## Definition

A priori a __locally compact topological group__ is a [[topological group]] $G$ whose underlying [[topological space]] is [[locally compact topological space|locally compact]].

Typically it is also assumed that $G$ is [[Hausdorff space|Hausdorff]]. (Notice that if not, then $G/\overline{\{1\}}$ is Hausdorff.).

One often says just "locally compact group".

## Properties

### In harmonic analysis

We take here locally compact groups $G$ to be also Hausdorff.

Locally compact topological groups are the standard object of study in classical abstract [[harmonic analysis]]. The crucial properties of locally compact groups is that they posses a left (right) [[Haar measure]] $\rho$ and that $L^1(\rho)$ has a structure of a
[[Banach algebra|Banach]] $*$-[[star-algebra|algebra]].

A left (right) [[Haar measure]] on a locally compact topological group is a nonzero [[Radon measure]] which is invariant under the left (right) multiplications by elements in the group. A topological subgroup $H$ of a locally compact topological group $G$ is itself locally compact (in [[induced topology]]) iff it is closed in $G$.

### Completeness

Again taking locally compact groups $G$ to be Hausdorff, such are [[complete space|complete]] both with respect to their left [[uniform space|uniformity]] and their right uniformity. For if $\{x_\alpha\}$ is a [[Cauchy net]] in $G$ and $U$ is a compact neighborhood of the identity $e$, then there is $\alpha$ so large that $x_\beta x_\alpha^{-1} \in U$ for all $\beta \geq \alpha$. Those elements converge to a point $x \in U$ since $U$ is compact, and the original net converges to $x \cdot x_\alpha$. A similar argument is used for the right uniformity.

### Pontrjagin duality

See at *[[Pontrjagin duality]]*.

### Kazhdan's property (T)

_[[Kazhdan's property (T)]]_

## Related concepts

* [[compact topological group]], [[compact Lie group]]

* [[maximal compact subgroup]], [[closed subgroup]]

* [[locally compact groupoid]]

* [[coset space coprojections admitting local sections]]

## References

* [[Linus Kramer]], *Locally Compact Groups*, 2017 ([pdf](https://www.uni-muenster.de/AGKramer/content/LCManuscript.pdf), [[Kramer_LocallyCompactGroups.pdf:file]])

* Gerald B. Folland, _A course in abstract harmonic analysis_, Studies in Adv. Math. CRC Press 1995

See also:

* Wikipedia, *[Locally compact group](https://en.wikipedia.org/wiki/Locally_compact_group)*

[[!redirects locally compact topological groups]]

[[!redirects locally compact group]]
[[!redirects locally compact groups]]