locally finite cover

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## Idea

A _locally finite cover_ is a [[cover]] which in a suitable sense looks _locally_ like a [[finite cover]].

## Definition

+-- {: .num_defn #LocallyFiniteCover}
###### Definition
**(locally finite cover)**

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

A [[cover]] $\{U_i \subset X\}_{i \in I}$ of $X$ by [[subsets]] of $X$  is called  _locally finite_  if it is a [[locally finite set of subsets]], hence
if for all points $x \in X$, there exists a  [[neighbourhood]] $U_x \supset \{x\}$ such that it [[intersection|intersects]] only [[finite number|finitely many]] elements of the cover, hence such that  $U_x \cap U_i \neq \emptyset$ for only a [[finite number]] of $i \in I$.

If $\{U_i \subset X\}_{i \in I}$ is an [[open cover]], then it is called a _locally finite open cover_.

=--

+-- {: .num_remark #AlternativeCharacterizations}
###### Remark
**(alternative characterizations of local finiteness)**

Let  $X$ be a [[topological space]] and let $\{U_i \to X\}_{i \in I}$ be a [[cover]] by [[subsets]]. Then the following are equivalent:

1. $\{U_i \subset X\}_{i \in I}$ is locally finite (def. \ref{LocallyFiniteCover});

1. there exist an _[[open cover]]_ $\{V_j \subset X\}_{j \in J}$ such that for each $j \in J$ there is a [[finite number]] of $i \in I$ that $V_j$ intersects $U_i$.

This is because the various $V_i$ constitute open neighbourhoods for all points $x \in X$.

Moreover, suppose that $\{V_j \subset X\}_{j \in J}$ is a cover by any subsets (not necessarily open), but that it is itself a [[locally finite set of subsets]]. Then if for all $j \in J$ there are a finite number of $i \in I$ such that $U_i$ intersects $V_j$, it follows again that also $\{U_i \subset X\}_{i \in I}$ is locally finite.

This is because by the local finiteness of $\{V_j \subset X\}_{j \in J}$ we have for every point $x \in X$ an open neighbourhood $O_x \supset \{x\}$ which intersects only a finite number of the $V_j$, and since each of these intersects only a finite number of the $U_i$, in total also $O_x$ can only intersect a finite number of the $U_i$.

=--

## Properties

The following says that if there exists a [[locally finite cover|locally finite]] [[refinement]] of a cover, then in fact there exists one with the same index set as the original cover.

+-- {: .num_lemma #LocallyFiniteRefinementImpliesLocallyFiniteCoverWithOriginalIndexSet}
###### Lemma
**(locally finite refinement induces locally finite cover with original index set)**

Let $(X,\tau)$ be a [[topological space]], let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]], and let $\{V_j \subset X\}_{j \in J}$, be a [[refinement]] to a [[locally finite cover]].

By definition of [[refinement]] we may [[axiom of choice|choose]] a [[function]]

$$
  \phi \colon J \to I
$$

such that

$$
  \underset{j \in J}{\forall}\left( V_j \subset U_{\phi(j)} \right)
  \,.
$$
Then  $\left\{ W_i \subset X \right\}_{i \in I}$ with

$$
  W_i
    \;\coloneqq\;
  \left\{
    \underset{j \in \phi^{-1}(\{i\})}{\cup} V_j
  \right\}
$$

is still a [[refinement]] of $\{U_i \subset X\}_{i \in I}$ to a locally finite cover.

=--

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

It is clear by construction that $W_i \subset U_i$, hence that we have a [[refinement]]. We need to show local finiteness.

Hence consider $x \in X$. By the assumption that $\{V_j \subset X\}_{j \in J}$ is locally finite, it follows that there exists an [[open neighbourhood]] $U_x \supset \{x\}$ and a [[finite set|finite]] [[subset]] $K \subset J$ such that

$$
  \underset{j \in J\backslash K}{\forall} \left( U_x \cap V_j = \emptyset \right)
  \,.
$$

Hence by construction

$$
  \underset{i \in I\backslash \phi(K)}{\forall} \left( U_x \cap W_i = \emptyset \right)
  \,.
$$

Since the [[image]] $\phi(K) \subset I$ is still a [[finite set]], this shows that $\{W_i \subset X\}_{i \in I}$ is locally finite.

=--

+-- {: .num_lemma}
###### Lemma
**([[shrinking lemma]])**

Let $(X,\tau)$ be a [[normal topological space]], and let $\{U_i \subset X\}_{i \in I}$ be a [[locally finite cover|locally finite]] [[open cover]]. Then there exists a shrinking to a locally finite open cover $\{V_i \subset X\}_{i \in I}$ whose [[topological closure|closures]] $Cl(-)$ are still contained in the original cover:

$$
  V_i \subset Cl(V_i) \subset U_i
  \,.
$$

=--

## Related concepts

* [[locally finite set of subsets]]

* [[locally finite cell complex]]

* [[cover]], [[open cover]],

* [[paracompact topological space]]

[[!redirects locally finite covers]]

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

[[!redirects locally finite closed cover]]
[[!redirects locally finite closed covers]]

[[!redirects locally finite family]]
[[!redirects locally finite families]]

[[!redirects locally finite]]


locally finite set of subsets

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## Definition

Given a [[topological space]] $X$, then a set of [[subset]] $\{S_i \subset X\}_{i \in I}$ is _locally finite_ if every point $x \in X$ [[intersection|intersects]] only a [[finite number]] of the $S_i$.

Often this property is considered for [[open covers]], see at _[[locally finite open cover]]_. But the condition also plays a role for collections of subsets which are not open or not covering, for instance in [[Michael's theorem]] ([Michael 53, theorem 1](#Michael53)).

## Related concepts

* [[countably locally finite set of subsets]]

* [[countably locally discrete set of subsets]]

* [[locally finite cover]]

* [[locally finite cell complex]]

## References

* {#Michael53} [[Ernest Michael]], _A note on paracompact spaces_, Proc. Amer. Math. Soc. 1953 ([jstor](http://www.jstor.org/stable/2032419), [pdf](http://www.renyi.hu/~descript/papers/Michael_parakompakt.pdf))

[[!redirects locally finite sets of subsets]]


locally finite type

> This article is about locally finite types in [[dependent type theory]]. For the notion of  [[homomorphisms]] between [[schemes]] which are locally of finite type, see [[morphism of finite type]].

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## Definition

In [[dependent type theory]], a **locally finite type** is a type $A$ such that for all elements $x:A$ and $y:A$, there is a natural number $n:\mathbb{N}$ and an [[equivalence of types]] between the [[identity type]] $x =_A y$ and the standard [[finite type]] $\mathrm{Fin}(n)$.

The type of all finite types is a [[univalent universe]] $Fin$; thus one could also refer to a locally finite type as a **locally $Fin$-small type**, since locally finite types are precisely the types that are [[locally small type|locally small]] relative to the univalent universe $Fin$ of finite types.

## Examples

\begin{example}
Every [[locally finite category|locally finite]] [[univalent category]] is a locally finite type.
\end{example}

\begin{example}
The locally finite [[h-sets]] are precisely the [[h-sets]] with [[decidable equality]]
\end{example}

## Properties

Since finite types are [[h-sets]], every locally finite type is an [[h-groupoid]], because its identity types are all h-sets. In particular, locally finite types are precisely the [[locally finite category|locally finite]] [[univalent categories]] which are [[groupoids]].

## See also

* [[finite set]], [[finite type]]

* [[FinSet]]

* [[locally finite category]]

* [[locally small type]]

* [[pi-finite type]]


locally finitely presentable category

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## Definition

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###### Definition

A **locally finitely presentable category** is an [[ℵ]]${}_0$-[[locally presentable category]].

=--

We spell out what this means:

An [[object]] $X$ of a [[category]] $C$ is said to be [[finitely presentable object|finitely presentable]] (sometimes called [[compact object|compact]] or 'finite') if the [[representable functor]] $C(X,-)$ is finitary, i.e., preserves [[filtered colimits]]. Write $C_{fp}$ for the full subcategory of $C$ consisting of the finitely presentable objects.

A category $C$ satisfying (any of) the following equivalent conditions is said to be __locally finitely presentable__ (or **lfp**):

1. $C$ has all small [[colimit|colimits]], the category $C_{fp}$ is [[essentially small category|essentially small]], and any object in $C$ is a [[filtered colimit]] of the canonical diagram of finitely presentable objects mapping into it.
1. $C$ is the category of models for an [[essentially algebraic theory]].  Here an 'essentially algebraic theory' is a small category $D$ with finite limits, and its category of 'models' is the category of finite-limit-preserving functors $D \to Set$. (See [[Gabriel--Ulmer duality]].)
1. $C$ is the category of models for a finite limit [[sketch]].
1. $C_{fp}$ has finite colimits, and the restricted [[Yoneda embedding]] $C\hookrightarrow [C_{fp}^{op},Set]$ identifies $C$ with the category of finite-limit-preserving functors $C_{fp}^{op} \to Set$.

Replacing "finite" by "of cardinality less than $\kappa$" everywhere, for some [[cardinal number]] $\kappa$, results in the notion of a [[locally presentable category]].

## Examples

* [[Set]], Graph, [[Pos]], [[Cat]], [[Ab]] are all lfp.

* [[Top]], [[FinSet]] are not lfp.

## Related pages

* [[locally strongly finitely presentable category]]

## References

* P. Gabriel, F. Ulmer, _Lokal präsentierbare Kategorien_, Springer Lect. Notes in Math. __221__ 1971 [Zbl0225.18004](https://zbmath.org/?q=an:0225.18004) [MR327863](https://mathscinet.ams.org/mathscinet-getitem?mr=327863)
* Jiří Adámek, Jiří Rosicky, _Locally presentable and accessible categories_, Cambridge University Press 1994.

Introductory account:

* [[Maru Sarazola]], *An introduction to locally finitely presentable categories* (2017) [[pdf](https://pi.math.cornell.edu/~maru/documents/locally_finitely_presentable_cats.pdf), [[Sarazola_LocallyPresentableCategories.pdf:file]]]

In additive context

* [[Henning Krause]], _Functors on locally finitely presented additive categories_, Colloq. Math. 75:1 (1998) [pdf](http://matwbn.icm.edu.pl/ksiazki/cm/cm75/cm75110.pdf)

If $V$ is a locally finitely presentable symmetric monoidal closed category then there is a bijection between exact $V$-localizations of the $V$-category of $V$-valued $V$-enriched presheaves on a $V$-category $C$
and $V$-enriched Grothendieck topologies on $C$:

* [[Francis Borceux]], Carmen Quinteiro, _A theory of enriched sheaves_, Cahiers Topologie Géom. Différentielle Catég. 37 (1996), no. 2, 145–162 [numdam](https://eudml.org/doc/91577) [MR1394507](https://mathscinet.ams.org/mathscinet-getitem?mr=1394507)
[[!redirects locally finitely presentable categories]]
[[!redirects lfp category]]
[[!redirects lfp categories]]

[[!redirects finitely presentable category]]
[[!redirects finitely presentable categories]]


locally free module

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## Idea

Under the [interpretation of modules as generalized vector bundles](http://ncatlab.org/nlab/show/module#RelationToVectorBundlesInIntroduction) a [[module]] being _locally free_ corresponds to the corresponding [[bundle]] being _locally trivial bundle_, hence a [[fiber bundle]].

Since a _trivial bundle_ corresponds to a [[free module]], a _locally free module_ is such that its [[localization of a module|localization]] to any [[maximal ideal]] is a [[free module]].

## Definition

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

An $R$-module $N$ over a [[Noetherian ring]] $R$ is called a **locally free module** if there is a [[Zariski topology|cover]] by [[ideals]] $I \hookrightarrow R$ such that the [[localization of a module|localization]] $N_I$ is a [[free module]] over the [[localization of a ring|localization]] $R_I$.

=--

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

For $R$ a [[commutative ring]], an $R$-[[module]] $N$ is called a **stalkwise free module** if for every [[maximal ideal]] $I \hookrightarrow R$ the [[localization of a module|localization]] $N_I$ is a [[free module]] over the [[localization of a ring|localization]] $R_I$.

=--

## Properties

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

Let $R$ be a ring and $N \in R$[[Mod]].

The following are equivalent

1. $N$ is [[finitely generated module|finitely generated]] and [[projective module|projective]],

1. $N$ is locally free, def. \ref{LocallyFreeModule} and locally finitely generated.

=--

For instance ([Clark, theorem 7.20](#Clark)).

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

For $R$ a [[Noetherian ring]] and $N$ a [[finitely generated module]] over $R$, $N$ is a locally free module precisely if it is a [[flat module]].

=--

By [Raynaud-Gruson, 3.4.6 (part I)](#RaynaudGruson)

## References

* Pete Clark, _Commutative algebra_ ([pdf](http://math.uga.edu/~pete/integral.pdf))
 {#Clark}

* Michel Raynaud, Laurent Gruson, _Critères de platitude et de projectivité_, Techniques de "platification" d'un module. Invent. Math. 13 (1971), 1–89.
 {#RaynaudGruson}

[[!redirects locally free modules]]


locally full sub-2-category

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## Definition

A __locally full sub 2-category__ is one whose embedding [[2-functor]] $C\to D$ is [[locally fully faithful 2-functor|locally fully faithful]].  This means that each $C(c_1,c_2)$ is a [[full subcategory]] of $D(c_1, c_2)$.

## Examples

* The sub-2-category of $Prof_{rep} \hookrightarrow$ [[Prof]] on all [[representable profunctor]]s is locally full.

* The sub-2-category of the 2-category $T Alg_l$ of algebras for a [[2-monad]] and [[lax morphisms]] between them contains, as a locally full sub-2-category, the 2-category $T Alg_p$ of algebras and pseudo morphisms (or of strict morphisms, if $T$ is strict).

Both of these examples are also [[wide subcategories]].  A wide and locally full sub-2-category is equivalent to an [[F-category]].  See also [[2-category equipped with proarrows]].

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locally fully faithful 2-functor

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## Definition

A [[2-functor]] $F\colon C \to D$ is **locally fully faithful** if each [[functor]] on [[hom-categories]]
$$ F \colon C(x,y) \to D(F x, F y) $$
is [[fully faithful functor|fully faithful]].

## Related pages

* [[2-category equipped with proarrows]]

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

[[!redirects locally fully faithful 2-functors]]

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locally graded category

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## Idea

The notion of **locally graded category** is a joint generalisation of [[enriched categories]], [[actegories]], and [[powered and copowered categories|powered categories]]. The idea is that we grade the morphisms of a category by a monoidal category $V$ in such a way that [[identity morphisms]] have grade labeled by the [[unit object]] $I$ and the [[composition|composite]] of morphisms is graded by the [[tensor product]] of the separate grades.

## Definition

Let $(V, \otimes, I)$ be a [[monoidal category]]. For small $V$, a **locally $V$-graded category** is a [[enriched category|category enriched]] over the [[presheaf category]] $([V^{op}, Set], \widehat\otimes, y_V)$, regarded with its monoidal structure given by [[Day convolution]].

Explicitly, a locally $V$-graded category $C$ comprises:

- a class $|C|$ of objects
- for all $v \in V$ and $c, c' \in |C|$, a set $C_v(c, c')$ of $v$-graded morphisms
- for every $v$-graded morphism $f : c \to c'$ and $p : v' \to v$, a $v'$-graded morphism $p^* f : c \to c'$
- a $I$-graded identity morphism $1_c$ for each object $c \in |C|$
- a $v \otimes v'$-graded composite morphism $gf : c \to c''$ for each $v$-graded morphism $f : c \to c'$ and $v'$-graded morphism $g : c' \to c''$

satisfying evident laws.

## Locally indexed categories

The presheaf category $[V^{op}, Set]$ is also [[cartesian monoidal category|cartesian monoidal]]. A category enriched in $([V^{op}, Set], \times, 1)$ is called **locally $V$-indexed** by [Levy (2019)](#Levy19).

When $V$ is [[cartesian monoidal category|cartesian monoidal]], the two concepts coincide ([Levy 2019](#Levy19)).

As we have stated the definitions, this is trivial, but both concepts have more elementary reformulations that avoid [[size issues]], for which this is a nontrivial theorem.

For example, with [Levy (2019), slide 21](#Levy19) this gives an abstract proof of the standard fact that [[simplicially enriched categories]] can be viewed as those [[simplicial objects]] in [[Cat]] which take value in [[identity-on-objects functors]] (e.g. [Riehl (2023), Prop. 1.2.3](#Riehl23)).

## References

Locally graded categories were introduced as **large $V$-categories** in:

* [[Richard Wood]], *Indicial methods for relative categories*, PhD thesis (1976) [[hdl:10222/55465](http://hdl.handle.net/10222/55465)]

* [[Richard Wood]],  *$V$-indexed categories*, in *Indexed Categories and Their Applications*, Lecture Notes in Mathematics **661** (1978) 126-140 [[doi:10.1007/BFb0061362](https://link.springer.com/chapter/10.1007/BFb0061362)]

See also

* [[Richard Wood]], *re: module for a category*, message to [[categories mailing list]] (2003) [[link](https://github.com/punkdit/categories/blob/26b751bbcbe6765fe447e805629ff6416f4c38b1/gmane/science/mathematics/categories/2425)]

For the terminology we use see:

* {#Levy19} [[Paul Blain Levy]], *Locally graded categories*, talk (2019) [slides:[pdf](https://www.cs.bham.ac.uk/~pbl/papers/locgrade.pdf), [[Levy-LocallyGradedCategories.pdf:file]]]

* [[Dylan McDermott]], [[Tarmo Uustalu]], _Flexibly graded monads and graded algebras_, in: *Mathematics of Program Construction MPC 2022*, Lecture Notes in Computer Science **13544**, Springer (2022) [[doi:10.1007/978-3-031-16912-0_4](https://dl.acm.org/doi/abs/10.1007/978-3-031-16912-0_4)]

For the example of [[simplicially enriched category]] example see for instance:

* {#Riehl18} [[Emily Riehl]], Prop. 1.2.3 in: _Homotopy coherent structures_, Expositions in Theory and Applications of Categories, **1** (2023) 1-31 [[arXiv:1801.07404](https://arxiv.org/abs/1801.07404), [tac:te1](http://www.tac.mta.ca/tac//expositions/articles/1/te1abs.html)]

[[!redirects locally graded categories]]
[[!redirects locally indexed category]]
[[!redirects locally indexed categories]]


locally groupoidal 2-category

A $2$-[[2-category|category]] is __locally groupoidal__ if each of its [[hom-category|hom-categories]] is a [[groupoid]].  In other words, a locally groupoidal $2$-category is a $(2,1)$-[[(2,1)-category|category]].

As the concept of $2$-category makes sense either [[strict 2-category|strictly]] or weakly (as a [[bicategory]]), so a locally groupoidal $2$-category might be either a locally groupoidal strict $2$-category or a locally groupoidal bicategory.  As for $2$-categories in general, it is really the strictness of the [[2-functor|functors]] between them that matters.


locally Hausdorff topological space

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## Definition

A [[topological space]] is _locally Hausdorff_ if every point has a [[neighbourhood]] (or, without loss of generality, an [[open neighbourhood]]) which is a [[Hausdorff topological space]] when equipped with the [[subspace topology]].

## References

* S. Niefield, _A note on the locally Hausdorff property_, Cahiers TGDC (1983) ([numdam](http://www.numdam.org/item?id=CTGDC_1983__24_1_87_0))

[[!redirects locally Hausdorff topological space]]
[[!redirects locally Hausdorff topological spaces]]

[[!redirects locally Hausdorff space]]
[[!redirects locally Hausdorff spaces]]


locally Heyting-algebraic 2-poset

[[!redirects locally Heyting-agebraic 2-posets]]

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## Definition ##

A [[2-poset]] $C$ is **locally Heyting-algebraic** if for every [[object]] $A:Ob(C)$ and $B:Ob(C)$, the [[hom-object]] $Hom(A,B)$ is a [[Heyting algebra]].

## Examples ##

The 2-poset [[Rel]] of [[sets]] and [[relations]] is locally Heyting-algebraic.

## See also ##

* [[2-poset]]

* [[Heyting algebra]]


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## Idea

An [[(∞,1)-site]] is **locally $\infty$-connected** if it has properties that ensure that  the [[(∞,1)-category of (∞,1)-sheaves]] over it is a [[locally ∞-connected (∞,1)-topos]]

## Definition

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

Call an [[(∞,1)-site]] $C$ **locally contractible** if every constant [[(∞,1)-presheaf]] on it is an [[(∞,1)-sheaf]] in the [[(∞,1)-topos]] over $C$.

=--

More explicitly, this means that every covering sieve $R$ on an object $U\in C$, regarded as a subcategory of $C/U$, is weakly contractible, i.e. its nerve $N(R)$ (which is just itself, if it is incarnated as a [[quasicategory]]) is contractible in the [[Kan-Quillen model structure]] on simplicial sets.  For the sheaf condition for a constant presheaf on $X\in \infty Gpd$ is that the map $Const(X)(U) = X \to \lim_R Const(X)$ is an equivalence, but $\lim_R Const(X) = Map(N(R),X)$, and this is equivalent to $X$ for all $X$ if and only if $N(R)$ is contractible as an $\infty$-groupoid.

## Properties

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

By the general notion of [[(∞,1)-colimit]] the constant $(\infty,1)$-presheaf functor has a left [[adjoint (∞,1)-functor]] given by taking colimits

$$
  Sh_{(\infty,1)}(C)
    \underoverset{\hookrightarrow}{\overset{L}{\longleftarrow}}{\bot}
  PSh_{(\infty,1)}(C)
    \stackrel{
       \overset{\lim_\to}{\longrightarrow}
    }
    {
       \underset{Const}{\leftarrow}
    }
  \infty Grpd
  \,.
$$

Since the [[(∞,1)-category of (∞,1)-sheaves]] sits by a [[full and faithful (∞,1)-functor]] inside presheaves and by assumption that every constant $(\infty,1)$-presheaf is an $(\infty,1)$-sheaf, this implies that we have also [[natural transformation|natural]] [[equivalence in an (∞,1)-category|equivalences]]

$$
  \begin{aligned}
     Sh(X, L Const S)
       &\simeq
     PSh(X, Const S)
     \\
       & \simeq
     \infty Grpd(\lim_\to X , S)
  \end{aligned}
  \,.
$$

=--

## Examples

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

Let $C$ be an 1-[[site]] such that every object $U$ has a [[split hypercover]] $Y \to U$ such that contracting all representables to points yields a weak equivalence. Equivalently, if the [[colimit]] functor $\lim_\to : [C^{op}, sSet] \to sSet$ sends this to a weak equivalence

$$
  \lim_\to Y \stackrel{\simeq}{\longrightarrow} \lim_\to U = *
  \,
$$

Then $C$ is locally $\infty$-connected.

=--

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

We may [[presentable (∞,1)-category|present]] $Sh_{(\infty,1)}(C)$ by the projective [[model structure on simplicial presheaves]] $[C^{op}, sSet]_{proj}$ [[Bousfield localization of model categories|left Bousfield localized]] at the [[Cech nerve]] projections $C(\coprod_i U_i) \to U$ for each [[covering]] family $\{U_i \to U\}$ in $C$.

It is immediate that we have a [[Quillen adjunction]] $(\underset{\rightarrow}{\lim} \dashv const)$ for the [[global model structure on simplicial presheaves]] on both sides. Now by the [recognition theorem for simplicial Quillen adjunctions](http://ncatlab.org/nlab/show/simplicial+Quillen+adjunction#Recognition) for this to descend to a Quillen adjunction on the local model structure it is sufficient that the left adjoint preserves the cofibrations of the local model structure and (already) that the right adjoint preserves the fibration objects. Since left [[Bousfield localization of model categories]] does not change the cofibrations, the first of these is immediate.

This means that to establish the claim it is now sufficient to show that constant simplicial presheaves already satisfy [[descent]] for a locally $\infty$-connected site. This is what we do now.

By the discussion of [[cofibrant resolution]] at [[model structure on simplicial presheaves]] we have that a [[split hypercover]] $Y \to U$ is a cofibrant [[resolution]] in $[C^{op}, sSet]_{proj, loc}$ of $U$.

For $S \in sSet$ a [[Kan complex]] let $Const S : C^{op} \to sSet$ the corresponding constant simplicial presheaf. This is fibrant in $[C^{op}, sSet]_{proj}$. Since every split hypercover is cofibrant, it follows that $Const S$ is an $\infty$-sheaf precisely if for all $U \in C$ and some split hypercover $Y \to U$ we have that the morphism on [[derived hom-space]]s

$$
  [C^{op}, sSet](U, Const S) \to [C^{op}, sSet](Y, Const S)
$$

is a weak equivalence (of [[Kan complex]]es, necessatily). But we have

$$
  [C^{op}, sSet](Y, Const S)
  \simeq
   sSet(\lim_\to Y, S)
$$

and

$$
  [C^{op}, sSet](U, Const S) \simeq S
  \,,
$$

so that the condition is that

$$
  S \to sSet(\lim_\to Y, S)
$$

is a weak equivalence. This is the case for all $S$ precisely if $\lim_\to Y$ is [[contractible]], which is precisely our assumption on $Y$.

=--

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

Let $X$ be a [[locally contractible topological space]]. Then $\hat Sh_{(\infty,1)}(C)$ is a [[locally ∞-connected (∞,1)-topos]].

=--

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

The [[category of open subsets]] $Op(X)$ is not in general a locally $\infty$-connected site according to the above definition. But there is another site of definition for $\hat Sh_{(\infty,1)}(X)$ which is: the full [[subcategory]] $cOp(X) \hookrightarrow Op(X)$  on the [[contractible]] [[open subset]]s.

=--

## Related concepts

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

  * [[connected topos]] / [[∞-connected (∞,1)-topos]]

  * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]] / **locally ∞-connected (∞,1)-site**

  * [[connected site]] / [[∞-connected (∞,1)-site]]

  * [[strongly connected site]] / [[strongly ∞-connected site]]

  * [[totally connected site]] / [[totally ∞-connected site]]

* [[local site]] / [[∞-local site]]

* [[cohesive site]], [[∞-cohesive site]]

[[!redirects locally ∞-connected (∞,1)-site]]
[[!redirects locally ∞-connected (∞,1)-sites]]
[[!redirects locally infinity-connected (infinity,1)-sites]]

[[!redirects locally ∞-connected site]]
[[!redirects locally infinity-connected site]]

[[!redirects locally ∞-connected sites]]
[[!redirects locally infinity-connected sites]]


locally internal category

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

* table of contents
{: toc}

## Idea

A _locally internal category_ is an analogue of a [[large category|large]] but [[locally small category]] relative to an [[elementary topos]], when that topos is thought of as generalizing the category of sets.

More generally, locally internal categories can be defined over any [[category with finite limits]].  The notion is best-behaved when $E$ is [[locally cartesian closed category|locally cartesian closed]] (for instance, in that case the [[codomain fibration]] of $E$ is an example).

## Definition
 {#Definition}

Given a [[category]] $E$ with [[finite limits]] and an [[object]] $X$ in $E$, one notices that the [[slice category]] $E/X$ is a [[symmetric monoidal category]] under [[fiber product]] in $E$.  Hence we can consider [[enriched category|categories enriched]] over $E/X$, i.e. $E/X$-categories.

A __locally internal category__ $C$ over $E$ is given by

* An $(E/X)$-[[enriched category]] $C_X$ for each object $X$ in $E$.  This is thought of as the category of $X$-[[indexed families]] of objects of $E$.

* For each morphism $f \colon X\to Y$ in $E$, an $(E/X)$-[[full subcategory|full embedding]] $\theta_f \colon f^* C_Y\to C_X$.  Here $f^* C_Y$ means the $(E/X)$-enriched category obtained by applying the symmetric monoidal functor $f^* \colon E/Y \to E/X$ to the [[hom-objects]] of the $(E/Y)$-enriched category $C_Y$, and an [[enriched functor]] is a "full embedding" if it induces [[isomorphisms]] on [[hom-objects]].

* $f \mapsto \theta_f$ is functorial up to [[coherence|coherent]] isomorphism.  This means certain [[commuting diagram|diagrams commute]] analogous to those of a [[pseudofunctor]], but with the functors $f^*$ applied at appropriate places to make them [[type checking|typecheck]].

Alternatively, a category locally internal to $E$ is a category [[enriched category|enriched in]] the [[bicategory]] [[Span|$Span(E)$]] admitting certain [[absolute colimits]] (see [Betti–Walters](#BettiWalters1985)).

## Properties

In the [[stack semantics]] of $E$, a locally internal category "looks like" an ordinary locally small category.

Locally internal categories can also be identified with [[Grothendieck fibrations]] or [[indexed categories]] over $E$ which satisfy a certain "representability" or "comprehensibility" condition:

A Grothendieck fibration $p: C \to E$ is called __locally small__ if, for every pair $A,B \in C$, there exists an object of $E_{pA \times pB}$,  $(x,y) : I \to pA \times pB$, and a morphism $f: x^*A \to y^*B \in C_I$, which is __terminal__, in the sense that given another such datum $(J,z,w,g)$, there is a unique map $u: J \to I$ so that $xu = z, yu = w$, and the coherence isomorphisms identify $u^*f$ with $g$. (This is [Elephant](#Elephant) B.1.3.12).

An indexed category $E \to \operatorname{CAT}$ is called locally small if the associated fibration is locally small.

If we also take care of the appropriate morphisms have the following:

+-- {: .un_remark}
###### Remark
(1)  The obvious forgetful functor from locally internal categories to $E$-indexed categories (equivalently, Grothendieck fibrations over $E$) is a [[full sub-2-category | fully faithful 2-functor]]. In particular, every [[indexed functor]] between locally internal categories is an [[enriched functor]]. [Elephant](#Elephant), Proposition B2.2.2.

(2a) Let $S$ be a [[locally cartesian closed category]], let $F:S\to S$ be an $S$-enriched functor whose underlying (ordinary) functor preserves pullbacks. Then $F$ extends to an $S$-[[indexed functor]].

 (2b) ([[Robert Pare]]) If this indexed functor preserves pullbacks (as an indexed functor) and if it induces the given enrichment, this extension is unique (up to a canonical isomorphism). [Elephant](#Elephant)  B2.2.8.
=--

## Related pages

* [[internal category]]
* [[indexed category]]
* [[enriched category]]
* [[enriched indexed category]]

## References

* J. Penon, _Catégories localement internes_, C. R. Acad. Sci. Paris __278__ (1974) A1577-1580 ([gallica](https://gallica.bnf.fr/ark:/12148/bpt6k6236784g/f355))

* Locally internal categories, Appendix in: P. Johnstone, Topos theory, 1977

* Chapter B2.2 of [[Sketches of an Elephant]]{#Elephant}

* {#BettiWalters1985} [[Renato Betti]], [[Robert F. C. Walters]], *Closed bicategories and variable category theory*, Universita degli Studi di Milano (1985), reprinted in: Reprints in Theory and Applications of Categories, **26** (2020) 1-27 $[$[tac:tr26](http://www.tac.mta.ca/tac/reprints/articles/26/tr26abs.html)$]$

[[!redirects locally internal categories]]


locally Klein geometry

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

##Idea

Generalising a [[Klein geometry]], a **locally Klein geometry** is a [[triple]] $(\Gamma, G,H)$  where  $G/H$ is a [[Klein geometry]] (not necessarily [[connected topological space|connected]]) and $\Gamma \subset G$ is a [[discrete group|discrete]] [[subgroup]] [[action|acting]] by left multiplication on the [[coset space]] $G/H$ as a group of [[covering space|covering transformations]] with the [[double coset space]], $\Gamma \backslash G/H$, [[connected]] ([Sharpe 97, Def. 3.10](#Sharpe97), [Sharpe 02, Def. 2.7](#Sharpe02)).

##Related concepts

* [[Clifford-Klein form]]

##References

* {#Sharpe97}  [[Richard Sharpe]], _Differential geometry: Cartan's generalization of Klein's Erlangen program_, Graduate Texts in Mathematics, vol. 166, Springer-Verlag, 1997.

* {#Sharpe02} [[Richard Sharpe]], _An introduction to Cartan Geometries_, in: [[Jan Slovák]], Martin Čadek (eds.): Proceedings of the 21st Winter School "Geometry and Physics". Circolo Matematico di Palermo, Palermo, 2002. Rendiconti del Circolo Matematico di Palermo, Serie II, Supplemento No. 69. pp. [61]--75 ([EuDML](https://eudml.org/doc/220395))

[[!redirects locally Klein geometries]]


locally modeled monoid

#Idea#

A _locally modeled monoid_ or,  _$R$-Ring_, is a generalized quantity in the sense of [[space and quantity]] which is modeled on a [[category of local models]] $R$.

#Definition#

Let $(R,U,L,A)$ be a [[category of local models]]. Then an **$R$-ring** or **monoid locally modeled on $R$** is a co-[[presheaf]]
$$
  R \to Set
$$
which preserves the [[limits]] of shape in $L$.

A morphism of such locally modeled monoids is a [[natural transformation]].

#Examples#

* for $R = Ring^{op}$, $R$-rings are just ordinary rings.
* for $R = $ [[CartSp]], $R$-rings are [[generalized smooth algebra]]s.

#References#

This is definition 1.1.6 of

* David Spivak, _Quasi-smooth derived manifolds_, PhD thesis, Berkeley (2007) ([pdf](http://www.uoregon.edu/~dspivak/files/thesis1.pdf))

where it appears as part of the discussion of [[derived smooth manifold]]s.

[[!redirects locally modelled monoid]]


locally monoidal (infinity,1)-operad

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

#Contents#
* table of contents
{:toc}

## Idea

A _locally monoidal $(\infty,1)$-operad_ (called a _coherent $(\infty,1)$-operad_ in ([Lurie](#Lurie))) is an [[(∞,1)-operad]] $\mathcal{O}$ whose [[∞-module over an ∞-algebra over an (∞,1)-operad|modules]] over $\mathcal{O}$-[[algebra over an (∞,1)-operad|algebras]] come equipped with a well behaved [[tensor product]]

## Definition

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

An [[(∞,1)-operad]] $\mathcal{O}^\otimes$ is **locally monoidal** if

1. it is unital;

1. the underlying [[(∞,1)-category]] $\mathcal{O}$ is an [[∞-groupoid]]

1. (some third condition).

=--

This is ([Lurie, def. 3.3.1.9](#Lurie)).

## Examples

Locally monoidal $(\infty,1)$-operads include

* [[Comm]]

  ([Lurie, example 3.3.1.12](#Lurie))

* [[Assoc]]

  ([Lurie, prop. 4.1.1.16](#Lurie))

* [[little k-cubes operad|little 0-cubes operad]]

  ([Lurie, example 3.3.1.13](#Lurie))

* [[little k-cubes operad]] for all $k \in \mathbb{N}$

  ([Lurie, theorem 5.1.1.1](#Lurie))

## References

Section 3.3.1 of

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#Lurie}

[[!redirects coherent (infinity,1)-operad]]
[[!redirects coherent (∞,1)-operad]]

[[!redirects coherent (∞,1)-operads]]
[[!redirects coherent (infinity,1)-operads]]

[[!redirects locally monoidal (∞,1)-operad]]

[[!redirects locally monoidal (∞,1)-operads]]
[[!redirects locally monoidal (infinity,1)-operads]]


locally multipresentable category

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

# Contents
* table of contents
{:toc}

## Idea

A _locally multipresentable category_ is similar to a [[locally presentable category]], but where we ask only for the existence of [[connected limits]] rather than all [[small limits]].

## Definition

+-- {: .num_defn #MultipresentableCategory}
###### Definition
**(locally multipresentable category)**

A [[category]] $\mathcal{C}$ is called **locally multipresentable** if

1. it is an [[accessible category]];

1. it has all [[connected limits]].

The second condition is equivalent to:

2'. it has all small [[multicolimits]].

=--

## Related concepts

* [[locally presentable category]]

* [[locally polypresentable category]]

* [[multicolimit]]

## References

The definition is due to

* [[Yves Diers]]. _Catégories localisables_. Diss. Paris 6 et Centre universitaire de Valenciennes et du Hainaut Cambrésis (1977) [[[Categories localisables.pdf:file]]]

[[!redirects locally multipresentable]]
[[!redirects locally multipresentable categories]]
[[!redirects multipresentable category]]
[[!redirects multipresentable categories]]


locally n-connected (n+1,1)-topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Topos theory
+-- {: .hide}
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

A [[topos]] may be thought of as a generalized [[topological space]]. Accordingly, the notions of

* [[locally connected space]]

* [[locally simply connected space]]

* locally $2$-connected space

* locally $n$-connected space

* [[locally contractible space]]

have analogs for [[topos]]es, [[(n,1)-toposes]] and [[(∞,1)-topos]]es

* [[locally connected topos]]

* [[locally simply connected (2,1)-topos]]

* **locally $n$-connected $(n+1,1)$-topos**

* **locally $\infty$-connected $(\infty,1)$-topos**

The numbering mismatch is traditional from [[topology]]; see [[n-connected space]].  It reads a bit better if we say _locally $n$-simply connected_ for _locally $n$-connected_, since _locally $1$-(simply) connected_ is _locally simply connected_, but being locally $n$-simply connected is still a property of an $(n+1,1)$-topos.

## Definitions

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

A [[(∞,1)-sheaf (∞,1)-topos]] $\mathbf{H}$ is called
**locally $\infty$-connected** if the (essentially unique)
[[global section]] [[(∞,1)-geometric morphism]]

$$
  (\Delta\dashv\Gamma):
  \mathbf{H} \xrightarrow{\Gamma}\infty\Grpd
$$

extends to an __[[essential geometric morphism]] $(\infty,1)$-geometric morphism__, i.e. there is a further [[left adjoint]] $\Pi$

$$
  (\Pi \dashv \Delta \dashv \Gamma) :
  \mathbf{H}
  \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}}
  \infty Grpd
  \,.
$$

If in addition $\Pi$ preserves the [[terminal object]] we say that $\mathbf{H}$ is an **[[∞-connected (∞,1)-topos]]**.

If $\Pi$ preserves even all [[finite limit|finite]] [[(∞,1)-product]]s we say that $\mathbf{H}$ is a [[strongly ∞-connected (∞,1)-topos]].

If $\Pi$ preserves even all [[finite limit|finite]] [[(∞,1)-limit]]s we say that $\mathbf{H}$ is a [[totally ∞-connected (∞,1)-topos]].

=--

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

In ([Lurie, section A.1](#Lurie)) this is called an $(\infty,1)$-topos
of **locally constant [[shape of an (infinity,1)-topos|shape]]**.

=--

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

For $\mathbf{H}$ a locally $\infty$-connected $(\infty,1)$-topos and $X \in \mathbf{H}$ an [[object]], we call $\Pi X \in $ [[∞Grpd]] the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] of $X$. The ([[categorical homotopy groups in an (∞,1)-topos|categorical]]) [[homotopy group]]s of $\Pi(X)$ we call the [[geometric homotopy groups in an (∞,1)-topos|geometric homotopy groups]] of $X$

$$
  \pi_\bullet^{geom}(X) := \pi_\bullet(\Pi (X))
  \,.
$$

=--

Similarly we have:

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

For $n \in \mathbb{N}$ an $(n+1,1)$-[[(n,1)-topos|topos]] $\mathbf{H}$ is called **locally $n$-connected** if the (essentially unique) [[global section]] geometric morphism is has an extra left adjoint.
=--

For $n = 0$ this reproduces the case of a [[locally connected topos]].

## Examples
{#LocallyContractibleExamples}

### Over locally $\infty$-connected sites

The follow proposition gives a large supply of examples.

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

Let $C$ be a [[locally ∞-connected (∞,1)-site]]/[[∞-connected (∞,1)-site]]. Then the  [[(∞,1)-category of (∞,1)-sheaves]] $Sh_{(\infty,1)}(C)$ is a
locally $\infty$-connected $(\infty,1)$-topos.
=--

See [[locally ∞-connected (∞,1)-site]]/[[∞-connected (∞,1)-site]] for the proof.

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

In ([SimpsonTeleman, prop. 2.18](#SimpsonTeleman))
is stated essentially what the above proposition asserts at the level of [[homotopy category|homotopy categories]]: if $C$ has contractible objects, then there exists a [[left adjoint]] $Ho(\Pi):Ho(Sh_{(\infty,1)}(C)) \to Ho(\infty Grpd)$.
=--

This includes the following examples.

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

The sites [[CartSp]]${}_{top}$ $CartSp_{smooth}$ $CartSp_{synthdiff}$ are locally $\infty$-connected.
The corresponding $(\infty,1)$-toposes are the [[cohesive (∞,1)-topos]]es [[ETop∞Grpd]], [[Smooth∞Grpd]] and [[SynthDiff∞Grpd]].
=--

### Over locally $n$-connected topological spaces
  {#OverLocallynConnectedTopologicalSpaces}

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

For $X$ a [[locally contractible space]], such that $Sh_{(\infty,1)}(X)$  is [[hypercomplete (infinity,1)-topos|hypercomplete]], $Sh_{(\infty,1)}(X)$ is a locally $\infty$-connected $(\infty,1)$-topos.

=--

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

The full subcategory $cOp(X) \hookrightarrow Op(X)$ of the [[category of open subsets]] on the contractible subsets is another site of definition for $Sh_{(\infty,1)}(X)$. And it is a [[locally ∞-connected (∞,1)-site]].
=--

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

For $X$ a [[locally contractible topological space]] we have that the
[[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos]] computes the correct [[homotopy type]] of $X$:

the image of $X$ as the [[terminal object in an (∞,1)-category|terminal object]] in $Sh_{(\inffty,1)}(C)$ under the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]-functor

$$
  \Pi : Sh_{(\infty,1)}(X) \to \infty Grpd
$$

is equivalent to the ordinary [[fundamental ∞-groupoid]] given by the [[singular simplicial complex]]

$$
  \Pi(X) \simeq Sing X
  \,.
$$
=--

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

By using the [[presentable (∞,1)-category|presentations]] of $Sh_{(\infty,1)}(X)$ by the [[model structure on simplicial presheaves]] as discussed at [[locally ∞-connected (∞,1)-site]] one finds that this boils down to the old Artin-Mazur theorem. More on this at [[geometric homotopy groups in an (∞,1)-topos]].
=--

### Locally $\infty$-connected over-$(\infty,1)$-toposes

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

For $\mathbf{H}$ a locally $\infty$-connected $(\infty,1)$-topos, also all its objects $X \in \mathbf{H}$ are locally $\infty$-connected, in that their [[petit topos|petit]] [[over-(∞,1)-toposes]] $\mathbf{H}/X$ are locally $\infty$-connected.

The two notions of fundamental $\infty$-groupoids of $X$ induced this way do agree, in that there is a natural equivalence
$$
  \Pi_X(X \in \mathbf{H}/X) \simeq \Pi(X \in \mathbf{H})
  \,.
$$
=--

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

By the general facts recalled at [[etale geometric morphism]] we have a composite [[essential geometric morphism]]

$$
  (\Pi_X \dashv \Delta_X \dashv \Gamma_X) :
  \mathbf{H}_{/X}
   \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{\X_*}{\to}}}
  \mathbf{H}
   \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}}
   \infty Grpd
$$

and $X_!$ is given by sending $(Y \to X) \in \mathbf{H}/X$ to $Y \in \mathbf{H}$.
=--

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

If in the above $X$ is contractible in that $\Pi X \simeq *$ then $\mathbf{H}/X$ is even an [[∞-connected (∞,1)-topos]].
=--

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

By the discussion there we need to check that $\Pi_X$ preserves the terminal object:

$$
  \Pi_X (X \to X) \simeq \Pi X_! (X \to X) \simeq \Pi X \simeq *
  \,.
$$
=--

## Properties
 {#LocInfConnProperties}

### Relation to slicing

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

Let $\mathcal{X}$ be an $(\infty,1)$-topos and $\{U_i\}_i$ a collection of [[objects]] such that

* the canonical morphism $\coprod_i U_i \to *$ out of their [[coproduct]] to the [[terminal object]] is an [[effective epimorphism in an (∞,1)-category|effective epimorphism]];

* all the [[slice-(∞,1)-toposes]] $\mathcal{X}_{/U_i}$ are locally $\infty$-connected.

Then also $\mathcal{X}$ itself is locally $\infty$-connected.

=--

This appears as ([Lurie, corollary A.1.7](#Lurie)).

### Relation to locally connected toposes
 {#RelationToLocallyConnectedToposes}

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

For $(\Pi \dashv \Delta \dashv \Gamma) :  \mathbf{H} \to \infty Grpd$ a locally $\infty$-connected $(\infty,1)$-topos, its underlying [[(n,1)-topos|(1,1)-topos]] $\tau_{\leq 0} \mathbf{H}$ is a [[locally connected topos]]. Moreover, if $\mathbf{H}$ is strongly connected (the extra left adjoint preserves finite products), then so is $\tau_{\leq 0} \mathbf{H}$.

=--

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

The [[global sections]] [[geometric morphism]]  $\Gamma \simeq \mathbf{H}(*,-)$ is given by homming out of the terminal object and hence preserves [[n-truncated object in an (infinity,1)-category|0-truncated]] objects by definition. Also, by the $(\Pi \dahsv \Delta)$-[[adjunction]] so does $\Delta$: for every $S \in Set \simeq \tau_{\leq }\infty Grpd \hookrightarrow \infty Grpd$ and every $X \in \mathbf{H}$ we have

$$
  \mathbf{H}(X, \Delta(S))
  \simeq
  \infty Grpd(\Pi(X), S)
  \simeq
  Set(\tau_{\leq 0} \Pi(X), S)
  \in
  Set \hookrightarrow \infty Grpd
  \,.
$$

Therefore by essential uniqueness of [[adjoints]] the restriction $\Delta|_{\leq 0} \colon Set \hookrightarrow \infty Grpd \stackrel{\Delta}{\to} \mathbf{H}$ has a [[left adjoint]] given by

$$
  \Pi_0 \coloneqq \tau_{\leq 0} \circ \Pi
  \,.
$$

Finally, by the discussion [here](n-truncated+object+of+an+%28infinity%2C1%29-category#PropertiesGeneral), $\tau_{\leq 0}$ preserves finite limits. Hence $\Pi_0$ does so if $\Pi$ does.

=--

## Further structures

The fact that the terminal geometric morphism is essential gives rise to various induced structures of interest. For instance it induces a notion of

* [[Whitehead tower in an (∞,1)-topos]].

For a more exhaustive list of extra structures see [[cohesive (∞,1)-topos]].

## Related concepts

* [[locally connected topos]] / **locally ∞-connected (∞,1)-topos**

  * [[connected topos]] / [[∞-connected (∞,1)-topos]]

  * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]], [[locally ∞-connected site]]

* [[connected site]]

* [[local site]]

* [[cohesive site]], [[(∞,1)-cohesive site]]

## References

Some discussion of the [[homotopy category of an (∞,1)-category|homotopy category]] of locally $\infty$-connected $(\infty,1)$-toposes is around proposition 2.18 of

* [[Carlos Simpson]], [[Constantin Teleman]], _de Rham's theorem for $\infty$-stacks_ ([pdf](http://math.berkeley.edu/~teleman/math/simpson.pdf))
{#SimpsonTeleman}

Under the term _locally constant shape_ the notion appears in section A.1 of

* {#Lurie} [[Jacob Lurie]], _[[Higher Algebra]]_

See also

* {#Hoyois13} [[Marc Hoyois]], _A note on Étale homotopy_, 2013 ([pdf](http://math.northwestern.edu/~hoyois/papers/etalehomotopy.pdf))

For related references see

* [[geometric homotopy groups in an (∞,1)-topos]]

* [[cohesive (∞,1)-topos]].

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locally nonconstant function

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#### Analysis
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# Contents
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{: toc}

## Definition

A function $f:\mathbb{R}_D \to \mathbb{R}_D$ is __locally nonconstant__ if for all [[Dedekind real numbers]] $x \lt y$ and $t \in \mathbb{R}_D$, there exists a Dedekind real number $z \in \mathbb{R}_D$ with $x \lt z \lt y$ and $\vert f(z) - t \vert \gt 0$.

## Examples

* Every [[monotonic function]] on the Dedekind real numbers is locally nonconstant.

* Every [[real polynomial function]] apart from the [[constant functions]] is locally nonconstant.

## See also

* [[intermediate value theorem]]

* [[monotonic function]]

* [[real polynomial function]]

* [[locally nonzero function]]

## References

* Auke Booij, *Extensional constructive real analysis via locators*, ([arXiv:1805.06781](https://arxiv.org/abs/1805.06781))


locally nonzero function

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# Contents
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## Definition

A function $f:\mathbb{R}_D \to \mathbb{R}_D$ is __locally nonzero__ if for all [[Dedekind real numbers]] $x \in \mathbb{R}_D$ and $y \in \mathbb{R}_D$ such that $x \lt y$, there exists a Dedekind real number $z \in \mathbb{R}_D$ with $x \lt z \lt y$ and $\vert f(z) \vert \gt 0$.

## Examples

* Every [[monotonic function]] on the Dedekind real numbers is locally nonzero.

* Every [[real polynomial function]] apart from the zero polynomial function is locally nonzero.

## See also

* [[intermediate value theorem]]

* [[monotonic function]]

* [[real polynomial function]]

* [[locally nonconstant function]]

## References

* [[Auke Booij]], *Extensional constructive real analysis via locators*, ([arXiv:1805.06781](https://arxiv.org/abs/1805.06781))

[[!redirects locally nonzero]]

[[!redirects locally nonzero functions]]


locally path-connected space

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#Contents#
* table of contents
{:toc}

## Definition

A [[topological space]] is called **locally path-connected** if it has a [[basis of the topology|basis]] of [[path-connected space|path-connected]] [[neighbourhoods]]. In other words, if for every point $x$ and [[neighbourhood]] $V \ni x$, there exists a path-connected neighbourhood $U \subset V$ that contains $x$.

## Properties

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

Let $X$ be a locally path connected space. Then the path connected component $P_x \subset X$ over any point $x \in X$ is an [[open set]].

=--

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

It is sufficient to show that every point $y \in P_x$ has a [[neighbourhood]] $U_y$ which is still contained in $P_x$. But by local path connectedness, $y$ has a neighbourhood $V_y$ which is path connected. It follows by concatenation of paths that $V_y \subset P_x$.

=--

A locally path-connected space is [[connected space|connected]] if and only if it is [[path-connected topological space|path-connected]].

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

The [[connected components]] of a locally path-connected space are the same as its path-connected components.

=--

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

A path connected component is always connected ([this lemma](connected+space#ConnectedPathConnectedSpace)), and in a locally path-connected space is it also open (lemma \ref{InLocallyPathConnectedSpaceThePathConnectedComponentsAreOpen}).
This means that every path-connected component is also connected.

Conversely, it is now sufficient to see that every connected component is path-connected. Suppose it were not, then it would be covered by
more than one disjoint non-empty path-connected components.
But by lemma \ref{InLocallyPathConnectedSpaceThePathConnectedComponentsAreOpen} these
would be all open. This would be in contradiction with the assumption that
$U$ is connected. Hence we have a [[proof by contradiction]].

=--

## Examples

+-- {: .num_example #LocallyPathConnectedEuclideanSpace}
###### Examples
**([[Euclidean space]] is locally path-connected)**

For $n \in \mathbb{N}$ then [[Euclidean space]] $\mathbb{R}^n$ (with its [[metric topology]]) is locally path-connected, since each [[open ball]] is [[path-connected topological space]].

Similarly the [[open intevals]], [[closed intervals]] and [[half-open intervals]], regarded as [[topological subspaces]] of the [[Euclidean space|Euclidean]] [[real line]], are locally path connected.

=--

+-- {: .num_example}
###### Example
**(open subspace of locally path-connected space is locally path connected)**

Every [[open subset|open]] [[subspace]] of a locally path connected topological space is itself locally path connected.

=--

+-- {: .num_example #LocallyPathConnectedCircle}
###### Example
**([[circle]] is locally path-connected)**

The [[Euclidean space|Euclidean]] [[circle]]

$$
  S^1 = \left\{ x \in \mathbb{R}^2 \;\vert\;  {\Vert x\Vert} = 1\right\}
  \subset \mathbb{R}^2
$$

is locally path connected.

=--

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

By definition of the [[subspace topology]] and the defining [[topological base]] of the [[Euclidean plane]], a [[base for a topology|base for the topology]] of $S^1$ is given by the [[images]] of [[open intervals]] under the [[local homeomorphism]]

$$
  (cos(-), sin(-)) \;\colon\; \mathbb{R}^1 \to S^1
  \,.
$$

But these open intervals are locally path connected by example \ref{LocallyPathConnectedEuclideanSpace}, and in fact they are, evidently [[path-connected topological space]].

=--

## Related concepts

The condition is a necessary assumption in the

* [[fundamental theorem of covering spaces]]

in the form of the condition for

* [[semi-locally simply connected topological spaces]]

[[!redirects locally path-connected space]]
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locally polypresentable category

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#### Compact objects
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# Contents
* table of contents
{:toc}

## Idea

A _locally polypresentable category_ is similar to a [[locally presentable category]], but where we ask only for the existence of [[wide pullbacks]] rather than all [[small limits]].

## Definition

+-- {: .num_defn #PolypresentableCategory}
###### Definition
**(locally polypresentable category)**

A [[category]] $\mathcal{C}$ is called **locally polypresentable** if

1. it is an [[accessible category]];

1. it has all [[wide pullbacks]].

The second condition is equivalent to:

2'. it has all small [[polycolimits]].

=--

## Related concepts

* [[locally presentable category]]

* [[locally multipresentable category]]

* [[polycolimit]]

## References

The definition is due to

* [[François Lamarche]]. _Modelling polymorphism with categories_ (1988)

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locally posetal 2-category

#Contents#
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##Definition

A $2$-[[2-category|category]] $C$ is __locally posetal__ or __locally partially ordered__ or __[[Pos]]-[[enriched category|enriched]]__ if every [[hom-category]] $C(x,y)$ is a [[partial order|poset]] - an object of the category [[Pos]] of partial orders.  One can also consider a __locally preordered__ $2$-category, where every hom-category is a proset (a [[preorder]]ed set); up to [[equivalence of categories|equivalence]] of $2$-categories, these aren\'t any more general.

Locally posetal $2$-categories are the usual model of [[2-posets]], aka [[(n,r)-category|(1,2)-categories]].  Just as the motivating example of a $2$-category is the $2$-category [[Cat]] of categories, so the motivating example of a $2$-poset is the $2$-poset [[Pos]] of posets.  If you interpret $\Pos$ as a [[full subcategory|full sub-2-category]] of $\Cat$, then it is indeed locally posetal.  Similarly, the $2$-category of prosets is a locally preordered $2$-category that is equivalent to $Pos$.

Compare the notion of [[partially ordered category]].  A locally partially ordered category is a category *[[enriched category|enriched]] over* the category [[Pos]] of [[poset]]s, while a partially ordered category is a category *[[internal category|internal]] to* $Pos$.  Similarly, a locally partially ordered category is a special kind of $2$-category, while a partially ordered category is a special kind of [[double category]].

## Examples
* [[Pos]]
* [[Rel]]

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locally positive locale

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## Definition

A [[locale]] $L$ is **locally positive** (or **overt** or **open**) if every [[open sublocale]] $O \in \mathcal{O}(L)$ is an indexed join of [[positive open sublocales]], or equivalently, if it has a base of positive open sublocales.

Other definitions of a locally positive locale include that the unique map $!:L \to 1$ is open, and that the product projection $\pi_2:L \times L' \to L'$ is open for every locale $L'$.

## See also

* [[locale]]

* [[positive locale]]

* [[overt space]]

* [[premetric locale]]

* [[geometric hyperdoctrine]]

## References

* [[Simon Henry]], Localic Metric spaces and the localic Gelfand duality ([arXiv:1411.0898v1](https://arxiv.org/abs/1411.0898v1))

* Graham Manuell, *Uniform locales and their constructive aspects*, ([arXiv:2106.00678](https://arxiv.org/abs/2106.00678))

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* table of contents
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## Idea

An [[(∞,1)-category]] is called _locally presentable_ if it has all small [[(∞,1)-colimits]] and its [[objects]] are [[generators and relations|presented]] under [[(∞,1)-colimits]] by a [[small set]] of [[small objects]].
This is the direct analog in [[(∞,1)-category]] theory of the notion of _[[locally presentable category]]_ in [[category theory]].

There is a wealth of equivalent ways to make precise what this means, which are listed [below](#Definition). Two particularly useful ones are:

1. A locally presentable $(\infty,1)$-category is an [[accessible (∞,1)-category]] that admits all small [[(∞,1)-colimits]].

1. The locally presentable $(\infty,1)$-categories $\mathcal{C}$ are precisely the [[accessible (∞,1)-functor|accessibly embedded]] [[localization of an (∞,1)-category|localizations]]/[[reflective sub-(∞,1)-category|reflections]] $\mathcal{C} \stackrel{\overset{}{\leftarrow}}{\hookrightarrow} PSh_\infty(K)$ of an [[(∞,1)-category of (∞,1)-presheaves]]. In particular, if the reflector of this reflection is a [[left exact (∞,1)-functor]], then $\mathcal{C}$ is an [[(∞,1)-topos]].

See also at _[[locally presentable categories - introduction]]_.

**Warning on terminology.** In [Lurie](#Lurie) the term _presentable $(\infty,1)$-category_ is used for what we call a _locally presentable $(\infty,1)$-category_ here, in order to be in line with the established terminology of _[[locally presentable category]]_ in ordinary [[category theory]].

**Terminological variant.** The term "$\kappa$-[[compactly generated (∞,1)-category]]" is sometimes used to mean "locally $\kappa$-presentable (∞,1)-category. See there for a discussion of usage differences.

## Definition
 {#Definition}

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

An [[(∞,1)-category]] $\mathcal{C}$ is called **locally presentable**
if

1. it is [[accessible (∞,1)-category|accessible]]

1. it has all small [[(∞,1)-colimits]].

=--

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

That $\mathcal{C}$ is locally presentable is equivalent to each of the following equivalent characterizations.

1. $\mathcal{C}$ is [[locally small (infinity,1)-category|locally small]], with all small [[(∞,1)-colimits]] such that there is a [[small set]] $S \hookrightarrow Obj(\mathcal{C})$ of [[small objects]] which generates all of $\mathcal{C}$ under [[(∞,1)-colimits]].

1.  $\mathcal{C}$ is the [[localization of an (∞,1)-category|localization]] of an [[(∞,1)-category of (∞,1)-presheaves]] $PSh_\infty(K)$ along an [[accessible (∞,1)-functor]]:

    there exists a [[small (∞,1)-category]] $K$ and a pair of [[adjoint (∞,1)-functors]]

    $$
      \mathcal{C} \stackrel{\overset{}{\leftarrow}}{\hookrightarrow} PSh_\infty(K)
    $$

    such that the [[right adjoint]] $\mathcal{C} \hookrightarrow PSh_\infty(K)$ is [[full and faithful (∞,1)-functor|full and faithful]] and [[accessible (∞,1)-functor|accessible]].

    (if here in addition $f$ is [[exact functor|left exact]] then $\mathcal{C}$ is an [[(∞,1)-category of (∞,1)-sheaves]] on $K$).

1. There exists a [[combinatorial simplicial model category]] $A$ and and [[equivalence of (infinity,1)-categories]] $\mathcal{C} \simeq L_W A$ with the [[simplicial localization]] of $A$.

   More explicitly: with $\mathcal{C}$ incarnated as a [[quasi-category]] there is an [[equivalence of quasi-categories]] $ \mathcal{C} \simeq N(A^\circ)$ of $\mathcal{C}$ with the [[homotopy coherent nerve]] of the full [[sSet]]-[[enriched category|enriched]] [[subcategory]] of $A$ on the [[bifibrant objects]].

1. $\mathcal{C}$ is [[accessible (infinity,1)-category|accessible]] and for every [[cardinal number|regular cardinal]] $\kappa$ the [[full sub-(∞,1)-category]] $\mathcal{C}^\kappa \hookrightarrow \mathcal{C}$ on the $\kappa$ [[compact object in an (∞,1)-category|compact objects]] admits $\kappa$-small [[(∞,1)-colimits]].

1. There exists a [[cardinal number|regular cardinal]] $\kappa$ such that $\mathcal{C}$ is $\kappa$-[[accessible (infinity,1)-category|accessible]] and $C^\kappa$ admits $\kappa$-small [[limit in quasi-categories|colimits]];

1. There exists a [[cardinal number|regular cardinal]] $\kappa$, a [[small (∞,1)-category]] $D$ with $\kappa$-small [[limit in quasi-categories|colimits]] and an equivalence $Ind_\kappa D \stackrel{\simeq}{\to} \mathcal{C}$ with the category of $\kappa$-[[ind-objects]] of $D$.

=--

This is [Lurie, Thm. 5.5.1.1 & Prop. A.3.7.6](#Lurie), following [Simpson 99](#Simpson99), [Dugger 00](#Dugger00); review in [Cisinski 19, Thm. 7.11.16, Rem. 7.11.17](#Cisinski19).

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

That [[localization of an (infinity,1)-category|localizations]] $\mathcal{C} \stackrel{\leftarrow}{\hookrightarrow} PSh_{(\infty,1)}(K)$ correspond to combinatorial simplicial model categories is essentially **[[Dugger's theorem]]** ([Dugger](#Dugger)): every [[combinatorial model category]] arises, up to Quillen equivalence, as the [[Bousfield localization of model categories|left Bousfield localization]] of the global projective [[model structure on simplicial presheaves]].

=--

Locally presentable $(\infty,1)$-categories have a number of nice
properties, and therefore it is of interest to consider as
morphisms between them only those [[(∞,1)-functor]]s that preserve
these properties. It turns out that it is useful to consider
_[[limit in a quasi-category|colimit]] preserving_ functors. By the [[adjoint (∞,1)-functor theorem]] these are precisely the functors that have a right [[adjoint (∞,1)-functor]].

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

Write [[Pr(∞,1)Cat]] $\subset$ [[(∞,1)Cat]] for the (non-full) [[sub-quasi-category|sub-(∞,1)-category]] of [[(∞,1)Cat]] (the collection of not-necessarily small $(\infty,1)$-categories) on

* those objects that are locally presentable $(\infty,1)$-categories;

* those morphisms that are colimit-preserving [[(∞,1)-functor]]s.

=--

This is [Lurie, def. 5.5.3.1](#Lurie).

This $(\infty,1)$-category $Pr(\infty,1)Cat$ in turn as special properties. More on that is at _[[symmetric monoidal (∞,1)-category of presentable (∞,1)-categories]]_.

## Properties

### Equivalent characterizations
 {#EquivalentCharacterizationsDetails}

We indicate stepts in the proof of prop. \ref{EquivalentCharacterizations}.

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

Let $f \colon \mathcal{C} \to \mathcal{D}$ be an [[(∞,1)-functor]] which exhibits $\mathcal{D}$ as an [[idempotent completion]] of $\mathcal{C}$. Let $\kappa$ be a [[regular cardinal]]. Then the induced functor on [[(∞,1)-categories of ind-objects]]

$$
  Ind_\kappa(f) \colon Ind_\kappa(\mathcal{C}) \to Ind_\kappa(\mathcal{D})
$$

is an [[equivalence of (∞,1)-categories]].

=--

This is ([Lurie, lemma 5.5.1.3](#Lurie)).

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

Let $L \colon \mathcal{C} \to \mathcal{D}$ be an [[(∞,1)-functor]] between [[(∞,1)-categories]] which have $\kappa$-[[filtered (∞,1)-colimits]], and let $R$ be a [[right adjoint|right]] [[adjoint (∞,1)-functor]] of $L$. If $R$ preserves $\kappa$-[[filtered (∞,1)-colimits]] then $L$ preserves $\kappa$-[[compact objects]].

=--

This is [Lurie, lemma 5.5.1.4](#Lurie).

(...)

### Stability under various constructions
 {#StabilityUnderVariousConstructions}

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

For $C$ a locally presentable $(\infty,1)$-category and $p : K \to C$ a [[diagram]] in $C$, also the [[over quasi-category|over (∞,1)-category]] $C_{/p}$ as well as the under-$(\infty,1)$-category $C_{p/}$ are locally presentable.

=--

This is [[Higher Topos Theory|HTT, prop. 5.5.3.10, prop. 5.5.3.11]].

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

Since [[Pr(∞,1)Cat]] admits all small limits, we obtain new locally presentable $(\infty,1)$-categories by forming limits over given ones. In particular the [[product]] of locally presentable $(\infty,1)$-categories is again locally presentable.

=--

### Limits and colimits
 {#LimitsAndColimits}

In the first definition of locally presentable $(\infty,1)$-category above only the existence of colimits is postulated. An important fact is that it follows automatically that also all small limits exist:

A [[representable functor]] $C^{op} \to \infty Grpd$ preserves [[limit in a quasi-category|limits]] (see [[(∞,1)-Yoneda embedding]]). If $C$ is locally presentable, then also the converse holds:

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

If $\mathcal{C}$ is a locally presentable $(\infty,1)$-category then an [[(∞,1)-functor]] $C^{op} \to \infty Grpd$ is a [[representable functor]]
precisely if it preserves [[limit in a quasi-category|limits]].

=--

This is [[Higher Topos Theory|HTT, prop. 5.5.2.2]].

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

We need to prove that a limit-preserving functor $F : C^{op} \to \infty Grpd$ is [[representable functor|representable]]. By the above characterizations we know that $C$ is an accessible localization of a presheaf category.

So consider first the case that $C = PSh(D)$ _is_ a presheaf category.  Write

$$
  f : D^{op} \stackrel{j^{op}}{\to} PSh(D)^{op} \stackrel{F}{\to} \infty Grpd
$$

for the precomposition of $F$ with the [[(∞,1)-Yoneda embedding]]. Then let

$$
  F' := Hom_{C}(-,f) : PSh(D)^{op} \to \infty Grpd
$$

the functor represented by $f$.

We claim that $F \simeq F'$, which proves that $F$ is represented by $F \circ j^{op}$: since both $F$ and $F'$ preserve limits (hence colimits as functors on $PSh(D)$) it follows from the fact that the Yoneda embedding exhibits the universal co-completion of $D$ that it is sufficient to show that $F \circ j^{op} \simeq F' \circ j^{op}$. But this is the case precisely by the statement of the full [[(∞,1)-Yoneda lemma]].

Now consider more generally the case that $C$ is a [[reflective sub-(∞,1)-category]] of $PSh(D)$. Let $L : PSh(D) \to C $ be the [[left adjoint]] reflector. Since it respects all colimits, the composite

$$
  F \circ L^{op} : PSh(D)^{op} \stackrel{L^{op}}{\to} C^{op} \stackrel{F}{\to} \infty Grpd
$$

respects all limits. By the above it is therefore represented by some object $X \in PSh(D)$.

By the general properties of [[reflective sub-(∞,1)-categories]], we have that $C$ is the full [[sub-(∞,1)-category]] of $PSh(D)$ on those objects that are [[local object]]s with respect to the morphisms that $L$ sends to equivalences. But $X$, since it presents $F \circ L^{op}$, is manifestly local in this sense and therefore also represents $F \circ L^{op}|_{C}$. But on $C$ the functor $L$ is equivalent to the identity, so that this is equivlent to $F$.

=--

This statement has the following important consequence:

+-- {: .num_cor}
###### Corollary

A locally presentable $(\infty,1)$-category $C$ has all small [[limit in a quasi-category|limits]].

=--

This is [[Higher Topos Theory|HTT, prop. 5.5.2.4]].

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

We may compute the limit after applying the [[(∞,1)-Yoneda embedding]] $j : C \to PSh_{(\infty,1)}(c)$. Since this is a [[full and faithful (∞,1)-functor]] it is sufficient to check that the limit computed in $PSh(C)$ lands in the essential image of $j$. But by the above lemma, this amounts to checking that the limit over limit-preserving functors is itself a limit-preserving functor. This follows using that limits of functors are computed objectwise and that generally limits commute with each other (see [[limit in a quasi-category]]):

to check for $I \to PSh(C)$ a diagram of limit-preserving functors that $\lim_i F_i$ is a functor that commutes with all limits, let $a : J \to C$ be a diagram and compute (verbatim as in ordinary category theory)

$$
  \begin{aligned}
    \lim_j (\lim_i F_i)(a_j)
    & \simeq
    \lim_j (\lim_i F_i(a_j))
    \\
    & \simeq
    \lim_i (\lim_j F_i(a_j))
    \\
    & \simeq
    \lim_i F_i(\lim a_j)
    \\
    & \simeq
    (\lim_i F_i)(\lim a_j)
  \end{aligned}
  \,.
$$

=--

### As $(\infty,1)$-categories presented by combinatorial simplicial model categories
 {#PresentedByCombinatorialSimplicialModelCategories}

By prop. \ref{EquivalentCharacterizations}
locally presentable $(\infty,1)$-categories are equivalently those [[(∞,1)-categories]] which are _presented_ by a [[combinatorial simplicial model category]] $C$ in that they are the full [[simplicially enriched category|simplicial subcategory]] $C^\circ \hookrightarrow C$ on [[bifibrant objects]] of $C$ (or, equivalently, the [[quasi-category]] associated to this [[simplicially enriched category]]).

(See also at *[[Ho(CombModCat)]]*).

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

Under this presentation, [[equivalence of (∞,1)-categories]] between locally presentable $(\infty,1)$-categories corresponds to [[zigzags]] of [[Quillen equivalences]] between presenting [[combinatorial simplicial model category|combinatorial simplicial model categories]]:

$C^\circ$ and $D^\circ$ are [[equivalence of (∞,1)-categories|equivalent as (∞,1)-categories]] precisely if they are connected by a [[zig-zag]] of [[simplicial Quillen adjunction|simplicial]] [[Quillen equivalences]]

$$
  C
  \stackrel{\leftarrow}{\to}
  \stackrel{\to}{\leftarrow}
  \stackrel{\leftarrow}{\to}
  \cdots
  D.
$$

=--

This is [Lurie, remark A.3.7.7](#Lurie).

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

Partly due to the fact that [[simplicial model category|simplicial model categories]] have been studied for a longer time -- partly because they are simply more tractable than [[(∞,1)-categories]] -- many $(\infty,1)$-categories are indeed handled in terms of such a presentation by a [[simplicial model category]].

The canonical example is the presentation of the [[(∞,1)-category of (∞,1)-sheaves]] on an ordinary (1-categorical) [[site]] $S$ by the simplicial [[model structure on simplicial presheaves|model category of simplicial presheaves]] on $S$.

=--

## Examples

The basic example is:

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

[[∞Grpd]] is locally presentable.

=--

([Lurie, example 5.5.1.8](#Lurie))

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

According to the discussion at [(∞,1)-colimit -- Tensoring with an ∞-groupoid](limit+in+a+quasi-category#Tensoring) every [[∞-groupoid]] is the colimit over itself of the functor contant on the point, the terminal $\infty$-groupoid. This is clearly compact, and hence generates [[∞Grpd]].

=--

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

An [[(∞,1)-topos]] is precisely a locally presentable $(\infty,1)$-category where the [[localization of an (∞,1)-category|localization]] functor also preserves finite limits.

=--

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

For $C$ and $D$ locally presentable $(\infty,1)$-categories, write $Func^L(C,D) \subset Func(C,D)$ for the full sub-$(\infty,1)$-category on left-adjoint $(\infty,1)$-functors. This is itself locally presentable

=--

This is [[Higher Topos Theory|HTT, prop 5.5.3.8]]

Notice that this makes the [[symmetric monoidal (∞,1)-category of presentable (∞,1)-categories]] _[[closed monoidal category|closed]]_ .

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

For $C$ an $(\infty,1)$-category with finite [[products]], the $(\infty,1)$-category $Alg_{(\infty,1)}(C)$ of algebras over $C$ regarded as an [[(∞,1)-algebraic theory]] is locally presentable.

=--

## Related concepts

* [[Ho(CombModCat)]]

[[!include locally presentable categories - table]]

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

## References

The theory of locally presentable $(\infty,1)$-categories was first implicitly conceived in terms of [[model category]] presentations in

* {#Simpson99} [[Carlos Simpson]],  *A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi* ([arXiv:math/9903167](http://arxiv.org/abs/math/9903167))

The full intrinsic $(\infty,1)$-categorical theory appears in

* {#Lurie} [[Jacob Lurie]], Section 5 of: _[[Higher Topos Theory]]_

with section A.3.7 establishing the relation to [[combinatorial model categories]] and [[Dugger's theorem]] in [[Higher Topos Theory|HTT, prop A.3.7.6]].

The statement of [[Dugger's theorem]] -- of which the characterization of locally presentable $(\infty,1)$-categories as localizations of $(\infty,1)$-presheaf categories is a variant -- is due to

* {#Dugger00} [[Dan Dugger]], _[[Combinatorial model categories have presentations]]_,  Adv. Math. 164 (2001), no. 1, 177-201 ([arXiv:math/0007068](http://arxiv.org/abs/math/0007068))

Further textbook account:

* {#Cisinski19} [[Denis-Charles Cisinski]], Section 7.11 in: _Higher category theory and homotopical algebra_,  Cambridge University Press 2019 ([doi:10.1017/9781108588737](https://doi.org/10.1017/9781108588737), [pdf](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf))

Exposition and review:

* topology@ub, *[Presentable infinity categories](http://www.ub.edu/topologia/seminars/Quasicategories_2019.html)*, seminar notes, 2019

On factorication of functors between presentable $\infty$-categories as [[coreflections]] followed by a tower of [[monadic functors]]:

* Lior Yanovski, *The Monadic Tower for $\infty$-Categories* ([arXiv:2104.01816](https://arxiv.org/abs/2104.01816))

Discussion [[category internal to an (infinity,1)-topos|internal to]] any [[(∞,1)-topos]]:

* [[Louis Martini]], [[Sebastian Wolf]], *Presentable categories internal to an ∞-topos* [[arXiv:2209.05103](https://arxiv.org/abs/2209.05103)]

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

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

[[!redirects presentable (infinity,1)-category]]

[[!redirects locally presentable infinity-category]]

[[!redirects locally presentable infinity1-category]]


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

This page collects material related to the book

* [[Jiří Adámek]], [[Jiří Rosický]]

  **Locally Presentable and Accessible Categories**

  London Mathematical Society Lecture Note Series, number 189,

  Cambridge University Press 1994

  [doi:10.1017/CBO9780511600579](https://doi.org/10.1017/CBO9780511600579)

  erratum: [[AdamekRosicky_LPAC_Erratum.pdf:file]]

on [[locally presentable category|locally presentable]] and [[accessible category|accessible]] [[categories]].  There is an erratum made available by the authors:

## Contents

1. **[[locally presentable categories]]**

1. **[[accessible categories]]**: see also [[sketch]]

1. **algebraic categories**: see [[algebraic category]],  [[equationally presentable category]], [[essentially algebraic theory]], and [[variety of algebras]]

1. **[[injectivity class]]es**: see also [[weak factorization system]]

1. **categories of models**: see [[internal logic]], [[theory]], [[essentially algebraic theory]], [[model]]

1. **[[Vopěnka's principle]]**

* Appendix: [[large cardinals]]

## Related concepts

[[!include locally presentable categories - table]]

category: reference

[[!redirects LPAC]]
[[!redirects Locally presentable and accessible categories]]