locally presentable categories - introduction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Model category theory
+-- {: .hide}
[[!include model category theory - contents]]
=--
#### $(\infty,1)$-Category theory
+-- {: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

> This page means to give an introduction to the notion of _[[locally presentable category]]_, and its related notions in [[higher category theory]] and survey some fundamental properties.

> Expected background of the reader:

> * For the first section _[Basic idea in category theory](#BasicIdea)_ the reader is assumed to be familiar with basic notions of [[category theory]] such as _[[presheaves]]_ and _[[colimits]]_.

> * For the section _[Basic idea in model category theory](#BasicIdeaInModelCategoryTheory)_ the reader is assumed to be familiar with basic notions in [[model category|model category theory]] such as _[[cofibrantly generated model categories]]_ and _[[homotopy colimits]]_.

> * For the third section _[Basic idea in (∞,1)-category theory](#BasicIdeaInInfinityCategoryTheory)_ the reader is assumed to be familiar with basic concepts of [[(∞,1)-category|(∞,1)-category theory]] such as _[[(∞,1)-categories of (∞,1)-presheaves]]_ and _[[(∞,1)-colimits]]_.

# Contents
* table of contents
{: toc}

## Basic idea in category theory
 {#BasicIdea}

The general idea is that a _[[locally presentable category]]_ is a [[large category]] [[generators and relations|generated from]] small data: from [[small objects]] under [[small colimits]].

### Generation from generators

The notion of _[[locally presentable category]]_ is, at least roughly, an analogue for [[categories]] of the notion of a [[finitely generated module]].

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

An [[abelian group]] $A$ is called _[[finitely generated module|finitely generated]]_ if there is a [[finite set|finite]] [[subset]]

$$
  \iota \colon S \hookrightarrow U(A)
$$

of the underlying set $U(A)$ of $A$, such that every element of $A$ is a sum of such generating elements.

=--

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

We always have the maximal such presentation where $S = U(A)$ is the whole underlying set and $\iota \colon F(U(A)) \to A$ is the [[counit of an adjunction|counit]] of the [[free-forgetful adjunction]]. But the presentation is all the more interesting/useful the _smaller_ $S$ is.

=--

Now, the [[categorification]] of "commutative sum" is _[[colimit]]_. Hence let now $\mathcal{C}$ be a category with all [[small colimits]].

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

We say a subclass $S \hookrightarrow Obj(\mathcal{C})$ of [[objects]] or equivalently the [[full subcategory]] $\mathcal{C}^0 \hookrightarrow \mathcal{C}$ on this subclass _generates_ $\mathcal{C}$ if every object in $\mathcal{C}$ is a [[colimit]] of objects in $\mathcal{C}^0$, hence the colimit over a [[diagram]] of the form

$$
  D \to \mathcal{C}^0 \hookrightarrow \mathcal{C}
  \,.
$$

=--

As before, such a presentation is all the more useful the "smaller" the generating data is. In order to grasp the various aspects of the notion of "smallness" in [[category theory]] we need to recall the notion of [[regular cardinal]].

### Small data

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

The [[cardinality]] $\kappa = {\vert S\vert}$ of a [[set]] $S$ is _[[regular cardinal|regular]]_ if every [[coproduct]]/[[disjoint union]] of sets of cardinality smaller than $\kappa$ and indexed by a set of cardinality smaller than $\kappa$ is itself of cardinality smaller than $\kappa$.

=--

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

The smallest [[regular cardinal]] is $\aleph_0 = {\vert \mathbb{N}\vert}$: every finite [[union]] of [[finite sets]] is itself a finite set. (See the entry on [[regular cardinal]]s for a discussion as to whether one might consider some finite cardinals as being `regular'.)

=--

We can now speak of objects that are "$\kappa$-small sums" using the notion of $\kappa$-[[filtered colimits]]:

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

For $\kappa$ a [[regular cardinal]], a $\kappa$-[[filtered category]] is one where every [[diagram]] of size $\lt \kappa$ has a [[cocone]].

=--

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

In an $\aleph_0$-[[filtered category]] every finite diagram has a cocone. This is equivalent to:

1. for every pair of objects there is a third object such that both have a morphism to it;

1. for every pair of [[parallel morphisms]] there is a morphism out of their codomain such that the two composites are equal.

=--

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

The [[tower diagram]] category $(\mathbb{N}, \leq)$

$$
  X_0 \to X_1 \to X_2 \to \cdots
$$

is filtered.

=--

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

For $\lambda \gt \kappa$ a bigger [[regular cardinal]], every $\lambda$-[[filtered category]] is in particular also $\kappa$-filtered.

=--

Using this we have the central definition now:

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

A $\kappa$-[[filtered colimit]] is a [[colimit]] over a $\kappa$-filtered diagram.

=--

A crucial characterizing property of $\kappa$-filtered colimits is the following:

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

A colimit in [[Set]] is $\kappa$-filtered precisely if it commutes with all $\kappa$-[[small limits]].

In particular a colimit in [[Set]] is filtered (meaning: $\aleph_0$-filtered) precisely if it commutes with all [[finite limits]].

=--

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

An [[object]] $A \in \mathcal{C}$ is a $\kappa$-[[compact object]] if it commutes with $\kappa$-filtered colimits, hence  if for $X \colon I \to \mathcal{C}$ any $\kappa$-[[filtered category|filtered]] [[diagram]], the canonical [[function]]

$$
  \underset{\to_i}{\lim} \mathcal{C}(A,X_i) \to \mathcal{C}(A, \underset{\to_i}{\lim} X_i)
$$

is a [[bijection]].

We say $X$ is a **[[small object]]** if it is $\kappa$-compact for _some_ [[regular cardinal]] $\kappa$.

=--

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

If $\lambda \gt \kappa$, then being $\lambda$-compact is a weaker condition than being $\kappa$-compact.

=--

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

The object $A$ commutes with the colimit over $I$ precisely if every morphism $A \to \underset{\to_i}{\lim} X_i $ lifts to a morphism $A \to X_j$ into one of the $X_j$. Schematically, depicting specifically a [[sequential colimit]], this means that we have:

$$
  \array{
    \cdots&\to&X_{j-1} &\to& X_j &\to& X_{j+1} &\to& \cdots
    \\
    &&&{}^{\mathllap{\exists \hat f}}\nearrow&\downarrow & \swarrow
    \\
    &&A& \stackrel{f}{\to} & \underset{\to}{\lim} X_i
  }
  \,.
$$

Hence $A$ is "small enough" such that mapping it into the sum of all the $X_i$ it always entirely lands inside one of the $X_i$.

=--

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

There is a close relation between the notion of "compact" as in, on the one hand, _[[compact topological space]]_ and _[[compact topos]] _, and on the other as in _[[compact object]]_ as above. This is mediated by proposition \ref{FilteredColimitsAndFiniteLimits}. But the relation is a bit more subtle and takes a bit more discussion than we maybe want to get into here.

=--

### Locally presentable category: generated from colimits over small objects

Using this we can now say:

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

A [[locally small category]] $\mathcal{C}$ is a _[[locally presentable category]]_ if it has all small [[colimits]] and there is a [[small set]] $S \hookrightarrow Obj(\mathcal{C})$ of [[small objects]] such that this generates $\mathcal{C}$, by def. \ref{GenerationOfCategoryFromSubcategoryUnderColimits}.

=--

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

The adjective "locally" in "[[locally presentable category]]" is to indicate that the condition is all about the [[objects]], only. There is a different notion of "presented category".

=--

There are a bunch of equivalent reformulations of the notion of locally presentable category. One of the more important ones we again motivate first by analogy with presentable modules:

### Generation exhibited by epimorphism from a free object

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

If an [[abelian group]] $A$ is generated by a set $S \hookrightarrow U(A)$ as in example \ref{FinitelyGeneratedAbelianGroup}, this means equivalently that there is an [[epimorphism]]

$$
  L \colon F(S) \to A
$$

from the [[free abelian group]] $F(S)$ generated by $S$, hence the group obtained by forming [[formal linear combination|formal sums]] of elements in $S$. Here the epimorphism sends formal sums to actual sums in $A$:

$$
  L(\sum_k s_k) \coloneqq \sum_k \iota(s_k)
  \,.
$$

=--

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

The [[categorification]] of the notion _[[free abelian group]]_ is the notion of _[[free cocompletion]]_ of a category $\mathcal{C}^0$: the [[category of presheaves]] $PSh(\mathcal{C}^0)$.

=--

Accordingly:

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

If a [[full subcategory]] $\iota \colon \mathcal{C}^0 \hookrightarrow \mathcal{C}$ generates $\mathcal{C}$ under colimits as in defn. \ref{GenerationOfCategoryFromSubcategoryUnderColimits}, then there is a [[functor]]

$$
  L \colon PSh(\mathcal{C}^0) \to \mathcal{C}
$$

which sends formal colimits to actual colimits in $\mathcal{C}$

$$
  L(\underset{\to_k}{\lim} s_k) \coloneqq \underset{\to_k}{\lim} \iota(s_k)
  \,.
$$

Here $L$ by construction preserves all colimits.

=--

Therefore conversely, given a colimit-preserving functor $L \colon PSh(\mathcal{C}^0) \to \mathcal{C}$ we want to say that it _locally presents_ $\mathcal{C}$ if $L$ is "suitably epi".

It turns out that "suitably epi" is to be the following:

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

A [[functor]] $L \colon PSh(\mathcal{C}^0) \to \mathcal{C}$ from the [[category of presheaves]] over a [[small category]] $\mathcal{C}^0$ is an **[[accessible functor|accessible]] [[localization]]** if

*  $L$ has a [[section]], hence a [[functor]] $R \colon \mathcal{C} \to PSh(\mathcal{C}^0)$ with a [[natural isomorphism]] $L\circ R \simeq id_{\mathcal{C}}$;

* such that

  1. $R$ is [[right adjoint]] to $L$;

  1. $R\circ L$ preserves $\kappa$-[[filtered colimits]].

=--

With this notion we have the following analog of the familiar statement that an abelian group is generated by $S$ precisely if there is an epimorphism $L \colon F(S) \to A$:

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

A category $\mathcal{C}$ is locally presentable according to def. \ref{LocallyPresentableCategory} precisely if it is an accessible localization, def. \ref{Localization},

$$
  L \colon PSh(\mathcal{C}^0) \to \mathcal{C}
$$

for some small category $\mathcal{C}^0$.

=--

This is due to ([Adámek-Rosický, prop 1.46](#AdamekRosicky)).

### Left exact localizations

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

A [[locally presentable category]] $\mathcal{C}$ is called a _[[topos]]_, precisely if the localization functor $L \colon PSh(\mathcal{C}^0) \to \mathcal{C} $ from theorem \ref{AdamekRosickyTheorem} in addition is a [[left exact functor]], meaning that it preserves [[finite limits]].

=--

## Summary and overview {#summary}

In summary the discussion [above](#BasicIdea) says that the notion of locally presentable categories sits in a sequence of notions as indicated in the row labeled "category theory" in the following table. The other rows are supposed to indicate that regarding a category as a [[(1,1)-category]] and simply varying in this story the parameters $(n,r)$ in "[[(n,r)-category]]" one obtains fairly straightforward analogs of the notion of locally presentable category in other fragments of [[higher category theory]]. These we discuss in more detail further below.

**Locally presentable categories:** [[large categories|Large categories]] whose [[objects]] arise from [[small object|small]] [[generators]] under [[small colimit|small]] [[relations]].

| [[(n,r)-categories]] ... | satisfying [[Giraud's axioms]] | inclusion of [[left exact functor|left exact]] [[localizations]] | [[generators and relations|generated]] under [[colimits]] from [[small objects]] | | [[localization]] of [[free cocompletion]] | | [[generators and relations|generated]] under [[filtered colimits]] from [[small objects]] |
| - | - | - | - | - | - | - | - |
| **[[(0,1)-category theory]]** | [[(0,1)-toposes]] | $\hookrightarrow$ | [[algebraic lattices]] | $\simeq$ [Porst's theorem](algebraic+lattice#RelationToLocallyFinitelyPresentableCategories) | [[subobject lattices]] in [[accessible functor|accessible]] [[reflective subcategories]] of [[presheaf categories]] | | |
| **[[category theory]]** | [[toposes]] | $\hookrightarrow$ | [[locally presentable categories]] | $\simeq$ [Adámek-Rosický's theorem](#AdamekRosickyTheorem) | [[accessible functor|accessible]] [[reflective subcategories]] of [[presheaf categories]] | $\hookrightarrow$ | [[accessible categories]] |
| **[[model category|model category theory]]** | [[model toposes]] | $\hookrightarrow$ | [[combinatorial model categories]] | $\simeq$ [Dugger's theorem](#DuggerTheorem) | [[left Bousfield localization]] of global [[model structures on simplicial presheaves]] | | |
| **[[(∞,1)-topos theory]]** | [[(∞,1)-toposes]] |$\hookrightarrow$ | [[locally presentable (∞,1)-categories]] | $\simeq$ <br/> [Simpson's theorem](#SimpsonTheorem) | [[accessible (∞,1)-functor|accessible]] [[reflective sub-(∞,1)-categories]] of [[(∞,1)-presheaf (∞,1)-categories]] | $\hookrightarrow$ |[[accessible (∞,1)-categories]] |

## Basic idea in model category theory
 {#BasicIdeaInModelCategoryTheory}

### Model structure on simplicial presheaves

The analog of a [[category of presheaves]] in [[model category]] theory is the [[model structure on simplicial presheaves]], which we now briefly indicate.

Write [[sSet]] for the category of [[simplicial sets]]. Here we always regard this as equipped with the standard [[model structure on simplicial sets]] $sSet_{Quillen}$.

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

For $C$ a [[small category]] write $[C^{op}, sSet]\simeq [C^{op}, Set]^{\Delta^{op}}$ for the category of [[simplicial presheaves]]. The **global projective [[model structure on simplicial presheaves]]** $[C^{op}, sSet]_{proj}$ has as

* [[weak equivalences]] the objectwise [[weak homotopy equivalences]] of simplicial sets

* [[fibrations]] the objectwise [[Kan fibrations]].

=--

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

Accordingly $[C^{op}, sSet]$ is a [[cofibrantly generated model category]] with generating (acyclic) cofibrations the [[tensoring]] of objects of $C$ with the generating (acyclic) cofibrations of $sSet_{Quillen}$.

=--

### Left Bousfield localization

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

Given a [[model category]] $[C^{op}, Set]_{proj}$ and set $\mathcal{S} \subset Mor([C^{op}, Set])$ of morphisms, the [[Bousfield localization of model categories|left Bousfield localization]] is the model structure with the same cofibrations and weak equivalences the $\mathcal{S}$-[[local morphisms]].

$$
 [C^{op}, Set]_{proj,\mathcal{S}}
  \stackrel{\overset{id}{\leftarrow}}{\underset{id}{\to}}
 [C^{op}, Set]_{proj}
  \,.
$$

=--

### Combinatorial model categories

The simple idea of the following definition is to say that the model category analog of _[[locally presentable category]]_ is simply a model structure on a locally presentable category.

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

A model category is a **[[combinatorial model category]]** if

1. the underlying category is a [[locally presentable category]];

1. the model structure is a [[cofibrantly generated model category]].

=--

### Dugger's theorem

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

Every [[combinatorial model category]], def. \ref{CombinatorialModelCategory}, is [[Quillen equivalence|Quillen equivalent]] to a [[Bousfield localization of model categories|left Bousfield localization]] of a global [[model structure on simplicial presheaves]] as in def. \ref{BousfieldLocalizationOfModelStructureOnSimplicialPresheaves}.

=--

See at [combinatorial model category - Dugger's theorem](combinatorial%20model%20category#DuggerTheorem).

## Basic idea in $(\infty,1)$-category theory
 {#BasicIdeaInInfinityCategoryTheory}

### $(\infty,1)$-Presheaves

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

For $\mathcal{C}$ and $\mathcal{D}$ two [[(∞,1)-categories]] and $\mathcal{C}_{s}, \mathcal{D}_s \in sSet$ two models as [[quasi-categories]], an [[(∞,1)-functor]] $F \colon \mathcal{C} \to \mathcal{D}$ is simply a homomorphism of simplicial set $\mathcal{C}_s \to \mathcal{D}_s$.

The [[(∞,1)-category of (∞,1)-functors]] $Func(\mathcal{C}, \mathcal{D})_s$ as a [[quasi-category]] is simply the [[hom object]] of simplicial set

$$
  Func(\mathcal{C}, \mathcal{D})_s = sSet(\mathcal{C}_s, \mathcal{D}_s)
  \in
   QuasiCat \hookrightarrow sSet
  \,.
$$

=--

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

For $\mathcal{D}$ an [[(∞,1)-category]], the **[[(∞,1)-category of (∞,1)-presheaves]]** on $\mathcal{D}$ is the functor category

$$
  PSh_\infty(\mathcal{D})
  =
  Func(\mathcal{D}^{op}, \infty Grpd)
$$

out of the [[opposite (∞,1)-category]] of $\mathcal{D}$ into the [[∞Grpd|(∞,1)-category of ∞-groupoids]].

=--

### Localizations of $(\infty,1)$-categories

The notions of [[adjoint functors]], [[full and faithful functors]] etc. have straightforward, essentially verbatim generalizations to $(\infty,1)$-categories:

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

A pair of [[(∞,1)-functors]]

$$
  C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}}
  D
$$

is a pair of **[[adjoint (∞,1)-functors]]**, if there exists a _unit transformation_  $\epsilon : Id_C \to R \circ L$ -- a morphism in the [[(∞,1)-category of (∞,1)-functors]] $Func(C,D)$ -- such that for all $c \in C$ and $d \in D$ the induced morphism

$$
  Hom_C(L(c),d)
   \stackrel{R_{L(c), d}}{\to}
  Hom_D(R(L(c)), R(d))
  \stackrel{Hom_D(\epsilon, R(d))}{\to}
  Hom_D(c,R(d))
$$

is an [[equivalence of ∞-groupoids]].

=--

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

An [[(∞,1)-functor]] $F \colon \mathcal{C} \to \mathcal{D}$ is a
**[[full and faithful (∞,1)-functor]]** if for all objects $X,Y \in \mathcal{C}$ the component

$$
  F_{X,Y} \colon \mathcal{C}(X,Y) \stackrel{\simeq}{\to} \mathcal{D}(F(X), F(Y))
$$

is an [[equivalence of ∞-groupoids]].

=--

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

A [[reflective sub-(∞,1)-category]] $\mathcal{C} \hookrightarrow \mathcal{D}$ is a [[full and faithful (∞,1)-functor]] with a left [[adjoint (∞,1)-functor]].

=--

### Locally presentable $(\infty,1)$-categories

We have then the essentially verbatim analog of the situation for ordinary categories:

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

An [[(∞,1)-category]] $\mathcal{C}$ is a **[[locally presentable (∞,1)-category]]** if there exists a [[small set]] of objects such that the [[full sub-(∞,1)-category]] $\mathcal{C}^0 \hookrightarrow \mathcal{C}$ on it generates $\mathcal{C}$ under [[(∞,1)-colimits]].

=--

And the equivalent characterization is now as before

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

An [[(∞,1)-category]] is a [[locally presentable (∞,1)-category]], def. \ref{LocallyPresentableInfinityCategory}, precisely if it is [[equivalence of (∞,1)-categories|equivalent]] to  [[localization of an (∞,1)-category|localization]], def. \ref{ReflectiveLocalization},

$$
  \mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}}
  PSh_\infty(\mathcal{C}^0)
$$

of an [[(∞,1)-category of (∞,1)-presheaves]], def. \ref{InfinityPresheaves}, such that $R \circ L$ preserves $\kappa$-[[filtered (∞,1)-colimits]] for some [[regular cardinal]] $\kappa$.

=--

This appears as [Lurie, theorem 5.5.1.1](#Lurie), attributed there to [[Carlos Simpson]].

### $(\infty,1)$-Toposes

As before, if a locally presentable $(\infty,1)$-category arises as the [[localization of an (∞,1)-category|localization]] $L \colon PSh_\infty(\mathcal{C}^0) \to \mathcal{C}$ of a [[left exact (∞,1)-functor]], then it is an [[(∞,1)-topos]].

### Presentation by combinatorial model categories

There is a close match between the theory of [[combinatorial model categories]] and [[locally presentable (∞,1)-categories]].

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

Every [[locally presentable (∞,1)-category]] arises, up to [[equivalence of (∞,1)-categories]], as the [[simplicial localization]] of a [[combinatorial model category]].

=--

This is part of [Lurie, theorem 5.5.1.1](#Lurie).

Accordingly, every [[simplicial Quillen adjunction]] between [[combinatorial model categories]] gives rise to a pair of [[adjoint (∞,1)-functors]] between the corresponding locally presentable $(\infty,1)$-categories.

Hence a [[left Bousfield localization]] of a [[model structure on simplicial presheaves]] presents a corresponding localization of an [[(∞,1)-category of (∞,1)-presheaves]] to a [[locally presentable (∞,1)-category]].

$$
  \array{
    \mathcal{C} &\stackrel{\overset{}{\leftarrow}}{\hookrightarrow}& PSh_\infty(C)
    \\
    \left[C^{op}, sSet\right]_{proj,\mathcal{S}}
    &\stackrel{\overset{id}{\leftarrow}}{\underset{id}{\to}}&
    [C^{op}, sSet]_{proj}
  }
$$

## References

The standard textbook for [[locally presentable categories]] is

* [[Jiří Adámek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_, Cambridge University Press, (1994)
 {#AdamekRosicky}

Decent accounts of [[combinatorial model categories]] include secton A.2.6 of

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

and

* [[Clark Barwick]], _On (Enriched) Left Bousfield Localization of Model Categories_ ([arXiv:0708.2067](http://arxiv.org/abs/0708.2067))

The standard text for [[locally presentable (∞,1)-categories]] is section 5 of [Lurie](#Lurie).


locally presentable categories - table

**Locally presentable categories:** [[cocomplete category|Cocomplete]] possibly-[[large categories]] generated under [[filtered colimits]] by [[small object|small]] [[generators]] under [[small colimit|small]] [[relations]].  Equivalently, [[accessible functor|accessible]] [[reflective localizations]] of [[free cocompletions]].  Accessible categories omit the cocompleteness requirement; toposes add the requirement of a [[left exact functor|left exact]] localization.

| $\phantom{A}$[[(n,r)-categories]]$\phantom{A}$ | $\phantom{A}$[[toposes]]$\phantom{A}$ | locally presentable | loc finitely pres | localization theorem | [[free cocompletion]] | accessible |
|--|--|--|--|--|--|--|
| **[[(0,1)-category theory]]** | [[locales]] | [[suplattice]] | [[algebraic lattices]] | [Porst's theorem](algebraic+lattice#RelationToLocallyFinitelyPresentableCategories) | [[powerset]] | [[poset]] |
| **[[category theory]]** | [[Grothendieck topos|toposes]] | [[locally presentable categories]] | [[locally finitely presentable categories]] | [Adámek-Rosický's theorem](locally+presentable+category#AsLocalizationsOfPresheafCategories) | [[presheaf category]] | [[accessible categories]] |
| **[[model category|model category theory]]** | [[model toposes]] | [[combinatorial model categories]] | | [[Dugger's theorem]] | global [[model structures on simplicial presheaves]] | n/a |
| **[[(∞,1)-category theory]]** | [[(∞,1)-toposes]] | [[locally presentable (∞,1)-categories]] | | [Simpson's theorem](locally+presentable+infinity-category#Definition) | [[(∞,1)-presheaf (∞,1)-categories]] | [[accessible (∞,1)-categories]] |


locally presentable 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 presentable category_ is a [[category]] which contains a [[small set]] $S$ of [[small objects]] such that every [[object]] is a nice [[colimit]] over objects in this set.

This says equivalently that a locally presentable category $\mathcal{C}$ is a [[reflective localization]] $\mathcal{C} \hookrightarrow PSh(S)$ of a [[category of presheaves]] over $S$. Since here $PSh(S)$ is the [[free colimit completion]] of $S$ and the localization imposes _relations_, this is a presentation of $\mathcal{C}$ by _[[generators and relations]]_, hence the name _(locally) presentable category_.

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

## Definition

There are many equivalent characterizations of _locally presentable categories_.  The following is one of the most intuitive, equivalent characterizations are discussed [below](#EquivalentCharacterizations).

+-- {: .num_defn #PresentableCategory}
###### Definition
**(locally presentable category)**

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

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

1. it has all small [[colimits]].

This means

1. $\mathcal{C}$ is a [[locally small category]];

1. $\mathcal{C}$ has all small [[colimits]];

1. there exists a [[small set]] $S \hookrightarrow Obj(\mathcal{C})$ of $\lambda$-[[compact]] objects that generates $\mathcal{C}$ under $\lambda$-[[filtered colimits]] for some [[regular cardinal]] $\lambda$.

=--

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

If follows that every [[object]] in a locally presentable category is a [[small object]].

=--

\begin{remark}
\label{Terminology}
The _locally_ in _locally presentable category_ refers to the fact that it is the _[[objects]]_ of (in) the category that are presentable, not the category as such (which is itself an [[object]] of [[Cat]]). This permits the distinction between, for instance, locally presentable categories, and [[finitely presentable objects]] of [[Cat]], which could be called "finitely presentable categories". In practice, however, it is common to drop "locally" from "locally presentable category" without modifying the meaning.
\end{remark}

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

Since a [[small object]] is one which is $\kappa$-[[compact object|compact]] for some $\kappa$, and any $\kappa$-compact object is also $\lambda$-compact for any $\lambda \gt \kappa$, it follows that there exists some $\kappa$ such that every object of the colimit-generating set $S$ is $\kappa$-compact.

=--

This provides a "stratification" of the class of locally presentable categories, as follows.

+-- {: .num_defn}
###### Definition
**(locally $\kappa$-presentable category)**

For $\kappa$ a [[regular cardinal]], a **locally $\kappa$-presentable category** is a locally presentable category, def. \ref{PresentableCategory}, such that the colimit-generating set $S$ may be taken to consist of $\kappa$-compact objects.

=--

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

Thus, a locally presentable category is one which is locally $\kappa$-presentable for _some_ [[regular cardinal]] $\kappa$ (hence also for every $\lambda\gt\kappa$).  In fact, in this case the fourth condition is redundant; once we know that there is a colimit-generating set consisting of $\kappa$-compact objects, it follows automatically that every object is $\lambda$-compact for some $\lambda$ (though there is no uniform upper bound on the required size of $\lambda$).  Moreover, colimit-generation is also stronger than necessary; it suffices to have a [[strong generator]] consisting of small objects.

=--

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

A locally ${\aleph}_0$-presentable category is called a
**[[locally finitely presentable category]]**.

=--

## Properties
 {#Properties}

### Equivalent characterizations
 {#EquivalentCharacterizations}

There are various equivalent characterizations of locally presentable categories.

#### As limit-preserving functor categories

+-- {: .num_prop}
###### Proposition
**(as limit sketches)**

Locally presentable categories are precisely the categories of [[sketch|models of limit-sketches]].

=--

This is [Adámek & Rosický (1994), corollary 1.52](#AdámekRosický94).

Restricted to locally finitely presentable categories this becomes:

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

Locally finitely presentable categories, def. \ref{LocallyFinitelyPresentable}, are equivalently the categories of
[[finite limit]] preserving functors $C \to Set$, for small finitely complete categories $C$.

=--

For the more detailed statement see below at _[Gabriel-Ulmer duality](#GabrielUlmerDuality)_. Equivalently this says that:

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

Locally finitely presentable categories
are equivalently [[models]] of finitary [[essentially algebraic theories]].

=--

#### As localizations of presheaf categories
 {#AsLocalizationsOfPresheafCategories}

+-- {: .num_prop #AsLocalizationOfPresheafCategories}
###### Proposition
**(as accessible reflective subcategories of presheaves)**

Locally presentable categories are precisely the [[accessible functor|accessibly embedded]] full [[reflective subcategories]]

$$
  (L \dashv i)
  :
  C \stackrel{\overset{L}{\leftarrow}}{\underset{i}{\hookrightarrow}}
  PSh(K)
$$

of [[categories of presheaves]] on some small category $K$.

=--

This appears as [Adámek & Rosický (1994), prop 1.46](#AdámekRosický94).

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

Here _accessibly embedded_ means that $C \hookrightarrow Psh(K)$ is an [[accessible functor]],  which in turn means that $C$ is closed in $Psh(K)$ under $\kappa$-[[filtered colimits]] for some [[regular cardinal]] $\kappa$.

=--

See also at _[[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]]_.

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

Locally presentable categories are [[complete category|complete]].

=--

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

A reflective subcategory of a complete category is complete, since [[monadic functor|monadic functors]] reflect limits, and the above proposition shows that any locally presentable category is a reflective subcategory of a presheaf category, which is complete.

=--

### Finite presentability and Gabriel--Ulmer duality
 {#GabrielUlmerDuality}

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

Write $Lex$ for the [[2-category]] of [[small categories]] with [[finite limits]], with finitely continuous (i.e., finite limit preserving) [[functors]] between them, and [[natural transformations]] between those.

Write $LFP$ for the [[2-category]] of locally finitely presentable categories, def. \ref{LocallyFinitelyPresentable}, [[right adjoint]] functors which preserve [[filtered colimits]], and natural transformations between them.

=--

+-- {: .num_prop}
###### Theorem
**([[Gabriel-Ulmer duality]])**

There is an [[equivalence of 2-categories]]

$$
  Lex^{op} \stackrel{\simeq}{\to} LFP
$$

$$
  C \mapsto Lex(C,Set)
$$

which sends a finitely complete category $C$ to the category of [[models]] of $C$, i.e., the category of [[left exact functors]] $C \to $ [[Set]].

=--

### Well-poweredness and well-copoweredness

* Every [[locally presentable category]] is well-powered, since it is a full reflective subcategory of a presheaf topos, so its subobject lattices are subsets of those of the latter.

* Every locally presentable category is also well-*copowered*.  This is shown in [Adámek & Rosický (1994), Prop. 1.57 & Thm. 2.49](#AdámekRosický94).

## Examples and applications

### Locally finitely presentable categories

We list examples of locally finitely presentable categories, def. \ref{LocallyFinitelyPresentable}.

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

The category [[Set]] of [[sets]] is locally finitely presentable.

For notice that every [[set]] is the [[directed colimit]] over the [[poset]] of all its [[finite set|finite]] [[subsets]].

Moreover, a set $S \in Set$ is a $\kappa$-[[compact object]] precisely if it has cardinality $|S| \lt \kappa$. So all finite sets are [[aleph|$\aleph_0$]]-compact.

Hence a a set of generators that exhibits $Set$ as a locally finitely complete category is given by the set containing one finite set of [[cardinality]] $n \in \mathbb{N}$ for all $n$.

=--

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

More generally, for $C$ any [[small category]] the [[category of presheaves]] $Set^C$ is locally finitely presentable.

This follows with [[Gabriel-Ulmer duality]]:
the [[lex completion|finite limit completion]] of $C$, $Lex(C)$, is also small, and $Set^C$ is [[equivalence of categories|equivalent]] to the category of finitely continuous functors $Lex(C) \to Set$.

=--

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

More generally still, if $A$ is locally finitely presentable and $C$ is [[small category|small]], then $A^C$ is locally finitely presentable.

To see this, embed $A$ as a [[accessible functor|finitely-accessible]] [[reflective subcategory]] of a [[presheaf topos]] $Set^B$, and then note that by [[2-functor|2-functoriality]] of $(-)^C$ we get $A^C$ as a finitely-accessible reflective subcategory of $Set^{B \times C}$.

=--

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

The category of [[algebra over a Lawvere theory|algebras of]] a [[Lawvere theory]], for example [[Grp]], is locally finitely presentable. A $T$-algebra $A$ is finitely presented if and only if the [[hom-functor]] $Alg_T(A, -)$ preserves [[filtered colimits]], and any $T$-[[algebra over an algebraic theory|algebra]] can be expressed as a filtered colimit of finitely presented algebras.

=--

\begin{example}
The category of [[coalgebras]] over a [[field]] $k$ is locally finitely presentable; similarly the category of commutative coalgebras over $k$ is locally finitely presentable.
\end{example}

\begin{example}
A [[poset]], regarded as a category, is locally finitely presentable if it is a complete [[lattice]] which is [[algebraic lattice|algebraic]] (each element is a directed [[join]] of finite elements).
\end{example}

\begin{remark}
**([[counter-examples]])**

* The category [[FinSet]] of _[[finite set|finite]]_ sets is *not* locally finitely presentable, as it does not have all countable colimits.

* The category [[Field]] of [[fields]] and field [[homomorphisms]] is *not* locally presentable, as it does not have all binary coproducts (for instance, there are none between fields of differing characteristics).

* [[TopologicalSpaces]] is not locally finitely presentable.

* The [[opposite category]] of a locally presentable category (in particular, a locally finitely presentable category) is _never_ locally presentable, unless it is a poset.  This is  [Gabriel-Ulmer, Satz 7.13](#GabrielUlmer).

\end{remark}

### Locally presentable categories

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

A [[poset]], considered as a category, is locally presentable precisely if it is a complete [[lattice]].

=--

\begin{example}\label{GrothAbCatsAreLocPresntbl}
Every [[Grothendieck abelian category]] is locally presentable [[Beke (200), Prop. 3.10](#Beke00), cf. [Krause (2015), Cor. 5.2](#Krause15)].

This implies in particular (by [this example](Grothendieck+category#ChainComplexesInGrothAbCat) at *[[Grothendieck abelian category]]*) that for $R$ a [[commutative ring]] ([[internalization|internal]] to any [[Grothendieck topos]]):

* the category $R$[[Mod]] of $R$-[[modules]] is locally presentable

* the category [[category of chain complexes|$Ch_\bullet(R Mod)$]] of [[chain complexes]] is locally presentable.

\end{example}

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

The following three examples, being [[presheaf categories]], are locally finitely presentable, thus _a fortiori_ locally presentable. They are important for the general study of [[(∞,1)-categories]].

* the category [[sSet]] of [[simplicial sets]];

* the category [[dSet]] of [[dendroidal sets]].

* for $C$ a [[small category]] the [[functor category]] $Funct(C,sSet)$ of [[simplicial presheaves]]

  (see Prop. \ref{PresheavesWithValuesInLocPresAreLocPres} below).

=--

More generally,

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

Every [[sheaf topos]] is locally presentable.

=--

This appears for instance as ([Borceux, prop. 3.4.16, page 220](#Borceux)). It follows directly with prop. \ref{AsLocalizationOfPresheafCategories} and using that every [[sheaf topos]] is an  accessibly embedded [[subtopos]] of a [[presheaf topos]] (see at _[[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]]_)

The main ingredient of a direct proof is:

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

For $C$ a [[site]] and $\kappa$ a [[regular cardinal]] strictly larger than the [[cardinality]] of $Mor(C)$, every $\kappa$-[[filtered colimit]] in the [[sheaf topos]] $Sh(C)$ is computed objectwise.

=--

This implies that all [[representable functor|representables]] in a [[sheaf topos]] are $\kappa$-[[compact objects]].

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

If $T$ is an [[accessible monad]] (a [[monad]] whose underlying [[functor]] is an [[accessible functor]]) on a locally presentable category $A$, then the category $A^T$ of [[algebras over a monad|algebras over the monad]] is locally presentable. In particular, if $A$ is locally presentable and $i: B \to A$ is a [[reflective subcategory]], then $B$ is locally presentable if $i$ is accessible.
=--

This appears in [Adámek & Rosický (1994), 2.78](#AdámekRosický94).

+-- {: .query}
This is actually somewhat subtle and gets into some transfinite combinatorics, from what I can gather.
=--

\begin{proposition}
\label{PresheavesWithValuesInLocPresAreLocPres}
Given

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

* $\mathcal{A}$ a locally presentable category

then also the [[functor category]] $Func(\mathcal{C}, \mathcal{A})$ is locally presentable.
\end{proposition}
This is [Adámek & Rosický (1994), Cor. 1.54](#AdámekRosický94)

See at _[Functor category -- Local presentability](http://ncatlab.org/nlab/show/functor+category#LocalPresentability)_ for more.

\begin{proposition}
A [[slice category]] of a locally presentable category
is again locally presentable.

\end{proposition}
This appears for instance as [Centazzo-Rosický-Vitale, remark 3](#CentazzoRosickyVitale).

\begin{proposition}
\label{LocallyPresentableGrothendieckConstruction}
**(locally presentable Grothendieck constructions)**
\linebreak
Given a [[pseudofunctor]] with values in [[CatAdj|$Cat_{Adj}$]] as
$$
  \array{
    \mathllap{
      \mathbf{C} \,\colon\,
      \;
    }
    Base &\longrightarrow& Cat
    \\
    \mathcal{X} &\mapsto& \mathbf{C}_{\mathcal{X}}
    \\
    \Big\downarrow\mathrlap{{}^{f}}
    &&
    \mathllap{^{f^\ast}}\Big\uparrow
    \Big\downarrow\mathrlap{{}^{f_!}}
    \\
    \mathcal{Y} &\mapsto& \mathbf{C}_{\mathcal{Y}}
  }
$$
such that for some [[regular cardinal]] $\kappa$

1. $Base$ is locally $\kappa$-presentable,

1. each $\mathbf{C}_{\mathcal{X}}$ is locally presentable,

1. $\mathbf{C}_{(-)}$ [[preserved colimit|preserves]] [[filtered colimit|$\kappa$-filtered]] [[2-limits]]

then also the [[Grothendieck construction]] $\int \mathbf{C}$ is locally presentable.
\end{proposition}
This follows, as explained in [MO:a/102083](https://mathoverflow.net/a/102083/381), from the analogous statement for [[accessible category|accessibility]] which appears as [Makkai & Paré (1989), Prop. 5.3.1. (4)](accessible+category#MakkaiParé1989).

### Combinatorial model categories

A [[combinatorial model category]] is a [[model category]] that is in particular a locally presentable category.

### Orthogonal subcategory problem

Given a class of morphisms $\Sigma$ in a locally presentable category, the answer to the [[orthogonal subcategory problem]] for $\Sigma^\perp$ is affirmative if $\Sigma$ is small, and is affirmative for any class $\Sigma$ assuming the large cardinal axiom known as [[Vopenka's principle]].

## Related concepts

* [[PrCat]], [[Pr(∞,1)Cat]], [[Ho(CombModCat)]]

* Another notion of "presentable category" is that of an _[[equationally presentable category]]_.

* Locally presentable categories are a special case of _[[locally bounded category|locally bounded categories]]_.

* [[class-locally presentable category]]

* [[locally strongly finitely presentable category]]

[[!include locally presentable categories - table]]

## References

### General

The definition is due to

* {#GabrielUlmer} [[Pierre Gabriel]], [[Friedrich Ulmer]], *Lokal präsentierbare Kategorien*, Springer LNM **221** (1971) [[doi:10.1007/BFb0059396](https://doi.org/10.1007/BFb0059396), [MR0327863](http://www.ams.org/mathscinet-getitem?mr=0327863)]

Textbook account:

* {#AdámekRosický94} [[Jiří Adámek]], [[Jiří Rosický]], *[[Locally presentable and accessible categories]]*, London Mathematical Society Lecture Note Series **189**, Cambridge University Press (1994) [[doi:10.1017/CBO9780511600579](https://doi.org/10.1017/CBO9780511600579)]

Review for the case of [[locally finitely presentable categories]]:

* [[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]]]

See also:

* {#CentazzoRosickyVitale} C. Centazzo, [[Jiří Rosický]], [[Enrico Vitale]], _A characterization of locally $\mathbb{D}$-presentable categories_ ([pdf](http://perso.uclouvain.be/enrico.vitale/ClaudiaJiri.pdf))

* {#Borceux} [[Francis Borceux]], _[[Handbook of Categorical Algebra]]: III Categories of Sheaves_ (proposition 3.4.16), page 220.

* {#Lurie} [[Jacob Lurie]], A.1.1 of: _[[Higher Topos Theory]]_

  > (where locally presentable categories are called [[presentable categories]]*)

On the example of [[Grothendieck abelian categories]]:

* {#Beke00} [[Tibor Beke]], *Sheafifiable homotopy model categories*, Math. Proc. Cambridge Philosophical Society **129** 3 (2000) 447-475 [[arXiv:math/0102087](https://arxiv.org/abs/math/0102087), [doi:10.1017/S0305004100004722](https://doi.org/10.1017/S0305004100004722)]

* {#Bird82} [[Greg Bird]], _Limits in 2-Categories of Locally-Presentable Categories [pdf](http://maths.mq.edu.au/~street/BirdPhD.pdf)]

* {#Krause15} [[Henning Krause]], _Deriving Auslander's formula_, Documenta Math. **20** (2015) 669-688 [[arXiv:1409.7051](https://arxiv.org/abs/1409.7051)]

### In enriched category theory
 {#ReferencesInEnrichedCategoryTheory}

Discussion of local presentability in [[enriched category theory]] (see also [references on enriched accessible categories](accessible+category#ReferencesInEnrichedCategoryTheory)):

* {#Kelly82} [[G. Max Kelly]], *Structures defined by finite limits in the enriched context, I*, Cahiers de topologie et géométrie différentielle **23** 1 (1982)  3-42 [[numdam:CTGDC_1982__23_1_3_0](http://archive.numdam.org/item/CTGDC_1982__23_1_3_0)]

See also:

* [MO:q/53470](https://mathoverflow.net/q/53470/381): *Enriched locally presentable categories*

[[!redirects locally presentable]]
[[!redirects locally presentable category]]
[[!redirects locally presentable categories]]
[[!redirects presentable category]]
[[!redirects presentable categories]]
[[!redirects locally representable category]]


locally pro-manifold

#Contents#
* table of contents
{:toc}

## Idea

A [[Fréchet manifold]] modeled on $\mathbb{R}^\infty \coloneqq \underset{\longleftarrow}{\lim}_n \mathbb{R}^n$ has the property that a [[smooth function]] out of it is _locally_ (on a [[neighbourhood]] of each point) given by a function that depends only on a [[finite number]] of the [[coordinate]] functions on $\mathbb{R}^\infty$ ([Michor 80, 9.5.9](#Michor80)). Hence _locally_ such an [[infinite-dimensional manifold]] looks like a [[pro-object]] in the category of finite-dimensional smooth manifolds.

The key examples of locally pro-manifolds are _[[jet bundles]]_ of [[fiber bundles]] of finite rank ([Saunders 89](#Saunders89), [Michor 80](#Michor80)).

## References

* {#Takens79} [[Floris Takens]], _A global version of the inverse problem of the calculus of variations_, J. Differential Geom. Volume 14, Number 4 (1979), 543-562. ([Euclid](https://projecteuclid.org/euclid.jdg/1214435235))

* {#Michor80} [[Peter Michor]], _Manifolds of differentiable mappings_, Shiva Publishing, Orpington, 1980.

* {#Saunders89} D. J. Saunders, _The Geometry of Jet Bundles, vol. 142 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1989.

* {#KhavkineSchreiber17} [[Igor Khavkine]], [[Urs Schreiber]], section 2.2 of _[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations: I. Jets and comonad structure]]_ ([arXiv:1701.06238](https://arxiv.org/abs/1701.06238))

[[!redirects locally pro-manifolds]]


locally proper map

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

#Contents#
* table of contents
{:toc}

## Idea

The relative version of a locally compact topological space is a continuous map that behaves like a 'family of locally compact spaces'.

## Definition

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

A [[continuous function]] $f  \colon X \to Y$ is called _locally proper_ if for any $x \in X$ and [[neighbourhood]] $V \ni x$ there is a neighbourhood $A\ni x$ with $A \subset V$ and a neighbourhood $U \ni f(x)$ such that $A \to U$ is a [[proper map]].

=--

## Examples

* The inclusion map of a locally closed subset $C\hookrightarrow X$ in a topological space $X$ is locally proper.

* A topological space $X$ is a [[locally compact space]] if and only if $X\to \ast$ is locally proper.

## Properties

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

Let $f\colon X\to Y$ and $g\colon Y\to Z$ be locally proper. Then $g\circ f$ is locally proper. If $W\to Y$ is any continuous function then $W\times_Y Z \to W$ is locally proper. (ie locally proper maps are closed under composition and stable under pullback. Hence they form a singleton [[coverage]])

=--

As a corollary, one has that for a locally compact space $X$, every locally closed
subspace of $X$ is locally compact.

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

Let $f\colon X\to Y$ be a continuous function. Then if

* there is an open covering $\mathcal{U} = \{U_i\}$ of $X$ such that $f\big|_{U_i} \colon U_i \to Y$ is locally proper for all $i$,

or

* there is an open covering $\mathcal{V} = \{V_j\}$ of $Y$ such that $f^{-1}(V_j) \to V_j$ is locally proper for all $j$,

then $f$ is locally proper. The converse implications also hold

=--

Recall that a continuous map $g\colon Y \to Z$ of topological spaces is called _separated_ if $Y\to Y\times_Z Y$ is a closed embedding.

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

If $f\colon X\to Y$ and $g\colon Y\to Z$ are continuous maps, $g$ is separated and $g\circ f$ is locally proper, then $f$ is locally proper

=--

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

Every continuous map from a [[locally compact]] [[Hausdorff space]] to a Hausdorff space is separated and locally proper.

=--

The following proposition generalises the well-known result that compact Hausdorff spaces are locally compact.

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

Every separated and proper map is locally proper

=--

The [[proper base change theorem]] also holds for locally proper maps if one uses [[direct image with compact support]] instead of ordinary [[direct image]]

## Related Entries

* [[proper map]]

* [[locally compact space]]

* [[direct image with compact support]]

## References

* Olaf M. Schnürer and Wolfgang Soergel, _Proper base change for separated locally proper maps_,  Rend. Sem. Mat. Univ. Padova **135** (2016) pp 223-250. doi:[10.4171/RSMUP/135-13](https://doi.org/10.4171/RSMUP/135-13), arXiv:[1404.7630](https://arxiv.org/abs/1404.7630)
* Günther Richter and Alexander Vauth, _Fibrewise sobriety_, in: Categorical structures and their applications. Proceedings of the North-West European category seminar, Berlin, Germany, March 28--29, 2003. River Edge, NJ: World Scientific. 265--283 (2004; Zbl 1065.18003). doi:[10.1142/9789812702418_0020](https://doi.org/10.1142/9789812702418_0020)

[[!redirects locally proper]]


locally regular category

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

## Idea

A *locally regular category* is a relative of a [[regular category]] in which the condition of [[finite limits]] is weakened to finite [[connected limits]].  It is so named because every [[slice category]] of a locally regular category is a regular category, although the converse is not quite true.

## Definition

A [[category]] $C$ is **locally regular** if

1. It has finite [[connected limits]] --- equivalently, it has [[pullbacks]] and [[equalizers]];

1. It has ([[extremal epi]], [[monomorphism|mono]]) factorizations which are stable under pullback; and

1. Every [[span]] factors as an extremal epi followed by a jointly-monic span.

## Examples

* Every [[regular category]] is locally regular.  Factorizations of spans may be obtained by factorizations of single morphisms into a binary product.

* The [[empty category]] is a locally regular category which is not regular.

* The category $LH$ of [[topological spaces]] (or [[locales]]) and [[local homeomorphisms]] is locally regular, but not regular.  Its slice categories are precisely the [[sheaf toposes]] of spaces (or locales).

* The [[syntactic category]] of an [[extensional type theory]] with [[dependent sum types]], [[identity types]], and [[propositional truncations]] is a [[locally regular category]]. If the dependent type theory also has [[dependent product types]], the syntactic category becomes [[locally cartesian closed category|locally cartesian closed]]. If the type theory also has a [[unit type]] its syntactic category becomes a [[regular category]].

## Properties

* A locally regular category is regular if and only if it has a [[terminal object]].

* The factorization axiom for spans implies, by [[induction]], a similar factorization property for nonempty finite [[cosinks]].  However, similar factorizations for empty cosinks (i.e. "supports") do not necessarily exist.

* Every locally regular category gives rise to a tabular [[allegory]] of binary relations (where we define a "binary relation" to mean a jointly-monic span).  For composition of relations, we require pullbacks and stable factorizations of spans.  For intersection of binary relations, we require equalizers.

* Conversely, every tabular allegory has a locally regular category of maps (left adjoints).  So locally regular categories are essentially the same as tabular allegories.

## References

* [[Peter Johnstone]], [[Sketches of an Elephant]], A3.2.7

[[!redirects locally regular categories]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $\mathcal{G}$ a [[geometry (for structured (∞,1)-toposes)]] a $\mathcal{G}$-[[structured (∞,1)-topos]] $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is _locally representable_ if it is locally equivalent to $Spec U$ for $U \in Pro(\mathcal{G})$ (the [[pro-objects in an (∞,1)-category]]), or $U \in \mathcal{G}$ itself if it is _locally finite presented_ .

This generalizes

* the notion of [[smooth manifold]] from [[differential geometry]];

* the notion of [[scheme]] from [[algebraic geometry]].

* etc.

## Definition

Let $\mathcal{G}$ be a [[geometry (for structured (∞,1)-toposes)]]. Write $\mathcal{G}_0$ for the underlying
discrete geometry. The identity functor

$$
  p : \mathcal{G}_0 \to \mathcal{G}
$$

is then a morphism of geometries.

Recall the notation $LTop(\mathcal{G})$ for the [[(∞,1)-category]] of $\mathcal{G}$-[[structured (∞,1)-topos]]es and [[geometric morphism]]s between them.

### Affine $\mathcal{G}$-schemes

+-- {: .num_theorem}
###### Theorem ( [[Structured Spaces|StSp]] 2.1.1 )

There is a pair of [[adjoint (∞,1)-functor]]s

$$
  p^* : LTop(\mathcal{G}) \stackrel{\leftarrow}{\to}
  LTop(\mathcal{G}_0)
  :
  \mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}}
$$

with $\mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}}$
 left adjoint to the canonical functor $p^*$ given by precomposition with $p$.

=--

+-- {: .num_remark}
###### Remark ( [[Structured Spaces|StSp]] p. 38 )

There is a canonical morphism

$$
  can :
  Pro(\mathcal{G})^{op}
  \to
  LTop(\mathcal{G}_0)
$$

=--

+-- {: .num_defn}
###### Definition ( affine $\mathcal{G}$-scheme, [[Structured Spaces|StSp]] 2.3.9)

Write $\mathbf{Spec}^{\mathcal{G}}$ for the [[(∞,1)-functor]]

$$
  \mathbf{Spec}^{\mathcal{G}} :
  Pro(\mathcal{G})^{op}
  \stackrel{can}{\to}
  LTop(\mathcal{G}_0)
  \stackrel{
    \mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}}
  }{\to}
  LTop(\mathcal{G})
  \,.
$$

A $\mathcal{G}$-[[structured (∞,1)-topos]] in the image of this functor is an **affine $\mathcal{G}$-scheme**.

=--

### $\mathcal{G}$-Schemes

+-- {: .num_defn}
###### Definition (geometric scheme, [[Structured Spaces|StSp]] 2.3.9)

Let $\mathcal{G}$ be a [[geometry (for structured (∞,1)-toposes)]].

A $\mathcal{G}$-[[structured (∞,1)-topos]] $(\mathcal{X},\mathcal{O}_{\mathcal{X}})$ is a **$\mathcal{G}$-scheme** if

* there exists a collection $\{U_i \in \mathcal{X}\}$

such that

* the $\{U_i\}$ cover $\mathcal{X}$ in that the canonical morphism $\coprod_i U_i \to {*}$ (with ${*}$ the [[terminal object]] of $\mathcal{X}$) is an [[effective epimorphism]];

* for every $U_i$ there exists an equivalence

  $$
   (\mathcal{X}/{U_i}, \mathcal{O}_{\mathcal{X}}|_{U_i})
   \simeq \mathbf{Spec}^{\mathcal{G}} A_i
  $$

  of structured $(\infty,1)$-toposes for some $A_i \in Pro(\mathcal{G})$ (in the [[(∞,1)-category]] of [[pro-object]]s of $\mathcal{G}$).

=--

+-- {: .num_defn}
###### Definition (pregeometric scheme, [[Structured Spaces|StSp]], 3.4.6)

For $\mathcal{T}$ a pregeometry, a $\mathcal{T}$-[[structured (infinity,1)-topos]]
$(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is a **$\mathcal{T}$-scheme** if
it is a $\mathcal{G}$-scheme for [[generalized the|the]] geometric envelope
$\mathcal{G}$ of $\mathcal{T}$.

This means that
for $f : \mathcal{T} \to \mathcal{G}$ [[generalized the|the]]  geometric envelope and
for $\mathcal{O}'_{\mathcal{X}}$
[[generalized the|the]] $\mathcal{G}$-structure on $\mathcal{X}$ such that
$\mathcal{O}_{\mathcal{X}} \simeq \mathcal{O}'_{\mathcal{X}} \circ f$, we have
that $(\mathcal{X}, \mathcal{O}'_{\mathcal{X}})$ is a $\mathcal{G}$-scheme.

=--

### Smooth $\mathcal{G}$-schemes

Let $\Tau$ be a [[pregeometry (for structured (∞,1)-toposes)]] and let $\Tau \hookrightarrow \mathcal{G}$ be an inclusion into an enveloping [[geometry (for structured (∞,1)-toposes)]].

We think of the objects of $\Tau$ as the _smooth_ test spaces -- for instance the [[cartesian product]]s of some affine line $R$ with itsef -- and of the objects of $\mathcal{G}$ as affine test spaces that may have singular points where they are not smooth.

The idea is that a _smooth_ $\mathcal{G}$-scheme is a $\mathcal{G}$-structured space that is locally not only equivalent to objects in $\mathcal{G}$, but even to the very nice -- "smooth" -- objects in $\mathcal{Tau}$.

+-- {: .num_defn}
###### Definition ( smooth $\mathcal{G}$-scheme, [[Structured Spaces|StSp]] 3.5.6)

With an envelope $\Tau \hookrightarrow \mathcal{G}$ fixed, a $\mathcal{G}$-scheme is called **smooth** if there the affine schemes $\mathbf{Spec}^{\mathcal{G}} A_i$ appearing in its definition may be chosen with $A_i$ in the image of the includion $\tau \hookrightarrow \mathcal{G}$.

=--

### Examples

#### Ordinary schemes

See the discussion at [[derived scheme]] for how ordinary [[scheme]]s are special cases of [[generalized scheme]]s.

#### Ordinary Deligne-Mumford stacks

See the discussion at [[derived Deligne-Mumford stack]] for how ordinary [[Deligne-Mumford stack]]s are special cases of [[derived Deligne-Mumford stack]]s.

#### Derived schemes

+-- {: .num_defn}
###### Definition (derived scheme, [[Structured Spaces]], 4.2.8)

Let $k$ be a commutative ring. Recall the pregoemtry $\mathcal{T}_{Zar}(k)$.

A **[[derived scheme]]** over $k$ is a $\mathcal{T}_{Zar}(k)$-scheme.

=--

#### Derived smooth manifolds

* [[derived smooth manifold]].

#### Derived Deligne-Mumford stacks

+-- {: .num_defn}
###### Definition (derived Deligne-Mumford stack, [[Structured Spaces]], 4.3.19)

Let $k$ be a commutative ring. Recall the pregeometry $\mathcal{T}_{et}(k)$

A **[[derived Deligne-Mumford stack]]** over $k$ is a $\mathcal{T}_{et}(k)$-scheme.

=--

#### Derived schemes with $E_\infty$-ring valued structure sheaves

The above [[derived scheme]]s have structure sheaves with values in [[simplicial object|simplicial]] commutative rings. There is also a notion of derived scheme whose structure sheaf takes values in [[E-infinity ring]]s. The theory of these is to be described in full detail in

* [[Jacob Lurie]], _[[Spectral Schemes]]_ .

An indication of some details is in

* [[Paul Goerss]], _[[Topological Algebraic Geometry - A Workshop]]_

See at _[[E-∞ scheme]]_ and _[[E-∞ geometry]]_.

## Related concepts

* [[geometry (for structured (∞,1)-toposes)]]

* [[structured (∞,1)-topos]]

* **locally representable structured (∞,1)-topos**

## References

Generalized schemes are definition 2.3.9 of

* [[Jacob Lurie]], _[[Structured Spaces|Derived Algebraic Geometry V - Structured Spaces]]_

The definition of affine $\mathcal{G}$-schemes (absolute spectra) is in section 2.2.

[[!redirects locally representable structured (∞,1)-topos]]

[[!redirects locally representable structured (∞,1)-toposes]]
[[!redirects locally representable structured (infinity,1)-toposes]]


locally ringed locale

## Idea

The notion of *locally ringed locales* is for [[locales]] what *[[locally ringed topological spaces]]* are for [[topological spaces]] (and [[locally ringed toposes]] for [[toposes]]).

[[!redirects locally ringed locales]]


locally ringed topological space

[[!redirects locally ringed space]]

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#### Topos Theory
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=--
#### Geometry
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[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A __locally ringed space__ is a [[ringed space]] $(X,\mathcal{O})$ such that the [[stalks]] of the [[structure sheaf]] $\mathcal{O}$ are [[local ring]]s.

A [[morphism]] of locally ringed spaces is a morphism of ringed spaces $(f,f^\sharp)\colon (X,\mathcal{O}_X)\to (Y,\mathcal{O}_Y)$, where $f:X\to Y$, such that $f_\sharp : f^*\mathcal{O}_Y \to \mathcal{O}_X$ is a morphism of [[local rings]] (that is, a map of rings which respects the [[maximal ideal]] (reflects invertibility)), where $f_\sharp$ is the [[adjunct]] of the [[comorphism]] $f^\sharp:\mathcal{O}_Y\to f_*\mathcal{O}_X$. Here we are considering $\mathcal{O}_X$ and $\mathcal{O}_Y$ to be local rings defined in the [[internal logic]] of the respective [[sheaf topos]]es, and $f^*\mathcal{O}_Y$ is an internal local ring because the [[inverse image]] functor $f^*$ preserves [[geometric logic]].

## Examples

\begin{example}\label{ExampleSchemes}
**([[schemes]])**
\linebreak
Historically, *[[schemes]]* are thought of as locally ringed spaces and this application of the notion to [[algebraic geometry]] motivated much of its development.

(However, already [Grothendieck (1973)](functorial+geometry#Grothendieck73) pointed out that it is often more frutiful to view [[schemes]] instead via their [[functor of points]], see at *[[functorial geometry]]* for more.)
\end{example}

\begin{example}\label{ExampleSmoothManifolds}
**([[smooth manifolds]])**
\linebreak
The [[category]] [[SmthMfd]] of [[smooth manifolds]] has a canonical [[fully faithful functor|embedding]] into the category of locally ringed spaces:

Indeed, any smooth manifold $M$ comes equipped with a [[sheaf]] of [[real numbers|real]]-valued [[smooth functions]] $\mathcal{C}^{\infty}_M$, and the [[maximal ideal]] in a given stalk consists precisely of [[germs]] of smooth functions which vanish at that point. Moreover, a [[smooth map]] $f \colon M \to N$ determines a [[comorphism]] $f^\sharp: \mathcal{C}^{\infty}_N \to f_* \mathcal{C}^{\infty}_M$ by the usual precomposition.

This inclusion of smooth manifolds into locally ringed spaces is [[fully faithful functor|fully faithful]]. For a proof, see Lucas Braune's comment at [Math.SE:511604](https://math.stackexchange.com/a/511604/58526).

(In fact, already the [[embedding of smooth manifolds into formal duals of R-algebras]] is [[fully faithful functor|fully faithful]], which means that all smooth manifolds are even "[[affine schemes|affine]]" locally ringed spaces, in a sense.)
\end{example}

## On the locality condition

The condition that all stalks $\mathcal{O}_{X,x}$ are local rings can be reformulated without referring to the points of $X$:

* The only open subset $U$ such that $\mathcal{O}_X(U)$ is the zero ring is $U = \emptyset$.
* Let $f, g \in \mathcal{O}_X(U)$ be local sections such that $f + g$ is invertible in $\mathcal{O}_X(U)$. Then there is an open covering $U = V \cup W$ such that $f|_V \in \mathcal{O}_X(V)$ and $g|_W \in \mathcal{O}_X(W)$ are invertible. (It's allowed that $V$ or $W$ are empty.)

## Related concepts

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

* [[ringed locale]], [[locally ringed locale]]

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

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

* [[locally algebra-ed topos]]

* [[structured (∞,1)-topos]]

## References
 {#References}

Textbook accounts:

* [[Siegfried Bosch]], p. 247-248 in *Algebraic Geometry and Commutative Algebra*, Universitext, Springer (2017) [[doi:10.1007/978-1-4471-4829-6](https://doi.org/10.1007/978-1-4471-4829-6)]

*  Ulrich Görtz, Torsten Wedhorn, p. 55 in: *Algebraic Geometry I: Schemes*, Springer (2020) [[doi:10.1007/978-3-658-30733-2](https://doi.org/10.1007/978-3-658-30733-2)]

See also:

* [[Aise Johan de Jong]], _[[The Stacks Project]]_, _Locally ringed spaces_ ([tag 01HA](http://stacks.math.columbia.edu/tag/01HA))
{#deJong}

* [[James Milne]], section 4 of _[[Lectures on Étale Cohomology]]_

* [[eom]], *[Ringed spaces](https://encyclopediaofmath.org/wiki/Ringed_space)*

See also references at *[[locally ringed topos]]*.

[[!redirects locally ringed topological spaces]]

[[!redirects locally ringed spaces]]


locally ringed topos

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#### Topos Theory
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#### Higher geometry
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _locally ringed topos_ is a [[locally algebra-ed topos]] for the [[theory of local rings]].

## Definition

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

A [[ringed topos]] $(X, \mathcal{O}_X)$ with [[point of a topos|enough points]] (such as the [[sheaf topos]] over a [[topological space]]) is a **locally ringed topos** if all [[stalk]]s $\mathcal{O}_X(x)$ are [[local ring]]s.

=--

This is a special case of the following equivalent definitions:

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

A **locally ringed topos** is a topos equipped with a commutative ring object (see [[ringed topos]]) that in addition satisfies the axioms

* $(0 = 1) \vdash false$
* $x + y = 1 \vdash \exists_z (x z = 1) \vee \exists_z (y z = 1)$

(note these are axioms for a [[geometric theory]], interpreted according to [[Kripke-Joyal semantics]] in a topos).

=--

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

A [[ringed topos]] $(X, \mathcal{O}_X)$ is a [[locally algebra-ed topos]] for the [[theory of local rings]]:

* a [[topos]] $X$

* equipped with a [[geometric morphism]]

  $$
    X \stackrel{\overset{\mathcal{O}_X}{\leftarrow}}{\underset{}{\to}}
    Sh(CRing^{op}_{fp}, Z )
  $$

  into the [[Zariski topos]], the  [[classifying topos]] for the [[theory of local rings]].

=--

## Properties

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

Definition \ref{InTermsOfStalks} is indeed a special case of def. \ref{InTermsOfClassifyingTopos}.

=--

This is for instance in [Johnstone (2002)](#Johnstone2002) and in [Lurie (2009), remark 2.5.11](#Lurie2009)

## Related concepts

* [[Cole's theory of spectrum]]

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

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

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

* [[locally algebra-ed topos]]

* [[locally algebra-ed (∞,1)-topos]]

## References

Original references

* [[Alexander Grothendieck]], [[Jean-Louis Verdier]], *Topos annelés, localisation dans les topos annelés*, Section 11 in Exposé iv in: [[M. Artin]] et al., *Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos*, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 ([[SGA 4]]), Lecture Notes in Mathematics __269__, Springer (1972) [[doi:10.1007/BFb0081551](https://link.springer.com/book/10.1007/BFb0081551), [pdf](http://www.cmls.polytechnique.fr/perso/laszlo/sga4/SGA4-1/sga41.pdf)]

* [[Monique Hakim]], Chapitre III in: *Topos annelés et schémas relatifs*, Ergebnisse der Mathematik und ihrer Grenzgebiete **64**, Springer (1972) [[doi:10.1007/978-3-662-59155-0](https://doi.org/10.1007/978-3-662-59155-0)]

See also:

* {#MacLaneMoerdijk} [[Saunders MacLane]], [[Ieke Moerdijk]], Section VIII.6 ofL _[[Sheaves in Geometry and Logic]]_ (1992)

* {#Johnstone2002} [[Peter Johnstone]], _[[Sketches of an Elephant]]_ (2002)

* {#Lurie2009} [[Jacob Lurie]], Section 2.5 of: _[[Structured Spaces]]_ (2009)

* {#deJong} [[Aise Johan de Jong]], Section 14.33 of: _[[The Stacks Project]]_ ([pdf](http://www.math.columbia.edu/algebraic_geometry/stacks-git/book.pdf)) ([project website](http://www.math.columbia.edu/algebraic_geometry/stacks-git/))

[[!redirects locally ringed toposes]]


locally small (infinity,1)-category

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#Contents#
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{:toc}

## Idea

The notion of _locally small $(\infty,1)$-category_ is the generalization of the notion of [[locally small category]] from [[category theory]] to [[(∞,1)-category]] theory.

## Definitions

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

A [[quasi-category]] $C$ is **locally small** if for all objects $x,y \in C$ the [[hom-object in a quasi-category|hom ∞-groupoid]] $Hom_C(x,y)$ is [[essentially small (infinity,1)-category|essentially small]].

=--

This appears as [[Higher Topos Theory|HTT, below prop. 5.4.1.7]].

## Properties

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

A [[quasi-category]] $C$ is locally small precisely if the following equivalent condition holds:

for every [[small set]] $S$ of objects in $C$, the full [[sub-quasi-category]] on $S$ is [[essentially small (∞,1)-category|essentially small]].

## References

This is the topic of section 5.4.1 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .

=--

[[!redirects locally small (∞,1)-category]]


locally small > history

< [[locally small]]
[[!redirects locally small -- history]]


locally small category

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#Contents#
* table of contents
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## Definition

A [[category]] is said to be **locally small** if each of its  [[hom-set|hom-sets]] is a [[small set]], i.e., is a [[set]] instead of a [[proper class]].  Local smallness is included by some authors in the definition of "category."

In other words, a locally small category is a $Set$-category, i.e. a category [[enriched category|enriched in]] the category [[Set]].

If [[Grothendieck universe]]s are used to handle [[size issues]], then one speaks of a **locally $U$-small** category if all [[hom-set|hom-sets]] are elements of $U$, or of a $U\Set$-category or simply of a $U$-category.

Compare with [[small category]]; a category is **small** if it is locally small and its set of objects is also a set.  (Some care must be taken if you want this definition to be [[principle of equivalence|equivalence-invariant]].)

## Remarks

Local smallness is an instance of a general scheme by which a category may be called "locally $P$" if all hom-sets satisfy property $P$.  This is more commonly used in [[enriched category theory]] where the [[hom-objects]] have more structure than a set and can support more interesting properties.

For instance, a topologically enriched category may be said to be _locally discrete_ if its [[hom-space]]s have [[discrete space|discrete]] topologies (hence it is essentially just an ordinary category).  Likewise, a [[2-category]] is said to be _locally discrete_ if its [[hom-category|hom-categories]] are [[discrete category|discrete]] (so it is essentially an ordinary category), _locally groupoidal_ if its hom-categories are [[groupoid|groupoids]], and so on.

However, this use of the word "locally" does not really have anything to do with the intuitive _geometrical_ meaning of "local," so it should not be taken too literally, especially when one is dealing with [[internal category|internal categories]] in spaces.  Furthermore, in other contexts a category is often said to be "locally $P$" if it has the completely unrelated property that all its [[over category|slice categories]] satisfy property $P$; see for instance [[locally cartesian closed category]].  Confusion is rarely created, however, because the properties $P$ that one is interested in applying to hom-sets are usually quite different from those that one applies to slice categories.

## Related concepts

* [[small category]] , **locally small category**

* [[large category]]

[[!redirects locally small]]
[[!redirects locally small categories]]


locally small object

#Contents#
* table of contents
{:toc}

## Definition

An object $A$ in a category $C$ is a __locally small object__ if the [[full subcategory]] of the [[slice category]] $C/A$ on the [[monomorphisms]] is [[essentially small category|essentially small]] (has a small skeleton). In other words, isomorphism classes of monomorphisms with target $A$ (=[[subobject]]s of $A$) form a set.

An object $B$ in a category $C$ is __colocally small__ if it is locally small in the dual category. In other words, (isomorphism classes of) quotient objects form a set.

A category is [[well-powered category|well-powered]] if its every object is locally small. In older literature and in the subject of abelian categories one sometimes says locally small category for a well-powered category, clashing with the weaker standard notion of a [[locally small category]] (all Hom-classes are sets).

## Literature

* (1.2 in) N. Popescu, *[[Abelian categories with applications to rings and modules]]*, London Mathematical Society Monographs 3, Academic Press 1973, xii + 470 pp
* Bo Stenström, _Rings of quotients_


locally small type

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#### Type theory
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## Definition ##

In [[homotopy type theory]], given a [[type universe]] $\mathcal{U}$, a [[type]] $A$ is **locally $\mathcal{U}$-small** if for all terms $a:A$ and $b:A$, the identity type $a =_A b$ is [[essentially small type|essentially $\mathcal{U}$-small]].

## See also ##

* [[type universe]]
* [[essentially small type]]
* [[type theoretic axiom of replacement]]
* [[locally finite type]]

## References ##

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: Section 2.19 of: *[[Symmetry]]* (2021) $[$[pdf](https://unimath.github.io/SymmetryBook/book.pdf)$]$


[[!redirects empty 210]]

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# Contents
* table of contents
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## Idea

A _locally strongly finitely presentable category_ is like a [[locally finitely presentable category]], but where the class of [[filtered colimits]] (respectively the class of [[finite limits]]) is replaced by the class of [[sifted colimits]] (respectively the class of [[finite products]]).

Locally strongly finitely presentable category are precisely those categories equivalent to [[varieties of algebras]].

## Related pages

- [[locally finitely presentable category]]
- [[sifted category]]

## References

*  [[Jiri Adamek]], [[Jiri Rosicky]], _On sifted colimits and generalized varieties_, TAC **8** (2001) pp.33-53. ([tac](http://www.tac.mta.ca/tac/volumes/8/n3/8-03abs.html))

*  [[Jiri Adamek]], [[Jiri Rosicky]], [[Enrico Vitale]], _What are sifted colimits?_, TAC __23__ (2010) pp. 251--260. ([tac](http://www.tac.mta.ca/tac/volumes/23/13/23-13abs.html))

* [[Jiri Adamek]], [[Jiri Rosicky]], [[Enrico Vitale]], _Algebraic Theories - a Categorical Introduction to General Algebra_ , Cambrige UP 2010. (ch. 2) ([draft](http://www.iti.cs.tu-bs.de/~adamek/algebraic_theories.pdf))


locally trivial category

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

For a [[category]] $C$ with a single [[isomorphism class]] of [[objects]], and a given object $a$, one can ask (assuming [[axiom of choice|Choice]]) for a map $\lambda:Obj(C) \to Mor(C)$ which picks out an [[isomorphism]] $\lambda_b:a \to b$ for all other objects $b$ of $C$. This is essentially a [[section]] of the restriction of the target map $t:s^{-1}(a) \to Obj(C)$.

For an [[internal category]], this is not always possible. But if one has a notion of local sections, such as given by a [[Grothendieck pretopology]], say, then we can ask when this can be done locally. Indeed, one can relax the condition on the isomorphism classes, and look at finding sections for each isomorphism class.

In the case of [[topological categories]] (perhaps only [[groupoid]]s), the existence of such local sections forces the space of isomorphism classes to be discrete.

**Note:** The terminology 'local' refers to local on the object of objects, not local in the usual sense of category theory, that is, for each hom-set.

## Definition

Let $(S,J)$ be a concrete [[site]] with [[pretopology]] $J$.

> Copy details below to here, rewriting for arbitrary concrete site.

## Special case of topological groupoids

For a topological (resp. Lie) category $X$, let $X_1^{iso}$ denote the subspace (resp. submanifold) of invertible arrows . (This always exists, by general abstract nonsense - I should look up the reference, it's in Bunge-Pare I think - [[David Roberts|DR]])

+-- {: .un_defn}
###### Definition
A topological groupoid $X_1 \rightrightarrows X_0$ is **locally trivial** if for every point $p\in X_0$ there is a neighbourhood $U$ of $p$ and a lift of the inclusion $\{p\} \times U \hookrightarrow X_0 \times X_0$ through $(s,t):X_1^{iso}\to X_0 \times X_0$.
=--

Here $X_1^{iso}$ are the morphisms in the [[core]] of $X$: the isomorphisms. Clearly for a [[Lie groupoid]] $X_1^{iso} = X_1$. It is simple to show from the definition that for a transitive Lie groupoid, $(s,t)$ has local sections. Ehresmann goes on to show a link between smooth [[principal bundles]] and transitive, locally trivial Lie groupoids.

>Details of this...

## Alternate characterisations

A locally discrete topological groupoid is weakly equivalent (as in chapter 1 of [[Fundamental Bigroupoids and 2-Covering Spaces]] to one with a discrete space of objects. There is in fact a canonical such space, the discrete space on the underlying set of the given space of objects.

> To be expanded...

[[!redirects locally trivial categories]]


locally univalent bicategory

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# Locally univalent bicategories
* table of contents
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## Definition

In an [[intensional type theory]] such as [[homotopy type theory]], a [[bicategory]] is **locally univalent** if every [[hom-category]] is a [[univalent category]].

More specifically, it is a bicategory $C$ for which the canonical functions
$$idtoiso^{2,1}(f, g):(f = g) \to inv2cell(f, g)$$
is an equivalence of types for all objects $a:Ob(C)$, $b:Ob(C)$, and morphisms $f:Mor_C(a,b)$ and $g:Mor_C(a,b)$.

### In set-level foundations

A locally univalent bicategory in [[set-level foundations]] is the same thing as a locally [[gaunt category|gaunt]] bicategory whose type of objects is a [[set]].

## See also

* [[bicategory]]

* [[univalent bicategory]]

## References

* [[Benedikt Ahrens]], Dan Frumin, Marco Maggesi, Niels van der Weide, _Bicategories in Univalent Foundations_ ([arXiv:1903.01152](https://arxiv.org/abs/1903.01152))

[[!redirects locally univalent bicategories]]


locally variational field theory

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## Idea

A [[Lagrangian field theory]] for fields defined by some [[field bundle]] is called _locally variational_ if its [[equations of motion]] is determined by a [[source form]] on the [[jet bundle]] which is _locally_, i.e. over some [[neighbourhood]] of any point of the base space, the [[Euler-Lagrange variational derivative]] of a [[Lagrangian density]] on this neighbourhood, but not necessarily globally so.

Some common examples of [[Lagrangian field theories]] are in fact globally variational (such as the plain [[scalar field]] or the uncharged [[Dirac field]]), but other common example of Lagrangian field theories are indeed only locally variational (see the _[Examples](#Examples)_ below).

## Example
 {#Examples}

Field theories with [[WZW terms]] and [[higher WZW terms]] are (only) locally variational: the WZW term is the pullback to the [[jet bundle]] of a [[circle n-bundle with connection]] on the base space (e.g. a [[line bundle with connection]], [[bundle gerbe with connection]], etc.) and the corresponding [[Euler-Lagrange form]] is proportional to the [[horizontal form]]-projection of the [[curvature]] $(n+1)$-form. This is locally [[exact differential form|exact]], but, crucially, not globally so, unless the [[principal infinity-bundle|higher bundle]] is [[flat infinity-bundle]].

Beware that this is not an exotic situation: already the Lagrangian density for the [[charged particle]], regarded as a field theory on its [[worldline]], coupled to an [[electromagnetic field]] with non-trivial [[magnetic charge]] ([[first Chern class]]) is of this form: the [[interaction]] term which gives the [[Lorentz force]] is an example of a non-trivial [[WZW term]] as above.

## References

Exposition is in

* {#Schreiber16} [[Urs Schreiber]], _[[schreiber:Higher Structures|Higher structures in Mathematics and Physics]]_, lecture at [Oberwolfach Workshop 1651a](https://www.mfo.de/occasion/1651a/www_view), 2016 Dec. 18-23

[[!redirects locally variational field theories]]

[[!redirects locally variational quantum field theory]]
[[!redirects locally variational quantum field theories]]


located subspace

# Located subspaces
* table of contents
{: toc}

## Idea

In [[constructive mathematics|constructive]] point-set [[topology]], there are a variety of notions of when a [[subspace]] is _located_.  This varies depending on what kind of space one is discussing, and even for a given kind of space, there are sometimes various weaker and stronger notions of locatedness.  Typically (if not always), these concepts are trivial in [[classical mathematics]].

A located subspace should be viewed as a _nice_ subspace, ruling out pathological counterexamples that exist for logical rather than topological reasons.  For example, given any space $X$ and any [[truth value]] $p$, there exists a subspace $\{x | p\}$, and these can seem very strange for undecidable $p$; but typically (if not always), this subspace will only be located if $p$ is true or false.

## Definitions

### In metric spaces

Given a [[metric space]] $X$, a [[subspace]] $S$ of $X$, and a [[point]] $x$ of $X$, we commonly define the __distance__ from $x$ to $S$ (or from $S$ to $x$) as an [[infimum]]:
$$ d(x,S) \coloneqq \inf_{y \in S} d(x,y) .$$
In [[constructive mathematics]], this infimum is generally an [[upper real number]], even if (as is usual) we require the distances $d(x,y)$ to always be a [[located real number]].

The subspace $S$ is __located__ (or __metrically located__ to be specific) if, for each point $x$, $d(x,S)$ is located.  (Although we are defining located subspaces in terms of located real numbers, this is backwards historically; the term was first used, by [[Jan Brouwer]], to refer to subsets.)

We should really speak of [[extended real number|extended]] upper/located real numbers, in case $S$ might be empty or we wish to apply the concept in an [[extended metric space]].  But at least if $X$ is [[bounded metric space|bounded]] and [[inhabited space|inhabited]], then a located subspace $S$ must be empty or inhabited.

Note that there is a smaller complete lattice than the upper reals that contains the located reals and in which we could take the infimum, namely the [[MacNeille real numbers]].  However, the inclusion of MacNeille reals into upper reals does not preserve infima (though it does preserve suprema), and in the definition of located subspace we really do want to assert that the *upper real* infimum, not the MacNeille real infimum, is a located real.  This is because upper real infima, unlike MacNeille real infima, have the property that if $\inf A \lt \varepsilon$ then there is an $a\in A$ such that $a\lt\varepsilon$, which is crucial for the desired behavior of locatedness.

### In uniform spaces

If $X$ is a [[uniform space]], [Bridges et. al.](#BridgesAL) define a subspace $S$ to be **almost located** if for any entourage $U$, there is an entourage $V$ such that $\neg V[S] \cup U[S] = X$.  In other words, every point $x\in X$ is either within $U$ of some point of $S$ or not within $V$ of any point of $S$.

Since this seems the most natural notion of locatedness for uniform spaces, one might naturally also call such a subspace **uniformly located**.  However, note that even if $X$ is a metric space, metric locatedness is strictly stronger than uniform locatedness; Bridges et. al. give an example showing that metric locatedness is not a uniform invariant (while uniform locatedness is of course).

### In topological spaces

Given a [[topological space]] $X$ and a [[subspace]] $S$ of $X$, $S$ is __located__ (or __topologically located__ to be specific) if, given any point $x$ and neighbourhood $U$ of $x$, either $U \cap S$ is [[inhabited subset|inhabited]] or some other neighbourhood $V$ of $x$ is [[disjoint subsets|disjoint]] with $S$.

See Section 7.3 of [Troelstra & van Dalen](#TvD).

## Examples

### Singletons

Any singleton subset of a metric space, or more generally any finite subset, is metrically located, since we can take the minimum of any finite set.  A singleton subset of a uniform space is uniformly located if and only if the uniform space is [[regular space|regular]].

### Covert subsets

+-- {: .un_theorem}
###### Theorem
Suppose $X$ is a [[uniformly regular space|uniformly regular]] uniform space and that (the underlying set of) $S\subseteq X$ is a [[covert set]].  Then $S$ is uniformly located in $X$.
=--

+-- {: .proof}
###### Proof
Since $X$ is uniformly regular, for any entourage $U$ there is an entourage $V$ such that $\neg V \cup U = X\times X$.  In particular, for any $x\in X$, for any $y\in S$, we have either $(x,y)\in U$ or $(x,y)\notin V$.  By covertness of $S$, it follows that either $(x,y)\in U$ for some $y\in S$, or $(x,y)\notin V$ for all $y\in S$, which is to say that $x\in \neg V[S] \cup U[S]$.
=--

### Compact subspaces

+-- {: .un_theorem}
###### Theorem
Suppose $X$ is a [[uniformly regular space|uniformly regular]] uniform space and that $S\subseteq X$ is [[compact space|compact]] in its induced topology.  Then $S$ is uniformly located in $X$.
=--

+-- {: .proof}
###### Proof
Given any entourage $U$, let $V$ and $W$ be entourages such that $\neg V \cup U = X\times X$ (by uniform regularity) and $W\circ W\circ W \subseteq V$.  Let $G$ be the interior of $\neg W$.  Then $\neg V \subseteq G$, since if $(x,y)\notin V$ then $W[x] \times W[y]$ is a neighborhood of $(x,y)$ contained in $\neg W$ (for if $x \approx_W w \approx_W z \approx_W y$ then $(x,y)\in V$, which is not the case, hence $(w,z)\notin W$).  Thus, $G\cup U = X\times X$.

Let $i: X\times S \to X\times X$ be the inclusion and $\pi : X\times S \to X$ the projection.  Then $i^* G \cup i^* U = X\times S$.  Since $X$ is uniformly regular, it is regular.  Thus $S$ is compact regular, and hence locally compact, so the locale product $X\times S$ is spatial.  Therefore $\pi$ is both an [[open map]] and a [[proper map]] of [[locales]].  We then have $i^* G \cup \pi^* \pi_! i^* U = X\times S$, hence $\pi_*(i^* G \cup \pi^* \pi_! i^* U) = X$, and so by properness $\pi_* i^* G \cup \pi_! i^* U = X$ as well.  In other words, for every $x\in X$ either $(x,y)\in G$ for all $y\in S$, or there exists a $y\in S$ such that $(x,y)\in U$.  Since $G\subseteq \neg W$, it follows that $S$ is almost located.
=--

## References

* [[Douglas Bridges]], Hajime Ishihara, Ray Mines, [[Fred Richman]], Peter Schuster, and Luminiţa Vîţă.  _Almost locatedness in uniform spaces_, Czechoslovak Mathematical Journal, Vol. 57 (2007), No. 1, 1–12, [web](http://dml.cz/dmlcz/128150)
  {#BridgesAL}

* [[Anne Sjerp Troelstra]] & [[Dirk van Dalen]], _Constructivism in Mathematics_ 2.
  {#TvD}

Various related notions of locatedness are studied in

* [[Frank Waaldijk]], *modern intuitionistic topology*, Ph.D. Thesis, [link](http://www.fwaaldijk.nl/modern%20intuitionistic%20topology.pdf)

[[!redirects located set]]
[[!redirects located sets]]
[[!redirects located subset]]
[[!redirects located subsets]]
[[!redirects located subspace]]
[[!redirects located subspaces]]

[[!redirects metrically located set]]
[[!redirects metrically located sets]]
[[!redirects metrically located subset]]
[[!redirects metrically located subsets]]
[[!redirects metrically located subspace]]
[[!redirects metrically located subspaces]]

[[!redirects uniformly located set]]
[[!redirects uniformly located sets]]
[[!redirects uniformly located subset]]
[[!redirects uniformly located subsets]]
[[!redirects uniformly located subspace]]
[[!redirects uniformly located subspaces]]

[[!redirects topologically located set]]
[[!redirects topologically located sets]]
[[!redirects topologically located subset]]
[[!redirects topologically located subsets]]
[[!redirects topologically located subspace]]
[[!redirects topologically located subspaces]]

[[!redirects almost located set]]
[[!redirects almost located sets]]
[[!redirects almost located subset]]
[[!redirects almost located subsets]]
[[!redirects almost located subspace]]
[[!redirects almost located subspaces]]


locator

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### Context
#### Analysis
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# Contents
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## Definition

### In the real numbers

A **locator** for a real number $c \in \mathbb{R}$ is an element of the indexed [[cartesian product]]

$$l \in \prod_{a \in \mathbb{Q}} \prod_{b \in \mathbb{Q}} \left([\{d \in \mathbb{R} \vert a \lt d \lt c\}] \uplus [\{d \in \mathbb{R} \vert c \lt d \lt b\}]\right)^{[\{d \in \mathbb{R} \vert a \lt d \lt b\}]}$$

where $A \uplus B$ is the [[disjoint union]] of two sets $A$ and $B$, $B^A$ is the [[function set|set of functions]] from $A$ to $B$, $[A]$ is the [[support of a set|support]] of $A$, and the indexed [[cartesian product]] is defined in [[set theory]] as

$$\prod_{a \in A} B(a) = \{f \in \left(\bigcup_{a\in A} B(a)\right)^A \vert \forall a, f(a) \in B(a) \}$$

for [[family of sets]] $(B(a))_{a \in A}$.

Equivalently, a **locator** is an element of the indexed [[cartesian product]]

$$l \in \prod_{a \in \mathbb{Q}} \prod_{b \in \mathbb{Q}} [\{d \in \mathbb{R} \vert a \lt d \lt c\}]^{[\{d \in \mathbb{R} \vert a \lt d \lt b\}]} \times [\{d \in \mathbb{R} \vert c \lt d \lt b\}]^{[\{d \in \mathbb{R} \vert a \lt d \lt b\}]}$$

### In arbitrary dense linear orders

Given a [[dense linear order]] $(A, \lt_A)$, the [[open interval]] in $A$ between elements $a \in A$ and $b \in B$ is the set

$$\mathrm{OpenInt}(a,b) = \{c \in A \vert a \lt c \lt b\}$$

Given a [[countable dense linear order]] $(B, \lt_B)$ such that $B \subseteq A$ and for all elements $a \in A$ and $b \in A$ where $a \lt b$, there exists $c \in B$ such that $a \lt c \lt b$, a **$B$-indexed locator** for an [[element]] $c \in A$ is an element of the indexed [[cartesian product]]

$$l \in \prod_{a \in B} \prod_{b \in B} \left([\mathrm{OpenInt}(a,c)] \uplus [\mathrm{OpenInt}(c,b)]\right)^{[\mathrm{OpenInt}(a,b)]}$$

## In dependent type theory

In dependent type theory, given a real number $r:\mathbb{R}$, a locator on the [[Dedekind real numbers]] is a [[dependent function]]

$$l:\prod_{p:\mathbb{Q}} \prod_{q:\mathbb{Q}} (p \lt q) \to ((p \lt r) + (r \lt q))$$

This above type is equivalent to the types

$$\prod_{p:\mathbb{Q}} \prod_{q:\mathbb{Q}} (p \lt q) \to \sum_{b:\mathrm{bool}} ((b = 1) \to (p \lt r)) \times ((b = 0) \to (r \lt q))$$
$$\prod_{p:\mathbb{Q}} \prod_{q:\mathbb{Q}} (p \lt q) \to \sum_{b:\mathrm{bool}} ((b = 0) \to (p \lt r)) \times ((b = 1) \to (r \lt q))$$

## Properties

* A locator is equivalent to having the structure of a [[Cauchy sequence]] with [[modulus of convergence]]. This is stronger than merely being a [[modulated Cauchy real number]].

* That every [[Dedekind real number]] has a $\mathbb{Q}$-indexed locator implies the weak [[limited principle of omniscience]].

## Principle of locators

Note: The "principle of locators" or "axiom of locators" are placeholder names for a principle or axiom which may or may not have an already existing name in the mathematics literature.

The **principle of locators** state that every real number $x:\mathbb{R}$ merely has a locator (i.e. the [[support]] of the locator has an element), and implies that the [[Cauchy real numbers]] is the [[terminal object|terminal]] [[Archimedean ordered field]], and coincides with the [[Dedekind real numbers]]. This is true in [[classical mathematics]], as is in [[constructive mathematics]] which also uses [[countable choice]].

However, in general constructive mathematics, while theorem 6.10.3 of [Booij 2020](#Booij20) states that a real number is a [[Cauchy real number]] if and only if it merely has a locator, not every real number is necessarily a Cauchy real number, so it is not necessarily true that every real number has a locator. In that case, this principle becomes the **axiom of locators**, which says that every element $x:\mathbb{R}$ of the [[terminal object|terminal]] [[Archimedean ordered field]] $\mathbb{R}$ merely has a locator, making it coincide with the [[Cauchy real numbers]]. We use terminal Archimedean ordered field since in [[predicative mathematics]] the [[Dedekind real numbers]] may not exist.

## Related concepts

* [[Dedekind cut]]
* [[Dedekind cut structure]]
* [[dense linear order]]
* [[lifting to locators]]

## References

* [[Auke Booij]], *Analysis in univalent type theory* (2020) $[$[etheses:10411]( 	http://etheses.bham.ac.uk/id/eprint/10411), [pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf)$]$

* [[Steve Vickers]], *Locators point-free* ([pdf](https://www.cs.bham.ac.uk//~sjv/locatorsPF.pdf))

[[!redirects locators]]

[[!redirects principle of locators]]
[[!redirects axiom of locators]]


Lochlainn O'Raifeartaigh

* [Wikipedia entry](http://en.wikipedia.org/wiki/Lochlainn_O%27Raifeartaigh)

category: people

[[!redirects L. O'Raifeartaigh]]


locus

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#### Geometry
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# Contents
* table of contents
{: toc}

## Idea

In [[mathematics]], by a *locus* one typically means a kind of [[subspace]] of [[points]] satisfying some prescribed conditions.

For example:

* a *[[critical locus]]* is a subspace of [[critical points]] in the [[domain]] of a [[differentiable function]];

* a *[[fixed locus]]* is a subspace of [[fixed points]] under a [[group action]];

* the term *algebraic locus* is sometimes used (e.g. [[Models for Smooth Infinitesimal Analysis|Moerdijk & Reyes 1991, p. 8]] or [arXiv:1604.07827, p. 1-2](https://arxiv.org/abs/1604.07827)) for what more commonly are called *[[varieties]]*, namely subspaces of [[solutions]] to [[polynomial]] [[equations]];

  similarly, there is a notion of *[[smooth locus]]*.

## Unrelated entries

Un-related to this standard use of "locus" is the notion of *[[Joyal locus]]* in ([[higher topos theory|higher]]) [[topos theory]].

## References

See also

* Wikipedia, *Locus (mathematics)*

[[!redirects locuses]]
[[!redirects loci]]


Loday-Pirashvili category

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#Contents#
* table of contents
{:toc}

## Idea

The LP category is a  [[category]] whose [[Lie algebra objects]] are [[Leibniz algebras]].

## Overview

Given any [[category]] $C$, one can define the [[arrow category]] $\Arr(C)$ of $C$, whose objects are morphisms in $C$ and whose morphisms are commutative squares. If $C$ is the category of [[vector space]]s (or some other $k$-linear [[closed monoidal category|closed]] [[symmetric monoidal category]] with [[equalizer]]s) one can define the _infinitesimal_ or _Loday--Pirashvili_ (LP) tensor product on the category of arrows, as well as an inner hom, equipping the category $\mathrm{Arr} C$ with a structure of a $k$-linear closed symmetric monoidal category.

The LP-tensor product is
$$(f:V_1\to V_0)\otimes (g:W_1\to W_0):= (V_0\otimes g + f\otimes W_0: V_0\otimes W_1 \oplus V_1\otimes W_0\to V_0\otimes W_0).$$
This is a truncation of the tensor product of [[chain complex]]es where $V_1\otimes W_1$ is dropped.

The inner hom is rather interesting:
$\mathbf{Hom}(f,g) = (p:\mathrm{Hom}_1(f,g)\to\mathrm{Hom}_0(f,g))$, where $\mathrm{Hom}_0(f,g)$ is the equalizer of two morphisms
$$
\mathrm{hom}(V_0,W_0)\oplus\mathrm{hom}(V_1,W_1)\to\mathrm{hom}(V_1,W_0),
$$
namely precomposing the first summand with $f$ and postcomposing the second summand with $g$ (where $\mathrm{hom}$ is the ordinary [[hom-set|inner hom]] in $C$), and where $\mathrm{Hom}_1(f,g)$ is the equalizer of two morphisms
$$\mathrm{Hom}_0(f,g)\oplus\mathrm{hom}(V_0,W_1)\to \mathrm{Hom}_0(f,g),$$
namely the identity and the map which replaces the lower component with the postcomposition by $g$ applied on $\mathrm{hom}(V_0,W_1)$ and keeps the upper component.
Finally, $p$ is the natural projection.

In the case of vector spaces this means that we have diagonal lifts in squares such that the lower square commutes but not necessarily the upper, i.e. $\mathrm{Hom}(f,g)$ is the space consisting of all triples $(u_1,u_0,\phi)$ where $u_1:V_1\to W_1$, $u_0:V_0\to W_0$ and $\phi:V_0\to W_1$
such that $g\circ u_1= u_0\circ f$ and $u_0=g\circ\phi$ while one does *not* require $\phi\circ f=u_1$.

There are a number of remarkable functors relating [[internalization|internal]] algebras in LP, [[Lie algebra]]s in LP etc., to or from some other categories of algebras. For example the categories of left [[Leibniz algebras]] and of right Leibniz algebras embed as full subcategories into the category of [[internal Lie algebras]] in LP. This embedding has an [[adjoint functor|adjoint]].
Notice that because of truncation, being a Lie algebra in LP is a bit less than a (strict) $2$-[[n-Lie algebra|Lie algebra]] (a requirement in degree $2$ is dropped).

## Literature and discussions

* {#LodayPirashvili98} [[Jean-Louis Loday]], [[Teimuraz Pirashvili]], _The tensor category of linear maps and Leibniz algebras_, Georg. Math. J. vol. 5, n.3 (1998) 263--276 ([doi:10.1023/B:GEOR.0000008125.26487.f3](https://link.springer.com/article/10.1023/B:GEOR.0000008125.26487.f3))

* [nCafe](http://golem.ph.utexas.edu/category/2007/10/loday_and_pirashvili_on_lie_2a.html)

[[!redirects Loday–Pirashvili category]]
[[!redirects Loday--Pirashvili category]]
[[!redirects Loday-Pirashvili tensor category]]
[[!redirects Loday–Pirashvili tensor category]]
[[!redirects Loday--Pirashvili tensor category]]