(infinity,n)Cat

[[!redirects (∞,n)Cat]]

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## Idea

The (large) $(\infty,n+1)$-category $(\infty,n)Cat$ is the collection of all (small) [[(∞,n)-categories]]:

* [[object]]s are [[(∞,n)-categories]];

* [[morphisms]] are [[(∞,n)-functor]]s;

* [[k-morphism]]s for $k \geq 2$ are $(k-1)$-[[transfor]]s.

There are various presentations for this. For general $n$ see for instance [this section](http://ncatlab.org/nlab/show/Theta-space#OfAll) at _[[Theta-space]]_. For low $n$ see the discussion at _[[(∞,1)Cat]]_ and _[[(∞,2)Cat]]_.

Often it is useful to consider just the maximal [[(∞,1)-category]] inside $(\infty,n)Cat$. This is what is presented by various [[model category]] structures on models for $(\infty,n)$-categories.

The discussion in ([BarwickSchommer-Pries](#BarwickSchommer-Pries)) shows that essentially all proposed models for $(\infty,n)Cat$ are in fact equivalent.

## Properties

### Automorphisms

The [[automorphism ∞-group]] of $(\infty,n)Cat$ is equivalent to $(\mathbb{Z}_2)^n$.

This is due to ([Barwick & Schommer-Pries](#BarwickSchommer-Pries)).

(See also at _[[duality]]_.)

## Related concepts

* [[Pos]]

* [[Set]]

* [[Grpd]], [[∞Grpd]]

* [[Cat]], [[Operad]]

* [[2Cat]]

* [[(∞,1)Cat]], [[(∞,1)Operad]]

* **$(\infty,n)Cat$**

## References

* [[Clark Barwick]], [[Chris Schommer-Pries]], _On the unicity of the homotopy theory of higher categories_ ([pdf](https://arxiv.org/pdf/1112.0040.pdf))
 {#BarwickSchommer-Pries}

[[!redirects InfinityCommaNCat]]

category: category


(infinity,r)-category > history

< [[(infinity,r)-category]]
[[!redirects (infinity,r)-category -- history]]


(n × k)-category

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## Idea

An _$(n \times k)$-category_ (read "n-by-k category") is an [[n-category]] [[internalization|internal]] to a $k$-category.  The term is "generic" in that it does not specify the level of strictness of the $n$-category and the $k$-category.

For example:

* A $(1 \times 0)$-category, as well as a $(0 \times 1)$-category, is precisely a [[category]].  More generally, $(n\times 0)$-categories and $(0\times n)$-categories are precisely $n$-categories.
* A $(1 \times 1)$-category is precisely a [[double category]] (either strict or weak).
* Generalizing to a 3rd axis, a $(1 \times 1 \times 1)$-category is precisely a [[triple category]], that is, a category internal to (categories internal to categories), i.e. a catgory internal to double categories, or a double category internal to categories --- which again could be strict or weak.
* An $(n \times 1)$-category is what [[Michael Batanin|Batanin]] calls a [[monoidal n-globular category]].

An $(n \times k)$-category has $(n + 1)(k + 1)$ kinds of cells.

Under suitable [[fibrant object|fibrancy]] conditions, a $(n \times k)$-category will have an underlying $(n + k)$-category (where here, $n + k$ is to be read arithmetically, rather than simply as notation). Fibrant $(1 \times 1)$-categories are known as [[framed bicategories]].

## Examples

* [[commutative ring|Commutative rings]], [[algebras]] and [[modules]] form a symmetric monoidal $(2 \times 1)$-category.
* [[conformal net|Conformal nets]] form a symmetric monoidal $(2 \times 1)$-category.

## Relationships

At least in some cases, if the structure is sufficiently strict or sufficiently fibrant, we can shift cells from $k$ to $n$.  For instance:

* A sufficiently strict $(1 \times 2)$-category canonically gives rise to a $(2 \times 1)$-category. (Cor. 3.11 in **[DH10](#DH10)**)

* Any double category (i.e. a $(1\times 1)$-category) has an underlying 2-category.

* A sufficiantly fibrant $(2\times 1)$-category has an underlying tricategory (i.e. $(3\times 0)$-category).

## Related concepts

* [[internal category]]

* [[higher category]]

* [[n-fold category]]

* [[(n,r)-category]]

## References ##

* [[Mike Shulman]], Constructing symmetric monoidal bicategories, arXiv preprint arXiv:1004.0993 (2010)

* [[Michael Batanin]], _Monoidal globular categories as a natural environment for the theory of weak $n$-categories_ , Advances in Mathematics 136 (1998), no. 1, 39–103.

The following paper contains some discussion on the relationship between various (weak) $(n \times k)$-categories for $n, k \leq 3$.

* {#DH10} [[Chris Douglas]], and [[Andre Henriques]], _Internal bicategories_ ([arXiv:1206.4284](https://arxiv.org/abs/1206.4284)) (2012)

There is some discussion on [this n-Category Café post](https://golem.ph.utexas.edu/category/2010/04/symmetric_monoidal_bicategorie.html) as well as [this one](https://golem.ph.utexas.edu/category/2011/03/liang_kong_on_levinwen_models.html).

[[!redirects n-by-k category]]
[[!redirects n-by-k categories]]


(n+1,1)-category of n-truncated objects

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## Idea

Given any [[object]] $X$ in any $(k,1)$-category $C$ for a [[natural number]] $k$, the $n$-[[truncated objects]] of $X$ for $n+1 \lt k$ form an $(n+1,1)$-category, an $(n+1)$-truncated $(k,1)$-category, (though in general it may be a [[large category|large]] one, cf. [[well-powered category]]). This is called the __$(n+1,1)$-category of $n$-truncated objects__ of $X$, or the __$n$-truncated object $(n+1,1)$-category__ of $X$.

## Properties

If $C$ is [[finitely complete category|finitely complete]], then the $n$-truncated objects form a finitely complete $(n+1,1)$-category, so we may speak of the __finitely complete $(n+1,1)$-category of $n$-truncated objects__.

In any [[coherent category|coherent]] $(k,1)$-category, then the $n$-truncated objects form a coherent $(n+1,1)$-category, so we may speak of the __coherent $(n+1,1)$-category of $n$-truncated objects__.

In any $(k,1)$-[[topos]], the $n$-truncated objects of $X$ form a $(n+1,1)$-topos, so we may speak of the __$(n+1,1)$-topos of $n$-truncated objects__.

The reader can probably think of other variations on this theme.

If $f : X \to Y$ is a morphism that has pullbacks along $n$-truncated morphisms, then pullback along $f$ induces a $(n+1,1)$-category morphism $f^* : Trunc_n(Y) \to Trunc_n(X)$. This is functorial in the sense that if $g : Y \to Z$ also has this property, then there is an inhabited equivalence $n+1$-groupoid $f^* \circ g^* \cong (g \circ f)^*$.

If $C$ has pullbacks of $n$-truncated morphisms, $Trunc_n$ is often used to denote the contravariant functor $C^{op} \to (n+1,1)Cat$ whose action on morphisms is $Trunc_n(f) = f^*$.

## Examples

* A [[(0,1)-category]] of (-1)-truncated objects of an object $X$ is a [[poset of subobjects]] of $X$.

* A [[well-pointed category|well-pointed]] [[Heyting category|Heyting]] [[pretopos]] of 0-truncated objects of the [[terminal object]] $1$ is a (possibly) [[predicative mathematics|predicative]] model of the [[category of sets]].

* A [[well-pointed category|well-pointed]] [[Heyting category|Heyting]] [[Π-pretopos]] of 0-truncated objects of the [[terminal object]] of $1$ is a (possibly) [[predicative mathematics|weakly predicative]] model of the [[category of sets]].

* A [[well-pointed category|well-pointed]] [[topos]] of 0-truncated objects of the [[terminal object]] $1$ is an [[predicative mathematics|impredicative]] model of the [[category of sets]].

* A [[well-pointed category|well-pointed]] [[W-topos]] of 0-truncated objects of the [[terminal object]] $1$ whose 0-truncated (-1)-[[connected object|connected]] [[morphisms]] all have [[sections]] is a model of [[ETCS]].

## Related concepts

* If one opts for the alternative definition that $n$-truncated objects _are_ $n$-truncated morphisms into the object (not equivalence large $n+1$-groupoids thereof), then one gets a $(n+1,1)$-[[precategory]] of $n$-truncated objects instead. In any case, the *$(n+1,1)$-category* of $n$-[[truncated objects]] $Trunc_n(X)$ in our sense is the $(n+1,1)$-categorical reflection of the $(n+1,1)$-precategory $TruncMor_n(X)$ of $n$-truncated objects in the alternative sense, and of course the reflection [[Rezk completion]] map $TruncMor_n(X) \to Trunc_n(X)$ is an [[equivalence]].

* [[poset of subobjects]]

* [[n-truncated object classifier]]

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(n,0)-category

An _$(n,0)$-category_ is an [[(n,r)-category]] that is an [[n-groupoid]].

By the general rules of $(n,r)$-[[(n,r)-category|categories]], an __$(n,0)$-category__ is an $\infty$-[[infinity-category|category]] such that
*  any $j$-morphism is an [[equivalence]], for $j \gt 0$;
*  any two parallel $j$-morphisms are equivalent, for $j \gt n$.

You can start from any notion of $\infty$-category, strict or weak; up to [[equivalence of categories|equivalence]], the result is the same as an [[n-groupoid]] with a corresponding level of strictness.

[[!redirects (n,0)-categories]]


(n,1)-category

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## Idea

The special case of an [[(n,r)-category]] for $r = 1$.

## Definition

An __$(n,1)$-category__, is an $n$-[[n-category|category]] $C$ that is __locally $(n-1)$-groupoidal__; that is, for any [[object]]s $x$ and $y$, the $(n-1)$-category $C(x,y)$ is an $(n-1)$-[[n-groupoid|groupoid]].

Equivalently it is an $(\infty,1)$-category for which the mapping spaces are all $(n-1)$-[[n-truncated object of an (infinity,1)-category|truncated]].

## Special cases:

*  A $(1,1)$-category is the same as a $1$-[[1-category|category]], which is an ordinary [[category]].
*  A $(2,1)$-category is a [[locally groupoidal 2-category|locally groupoidal]] $2$-[[2-category|category]].
*  An $(\infty,1)$-[[(infinity,1)-category|category]] can be understood as a [[quasi-category]] or in many other ways.

## Extra stuff, structure, property

* An $(n,1)$-category with the analogous properties of a [[topos]] is an [[(n,1)-topos]].

## Examples

The canonical example of an $(n+1,1)$-category is [[nGrpd]].

## Related concepts

* [[0-category]], [[(0,1)-category]]

* [[category]]

* [[2-category]]

* [[3-category]]

* [[n-category]]

* [[(∞,0)-category]]

* **(n,1)-category**

* [[(∞,1)-category]]

* [[(∞,2)-category]]

* [[(∞,n)-category]]

* [[(n,r)-category]]

## References

In Section 11 of

* [[Charles Rezk]], _A cartesian presentation of weak n-categories_, [arXiv:0901.3602](https://arxiv.org/abs/0901.3602)

the author describes a presentation of $(n,1)$-categories by a [[left Bousfield localization]] of the [[model structure]] presenting [[complete Segal spaces]].

[[!redirects (n,1)-categories]]


(n,1)-sheaf

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## Definition

For $n \in \mathbb{Z}$, $-2 \leq n \lt \infty$,
an _$(n,1)$-sheaf_ is an [[n-truncated]] [[(∞,1)-sheaf]]: an [[(∞,1)-sheaf]] with values in [[n-groupoids]].

This is also sometimes called an _$(n-1)$-stack_.

## Examples

* [[(0,1)-sheaf]] = [[ideal]] (of a [[poset]]/[[proset]])

* [[(1,1)-sheaf]] = [[sheaf]]

* [[(2,1)-sheaf]] = [[stack]] (of groupoids)

## Related concepts

* [[n-truncated object in an (∞,1)-category]]

* [[model structure for homotopy n-types]]

[[!redirects (n,1)-sheaves]]
[[!redirects (n+1,1)-sheaf]]

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(n,1)-topos

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## Idea

A (Grothendieck) $(n,1)$-topos is the [[(n,r)-category|(n,1)-category]] version of a [[Grothendieck topos]]: a collection of [[n-groupoid|(n-1)-groupoid]]-valued sheaves on an $(n,1)$-categorical site.

Notice that an [[∞-stack]] on an ordinary (1-categorical) [[site]] that takes values in [[∞-groupoid]]s which happen to by 0-[[truncated]], i.e. which happen to take values just in [[Set]] $\hookrightarrow$ [[∞Grpd]] is the same as an ordinary [[sheaf]] of sets.

This generalizes: every $(n,1)$-topos arises as the full [[sub-quasi-category|(∞,1)-subcategory]] on $(n-1)$-[[truncated]] objects in an [[(∞,1)-topos]] of $\infty$-stacks on an [[(n,1)-category]] site.

## Definition

Recall that

* a 1-[[Grothendieck topos]] is precisely an [[accessible functor|accessible]] [[geometric embedding]] into a [[category of presheaves]] $PSh(C)$ on some [[small category]] $C$

  $$
    Sh(C) \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PSh(C)
    \,.
  $$

* a [[(∞,1)-topos]] (of [[∞-stack]]s/[[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaves]]) is precisely an [[accessible (∞,1)-functor|accessible]] [[reflective (∞,1)-subcategory|geometric embedding]] into a [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(C)$ on some small [[(∞,1)-category]] $C$:

  $$
    Sh_{(\infty,1)}(C)
    \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow}
    PSh_{(\infty,1)}(C)
    \,.
  $$

Accordingly now,

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

An **$(n,1)$-topos** $\mathcal{X}$ is an accessible left exact [[localization of an (∞,1)-category|localization]] of the full [[sub-quasi-category|(∞,1)-subcategory]] $PSh_{\leq n-1}(C) \subset PSh_{(\infty,1)}(C)$ on $(n-1)$-[[truncated]] objects in an [[(∞,1)-category of (∞,1)-presheaves]] on a small [[(∞,1)-category]] $C$:

$$
  \mathcal{X}
   \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow}
   PSh_{\leq n-1}(C)
   \,.
$$

=--

This appears as [[Higher Topos Theory|HTT, def. 6.4.1.1]].

## Properties

Write [[(∞,1)-Topos]] for the [[(∞,1)-category]] of [[(∞,1)-topos]] and [[(∞,1)-geometric morphisms]]. Write $(n,1)Topos$ for the [[(n,1)-category|(n+1,1)-category]] of $(n,1)$-toposes and geometric morphisms between these.

The following proposition asserts that when passing to the $(n,1)$-topos of an [[(∞,1)-topos]] $\mathcal{X}$, only the [[n-localic (∞,1)-topos|n-localic]] "Postnikov stage" of $\mathcal{X}$ matters.

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

Every $(n,1)$-topos $\mathcal{Y}$ is the [[(n,1)-category]] of $(n-1)$-[[truncated]] objects in an [[n-localic (∞,1)-topos]] $\mathcal{X}_n$

$$
  \tau_{n-1} X_n  \stackrel{\simeq}{\to} \mathcal{Y}
  \,.
$$

=--

This is ([[Higher Topos Theory|HTT, prop. 6.4.5.7]]).

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

For any $0 \leq m \leq n \leq \infty$, $(m-1)$-truncation induces a [[localization of an (∞,1)-category|localization]]

$$
  Topos_{(m,1)} \stackrel{\overset{\tau_{m-1}}{\leftarrow}}{\hookrightarrow}
  Topos_{n,1}
$$

that identifies $Topos_{(m,1)}$ equivalently with the full subcategory of $m$-localic $(n,1)$-toposes.

=--

(This is [[Higher Topos Theory|6.4.5.7]] in view of the following remarks.)

## Examples

### $(2,1)$-Toposes

If $E$ is a [[(2,1)-topos]] in which every object is *covered* by a 0-[[truncated]] object, then $E$ is equivalent to the category of (2,1)-sheaves on a 1-site (rather than merely a (2,1)-site, as is the case for general (2,1)-topoi), and is thus canonically *associated* to a 1-topos, namely the category of 1-sheaves on that same 1-site.  And in fact, $E$ can be recovered from this 1-topos as the category of (2,1)-sheaves for its canonical topology.

See [[michaelshulman:truncated 2-topos]] for more.

## Related concepts

[[!include flavors of higher toposes -- list]]

## References

Section 6.4 of

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

[[!redirects (n,1)-toposes]]
[[!redirects (n,1)-topoi]]


(n,n)-category

Following the pattern of the notion of [[(n,r)-category]], an _$(n,n)$-category_ is a [[higher category theory|higher category]] with non-trivial cells of at most dimension $n$ and none of them guaranteed to be reversible.

So this is what is usually simply called an [[n-category]].

Note that it is possible to go on to an $(n,n+1)$-category, or $(n+1)$-[[n-poset|poset]].  You can either consider than the $n$-cells are ordered, or else consider that there are irreversible $(n+1)$-cells which are indistinguishable.  (Reversible indistinguishable $(n+1)$-cells are all identities and so might as well not exist.)


(n,r)-category

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## Idea

An $(n,r)$-category is a [[higher category theory|higher category]] such that, essentially:

* all [[k-morphisms]] for $k \gt n$ are trivial.

* all [[k-morphisms]] for $k \gt r$ are reversible.

Put another way: given a sequence of (higher) categories $C_0, C_1, ..., C_n$ in which each $C_{i+1}$ is of the form $Hom(A, B)$ for some $0$-cells $A$ and $B$ from $C_i$, let us say that $C_n$ is a depth-$n$ Hom-category of $C_0$. (We can also cleanly extend this notion to depth-$\infty$ Hom-categories, by taking the position that there are none). An $(n, r)$-category, then, is one in which every depth-$r$ Hom-category is an $\infty$-groupoid, and, furthermore, every depth-$(n+2)$ Hom-category is a [[point]]. (The appearance of $n+2$ here rather than $n$ allows us to make sense of this definition even when $n$ is as low as $-2$, and suggests that perhaps, had history gone differently, the conventions would be to number these differently.)

So $(n,r)$-categories are a generalisation of both $n$-[[n-category|categories]] and $n$-[[n-groupoid|groupoids]], covering all of the ground in between (and a bit beyond). As $n$ increases, there are many more possibilities, until there are infinitely many kinds of $(\infty,r)$-[[(infinity,n)-category|categories]].

## Definition
{#Definition}

Given a notion of [[infinity-category|$\infty$-category]] (as weak or strict as you like), then an **$(n,r)$-category** can be defined to be an $\infty$-category such that

* any $j$-morphism is an [[equivalence]], for $j \gt r$;
* any two [[parallel morphisms|parallel]] $j$-morphisms are equivalent, for $j \gt n$.

As explained below, we may assume that $n \geq -2$ and $0 \leq r \leq \max(0, n + 1)$.

For finite $r$, we can also define this inductively in terms of [[(∞,r)-categories]] as follows:

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

For $-2 \leq n \leq \infty$, an **[[(n,0)-category]]** is an [[∞-groupoid]] that is [[n-truncated]]: an [[n-groupoid]].

For $0 \lt r \lt \infty$, an **(n,r)-category** is an [[(∞,r)-category]] $C$ such that for all [[object]]s $X,Y \in C$ the $(\infty,r-1)$-categorical [[hom-object]] $C(X,Y)$ is an $(n-1,r-1)$-category.
=--

(Even for $r = \infty$, this definition makes sense, taking $\infty - 1$ to be $\infty$, as long as we know that an $(-1,\infty)$-category is the same thing as a $(-1,0)$-category.  But this may be overkill.)

You can also start with a notion of $n$-[[n-poset|poset]], then define an $(n,r)$-category to be an $(n+1)$-poset such that any $j$-morphism is an [[equivalence]] for $j \gt r$. Or, for $r \leq n$, you can start with a notion of $n$-[[n-category|category]], then define an $(n,r)$-category to be an $n$-category such that any $j$-morphism in an [[equivalence]] for $j \gt r$.

To interpret the first definition above correctly for low values of $j$, we must assume that all objects ($0$-morphisms) in a given $\infty$-category are parallel, which leads us to speak of the two $(-1)$-morphisms that serve as their common source and target and to accept any object as an equivalence between these. In particular, any $j$-morphism is an equivalence for $j \lt 1$, so if $r = 0$, then the condition is satisfied for any smaller value of $r$. Thus, we assume that $r \geq 0$.

To say that parallel $(-1)$-morphisms must be equivalent is meaningful; it requires that there be an object. One can continue to $(-2)$-morphisms and so on, but there is nothing to vary about these; so we assume that $n \geq -2$. In other words, a $(-2)$-[[(-2)-category|category]] will automatically be an $n$-category for any smaller value of $n$.

If any two parallel $j$-morphisms are equivalent, then any $j$-morphism between equivalent $(j-1)$-morphisms is an equivalence (being parallel to an identity for $j \gt 0$ and automatically for $j \lt 1$). Accordingly, any $(n,r)$-category for $r \gt n + 1$ is also an $(n,n+1)$-category. Thus, we assume that $r \leq n + 1$ (except when $n = -2$, where it would conflict with the convention $r \geq 0$ and so we simply take $r = 0$.)

## Homotopy-theoretic relation {#HomtopyTheory}

From the point of view of [[homotopy theory]], the notion of $(n,r)$-categories may be understood as a combination of the notion of [[homotopy n-type]] and that of [[directed space]].

Recall that an [[(∞,0)-category]] is an [[∞-groupoid]]. In light of the [[homotopy hypothesis]] -- that identifies $\infty$-groupoids with (nice) [[topological spaces]] and [[n-groupoids]] with [[homotopy n-types]] -- and in view of the notion of [[directed space]], the following terminology is suggestive:

+-- {: .standout}

An $(n,r)$-category is an $r$-directed homotopy $n$-type.

=--

Here we read

* _$0$-directed_ as _undirected_

and

* _$1$-directed_ as _directed_ .

Then, indeed, we have for instance that

* a [[(1,0)-category]] is an undirected 1-type: a 1-[[groupoid]],

* a [[(n,r)-category|(2,0)-category]] is an undirected 2-type: a [[2-groupoid]],

* etc.

* a [[(n,n)-category|(1,1)-category]] is **directed 1-type** : a [[category]],

* an [[(n,n)-category]] is an $n$-directed $n$-type: an [[n-category]],

* etc.

* an [[(∞,0)-category]] is an undirected space: an [[∞-groupoid]],

* an [[(∞,1)-category]] is a [[directed space]]: a [[quasi-category]],

* an [[(∞,n)-category]] is an $n$-directed space

* etc.

## Special cases

An [[(n,n)-category]] is simply an $n$-[[n-category|category]]. An $(n,n+1)$-category is an $(n+1)$-[[n-poset|poset]]. Note that an $\infty$-category and an $\infty$-poset are the same thing. An $(n,0)$-category is an $n$-[[n-groupoid|groupoid]].  Even though they have no special name, $(n,1)$-[[(n,1)-category|categories]] are widely studied.

For low values of $n$, many of these notions coincide.  For instance, a $0$-[[0-groupoid|groupoid]] is the same as a $0$-[[0-category|category]], namely a [[set]].  And $(-1)$-[[(-1)-groupoid|groupoid]], $(-1)$-[[(-1)-category|category]], and $0$-[[0-poset|poset]] all mean the same thing (namely, a [[truth value]]) while $(-2)$-[[(-2)-groupoid|groupoid]], $(-2)$-[[(-2)-category|category]], and $(-1)$-[[(-1)-poset|poset]] likewise all mean the same thing (namely, the [[point]]).

Of particular importance is the case where $n = \infty$. See

* [[(∞,n)-category]] .

### Topos cases

An analogous systematics exists for $(n,r)$-categories that in additions have the property of being a [[topos]] or [[higher topos theory|higher topos]].

* a [[(0,1)-topos]] is a [[Heyting algebra]]

  * in particular, a $(0,1)$-[[Grothendieck topos]] is a [[locale]]

* a $(1,1)$-topos is a [[topos]]

* an [[(∞,1)-topos]] is what [[Higher Topos Theory]] calls an $\infty$-topos

## The periodic table

There is a [[periodic table]] of $(n,r)$-categories:

$n$→
$r$↓
  $-2$
  $-1$
  $0$
  $1$
  $2$
  ...
  $\infty$

$0$
  [[point]]
  [[truth value]]
  [[set]]
  [[groupoid]]
  [[2-groupoid]]
  ...
  [[∞-groupoid]]

$1$
  "
  "
  [[partial order|poset]]
  [[category]]
  [[(2,1)-category]]
  ...
  [[(∞,1)-category]]

$2$
  "
  "
  "
  [[2-poset]]
  [[2-category]]
  ...
  [[(∞,2)-category]]

$3$
  "
  "
  "
  "
  [[3-poset]]
  ...
  [[(∞,3)-category]]

⋮
  ⋮
  ⋮
  ⋮
  ⋮
  ⋮
  ⋱
  ⋮

$\infty$
  point
  truth value
  poset
  2-poset
  3-poset
  ...
  [[(∞,∞)-category]]/[[∞-poset]]

## Models for weak (n,r)-categories

There are various [[model category]] models for collections of $(n,r)$-categories.

* The standard [[model structure on simplicial sets]] models [[∞-groupoid|(∞,0)-categories]].

* The Joyal-[[model structure on simplicial sets]] models [[(∞,1)-category|(∞,1)-categories]].

* The [[Charles Rezk]]-model structure for [[Theta space]]s models general $(n,r)$-categories.

## Related concepts

* [[directed (n,r)-graph]]

* [[(n,r)-site]]

* [[0-category]], [[(0,1)-category]]

* [[category]]

* [[2-category]]

* [[3-category]]

* [[n-category]]

* [[(∞,0)-category]]

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

* [[(∞,1)-category]]

* [[(∞,2)-category]]

* [[(∞,n)-category]]

* **(n,r)-category**

* [[(n × k)-category]], [[n-fold category]]

## References

* [[John Baez]], [[Michael Shulman]], _Lectures on n-Categories and Cohomology_, ([arXiv:math/0608420](https://arxiv.org/abs/math/0608420)), in particular pp. 34-36.

[[!redirects (n,r) categories]]
[[!redirects (n,r) category]]
[[!redirects (n,r)-categories]]
[[!redirects (n,r)-category]]
[[!redirects (n, r)-category]]
[[!redirects (n, r)-categories]]


(n,r)-congruence

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Internal $(\infty,1)$-Categories
+--{: .hide}
[[!include internal infinity-categories contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In general, the idea is that an $k$-congruence in an [[n-category]] $K$, where $k\le n$, is an "[[internal category|internal]] $(k-1)$-category" in $K$.

Here we consider the case $m\le 2$, although we allow $n$ and $m$ to be of the form $(r,s)$; see _[[(n,r)-category]]_.

## Definition

+--{: .num_defn}
###### Definition
Let $D$ be a [[2-congruence]] in a [[2-category]] $K$.

* $D$ is a **(2,1)-congruence** if it is an internal groupoid, i.e. there is a map $D_1\to D_1$ providing "inverses".
* It is a **(1,2)-congruence** if $D_1\to D_0\times D_0$ is ff.
* It is a **1-congruence** if it is both a (2,1)-congruence and a (1,2)-congruence.
* it is a **(0,1)-congruence** if $D_1\to D_0\times D_0$ is an equivalence.
=--

Note that in a 1-category,

* a 2-congruence is just an internal category (a 1-category),
* a (2,1)-congruence is an internal groupoid (a (1,0)-category),
* a (1,2)-congruence is an internal poset (a (0,1)-category), and
* a 1-congruence is an internal equivalence relation (a 0-category).

Of course, a (0,1)-congruence in any 2-category is completely determined by any object $D_0$.

+--{: .num_theorem}
###### Theorem
Let $q:X\to Y$ be a morphism in $K$.  If $Y$ is $n$-truncated for $n\ge -1$, then $ker(q)$ is an $(n+1)$-congruence.  This means that:

1. If $Y$ is [[groupoidal object|groupoidal]], then $ker(q)$ is a (2,1)-congruence.
1. If $Y$ is [[posetal object|posetal]], then $ker(q)$ is a (1,2)-congruence.
1. If $Y$ is [[discrete object|discrete]], then $ker(q)$ is a 1-congruence.
1. If $Y$ is [[nLab:subterminal object|subterminal]], then $ker(q)$ is a (0,1)-congruence.

In all these cases the converse is true if $K$ is [[regular 2-category|regular]] and $q$ is [[nLab:eso morphism|eso]].
=--

+--{: .proof}
###### Proof
The forward directions are fairly obvious; it is the converses which take work.  Suppose first that $ker(q)$ is a (2,1)-congruence, and let $\alpha: f \to g: X \rightrightarrows Y$ be any 2-cell.  Pulling back the eso $q$ along $f$ and $g$ gives $P_1\to T$ and $P_2\to T$; let $r:P \to T$ be the pullback $P_1\times_X P_2$.  Since $K$ is regular, $r$ is eso.  By definition of kernels, the 2-cell $\alpha r$ corresponds to a map $P\to (q/q)$.  But $(q/q)\rightrightarrows C$ is a (2,1)-congruence, so composing this map with the "inverse" map $(q/q)\to(q/q)$ gives another map $P\to (q/q)$, and thereby another 2-cell $f r \to g r$ which is inverse to $\alpha r$.  Finally, since $r$ is eso, precomposing with it reflects invertibility, so $\alpha$ must also be invertible.  Thus $Y$ is groupoidal.

Now suppose that $ker(q)$ is a (1,2)-congruence, and let $\alpha,\beta: f\to g: T\to Y$ be two parallel 2-cells.  With notation as in the previous paragraph, the 2-cells $\alpha r$ and $\beta r$ correspond to morphisms $P\rightrightarrows (q/q)$ which become isomorphic in $X$.  But since $(q/q)\rightrightarrows X$ is a (1,2)-congruence, this implies that the two maps $P\rightrightarrows (q/q)$ are isomorphic, and hence $\alpha r = \beta r$.  And since $r$ is eso, precomposing with it is faithful, so $\alpha=\beta$; thus $Y$ is posetal.

The discrete case follows by combining the posetal and groupoidal cases, so it remains to show that if $ker(q)$ is a (0,1)-congruence then $Y$ is subterminal.  We know it is discrete, so it suffices to show that given two $f,g:T \rightrightarrows Y$ we have a 2-cell $f\to g$.  Continuing with the same notation, and letting $h,k:P\to X$ be the induced maps with $q h \cong f r$ and $q k \cong g r$, we have $(h,k):P\to X\times X = (q/q)$, and therefore the 2-cell defining the fork $(q/q) \;\rightrightarrows\;X \overset{q}{\to} Y$ gives us a 2-cell $q h \to q k$ and therefore $f r \to g r$.  Now $r$ is the quotient of its kernel, so for this 2-cell to induce a 2-cell $f\to g$ it suffices for it to be an action 2-cell for the actions of $ker(r)$ on $f r$ and $g r$; but this is automatic since we know $Y$ to be posetal.  Thus we have a 2-cell $f\to g$ as desired, so $Y$ is subterminal.
=--

## Related concepts

* [[congruence]] [[2-congruence]]

* [[internal category]], [[internal category in an (infinity,1)-category]]

## References

* [[Mike Shulman]], _[[michaelshulman:n-congruence]]_

[[!redirects (n,r)-congruences]]


(n,r)-site

An [[(n,r)-site]] is an [[(n,r)-category]] equipped with a notion of [[coverage]].

## Examples

* [[(0,1)-site]]

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

* [[(2,1)-site]], [[(2,2)-site]]

* [[(∞,1)-site]]

[[!redirects (n,r)-site]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The $n$-connected/$n$-truncated factorization system is an [[orthogonal factorization system in an (∞,1)-category]],  specifically in an [[(∞,1)-topos]], that generalizes the _relative [[Postnikov systems]]_ of [[∞Grpd]]: it factors any morphism through its _[[n-image|(n+2)-image]]_ by an _[[n-epimorphism|(n+2)-epimorphism]]_ followed by an _[[n-monomorphism|(n+2)-monomorphism]]_.

As $n$ ranges through $(-1), 0, 1, 2, 3, \cdots$ these factorization systems form an [[k-ary factorization system|∞-ary factorization system]].

## Definitions

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

Let $\mathbf{H}$ be an [[(∞,1)-topos]]. For all $(-2) \leq n \lt \infty$ the [[class]] of [[n-truncated morphisms in an infinity,1-category|n-truncated morphisms]]  in $\mathbf{H}$ forms the right class in a [[orthogonal factorization system in an (∞,1)-category]]. The left class is that of [[n-truncated morphisms in an infinity,1-category|n-connected morphisms]]  in $\mathbf{H}$.

=--

This appears as a remark in [[Higher Topos Theory|HTT, Example 5.2.8.16]]. A construction of the factorization in terms of a [[model category]] [[presentable (∞,1)-category|presentation]] is in ([Rezk, prop. 8.5](#Rezk)).

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

For $n = -1$ this says that [[effective epimorphisms in an (∞,1)-category]] have the [[left lifting property]] against [[monomorphisms in an (∞,1)-category]]. Therefore one may say that the effective epimorphisms in an $(\infty,1)$-topos are the [[strong epimorphisms]].

=--

## Properties

### Stability

\begin{prop}\label{nConnectedTruncatedIsStable}
For all $n$, the $n$-connected/$n$-truncated factorization system is [[stable factorization system|stable]]: the class of [[n-connected]] morphisms is preserved under [[(∞,1)-pullback]].

\end{prop}

This appears as ([Lurie, prop. 6.5.1.16(6)](#Lurie)).

It follows that:
\begin{prop}\label{HomotopyPullbackPreservesNImageFactorization}
For all $n$, [[n-images]] are preserved by [[(∞,1)-pullback]]
\end{prop}
\begin{proof}
Let $X \xrightarrow{f} Y$ with $n$-image $im_n(f)$.
By the [[pasting law]] its [[(infinity,1)-pullback|$\infty$-pullback]] along any $g \,\colon\, X' \xrightarrow{\;} X$ may be decomposed as two consecutive $\infty$-pullbacks:

\begin{tikzcd}
  g^\ast X
  \ar[r]
  \ar[d]
  \ar[dr, phantom, "{\mbox{\tiny(pb)}}"]
  \ar[
    dd,
    bend right=50,
    "{g^\ast f}"{swap}
  ]
  &
  X
  \ar[d, "{\mbox{\tiny $n$-cnct}}"{description}]
  \ar[
    dd,
    bend left=50,
    "{f}"
  ]
  \\
  \mathrm{im}_n(g^\ast f)
  \ar[r]
  \ar[d]
  \ar[dr, phantom, "{\mbox{\tiny(pb)}}"]
  &
  \mathrm{im}_n(f)
  \ar[d, "{\mbox{\tiny $n$-trnct}}"{description}]
  \\
  Y'
  \ar[
    r,
    "g"{swap}
  ]
  &
  Y
\end{tikzcd}

Here the pullback of the $n$-truncated map in again $n$-truncated since the right class of any orthogonal factorization system is stable under pullback. The analogous statement holds also for the $n$-connected map by Prop. \ref{nConnectedTruncatedIsStable}. Therefore the pullback of $im_n(f)$ along $g$ is indeed $im_n(g^\ast f)$, as shown.

\end{proof}

## Examples

### The case $n = -2$

A [[(-2)-truncated]] [[morphism]] is precisely an [[equivalence in an (∞,1)-category]] (see there or [[Higher Topos Theory|HTT, example 5.5.6.13]]).

Moreover, every morphism is [[(-2)-connected]].

Therefore for $n = -2$ the $n$-connected/$n$-truncated factorization system says (only) that equivalences have inverses, unique up to coherent homotopy.

### The case $n = -1$

A [[(-1)-truncated]] morphism is precisely a [[monomorphism in an (∞,1)-category|full and faithful morphism]].

A [[(-1)-connected]] morphism is one whose [[homotopy fiber]]s are [[inhabited]].

In [[∞Grpd]] a morphism between _[[0-truncated]]_ [[object]]s ([[set]]s)

* is full and faithful precisely if it is an [[injection]];

* has non-empty fibers precisely if it is an [[epimorphism]].

Therefore between 0-truncated objects the (-1)-connected/(-1)-truncated factorization system is the [[epi/mono factorization system]] and hence [[image]] factorization.

Generally, the (-1)-connected/(-1)-truncated factorization is through the **$\infty$-categorical [[1-image]]**, the _[[homotopy image]]_ (see there for more details).

### The case $n = 0$

Let $X,Y$ be two [[groupoid]]s ([[homotopy n-type|homotopy 1-types]]) in [[∞Grpd]].

A morphism $X \to Y$ is [[0-truncated]] precisely if it is a [[faithful functor]].

A morphism $X \to Y$ is [[0-connected]] precisely if it is a [[full functor]] and an [[essentially surjective functor]]: a [[essentially surjective and full functor]]

Therefore on homotopy 1-types the 0-connected/0-truncated factorization system is the [[(eso+full, faithful) factorization system]].

## References

The general abstract statement is in

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

A [[model category]]-theoretic discussion is in section 8 of

* {#Rezk} [[Charles Rezk]], _Toposes and homotopy toposes_ ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf))

Discussion in [[homotopy type theory]] is in

* [[Univalent Foundations Project]], section 7.6 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

[[!redirects n-connected/n-truncated factorization system]]
[[!redirects (n-connected, n-truncated) factorization system]]

[[!redirects (n-epi, n-mono) factorization system]]
[[!redirects (n-epi, n-mono) factorization systems]]


(p,q)-string

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[type II string theory]]: a [[bound state]] of $p$-[[F1-branes]] and $q$ [[D1-branes]].

Under [[S-duality]] the [[modular group]] $SL_2(\mathbb{Z})$ acts on these $(p,q)$-tuples and mixes the $F1$- and the $D1$-brane states.

[[!include F-branes -- table]]

## Related concepts

* [[S-duality]]

* [[F-theory]]

[[!include brane bound states -- table]]

## References

* [[Igor Bandos]], _Superembedding Approach and S-Duality. A unified description of superstring and super-D1-brane_, Nucl.Phys.B599:197-227,2001 ([arXiv:hep-th/0008249](http://arxiv.org/abs/hep-th/0008249))

[[!redirects (p,q)-strings]]


(p,q)5-brane

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[type IIB string theory]] there are [[bound states]] of [[D5-branes]] with [[NS5-branes]]. The bound state of $p \in \mathbb{Z}$ D5-branes with $q \in\mathbb{Z}$ NS5-branes is then called a _$(p,q)$-fivebrane_ or similar.

As $p$ and $q$ varies the species of $(p,q)$5-branes form the [[lattice in a vector space|lattice]] $\mathbb{Z}^2$ and are naturally [[action|acted]] on by the [[S-duality]] group [[SL(2,Z)]].

## Properties

### $(p,q)$-Brane webs

Label local [[coordinate functions]] $x^a$ on 10d [[Minkowski spacetime]] $\mathbb{R}^{9,1}$  by $0123455'6789$ and write $v_a \coloneqq \partial_{x^a}$ for the corresponding [[vector field]]

Consider a $(1,0)$5-brane (a [[D5-brane]]) along the [[multivector field]]
$v_0 v_1 v_2 v_3 v_4 v_5$  and a $(0,1)$5-brane (an [[NS-brane]]) along $v_0 v_1 v_2 v_3 v_4 v_{5'}$

| $a =$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $5'$ | $6$ | $7$ | $8$ | $9$ |
|-------|-----|-----|-----|-----|-----|-----|------|-----|-----|-----|-----|
|  D5   | --- | --- | --- | --- | --- | --- |      |     |     |     |     |
|  NS5  | --- | --- | --- | --- | --- |     |  --- |     |     |     |     |

Charge conservation implies that at the [[brane intersection]] of the two a $(1,1)$5-brane emerges stretched diagonally along $v_5 + v_{g'}$, i.e. along the [[multivector field]] $v_0 v_1 v_2 v_3 v_4 (v_5 + v_{5'})$

[Aharony-Hanany 97, Sec. 3](#AharonyHanany97)

<img src="https://ncatlab.org/nlab/files/D5NS5Junction.jpg" width="500"/>

The [[worldvolume]]-[[quantum field theory]] at the [[brane intersection]] point is a [[geometric engineering of quantum field theory|geometric engineering]] of [[D=5 N=1 SYM]] .

The [[T-duality|T-dual]] perspective are [[D4/NS5-brane webs]] ([Witten 97](#Witten97)).

The result of connecting several such [[brane intersections]] are called _$(p,q)$-5brane webs_ ([Aharony-Hanany-Krol 97](#AharonyHananyKrol97)).

## Related entries

* [[(p,q)-string]]

* [[D=5 N=1 SYM]]

[[!include brane bound states -- table]]

[[!include F-branes -- table]]

## References

The original articles are:

* {#AharonyHanany97} [[Ofer Aharony]], [[Amihay Hanany]], _Branes, Superpotentials and Superconformal Fixed Points_, Nucl. Phys. B504:239-271, 1997 ([arXiv:hep-th/9704170](https://arxiv.org/abs/hep-th/9704170))

* {#AharonyHananyKrol97} [[Ofer Aharony]], [[Amihay Hanany]], [[Barak Kol]], _Webs of $(p,q)$ 5-branes, Five Dimensional Field Theories and Grid Diagrams_, JHEP 9801:002,1998 ([arXiv:hep-th/9710116](http://arxiv.org/abs/hep-th/9710116))

* [[Oren Bergman]], Gabi Zafrir, _Lifting 4d dualities to 5d_, JHEP04 (2015) 141 ([arXiv:1410.2806](https://arxiv.org/abs/1410.2806))

The [[T-duality|T-dual]] perspective are [[D4/NS5-brane webs]]:

* {#Witten97} [[Edward Witten]], _Solutions Of Four-Dimensional Field Theories Via M Theory_, Nucl. Phys. B500:3-42, 1997 ([arXiv:hep-th/9703166](https://arxiv.org/abs/hep-th/9703166))

Further intersection with [[orientifolds]]:

* [[Amihay Hanany]], [[Alberto Zaffaroni]], _Issues on Orientifolds: On the brane construction of gauge theories with $SO(2n)$ global symmetry_, JHEP 9907 (1999) 009 ([arXiv:hep-th/9903242](https://arxiv.org/abs/hep-th/9903242))

* Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Masato Taki, Futoshi Yagi, _More on 5d descriptions of 6d SCFTs_, JHEP10 (2016) 126 ([arXiv:1512.08239](https://arxiv.org/abs/1512.08239))

* Gabi Zafrir, _Brane webs in the presence of an $O5^-$-plane and 4d class S theories of type D_, JHEP07 (2016) 035 ([arXiv:1602.00130](https://arxiv.org/abs/1602.00130))

* [[Taro Kimura]], Rui-Dong Zhu, Section 2 and 3  of _Web Construction of ABCDEFG and Affine Quiver Gauge Theories_ ([arXiv:1907.02382](https://arxiv.org/abs/1907.02382))

On $(p,q)$-5-branes as [[defect branes]]:

* Tetsuji Kimura, _Defect $(p,q)$ Five-branes_, Nucl.Phys. B893 (2015) 1-20 ([arXiv:1410.8403](https://arxiv.org/abs/1410.8403), [doi:10.1016/j.nuclphysb.2015.01.023](https://doi.org/10.1016/j.nuclphysb.2015.01.023))

[[!redirects (p,q)5-branes]]

[[!redirects (p,q)5-brane web]]
[[!redirects (p,q)5-brane webs]]

[[!redirects D4/NS5-brane webs]]


> This page is about object classifier objects in [[(∞,1)-toposes]]. For the unrelated notion of the [[classifying topos]] of the [[theory of objects]] see at _[[classifying topos for the theory of objects]]_.

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

#Contents#
* table of contents
{:toc}

## Idea

A crucial ingredient in a [[topos]] is a [[subobject classifier]]. From the point of view of [[homotopy theory]], that this has to do with [[subobjects]] turns out to be a coincidence of low dimensions: subobjects are [[(-1)-truncated]] morphisms.

As discussed also at [[stuff, structure, property]], the classifying objects in [[higher topos theory]] classify more general morphisms.

When one passes all the way to [[∞-toposes]], there should be objects that classify _all_ morphisms, subject to some bound on size. This is made precise in the context of [[(∞,1)-topos theory]].

One way to characterize an [[(∞,1)-topos]] is as

* a [[presentable (∞,1)-category]]

* with [[universal colimits]]

* such that for all sufficiently large [[regular cardinal]]s $\kappa$ there is a **classifying object** for the class of all $\kappa$-compact morphisms in $X$.

This statement is originally due to [[Charles Rezk]]. It is reproduced as ([[Higher Topos Theory|Lurie HTT, theorem 6.1.6.8]]).

In terms of [[homotopy type theory]] these object classifiers are _[[types of types]]_. See there for more details and see at _[[relation between category theory and type theory]]_.

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

An [[object classifier]] is a (small) _self-reflection_ of the $(\infty, 1)$-topos inside itself ([[type of types]], internal [[universe]]). It possesses an [[category object in an (infinity,1)-category|internal (∞,1)-topos ]] structure. See also ([[Science of Logic#WesenAlsReflexionInIhmSelbst|WdL, book 2, section 1]]).

=--

## Definition
 {#Definition}

Let $\mathcal{C}$ be a [[(∞,1)-category]], and let $S$ be a class of [[1-morphisms]] of $\mathcal{C}$ which is stable under [[(∞,1)-pullback]]. Then an **$S$-classifier** is a [[terminal object in an (∞,1)-category|terminal object]] in the sub-[[arrow category|category of arrows]] $\mathcal{C}^{\Delta^1}$ of $S$ whose morphisms are [[(∞,1)-pullback]] squares in $\mathcal{C}$.

Explicitly, an $S$-classifier consists of

* a morphism $\widehat {S Type} \longrightarrow S Type$ in $S$

* such that for each $X \stackrel{p}{\to} B$ in $S$, there exists an essentially unique [[(∞,1)-pullback]] square of the form

  $$
    \array{
       X &\longrightarrow& \widehat {S Type}
       \\
       \downarrow^{\mathrlap{p}} && \downarrow
       \\
       B &\stackrel{'X'}{\longrightarrow}& S Type
    }
    \,.
  $$

  in $\mathcal{C}$

Here $'X'$ may be called the _name_, or the _[[classifying morphism]]_, or the _[[modulating morphism]]_ or the _internal reflection_ of $X$ over $B$,

For example,

* When $S$ is the class of **all monomorphisms** in $C$, an $S$-classifier is called a **subobject classifier**. For instance, every topos has a subobject classifier.

* When $S$ is the class of **all faithful morphisms** in $C$, an $S$-classifier is called a **discrete object classifier**. For instance, every (2,1)-topos has a discrete object classifier.

* When $S$ is the class of **all morphisms** in $C$, an $S$-classifier is called an **object classifier**. However, due to size issues, interesting categories tend not to have such objects, which is one reason to be interested in the next example:

* When $S$ is the class of **all relatively $\kappa$-compact morphisms** (for some regular cardinal $\kappa$--see below for the definition), an $S$-classifier is called a **$\kappa$-compact-object classifier**.

**Note on terminology:** In all cases, the "things" classified by an "(adjectives) object classifier" are _arrows_ -- this is no different from the most famous case of _[[subobject classifiers]]_, which classify _monos_. For each object $X$, a _subobject classifier_ classifies the _subobjects of $X$_. For each object $X$, an _object classifier_ classifies the _objects over $X$_.

So with that $\kappa$ fixed, we may write

$$
  \array{
    \widehat Type
    \\
    \downarrow
    \\
    Type
  }
$$

for such a "universal bundle of $\kappa$-small objects". Intuitively this is easy to describe: a point in $Type$ corresponds to a $\kappa$-small object, hence is the "name" $'X'$ or "code for" a $\kappa$-small object, and the [[fiber]] in $\widehat Type$ over that point is the very object $X$ itself.

If one gives the projection of the universal object bundle $\widehat Type \to Type$ a name, such as $El$, and writes $El^{-1}(-)$ for its preimages then $X \simeq El^{-1}('X')$. This is, with the ${(-)}^{-1}$-suppressed, the notation used at _[Type universes a la Tarski](type+of+types#TarskiStyle)_.

## Details

### n-truncated object classifier

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

Let $C$ be an [[(∞,1)-category]] and $S \in C_1$ a class of [[morphisms]] that is stable under [[(∞,1)-pullback]] in $C$.

Let $Cod_C$ be the [[codomain fibration]] of $X$, i.e. the [[(∞,1)-category of (∞,1)-functors]]

$$
  Cod_C := Func(\Delta[1], C)
$$

equipped with the [[Cartesian fibration]] $Cod_C \to C$ induced from the endpoint inclusion $\Delta[0] \to \Delta[1]$.

Write

* $Cod_C^S$ for the full [[sub-(∞,1)-category]] of $Cod_C$ on the object in $S$;

* $Cod_C^{(S)}$ the non-full subcategory whose objects are the elements of $S$, and whose morphisms are squares that are pullback diagrams.

Then evaluation at $\Delta[0] \to \Delta[1]$ yields

* a [[Cartesian fibration]] $Cod_C^S \to C$;

* a [[right fibration]] $Cod_C^{(S)} \to C$.

We say a morphism $f :x \to y$ in $C$ _classifies_ $S$ -- or simply that $y$ classifies $S$ -- if it is the [[terminal object]] in $Cod_C^{(S)}$.

=--

This is [[Higher Topos Theory|HTT, notation 6.1.3.4]] and [[Higher Topos Theory|HTT, def. 6.1.6.1 ]].

#### Subobject classifier {#DetailsSubObjClassf}

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

A **subobject classifier** for $C$ is an object that classifies the class $S$ of [[monomorphism in an (∞,1)-category|monomorphisms]]/[[(-1)-truncated]] morphisms in $C$.

=--

This is ([[Higher Topos Theory|HTT, def. 6.1.6.1 ]]).

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

The $(\infty,1)$-category [[∞Grpd]] has a a subobject classifier: the [[0-groupoid]]/[[set]] $\{\emptyset,*\}$ with two elements (the two [[(-1)-truncated]] $\infty$-groupoids).

=--

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

Every [[(∞,1)-topos]] has a [[subobject classifier]].

=--

This appears as ([[Higher Topos Theory|HTT, prop. 6.1.6.3]]) and the remark below that.

#### Discrete object classifier

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

A **discrete object classifier** for $C$ is an object that classifies the class $S$ of [[faithful morphism in an (∞,1)-category|faithful morphism]]/[[(0)-truncated]] morphisms in $C$.

=--

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

The $(\infty,1)$-category [[∞Grpd]] has a a discrete object classifier: the [[1-groupoid]]/[[groupoid]] [[Set]] whose elements are sets ([[0-truncated]] $\infty$-groupoids).

=--

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

Every [[(∞,1)-topos]] has a [[discrete object classifier]].

=--

### Object classifier {#DetailsObjClassf}

**Remark/Warning.** The point of having [[subobjects]] and hence [[monomorphisms]] classified by an object in an ordinary [[topos]] may be thought of as being solely due to the fact that in a 1-[[topos]], any object necessarily classifies a _[[poset]]_ i.e. a [[(0,1)-category]] of morphisms, and the point of subobjects/monomorphisms of a given object is that they do not have [[automorphisms]].

In an $(\infty,1)$-topos we thus expect an object that classifies _all_ morphisms, in that the assignment

$$
  c \mapsto Core(C_{/c})
$$

of an object $c\in C$ to the [[core]] of its [[over quasi-category|over (∞,1)-category]] yields a [[(∞,1)-functor]] $C^{op} \to \infty Grpd$ that is [[representable functor|representable]].

Indeed, this is _essentially_ the case -- up to size issues, that the following definitions take care of.

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

For $\kappa$ some [[cardinal]], say a morphism $f : x \to y$ in $C$ is **[[relatively k-compact morphism in an (infinity,1)-category|relatively k-compact]]** if for all [[(∞,1)-pullbacks]] along $h : y' \to y$ to $\kappa$-[[compact object in an (∞,1)-category|compact object]]s, $y'$, the pulled back object $h^* x'$ is itself a $\kappa$-compact object.

=--

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

A [[presentable (∞,1)-category]] $C$  is an [[(∞,1)-topos]] precisely if

1. it has [[universal colimits]];

1. for sufficiently large regular [[cardinal]]s $\kappa$, $C$ has a classifying object for relatively $\kappa$-compact morphisms.

=--

This is due to [[Charles Rezk]]. The statement appears as [[Higher Topos Theory|HTT, theorem 6.1.6.8]].

The proof essentially consists of showing that by the [[adjoint functor theorem]] (specifically, the [[representable functor theorem]]), the existence of object classifiers is equivalent to [[continuous functor|continuity]] of the [[core]] [[self-indexing]] $C^{op} \to \infty Gpd$ defined by $x\mapsto Core(C/x)$.  In the presence of universal colimits, this latter condition is equivalent to all colimits being [[van Kampen colimits]], which in turn yields the connection to the Giraud-type exactness properties.

## Examples

### Object classifier in $\infty Grpd$
  {#ObjectClassifierInInfinityGroupoid}

We discuss that the $\kappa$-small object classifier in the $(\infty,1)$-topos
[[∞Grpd]] of [[∞-groupoids]] is itself the [[core]] of the [[(∞,1)-category]] $\infty Grpd_\kappa$ of $\kappa$-small $\infty$-groupoids. Observing that the connected components of this are the [[delooping]] $B Aut(F)$ of the [[automorphism ∞-group]] of a given [[homotopy type]] $[F]$, and using that [[∞Grpd]] is [[presentable (∞,1)-category|presented]] by [[Top]] $\simeq$ [[sSet]] (see also at _[[homotopy hypothesis]]_) this recovers classical theorems about the classification of [[fibrations]] in simplicial sets/topological spaces by a [[universal Kan fibration]], as listed in the [References](http://ncatlab.org/nlab/show/associated+infinity-bundle#References) at _[[associated ∞-bundle]]_.

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

The $\kappa$-compact object classifier in [[∞Grpd]]
is

$$
  Type_\kappa := Core(\infty Grpd_\kappa)
  \,,
$$

the [[core]] of the [[full sub-(∞,1)-category]] of [[∞Grpd]] on the $\kappa$-[[small ∞-groupoids]].

The corresponding [[universal bundle]] is presented by the map of [[simplicial sets]]

$$
   \widehat Type_\kappa \to Type_\kappa
$$

which is the [[pullback]] of simplicial sets

$$
  \array{
    \widehat Type_\kappa &\to& Z_{\infty Grpd}
    \\
    \downarrow && \downarrow
    \\
    Type_\kappa &\to& \infty Grpd
  }
$$

of the [[universal right fibration]] along the defining inclusion of (the [[Kan complex]] presenting) $Type_\kappa$.

=--

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

In [[∞Grpd]] the relatively $\kappa$-compact morphisms, $X \to Y$, def. \ref{RelativelyKappaCompact}, are precisely those all whose [[homotopy fibers]]

$$
  X_{y} := X \times_{Y} \{y\}
$$

over all [[objects]] $y \in Y$ are $\kappa$-[[small infinity-groupoids]].

=--

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

We may write $Y$ as an [[(∞,1)-colimit]] over itself (see there)

$$
  Y \simeq {\lim_{\to}}_{y \in Y} \{y\}
$$

and then use the fact that [[∞Grpd]] -- being an [[(∞,1)-topos]] -- has
[[universal colimits]], to obtain the [[(∞,1)-pullback]] diagram

$$
  \array{
    {\lim_{\to}}_{y \in Y} X_y
      &\stackrel{\simeq}{\to}
     & X
    \\
    \downarrow && \downarrow
    \\
    {\lim_{\to}}_{y \in Y} \{y\}
    &\stackrel{\simeq}{\to}&
    Y
  }
$$

exhibiting $X$ as an $(\infty,1)$-colimit of $\kappa$-small objects
over $Y$. By stability of $\kappa$-compact objects under
$\kappa$-small colimits (see [here](http://ncatlab.org/nlab/show/compact+object+in+an+%28infinity%2C1%29-category#StabilityUnderColimits))
it follows that $X$ is $\kappa$-compact if $Y$ is.

=--

+-- {: .proof}
###### Proof of the proposition

Since [[right fibrations]] are stable under pullback (see [here](http://ncatlab.org/nlab/show/right%2Fleft+Kan+fibration#PreservationByPullback)), this is still a right fibration. Since, up to equivalence, every morphism into a [[Kan complex]] is a
right fibration (see [here](http://ncatlab.org/nlab/show/right%2Fleft+Kan+fibration#OverKanComplex)), and since every
morphism out of a Kan complex into $\infty Grpd_\kappa$ factors
through the core $Type_\kappa$ it follows that
$Type_\kappa$ classifies all morphisms $X \to Y$ in [[∞Grpd]] whose [[homotopy fibers]]

$$
  X_y \simeq X \times_Y \{y\}
$$

are $\kappa$-[[compact object in an (∞,1)-category|compact]].

The claim then follows with lemma \ref{RelativelyCompactInInfinityGroupods}.

=--

### Object classifier in presheaf $(\infty,1)$-toposes

Let $C$ be an [[(∞,1)-category]] and $\mathbf{H} = PSh_{\infty}(C)$ the [[(∞,1)-category of (∞,1)-presheaves]] over $C$.

By the [[(∞,1)-Yoneda lemma]],
the $\kappa$-compact object classifier here should be
the presheaf which assigns to $U \in C$ the
$\infty$-groupoid of relatively $\kappa$-compact
morphisms $X \to U$ in $PSh_\infty(C)$.

## Related concepts

* [[univalent foundations for mathematics]]

* [[type of propositions]], [[subobject classifier]], [[partial map classifier]]

* [[type of sets]], [[category of sets]], [[discrete object classifier]]

* [[type of types]], [[univalence]]

* [[classifying morphism]]

* [[Awodey's conjecture]]

## References

The general notion is due to:

* [[Jacob Lurie]], section 6.1.6 of _[[Higher Topos Theory]]_ (2009)

The object classifier in the archetypical special case of the $\infty$-topos [[∞Grpd]] of [[infinity-groupoids|$\infty$-groupoids]], seen in the [[classical model structure on simplicial sets]], is discussed in

* {#KapulkinLumsdaineVoevodsky12} [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], [[Vladimir Voevodsky]], *Univalence in simplicial sets* [[arXiv:1203.2553](http://arxiv.org/abs/1203.2553)]

* {#KapulkinLumsdaine21} [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], *The Simplicial Model of Univalent Foundations (after Voevodsky)*, Journal of the European Mathematical Society **23** (2021) 2071–2126  $[$[arXiv:1211.2851](https://arxiv.org/abs/1211.2851), [doi:10.4171/jems/1050](https://doi.org/10.4171/jems/1050)$]$

as [[categorical semantics]] for [[univalence|univalent]] [[type universes]] in [[homotopy type theory]].

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(∞%2C1)-category > history

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[[!redirects (∞,1)-category/history]]
[[!redirects (?%2C1)-category -- history]]


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(∞,1)-algebra over an (∞,1)-operad > history

> [[infinity-algebra over an (infinity,1)-operad]]


(∞,1)-algebraic theory

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

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

## Idea

In as far as an [[algebraic theory]] or [[Lawvere theory]] is nothing but a [[small category]] with finite [[product]]s and an [[algebra]] for the theory a product-preserving [[functor]] to [[Set]], the notion has an evident generalization to [[higher category theory]] and in particular to [[(∞,1)-category]] theory.

## Definition

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

An **$(\infty,1)$-Lawvere theory** is (given by a syntactic $(\infty,1)$-category that is) an [[(∞,1)-category]] $C$ with finite [[limit in a quasi-category|(∞,1)-product]]s. An $(\infty,1)$-algebra for the theory is an [[(∞,1)-functor]] $C \to $ [[∞Grpd]] that preserves these products.

The $(\infty,1)$-category of [[∞-algebras over an (∞,1)-algebraic theory]] is the full [[sub-(∞,1)-category]]

$$
  Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C^{op})
$$

of the [[(∞,1)-category of (∞,1)-presheaves]] on $C^{op}$ on the product-preserving $(\infty,1)$-functors

=--

In a full $(\infty,1)$-category theoretic context this appears as [[Higher Topos Theory|HTT, def. 5.5.8.8]]. A definition in terms of [[simplicially enriched categories]] and the [[model structure on sSet-categories]] to present $(\infty,1)$-categories is in [Ros]((http://www.math.muni.cz/~rosicky/papers/infty6.pdf). The introduction of that article lists further and older occurences of this definition.

## Properties
 {#Properties}

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

Let $C$ be an [[(∞,1)-category]] with finite [[limit in a quasi-category|products]]. Then

* $Alg_{(\infty,1)}(C)$ is an [[accessible (∞,1)-functor|accessible]] [[localization of an (∞,1)-category|localization]] of the [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(C^{op})$ (on the [[opposite (∞,1)-category|opposite]]).

  So in particular it is a [[locally presentable (∞,1)-category]].

* $Alg_{(\inft)}$ is a [[compactly generated (∞,1)-category]].

* The $(\infty,1)$-[[Yoneda embedding]] $j : C^{op} \to PSh_{(\infty,1)}(C^{op})$ factors through $Alg_{(\infty,1)}(C)$.

* The full subcategory $Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C)$ is stable under [[sifted colimit]]s.

=--

This is [[Higher Topos Theory|HTT, prop. 5.5.8.10]].

## Models {#Models}

There are various [[model category]] [[presentable (∞,1)-category|presentations]] of $Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C^{op})$.

Recall that the [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(C^{op})$ itself is modeled by the [[model structure on simplicial presheaves]]

$$
  PSh_{(\infty,1)}(C^{op})
  \simeq
  [T, sSet]^\circ
  \,,
$$

where we regard $T$ as a [[Kan complex]]-[[enriched category]] and have on the right the [[sSet]]-[[enriched functor category]] with the projective or injective model structure, and $(-)^\circ$ denoting the full enriched subcategory on fibrant-cofibrant objects.

This says in particular that every weak $(\infty,1)$-functor $f : T \to \infty \mathrm{Grp}$ is equivalent to a _rectified_ on $F : T \to KanCplx$. And $f \in PSh_{(\infty,1)}(C^{op})$ belongs to $Alg_{(\infty,1)}(C)$ if $F$ preserves finite products _weakly_  in that for $\{c_i \in C\}$ a finite collection of objects, the canonical natural morphism

$$
  F(c_1 \times \cdots\times \c_n)
  \to
  F(c_1) \times \cdots \times F(c_n)
$$

is a [[homotopy equivalence]] of [[Kan complex]]es.

If $T$ is an ordinary category with products, hence an ordinary [[Lawvere theory]], then such a functor is called a **[[homotopy T-algebra]]**. There is a model category structure on these (see there).

We now look at model category structure on _strictly_ product preserving functors $C \to sSet$, which gives an equivalent model for $Alg_{(\infty,1)}(C)$. See [[model structure on simplicial T-algebras]].

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

Let $C$ be a [[category]] with finite [[product]]s, and let $sTAlg \subset Func(C,sSet)$ be the [[full subcategory]] of the [[functor category]] from $C$ to [[sSet]] on those functors that preserve these products.

Then $sAlg(C)$ carries the structure of a [[model category]] $sAlg(C)_{proj}$ where the weak equivalences and the fibrations are objectwise those in the standard [[model structure on simplicial sets]].

=--

This is due to ([Quillen](#Quillen)).

The inclusion $i : sAlg(C) \hookrightarrow sPSh(C^{op})_{proj}$ into the projective [[model structure on simplicial presheaves]] evidently preserves fibrations and acylclic fibrations and gives a [[Quillen adjunction]]

$$
  sAlg(C)_{proj}
  \stackrel{\leftarrow}{\underset{i}{\hookrightarrow}}
  sPSh(C^{op})
  \,.
$$

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

The total right [[derived functor]]

$$
  \mathbb{R}i : Ho(sAlg(C)_{proj}) \to Ho(sPSh(C^{op})_{proj})
$$

is a [[full and faithful functor]] and an object $F \in sPSh(C^{op})$ belongs to the [[essential image]] of $\mathbb{R}i$ precisely if it preserves products up to [[weak homotopy equivalence]].

=--

This is due to ([Bergner](#Bergner)).

It follows that the natural $(\infty,1)$-functor

$$
  (sAlg(C)_{proj})^\circ \stackrel{}{\to}
  PSh_{(\infty,1)}(C^{op})
$$

is an [[equivalence of quasi-categories|equivalence]].

A comprehensive statement of these facts is in [[Higher Topos Theory|HTT, section 5.5.9]].

## Examples

### Simplicial 1-algebras

For $T$ (the [[syntactic category]] of) an ordinary [[algebraic theory]] (a [[Lawvere theory]]) let $T Alg$ be the category of its ordinary algebras, the ordinary product-preserving functors $T \to Set$.

We may regard $T$ as an $(\infty,1)$-category and consider its $(\infty,1)$-algebras. By the above discussion, these are modeled by product-presering functors $T \to sSet$. But this are equivalently [[simplicial object]]s in $T$-algebras

$$
  [T, sSet]_\times \simeq T Alg^{\Delta^{op}}
  \,.
$$

There is a standard [[model structure on simplicial T-algebras]] and we find that simplicial $T$-1-algebras model $T$-$(\infty,1)$-algebras.

### Homotopy $T$-algebras

For $T$ an ordinary Lawvere theory, there is also a model category structure on ordinary functors $T \to sSet$ that preserve the products only up to weak equivalence. Such functors are called [[homotopy T-algebra]]s.

This model structure is equivalent to the [[model structure on simplicial T-algebras]] (see [[homotopy T-algebra]] for details) but has the advantage that it is a left [[proper model category]].

### Simplicial theories

There is a notion of _simplicial algebraic theory_ that captures some class of $(\infty,1)$-algebraic theories. For the moment see section 4 of ([Rezk](#Rezk))

### Structure-$(\infty,1)$-sheaves

A pre[[geometry (for structured (∞,1)-toposes)]] is a (multi-sorted) $(\infty,1)$-algebraic theory. A _structure $(\infty,1)$-sheaf_ on an [[(∞,1)-topos]] $\mathcal{X}$ in the sense of [[structured (∞,1)-topos]]es is an $\infty$-algebra over this theory

$$
  \mathcal{O} : \mathcal{T} \to \mathcal{X}
$$

in the $(\infty,1)$-topos $\mathcal{X}$ -- a special one satisfying extra conditions that make it indeed behave like a sheaf of _function algebras_ .

### Symmetric monoidal $(\infty,1)$-Categories and $E_\infty$-algebras {#EInfty}

There is a $(2,1)$-algebraic theory whose algebras in
[[(∞,1)Cat]] are [[symmetric monoidal (∞,1)-categories]].  Hence monoids in these algebras are [[E-∞ algebra]]s (see [[monoid in a monoidal (∞,1)-category]]).

This is in ([Cranch](#Cranch)). For more details see
[[(2,1)-algebraic theory of E-infinity algebras]].

## Related concepts

* [[algebraic theory]] / [[Lawvere theory]] / [[essentially algebraic theory]]

  * [[2-Lawvere theory]]

  *  **algebraic $(\infty,1)$-theory** / [[essentially algebraic (∞,1)-theory]]

     * [[simplicial Lawvere theory]]

* [[monad]] / [[(∞,1)-monad]]

* [[operad]] / [[(∞,1)-operad]]

## References

The model structure presentation for the $(\infty,1)$-category of  $(\infty,1)$-algebras goes back all the way to

* [[Dan Quillen]], _Homotopical Algebra_ Lectures Notes in Mathematics 43, SpringerVerlag, Berlin, (1967)
{#Quillen}

A characterization of $(\infty,1)$-categories of $(\infty,1)$-algebras in terms of [[sifted colimit]]s is given in

* J. Rosicky _On homotopy varieties_ ([pdf](http://www.math.muni.cz/~rosicky/papers/infty6.pdf))

using the incarnation of $(\infty,1)$-categories as [[simplicially enriched categories]].

An $(\infty,1)$-categorical perspective on these homotopy-algebraic theories is given in

* [[Andre Joyal]], _The theory of quasi-categories and its applications_, lectures at CRM Barcelona February 2008, draft [hc2.pdf](http://www.crm.cat/HigherCategories/hc2.pdf#page=44)_

from page 44 on.

A detailed account in the context of a general theory of [[(∞,1)-category of (∞,1)-presheaves]] is the context of section 5.5.8 of

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

The [[model category]] [[presentable (infinity,1)-category|presentations]] of $(\infty,1)$-algebras is studied in

* [[Charles Rezk]], _Every homotopy theory of simplicial algebras admits a proper model_ ([math/0003065](http://arxiv.org/abs/math/0003065)) ,
{#Rezk}

where it is shown that every such model is [[Quillen equivalence|Quillen equivalent]] to a left [[proper model category]]. The article uses a monadic definition of $(\infty,1)$-algebras.

A discussion of [[homotopy T-algebra]]s and their strictification is in

* [[Bernard Badzioch]], _Algebraic theories in homotopy theory_ Annals of Mathematics, 155 (2002), 895-913 ([JSTOR](http://www.jstor.org/stable/3062135))

and for multi-sorted theories in

* [[Julie Bergner]], _Rigidification of algebras over multi-sorted theories_ , Algebraic and Geometric Topoogy 7, 2007.
{#Bergner}

A discussion of [[E-∞ algebra]]-structures in terms of $(\infty,1)$-algebraic theories is in

* [[James Cranch]], _Algebraic Theories and $(\infty,1)$-Categories_ ([arXiv](http://arxiv.org/abs/1011.3243))
{#Cranch}

See also

* [MO discussion](http://mathoverflow.net/questions/118500/what-is-a-simplicial-commutative-ring-from-the-point-of-view-of-homotopy-theory)

Expressed as a higher form of [[Lawvere theory]] see

* [[John D. Berman]], _Higher Lawvere theories_ ([arXiv:1903.02991](https://arxiv.org/abs/1903.02991))

[[!redirects (infinity,1)-algebraic theory]]
[[!redirects (infinity,1)-algebraic theories]]
[[!redirects (∞,1)-algebraic theories]]

[[!redirects algebraic (∞,1)-theory]]
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[[!redirects infinity1-algebraic theory]]


(∞,1)-category of (∞,1)-categories > history

< [[(∞,1)-category of (∞,1)-categories]]


(∞,1)-category of (∞,1)-modules

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

#Contents#
* table of contents
{:toc}

## Idea

For $R \in CRing_\infty$ an [[E-∞ ring]], the [[(∞,1)-modules]] over $R$ with [[homomorphisms]] between them form an [[(∞,1)-category]], the _$(\infty,1)$-category of $(\infty,1)$-modules over $R$_.

## Properties

### Compact generation, dualizable and perfect objects
 {#CompactGeneration}

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

Let $R$ be an [[A-∞ ring]]. The (∞,1)-category of ∞-modules $R Mod$ is a [[compactly generated (∞,1)-category]] and the [[compact object in an (∞,1)-category|compact objects]] coincide with the [[perfect modules]]

If $R$ is commutative ([[E-∞ ring|E-∞]]) then the perfect modules (and hence the compact objects) also coincide with the [[dualizable objects]].

=--

The first statement is ([[Higher Algebra|HA, prop. 7.2.4.2]]), the second ([[Higher Algebra|HA, prop. 7.2.4.4]]). For [[chain complexes]] this also appears as ([BFN 08, lemma 3.5](#BFN08)).

### Stable Dold-Kan correspondence
 {#StableDoldKan}

For $R$ an ordinary [[ring]], write $H R$ for the corresponding [[Eilenberg-MacLane spectrum]].

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

For $R$ any [[ring]] (or [[ringoid]], even) there is a [[Quillen equivalence]]

$$
  H R Mod \simeq Ch_\bullet(R Mod)
$$

between model structure on  $H R$-module spectra and the [[model structure on chain complexes]] (unbounded) of ordinary $R$-[[module]]s.

This presents a corresponding [[equivalence of (∞,1)-categories]]. If $R$ is a commutative ring, then this is an equivalence of [[symmetric monoidal (∞,1)-categories]].

=--

This equivalence on the level of [[homotopy categories]] is due to ([Robinson](#Robinson)). The refinement to a Quillen equivalence is ([SchwedeShipley, theorem 5.1.6](#SchwedeShipley)). See also the discussion at _[[stable model categories]]_. A direct description as an equivalence of $(\infty,1)$-categories appears as ([Lurie, theorem 7.1.2.13](#Lurie)).

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

This is a stable version of the [[Dold-Kan correspondence]].

=--

See at __[[algebra spectrum]]_ for the corresponding statement for $H R$-algebra spectra and [[dg-algebras]].

### Periodicity

For $E$ a [[periodic ring spectrum]], then $E Mod$ ought to inherit a $\mathbb{Z}/2\mathbb{Z}$-[[∞-action]]. See at _[periodic ring spectrum -- Periodicity of modules](periodic+ring+spectrum#PeriodicityOnModules)_
## Related concepts

* [[Mod]]

* [[2Mod]]

* [[nMod]]

## References

Modules over algebras over an arbitrary [[(∞,1)-operad]] are discussed in  section 3.3 of

* [[Jacob Lurie]], _[[Higher Algebra]]_

Modules specifically over [[A-∞ algebras]] are discussed in section 4.2 there.

The equivalence between the [[homotopy categories]] of  $H R$-module spectra and $Ch_\bullet(R Mod)$ is due to

* Alan Robinson, _The extraordinary derived category_ , Math. Z. 196 (2) (1987) 231–238.
 {#Robinson}

The refinement of this statement to a [[Quillen equivalence]] is due to

* [[Stefan Schwede]], [[Brooke Shipley]], _Stable model categories are categories of modules_ , Topology 42 (2003), 103-153 ([pdf](http://www.math.uic.edu/~bshipley/classTopFinal.pdf))
 {#SchwedeShipley}

Discussion in the context of [[derived algebraic geometry]] includes

* {#BFN08} [[David Ben-Zvi]], [[John Francis]], [[David Nadler]], section 3.1 of _Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry_, J. Amer. Math. Soc. 23 (2010), no. 4, 909-966 ([arXiv:0805.0157](http://arxiv.org/abs/0805.0157))

[[!redirects (∞,1)-categories of (∞,1)-modules]]

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

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

[[!redirects (∞,1)Mod]]


(∞,1)-comparison lemma

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

# The $(\infty,1)$-comparison lemma
* table of contents
{: toc}

## Idea

The **(∞,1)-comparison lemma** says that, under certain conditions, a functor between  [[(∞,1)-sites]] induces an equivalence between the categories of [[(∞,1)-sheaves]] on the sites.

In [this paper](#Hoyois2013), Lemma C.3, Hoyois proves the following comparison lemma.

+-- {: .num_theorem #Hoyois2013}
###### Lemma

Let $D$ be a [[locally small]] [[(∞,1)-category]], $C$ a [[small]] (∞,1)-category, and $u : C \to D$ a fully faithful functor. Let $\tau$ and $\rho$ be quasi-topologies on $C$ and $D$, respectively. Suppose that:

a. Every $\tau$-[[sieve]] is generated by a cover $\{U_i \to X\}$ such that:

  1. the fiber products $U_{i_0} \times_X \cdots\times_X U_{i_n}$ exist and are preserved by $u$;

  2. $\{u(U_i) \to u(X)\}$ is a $\bar\rho$-cover.

b. For every $X \in C$ and every $\rho$-sieve $R \hookrightarrow u(X)$, $u^*(R) \hookrightarrow X$ is a $\bar\tau$-sieve in $C$.

c. Every $X \in D$ admits a $\bar\rho$-cover $\{U_i \to X\}$ such that the fiber products $U_{i_0} \times_X \cdots \times_X U_{i_n}$ exist and belong to the essential image of $u$.

Then the adjunction $u^* \dashv u_*$ restricts to an equivalence of ∞-categories $Shv_\rho(D) \simeq Shv_\tau(C)$.
=--
Here, a _quasi-topology_ is a collection of sieves closed under pullback and $\bar\tau$ is the coarsest topology containing a quasi-topology $\tau$. The stability under pullback ensures that $Shv_\tau(C)=Shv_{\bar\tau}(C)$.

## Generalization

It seems difficult to find a useful generalization not assuming the existence of some pullbacks. For the conclusion of the [lemma](#Hoyois2013), the following conditions (b is unchanged) are both necessary and sufficient:

a. For every $\tau$-sieve $U \hookrightarrow X$, $a_\rho u_!(U \to X)$ is an equivalence.

b. For every $X \in C$ and every $\rho$-sieve $R \hookrightarrow u(X)$, $u^*(R) \hookrightarrow X$ is a $\bar\tau$-sieve in $C$.

c. For every $X \in D$, its image in $Shv_\rho(D)$ belongs to the smallest subcategory generated by the image of $C$ under colimits.

We can take these conditions a and b to define, respectively, the notions of [[cover-preserving functor]] (continuous functor) and [[comorphism of sites]] (cocontinuous functor) for (∞,1)-sites.

## Related concepts

* [[comparison lemma]]

* [[dense subsite]]

## References

* [[Marc Hoyois]], _A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula_. [arXiv:1309.6147](https://arxiv.org/abs/1309.6147)
  {#Hoyois2013}


(∞,1)-end

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

#Contents#
* table of contents
{: toc}

## Idea

This is the generalization of the notion of *[[ends]]* from [[category theory]] to [[(∞,1)-category theory]].

## Definition

### End of a space-valued functor

Given an [[(∞,1)-functor]] $F \colon J^{op} \times J \to \infty Gpd$ to [[∞Groupoids]] from the [[product category|product]] of a [[locally small (∞,1)-category]] $J$ with its [[opposite (∞,1)-category]], we define its *end* to be the [[(∞,1)-categorical hom-space]] in the [[(∞,1)-category of (∞,1)-functors]] from the [[hom-functor]] of $J$ to $F$:

$$
  \int_J F
    \coloneqq
  \infty Gpd^{J^{op} \times J}
  \big(
    J(-,-), F(-,-)
  \big)
$$

### Ends and coends as a weighted (co)limit

More generally, given an [[(∞,1)-functor]] $F \colon J^{op} \times J \to C$ to [[∞Groupoids]] to any [[locally small (∞,1)-category]] $C$,
we define its (co)end, when it exists, as the (co)[[representing object]] of

$$
  C\left(c, \int_J F \right)
   \simeq
  \int_J C(c, F)
    \simeq
  \infty Gpd^{J^{op} \times J}
  \big(
    J(-,-), C(c, F(-,-))
  \big)
$$
$$
  C\left(\int^J F, c \right) \simeq
  \int_{J^{op}} C(F, c) \simeq
  \infty Gpd^{J \times J^{op}}( J^{op}(-,-), C(F(-,-), c) )
$$

When $C = \infty Gpd$, this definition agrees with the previous.

### As (co)limits over the twisted arrow category

Let $F : J^{op} \times J \to C$ be a functor between (∞,1)-categories.
We can express the (co)end as (co)limits over the [[twisted arrow (∞,1)-category]] by

$$
  \int_J F \simeq \lim \left( Tw(J) \xrightarrow{(src, tgt)} J^{op} \times J
  \xrightarrow{F} C \right)
$$
$$
  \int^J F \simeq \colim \left( Tw(J^{op})^{op} \xrightarrow{(src, tgt)}  J^{op} \times J
  \xrightarrow{F} C \right)
$$

The second of these, for example, follows from the fact the projection
$p : Tw(J^{op}) \to J \times J^{op}$ is the left fibration classified by $J^{op}(-,-)$, and so we can write the hom-functor as the left Kan extension $p_!(1) = J^{op}(-,-)$ and take an adjoint transpose.

## Properties

+-- {: .num_prop }
###### Proposition (Fubini)

Given a functor $F : J^\op \times J \times K^\op \times K \to C$, if the relevant ends/coends exist, we have natural equivalences
$$
\int_J \int_K F \simeq \int_{J \times K} F
\qquad \qquad
\int^J \int^K F \simeq \int^{J \times K} F
$$

=--
+-- {: .proof}
###### Proof
For ends, using the fact $Tw$ preserves products we can compute
$$
  \begin{aligned}
  \int_J \int_K F
   &\simeq \lim_{j \to j' \in Tw(J)} \lim_{k \to k' \in Tw(K)} F(j,j',k,k')
   \\&\simeq \lim_{(j,k) \to (j',k') \in \Tw(J \times K)} F(j,j',k,k')
   \\&\simeq \int_{J \times K} F
  \end{aligned}
$$
The same argument applies for coends.
=--

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

Given functors $F,G : X \to Y$ between two (∞,1)-categories, the space of natural transformations between them is given by the end

$$ Y^X(F, G) \simeq \int_{x \in X} Y(F(x), G(x)) $$
=--

This is proposition 5.1 of [Gepner-Haugseng-Nikolaus 15](#GepnerHaugsengNikolaus15).

## Related concepts

* [[twisted arrow (∞,1)-category]]
* [[lax (∞,1)-colimit]]

#References

* {#GepnerHaugsengNikolaus15} [[David Gepner]], [[Rune Haugseng]], [[Thomas Nikolaus]], _Lax colimits and free fibrations in $\infty$-categories_ ([arXiv:1501.02161](http://arxiv.org/abs/1501.02161))

[[!redirects (∞,1)-ends]]
[[!redirects (∞,1)-coend]]
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