(∞,1)-limit

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of [[limit]] and [[colimit]] generalize from [[category theory]] to [[(∞,1)-category theory]]. One model for [[(∞,1)-categories]] are [[quasi-categories]]. This entry discusses limits and colimits in quasi-categories.

## Definition

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

For $K$ and $C$ two [[quasi-category|quasi-categories]] and $F : K \to
C$ an [[(∞,1)-functor]] (a morphism of the underlying [[simplicial set]]s) , the **limit** over $F$ is, if it exists, the [[terminal object in a quasi-category|quasi-categorical terminal object]] in the [[over quasi-category]] $C_{/F}$:

$$
  \underset{\leftarrow}{\lim} F \coloneqq TerminalObj(C_{/F})
$$

(well defined up to a contractible space of choices).

A **colimit** in a quasi-category is accordingly a limit in the [[opposite quasi-category]].

=--

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

Notice from the discussion at [[join of quasi-categories]] that there are two definitions -- denoted $\star$ and $\diamondsuit$ -- of join, which yield results that differ as simplicial sets, though are equivalent as quasi-categories.

The notation $C_{/F}$ denotes the definition of [[over quasi-category]] induced from $\star$, while the notation $C^{/F}$ denotes that induced from $\diamondsuit$. Either can be used for the computation of limits in a quasi-category, as for quasi-categorical purposes they are weakly equivalent.

So we also have

$$
  \underset{\leftarrow}{\lim} F \coloneqq TerminalObj(C^{/F})
  \,.
$$

See [[Higher Topos Theory|HTT, prop 4.2.1.5]].

=--

## Properties

### In terms of slice $\infty$-categories
 {#InTermsOfSlices}

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

Let $\mathcal{C}$ be a [[quasi-category]] and let
$f \colon K \to \mathcal{C}$ be a [[diagram]] with $(\infty,1)$-colimiting [[cocone]] $\tilde f \colon K \star \Delta^0 \to \mathcal{C}$. Then the induced map of [[slice (∞,1)-categories|slice quasi-categories]]

$$
  \mathcal{C}_{/\tilde f} \to \mathcal{C}_{f}
$$

is an [[acyclic Kan fibration]].

=--

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

For $F \colon \mathcal{K} \to \mathcal{C}$ a diagram in an $(\infty,1)$-category and $\underset{\leftarrow}{\lim} F$ its limit, there is a natural [[equivalence of (∞,1)-categories]]

$$
  \mathcal{C}_{/F} \simeq \mathcal{C}_{/\underset{\leftarrow}{\lim} F}
$$

between the [[slice (∞,1)-categories]] over $F$ (the $(\infty,1)$-category of $\infty$-[[cones]] over $F$) and over just $\underset{\leftarrow}{\lim}F$.

=--

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

Let $\tilde F \colon \Delta^0 \star \mathcal{K} \to \mathcal{C}$
be the limiting cone. The canonical cospan of $\infty$-functors
$$
  \ast \to \Delta^0 \star \mathcal{K} \leftarrow \mathcal{K}
$$

induces a span of slice $\infty$-categories

$$
  \mathcal{C}_{/\underset{\leftarrow}{\lim}F}
  \leftarrow
  \mathcal{C}_{/\tilde F}
  \rightarrow
  \mathcal{C}_{/F}
  \,.
$$

The right functor is an equivalence by prop. \ref{SlicingOverLimitingCone}. The left functor is induced by restriction along an op-[[final (∞,1)-functor]] (by the Examples discussed there) and hence is an equivalence by the discussion at _[[slice (∞,1)-category]]_ ([Lurie, prop. 4.1.1.8](#Lurie)).

=--

This appears for instance in ([Lurie, proof of prop. 1.2.13.8](#Lurie)).

### In terms of $\infty$-Hom adjunction

The definition of the limit in a quasi-category in terms of terminal objects in the corresponding [[over quasi-category]] is well adapted to the particular nature the incarnation of $(\infty,1)$-categories by quasi-categories. But more intrinsically in $(\infty,1)$-category theory, it should be true that there is an [[adjunction]] characterization of
$(\infty,1)$-limits : limit and colimit, should be (pointwise or global) [[right adjoint|right]] and [[left adjoint|left]]
[[adjoint (infinity,1)-functor]] of the constant diagram $(\infinity,1)$-functor,
$const : C \to Func(K,C)$.

$$
  (colim \dashv const \dashv lim)
  :
  Func(K,C)
  \stackrel{\overset{lim}{\to}}{\stackrel{\overset{const}{\leftarrow}}
  {\underset{colim}{\to}}}
  Func(*,C) \simeq C
  \,.
$$

By the discussion at [[adjoint (∞,1)-functor]] ([[Higher Topos Theory|HTT, prop. 5.2.2.8]]) this requires exhibiitng a morphism $\eta : Id_{Func(K,C)} \to const colim$ in $Func(Func(K,C),Func(K,C))$ such that for every $f \in Func(K,C)$ and $Y \in C$
the induced morphism

$$
  Hom_{C}(colim(f),Y) \to Hom_{Func(K,C)}(const colim(f), const Y)
  \stackrel{Hom(\eta, const Y)}{\to}
  Hom_{Func(K,Y)}(f, const Y)
$$

is a weak equivalence in $sSet_{Quillen}$.

But first consider the following pointwise characterization.

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

Let $C$ be a [[quasi-category]], $K$ a [[simplicial set]]. A [[co-cone]] [[diagram]]
$\bar p : K \star \Delta[0] \to C$ with cone point $X \in C$
is a [[colimit|colimiting]] [[diagram]] (an initial object in
$C_{p/}$) precisely if for every object $Y \in C$ the morphism

$$
  \phi_Y : Hom_C(X,Y) \to Hom_{Func(K,C)}(p, const Y)
$$

induced by the morpism $ p \to const X$ that is encoded by $\bar p$ is an
equivalence (i.e. a [[homotopy equivalence]] of [[Kan complex]]es).

=--

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

This is [[Higher Topos Theory|HTT, lemma 4.2.4.3]].

The key step is to realize that $Hom_{Func(K,C)}(p, const Y)$ is given (up to equivalence) by
the [[pullback]] $C^{p/} \times_C \{Y\}$ in [[sSet]].

Here is a detailed way to see this,
using the discussion at [[hom-object in a quasi-category]].

We have that $Hom_{Func(K,C)}(p, const Y)$ is given by
$(C^K)^{p/} \times_{C^K} const Y$. We compute

$$
  \begin{aligned}
    ((C^K)^{p/} \times_{C^K} const Y)_n
    & =
    Hom_{{\Delta[0]}/sSet}( \Delta[0] \diamondsuit \Delta[n] , C^K ) \times_{(C^K)_n} \{const Y\}
    \\
    & =
    Hom_{{\Delta[0]}/sSet}( \Delta[0] \coprod_{\Delta[n]} \Delta[n] \times \Delta[1] , C^K )
    \times_{(C^K)_n} \{const Y\}
    \\
    & =
    \{p\}
      \times_{Hom(\Delta[0],C^K)}
    Hom(\Delta[0], C^K)
      \times_{Hom(\Delta[n], C^K)}
    Hom(\Delta[n] \times \Delta[1], C^K)
      \times_{Hom(\Delta[n], C^K)}
    \{const Y\}
    \\
    & =
    \{p\}
      \times_{Hom(K,C)}
    Hom(K,C)
      \times_{Hom(\Delta[n]\times K,C)}
    Hom(\Delta[n]\times K \times \Delta[1], C)
      \times_{Hom(\Delta[n]\times K, C)}
    Hom(\Delta[n],C)
      \times_{\Delta[n],C}
    \{Y\}
    \\
    &=
    \{p\}
      \times_{Hom(K,C)}
    Hom(K \diamondsuit \Delta[n], C)
      \times_{Hom(\Delta[n],C)}
    \{Y\}
    \\
    &=
    (C^{p/}\times_C \{Y\})_n
  \end{aligned}
$$

Under this identification, $\phi_Y$ is the morphism

$$
  \left(
  C^{X/}
   \stackrel{\phi'}{\to}
  C^{\bar p/}
    \stackrel{\phi''}{\to}
  C^{p/}
  \right)
  \times_C \{Y\}
  \,,
$$

in [[sSet]] where $\phi'$ is a section of the map $C^{\bar p/} \to C^{X/}$,
(which one checks is an acyclic [[Kan fibration]]) obtained by choosing composites
of the co-cone components with a given morphism $X \to Y$.

The morphism $\phi''$ is a [[left fibration]] (using [[Higher Topos Theory|HTT, prop. 4.2.1.6]])

One finds that the morphism $\phi''$ is a [[left fibration]].

The strategy for the completion of the proof is: realize that the first condition
of the proposition is equivalent to $\phi''$ being an acyclic Kan fibration, and the
second statement equivalent to $\phi''_Y$ being an acyclic Kan fibration, then
show that these two conditions in turn are equivalent.

=--

### In terms of products and equalizers

A central theorem in ordinary [[category theory]] asserts that a [[category]] has [[limit]]s already if it has [[product]]s and [[equalizer]]s. The analog statement is true here:

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

Let $\kappa$ be a [[regular cardinal]]. An [[(∞,1)-category]] $C$ has all $\kappa$-small limits precisely if it has [[equalizer]]s and $\kappa$-small [[product]]s.

=--

This is [[Higher Topos Theory|HTT, prop. 4.4.3.2]].

### In terms of homotopy limits {#TermsOfHomotopy}

The notion of [[homotopy limit]], which exists for [[model categories]] and in particular for [[simplicial model categories]] and in fact in all plain [[Kan complex]]-[[enriched categories]] -- as described in more detail at [[homotopy Kan extension]] -- is supposed to be a model for $(\infty,1)$-categorical limits. In particular, under sending the Kan-complex enriched categories $C$ to quasi-categories $N(C)$ using the [[homotopy coherent nerve]] functor, homotopy limits should precisely corespond to quasi-categorical limits. That this is indeed the case is asserted by the following statements.

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

Let $C$ and $J$ be [[Kan complex]]-[[enriched categories]] and let $F : J \to C$ be an [[sSet]]-[[enriched functor]].

Then a [[cocone]] $\{\eta_i : F(i) \to c\}_{i \in J}$ under $F$ exhibits the object $c \in C$ as a [[homotopy colimit]] (in the sense discussed in detail at [[homotopy Kan extension]]) precisely if the induced morphism of quasi-categories

$$
  \bar {N(F)} : N(J)^{\triangleright} \to N(C)
$$

is a quasi-categorical colimit [[diagram]] in $N(C)$.

=--

Here $N$ is the [[homotopy coherent nerve]], $N(J)^{\triangleright}$ the [[join of quasi-categories]] with the point, $N(F)$ the image of the simplicial functor $F$ under the homotopy coherent nerve and $\bar{N(F)}$ its extension to the join determined by the cocone maps $\eta$.

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

This is [[Higher Topos Theory|HTT, theorem 4.2.4.1]]

A central ingredient in the proof is the fact, discused at [[(∞,1)-category of (∞,1)-functors]] and at [[model structure on functors]], that [[sSet]]-[[enriched functor]]s do model [[(∞,1)-functor]]s, in that for $A$ a [[combinatorial simplicial model category]], $S$ a [[quasi-category]] and $\tau(S)$ the corresponding $sSet$-category under the left adjoint of the [[homotopy coherent nerve]], we have an [[equivalence of quasi-categories]]

$$
  N(([C,A]_{proj})^\circ)
  \simeq
  Func(S, N(A^\circ))
$$

and the same is trued for $A$ itself replaced by a [[chunk of a model category|chunk]] $U \subset A$.

With this and the discussion at [[homotopy Kan extension]], we find that the cocone components $\eta$ induce for each $a \in [C,sSet]$ a [[homotopy equivalence]]

$$
  C(c,a) \stackrel{}{\to}
  [J^{op}, C](j F, const a)
$$

which is hence equivalently an equivalence of the corresponding [[hom-object in a quasi-category|quasi-categorical hom-objects]]. The claim follows then from the above discussion of characterization of (co)limits in terms of $\infty$-hom adjunctions.

=--

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

The quasi-category $N(A^\circ)$ [[presentable (∞,1)-category|presented]] by a [[combinatorial simplicial model category]] $A$ has all small quasi-categorical limits and colimits.

=--

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

This is [[Higher Topos Theory|HTT, 4.2.4.8]].

It follows from the fact that $A$ has
(pretty much by definition of [[model category]] and [[combinatorial model category]])
all [[homotopy limit]]s and [[homotopy colimit]]s (in fact all [[homotopy Kan extension]]s)
by the above proposition.

=--

Since $(\infty,1)$-categories equivalent to those of the form $N(A^\circ)$ for $A$ a [[combinatorial simplicial model category]] are precisely the [[locally presentable (∞,1)-categories]], it follows from this in particular that every locally presentable $(\infty,1)$-category has all limits and colimits.

### Commutativity of limits {#CommutativityOfLimits}

The following proposition says that if for an $(\infty,1)$-functor $F : X \times Y \to C$ limits (colimits) over each of the two variables exist separately, then they commute.

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

Let $X$ and $Y$ be [[simplicial set]]s and $C$ a [[quasi-category]]. Let $p : X^{\triangleleft} \times Y^{\triangleleft} \to C$ be a [[diagram]]. If

1. for every object $x \in X^{\triangleleft}$ (including the cone point) the restricted diagram $p_x : Y^{\triangleleft} \to C$ is a limit diagram;

1. for every object $y \in Y$ (not including the cone point) the restricted diagram $p_y : X^{\triangleleft} \to C$ is a limit diagram;

then, with $c$ denoting the cone point of $Y^{\triangleleft}$, the restricted diagram, $p_c : X^{\triangleleft} \to C$ is also a limit diagram.

=--

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

This is [[Higher Topos Theory|HTT, lemma 5.5.2.3]]

=--

In other words, suppose that $\lim_x F(x,y)$ exists for all $y$ and $\lim_y F(x,y)$ exists for all $x$ and also that $\lim_y (\lim_x F(x,y))$ exists, then this object is also $\lim_x \lim_y F(x,y)$.

## Examples

### $\infty$-Limits of special shape

#### Coproduct

...

#### Pullback / Pushout

See also [[(∞,1)-pullback]].

The non-degenerate cells of the [[simplicial set]] $\Delta[1] \times \Delta[1]$ obtained as the [[cartesian product]] of the simplicial 1-[[simplex]] with itself look like

$$
  \array{
    (0,0) &\to& (1,0)
    \\
    \downarrow &\searrow& \downarrow
    \\
    (0,1) &\to& (1,1)
  }
$$

A **sqare** in a [[quasi-category]] $C$ is an image of this in $C$, i.e. a morphism

$$
  s : \Delta[1] \times \Delta[1] \to C
  \,.
$$

The simplicial square $\Delta[1]^{\times 2}$ is [[isomorphism|isomorphic]], as a [[simplicial set]], to the [[join of simplicial sets]] of a 2-[[horn]] with the point:

$$
  \Delta[1] \times \Delta[1]
  \simeq
  \{v\} \star \Lambda[2]_2
  =
  \left(
  \array{
    v &\to& 1
    \\
    \downarrow &\searrow& \downarrow
    \\
    0 &\to& 2
  }
  \right)
$$

and

$$
  \Delta[1] \times \Delta[1]
  \simeq
  \Lambda[2]_0 \star \{v\}
  =
  \left(
  \array{
    0&\to& 1
    \\
    \downarrow &\searrow& \downarrow
    \\
    2 &\to& v
  }
  \right)
  \,.
$$

If a square $\Delta[1] \times \Delta[1] \simeq \Lambda[2]_0 \star \{v\} \to C$ exhibits $\{v\} \to C$ as a colimit over $F : \Lambda[2]_0 \to C$, we say the colimit

$$
  v := \lim_\to F := F(1) \coprod_{F(0)} F(2)
$$

is [[generalized the|the]] **pushout** of the diagram $F$.

##### Pasting law of pushouts {#PushoutPasting}

We have the following $(\infty,1)$-categorical analog of the familiar [pasting law of pushouts](http://ncatlab.org/nlab/show/pullback#Pasting) in ordinary [[category theory]]:

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

A pasting diagram of two squares is a morphism

$$
  \Delta[2] \times \Delta[1] \to C
  \,.
$$

Schematically this looks like

$$
  \array{
     x &\to& y &\to& z
     \\
     \downarrow && \downarrow && \downarrow
     \\
     x' &\to& y' &\to& z'
  }
  \,.
$$

If the left square is a pushout diagram in $C$, then the right square is precisely if the outer square is.

=--

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

A proof appears as [[Higher Topos Theory|HTT, lemma 4.4.2.1]]

=--

#### Coequalizer

...

#### Quotients

* [[infinity-quotient]]

#### Tensoring and cotensoring with an  $\infty$-groupoid {#Tensoring}

##### Recap of the 1-categorical situation

An ordinary [[category]] with [[limit]]s is canonically [[power|cotensored]] over [[Set]]:

For $S, T \in $ [[Set]] and $const_T : S \to Set$ the [[diagram]] parameterized by $S$ that is constant on $T$, we have

$$
  \lim_{\leftarrow} \, const_T \simeq T^S
  \,.
$$

Accordingly the cotensoring

$$
  (-)^{(-)} : Set^{op} \times C \to C
$$

is defined by

$$
  c^S := \lim_{\leftarrow} (S \stackrel{const_c}{\to} C)
  = \prod_{S} c
  \,.
$$

And by continuity of the [[hom-functor]] this implies the required natural isomorphisms

$$
  Hom_C(d,c^S) = Hom_C(d, {\lim_{\leftarrow}}_S c) \simeq {\lim_{\leftarrow}}_S
  Hom_C(d,c) \simeq Set(S,Hom_C(d,C))
  \,.
$$

Correspondingly if $C$ has [[colimit]]s, then the [[copower|tensoring]]

$$
  (-) \otimes (-) : Set \times C \to C
$$

is given by forming [[colimit]]s over constant diagrams: $S \otimes c := {\lim_{\to}}_S c$, and again by continuity of the [[hom-functor]] we have the required natural isomorphism

$$
  Hom_C(S \otimes c, d) =
  Hom_C({\lim_{\to}}_S c,d)
  \simeq
  {\lim_{\leftarrow}}_S Hom_C(c,d)

  \simeq
  Set(S,Hom_C(c,d))
  \,.
$$

Of course all the colimits appearing here are just [[coproduct]]s and all limits just [[product]]s, but for the generalization to $(\infty,1)$-categories this is a misleading simplification, it is really the notion of limit and colimit that matters here.

##### Definition

We expect for $S, T \in $ [[∞Grpd]] and for $const_T : S \to \infty Grpd$ the constant diagram, that

$$
  \lim_{\leftarrow} \, const_T \simeq T^S
  \,,
$$

where on the right we have the [[internal hom]] of $\infty$-groupoids, which is modeled in the [[model structure on simplicial sets]] $sSet_{Quillen}$ by the fact that this is a [[closed monoidal category|closed]] [[monoidal category]].

Correspondingly, for $C$ an $(\infty,1)$-category with colimits, it is [[copower|tensored]] over [[∞Grpd]] by setting

$$
  (-)\otimes (-) : \infty Grpd \times C \to C
$$

$$
  S \otimes c := {\lim_{\to}}_S c
  \,,
$$

where now on the right we have the $(\infty,1)$-categorical colimit over the constant diagram $const : S \to C$ of shape $S$ on $c$.

Then by the $(\infty,1)$-continuity of the hom, and using the above characterization of the [[internal hom]] in $\infty Grpd$ we have the required natural equivalence

$$
  Hom_C(S \otimes c, d)
  =
  Hom_C({\lim_{\to}}_S c, d)
  \simeq
  {\lim_{\leftarrow}}_S Hom_C(c,d)
  \simeq
  \infty Grpd(S,Hom_C(c,d))
  \,.
$$

The following proposition should assert that this is all true

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

The $(\infty,1)$-categorical colimit ${\lim_{\to}} c$ over the diagram of shape $S \in \infty Grpd$ constant on $c \in C$ is characterized by the fact that it induces natural equivalences

$$
  Hom_C({\lim_{\to}}_S c, d)
  \simeq
  \infty Grpd(S, Hom_C(c,d))
$$

for all $d \in C$.

=--

This is essentially [[Higher Topos Theory|HTT, corollary 4.4.4.9]].

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

Every [[∞-groupoid]] $S$ is the $(\infty,1)$-colimit in [[∞Grpd]] of the constant diagram on the [[point]] over itself:

$$
  S \simeq {\lim_{\to}}_S const_*
  \,.
$$

=--

This justifies the following definition

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

For $C$ an $(\infty,1)$-category with colimits, the **tensoring of $C$ over $\infty Grpd$** is the $(\infty,1)$-functor

$$
  (-) \otimes (-) : \infty Grpd \times C \to C
$$

given by

$$
  S \otimes c = \lim_{\to} (const_c : S \to C)
  \,.
$$

=--

See [[Higher Topos Theory|HTT, section 4.4.4]].

##### Models {#ModelsForTensoring}

We discuss models for $(\infty,1)$-(co)limits in terms of ordinary [[category theory]] and [[homotopy theory]].

\begin{lemma}
If $C$ is [[presentable (infinity,1)-category|presented]] by a [[simplicial model category]] $A$, in that $C \simeq A^\circ$, then the
$(\infty,1)$-tensoring and $(\infty,1)$-cotensoring of $C$ over [[∞Grpd]]
is modeled by the ordinary [[copower|tensoring]] and [[power|powering]]
of $A$ over [[sSet]]:

For $\hat c \in A$ cofibant and representing an object $c \in C$ and for $S \in sSet$ any simplicial set, we have an equivalence

$$
  c \otimes S \simeq \hat C \cdot S
  \,.
$$

\end{lemma}

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

The powering in $A$ satisfies the [[natural isomorphism]]

$$
  sSet(S,A(\hat c,\hat d)) \simeq A(\hat c \cdot S, \hat d)
$$

in [[sSet]].

For $\hat c$ a cofibrant and $\hat d$ a fibrant representative, we have that both sides here are [[Kan complex]]es that are equivalent to the corresponding [[derived hom space]]s in the corresponding $(\infty,1)$-category $C$, so that this translates into an equivalence

$$
  Hom_C(c \cdot S, d)
  \simeq
  \infty Grpd(S, Hom_C(c,d))
  \,.
$$

The claim then follows from the above proposition.

=--

### Limits in over-$(\infty,1)$-categories {#InOvercategories}

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

For $C$ an $(\infty,1)$-category, $X : D \to C$ a diagram, $C/X$ the [[over-(∞,1)-category]] and $F : K \to C/X$ a diagram in the over-$(\infty,1)$-category, then the [[(∞,1)-limit]] $\lim_{\leftarrow} F$ in $C/X$ coincides with the $(\infty,1)$-limit $\lim_{\leftarrow} F/X$ in $C$.

=--

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

Modelling $C$ as a [[quasi-category]] we have that $C/X$ is given by the [[simplicial set]]

$$
  C/X : [n] \mapsto Hom_X([n] \star D, C)
  \,,
$$

where $\star$ denotes the [[join of simplicial sets]]. The limit $\lim_{\leftarrow} F$ is the initial object in $(C/X)/F$, which is the quasi-category given by the simplicial set

$$
  (C/X)/F : [n] \mapsto Hom_{F}( [n] \star K, C/X)
  \,.
$$

Since the join preserves colimits of simplicial sets in both arguments, we can apply the [[co-Yoneda lemma]] to decompose $[n] \star K = {\lim_{\underset{{[r] \to [n]\star K}}{\to}}}  [r]$, use that the [[hom-functor]] sends colimits in the first argument to limits and obtain

$$
  \begin{aligned}
     Hom([n] \star K, C/X)
     &\simeq
     Hom( {\lim_{\to}}_r [r], C/X)
     \\
     & \simeq {\lim_{\leftarrow}}_r Hom([r], C/X)
     \\
     & \simeq {\lim_{\leftarrow}}_r Hom_F( [r] \star D, C )
     \\
     & \simeq  Hom_F({\lim_{\to}}_r ([r] \star D), C )
     \\
     & \simeq  Hom_F( ({\lim_{\to}}_r[r]) \star D, C )
     \\
     & \simeq Hom_F(([n] \star K) \star D, C)
     \\
     & \simeq Hom_F([n] \star (K \star D), C)
  \end{aligned}
  \,.
$$

Here $Hom_F([r]\star D,C)$ is shorthand for the hom in the (ordinary) [[under category]] $sSet^{D/}$ from the canonical inclusion $D \to [r] \star D$ to $X : D \to C$. Notice that we use the 1-categorical analog of the statement that we are proving here when computing the colimit in this under-category as just the colimit in $sSet$. We also use that the [[join of simplicial sets]], being given by [[Day convolution]] is an associative tensor product.

In conclusion we have an isomorphism of simplicial sets

$$
  (C/X)/F \simeq C/(X/F)
$$

and therefore the initial objects of these quasi-categories coincide on both sides. This shows that $\lim_{\leftarrow} F$ is computed as an initial object in $C/(X/F)$.

=--

### Limits and colimits with values in $\infty Grpd$ {#WithValInooGrpd}

Limits and colimits over a [[(∞,1)-functor]] with values in the [[(∞,1)-category]] [[∞-Grpd]] of [[∞-groupoids]] may be reformulated in terms of the  [[universal fibration of (∞,1)-categories]], hence in terms of the [[(∞,1)-Grothendieck construction]].

Let [[∞Grpd]] be the [[(∞,1)-category]] of [[∞-groupoid]]s. Let the [[(∞,1)-functor]] $Z|_{Grpd} \to \infty Grpd^{op}$ be the [[universal fibration of (infinity,1)-categories|universal ∞-groupoid fibration]] whose fiber over the object denoting some $\infty$-groupoid is that very $\infty$-groupoid.

Then let $X$ be any [[∞-groupoid]] and

$$
  F : X \to \infty Grpd
$$

an [[(∞,1)-functor]]. Recall that the [[Cartesian fibration|coCartesian fibration]] $E_F \to X$ classified by $F$ is the pullback of the [[universal fibration of ∞-groupoids]] $Z|_{Grpd}$ along F:

$$
  \array{
    E_F &\to& Z|_{Grpd}
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{F}{\to}& \infty Grpd
  }
$$

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

Let the assumptions be as above. Then:

* The $\infty$-colimit of $F$ is equivalent to the [[(∞,1)-Grothendieck construction]] $E_F$:

  $$
    \underset{\longrightarrow}{\lim} F \simeq E_F
  $$

* The $\infty$-limit of $F$ is equivalent to the [[∞-groupoid of sections]] of $E_F \to X$

  $$
    \underset{\longleftarrow}{\lim} \simeq \Gamma_X(E_F)
    \,.
  $$

=--

The statement for the colimit is corollary 3.3.4.6 in [[Higher Topos Theory|HTT]]. The statement for the limit is corollary 3.3.3.4.

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

The form of the statement in prop. \ref{InfinityGroupoidalCoLimitsViaIntegrationAndSlicing} is the special case of the general form of [[internal (co-)limits]], here internal to the [[(∞,1)-topos]] [[∞Grpd]] with $Core(\inftyGrpd_{small})$ its small [[object classifier]]. See at _[internal (co-)limit -- Groupoidal homotopy (co-)limits](internal+%28co-%29limit#ExamplesInfinityGroupoidal)_ for more on this.

=--

### Limits and colimits with values in $(\infty,1)$Cat
 {#ColimitsInInfinityCat}

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

For $F : D \to $ [[(∞,1)Cat]] an [[(∞,1)-functor]], its $\infty$-colimit is given by forming the [[(∞,1)-Grothendieck construction]] $\int F$ of $F$ and then inverting the [[Cartesian morphism]]s.

Formally this means, with respect to the [[model structure for Cartesian fibrations]] that there is a [[natural isomorphism]]

$$
  \lim_\to F \simeq (\int F)^\sharp
$$

in the [[homotopy category]] of the presentation of $(\infty,1)$-category by [[marked simplicial sets]].

=--

This is [[Higher Topos Theory|HTT, corollary 3.3.4.3]].

For the special case that $F$ takes values in ordinary categories see also at [[2-limit]] the section [2-limits in Cat](http://ncatlab.org/nlab/show/2-limit#2ColimitsInCat).

### Limits in $\infty$-functor categories

For $C$ an ordinary [[category]] that admits small [[limit]]s and [[colimit]]s, and for $K$ a [[small category]], the [[functor category]] $Func(D,C)$ has all small limits and colimits, and these are computed objectwise. See [[limits and colimits by example]]. The analogous statement is true for an [[(∞,1)-category of (∞,1)-functors]].

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

Let $K$ and $C$ be [[quasi-categories]], such that $C$ has all [[limit in a quasi-category|colimits]] indexed by $K$.

Let $D$ be a small quasi-category. Then

* The [[(∞,1)-category of (∞,1)-functors]] $Func(D,C)$ has all $K$-indexed colimits;

* A morphism $K^\triangleright \to Func(D,C)$ is a colimiting cocone precisely if for each object $d \in D$ the induced morphism $K^\triangleright \to C$ is a colimiting cocone.

=--

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

This is [[Higher Topos Theory|HTT, corollary 5.1.2.3]]

=--

## Related concepts

* [[limit]], [[internal limit]]

* [[2-limit]]

* **$(\infty,1)$-limit

  * [[finite (∞,1)-limit]], [[relative (∞,1)-limit]]

  * [[homotopy limit]]

  * [[lax (∞,1)-colimit]]

## References

### General

The definition of limit in quasi-categories is due to

* [[André Joyal]], _Quasi-categories and Kan complexes_ Journal of Pure and Applied Algebra 175 (2002), 207-222.

A brief survey is on page 159 of

* [[André Joyal]], _The theory of quasicategories and its applications_ lectures at [Simplicial Methods in Higher Categories](http://www.crm.es/HigherCategories/), ([pdf](http://www.crm.cat/HigherCategories/hc2.pdf))

A detailed account is in [definition 1.2.13.4, p. 48](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=48) in

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

* [[Denis-Charles Cisinski]], _Do homotopy limits compute limits in the associated quasicategory in the non-combinatorial model category case?_, [MO/176111/176142](http://mathoverflow.net/questions/176111/do-homotopy-limits-compute-limits-in-the-associated-quasicategory-in-the-non-com/176142#176142).

A discussion of [[weighted limit|weighted]] $(\infty,1)$-limits is in

* Martina Rovelli, _Weighted limits in an (∞,1)-category_, 2019, [arxiv:1902.00805](https://arxiv.org/abs/1902.00805)

A discussion of free colimit completion constructions is in

* [[Charles Rezk]], _Free colimit completion in ∞-categories_ ([arXiv:2210.08582](https://arxiv.org/abs/2210.08582))

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

* [[Louis Martini]], [[Sebastian Wolf]], *Limits and colimits in internal higher category theory* [[arXiv:2111.14495](https://arxiv.org/abs/2111.14495)]

### In homotopy type theory

A formalization of some aspects of $(\infty,1)$-limits in terms of [[homotopy type theory]] is [[Coq]]-coded in

* [[Guillaume Brunerie]], _[HoTT/Coq/Limits](https://github.com/guillaumebrunerie/HoTT/tree/master/Coq/Limits)_

See also

* [[Egbert Rijke]], _Homotopy Colimits and a Descent Theorem_, March 14, 2013 ([video](http://video.ias.edu/univalent-1213-0314-EgbertRijke))

[[!redirects limit in a quasi-category]]
[[!redirects limits in a quasi-category]]
[[!redirects colimit in a quasi-category]]
[[!redirects colimits in a quasi-category]]
[[!redirects limit in quasi-categories]]
[[!redirects limits in quasi-categories]]
[[!redirects colimit in quasi-categories]]
[[!redirects colimits in quasi-categories]]
[[!redirects limit in a quasicategory]]
[[!redirects colimit in a quasicategory]]
[[!redirects limit in quasicategories]]
[[!redirects limits in quasicategories]]
[[!redirects colimit in quasicategories]]
[[!redirects colimits in quasicategories]]
[[!redirects (infinity,1)-limit]]
[[!redirects (infinity,1)-colimit]]
[[!redirects (infinity,1)-limits]]
[[!redirects (infinity,1)-colimits]]

[[!redirects (∞,1)-limit]]
[[!redirects (∞,1)-colimit]]
[[!redirects (∞,1)-limits]]
[[!redirects (∞,1)-colimits]]

[[!redirects (∞,1)-product]]
[[!redirects (∞,1)-coproduct]]
[[!redirects (∞,1)-products]]
[[!redirects (∞,1)-coproducts]]

[[!redirects (∞,1)-pushout]]
[[!redirects (∞,1)-pushouts]]

[[!redirects limit in an (infinity,1)-category]]
[[!redirects limits in an (infinity,1)-category]]
[[!redirects limits in (infinity,1)-categories]]
[[!redirects limit in an (∞,1)-category]]
[[!redirects limits in an (∞,1)-category]]
[[!redirects limits in (∞,1)-categories]]

[[!redirects colimit in an (infinity,1)-category]]
[[!redirects colimits in an (infinity,1)-category]]
[[!redirects colimits in (infinity,1)-categories]]
[[!redirects colimit in an (∞,1)-category]]
[[!redirects colimits in an (∞,1)-category]]
[[!redirects colimits in (∞,1)-categories]]

[[!redirects (∞,1)-powering]]
[[!redirects (∞,1)-tensoring]]
[[!redirects (∞,1)-copower]]

[[!redirects infinity1-limit]]
[[!redirects infinity1-colimit]]


(∞,1)-local geometric morphism

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

#Contents#
* table of contents
{:toc}

## Idea

The analog in the context of [[(∞,1)-topos]] theory of a [[local geometric morphism]] in [[topos theory]].

## Definition

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

A **local (∞,1)-geometric morphism** $f : \mathbf{H} \to \mathbf{S}$ between [[(∞,1)-topos]]es $\mathbf{H},\mathbf{S}$ is

* an [[(∞,1)-geometric morphism]]

  $$
    (f^* \dashv f_*)
     :
      \mathbf{H} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}
    {\to}}  \mathbf{S}
  $$

* such that

  1. a further [[right adjoint]]
     $f^! : \mathbf{S} \to \mathbf{H}$
     to the [[direct image]] functor exists:

     $$
       (f^* \dashv f_* \dashv f^!) :
       \mathbf{H}
        \stackrel{\overset{f^*}{\leftarrow}}
        {\stackrel{\underset{f_*}{\to}}{\underset{f^!}{\leftarrow}}}
       \mathbf{S}
     $$

   1. and $f$ is a [[∞-connected (∞,1)-geometric morphism]].

If $f : \mathbf{H} \to \mathbf{S}$ is the [[global section]] [[(∞,1)-geometric morphism]] in the [[over-(∞,1)-category]] [[Topos]]$/\mathbf{S}$, then we say that $\mathbf{H}$ is a **local $(\infty,1)$-topos** over $\mathbf{S}$.

=--

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

If $\mathbf{S} = $ [[∞Grpd]] then the extra condition that $f$ is [[∞-connected (∞,1)-geometric morphism]] is automatic (see [Properties -- over ∞Grpd](#OverInfGrpd)).

=--

## Properties {#Properties}

### Over $\infty Grpd$ {#OverInfGrpd}

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

Every local $(\infty,1)$-topos over [[∞Grpd]] has [[homotopy dimension]] $\leq 0$.

=--

See [[homotopy dimension]] for details.

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

If an [[(∞,1)-geometric morphism]] $f : \mathbf{H} \to $ [[∞Grpd]] has an extra [[right adjoint]] $f^!$ to its [[direct image]], then $\mathbf{H}$ is an [[∞-connected (∞,1)-topos]].

=--

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

By the general properties of [[adjoint (∞,1)-functor]]s it is sufficient to show that $f_! f^* \simeq Id$. To see this, we use that every [[∞-groupoid]] $S \in $ [[∞Grpd]] is the [[(∞,1)-colimit]] (as discussed there) over itself of the [[(∞,1)-functor]] constant on the point: $S \simeq {\lim_\to}_{S} *$.

The [[left adjoint]] $f^*$ preserves all [[(∞,1)-colimit]]s, but if $f_*$ has a right adjoint, then it does, too, so that for all $S$ we have

$$
  f_* f^* {\lim_\to}_S  * \simeq {\lim_\to}_S f_* f^* *
  \,.
$$

Now $f_*$, being a [[right adjoint]] preserves the [[terminal object in an (∞,1)-category|terminal object]] and so does  $f^*$ by definition of [[(∞,1)-geometric morphism]]. Therefore

$$
  \cdots \simeq {\lim_\to}_S * \simeq S
  \,.
$$

=--

### Concrete objects

Every local $(\infty,1)$-geometric morphism induces a notion of [[concrete (∞,1)-sheaves]]. See there for more (also see [[cohesive (∞,1)-topos]]).

## Examples

### Local over-$(\infty,1)$-toposes

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

Let $\mathbf{H}$ be any [[(∞,1)-topos]] (over [[∞Grpd]]) and let $X \in \mathbf{H}$ be an [[object]] that is [[small-projective]]. Then the [[over-(∞,1)-topos]] $\mathbf{H}/X$ is local.

=--

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

We check that the [[global section]] [[(∞,1)-geometric morphism]] $\Gamma : \mathbf{H}/X \to $ [[∞Grpd]] preserves [[(∞,1)-colimit]]s.

The functor $\Gamma$ is given by the [[hom-functor]] out of the [[terminal object]] of $\mathcal{H}/X$, this is $(X \stackrel{Id}{\to} X)$:

$$
  \Gamma : (A \stackrel{f}{\to} X) \mapsto Hom_{\mathbf{H}/X}(Id_X, f)
  \,.
$$

The [[derived hom-space|hom-∞-groupoid]]s in the [[over-(∞,1)-category]] are (as discussed there) [[homotopy fiber]]s of the hom-sapces in $\mathbf{H}$: we have an [[(∞,1)-pullback]] diagram

$$
  \array{
      \mathbf{H}/X(Id_X, (A \to X)) &\to&
       \mathbf{H}(X,A)
      \\
      \downarrow && \downarrow^{f_*}
      \\
      * &\stackrel{Id_X}{\to}& \mathbf{H}(X,X)
  }
  \,.
$$

Overserve that [[(∞,1)-colimit]]s in the [[over-(∞,1)-category]] $\mathbf{H}/X$ are computed in $\mathbf{H}/X$.

$$
  {\lim_{\to}}_i (A_i \stackrel{f_i}{\to} X)
  \simeq
  ({\lim_\to}_i A_i) \to X
  \,.
$$

If $X$ is [[small-projective]] then by definition we have

$$
  \mathbf{H}(X, {{\lim}_\to}_i A_i)
  \simeq
  {\lim_\to}_i \mathbf{H}(X, A_i)
  \,,
$$

Inserting all this into the above $(\infty,1)$-pullback gives the $(\infty,1)$-pullback

$$
  \array{
      \mathbf{H}/X(Id_X, {\lim_\to}_i (A_i \to X)) &\to&
       {\lim_\to}_i \mathbf{H}(X, A_i)
      \\
      \downarrow && \downarrow^{f_*}
      \\
      * &\stackrel{Id_X}{\to}& \mathbf{H}(X,X)
  }
  \,.
$$

By [[universal colimits]] in the [[(∞,1)-topos]] [[∞Grpd]], this [[(∞,1)-pullback]] of an  [[(∞,1)-colimit]] is the $(\infty,1)$-colimit of the separate pullbacks, so that

$$
  \Gamma({\lim_\to}_i (A_i \to X)))
  \simeq
  \mathbf{H}/X(Id_X, {\lim_\to}_i (A_i \to X))
   \simeq
  {\lim_\to}_i \mathbf{H}/X(Id_X,(A_i \to X))
  \simeq
  {\lim_\to}_i \Gamma(A_i \to X)
  \,.
$$

So $\Gamma$ does commute with colimits if $X$ is [[small-projective]]. Since all [[(∞,1)-topos]]es are [[locally presentable (∞,1)-categories]] it follows by the [[adjoint (∞,1)-functor]] that $\Gamma$ has a [[right adjoint|right]] [[adjoint (∞,1)-functor]].

=--

## Related concpepts

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

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

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

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

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

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

and

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

* [[connected site]]

* [[local site]]

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

## References

The 1-categorical notion is discussed in

* [[Peter Johnstone]], [[Ieke Moerdijk]], _Local maps of toposes_  Proc. London Math. Soc.  (1989)   s3-58  (2):  281-305.  ([pdf](http://plms.oxfordjournals.org/content/s3-58/2/281.full.pdf+html))

[[!redirects local (∞,1)-geometric morphisms]]
[[!redirects local (∞,1)-topos]]
[[!redirects local (∞,1)-toposes]]
[[!redirects local (∞,1)-topoi]]

[[!redirects local (infinity,1)-topos]]
[[!redirects local (infinity,1)-toposes]]


(∞,1)-logic

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Type theory
+--{: .hide}
[[!include type theory - contents]]
=--
#### Foundations
+--{: .hide}
[[!include foundations - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

As a [[category]] has [[internal logic]], so a [[(∞,1)-category]] has internal _$(\infty,1)$-logic_

## Examples

See

* [[internal logic of an (∞,1)-topos]]

## Related concepts

* [[type theory]], [[logic]]

* [[2-type theory]], [[2-logic]]

* [[(∞,1)-type theory]], **(∞,1)-logic**


(∞,1)Operad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### [[categories of categories - contents|categories of categories]]
+-- {: .hide}
[[!include categories of categories - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}

## Idea

The [[(∞,2)-category]] of [[(∞,1)-operads]]. The [[operad|operadic]] generalization of [[(∞,1)Cat]].

## Properties

[[!include table - models for (infinity,1)-operads]]

## Related concepts

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

* [[2Cat]]

* [[(∞,1)Cat]], [[(∞,1)Operad]]

* [[(∞,n)Cat]]

category: category

[[!redirects (infinity,1)Operad]]


(∞,2)-category of cobordisms

## Idea

An [[(∞,n)-category of cobordisms]] for $n = 2$.

## Related concepts

* [[2d TQFT]], [[TCFT]]

* [[Calabi-Yau object]]

[[!redirects (infinity,2)-category of cobordisms]]


(∞,2)-category theory

## Idea

The theory of [[(∞,2)-categories]]...

## Related concepts

[[!include table of category theories]]


(∞,2)-functor

## Idea

[[(infinity,n)-functor]] for $n = 2$

## Related entries

* [[2d TQFT]], [[TCFT]]

[[!redirects (∞,2)-functors]]

[[!redirects (infinity,2)-functor]]
[[!redirects (infinity,2)-functors]]


(∞,n)-category theory

## Idea

The theory of [[(∞,n)-categories]]

## Related concepts

[[!include table of category theories]]


0-bundle

## Idea

In the context of [[principal infinity-bundles]] a 0-bundle is equivalently a [[function]].

## Examples

* [[prequantum 0-bundle]].

[[!redirects 0-bundles]]


0-category

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

# Contents
* table of contents
{: toc}

## Definition

A __$0$-category__ (or __$(0,0)$-category__) is, up to [[equivalence of categories|equivalence]], the same as a [[set]] (or [[class]]).

+-- {: .num_remark}
###### Remark
Although this terminology may seem strange at first, it simply follows the logic of $n$-[[n-category|categories]] (and $(n,r)$-[[(n,r)-category|categories]]).  To understand these, it is very helpful to use [[negative thinking]] to see sets as the beginning of a sequence of concepts: sets, [[category|categories]], [[2-category|2-categories]], [[3-category|3-categories]], etc.  Doing so reveals patterns such as the [[periodic table]]; it also sheds light on the theory of [[homotopy groups]] and [[n-stuff]].

For example, there should be a $1$-category of $0$-categories; this is the [[Set|category of sets]]. Then a [[enriched category|category enriched]] over this is a $1$-category (more precisely, a [[locally small category]]). Furthermore, an enriched groupoid is a [[groupoid]] (or $1$-groupoid), so a $0$-category is the same as a [[0-groupoid]].

To some extent, one can continue to define a [[(-1)-category]] to be a [[truth value]] and a [[(-2)-category]] to be a triviality (that is, there is exactly one). These don\'t fit the pattern perfectly; but the concepts of [[(-1)-groupoid]] and [[(-2)-groupoid]] for them do work perfectly, as does the concept of [[0-poset]] for a truth value.
=--

+-- {: .num_remark}
###### Remark
Interpreted literally, $0$-category or $(0, 0)$-category would be an $\infty$-category such that every $j$-cell for $j \gt 0$ is an equivalence, and any two such $j$-cells that are parallel are equivalent. The picture that apparently emerges from this description might suggest a [[symmetric proset]], a set equipped with an [[equivalence relation]], or something even more complicated than that. One *could* thus say that a $0$-category is a symmetric proset, when considered just up to [[isomorphism of categories|isomorphism]]. But it is more appropriate in higher category theory to consider these things up to equivalence rather than up to isomorphism; when we do this, a $0$-category is equivalent to a set again.
=--

## Related concepts

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

* [[category]]

* [[2-category]]

* [[3-category]]

* [[n-category]]

* [[(∞,0)-category]]

* [[(∞,1)-category]]

* [[(∞,2)-category]]

* [[(∞,n)-category]]

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

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


0-dimensional TQFT

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

## Idea

A _0-dimensional TQFT_ is a [[TQFT]] regarded in the sense of [[FQFT]] as a [[representation]] of the [[category]] of 0-dimensional [[cobordism]]s.

This degenerate case turns out to exhibit a nontrivial amount of interesting information, in particular if regarded in the context of _super_ QFT.

## Definition

### 0-Dimensional Cobordisms

The [[category]] $Cob_0$ of 0-dimensional [[cobordism]]s is the [[symmetric monoidal category]] $Cob_0$ having the $-1$-dimensional [[manifold]] $\emptyset$ as the only [[object]] and [[isomorphism]] classes of compact $0$-dimensional manifolds as morphisms. Clearly $Cob_0$ is equivalent to $\mathbf{B}\mathbb{N}$

### 0-Dimensional TQFT

A **0-dimensional [[TQFT]]** (with values in $\mathbb{Z}$-[[module]]s) is a [[monoidal functor]]

$$
  Z\colon Cob_0\to \mathbb {Z} Mod
  \,.
$$

By definition of [[monoidal functor]], one has $Z({\emptyset})=\mathbb{Z}$ and so $Z$ is completely (and freely) determined by the assignment $Z(\{pt\}\in End_\mathbb{Z}(\mathbb{Z})=\mathbb{Z}$. In other words, the space of 0-dimensional TQFTs is $\mathbb{Z}$.

### Over a manifold

One can consider TQFTs with a target manifold $X$: all bordisms are required to have a map to $X$.

In dimension $0$, morphisms in $Cob_0(X)$ are the topological [[monoid]] $\bigcup_{n\geq 1} Sym^n(X)$. In particular, continuous tensor functors from $Cob_0(X)$ to $\mathbb{Z}$-modules are naturally identified with degree 0 [[integral cohomology]] $H^0(X;\mathbb{Z})$.

### Extended version

The picture becomes more interesting if one goes from topological field theory to [[extended topological quantum field theory]]. Indeed, from this point of view, to the $-1$-dimensional vacuum is assigned the symmetric monoidal [[0-category]] $\mathbb{Z}$, and consequently, the infinity-version of the space of all $0$-dimensional TQFTs is the [[Eilenberg-Mac Lane spectrum]]. It follows that the space of extended $0$-dimensional TQFTs with target $X$ (taking values in $\mathbb{Z}$-modules) is the graded [[integral cohomology]] ring $H^*(X;\mathbb{Z})$.

### Super version

From the [[differential geometry]] point of view, a relation between de Rham cohomology of a smooth manifold $X$ and $0$-dimensional functorial field theories arises if one moves from topological field theory to $(0|1)$-supersymmetric field theory, see [[Axiomatic field theories and their motivation from topology]].

> It would be interesting to describe a direct connection between the extended and the susy theory; it should parallel the usual Cech-de Rham argument


0-functor

As a $0$-[[0-category|category]] is simply a [[set]], so a __$0$-functor__ is simply a [[function]].  See also $n$-[[n-functor|functor]].

[[!redirects 0-functor]]
[[!redirects 0-functors]]


0-groupoid

#Contents#
* table of contents
{:toc}

## Idea

A _0-groupoid_ or **[[homotopy n-type|0-type]]** is a [[set]]. This terminology may seem strange at first, but it is very helpful to see sets as the beginning of a sequence of concepts: sets, [[groupoid|groupoids]], [[2-groupoid|2-groupoids]], [[3-groupoid|3-groupoids]], etc. Doing so reveals patterns such as the [[periodic table]]. (It also sheds light on the theory of [[homotopy group]]s and [[n-stuff]].)

For example, there should be a $1$-groupoid of $0$-groupoids; this is the [[underlying groupoid]] of the [[Set|category of sets]]. Then a [[enriched category|groupoid enriched]] over this is a $1$-groupoid (more precisely, a [[locally small category|locally small groupoid]]). Furthermore, an enriched category is a [[category]] (or $1$-category), so a $0$-groupoid is the same as a [[0-category]].

One can continue to define a [[(−1)-groupoid]] to be a [[truth value]] and a [[(−2)-groupoid]] to be a triviality (that is, there is exactly one).

## Related concepts

[[!include homotopy n-types - table]]

[[!redirects 0-groupoids]]

[[!redirects 0-type]]
[[!redirects 0-types]]


0-morphism

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

#Contents#
* table of contents
{:toc}

## Definition

Following the patterns of [[k-morphism]]s in a [[category]] or [[higher category]] it makes sense to say
**$0$-morphism** for an _[[object]]_ of that category.

## Related concepts

* [[object]]

* [[morphism]]

* [[2-morphism]]

* [[k-morphism]]

[[!redirects 0-morphisms]]


0-object > history

< [[0-object]]
[[!redirects 0-object -- history]]


0-poset

A __0-poset__ is a [[truth value]]. Compare the concept of $1$[[1-poset|poset]] (a [[partial order|poset]]) and $(-1)$-[[(-1)-poset|poset]] (which is trivial); compare also with $(-1)$-[[(-1)-category|category]] and $0$-[[0-groupoid|groupoid]], which mean the same thing for different reasons.

The point of 0-posets is that they complete some patterns in the [[periodic table]] of $n$-categories, in particular the progression of $n$-[[n-poset|posets]].

For example, there should be a $0$-[[0-category|category]] of $0$-posets; a $0$-category is simply a [[set]], and this set is the set of truth values, classically
$$
  (-1)Pos := \{\bot, \top\}
  \,.
$$
Actually, we should expect the $0$-category of $0$-posets to be a $1$-poset; this is simply a poset, and indeed truth values do form a poset (where $\bot \leq \top$).

If we equip the category of $0$-posets with its [[cartesian monoidal category|monoidal cartesian structure]] (which is [[logical conjunction|conjunction]], the logical AND operation), then an $\infty$-[[enriched category|category enriched]] over this should be a $1$-poset; and indeed it is (up to [[equivalence of categories]]) a poset (although up to isomorphism only, a category enriched over truth values under conjunction is actually a set equipped with a [[preorder]]).

See [[(−1)-category]] for references on this sort of [[negative thinking]].


0-sphere

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

#Contents#
* table of contents
{:toc}

## Idea

The _0-sphere_ $S^0$ is the [[n-sphere]] for $n = 0$: The [[disjoint union]] of two [[points]],

$$
  S^0 \;\;\simeq\;\; \ast \sqcup \ast
  \,.
$$

This definition (even if possibly not be familiar from standard texts of [[point-set topology]]) is certainly justified by the fact that the 2-element set is clearly:

* the [[unit sphere]] inside the [[real line]] $\mathbb{R}^1$,

  and as such part of the general pattern that the [[n-sphere]] is the [[unit sphere]] inside $\mathbb{R}^{n+1}$;

* the [[topological space]]/[[homotopy type]] whose [[suspension]] is equivalently the [[1-sphere]] (the [[circle]]):

  $$
    S^1 \,\simeq\, \mathrm{S} S^0
    \,,
  $$

  thus being the base case of the [[induction|inductive]] definition of the [[n-spheres|$n$-spheres]]:

  $$
    S^{n+1}
      \;\simeq\;
    \mathrm{S} S^{n}
      \;\simeq\;
    \mathrm{S} \mathrm{S} S^{n-1}
      \;\simeq\;
    \mathrm{S}^{n+1} S^0
    \,.
  $$

In fact, with [[suspension]]  $\mathrm{S}X$ understood as the [[homotopy pushout]] of the [[terminal object|terminal]] [[map]] $X \to \ast$ along itself, one has also

$$
  S^0 \;\coloneqq\; \mathrm{S} S^{-1}
$$

for $S^{-1} \,\coloneqq\, \varnothing$ the [[empty set]].

These spheres of degenerate dimensions play an imprtant role in [[homotopy theory]], see for instance the [[cofibrantly generated model category|generating cofibrations]] in the [[classical model structure on topological spaces]] ([here](classical+model+structure+on+topological+spaces#TopologicalGeneratingCofibrations)).

Of course, the underlying object of $S^0$, simple as it is, plays various other roles, too:

* in [[topos theory]], the two element set plays the role of the [[subobject classifier]] among [[Sets]];

* in [[formal logic]] this is also known as the [[classical mathematics|classical]] *[[boolean domain]]*: the [[set]] of [[classical mathematics|classical]] [[truth values]];

  and in [[data type|data]]-[[type theory]] one also speaks of the [[type]] $Bit$ of [[bits]];

* that same type regarded in [[homotopy type theory]] again plays the role of the $0$-sphere in the sense of *[[sphere types]]* -- see also at *[boolean domain -- In HoTT](boolean+domain#InHomotopyTypeTheory)*;

* in [[category theory]] the [[disjoint union]] of two copies of the [[terminal object]] (be it the 2-element [[set]] or the corresponding [[discrete category]], etc.) is often denoted "$\mathbf{2}$"

  (cf. e.g. at *[[Stone duality]]*).

## Related concepts

* [[1-sphere]]

* [[2-sphere]]

* [[4-sphere]]

* [[7-sphere]]

[[!redirects 0-spheres]]

[[!redirects (-1)-sphere]]
[[!redirects (-1)-spheres]]


1-category

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[higher category theory]] one sometimes needs, for emphasis, to say _1-category_ for _[[category]]_.

## Details

Fix a meaning of $\infty$-[[infinity-category|category]], however weak or strict you wish. Then a __$1$-category__ is an $\infty$-category such that every [[2-morphism]] is an [[equivalence]] and all [[parallel pairs]] of [[k-morphism|j-morphisms]] are equivalent for $j \geq 2$. Thus, up to equivalence, there is no point in mentioning anything beyond $1$-morphisms, except whether two given parallel $1$-morphisms are equivalent.

If you rephrase equivalence of $1$-morphisms as [[equality]], which gives the same result up to [[equivalence of categories|equivalence]], then all that is left in this definition is a [[category]].  Thus one may also say that a __$1$-category__ is simply a category.

The point of all this is simply to fill in the general concept of $n$-[[n-category|category]]; nobody thinks of $1$-categories as a concept in their own right except simply as categories.

## Related concepts

The notions of $1$-[[1-groupoid|groupoid]] and $1$-[[1-poset|poset]] are defined on the same basis.

[[!redirects 1-categories]]

[[!redirects 1‑categories]]


1-category equipped with relations

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

## Idea ##

A [[2-category equipped with proarrows]] is a 2-category together with a 2-category of "proarrows" which are intended to generalize the arrows of $K$ in the same way that [[profunctors]] generalize the [[functors]] in [[Cat]].  Since profunctors are a [[categorification]] of [[relations]], it is natural to think of decategorifying such equipments to give a structure on a 1-category that equips it with "relations".  We call this structure a *1-category equipped with relations*.

## Definition ##

### (1,2)-categories equipped with proarrows

Recall that a *2-category equipped with proarrows* (aka "proarrow equipment" or "equipment") can be defined as a certain sort of [[double category]], with $\mathcal{V}(\underline{K}) = K$.  If, in such a double category, any two squares with the same boundary are equal, we say that it is is a **(1,2)-category equipped with proarrows**, or a **(1,2)-category proarrow equipment**.  This is equivalent to requiring that the 2-category of proarrows (and hence also the underlying 2-category of arrows) is [[locally posetal 2-category|locally posetal]], i.e. a (1,2)-category.

For example, if $V$ is any [[quantale]], then $V Cat$ is naturally a (1,2)-category equipped with proarrows.  In particular, taking $V=\mathbb{2}$, we have a (1,2)-category proarrow equipment whose objects are [[preorders]].

### 1-categories equipped with relations

A **1-category equipped with relations** is a (1,2)-category equipped with proarrows, regarded as a double category $\underline{K}$, together with an [[involution]] $\underline{K}^{h op} \cong \underline{K}$ which is (isomorphic to) the identity on objects and (vertical) arrows.  Here $\underline{K}^{h op}$ denotes the horizontal opposite of a double category obtained by reversing the horizontal (pro-)arrows but not the vertical ones.  We also call this structure a **relation equipment** or a **1-category proarrow equipment**.

In particular, the definition implies that we have an involution $K \cong K^{co}$ which is the identity on objects and arrows, which for a (1,2)-category means that $K$ is actually (equivalent to) a 1-category.  Note though that the 2-category of proarrows (which we now call "relations") is still (like [[Rel]]) a (1,2)-category, not necessarily a 1-category.

For example, for any quantale $V$, the sub-2-category of $V Cat$ consisting of the *symmetric* $V$-categories (those where $A(x,y) = A(y,x)$) is a 1-category equipped with relations.  In particular, for $V=\mathbb{2}$, we have the relation equipment $\underline{Rel}$ of sets, functions, and binary relations.

In general, we can think of a relation equipment as generalizing some of the properties of $\underline{Rel}$.  For instance, internal relations in any [[regular category]] also form a relation equipment.

## Cartesian 1-categories equipped with relations ##

It is proven in

* Carboni, Kelly, Wood, "A 2-categorical approach to change of base and geometric morphisms, I" ([PDF](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1991__32_1/CTGDC_1991__32_1_47_0/CTGDC_1991__32_1_47_0.pdf))

that a (1,2)-category is a [[cartesian bicategory]] precisely when it is a [[cartesian object]] in a suitable 2-category of proarrow equipments (where we make a bicategory $M$ into an equipment by taking the proarrows to be those of $M$ and the arrows to be the "maps" in $M$, i.e. the morphisms having right adjoints).  Here is a rough sketch of the argument, using the double-category description of equipments.

+-- {: .un_theorem}
###### Theorem
Let $\underline{K}$ be a 1-category equipped with relations, which is a [[cartesian object]] in the 2-category of relation equipments (that is, it is a **cartesian relation equipment**).  Then $\mathcal{H}(\underline{K})$ is a cartesian bicategory.
=--
+-- {: .proof}
###### Proof
That $\underline{K}$ is a cartesian object means, in particular, that it is a [[pseudomonoid]] in the 2-category of equipments.  By lifting the coherence data from arrows to representable proarrows, it follows that $\mathcal{H}(\underline{K})$ is a monoidal 2-category.  Being a cartesian object also gives a cartesian product on objects and proarrows, with diagonals $\Delta\colon X\to X\times X$, and lifting these arrows to representable proarrows $\Delta_\bullet$ and $\Delta^\bullet$ gives each object a commutative monoid and comonoid structure.  Now for any proarrow $\phi\colon X\to Y$, the square
$$\array{X & \overset{\phi}{\to} & Y\\
  ^\Delta \downarrow & \Downarrow & \downarrow^\Delta\\
  X\times X& \underset{\phi\times \phi}{\to} & Y\times Y}$$
in $\underline{K}$ induces 2-cells, i.e. inequalities, $\Delta_\bullet \phi \le (\phi\times\phi)\Delta_\bullet$ and $\phi \Delta^\bullet  \le \Delta^\bullet(\phi\times\phi)$.
=--

A [[bicategory of relations]] is a (1,2)-category which is a cartesian bicategory, and which also satisfies some additional conditions.  We can also construct this structure starting from a relation equipment.

+-- {: .un_theorem}
###### Theorem
Let $\underline{K}$ be a relation equipment satisfying the hypotheses of the previous theorem, and suppose in addition that every proarrow $\phi\colon x\nrightarrow y$ in $\underline{K}$ can be written as $f_\bullet g^\bullet$ for some (vertical) arrows $f$ and $g$.  (That is, "tabulations" in a certain sense exist.)  Then $\mathcal{H}(\underline{K})$ is a [[bicategory of relations]].
=--
+-- {: .proof}
###### Sketch of Proof
We first verify the axiom $\Delta^\bullet \Delta_\bullet = 1$.  Since $\Delta^\bullet \Delta_\bullet$ is the restriction of $1_{X\times X}$ along $\Delta$ on both sides, it suffices to show that
$$\array{X & \overset{1_X}{\to} & X\\
  ^\Delta\downarrow &\Downarrow& \downarrow^\Delta\\
  X\times X& \underset{1_{X\times X}}{\to} & X\times X}$$
is a cartesian 2-cell in $\underline{K}$.  But if we have any other square
$$\array{A & \overset{\phi}{\to} & B\\
  ^{(f,g)}\downarrow &\Downarrow& \downarrow^{(h,k)}\\
  X\times X& \underset{1_{X\times X}}{\to} & X\times X}$$
then $(f,g)$ factoring through $\Delta$ means that $f=g$, and likewise $h=k$.  Composing the given square  with the projection
$$\array{X\times X & \overset{1_{X\times X}}{\to} & X\times X\\
  \downarrow &\Downarrow & \downarrow\\
  X& \underset{1_X}{\to} & X}$$
(which comes from being a cartesian object in $Equipments$), we obtain a square
$$\array{A & \overset{\phi}{\to} & B \\
  ^f\downarrow &\Downarrow & \downarrow^g\\
  X& \underset{1_X}{\to} & X}$$
which factors the given square through the putative cartesian one.  The factorization is unique since all 2-cells are unique.

We now verify the Frobenius axiom $\Delta^\bullet \Delta_\bullet = (1\times \Delta_\bullet)(\Delta^\bullet \times 1)$.  Since $\Delta$ is associative, we have a square
$$\array{X & \overset{1_X}{\to} & X\\
  ^\Delta\downarrow && \downarrow^\Delta\\
  X\times X & \Downarrow & X\times X\\
  ^{1\times \Delta}\downarrow && \downarrow^{\Delta\times 1}\\
  X\times X\times X & \underset{1_{X\times X\times X}}{\to} & X\times X\times X}$$
and therefore a square
$$\array{X\times X & \overset{\Delta^\bullet \Delta_\bullet}{\to} & X\times  X\\
  ^{1\times \Delta}\downarrow && \downarrow^{\Delta\times 1}\\
  X\times X\times X & \underset{1_{X\times X\times X}}{\to} & X\times X\times X}$$
and it suffices to show that this is a cartesian 2-cell.  So suppose given a square
$$\array{A & \overset{\phi}{\to} & B\\
  ^{(f,g,g)}\downarrow & \Downarrow & \downarrow^{(h,h,k)}\\
  X\times X\times X & \underset{1_{X\times X\times X}}{\to} & X\times X\times X.}$$
The fact that $g$ and $h$ appear twice is equivalent to saying that the left and right boundaries of this square factor through $1\times\Delta$ and $\Delta\times 1$, respectively.  Now by assumption, $\phi = u_\bullet v^\bullet$ for some $u\colon C\to B$ and $v\colon C\to A$.  Thus our square is equivalent to one
$$\array{C & \overset{1_C}{\to} & C\\
  ^{(f v,g v,g v)}\downarrow & \Downarrow & \downarrow^{(h u,h u,k u)}\\
  X\times X\times X & \underset{1_{X\times X\times X}}{\to} & X\times X\times X.}$$
But this is just a 2-cell in the vertical category $K$, which is a 1-category; hence we have $(f v,g v, g v) = (h u, h u, k u)$ and thus $f v = h u = g v = k u$.  Calling their common value $m$, we thus have a composite square
$$\array{C & = & C & = & C\\
  ^{(m,m)}\downarrow && \downarrow^{m} && \downarrow^{(m,m)}\\
  X\times X & \underset{\Delta^\bullet }{\to} & X & \underset{\Delta_\bullet}{\to} & X\times X}$$
(since $\Delta m = (m,m)$) which gives us the desired factorization.  The other Frobenius axiom is, of course, dual.
=--

+-- {: .un_corollary}
###### Corollary
If $\underline{K}$ is a relation equipment satisfying the hypotheses of the theorem, then $\mathcal{H}(\underline{K})$ is an [[allegory]].
=--
+-- {: .proof}
###### Proof
It is shown [here](http://rfcwalters.blogspot.com/2009/10/categorical-algebras-of-relations.html) that any bicategory of relations is an allegory.
=--

## See also ##

Other attempted axiomatizations of the same idea "something that acts like the category of relations in a regular category" include:

* [[allegory]]

* [[bicategory of relations]]

## References

A comparison of "regular proarrow equipments" with "regular [[fibrations]] of [[subobjects]]" is in

* [[Finn Lawler]], *Fibrations of predicates and bicategories of relations*, [arXiv](http://arxiv.org/abs/1502.08017)

A study of equipments of relations from a double-categorical viewpoint, and a characterization of those that arise from some [[factorization system]], is in

* [[Michael Lambert]], *Double categories of relations*, [arxiv](https://arxiv.org/abs/2107.07621) 2021

[[!redirects (1,2)-category equipment]]
[[!redirects (1,2)-category equipped with proarrows]]
[[!redirects 1-category equipped with proarrows]]
[[!redirects relation equipment]]
[[!redirects relation equipments]]
[[!redirects 1-category relation equipment]]
[[!redirects 1-category equipment]]
[[!redirects 1-category equipments]]
[[!redirects 1-category proarrow equipment]]
[[!redirects (1,2)-category proarrow equipment]]


1-dimensional Chern-Simons theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

### Idea

By the general mechanism of [[schreiber:∞-Chern-Simons theory]], every [[invariant polynomial]] of total degree 2 induces a 1-dimensional Chern-Simons-like theory.

## Examples

### For the first Chern class

By the general mechanism of [[schreiber:∞-Chern-Simons theory]] there is a Chern-Simons action functional associated to the first [[Chern class]], or rather to the corresponding [[invariant polynomial]], which is simply the [[trace]] map on the [[unitary Lie algebra]]

$$
  tr : \mathfrak{u}(n) \to \mathbb{R}
  \,.
$$

This yields an [[action functional]] for a 1-dimensional [[QFT]] as follows:

The [[configuration space]] over a 1-dimensional $\Sigma$ is the [[groupoid of Lie algebra valued 1-forms]] $\Omega^1(\Sigma, \mathfrak{u})$. After identifying $\Sigma \subset \mathbb{R}$ this may be identified with the space of $\mathfrak{u}(n)$-valued functions.

The action functional is simply the [[trace]] operation

$$
  S_{CS}(\phi) = \int_\Sigma tr(\phi)
  \,.
$$

Degenerate as this situation is, it can be useful to regard the [[trace]] as a Chern-Simons action functional.

* Arguments for a role in large $N$ gauge theory are in ([Nair 06](#Nair)).

* The _[[spectral action]]_  is of this form.

### For a group character, on a coadjoint orbit

For $G$ a suitable [[Lie group]] (compact, semi-simple and simply connected) the [[Wilson loops]] of $G$-[[principal connections]] are equivalently the [[partition functions]] of a 1-dimensional Chern-Simons theory.

This appears famously in the formulation of [[Chern-Simons theory]] [with Wilson lines](Chern-Simons+theory#WithWilsonLineObservables). More detailes are at _[[orbit method]]_.

### For a symplectic Lie 0-algebroid

A [[symplectic manifold]] regarded as a [[symplectic Lie n-algebroid]] with $n = 0$ induces a 1d Chern-Simons theory whose [[Chern-Simons form]] is a Liouville form of the symplectic form.

This case is discussed in ...

## Related concepts

* [[higher dimensional Chern-Simons theory]]

  * **1d Chern-Simons theory**

  * [[2d Chern-Simons theory]]

  * [[3d Chern-Simons theory]]

  * [[4d Chern-Simons theory]]

  * [[5d Chern-Simons theory]]

  * [[6d Chern-Simons theory]]

  * [[7d Chern-Simons theory]]

  * [[AKSZ sigma-models]]

  * [[string field theory]]

  * [[infinite-dimensional Chern-Simons theory]]

## References

### For the first Chern class

A discussion of 1d CS theory in the context of large $N$-gauge theory is in

* V.P. Nair, _The Matrix Chern-Simons One-form as a Universal Chern-Simons Theory_ 	Nucl.Phys.B750:289-320,2006 ([arXiv:hep-th/0605007](http://arxiv.org/abs/hep-th/0605007))
  {#Nair}

An exposition of this theory formulated via an [[extended Lagrangian]] in [[higher geometric quantization]] is in section 1 of

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_

Further discussion is in section 5.7 of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

### For a symplectic Lie 0-algebroid

A 1d Chern-Simons theory with target a [[cotangent bundle]] is discussed in

* [[Ryan Grady]], [[Owen Gwilliam]], _One-dimensional Chern-Simons theory and the $\hat{A}$ genus_ ([arXiv:1110.3533](http://arxiv.org/abs/1110.3533))
  {#GradyGwilliam}

[[!redirects 1-dimensional Chern-Simons theories]]

[[!redirects 1d Chern-Simons theory]]
[[!redirects 1d Chern-Simons theories]]


1-functor

As a $1$-[[1-category|category]] is simply a [[category]], so a __$1$-functor__ is simply a [[functor]].  See also $n$-[[n-functor|functor]].

[[!redirects 1-functor]]
[[!redirects 1-functors]]


1-groupoid

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

#Contents#
* table of contents
{:toc}

## Idea

By _1-groupoid_ one means -- for emphasis -- a _[[groupoid]]_ regarded as an _[[∞-groupoid]]_.

A quick way to make this precise is to says that a 1-groupoid is a [[Kan complex]] which is [[equivalence|equivalent]] ([[homotopy equivalence|homotopy equivalent]]) to the [[nerve]] of a groupoid: a [[simplicial coskeleton|2-coskeletal]] Kan complex. More abstractly this is a [[truncated object of an (∞,1)-category|1-truncated]] [[∞-groupoid]].

More generally and more vaguely: Fix a meaning of $\infty$-[[infinity-groupoid|groupoid]], however weak or strict you wish. Then a __$1$-groupoid__ is an $\infty$-groupoid such that all [[parallel pairs]] of $j$-morphisms are [[equivalence|equivalent]] for $j \geq 2$. Thus, up to equivalence, there is no point in mentioning anything beyond $1$-morphisms, except whether two given parallel $1$-morphisms are equivalent. If you rephrase equivalence of $1$-morphisms as [[equality]], which gives the same result up to [[equivalence of categories|equivalence]], then all that is left in this definition is a [[groupoid]].  Thus one may also say that a __$1$-groupoid__ is simply a groupoid.

The point of all this is simply to fill in the general concept of $n$-[[n-groupoid|groupoid]]; nobody thinks of $1$-groupoids as a concept in their own right except simply as groupoids.  Compare $1$-[[1-category|category]] and $1$-[[1-poset|poset]], which are defined on the same basis.

## Related concepts

[[!include homotopy n-types - table]]

[[!redirects 1-groupoids]]


1-morphism

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

* [[object]]

* [[morphism]] / **1-morphism**

  [[horizontal morphism]], [[vertical morphism]]

* [[2-morphism]]

* [[3-morphism]]

* [[k-morphism]]

***

A **1-morphism** is a [[morphism]] in an [[n-category]]. One says 1-morphism for emphasis as a special case of [[k-morphism]].

[[!redirects 1-morphism]]
[[!redirects 1-morphisms]]


1-poset

Fix a meaning of $\infty$-[[infinity-category|category]], however weak or strict you wish. Then a __$1$-poset__ is an $\infty$-category such that every $2$-morphism is an [[equivalence]] and all [[parallel pair]]s of $j$-morphisms are equivalent for $j \geq 1$. Thus, up to equivalence, there is no point in mentioning anything beyond $1$-morphisms, not even whether two given parallel $1$-morphisms are equivalent. Up to [[equivalence of categories|equivalence]], therefore, all that is left in this definition is a [[partial order|poset]].  Thus one may also say that a __$1$-poset__ is simply a poset.

The point of all this is simply to fill in the general concept of $n$-[[n-poset|poset]]; nobody thinks of $1$-posets as a concept in their own right except simply as posets.  Compare $1$-[[1-category|category]] and $1$-[[1-groupoid|groupoid]], which are defined on the same basis.


1-topos

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

#Contents#
* table of contents
{:toc}

## Idea

A __$1$-topos__, or __$(1,1)$-topos__, is simply a [[topos]] in the usual sense of the word.  The prefix $1$- may be added when also discussing [[higher category theory|higher categorical]] types of topoi in [[higher topos theory]] such as [[2-topos]], $(\infty,1)$-[[(infinity,1)-topoi|topos]], or even $\infty$-[[infinity-topos|topos]].  Compare that a [[(0,1)-topos]] is a [[Heyting algebra]].

Similarly, a __Grothendieck $1$-topos__, or __Grothendieck $(1,1)$-topos__, is simply a [[Grothendieck topos]].  Compare that a [[Grothendieck (0,1)-topos]] is a [[frame]] (or [[locale]]).

Note that a 1-topos is not exactly a particular sort of [[2-topos]] or $\infty$-topos, just as a [[Heyting algebra]] is not a particular sort of 1-topos.  The (1,2)-category of locales (i.e. (0,1)-topoi) embeds fully in the 2-category of Grothendieck 1-topoi by taking sheaves, but a locale is not identical to its topos of sheaves (and in fact no nontrivial 1-topos can be a poset), in that the following [[diagram]] of [[functors]] can *not* be filled by a [[natural isomorphism]]:

\begin{tikzcd}
  \mathrm{Top}_{(0,1)}
  \ar[rr, "\mathrm{Sh}(-)"]
  \ar[d, "{\mathrm{forget}}"{pos=.9,yshift=5pt,sloped}]
  \ar[drr, phantom, "{\#}"]
  &&
  \mathrm{Top}_{(1,1)}
  \ar[d, "{\mathrm{forget}}"{pos=.9,yshift=5pt,sloped}]
  \\
  \mathrm{Cat}_{(0,1)}
  \ar[rr, hook]
  &&
  \mathrm{Cat}_{(1,1)}
\end{tikzcd}

Likewise, one expects every Grothendieck 1-topos to give rise to a 2-topos or $\infty$-topos of [[stacks]], hopefully producing a full embedding of some sort.

## Related concepts

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

[[!redirects (1,1)-topos]]
[[!redirects Grothendeick 1-topos]]
[[!redirects Grothendeick (1,1)-topos]]

[[!redirects 1-toposes]]
[[!redirects 1-topoi]]


10d supergravity

[[supergravity]] in [[dimension]] 10.

* [[type II supergravity]], [[double field theory]]

* [[heterotic supergravity]]


10j symbol

In the context of fixed [[triangulations]] of spacetime arising in [[spin foam models]], the [[10j symbol]] is a spin network that arises in the partition function for the [[Barrett-Crane model]] of [[Riemannian quantum gravity]]. [[Greg Egan]], [[J. Daniel Christensen]], and [[John Baez]] formulated an algorithm in C++ to describe these symbols that arise in this discrete [[spin foam model]].

##Definition

A [[10j symbol]] is a spin(4) [[spin network]] containing five vertices and ten edges, where the edges are labels by [[spins]]. This symbol can be understood as a function which receives ten spins as input and produces a complex number as output, this process underpins the calculation of the partition function for the [[Barrett-Crane model]], which depends on the contraction of tensors at the vertices.

##References

* J. Daniel Christensen and Greg Egan, _An  Efficient  Algorithm  for the  Riemannian 10j Symbols._ [arxiv:gr-qc/0110045v3](http://arxiv.org/abs/gr-qc/0110045v3)

* Baez, John C., J. Daniel Christensen, and Greg Egan. _Asymptotics of 10j Symbols_ Classical and Quantum Gravity 19, no. 24 (December 21, 2002): 6489–6513. [https://doi.org/10.1088/0264-9381/19/24/315.](https://arxiv.org/abs/gr-qc/0208010)


11-dimensional Chern-Simons theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The most basic version of [[higher dimensional Chern-Simons theory]] over a ([[compact topological space|compact]]) [[smooth manifold]] $X$ of [[dimension]] 11 has as [[field (physics)|fields]] [[cocycles]] $\hat D \colon X \to \mathbf{B}^5 U(1)_{conn}$ in degree-5 [[ordinary cohomology]] and whose [[action functional]] is given by the [[fiber integration in ordinary differential cohomology]] of the [[cup product in ordinary differential cohomology]] of such a field with itself:

$$
  \hat D \mapsto \exp\left(\tfrac{i}{\hbar}\int_X \hat D \cup \hat D\right)
  \,.
$$

This is the direct generalization of $U(1)$-[[3d Chern-Simons theory]] of the abelian [[7d Chern-Simons theory]], and all three are related by [[holography]] to the [[self-dual higher gauge field]] in dimension 2,6, and 10, respectively.

However, for applications in [[string theory]] more refined versions of these theories matter. For instance in 7d  the full [[6d (2,0)-superconformal QFT]] contains not just a single abelian [[higher self-dual gauge field]] and accordingly the corresponding [[7d Chern-Simons theory]] is richer, namely is, by [AdS7/CFT6](#AdS-CFT#AdS7CFT6), the [[KK-compactification]] of [[11-dimensional supergravity]] on $S^4$. Similarly, in 10-dimensions the [[RR-field]] of [[type II superstring theory]] is a [[higher self-dual gauge field]] whose quantization law is of the form that makes it qualify ([Moore-Witten 99](#MooreWitten99)) as the holographic boundary theory of an 11d Chern-Simons theory. However, as a configuration of the [[RR-field]] is a [[cocycle]] in [[twisted differential K-theory]], so there should be an 11d Chern-Simons theory given ([Belov-Moore 06](#BelovMoore06)) by the [[fiber integration in differential cohomology]] of the [[cup product in differential cohomology]] in K-theory.

## Related concepts

* [[11d supergravity]]

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

  * [[2d Chern-Simons theory]]

  * [[3d Chern-Simons theory]]

  * [[4d Chern-Simons theory]]

  * [[5d Chern-Simons theory]]

  * [[6d Chern-Simons theory]]

  * [[7d Chern-Simons theory]]

## References

The [[self-dual higher gauge field]] nature (see there for more) in terms of a [[quadratic form]] on [[differential K-theory]] is discussed originally around

* {#MooreWitten99} [[Gregory Moore]], [[Edward Witten]], _Self-Duality, Ramond-Ramond Fields, and K-Theory_, JHEP 0005:032 (2000) [[arXiv:hep-th/9912279](http://arxiv.org/abs/hep-th/9912279), [doi:10.1088/1126-6708/2000/05/032](https://doi.org/10.1088/1126-6708/2000/05/032)]

and ([Freed 00](#Freed00)) for [[type I superstring theory]], and for [[type II superstring theory]] in

* {#Wittem} [[Edward Witten]], _Duality Relations Among Topological Effects In String Theory_, JHEP 0005:031,2000 ([arXiv:hep-th/9912086](http://arxiv.org/abs/hep-th/9912086))

* {#FreedHopkins00} [[Daniel Freed]], [[Michael Hopkins]], _On Ramond-Ramond fields and K-theory_, JHEP 0005 (2000) 044 ([arXiv:hep-th/0002027](http://arxiv.org/abs/hep-th/0002027))

* {#FMW00} D. Diaconescu, [[Gregory Moore]], [[Edward Witten]], _$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory_, Adv.Theor.Math.Phys.6:1031-1134,2003 ([arXiv:hep-th/0005090](http://arxiv.org/abs/hep-th/0005090)), summarised in _A Derivation of K-Theory from M-Theory_ ([arXiv:hep-th/0005091](http://arxiv.org/abs/hep-th/0005091))

with more refined discussion in [[twisted differential K-theory|twisted differential]] [[KR-theory]] in

* {#DFM09} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))

See at _[[orientifold]]_ for more on this. The relation to [[11d Chern-Simons theory]] is made manifest in

* {#BelovMooreII} Dmitriy Belov, [[Greg Moore]], _Type II Actions from 11-Dimensional Chern-Simons Theories_ ([arXiv:hep-th/0611020](http://arxiv.org/abs/hep-th/0611020))

Review is in

* {#Szabo12} [[Richard Szabo]], section 3.6 and 4.6 of _Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology_ ([arXiv:1209.2530](http://arxiv.org/abs/1209.2530))

[[!redirects 11d Chern-Simons theory]]


11d supergravity Lie 3-algebra

[[!redirects supergravity Lie 3-algebra]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The _supergravity Lie 3-algebra_ $\mathfrak{sugra}(10,1)$ or [[M2-brane]] extension $\mathfrak{m}2\mathfrak{brane}$ is a [[super L-∞ algebra]]  that is a shifted [[∞-Lie algebra cohomology|extension]]

$$
  0 \to b^2 \mathbb{R} \to
  \mathfrak{sugra}(10,1) \to
  \mathfrak{siso}(10,1)
  \to
  0
$$

of the [[super Poincare Lie algebra]] $\mathfrak{siso}(10,1)$ in 10+1 dimensions induced by the exceptional degree 4-[[Lie algebra cohomology|super Lie algebra cocycle]]

$$
  \mu_4 = \bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b
  \;\;
  \in CE(\mathfrak{siso}(10,1))
  \,.
$$

This is the same mechanism by which the [[String Lie 2-algebra]] is a shifted central extension of $\mathfrak{so}(n)$.

## Properties

### The Chevalley-Eilenberg algebra

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

The [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{sugra}(10,1))$ is generated on

* elements $\{e^a\}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$

* a single element $c$ of degree $(3,even)$

* and elements $\{\psi^\alpha\}$ of degree $(1,odd)$

with the differential defined by

$$
  d_{CE} \, \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}
$$

$$
  d_{CE} \, e^{a } = \omega^a{}_b \wedge e^b + \frac{i}{2}\bar \psi \Gamma^a \psi
$$

$$
  d_{CE} \, \psi = \frac{1}{4} \omega^{ a b} \Gamma_{a b} \psi
$$

$$
  d_{CE} \, c = \frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b
  \,.
$$

=--

> (fill in details)

### Hidden super Lie 1-algebra

At the end of ([D'Auria-Fre 82](#DAuriaFr82)) the authors ask for a super Lie 1-algebra $\mathfrak{g}$, equipped with a degree-3 element $A$ in its [[Chevalley-Eilenberg algebra]], and equipped with a homomorphism $p\colon \mathfrak{g}\longrightarrow \mathbb{R}^{10,1\vert \mathbf{32}}$ such that the pullback of the 4-cocycle $\mu_4$ along $p$ is trivialized by $A$:

$$
  p^\ast \mu_4 = d_{CE}A
  \,.
$$

In the [[model structure on L-infinity algebras|homotopy theory of L-infinity algebra]] this means that

$$
  \array{
     \mathfrak{g} &\longrightarrow& \ast
     \\
     {}^{\mathllap{p}}
     \downarrow
       &\swArrow_{\mathrlap{A}}&
     \downarrow
     \\
     \mathbb{R}^{10,1\vert \mathbf{32}}
       &\underset{\mu_4}{\longrightarrow}&
     \mathbf{B}^3 \mathbb{R}
  }
  \,.
$$

Compare this to the characterization of $\mathfrak{sugra}(10,1)$ as the [[homotopy fiber]] of $\mu_4$, hence as the _universal_ solution to this situation

$$
  \array{
     \mathfrak{sugra}(10,1) &\longrightarrow& \ast
     \\
     \downarrow
       &\swArrow_{}&
     \downarrow
     \\
     \mathbb{R}^{10,1\vert \mathbf{32}}
       &\underset{\mu_4}{\longrightarrow}&
     \mathbf{B}^3 \mathbb{R}
  }
  \,.
$$

In any case, in ([D'Auria-Fre 82](#DAuriaFr82)) possible choices for $p \colon \mathfrak{g} \to \mathbb{R}^{10,1\vert\mathbf{32}}$ are found.

Curiously, the bosonic [[body]] of $\mathfrak{g}$ is such that when adapted to a compactification to 4d, then it is the [[exceptional tangent bundle]] on which the [[U-duality]] group [[E7]] has a canonical action.

In ([BAIPV 04](#BAIPV04)) these solutions are shown to extend to a 1-parameter family of solutions. Further comments are in ([Andrianopoli-D'Auria-Ravera 16](#AndrianopoliDAuriaRavera16)).

### Relation to M5-brane action functional

The supergravity Lie 3-algebra carries a 7-cocycle (the one that induces the [[supergravity Lie 6-algebra]]-extension of it). The corresponding WZW term is that of the [[M5-brane]] in its [[Green-Schwarz action functional]]-like formulation.

[[!include brane scan]]

### Relation to the 11-dimensional polyvector super Poincaré-algebra
 {#Polyvector}

#### Via derivations

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

Let $\mathfrak{der}(\mathfrak{sugra}(10,1))$ be the [[automorphism ∞-Lie algebra]] of $\mathfrak{sugra}(10,1)$. This is a [[dg-Lie algebra]]. Write
$\mathfrak{der}(\mathfrak{sugra}(10,1))_0$ for the ordinary [[Lie algebra]] in degree 0.

This is [[isomorphic]] to the polyvector-extension of the [[super Poincaré Lie algebra]] (see there) in $d = 10+1$ -- the
"[[M-theory super Lie algebra]]" -- with "2-brane central charge": the Lie algebra spanned by generators
$\{P_a, Q_\alpha, J_{a b}, Z^{a b}\}$ and graded Lie brackets those of the [[super Poincaré Lie algebra]] as well as

$$
  [Q_\alpha, Q_\beta] = i (C \Gamma^a)_{\alpha \beta} P_a + (C \Gamma_{a b})Z^{a b}
$$

$$
  [Q_\alpha, Z^{a b}] = 2 i (C \Gamma^{[a})_{\alpha \beta}Q^{b]\beta}
$$

etc.

=--

This observation appears implicitly in ([Castellani 05, section 3.1](#Castellani05)), see ([FSS 13](#FSS13)).

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

With the presentation of the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{sugra}(10,1))$ as in prop. \ref{TheCEAlgebra} above, the generators are identified with [[derivation]]s on $CE(\mathfrak{sugra}(10,1))$ as

$$
  P_a = [d_{CE}, \frac{\partial}{\partial e^a} ]
$$

and

$$
  Q_\alpha = [d_{CE}, \frac{\partial}{\partial \psi^\alpha} ]
$$

and

$$
  J_{a b} = [d_{CE}, \frac{\partial}{\partial \omega^{a b}} ]
$$

and

$$
  Z^{a b} = [d_{CE}, e^a \wedge e^b \wedge \frac{\partial}{\partial c}]
$$

etc. With this it is straightforward to compute the commutators.
Notably the last term in

$$
  [Q_\alpha, Q_\beta] = i (C \Gamma^a)_{\alpha \beta} P_a + (C \Gamma_{a b})Z^{a b}
$$

arises from the contraction of the 4-cocycle $\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b$ with $\frac{\partial}{\partial \psi^\alpha}\wedge \frac{\partial}{\partial \psi^\beta}$.

=--

#### Via the Heisenberg Lie 3-algebras

(...)

## Applications

The field configurations of 11-dimensional [[supergravity]] may be identified with [[∞-Lie algebra-valued forms]] with values in $\mathfrak{sugra}(10,1)$.
See [[D'Auria-Fre formulation of supergravity]].

## Related concepts

[[supergravity Lie 6-algebra]] $\to$ **supergravity Lie 3-algebra** $\to$ [[super Poincaré Lie algebra]]

* [[4d supergravity Lie 2-algebra]]

* [[extended supersymmetry]]

* [[type II supergravity Lie 2-algebra]]

* [[type II supersymmetry algebra]]

* [[M-theory supersymmetry algebra]]

## References

The [[Chevalley-Eilenberg algebra]] of $\mathfrak{sugra}(10,1)$ first appears in (3.15) of

* {#DAuriaFr82} [[Riccardo D'Auria]], [[Pietro Fré]], _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982)

and later in the textbook

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], _[[Supergravity and Superstrings - A Geometric Perspective]]_

The manifest interpretation of this as a [[Lie 3-algebra]] and the supergravity field content as [[∞-Lie algebra valued forms]] with values in this is mentioned in

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:L-∞ algebra connections]]_

* {#FSS13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_

A systematic study of the super-[[Lie algebra cohomology]] involved is in

* [[John Baez]], [[John Huerta]], _Division algebras and supersymmetry I_ ([arXiv:0909.0551](http://arxiv.org/abs/0909.0551))

* [[John Baez]], [[John Huerta]], _Division algebras and supersymmetry II_ ([arXiv:1003.34360](http://arxiv.org/abs/1003.3436))

See also [[division algebra and supersymmetry]].

Further discussion of its "hidden" super Lie algebra includes

* {#BAIPV04} [[Igor Bandos]], [[José de Azcárraga]], [[José M. Izquierdo]], Moises Picon, Oscar Varela, _On the underlying gauge group structure of $D=11$ supergravity_, Phys. Lett. B **596** (2004) 145-155 [[arXiv;hep-th/0406020](http://arxiv.org/abs/hep-th/0406020)]

* [[Igor Bandos]], [[Jose de Azcarraga]], Moises Picon, Oscar Varela, _On the formulation of $D=11$ supergravity and the composite nature of its three-from field_, Annals Phys. **317** (2005) 238-279 [[arXiv:hep-th/0409100](https://arxiv.org/abs/hep-th/0409100)]

* {#AndrianopoliDAuriaRavera16} [[Laura Andrianopoli]], [[Riccardo D'Auria]], [[Lucrezia Ravera]], _Hidden Gauge Structure of Supersymmetric Free Differential Algebras_ [[arXiv:1606.07328](https://arxiv.org/abs/1606.07328)]

Further review:

* [[Lucrezia Ravera]], *On the hidden symmetries of $D=11$ supergravity* [[arXiv:2112.00445](https://arxiv.org/abs/2112.00445)]

The computation of the automorphism Lie algebra of $\mathfrak{sugra}(10,1)$ is in

* {#Castellani05} [[Leonardo Castellani]], _Lie derivatives along antisymmetric tensors and the M-theory superalgebra_, J. Phys. Math. **3** (2011) 1-7 [[arXiv:hep-th/0508213](http://arxiv.org/abs/hep-th/0508213)]

A similar argument with more explicit use of the Lie 3-algebra as underlying the [[Green-Schwarz action functional|Green-Schwarz-like action functional]] for the [[M5-brane]] is in

* [[Dmitri Sorokin]], [[Paul Townsend]], _M-theory superalgebra from the M-5-brane_, Phys. Lett. B **412** (1997) 265-273 [[arXiv:hep-th/9708003](http://arxiv.org/abs/hep-th/9708003)]

[[!redirects m2brane]]


120-cell

#Contents#
* table of contents
{:toc}

## Idea

...one of the [[regular polytopes]] in [[dimension]] 4...

...hence a higher dimensional analogs of the [[Platonic solids]]...

\begin{centre}
  \begin{imagefromfile}
    "file_name": "Chilton120Cell.jpg",
    "width": 360
  \end{imagefromfile}
\end{centre}

> graphics grabbed from [Stillwell 01](#Stillwell01)

## Related concepts

* [[24-cell]]

* [[600-cell]]

## References

* {#Stillwell01} John Stillwell, _The story of the 120-Cell_, Notices of the American Mathematical Society 2001 ([pdf](https://www.ams.org/notices/200101/fea-stillwell.pdf))

See also

* Wikipedia, _[120-cell](https://en.wikipedia.org/wiki/120-cell)_


13-sphere

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

#Contents#
* table of contents
{:toc}

## Idea

The [[n-sphere]] of [[dimension]] $n = 13$.

## Properties

### AKM-Theorem

In higher-dimensional analogy to the [[AKM-theorem]] we have:

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

The [[13-sphere]] is the [[quotient space]] of the (right-)[[octonionic projective plane]] by the left multiplication [[action]] by [[Sp(1)]]:

$$
  \mathbb{O}P^2 / \mathrm{Sp}(1)
  \simeq
  S^{13}
$$

=--

([Atiyah-Berndt 02](#AtiyahBerndt02))

## References

* {#AtiyahBerndt02} [[Michael Atiyah]], [[Jürgen Berndt]], in:  Surv. Differ. Geom. VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck (International Press, Somerville, MA, 2003) 1-27 ([arXiv:math/0206135](https://arxiv.org/abs/math/0206135), [doi:10.4310/SDG.2003.v8.n1.a1](https://dx.doi.org/10.4310/SDG.2003.v8.n1.a1))


15-sphere

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

#Contents#
* table of contents
{:toc}

## Idea

The [[sphere]] of [[dimension]] 15.

## Properties

### Octonionic Hopf fibration
 {#OctonionicHopfFibration}

The 15-sphere participates in the [[octonionic Hopf fibration]], the analog of the complex [[Hopf fibration]] with the field of [[complex numbers]] replaced by the division ring of [[octonions]] $\mathbb{O}$.

$$
  \array{
    S^7 &\hookrightarrow& S^15
    \\
    && \downarrow^\mathrlap{p}
    \\
    && S^8
  }
$$

Here the idea is that $S^{15}$ can be construed as $\{(x, y) \in \mathbb{O}^2: {|x|}^2 + {|y|}^2 = 1\}$, with $p$ mapping $(x, y)$ to $x/y$ as an element in the [[projective line]] $\mathbb{P}^1(\mathbb{O}) \cong S^8$, with each [[fiber]] a [[torsor]] parametrized by octonionic [[scalars]] $\lambda$ of unit [[norm]] (so $\lambda \in S^7$).

### Other properties

* $S^{15}$ is the only sphere that admits three homogeneous Einstein metrics.
* It is the only sphere that appears as a regular orbit in three cohomogeneity one actions on projective spaces, namely of $SU(8)$, $Sp(4)$ and $Spin(9)$ on $\mathbb{C}P^8$, $\mathbb{H}P^4$ and $\mathbb{O}P^2$, respectively ([OPPV, p. 1](#OPPV))

## References

* {#OPPV} Liviu Ornea, Maurizio Parton, Paolo Piccinni, Victor Vuletescu, _Spin(9) geometry of the octonionic Hopf fibration_, ([arXiv:1208.0899](http://arxiv.org/abs/1208.0899))


1d Dijkgraaf-Witten theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

## Idea

The 1-dimensional analog of 3d [[Dijkgraaf-Witten theory]].

For the moment a detailed discussion is at _[[local prequantum field theory]]_ in the section _[1d Dijkgraaf-Witten theory](prequantum+field+theory#1dDWTheory)_.

[[!redirects 1-dimensional Dijkgraaf-Witten theory]]


1d WZW model

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Wess-Zumino-Witten theory
+--{: .hide}
[[!include infinity-Wess-Zumino-Witten theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[infinity-Wess-Zumino-Witten theory - contents|higher dimensional WZW models]] the following 1-dimensional [[sigma-models]] are seen to be examples:

* the free [[non-relativistic particle]];

* the massive [[Green-Schwarz action functional|Green-Schwarz]] [[superparticle]] (the [[D0-brane|D0]] [[super p-brane]]).

([Azcarraga-Izqierdo 95, section 8.3 and 8.7](#AzcarragaIzqierdo)) .

## Examples

### Free massive non-relativistic particle

Write

$$
  H \coloneqq G/R
$$

for the [[coset]] obtained as the [[quotient]] of the [[Galilei group]] in some [[dimension]] $d$ by the [[rotation group|group of]] [[rotations]]. This $H$ has a canonical global [[coordinate chart]] $(t,x, \dot x)$. We may regard it as the first order [[jet bundle]] to the [[bundle]] $\mathbb{R}^d \times \mathbb{R} \to \mathbb{R}$ whose [[sections]] are [[trajectories]] in [[Cartesian space]] $\mathbb{R}^d$ (the [[field bundle]] for the 1d [[sigma-model]] with [[target space]] $\mathbb{R}^d$).

Among the $H$-[[left invariant differential 2-forms]] on $H$ is

$$
  \omega_m \coloneqq m (d_{dR} x - \dot x d_{dR} t) \wedge d_{dR} \dot x
$$

for some $m \in \mathbb{R}$ (where a contraction of vectors is understood).

This is a representative of a degree-2 [[cocycle]] in the [[Lie algebra cohomology]] of $Lie(H)$. We may regard this as the [[curvature]] of a [[connection on a bundle|connection]] [[differential 1-form|1-form]]

$$
  A \coloneqq m \dot x (d_{dR} x - \frac{1}{2} \dot x d_{dR} t)
  \,.
$$

Hence the value of the [[action functional]] of the corresponding 1d pure (topological) WZW model on a field configuration is

$$
   m \int_{\Sigma_1} \dot  x (d_{dR} x - \frac{1}{2} \dot x d_{dR} t)
   =
   m \int_{\Sigma_1} \frac{\partial L}{ \partial \dot x}(d x - \dot x)+ L d t
  \,,
$$

where $L(x, \dot x) d t = \frac{1}{2}m \dot x^2 d t$ is the [[Lagrangian]] of the the free [[non-relativistic particle]] of [[mass]] $m$.

Evaluated on [[jet prolongations]] of [[sections]] of the [[field bundle]], for which the relation $d x = \dot x d t$ holds, then the first term of this expression vanishes and so the resulting WZW-type [[action functional]] is that of the free [[non-relativistic particle]].

See ([Azcarraga-Izqierdo, section 8.3](#AzcarragaIzqierdo)) for a useful account.

## References

* {#AzcarragaIzqierdo} [[José de Azcárraga]], J. Izqierdo, sections 8.3 and 8.7 of  _[[Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics]]_, Cambridge monographs of mathematical physics, (1995)

[[!redirects 1d WZW models]]


1lab

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

\tableofcontents

## Idea

The **1lab** is an online reference maintained by [[Amélia Liao]] primarily for [[category theory]] done in [[univalent type theory|univalent]] [[cubical type theory]] implemented in [[cubical Agda]].

Web:

* [1lab website](https://1lab.dev/)

* [Github page for the 1lab](https://github.com/plt-amy/1lab)

The name "1lab" is apparently a reference to (the name of) the "[[nLab|$n$Lab]]", with the specification $n = 1$ meant to highlight this focus on [[univalent category|type-theoretic]] [[1-categories]] (as opposed to [[higher category theory|higher categories]], whose type-theoretic formulation remains underdeveloped, generally).

## Related entries

[[!include proof assistants and formalization projects -- list]]


1T relation

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

#Contents#
* table of contents
{:toc}

## Idea

In [[knot theory]], by the _1-term relations_, or _1T relations_ for short, one means the [[relation]] on the [[linear span]] of the [[set]] of [[chord diagrams]] which relates every chord diagram with an "isolated chord" (one not intersecting any other chord in the diagram) to [[zero]].

The [[quotient space]] of the [[linear span]] of [[chord diagrams]] by the 1T and by the [[4T relations]] is the [[domain]] for _[[unframed weight systems]]_ on chord diagrams. If one omits the 1T relation and imposes only the [[4T relation]], one speaks of _[[framed weight systems]]_.

## Related concepts

[[!include chord diagrams and weight systems -- table]]

## References

Original articles

* {#BarNatan95} [[Dror Bar-Natan]], _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))

  (for [[horizontal chord diagrams]])

Textbook accounts

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Section 4 of:  _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv/1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))

* [[David Jackson]], [[Iain Moffat]], Section 11 of: _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))

Lecture notes:

* {#BarNatanStoimenow97} [[Dror Bar-Natan]], Alexander Stoimenow, _The Fundamental Theorem of Vassiliev Invariants_ ([arXiv:q-alg/9702009](https://arxiv.org/abs/q-alg/9702009))

[[!redirects 1T relations]]

[[!redirects 1T-relation]]
[[!redirects 1T-relations]]

[[!redirects 1-term relation]]
[[!redirects 1-term relations]]


2-adjunction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **2-adjunction** is a common name for various kinds of adjunctions in [[2-category theory]]; not only adjunctions between [[2-categories]] themselves, but more generally adjunctions within an arbitrary [[3-category]].  Since there are various different levels of strictness at which one works in 2-category theory, there are various different kinds of 2-adjunction, including:

* A [[strict 2-adjunction]], involving [[strict 2-categories]], [[strict 2-functors]], [[strict 2-natural transformations]], and on-the-nose [[triangle identities]].  This can be generalized to an arbitrary [[strict 3-category]].
* A [[biadjunction]], involving [[weak 2-categories]] ([[bicategories]]), weak 2-functors ([[pseudofunctors]]), [[pseudonatural transformations]], and triangle identities up to isomorphism.  This can be generalized to an arbitrary [[tricategory]].
* The word [[pseudoadjunction]] is sometimes used interchangeably with "biadjunction", but often it refers specifically to a notion definable internal to a [[Gray-category]].  When specialized to the canonical Gray-category $Gray$, this produces a notion involving strict 2-categories and strict 2-functors, but *pseudo* natural transformations, and triangle identities up to isomorphism.
* A [[lax 2-adjunction]] involves triangle identities only up to noninvertible transformation, and perhaps [[lax 2-functors]] and/or [[lax 2-natural transformations]] as well.  If only the triangle identities are lax, this can be defined at any of the above levels of strictness and internalized in any of the above ways; but if the functors or transformations are lax, then it doesn't fit very easily into a well-known abstract 3-categorical structure, since there is no 3-category (as usually understood) including lax functors or transformations.

There are also more specialized kinds of 2-adjunction, such as

* An [[idempotent 2-adjunction]] is a categorification of an [[idempotent adjunction]].
* A [[lax-idempotent 2-adjunction]] is a weaker sort of categorification, relevant to [[lax-idempotent 2-monads]].

## Related concepts

* [[adjoint functor]], [[adjoint triple]], [[adjoint quadruple]]

* [[proadjoint]], [[Hopf adjunction]]

* **2-adjunction**

  [[biadjunction]], [[pseudoadjunction]], [[lax 2-adjunction]]

  [[strict adjoint 2-functor]]

  [[lax-idempotent 2-adjunction]]

* [[adjoint (infinity,1)-functor]]

* [[(∞,n)-category with adjoints]]

[[!redirects 2-adjointness]]
[[!redirects 2-adjunctions]]

[[!redirects adjoint 2-functor]]
[[!redirects 2-adjoint]]
[[!redirects 2-adjoints]]
[[!redirects left 2-adjoint]]
[[!redirects left 2-adjoints]]
[[!redirects right 2-adjoint]]
[[!redirects right 2-adjoints]]