stratified category

# Contents
* table of contents
{: toc}

##Idea

The term *stratified category* is used to refer to a [[stable (∞, 1)-category]] which is stratified along a [[poset]] ([Glasman 15, Def. 3.5](#Glasman15)).

## Related concepts

* [[poset-stratified space]]

* [[stratified space]]

* [[stratifolds]]

## References

* {#Glasman15} [[Saul Glasman]], _Stratified categories, geometric fixed points and a generalized Arone-Ching theorem_, ([arXiv:1507.01976](https://arxiv.org/abs/1507.01976))

* [[David Ayala]], [[Aaron Mazel-Gee]], [[Nick Rozenblyum]], _Stratified noncommutative geometry_, ([arXiv:1910.14602](https://arxiv.org/abs/1910.14602))

* Grigory Kondyrev, [[Aaron Mazel-Gee]], [[Jay Shah]], _Dualizable objects in stratified categories and the 1-dimensional bordism hypothesis for recollements_, ([arXiv:2103.15785](https://arxiv.org/abs/2103.15785))


stratified homotopy hypothesis

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#### Homotopy theory
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#### Higher category theory
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#Contents#
* table of contents
{:toc}

## Idea

Where the [[homotopy hypothesis]] is the assertion that [[∞-groupoids]] are equivalent to [[topological spaces]] (considered modulo [[weak homotopy equivalence]]), in [AFR15](#AFR15) the authors construct an equivalence for [[(∞,1)-categories]]. They do so in terms of what are called **striation sheaves**, which are sheaves on conically smooth [[poset-stratified space|stratified spaces]] satisfying a certain [[descent]] condition.

The construction relies on a fully faithful embedding of conically smooth stratified spaces, and conically smooth maps among them, into $(\infty, 1)$-categories via the exit-path functor, which maps stratified spaces to paths within them that once they leave a stratum do not re-enter it.

In [Haine18](#Haine18) the author presents a completely self-contained proof of a precise form of the stratified homotopy hypothesis.

##References

* {#AFR15} [[David Ayala]], [[John Francis]], [[Nick Rozenblyum]], *A stratified homotopy hypothesis*, [arXiv](http://arxiv.org/abs/1502.01713)

* {#Haine18} [[Peter J. Haine]], _On the homotopy theory of stratified spaces_ ([arXiv:1811.01119](https://arxiv.org/abs/1811.01119))


stratified simplicial set

#Idea#

A _stratified simplicial set_ is a [[simplicial set]] equipped with information about which of its [[simplex|simplices]] are to be regarded as being _[[thin element|thin]]_ in that they are like identities or at least like equivalences in a [[higher category theory|higher category]].

The theory of [[simplicial weak ∞-categories]] is based on stratified simplicial sets.

# Definition #

A **stratification** of a [[simplicial set]] $X : \Delta^{op} \to Set$ is a subset $t X \subset \coprod_{[n]} X_n$ of its set of simplices (not in general a simplicial subset!) such that

* no 0-simplex of $X$ is in $t X$;

* every degenerate simplex in $X$ is in $t X$.

A **stratified simplicial set** is a pair $(X, t X)$ consisting of a simplicial set $X$ and a stratification $t X$ of $X$.

The elements of $t X$ are called the **thin** simplices of $X$.

For $(X, t X)$ and $(Y, t Y)$ stratified simplicial sets, a morphism $f : X \to Y$ of simplicial sets is set to be a **stratified map** if it respects thin cells in that

$$
  f(t X ) \subset t Y
  \,.
$$

The [[category]] of stratified simplicial sets and stratified maps between them is usually denoted $Strat$.

This category is a [[quasitopos]].  Hence, in particular, it is [[cartesian closed category|cartesian closed]].

#Examples#

* Every simplicial set gives rise to a stratified simplicial set

  * using the **maximal stratification**: all simplices of dimension >0 are regarded as thin;

  * using the **minimal stratification**: only degenerate simplices are thin.

  These two stratifications give [[left adjoint|left]] and [[right adjoints]] to the [[forgetful functor]] from stratified simplicial sets to simplicial sets.

* The **standard thin** $n$-**simplex** is obtained from $\Delta[n]$ by making its only non-degenerate $n$-simplex thin.

* The $k$th **standard admissible** $n$-**simplex** $\Delta^a_k[n]$, defined for $n \geq 2$, $0 \lt k \lt n$, is obtained from $\Delta[n]$ by making all simplices $\alpha \colon [m] \to [n]$ with $k-1,k,k+1 \in$ im$(\alpha)$ thin.

* The **standard admissible** $(n-1)$-**dimensional** $k$-**horn** $\Lambda^a_k[n]$, defined for $n \geq 2$, $0 \lt k \lt n$, is the pullback of the stratified simplicial set $\Delta^a_k[n]$.

* A [[complicial set]] is a stratified simplicial set satisfying certain extra conditions. Complicial sets are precisely those simplicial sets which arise (up to isomorphism) as the [[oriental|∞-nerve]] $N(C)$ of a [[strict ∞-category]] $C$, where the thin cells are the images of the identity cells of $C$.

* A [[simplicial set]] is a [[Kan complex]] precisely if its maximal stratification makes it a [[weak complicial set]].

#The category of stratified simplicial sets#

There are several [[tensor products]] on the category $Strat$ of stratified simplicial sets that make it a [[monoidal category]].

## Strat with the Verity-Gray tensor product ##

Consider the monoidal category $(Strat, \otimes)$ where $\otimes$ is the [[Verity-Gray tensor product]].

(Notice that this is not [[closed monoidal category|closed]], as far as I understand.)

Using the canonical stratification of [[oriental|∞-nerves]] on [[strict ∞-categories]] as [[complicial sets]], the $\omega$-nerve is a functor

$$
  N : Str \omega Cat \to Strat
  \,.
$$

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###### Proposition
The functor $N : Str \omega Cat \to Strat$ has a [[left adjoint]] $F : Strat \to Str \omega Cat$ which is a [[strong monoidal functor]].
=--

Or so it is claimed on [slide 60](http://www.mat.uc.pt/~categ/ct2007/slides/verity.pdf#page=60) of
[Ver07](http://www.mat.uc.pt/~categ/ct2007/slides/verity.pdf)

#References#

A useful quick introduction is the beginning of these slides:

* [[Dominic Verity]], _Weak Complicial sets and internal quasi-categories_ ([pdf slides](http://www.mat.uc.pt/~categ/ct2007/slides/verity.pdf))

[[!redirects stratified simplicial sets]]


stratified space

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### Context
#### Topology
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#### Geometry
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[[!include higher geometry - contents]]
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# Contents
* table of contents
{: toc}

## Idea

The term *stratified space* usually refers to a [[topological space]] equipped with the structure of a [[stratification]], which decomposes the space into [[topological subspaces]] called *strata*.

Often strata are required to be "nice" in some sense (for instance, one may require strata to be [[manifolds]], while the stratified space itself need not be a manifold), and the way that strata are linked together is typically controlled by additional conditions. More generally, the term "stratified space" may refer to any notion of [[spaces]] equipped with some type of stratification structure.

Examples are ubiquitous, canonical stratifications are carried by [[polyhedra]], [[cell complexes]], [[algebraic varieties]], [[orbit spaces]] of many [[group actions]] on [[manifolds]], [[mapping cylinders]] of maps between manifolds, and [[moduli stacks of formal groups]] (stratified by [[height of a formal group|height of formal groups]]).

## Variants

There are many different definitions of stratifications on spaces:

* stratifications from [[filtered topological space|filtrations]],

* [[poset-stratified space|poset-stratifications]],

* [[Whitney stratifications]] (cf. the [Wikipedia](https://en.wikipedia.org/wiki/Whitney_conditions)),

* [[Thom-Mather stratifications]] (see e.g. [Wiki entry](https://en.wikipedia.org/wiki/Thom%E2%80%93Mather_stratified_space)),

* [[stratifold|stratifolds]],

* [[orbifold|orbifolds]],

* homotopical stratifications (as developed by [Quinn](#Quinn)),

* conically smooth stratifications (as developed by [Ayala-Francis-Tanaka](#AFT)),

* etc. ...

## Basic features

### Stratification structures
 {#StratificationStructures}

Most of the above variants share the same basic structure, captured by the following definitions.

\begin{terminology}\label{ExitPathPreorder}
**(exit path preorder)** \linebreak
Given a decomposition $f$ of a [[topological space]] $X$ into [[connected space|connected]] [[subspace#topological_subspaces|subspaces]] (denoted, in the following, by lower-case letters such as $s$), the *exit path preorder* $\mathsf{Exit}(f)$ of $f$ is the [[preorder]] of subspaces $s$ in $f$ with a generating arrow $s \to r$ whenever the [[topological closure]] of $r$ intersects $s$ non-trivially.
\end{terminology}

\begin{defn}\label{Stratification}
A **stratification** $f$ on $X$ is a decomposition of $X$ such that $\mathsf{Exit}(f)$ (Trm. \ref{ExitPathPreorder}) is an actual [[poset]].
\end{defn}

The subspaces in a stratification are also called *strata*. The [[opposite category|opposite]] poset $\mathsf{Exit}(f)^{\mathrm{op}}$ is also called the *entrance path poset* and denoted by $\mathsf{Entr}(f)$.

\begin{rmk}\label{CharacteristicMap}
**(characteristic map)** \linebreak
Given a stratification $f$ on $X$, there is a map $X \to \mathrm{Exit}(f)$ mapping points $x \in s$ to $s \in \mathrm{Exit}(f)$, which is sometimes called the **characteristic map** of $f$, and (abusing notation) denoted by $f \colon X \to \mathrm{Exit}(f)$.
\end{rmk}

The characteristic map (Rmk. \ref{CharacteristicMap}) need not be [[continuous map|continuous]] in general (unless the stratification is locally finite, see Rmk \ref{ContinuityOfCharacteristicMap} below); it is, however, continuous on the [[preimage]] of any finite [[full functor|full]] subposet of the exit path poset. It is often convenient to construct stratifications by constructing their characteristic map.

\begin{example}\label{CWStratification}
**(Stratification of CW complexes)**\linebreak
The following are stratifications of the 1- and 2-spheres:
[[SphereStratification.jpg:pic]]

In general, a stratification of a CW complex $X$ is obtained by taking the strata to be the $n$-cells of $X$ (for each $n$), but with the $(n-1)$-skeleton $X_{n-1}$ removed.
\end{example}

\begin{example}\label{PosetStratification}
**(Poset stratifications)** \linebreak
Let $X$ be a [[topological space]], $P$ a [[poset]], and $f \colon X \to P$ a [[continuous map]] (in other words, $f$ is a $P$-[[poset-stratified space|stratification of $X$]]). This determines a stratification $\mathrm{c}(f)$ of $X$ (in the sense of Def. \ref{Stratification} above) whose strata are the [[connected components]] of the [[preimages]] $f^{-1}(x)$, $x \in P$. The map $f$ factors uniquely through the characteristic map $\mathrm{c}(f) \colon X \to \mathsf{Exit}(\mathrm{c}(f))$ by a [[conservative functor|conservative]] poset map $\mathsf{Exit}(\mathrm{c}(f)) \to P$. (Such (characteristic,conservative)-factorizations are essentially unique.)
\end{example}

 The example shows that any poset-stratification determines a unique stratification. However, many poset-stratifications may determine the same stratification in this way.

\begin{example}\label{FilteredSpaces}
**(Filtered spaces)** Any [[filtered topological space]] $X_0 \subset X_1 \subset ... \subset X_n$ in which each $X_i$ is a [[closed subspace]] of $X_{i+1}$ defines a [[continuous map]] $X \to [n] = (0 \to 1 \to ... \to n)$ mapping points in $X_{i+1} \setminus X_{i}$ to $i$, and thus a stratification by Exp. \ref{PosetStratification}. As a concrete instance of this example, the filtration by skeleta of any [[cell complex]] defines the "stratification by cells" in this way.
\end{example}

\begin{example}\label{TrivialStratification}
**(Trivial stratification)** \linebreak
Every [[topological space]] $U$ is trivially stratified (in the sense of Def. \ref{Stratification}) with strata being the [[connected components]] of $U$.
\end{example}

\begin{rmk}\label{ContinuityOfCharacteristicMap}
**(Continuity of characteristic map)** \linebreak
Let $(X,f)$ be a stratified space according to Def. \ref{Stratification}. One says that the stratification $f$ is "locally finite" if each stratum $s$ of $f$ has an [[open neighborhood]] in $X$ which only contains finitely many strata. (If $(X,f)$ satisfies the frontier condition (Rmk. \ref{FrontierCondition} below) then, equivalently, $(X,f)$ is locally finite iff each point $x \in X$ has an [[open neighborhood]] intersecting only finitely many strata.) If $f$ is locally finite, then the characteristic map $f \colon X \to \mathsf{Exit}(f)$ (Def. \ref{CharacteristicMap}) is a [[continuous map]].
\end{rmk}

\begin{rmk}\label{FrontierCondition}
**(Openness of characteristic map)** \linebreak
Let $(X,f)$ be a stratified space (Def. \ref{Stratification}). One says the stratification $f$ satisfies the "frontier condition" (or, as an adjective, that it is "frontier-constructible") if, for any two strata $s, r$, the [[topological closure]] $\overline r$ [[intersection|intersects]] $s$ non-trivially, then $s \subset \overline r$. The stratification $f$ is frontier-constructible iff the characteristic map $f \colon X \to \mathsf{Exit}(f)$ (Def. \ref{CharacteristicMap}) is an [[open map]].
\end{rmk}

It is generally very reasonable to assume stratifications to be locally finite and frontier-constructible.

### The category of stratifications

\begin{defn}\label{StratifiedMap}
A **stratified map** $F \colon (X,f) \to (Y,g)$ of stratified spaces (Def. \ref{Stratification}) is a [[continuous map]] $F \colon X \to Y$ which factors through the characteristic maps $f$ and $g$ (Def. \ref{CharacteristicMap}) by a (then necessarily unique) map, denoted by $\mathsf{Exit}(F) \colon \mathsf{Exit}(f) \to \mathsf{Exit}(g)$.
\end{defn}

Stratified spaces and their maps (Def. \ref{StratifiedMap}) form a [[category]] $\mathbf{Strat}$ of stratification. The construction of exit path posets (Trm. \ref{ExitPathPreorder}) yields a [[functor]] $\mathsf{Exit} : \mathbf{Strat} \to \mathbf{Pos}$ ([[formal duality|dually]], using $(-)^{\mathrm{op}} : \mathbf{Pos} \to \mathbf{Pos}$ one obtains the entrance path poset functor $\mathsf{Entr} \colon \mathbf{Strat} \to \mathbf{Pos}$). The functor has a right inverse, as follows.

\begin{defn}\label{ClassifyingStratification}
Every poset $P$ has a **classifying stratification** $\parallel P \parallel$ (also called the *stratified realization* of $P$), whose [[underlying]] [[topological space]] is the [[classifying space]] $\left|   P \right|$ of $P$ (i.e. the [[topological realization]] of the [[simplicial nerve]] of $P$), and whose characteristic map (Def. \ref{CharacteristicMap}) is the map $\left| P \right| \to P$ that maps $\left| P^{\leq x} \right| \setminus \left| P^{\lt x} \right|$ to $x$ (here, the full subposets $P^{\leq x} = \{y \leq x\}$ and $P^{\lt x} = \{y \lt x\}$ of $P$ are the "lower" resp. "strict lower closures" of an element $x$ in $P$). Moreover, given a poset map $F \colon P \to Q$, the realization of its nerve yields a stratified map ${\parallel F \parallel} \colon {\parallel P \parallel} \to {\parallel Q \parallel}$. We obtain a [[functor]] ${\parallel - \parallel} \colon \mathbf{Pos} \to \mathbf{Strat}$.
\end{defn}

Every classifying stratification is frontier-constructible (Rmk. \ref{FrontierCondition}).

It makes sense to further terminologically distinguish maps of stratifications as follows.

{#stratified_map_types}
\begin{terminology}
**(Types of stratified maps)** \linebreak
A stratified map $F \colon (X,f) \to (Y,g)$  (Def. \ref{StratifiedMap}) is called:

* a **substratification** if $F \colon X \to Y$ is a [[topological subspace]] and $\mathsf{Exit}(F)$ is [[conservative functor|conservative]]; if, moreover, $X = g^{-1} \circ \mathsf{Exit}(F) \circ f (X)$ then one says the substratification is **constructible**;

* a **coarsening** if $F \colon X \to Y$ is a [[homeomorphism]] (to emphasize the opposite process, one also calls $F$ a **refinement**);

* a **stratified homeomorphism** (or **stratified iso**) if $F \colon X \to Y$ is a [[homeomorphism]] of [[topological spaces]] and $\mathsf{Exit}(F)$ is an [[isomorphism]] of [[posets]].

* {#stratified_bundle} a **stratified bundle** if points $x \in s$ for $s$ a stratum of $(Y,g)$ admit [[open neighbourhoods]] $U_x \subset s$ such that, up to stratified iso, $F$ restricts on $F^{-1}(U_x)$ to a projection $U_x \times f_x \to U_x$ (see Def. \ref{defn_strat_product}) for some 'fiber' stratification $(F^{-1}(x),f_x)$.

\end{terminology}

{#constructible_strat_bun}
\begin{rmk} (Constructible bundles). The preceding definition of stratified bundles is rather weak, and in practice one often finds bundles that satisfy stronger conditions, such as constructibility: a *constructible* stratified bundle is a bundle that, up to bundle isomorphism, can be classified by functorial information associated to the fundamental category (see Def. \ref{defn_exit_path_cat}) of its stratified base space (cf. [Sec. 6.3, Ayala-Francis-Rozenblyum 2015](#AyalaFrancisRozenblyum15)).
\end{rmk}

## Fundamental categories

Just as spaces have [[fundamental infinity-groupoid|fundamental]] $\infty$-[[fundamental infinity-groupoid|groupoids]], stratified spaces also have "[[fundamental category|fundamental categories]]". However, the role of *sets* for spaces is now played by *posets*: the following table illustrates the analogy. (The table is further explained below.)

| base concept | $\infty$-concept | presentation |
| ----- | ----- | ----- |
| [[set|sets]] $\simeq$ $(0,0)$-[[0-category|categories]] | ∞-sets $\simeq$ [[topological space|spaces]] | sets with w.e. |
| [[poset|posets]] $\simeq$ $(0,1)$-[[(0,1)-category|categories]] | ∞-posets $\simeq$ **stratified spaces** | posets with w.e. |
| [[category|categories]] $=$ $(1,1)$-categories | [[(∞,1)-category|∞-categories]] | [[categories with weak equivalences|categories with w.e.]] |

In the table, an "$\infty$-X" is intuitively to be understood as an [[(∞,∞)-category]] which admits a [[conservative functor]] to an X, where X can e.g. stand for "set", "poset", or "category". Yet more generally, X can be an [[(n,r)-category]] for $n,r \lt \infty$. A "set with weak equivalences" means a poset with weak equivalences in which each arrow is a weak equivalence. The left column is related to the middle column by an "$\infty$-zation functor" (which simply includes $1$-structures into $\infty$-structures), and the middle and right columns are related by an "[[category with weak equivalences#InfCatPres|∞-localization functor]]" (which should be a weak equivalence in some sense).

In order to make the above precise, one must work with sufficiently convenient stratifications. We describe two simple ways of constructing/presenting **fundamental $\infty$-posets** below.

### For conical stratificiations {#conical_strat}

\begin{defn} \label{defn_strat_product} Given stratifications $(X,f)$ and $(Y,g)$ their **product** is the stratification of $X \times Y$ with characteristic map $f \times g$.
\end{defn}

\begin{defn} Given a stratification $(X,f)$, the **stratified (open) cone** $(\mathrm{cone}(X), \mathrm{cone}(f))$ stratifies the topological open cone $\mathrm{cone}(X) = X \times [0,1) / X \times \{0\}$ by the product $(X,f) \times (0,1)$ away from the cone point $\{0\}$ (here, the open interval $(0,1)$ is trivially stratified), and by setting the cone point $\{0\}$ to be its own stratum.
\end{defn}

\begin{defn} A **conical** stratification $(X,f)$ is a stratification in which each point $x \in X$ has a neighborhood (i.e. an open substratification) that is a stratified product $U \times (\mathrm{cone}(Z),\mathrm{cone}(l))$ for some "link" stratification $(Z,l)$ and such that $x \in U \times \{0\}$.
\end{defn}

Every conical stratification is frontier-constructible.

\begin{defn} \label{defn_exit_path_cat} Given a conical stratification $(X,f)$, then [Lurie](#LurieHA) constructs **exit path $\infty$-category** $\mathcal{E}\mathrm{xit}(f)$ as a quasicategory: the $k$-simplices of the quasicategory $\mathcal{E}\mathrm{xit}(f)$ are precisely stratified maps $\parallel [k]\parallel \to (X,f)$, where $[k] = (0 \to 1 \to ... \to k)$.
\end{defn}

\begin{rmk} One dually defines the **entrance path $\infty$-category** $\mathcal{E}\mathrm{ntr}(f)$ with $k$-simplices $\parallel [k]^{\mathrm{op}} \parallel \to (X,f)$.
\end{rmk}

{#exit_entrance_conv}
\begin{terminology} An **exit path** in a stratified space $(X,f)$ is a stratified map $f : \parallel [1] \parallel \to (X,f)$; we say $f$ *starts* $f(0)$ and *ends* at $f(1)$ (dually, we say $f$ is an **entrance path** which starts at $f(1)$ and ends at $f(0)$).
\end{terminology}

The construction translates from "stratified spaces" to
"$\infty$-posets" in the above table: the conservative functor $\mathcal{E}\mathrm{xit}(f) \to \mathsf{Exit}(f)$ takes objects $\parallel [0]\parallel \to X$ to the stratum $s \in \mathsf{Exit}(f)$ that their image lies in.

### For regular stratifications {#regular_strat}

\begin{defn} A stratification $(X,f)$ is **regular** if it admits a refinement $\parallel P \parallel \to (X,f)$ by the stratified realization of some poset $P$.
\end{defn}

\begin{defn} Given a regular stratification $(X,f)$ and a refinement $F : \parallel P\parallel \to (X,f)$, one can construct the **presented exit path $\infty$-category** $\mathcal{PE}\mathrm{xit}(f)$ as a poset with weak equivalences with underlying poset $P$ and weak equivalences $\mathsf{Exit}(F)^{-1}(\mathrm{id})$.
\end{defn}

(Showing that this construction is, in an appropriate sense, independent of the choice of $P$ requires a bit more work...)

\begin{rmk} \label{rmk_cellulable} A special case of the definition asks for refinements by regular cell complexes (or simplicial complexes) in place of classifying stratifications of posets, in which case one speaks of **cellulable** (resp. **triangulable**) stratifications.
\end{rmk}

The construction translates between "stratified spaces" and "posets with weak equivalences" in the above table. Since spaces are trivially stratified spaces, this specializes to a translation between "spaces" and "sets with weak equivalences".

## Related concepts

* [[stratification]]

* [[piecewise flat spacetime]]

## References

A notion of purely topologically stratified sets was introduced in

* [[Frank Quinn]], _Homotopically stratified sets_, J. Amer. Math. Soc. 1 (1988), 441--499. MR 89g:57050
 {#Quinn}

Notions of smoothly stratified spaces were considered by

* H. Whitney, _Local properties of analytic varieties_, Differentiable and combinatorial topology (S. Cairns, ed.), Princeton Univ. Press, Princeton, 1965, pp. 205--244. MR 32:5924

* R. Thom, _Ensembles et morphismes stratifiés_, Bull. Amer. Math. Soc. 75 (1969), 240--282.
MR 39:970

* J. Mather, _Notes on topological stability_, Harvard Univ., Cambridge, (1970)

Locally conelike stratified spaces have been considered  in

* L. Siebenmann, _Deformations of homeomorphisms on stratified sets_, Comment. Math. Helv. 47 (1971), 123--165. MR 47:7752

These are all special cases of ([Quinn's definition](#Quinn)).

Discussion of [[differential operators]] on stratified spaces is in

* R. B. Melrose,  _Pseudodifferential operators, corners and singular limits_, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 217--234, Math. Soc.

* Pierre Albin, Eric Leichtnam, Rafe Mazzeo, Paolo Piazza, _The signature package on Witt spaces, I. Index classes_ ([arXiv:0906.1568v2](http://uk.arxiv.org/abs/0906.1568v2))

See also

* Bruce Hughes, _Geometric topology of stratified spaces_ ([pdf](http://aimsciences.org/journals/pdfsnews.jsp?paperID=2468&mode=full))

Discussion of the [[fundamental category]] of a (Whitney‑)stratified space is in

* [[Jonathan Woolf]], _Transversal homotopy theory_ [[arXiv:0910.3322](http://arxiv.org/abs/0910.3322)]

* [[Markus Banagl]], *Topological invariants of stratified spaces*, Springer Monographs in Mathematics, Springer (2000) [[doi:10.1007/3-540-38587-8](https://doi.org/10.1007/3-540-38587-8)]

A [[homotopy hypothesis]] for stratified spaces is discussed in

* {#AyalaFrancisRozenblyum15} [[David Ayala]], [[John Francis]], [[Nick Rozenblyum]], *A stratified homotopy hypothesis*, [arXiv](http://arxiv.org/abs/1502.01713)

* [[Peter J. Haine]], _On the homotopy theory of stratified spaces_ ([arXiv:1811.01119](https://arxiv.org/abs/1811.01119))

The former is based on the notion of stratifications developed in

* {#AFT} David Ayala, John Francis, and Hiro Lee Tanaka. _Local structures on stratified spaces_. Advances in Mathematics 307 (2017): 903-1028.

An earlier paper on exit paths is

* [[David Treumann]], *[Exit paths and constructible stacks](https://www2.bc.edu/david-treumann/epacs.pdf)*, Compositio Math. 145 (2009), 1504–1532

Poset-stratified spaces and the conicality condition, as well as the construction of the fundamental $\infty$-poset of a conical stratification as a quasicategory, first appeared in:

* {#LurieHA} [[Jacob Lurie]], *[[Higher Algebra]]*, 2012, in particular Appendix A ([pdf](https://www.math.ias.edu/~lurie/papers/HA.pdf))

Stratified spaces are used to develop a graphical calculus for Gray categories in:

* Benjamin Taylor Hummon, *[Surface Diagrams for Gray-Categories](https://escholarship.org/uc/item/5b24s9cc)*, PhD thesis.

[[!redirects stratified space]]
[[!redirects stratified spaces]]

[[!redirects stratum]]
[[!redirects strata]]


stratifold

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###Context###
#### Geometry
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[[!include higher geometry - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

Stratifolds are a generalization of smooth [[manifolds]] -- a notion of [[generalized smooth space]] --  which were introduced by [[Matthias Kreck|Kreck]]; see his [lecture notes](#Kreck).

Stratifolds comprise one of the many variants of the concept of a _[[stratified space]]_, and they may include some types of singularities.

## Definition

Stratifolds form a class of [[differential module|differential modules]], which are pairs $(S,C)$ of a [[topological space]] $S$ together with a subalgebra $C$ of the algebra of [[continuous function|continuous]] real-valued functions $S\to R$ such that

1. $C$ is locally detectable, i.e. for all continuous functions $f: S\to R$, $f$ is in $C$ iff for every $x\in S$ there exist an open neighborhood $U\ni x$ and $g\in C$ such that $g|_U = f|_U$,

1. let $f_1,\ldots,f_n$ be elements of $C$ and $\phi : R^n\to R$ a smooth function. Then $\phi \circ (f_1,\ldots,f_n) : S\to R$ is in $C$.

Local detectability is equivalent to requiring that $C$ is an algebra of global sections of a given subsheaf of the [[sheaf]] of all continuous functions on $S$; in particular [[germ]]s at every point can be defined.

For a [[manifold]] $C= C^\infty(S)$. For a differentiable space $S = (S,C)$ a [[tangent space]] $T_x S$ can be defined at each $x\in S$. Define $S^i = \{x\in S | \mathrm{dim} T_x S = i\}$. By construction, $S$ decomposes into a [[disjoint union]] $S = \biguplus_{i=0}^\infty S^i$. There is an induced stratifold structure on the topological subspace $S^i\subset S$, which we denote by $(S^i, C(S^i))$.

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

A __$k$-dimensional stratifold__ $(S,C)$ is a differential space such that

* $S$ is _locally compact [[Hausdorff space]] with countable basis,

* $T_x S \leq k$ for all $x\in S$ (i.e. $S = \bigcup_{i=0}^k S^i$),

* $(S^i,C(S^i))$ is isomorphic to a smooth [[manifold]],

* the restriction map $C(S)\to C(S^i)$ induces an isomorphism of [[stalk]]s of germs $C(S)_x\to C(S_i)_x = C^\infty(S^i)_x$ in all points $x\in S^i$,

* for all $y\in S$, and all $U\ni y$ open, there is a "bump function" $\rho\in C$ nonvanishing at $y$, but whose support is contained in $U$.

=--

## References

* [The stratifold page](https://www.hcm.uni-bonn.de/homepages/prof-dr-matthias-kreck/the-stratifold-page/)

* [[Matthias Kreck]], _Differential Algebraic Topology_ ([pdf](http://www.hausdorff-research-institute.uni-bonn.de/files/kreck-DA24_08_07.pdf))
 {#Kreck}

[[!redirects stratifolds]]


streak

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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition ##

In [[constructive mathematics]], given a $\sigma$-locale $\Sigma$ whose poset of opens $O(\Sigma)$ is a [[sigma-frame|$\sigma$-frame]], a **$\Sigma$-streak** is an [[archimedean difference protoring]] $M$ such that the [[strict total order]] $\lt:M \times M \to \Omega$ [[factors]] into $\lt^{'}:M \times M \to \Sigma$ and the canonical [[monotone]] $i:\Sigma \to \Omega$, where $\Omega$ is the [[subobject classifier|subset classifier]]. $\Sigma$ is typically called the **[[open subset]] classifier**.

## Examples ##

* For any $\sigma$-locale $\Sigma$, the [[Dedekind real numbers]] are the [[terminal object|terminal]] $\Sigma$-streak.

* For any $\sigma$-locale $\Sigma$, the [[Sigma-Dedekind real numbers|$\Sigma$-Dedekind real numbers]] are a $\Sigma$-streak.

## See also ##

* [[archimedean difference protoring]]

* [[Dedekind cut]]

* [[Dedekind real numbers]]

* [[number line rig]]

## References

* Davorin Lešnik, Synthetic Topology and Constructive Metric Spaces, ([arxiv:2104.10399](https://arxiv.org/abs/2104.10399))

[[!redirects streaks]]

[[!redirects sigma streak]]
[[!redirects sigma-streak]]
[[!redirects sigma streaks]]
[[!redirects sigma-streaks]]

[[!redirects Sigma streak]]
[[!redirects Sigma-streak]]
[[!redirects Sigma streaks]]
[[!redirects Sigma-streaks]]


stream

# Streams
* table of contents
{: toc}

## Idea

There are two notions of *stream* in use, as an [[infinite]] [[sequence]] or as a potentially infinite list.

Accounts which take streams to be...

* infinite sequences: [Rutten05](#Rutten05), [ACS15](#ACS15)

* potentially infinite list: [Jacobs16](#Jacobs16)

## Definitions

### Explicit definitions

Let $A$ be a [[set]]. Then

* the set of streams of $A$ in the sense of infinite sequences is the set of functions $A^\mathbb{N}$ from the [[natural numbers]] to $A$

* the set of streams of $A$ in the sense of potentially infinite list $A_*$ is

  * the disjoint union of the set $A^\mathbb{N}$ of infinite sequences in $A$ and the set $A^* \coloneqq \biguplus_{n \in \mathbb{N}} A^n$ of finite lists $A$.

  * the set of all infinite sequences of the disjoint union $A \uplus \{*\}$ such that $a_{i + 1} = *$ if $a_i = *$ for any natural number $i$. Finite lists are identified with a finite list of elements of $A$ followed by all $*$s, while infinite sequences is identified with a sequence of all elements of $A$.

$$A_* \coloneqq \{a \in (A \uplus \{*\})^\mathbb{N} \vert \forall i \in \mathbb{N}.(a_i = *) \implies (a_{i + 1} = *)\}$$

In [[constructive mathematics]] only the second definition of potentially infinite list is a valid definition of stream. In [[dependent type theory]], the two definitions of potentially infinite list also make sense

$$A_* \coloneqq \mathbb{N} \to A + \sum_{n:\mathbb{N}} (\mathrm{Fin}(n) \to A)$$

$$A_* \coloneqq \sum_{a:\mathbb{N} \to (A + 1)} \prod_{i:\mathbb{N}} (a(i) = \mathrm{inr}(*)) \to (a(i + 1) = \mathrm{inr}(*))$$

but if the type $A$ is not a set, $A_*$ will not be a [[set]] either.

### Coalgebraic definitions

Let $A$ be a [[set]] (or [[object]] of any [[category]] $C$ where the following makes sense). Then

* the set of streams of $A$ in the sense of infinite sequences is the final coalgebra of the endofunctor $X \mapsto A \times X$.

* the set of streams of $A$ in the sense of potentially infinite sequences is the final coalgebra of the endofunctor $X \mapsto 1 + A \times X$.

## Examples

* An [[extended natural number]] is an element of $1_*$ (much as a [[natural number]] is an element of $1^*$).  Classically, such an element is either a [[natural number]] or [[infinity]] (the unique element of $1^{\mathbb{N}}$).

## Related concepts

* [[writer monad]]

* [[cofree coalgebra]]

## References

* {#Rutten05} [[Jan Rutten]], *A coinductive calculus of streams*, Mathematical Structures in Computer Science, Mathematical Structures in Computer Science, Volume 15, Issue 1, February 2005, pp. 93-147 ([doi:10.1017/S0960129504004517](https://doi.org/10.1017/S0960129504004517))

* {#ACS15} [[Benedikt Ahrens]], [[Paolo Capriotti]], [[Régis Spadotti]], *Non-wellfounded trees in Homotopy Type Theory*, in 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 17-30, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015) ([doi:10.4230/LIPIcs.TLCA.2015.17](https://doi.org/10.4230/LIPIcs.TLCA.2015.17), [arXiv:1504.02949](https://arxiv.org/abs/1504.02949))

* {#Jacobs16} [[Bart Jacobs]], *Introduction to Coalgebra Towards Mathematics of States and Observation*, Cambridge Tracts in Theoretical Computer Science. Cambridge: Cambridge University Press; 2016:i-iv. (ISBN:9781316823187, [doi:10.1017/CBO9781316823187](https://doi.org/10.1017/CBO9781316823187))

[[!redirects stream]]
[[!redirects streams]]


Street fibration

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###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

A **Street fibration**, or **weak fibration**, is a generalization of a [[Grothendieck fibration]] which obeys the [[principle of equivalence]], i.e. covariant under [[equivalence of categories]].  In [[Cat]], almost all Street fibrations which arise in practice are actually Grothendieck fibrations, so the extra generality is not very useful; since it is also fairly cumbersome, it is rarely used.  However, when working with [[anafunctors]] or internally to a [[bicategory]] (a weak [[2-category]]), one is essentially forced to work with Street fibrations instead; see below.

## Definition

Let $p\colon E\to B$ be a [[functor]].  The notion of when a morphism of $E$ is [[cartesian morphism|cartesian]] is unchanged from the version for [[Grothendieck fibrations]].  However, now we say that $p$ is a **Street fibration** if for any $f:a\to b$ in $B$ and $e\in E_b$ there exists a cartesian $\phi:e'\to e$ and an _isomorphism_ $h:p(e') \cong a$ such that $f\circ h = p(\phi)$.

An [[internalization|internalized]] version of this definition can be given in any [[2-category]]; see [[fibration in a 2-category]].  The above definition corresponds to the special case of the 2-category [[Cat]].

### In terms of adjoint inverses

Back in the 1960s, J. Gray characterized fibrations as the functors $p\colon E\to B$
where for every $a \in E$ the [[slice functor]] $p/a \colon E/a \to B/p(a)$ has a [[right
adjoint]] [[right inverse]] $c_a$. One avoids violating the [[principle of equivalence]] if one just requires that $c_a$ is
[[fully faithful functor|full and faithful]], i.e. the [[counit of the adjunction]] $p/a \dashv c_a$ is an isomorphism. Explicating this condition in elementary terms, one arrives at the above definition.

A simpler version of this characterization is the following.

+--{: .un_lemma}
###### Lemma
A functor $p \colon E \to B$ is a [[cleavage|cloven]] Street fibration if and only if the canonical functor $i \colon E \to B/p$ has a right adjoint $r$ in $Cat / B$.
=--
+-- {: .proof }
###### Proof
First, recall that the 2-category $Cat/B$ has objects the functors $C \to B$, as morphisms the triangles
$$\array{C & \xrightarrow{h} & C' \\ & f \searrow \cong \swarrow g &  \\  & B  & }$$
which commute up to specified isomorphism, and as 2-cells the natural transformations $\alpha : h_1 \to h_2$ which commute with the specified isomorphisms (i.e. $ f \xrightarrow{\cong} g h_1 \xrightarrow{g \alpha} g h_2 \xrightarrow{\cong} f$ is the identity).

Next, recall that the [[comma category]] $B/p$ has objects the triples $(x, e, k)$, with $k \colon x \to p(e)$ a morphism in $B$.  Let $\pi \colon B/p \to B$ denote the projection $(x, e, k) \mapsto x$.  The canonical morphism $i\colon p \to B/p$ in $Cat/B$ sends $e\in E$ to the triple $(p(e),e,1_{p(e)})$.

Somewhat imprecisely, seeing both categories $E$ and $B/p$ as sitting over $B$ means that functors between those should be the identity (up to specified isomorphism) on the $b$ component, and natural transformations should have the identity as their $b$ component (modulo the specified isomorphisms).

Now, to give an adjunction $i \dashv r$ in $Cat$, it suffices to give, for each object $k \colon x \to p(e)$ in $B/p$, an object $r(x,e,k) = r(k)$ in $E$ and an arrow $i(r(k)) = (p(r(k)), r(k), 1_{p(r(k))}) \to (x,e,k)$ in $B/p$ that is [[universal arrow|universal]] from $i$ to $k$.  Such an arrow consists of a pair of morphisms $\theta_k\colon p(r(k)) \to x$ in $B$ and $\epsilon_k\colon r(k) \to e$ in $E$ such that
$$
\array{
  p(r(k)) & \overset{1}{\to} & p(r(k)) \\
  \mathllap{\theta} \downarrow & & \downarrow \mathrlap{p(\epsilon_k)} \\
  x & \underset{k}{\to} & p(e)
}
$$
commutes, i.e. such that $p(\epsilon_k) = k \circ \theta_k$.

Note that since $x = \pi(x,e,k)$, the morphisms $\theta$ supply a natural transformation $\theta\colon p r \to \pi$, not necessarily invertible.  By general [[doctrinal adjunction]] arguments, in order for $i \dashv r$ to be an adjunction in $Cat/B$, it is necessary and sufficient that $\theta$ be invertible (and we then use it to make $r$ into a morphism in $Cat/B$).

Now, the universal property of $(\theta_k,\epsilon_k)$ tells us that for any other morphism in $B/p$ from some $i(y)$ to $k$, i.e., for any $y$ and any pair $(f,g)$ making the square
$$
\array{
  p(y) & \stackrel{1}{\to} & p(y) \\
  \mathllap{f} \downarrow & & \downarrow \mathrlap{p(g)} \\
  x & \underset{k}{\to} & p(e)
}
$$
commute, there is a unique map $h \colon y \to r(k)$ in $B$ such that the above square factors in $B/p$ as
$$
\array{
  p(y) & \stackrel{1}{\to} & p(y) \\
  \mathllap{p(h)} \downarrow &  & \downarrow \mathrlap{p(h)} \\
  \mathllap{p(r(k))} & \stackrel{1}{\to} & \mathrlap{p(r(k))}\\
  \mathllap{\theta_k} \downarrow & & \downarrow \mathrlap{p(\epsilon_k)} \\
  x & \underset{k}{\to} & p(e).
}
$$
In other words, the universal property provides a unique $h$ such that $\epsilon_k \circ h = g$ and $\theta_k \circ p(h) = f$, which exactly asserts that $\epsilon_k$ is a [[cartesian arrow]].

Thus, the existence of a right adjoint to $i$ means precisely that for each morphism $k \colon x \to p(e)$, a choice is given of a cartesian lift of $k$ up to isomorphism.  In turn, this means that $p$ is a cloven fibration.
=--

### Other definitions

Most other characterizations of Grothendieck fibrations can also be given in a way which does not violate the [[principle of equivalence]] to characterize Street fibrations.

## Comparison to Grothendieck fibrations

Every Grothendieck fibration is a Street fibration, as is every [[equivalence of categories]] (the latter are not in general Grothendieck fibrations).  Since Street fibrations are closed under composition, any functor equivalent to a Street fibration is also a Street fibration (this is the "covariance" property).  As remarked above, however, almost all functors in [[Cat]] which one wants to treat as "fibrations" are actually Grothendieck fibrations.  Moreover, in $Cat$ every Street fibration is the composite of an equivalence with a Grothendieck fibration: the latter can be taken to be a [[fibrant replacement]] of it in the [[canonical model structure]] on Cat.  (In fact, a Street fibration is a Grothendieck fibration precisely when it is also a [[isofibration]].)

For these reasons, Street fibrations in $Cat$ are little-studied.  The fact that they exist, and that any Street fibration is equivalent to a Grothendieck one, can be useful in assuaging philosophical worries about violation of the [[principle of equivalence]] in the notion of Grothendieck fibration, but in practice in $Cat$ one usually wants to use Grothendieck fibrations.

However, when internalizing in other 2-categories, covariance becomes more important.  This is most obvious when the 2-category in which one wishes to internalize is only a weak 2-category (a [[bicategory]]), in which it does not even make much sense to internalize the notion of Grothendieck fibration.  (This is even the case in [[Cat]] if we define it using [[anafunctors]], as is often important to do in the absence of the [[axiom of choice]].)  However, even when internalizing in a [[strict 2-category]], if the strict 2-category itself is "only defined up to [[biequivalence]]," then one is again compelled to use internal Street fibrations, since they are "covariant under biequivalence" of the containing bicategory, whereas Grothendieck fibrations are not.

For example, this is arguably the case in the 2-category [[Topos]], where we have to make the arbitrary choice of whether to define the [[geometric morphism|morphisms]] to be [[left exact functor|left-exact]] left [[adjoint functor|adjoints]] $f^*$, right adjoints $f_*$ having a left-exact left adjoint, or adjoint pairs $(f^*,f_*)$ in which $f^*$ is left exact.  The three choices give three strict 2-categories which are all biequivalent, but not strictly 2-equivalent; thus it is more sensible to consider internal Street fibrations in $Topos$ than strict Grothendieck ones.

## Properties

Since Street fibrations are the "minimal covariant modification" of Grothendieck ones, any property enjoyed by Grothendieck fibrations which is itself covariant under equivalence will also be possessed by Street fibrations.

One thing to note is that in general, when working with Street fibrations, one must always replace [[fibers]] with [[essential fibers]].  In particular, the lifting property of a Street fibration is insufficient for a morphism $u\colon J\to I$ in $B$ to induce a functor $u^*\colon E^I\to E^J$ between strict fibers, but such a functor is induced between essential fibers.  In this way we can reconstruct a [[pseudofunctor]] $B^{op}\to Cat$ from any Street fibration, and show that the [[2-category]] of Street fibrations is equivalent to that of such pseudofunctors (though not, as in the case of Grothendieck fibrations, [[strict 2-equivalence of 2-categories|strictly 2-equivalent]]).

Just as every Grothendieck fibration is a [[strictly exponentiable functor]], i.e. an [[exponential object|exponentiable morphism]] in the [[strict 2-category]] [[Cat]], every Street fibration is a non-strictly exponentiable functor, i.e. an exponentiable morphism in the weak [[2-category]] Cat.

## Examples

Of course, any Grothendieck fibration is a Street fibration, but the interest is in those which are not Grothendieck.  Some of the applications are internal to 2-categories other than [[Cat]].

* The original motivation was to consider fibrations (and opfibrations, and [[two-sided fibrations]]) in $V Cat^{op}$ as a means to construct [[Prof]].

* Street fibrations in [[Topos]] are discussed in the paper of Johnstone below.

Street fibrations in [[Cat]] also arise in mathematics, though often under different names.  The following examples were pointed out by [[George Janelidze]], [[Steve Lack]], and [[Ross Street]].

* The notion of "admissible object" in [[categorical Galois theory]] can be expressed in terms of a functor being (partially) a Street fibration, not generally Grothendieck.

* Any [[left exact functor|left exact]] (or, more generally, "semi-left exact") [[reflection]] is a Street fibration, not generally Grothendieck.

## References

The notion was originally defined in

* [[Ross Street]], *Fibrations in bicategories*, Cahiers de topologie et géométrie différentielle catégoriques, tome 21, no 2 (1980), 111-160.  ([pdf](http://www.numdam.org/article/CTGDC_1980__21_2_111_0.pdf))

although there it was given in a very abstract form.  A more explicit form like that given above can be found in

* Ross Street, *Conspectus of variable categories*, Journal of Pure and Applied Algebra, Volume 21, Issue 3 (1981), 307-338. ([pdf](https://www.sciencedirect.com/science/article/pii/0022404981900219/pdf?md5=3d791b95c31feb34887b5144975c2f02&pid=1-s2.0-0022404981900219-main.pdf))

Some alternate characterizations, and applications in [[Topos]], are discussed in

* [[Peter Johnstone]], *Fibrations and partial products in a 2-category*.

[[!redirects Street fibration]]
[[!redirects Street fibrations]]
[[!redirects weak fibration]]
[[!redirects weak fibrations]]
[[!redirects Street opfibration]]
[[!redirects Street opfibrations]]
[[!redirects weak opfibration]]
[[!redirects weak opfibrations]]


StreetCBS

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
=--
=--

These two papers

* [[Ross Street]], _Two-dimensional sheaf theory_ J. Pure Appl. Algebra 23 (1982), no. 3, 251-270, [MR83d:18014](http://www.ams.org/mathscinet-getitem?mr=644277), doi

* [[Ross Street]], _[[zoranskoda:Characterization of Bicategories of Stacks]]_, MR0682967 (84d:18006), p. 282-291 in: Category theory (Gummersbach 1981) Springer LNM __962__, 1982

basically supply (via observations by [[Mike Shulman]], see _[[michaelshulman:StreetCBS]]_) definitions on the $n$Lab of [[regular 2-category]], [[coherent 2-category]], [[exact 2-category]], [[2-pretopos]], [[2-site]], [[2-sheaf]], and [[Grothendieck 2-topos]], as well as the proof of the [[2-Giraud theorem]].  The definitions we are interested in are the [[bicategory|bicategorical]] ones, which appear in the second paper, but the only proofs are in the first paper which does a strict version for strict 2-categories.  Note in particular that where he uses the prefix "bi‑" we use either "2‑" or no prefix; thus our 2-sites are what he calles bisites, etc.  There are also a couple of mistakes in the definitions and proofs which we have tried to correct in the relevant $n$Lab entries.

category: reference


stress tensor

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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## In continuum mechanics / solid state physics

In [[elasticity theory]] (in the context of [[continuum mechanics]]/[[solid state physics]]) a _stress tensor_ expresses [[forces]] acting inside a [[solid body]].

There is a corresponding [[strain tensor]] which expresses the deformation of the body in reaction to these forces.

## Related concepts

* [[stress-energy tensor]]

## References

* {#LandauLifshitz59} [[Lev Landau]], [[Evgeny Lifshitz]], I.§2 of _Theory of Elasticity_, part VII of  _[[Course of Theoretical Physics]]_, 1959, 1970


stress-energy tensor

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#### Physics
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[[!include physicscontents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

In a [[model (in theoretical physics)]] of [[gravity]] coupled to other fields, the _energy momentum tensor_ or _stress energy tensor_ T is the [[variational calculus|variation]] of the matter part $S_{fields}$ in the [[Einstein-Hilbert action]] with respect to the metric, $\delta S_{fields} = T \delta g$.

It is a measure for the [[energy]] and [[momentum]] density of the matter contribution to the theory.

Since the variation of the [[Einstein-Hilbert action]] $S_{EH}$ is the [[Einstein tensor]] $G$, the [[equations of motion]] of [[gravity]], hence [[Einstein's equations]] assert that (up to constant prefactors)

$$
  G = T
  \,.
$$

## Properties

### Renormalization and Vacuum energy

The [[renormalization]] freedom in [[perturbative QFT|perturbative]] [[quantization]] of [[gravity]] ([[perturbative quantum gravity]]) induces freedom in the choice of [[vacuum expectation value]] of the stress-energy tensor and hence in the [[cosmological constant]].

For details see [there](cosmological+constant#InPerturbativeQuantumGravity).

## Details

* For the [[scalar field]]: see at _[[geometry of physics -- A first idea of quantum field theory]]_ [this example](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#ScalarFieldEnergyMomentum)

## Related concepts

* [[stress tensor]]

* [[energy-momentum relation]]

* [[Dirac current]]

[[!redirects energy momentum tensor]]
[[!redirects energy-momentum tensor]]

[[!redirects energy-momentum]]


strict (2,1)-category

+-- {: .rightHandSide}
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### Context
#### Higher category theory
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[[!include higher category theory - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

A *strict (2,1)-category* is a [[2-category]] which is both a *[[strict 2-category]]* as well as a [[(2,1)-category]], hence which is a [[Grpd]]-[[enriched category]], referred to as a *g.e. category* in original articles ([Fantham & Moore 1983](#FanthamMoore83)).

## Examples

\begin{example}\label{Homotopy2CategoryOfTopologicalSpaces}
**(homotopy 2-category of topological spaces)**
\linebreak
  There is a strict $(2,1)$-category whose underlying [[1-category]] is that of [[TopologicalSpaces]] and whose [[2-morphisms]] are the [[higher homotopy]]-classes of [[homotopies]] between [[continuous functions]].

If one restricts to topological spaces which admit the structure of [[CW-complexes]], then this is [[equivalence of 2-categories|equivalently]] the [[homotopy 2-category]] of the [[(infinity,1)-category|$(\infty,1)$-category]] of the [[simplicial localization]] of [[TopologicalSpaces]] at the [[weak homotopy equivalences]] ([[equivalence of (infinity,1)-categories|equivalently]] that of [[infinity-groupoids|$\infty$-groupoids]]), hence is the [[(2,1)-category]]-enhancement of the [[classical homotopy category]].

Since higher homotopy-classes of homotopies were also known as "tracks", some authors (following [Baues 1991, p. 300](#Baues91))
say "track category", not just for this example, but for $Grpd$-enriched categories generally.
\end{example}

## References
 {#References}

Original discussion of $Grpd$-enriched categories motivated from the [[homotopy 2-category]] of [[topological spaces]]:

* {#FanthamMoore83} [[Peter H. H. Fantham]], [[Eric J. Moore]], *Groupoid Enriched Categories and Homotopy Theory*, Canadian Journal of Mathematics **35** 3 (1983) 385-416 ([doi:10.4153/CJM-1983-022-8](https://doi.org/10.4153/CJM-1983-022-8))

In this context, some authors use the term "track category" for "$Grpd$-enriched category" (referring to their [[2-morphisms]], since "track" is a term for relative [[higher homotopy]]-classes of [[homotopies]]), following:

* {#Baues91} Chapter VI of: [[Hans-Joachim Baues]], *Combinatorial Homotopy and 4-Dimensional Complexes*, Expositions in Mathematics, De Gruyter 1991 ([doi:10.1515/9783110854480](https://doi.org/10.1515/9783110854480))

see also

* [[Klaus Heiner Kamps]], [[Tim Porter]], Sec. IV.1 of: _Abstract Homotopy and Simple Homotopy Theory_, World Scientific 1997 ([doi:10.1142/2215](https://doi.org/10.1142/2215), [GoogleBooks](http://books.google.de/books?id=7JYKxInRMdAC&dq=Porter+Kamps&printsec=frontcover&source=bl&ots=uuyl_tIjs4&sig=Lt8I92xQBZ4DNKVXD0x76WkcxCE&hl=de&sa=X&oi=book_result&resnum=3&ct=result#PPP1,M1))

For example, [[Toda brackets]] have a neat desciption in the [[homotopy 2-category]] of topological spaces regarded (Ex. \ref{Homotopy2CategoryOfTopologicalSpaces}) as a strict (2,1)-category (hence: "track category"):

* {#HardieKampsKieboom99} [[Keith Hardie]], [[Klaus Heiner Kamps]], [[Rudger  Kieboom]], _Higher homotopy groupoids and Toda brackets_, Homology Homotopy Appl. Volume 1, Number 1 (1999), 117-134 ([euclid:hha/1139840198](https://projecteuclid.org/euclid.hha/1139840198))

* {#HardieMarcumOda01} [[Keith Hardie]], [[Howard Marcum]], [[Nobuyuki Oda]], _Bracket operations in the homotopy theory of a 2-category_, Rend. Ist. Mat. Univ. Trieste 33, 19–70 (2001) ([rendiconti:33/02](https://rendiconti.dmi.units.it/volumi/33/02.pdf))

* {#HardieKampsMarcum02} [[Keith Hardie]], [[Klaus Heiner Kamps]], [[Howard Marcum]], _The Toda bracket in the homotopy category of a track bicategory_, Journal of Pure and Applied Algebra Volume 175, Issues 1–3, 8 November 2002, Pages 109-133 (doi:10.1016/S0022-4049(02)00131-7)

* [[Howard Marcum]], [[Nobuyuki Oda]], _Long Box Bracket Operations in Homotopy Theory_, Appl Categor Struct 19, 137–173 ([doi:10.1007/s10485-009-9186-3](https://doi.org/10.1007/s10485-009-9186-3))

Often strict (2,1)-categories are discussed in the broader context of [[strict 2-categories]], i.e. [[Cat]]-[[enriched categories]], and not explicitly identified as the special case that they are, e.g. in:

* [[Susan B. Niefield]], [[Dorette A. Pronk]], *Internal groupoids and exponentiability*, Cahiers de topologie et géométrie différentielle catégoriques, [**LX** 4 (2019)](http://cahierstgdc.com/index.php/volume-lx/) ([pdf](http://cahierstgdc.com/wp-content/uploads/2019/10/Niefield-Pronk-LX-4.pdf))

[[!redirects strict (2,1)-categories]]

[[!redirects Grpd-enriched category]]
[[!redirects Grpd-enriched categories]]

[[!redirects g.e. category]]
[[!redirects g.e. categories]]

[[!redirects ge category]]
[[!redirects ge categories]]

[[!redirects ge-category]]
[[!redirects ge-categories]]

[[!redirects track category]]
[[!redirects track categories]]


strict 2-category

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# Strict $2$-categories
* table of contents
{: toc}

## Idea
 {#Idea}

The concept of *strict 2-categories* is the simplest generalization of that of [[categories]] to the *[[n-categories|$n$-categories]]* of [[higher category theory]]. It is a one-step [[vertical categorification|categorification]] of the concept of a *[[category]]* with strict choice of structure morphisms.

More concretely, a strict 2-category is a [[directed n-graph|directed 2-graph]] equipped with [[horizontal composition|horizontal-]] and [[vertical composition]] of adjacent 1-cells ([[1-morphisms]]) and 2-cells ([[2-morphisms]]), respectively, which is strictly [[unitality|unital]] and [[associativity|associative]] in both directions, and such that both types of composition are compatible (the "[[interchange law]]").

A quick way of making this precise, is to say that strict 2-categories are [[Cat]]-[[enriched categories]], see [below](#DefinitionStrictTwoCategories).

The term _2-category_ implicitly refers to a [[geometric shapes for higher structures|globular]] structure.  By contrast, [[double category|double categories]] are based on [[cubes]] instead.  The two notions are closely related, however: every strict 2-category gives rise to several strict double categories, and every double category has several [[underlying]] 2-categories.

Notice that _[[double category]]_ is another term for _[[n-fold category|2-fold category]]_.  Strict 2-categories may be identified with those strict 2-fold/double categories whose category of vertical morphisms is [[discrete category|discrete]], or those whose category of horizontal morphisms is discrete.

(And similarly, strict globular [[n-category|n-categories]] may be identified with those [[n-fold category|n-fold categories]] for which all cube faces "in one direction" are discrete. A similar statement for weak $n$-categories is to be expected, but little seems to be known about this.)

## Definition

### Strict 2-categories
 {#DefinitionStrictTwoCategories}

A _strict 2-category_, often called simply a _2-category_, is a [[enriched category|category enriched over]] [[Cat]], where [[Cat]] is regarded as the [[1-category]] of [[strict categories]] with [[functors]] between them and equipped with the [[cartesian monoidal category|cartesian monoidal]]-[[structure]] given by forming [[product categories]].

### Strict 2-groupoids

Similarly, a [[strict 2-groupoid]] is a [[enriched groupoid|groupoid enriched]] over the [[1-category]] [[Grpd]]. This is also called a *[[geometric shapes for higher structures|globular]] strict 2-groupoid*, to emphasise the underlying geometry.

The category of strict 2-groupoids is [[equivalence of categories|equivalent]] to the category  of [[crossed modules]] over groupoids. It is also equivalent to the category of (strict) [[double groupoids]] "with connections".

They are also special cases of strict globular omega-groupoids, and the category of these is equivalent to the category of [[crossed complex]]es.

### Details

Working out the meaning of *[[Cat]]-[[enriched category]]* ([above](#DefinitionStrictTwoCategories)), we find that a strict 2-category $K$ is given by

* a [[class]] $ob K$ of [[objects]] $a,b,c,\ldots$, together with

* a [[hom-category]] $K(a,b)$ for each $a,b$, and

* [[functors]]

  * ([[identity morphisms]]) $1_a : \mathbf{1} \to K(a,a)$

  * ([[composition]]) $comp : K(b,c) \times K(a,b) \to K(a,c)$

  for each $a,b,c$, satisfying [[associativity]] and [[unitality]] [[axioms]] (as given at *[[enriched category]]*).

As for ordinary ($Set$-enriched) [[categories]], an object $f \in K(a,b)$ is called a _[[1-morphism]]_ or _1-cell_ from $a$ to $b$ and written $f:a\to b$ as usual.  But given $f,g:a\to b$, it is now possible to have non-trivial arrows $\alpha:f\to g \in K(a,b)$, called *[[2-morphisms]]* or _2-cells_ from $f$ to $g$ and written as $\alpha : f \Rightarrow g$.  Because the hom-objects $K(a,b)$ are by definition categories, 2-cells carry an associative and unital operation called _vertical composition_.  The identities for this operation, of course, are the identity 2-cells $1_f$ given by the category structure on $K(a,b)$.

The functor $comp$ gives us an operation of _horizontal_ composition on 2-cells.  Functoriality of $comp$ then says that given $\alpha : f \Rightarrow g : a\to b$ and $\beta : f' \Rightarrow g' : b\to c$, the composite $\comp(\beta,\alpha)$ is a 2-cell $\beta \alpha : f'f \Rightarrow g'g : a \to c$.  Note that the boundaries of the composite 2-cell are the composites of the boundaries of the components.

We also have the _[[interchange law]]_ (also called _Godement law_ or _middle 4 interchange law_): because $comp$ is a functor it commutes with composition in the hom-categories, so we have (writing [[vertical composition]] with $\circ$ and [[horizontal composition]] as juxtaposition):
$$
 (\beta' \circ \beta)(\alpha' \circ \alpha) = (\beta' \alpha') \circ (\beta \alpha)
$$

The [[axioms]] for [[associativity]] and [[unitality]] of $comp$ ensure that [[horizontal composition]] behaves just like [[composition]] of [[morphisms]] in a [[1-category]].  In particular, the action of $comp$ on objects $f,g$ of [[hom-categories]] (i.e. 1-cells of $K$) is the usual composite of morphisms.

### More details
{#detailedDefn}

(See also the section below on [[sesquicategories]], which provide a conceptual package for the [[stuff, structure, property|stuff and structure]] described below.)

In even more detail, a strict $2$-category $K$ consists of *stuff*:

*  a collection $Ob K$ or $Ob_K$ of _[[objects]]_ or _$0$-cells_,

*  for each object $a$ and object $b$, a collection $K(a,b)$ or $Hom_K(a,b)$ of _[[1-morphisms]]_ or _$1$-cells_ $a \to b$, and

*  for each object $a$, object $b$, morphism $f\colon a \to b$, and morphism $g\colon a \to b$, a collection $K(f,g)$ or $2 Hom_K(f,g)$ of _[[2-morphisms]]_ or _$2$-cells_ $f \Rightarrow g$ or $f \Rightarrow g\colon a \to b$,

that is equipped with the following *[[structure]]*:

*  for each object $a$, an _[[identity morphism|identity]]_ $1_a\colon a \to a$ or $\id_a\colon a \to a$,

*  for each $a,b,c$, $f\colon a \to b$, and $g\colon b \to c$, a _composite_ $f ; g\colon a \to c$ or $g \circ f\colon a \to c$,

*  for each $f\colon a \to b$, an _identity_ $1_f\colon f \Rightarrow f$ or $\Id_f\colon f \Rightarrow f$,

*  for each $f,g,h\colon a \to b$, $\eta\colon f \Rightarrow g$, and $\theta\colon g \Rightarrow h$, a _[[vertical composition|vertical composite]]_ $\theta \bullet \eta\colon f \Rightarrow h$,

*  for each $a,b,c$, $f\colon a \to b$, $g,h\colon b \to c$, and $\eta\colon g \Rightarrow h$, a _left [[whiskering]]_ $\eta \triangleleft f\colon g \circ f \Rightarrow h \circ f$, and
*  for each $a,b,c$, $f,g\colon a \to b$, $h\colon b \to c$, and $\eta\colon f \Rightarrow g$, a _right [[whiskering]]_ $h \triangleright \eta \colon h \circ f \Rightarrow h \circ g$,

satisfying the following properties:

1.  for each $f\colon a \to b$, the composites $f \circ \id_a$ and $\id_b \circ f$ each equal $f$,

1.  for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d$, the composites $h \circ (g \circ f)$ and $(h \circ g) \circ f$ are equal,
1.  for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\eta \bullet \Id_f$ and $\Id_g \bullet \eta$ both equal $\eta$,
1.  for each $f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h \overset{\iota}\Rightarrow i\colon a \to b$, the vertical composites $\iota \bullet (\theta \bullet \eta)$ and $(\iota \bullet \theta) \bullet \eta$ are equal,
1.  for each $a \overset{f}\to b \overset{g}\to c$, the [[whiskerings]] $\Id_g \triangleleft f$ and $g \triangleright \Id_f$ both equal $\Id_{g \circ f }$,
1.  for each $\eta\colon f \Rightarrow g\colon a \to b$, the [[whiskerings]] $\eta \triangleleft \id_a$ and $\id_b \triangleright \eta$ equal $\eta$,
1.  for each $f\colon a \to b$ and $g \overset{\eta}\Rightarrow h \overset{\theta}\Rightarrow i\colon b \to c$, the vertical composite $(\theta \triangleleft f) \bullet (\eta \triangleleft f)$ equals the [[whiskering]] $(\theta \bullet \eta) \triangleleft f$,
1.  for each $f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h\colon a \to b$ and $i\colon b \to c$, the vertical composite $(i \triangleright \theta) \bullet (i \triangleright \eta)$ equals the [[whiskering]] $i \triangleright (\theta \bullet \eta)$,
1.  for each $a \overset{f}\to b \overset{g}\to c$ and $\eta\colon h \Rightarrow i\colon c \to d$, the left [[whiskerings]] $\eta \triangleleft (g \circ f)$ and $(\eta \triangleleft g) \triangleleft f$ are equal,
1.  for each $\eta\colon f \Rightarrow g\colon a \to b$ and $b \overset{h}\to c \overset{i}\to d$, the right [[whiskerings]] $i \triangleright (h \triangleright \eta)$ and $(i \circ h) \triangleright \eta$ are equal,
1.  for each $f\colon a \to b$, $\eta\colon g \Rightarrow h\colon b \to c$, and $i\colon c \to d$, the [[whiskerings]] $i \triangleright (\eta \triangleleft f)$ and $(i \triangleright \eta) \triangleleft f$ are equal, and
1.  for each $\eta\colon f \Rightarrow g\colon a \to b$ and $\theta\colon h \Rightarrow i\colon b \to c$, the vertical composites $(i \triangleright \eta) \bullet (\theta \triangleleft f)$ and $(\theta \triangleleft g) \bullet (h \triangleright \eta)$ are equal.

The construction in the last axiom is the _[[horizontal composition|horizontal composite]]_ $\theta \circ \eta\colon h \circ f \to i \circ g$.  It is possible (and probably more common) to take the [[horizontal composition|horizontal composite]] as basic and the [[whiskerings]] as derived operations.  This results in fewer, but more complicated, axioms.

## 2-categories as sesquicategories
 {#AsSesquicategories}

The fine-grained description in the [previous subsection](#detailedDefn) can be concisely repackaged by saying that a 2-category is a [[sesquicategory]] that satisfies the [[interchange axiom]], i.e., the last axiom (12) which gives the [[horizontal composition]]-construction. This description is essentially patterned after the "five rules of functorial calculus" introduced by [Godement (1958)](#Godement58) for the special case [[Cat]].

So to say it again, but a little differently: a sesquicategory consists of a category $K$ (giving the $0$-cells and $1$-cells) together with a functor

$$
  K(-, -)
  \;\colon\;
  K^{op} \times K
    \longrightarrow
  Cat
$$

such that composing $K(-, -)$ with the functor $ob: Cat \to Set$ (the one sending a category to its set of objects) gives $\hom_K: K^{op} \times K \to Set$, the hom-functor for the category $K$. So: for $0$-cells $a, b$, the objects of the category $K(a, b)$ are $1$-cells $f \in \hom_K(a, b)$. The morphisms of $K(a, b)$ are $2$-cells (with $0$-source $a$ and $0$-target $b$). Composition within the category $K(a, b)$ corresponds to [[vertical composition]].

For each object $a$ of $K$ and each morphism $h: b \to c$ of $K$, there is a functor $K(a, h): K(a, b) \to K(a, c)$. This is *right [[whiskering]]*; it sends a [[2-morphism|2-cell]] $\eta$ (a morphism of $K(a, b)$) to a morphism $h \triangleright \eta$ of $K(b, c)$. Similarly, for each object $c$ and morphism $f: a \to b$, there is a functor $K(f, c): K(b, c) \to K(a, c)$. This is *left [[whiskering]]*; it sends a [[2-morphism|2-cell]] $\eta$ (a morphism of $K(b, c)$) to a morphism $\eta \triangleleft f$ of $K(a, c)$.

The long list of compatibility properties enumerated in the previous subsection, all except the last, are concisely summarized in the definition of sesquicategory as recalled above. For example, property (8) just says that left [[whiskering]] preserves [[vertical composition]], as it must since it is a [[functor]] (a morphism in $Cat$).

In summary, a sesquicategory consists of "stuff" and structure as described in the [previous subsection](/nlab/show/strict+2-category#detailedDefn), satisfying properties 1-11. A 2-category is then a sesquicategory that further satisfies the [[interchange axiom]] (12). Some further illumination of this point of view can be obtained by contemplating [[string diagrams]] for 2-categories, where the [[interchange axiom]] corresponds to [[isotopies]] of (planar, progressive) string diagrams during which the relative heights of nodes labeled by 2-cells are interchanged.

## Remarks

* Strict 2-categories are the same as a [[strict omega-categories]] which are trivial in degree $n \geq 3$.

* This is to be contrasted with a _weak 2-category_ called a [[bicategory]]. In a strict 2-category composition of 1-morphisms is strictly associative and composition with [[identity morphism]]s strictly satisfies the required identity law. In a weak 2-category these laws may hold only up to coherent 2-morphisms.

## In dependent type theory

In [[dependent type theory]], there are multiple notions of a "strict 2-category", because there are multiple notions of a [[category]]:

* A strict 2-category could be a [[precategory]] enriched over the [[cartesian monoidal category]] [[Cat]] of [[strict categories]]
* A strict 2-category could be a [[strict category]] enriched over the cartesian monoidal category Cat of strict categories
* A strict 2-category could be a [[univalent category]] enriched over the cartesian monoidal category Cat of strict categories

The first definition is a naive translation of "strict 2-category" from set theory to dependent type theory, but in the absence of [[axiom K]] or a similar axiom, these strict 2-categories behave differently from the strict 2-categories as defined in [[set theory]]. The second definition adds a [[0-truncation]] condition to the type of objects to ensure that the strict 2-categories actually behave like the strict 2-categories in set theory. The third definition satisfies the [[principle of equivalence]]: equality of objects is the same as isomorphism of strict categories, and ensures that strict 2-categories are [[h-groupoids]].

## Related concepts

* [[strict 2-groupoid]]

* [[strict 2-functor]]

* [[strict 2-natural transformation]]

* [[strict adjoint 2-functor]]

* [[strict 2-group]]

## History

As intimated above, the essential rules which abstractly govern the behavior of functors and natural transformations and their various compositions were made explicit by [Godement (1958)](#Godement58), in his "five rules of functorial calculus". He did not however go as far as use these rules to define the abstract notion of 2-category; this step was taken later by [Bénabou (1965)](#Bénabou65). In any event, the primitive compositional operations in [Godement (1958)](#Godement58) were what we call *[[vertical composition]]* and *[[whiskering]]*, with [[horizontal composition]] of [[natural transformations]] being a derived operation (made unambiguous in the presence of the interchange axiom). Indeed, [[horizontal composition]] is often called the *Godement product*.

A few years after introducing 2-categories, Bénabou introduced the more general notion of *[[bicategories]]*.

Literature references for the abstract notion of [[sesquicategory]], a structure in which vertical compositions and whiskerings are primitive, do not seem to be abundant, but they are mentioned for example in [Street (1996)](#Street96) together with the observation that 2-categories are special types of sesquicategories (page 535).

## References
 {#References}

The "five rules of functorial calculus" ([above](#AsSesquicategories)) were formulated (as: *cinq règles de calcul fonctoriel*) in:

* {#Godement58} [[Roger Godement]], Appendice (pp. 269) of:  *Topologie algébrique et theorie des faisceaux*, Actualités Sci. Ind. **1252**, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), [[Godement-TopologieAlgebrique.pdf:file]]]

The definition of "[[double categories]]" (of which, at least in hindsight, strict 2-categories are an immediate special case) is due to:

* {#Ehresmann63} [[Charles Ehresmann]], *Catégories double et catégories structurées*, C.R. Acad. Paris 256 (1963) 1198-1201 [[[Ehresmann-CategoriesDoubles.pdf:file]], [gallica](https://gallica.bnf.fr/ark:/12148/bpt6k3208j/f1246)]

also discussed (according to [reviewers](https://www.jstor.org/stable/2314770) who have seen the text) in:

* [[Charles Ehresmann]], *Catégories et structures*, Dunod, Paris, (1965)

However, Ehresmann did not isolate the notion of strict 2-categories as such.

But apparently inspired by [Ehresmann (1963)](#Ehresmann63) the actual definition of strict 2-categories is due to:

* {#Bénabou65} [[Jean Bénabou]], Example (5) of: *Catégories relatives*, C. R. Acad. Sci. Paris **260** (1965) 3824-3827 [[gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4019v/f37.item)]

  > (conceived as [[Cat]]-[[enriched categories]] and under the name "2-categories")

* {#Maranda65} [[Jean-Marie Maranda]], Def. 1 in: *Formal categories*, Canadian Journal of Mathematics **17** (1965) 758-801 [[doi:10.4153/CJM-1965-076-0](https://doi.org/10.4153/CJM-1965-076-0), [pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A7C463460EB8CAC64C2CA340F870CF80/S0008414X00039729a.pdf/formal-categories.pdf)]

  > (conceived as [[Cat]]-[[enriched categories]] and under the name "categories of the second type")

* {#EK65} [[Samuel Eilenberg]], [[G. Max Kelly]], p. 425 of: *Closed Categories*,  in:  [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

  > (expressed entirely in components and under the name *hypercategories*)

and the notion is invoked for various purposes (such as in speaking of [[Cat]] as a 2-category) in:

* {#Bénabou68} [[Jean Bénabou]], *Structures algébriques dans les catégories*, Cahiers de topologie et géométrie différentielle, **10** 1 (1968) 1-126 [[doi:CTGDC_1968__10_1_1_0](http://www.numdam.org/item/?id=CTGDC_1968__10_1_1_0)]

Exposition and review:

* {#Street96}  [[Ross Street]], p. 535 in: *Categorical Structures*, in Handbook of Algebra Vol. 1 (ed. M. Hazewinkel), Elsevier Science, Amsterdam (1996) [doi:10.1016/S1570-7954(96)80019-2, [pdf](http://maths.mq.edu.au/~street/45.pdf), [[Street-CategoricalStructures.pdf:file]] [ISBN:978-0-444-82212-3](https://shop.elsevier.com/books/handbook-of-algebra/hazewinkel/978-0-444-82212-3)]

* [[Saunders MacLane]], §XII.3 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#Lack10} [[Steve Lack]], _A 2-categories companion_,  In: Baez J., May J. (eds.) *[[Towards Higher Categories]]*. The IMA Volumes in Mathematics and its Applications, vol 152. Springer 2010 ([arXiv:math.CT/0702535](http://arxiv.org/abs/math.CT/0702535), [doi:10.1007/978-1-4419-1524-5_4](https://doi.org/10.1007/978-1-4419-1524-5_4))

* {#Richter19} [[Birgit Richter]], Section 9.5 of: _From categories to homotopy theory_, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press 2020 ([doi:10.1017/9781108855891](https://doi.org/10.1017/9781108855891), [book webpage](https://www.math.uni-hamburg.de/home/richter/catbook.html), [pdf](https://www.math.uni-hamburg.de/home/richter/bookdraft.pdf))

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], Section 2.3 of: _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))

See also

* Wikipedia, *[Strict 2-category](https://en.wikipedia.org/wiki/Strict_2-category)*

The special case of [[strict (2,1)-categories]]:

* [[Peter H. H. Fantham]], [[Eric J. Moore]], *Groupoid Enriched Categories and Homotopy Theory*, Canadian Journal of Mathematics **35** 3 (1983) 385-416 ([doi:10.4153/CJM-1983-022-8](https://doi.org/10.4153/CJM-1983-022-8))

[[!redirects strict 2-category]]
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strict 2-equivalence of 2-categories

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A **strict 2-equivalence** between [[strict 2-categories]] is an [[equivalence of categories|equivalence]] of [[Cat]]-[[enriched category|enriched]] categories.  Thus, it consists of a pair of [[strict 2-functors]] and [[strict 2-natural transformation|strict 2-natural isomorphisms]] relating their composites to identities.

Traditionally this is called simply a *2-equivalence*, with the word *biequivalence* used for [[equivalence of 2-categories|the weaker notion]].  On the nLab we tend to say simply *equivalence* for the weaker notion, in line with our general philosophy that weak higher categories deserve the unadorned names.

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strict 2-functor

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## Idea

A _strict 2-functor_ is a [[2-functor]] between [[strict 2-categories]] which strictly respects the [[composition]] operation of [[1-morphisms]] and [[2-morphisms]]. This is in contrast to _[[weak 2-functors]]_ (also called _[[pseudofunctors]]_) which may respect the composition of [[1-morphisms]] only up to [[natural isomorphism]] (hence up to [[2-morphisms]] in the [[2-category of categories]]).

While the generally "correct" concepts in [[2-category theory]] are _[[weak 2-categories]]_ with _[[weak 2-functors]]_ between them, it is often useful to recognize strict 2-functors among weak 2-functors, or to replace, up to [[eqivalence of 2-categories]] and [[pseudonatural transformation]] , weak 2-functors by strict ones.

## Definition

Let [[Cat]] be the _[[1-category]]_ of [[categories]], regarded as a [[Bénabou cosmos]] for [[enriched category theory]] via its [[cartesian closed category]]-[[structure]].

Notice that [[strict 2-categories]] may be identified with [[Cat]]-[[enriched categories]]. Under this identification, strict 2-functors are [[Cat]]-[[enriched functors]].

## Related concepts

* [[strict 2-category]]

* [[strict 2-groupoid]]

* [[strict 2-group]]

* [[strict 2-natural transformation]]

* [[strict adjoint 2-functor]]

* [[2-functor]], [[double functor]]

* [[n-functor]]

[[!redirects strict 2-functors]]


strict 2-group

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## Idea

The notion of strict 2-group is a strict [[vertical categorification]] of that of [[group]].

A strict 2-group is a [[group object]] [[internalization|internal]] to the [[category]] [[Grpd]] of [[groupoid]]s (regarded as an ordinary category, not as a [[2-category]]).

This means that it is a groupoid $G$ equipped with a product [[functor]] $\cdot : G \times G \to G$ that behaves like the product in a [[group]], in that it is unital and associative and such that there are inverses under multiplication.

More general [[2-group]]s correspond to group objects in the [[2-category]] incarnation of [[Grpd]]. For them associativity, inverses etc have to hold and exist only up to coherent natural isomorphism. So strict 2-groups are particularly rigid incarnations of [[2-group]]s.

We may think of any 2-group $G$ in terms of its [[delooping]] $\mathbf{B}G$, a [[2-groupoid]] with a single object, with morphisms the objects of $G$ and [[k-morphism|2-morphism]]s the morphisms of $G$. If $G$ is a strict 2-group, then $\mathbf{B}G$ is a strict 2-groupoid. This is often a useful point of view. In particular, the general strictification result of [[bicategory|bicategories]] implies that any such 2-groupoid is equivalent to a strict one. So, up to the right notion of equivalence, strict 2-groups already exhaust all 2-groups; we just have to take care to allow for *homomorphisms* of these $2$-groups to be weak. (However, this theorem may not apply to structured $2$-groups, such as [[Lie 2-group]]s.)

Strict 2-groups are also equivalently encoded in terms of [[crossed module]]s $(G_2 \to G_1)$ of ordinary groups: $G_1$ is the group of [[object]]s of the groupoid $G$ and $G_2$ the group of [[morphism]]s in $G$ whose source is the neutral element in $G_1$.

In applications it is usually useful to pass back and forth between the 2-groupoid incarnation of strict 2-groups and their incarnation as crossed modules. The first perspective makes transparent many constructions, while the second perspective gives a useful means to do computations with 2-groups. The translation between the two points of view is described in detail below.

##Definitions

A **strict [[2-group]]** is equivalently:

* an [[internal category]] in [[Grp]],
* an [[internal groupoid]] in [[Grp]],
* an internal [[group object]] in [[Cat]],
* an internal [[group object]] in [[Grpd]],
* a [[strict 2-groupoid]] with one object.

## Expanding the definition

We examine the first definition in more detail.

Copying and adapting from the entry on general internal categories we have:

A [[internal category]] in [[Grp]] is

* a collection of group homomorphisms of the form
  $$
    C_1 \stackrel{s,t}{\to} C_0 \stackrel{i}{\to} C_1
  $$
such that the composites $s\cdot i$ and $t\cdot i$ are the identity morphisms on $C_0$, and
  such that, writing  $C_1 \times_{t,s} C_1$ for the pullback,
  $$
    \array{
      C_1 \times_{t,s} C_1 &\to& C_1
      \\
      \downarrow && \downarrow^{t}
      \\
      C_1 &\stackrel{s}{\to}& C_0
    }
  $$
  there is, in addition,  a homomorphism
  $$
     C_1 \times_{t,s} C_1 \stackrel{comp}{\to} C_1
  $$
  "respecting $s$ and $t$";

 * such that the _composition_ $comp$ is associative
   and unital with respect to $i$ "in the obvious way".

### In terms of strict 2-groupoids

Every strict 2-group $G$ defines a strict [[2-groupoid]] $\mathbf{B}G$ -- called its [[delooping]] -- defined by the fact that

* $\mathbf{B}G$ has a single object $\bullet$;

* The [[hom-object|hom-groupoid]] $\mathbf{B}G(\bullet,\bullet) = G$ is the 2-group $G$ itself, regarded as a [[groupoid]];

* the horizontal composition in $\mathbf{B}G$ is given by the group product operation on $G$.

Conversely, every strict 2-groupoid with a single object $\bullet$ defines a 2-group this way.

Beware, however, as discussed in detail at [[crossed module]], that (strict) 2-groups and (strict) one-object 2-groupoids, live is somewhat different 2-categories. If one wants to really _identify_ $\mathbf{B}G$ in a way that respects morphisms between these objects, one needs to think of $\mathbf{B}G$ as a [[pointed object]] equipped with its unique pointing ${*} \to \mathbf{B}G$.

### In terms of crossed modules {#InTermsOfCrossedModules}

We describe how a [[crossed module]]

$$
  [\mathbf{B}G] = (G_2 \stackrel{\delta}{\to} G_1)
$$

with [[action]]

$$
  \alpha
  \,\colon\,
  G_1
    \longrightarrow
  Aut(G_2)
$$

encodes a strict one-object 2-groupoid $\mathbf{B}G$, and hence a strict 2-group $G$.

There are four isomorphic but different ways to construct $\mathbf{B}G$ from $[\mathbf{B}G]$, which differ by whether the composition of 1-morphisms and of 1-morphisms with 2-morphisms in $\mathbf{B}G$ is taken to correspond to the product in the groups $G_1$ and $G_2$, respectively, or in their opposites.

In concrete computations it happens that not all of these choices directly yield the expected formulas in terms of classical group theory from a given diagrammatics involving $\mathbf{B}G$. While all choices will be isomorphic, some will be more convenient. Therefore often it matters which one of the four choices below one takes in order to get a streamlined translation between diagrammatics and formulas. For concrete examples of this phenomenon in practice see [[nonabelian group cohomology]] and [[gerbe]].

We now define the one-object strict [[2-groupoid]] $\mathbf{B}G$ from the crossed module $(\delta : G_2 \to G_1)$ with action $\alpha : G_1 \to Aut(G_2)$.

* $\mathbf{B}G$ has a single [[object]] $\bullet$;

* The set of [[k-morphism|1-morphisms]] of $\mathbf{B}G$ is the group $G_1$:

  $$
    1Mor_{\mathbf{B}G}(\bullet, \bullet) := G_1
    \,.
  $$

  For $g \in G_1$ we write $\bullet \stackrel{g}{\to} \bullet$ for the corresponding 1-morphism in $\mathbf{B}G$;

* Compositition of 1-morphisms is given by the product operation in $G_1$. There are two choices for the order in which to form the product.

  * **(convention F)** horizontal composition is given by

    $$
      (\bullet \stackrel{g_1}{\to} \bullet
      \stackrel{g_2}{\to} \bullet)
      \;\;
      =
      \;\;
      (\bullet \stackrel{g_1 g_2}{\to} \bullet)
    $$

  * **(convention B)** horizontal composition is given by

    $$
      (\bullet \stackrel{g_1}{\to} \bullet
      \stackrel{g_2}{\to} \bullet)
      \;\;
      =
      \;\;
      (\bullet \stackrel{g_2 g_1}{\to} \bullet)
    $$

* The [[set]] of [[k-morphism|2-morphism]]s of $\mathbf{B}G$ is the cartesian [[product]] $G_1 \times G_2$ where

  * the source operation is projection on the first factor

    $$
      s
        \coloneqq
      p_1
      \,\colon\,
      G_1 \times G_2 \to G_1
    $$

  * the target operation on morphisms starting at the identity morphism is the boundary map $\delta : G_2 \to G_1$ of the crossed module combined with the product in $G_1$

    $$
      t|_{{Id}\times G_2} = \delta
    $$

  So in diagrams this means that a 2-morphism
  corresponding to $(Id, h) \in G_1 \times G_2$ is labelled as

  $$
    \array{
       &  \nearrow \searrow^{\mathrlap{Id}}
       \\
      \bullet &\Downarrow^{\mathrlap{h}}& \bullet
      \\
       &  \searrow \nearrow_{\mathrlap{\delta(h)}}
    }
    \,.
  $$

  The target of general 2-morphisms labeled by $h$ and starting at some $g$ is either $\delta(h)g $ or $g \delta(h)$, depending on the choice of conventions discussed in the following.

* Horizontal composition of 1-morphisms with 2-morphisms ("whiskering") is determined by the rule

  * **(convention R)

    $$
      \array{
        & \nearrow \searrow^{Id}
        \\
        \bullet &\Downarrow^h& \bullet
        &\stackrel{g}{\to}&
        \bullet
        \\
        & \searrow \nearrow
      }
      \;\;
      :=
      \;\;
      \array{
        & \nearrow \searrow^{g}
        \\
        \bullet &\Downarrow^h& \bullet
        \\
        & \searrow \nearrow
      }
    $$

    $$
      \array{
       && & \nearrow \searrow^{Id}
        \\
        \bullet
        &\stackrel{g}{\to}&
        \bullet &\Downarrow^h& \bullet
        \\
        && & \searrow \nearrow
       }
      \;\;
      :=
      \;\;
      \array{
        & \nearrow \searrow^{g}
        \\
        \bullet &\Downarrow^{\alpha(g)(h)}& \bullet
        \\
        & \searrow \nearrow
      }
    $$

  * **(convention L)

    $$
      \array{
       && & \nearrow \searrow^{Id}
        \\
        \bullet
        &\stackrel{g}{\to}&
        \bullet &\Downarrow^h& \bullet
        \\
        && & \searrow \nearrow
       }
      \;\;
      :=
      \;\;
      \array{
        & \nearrow \searrow^{g}
        \\
        \bullet &\Downarrow^h& \bullet
        \\
        & \searrow \nearrow
      }
    $$

    $$
      \array{
        & \nearrow \searrow^{Id}
        \\
        \bullet &\Downarrow^h& \bullet
        &\stackrel{g}{\to}&
        \bullet
        \\
        & \searrow \nearrow
      }
      \;\;
      :=
      \;\;
      \array{
        & \nearrow \searrow^{g}
        \\
        \bullet &\Downarrow^{\alpha(g)(h)}& \bullet
        \\
        & \searrow \nearrow
      }
    $$

* Horizontal composition of 2-morphisms
  starting at the identity 1-morphism is fixed by the
  convention chosen for composition of 1-morphisms

  * in **convention F**

    $$
      \array{
        & \nearrow \searrow^{\mathrlap{Id}} &
        &
         \nearrow \searrow^{\mathrlap{Id}}
        \\
        \bullet &\Downarrow^{h_1}&
        \bullet
        &\Downarrow^{h_2}&
        \bullet
        \\
        & \searrow \nearrow && \searrow \nearrow
      }
      \;\;\;
      =
      \;\;\;
      \array{
        & \nearrow \searrow^{\mathrlap{Id}}
        \\
        \bullet &\Downarrow^{h_1 h_2}&
        \bullet
        \\
        & \searrow \nearrow
      }
    $$

  * in **convention B**

    $$
      \array{
        & \nearrow \searrow^{\mathrlap{Id}} &
        &
       \nearrow \searrow^{\mathrlap{Id}}
        \\
        \bullet &\Downarrow^{h_1}&
        \bullet
        &\Downarrow^{h_2}&
        \bullet
        \\
        & \searrow \nearrow && \searrow \nearrow
      }
      \;\;\;
      =
      \;\;\;
      \array{
        & \nearrow \searrow^{\mathrlap{Id}}
        \\
        \bullet &\Downarrow^{h_2 h_1}&
        \bullet
        \\
        & \searrow \nearrow
      }
    $$

  Notice that this is compatible with the source-target maps due to the fact that that $\delta$ is a group homomorphism.

* With these choices made, all other compositions are now fixed by use of the exchange law:

* Vertical composition of composable 2-morphisms is given, on the labels, by the product in $G_2$, in the following order

  **(in convention L B)**

  $$
    \array{
       & \nearrow &\Downarrow^{\mathrlap{h_1}}& \searrow
       \\
       \bullet &&\to&& \bullet
       \\
       & \searrow &\Downarrow^{\mathrlap{h_2}}& \nearrow
    }
    \;\;\;
    =
    \;\;\;
    \array{
      & \nearrow \searrow
      \\
      \bullet
      &\Downarrow^{\mathrlap{h_2 h_1}}& \bullet
      \\
      & \searrow \nearrow
    }
    \,.
  $$

## Examples

### From crossed modules

By the above, every [[crossed module]] gives an example of a 2-group.

But the nature of some strict 2-groups is best understood by genuinely regarding them as 2-categorical structures. This is true notably for the example of the automorphism 2-groups, discussed below. These, too, of course are equivalenly encoded by crossed modules, but that may hide a bit their structural meaning.

### Automorphism 2-groups

For $a$ any [[object]] in a [[strict 2-category]] $C$, there is the strict **automorphism 2-group** $Aut_C(a)$ whose

* objects are 1-[[isomorphism]]s $a \to a$ in $C$;

* morphisms are 2-isomorphisms between these 1-isomorphisms.

In particular, for $K$ a [[group]] and $\mathbf{B}K$ its [[delooping]] [[groupoid]], we have the automorphism 2-group of $\mathbf{B}K$ in the 2-category [[Grpd]]. This is usually called the **automorphism 2-group of the group $K$**

$$
  AUT(K) := Aut_{Grpd}(\mathbf{B}K)
  \,.
$$

Its objects are the ordinary [[automorphism]]s of $K$ in [[Grp]], while its 2-morphisms go between two automorphisms that differ by an inner automorphism.

Accordingly, the [[crossed module]] corresponding to the 2-group $AUT(K)$ is

$$
  [AUT(K)] =
  \left(
    \array{
      K &\stackrel{Ad}{\to}& Aut(K)
      \\
    }
  \right)
  \,,
$$

where the boundary map is the one that sends each element $k \in K$ to the inner automorphism given by conjugation with $k$:

$$
  Ad(k) : q \mapsto k q k^{-1}
  \,.
$$

### From congruence relations

Perhaps the _simplest example_ of such a structure is a
[[congruence|congruence relation]] on a group $G$. If $\sim$ is a congruence relation on $G$, then we form the 2-group by setting $C_0 = G$ and $C_1$ to be the group of pairs $(a,b)$ with $a\sim b$.  That this is a group follows from the definition of congruence given in the above reference. The two maps $s$ and $t$ are defined by $s(a,b) = a$, $t(a,b) = b$, whilst $i(a) = (a,a)$. The pullback is a subgroup of $C_1\times C_1$ given by all 'pairs of pairs' $((a,b),(b,c))$ and the composition homomorphism sends such a pair to $(a,c)$.  The other properties are easy to check.

Any congruence relation corresponds to a [[normal subgroup]], given by those elements $a$ that are congruent to the identity element of $G$, so that $e\sim a$. Likewise given a normal subgroup $N$ of $G$ you get a congruence, with $a \sim b$ iff $b^{-1} a$ (or equivalently, $a b^{-1}$) belongs to $N$.

## Related concepts

* [[2-group]]

* [[strict 2-category]], [[strict 2-groupoid]]

## References

For more see the references at *[[2-group]]* and at *[[crossed module]]*.

The equivalence between strict 2-groups and [[crossed modules]]:

* [[Ronnie Brown]] and C. Spencer, _G-groupoids, crossed modules and the fundamental groupoid of a topological group_, Proc. Kon. Ned. Akad. v. Wet **79** (1976) 296-302.

Textbook account:

* [[Saunders MacLane]], §XII.8 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

[[!redirects strict 2-groups]]


strict 2-groupoid

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# Strict $2$-groupoids
* table of contents
{: toc}

## Idea

The general notion of [[2-groupoid]] is also called _weak $2$-groupoid_ to distinguish from the special case of strict $2$-groupoids.

## Definitions

A __strict $2$-groupoid__ is equivalently:

*  a [[strict 2-category]] in which both [[1-morphisms]] and [[2-morphisms]] are [[isomorphism|invertible]],

*  a [[Grpd]]-[[enriched category]] in which all [[1-morphisms]] are strictly invertible,

* a [[Grpd]]-[[enriched groupoid]],

*  a [[strict omega-groupoid]] that is $2$-[[truncated]] (i.e. in which all [[k-morphism]]s for $k \gt 2$ are [[identities]]).

## Properties

A strict 2-groupoid is in particular a [[strict 2-category]] and a [[2-groupoid]], but a 2-groupoid that is a strict 2-category need not be a strict 2-groupoid, since its 1-morphisms might only be weakly invertible.

Strict $2$-groupoids embed into all $2$-groupoids (modeled by [[bigroupoids]]) by regarding a strict $2$-category as a special case of a [[bicategory]].  They embed into all $2$-groupoids modeled as [[Kan complex]]es via the [[omega-nerve]].

A strict 2-groupoid can also be identified with a [[crossed complex]] of the form $(G_2 \to G_1 \stackrel{\longrightarrow}{\longrightarrow} G_0)$.

Strict 2-groupoids still model all [[homotopy 2-types]]. See also at _[homotopy hypothesis -- for homotopy 2-types](homotopy+hypothesis#For2Groupoids)_.

## Examples

### Fundamental 2-groupoid

The [[fundamental 2-groupoid]] of a [[topological space]] with [[thin homotopy|thin]] [[homotopy classes]] of paths ([Hardie, Kapms & Kieboom (2000)](#HardieKapmsKieboom00)).

Similarly the strict [[path 2-groupoid]] of a [[manifold]].

### Delooping of strict 2-groups

The [[deloopings]] of [[strict 2-groups]] are the strict 2-groupoids with a single object.

About the simplest example of such an object then comes from a [[group homomorphism]]

$$\phi: H\to G$$
as follows.

Just as a function between sets, $f : X\to Y$ defines as an equivalence relation on $X$ by $x_0\sim x_1$ if and only if $\phi x_0 = \phi x_1$, so here we get an equivalence relation on the group $H$.  That equivalence relation is a [[congruence]] so is an internal equivalence relation, that is, it is internal to the category of groups.  An equivalence relation is also a [[groupoid]] in a well known way, and here we get an internal groupoid within the category of groups.  There will be a group of objects, namely $H$, and a group of arrows, given by $H \times_G H$, the pullback of $\phi$ along itself. Working this out, clearly this group consists of the pairs $(h_0,h_1)$ of elements having the same image in $G$.Such a pair goes from $h_0$ to $h_1$. The composition is then very simple:
$$(h_0,h_1)\star (h_1,h_2) = (h_0,h_2)$$
and inverses are given by swapping the two entries, $(h_0,h_1)^{-1} = (h_1,h_0)$. All this is happening within that same group theoretic context and there is a group multiplication on each of the sets given by $(h_0,h_1).(h^\prime_0,h^\prime_1) = (h_0h^\prime_0,h_1h^\prime_1)$. With this the maps giving the source and target of an 'arrow' are homomorphisms as is the composition.

## Related concepts

* [[strict 2-category]], [[strict 2-functor]]

* [[strict 2-group]]

* [[n-groupoid]], [[infinity-groupoid]]

## References

See also the references at *[[strict 2-category]]*.

The [[fundamental 2-groupoid]] of a topological space with [[thin homotopy|thin]] [[homotopy classes]] of paths:

* {#HardieKapmsKieboom00} [[Keith A. Hardie]], [[Klaus H. Kamps]], [[Rudger Kieboom]], *A homotopy 2-groupoid of a Hausdorff space*. Papers in honour of Bernhard Banaschewski (Cape Town, 1996). Appl. Categ. Structures **8** (2000) 209-234 [[doi:10.1023/A:1008758412196](https://doi.org/10.1023/A:1008758412196)]

[[!redirects strict 2-groupoid]]
[[!redirects strict 2-groupoids]]