walking equivalence

\tableofcontents

\section{Idea}

The _walking equivalence_ (as in "[[walking structure]]") or _free-standing equivalence_ is the [[2-category]] (in fact a [[(2,1)-category]]) which 'represents' [[equivalence|equivalences]] in a 2-category. It is a [[categorification]] of the [[free-standing isomorphism]], though not the only one: the [[walking adjoint equivalence]] is another.

Roughly speaking, it is the minimal 2-category which contains a 1-arrow $f$ with an inverse-up-to-isomorphism $g$, that is to say, which contains in addition to $g$ a [[2-isomorphism]] between $g f$ and the identity, and a 2-isomorphism between $f g$ and the identity.

The _walking semi-strict equivalence_ is the same except that either $f g$ or $g f$ is required to be equal on the nose to the identity.

\section{Definitions}

\begin{defn} \label{DefinitionFreeStandingEquivalence} Let $F_{\leq 1}$ be the [[free category]] on the [[directed graph]] with exactly two objects $0$ and $1$, an arrow $i: 0 \rightarrow 1$, and an arrow $i^{-1}: 1 \rightarrow 0$. Let $F$ be the free strict 2-category on the 2-truncated [[reflexive globular set]] whose 1-truncation is the underlying reflexive directed graph of $F_{\leq 1}$, and which in addition has a 2-arrow $\iota_{0}: i^{-1} \circ i \rightarrow id(0)$, a 2-arrow $\iota_{0}^{-1}: id(0) \rightarrow i^{-1} \circ i$, a 2-arrow $\iota_{1}: i \circ i^{-1} \rightarrow id(1)$, and a 2-arrow $\iota_{1}^{-1}: i \circ i^{-1} \rightarrow id(1)$.

The _free-standing equivalence_ is the quotient $\mathcal{E}$ of $F$ by the relation on 2-arrows generated by forcing the equations $\iota_{0}^{-1} \circ \iota_{0} = id$, $\iota_{0} \circ \iota_{0}^{-1} = id$, $\iota_{1}^{-1} \circ \iota_{1} = id$, and $\iota_{1} \circ \iota_{1}^{-1} = id$ to hold. \end{defn}

\begin{rmk} The [[1-morphism|arrow]] $i: 0 \rightarrow 1$ is an [[equivalence]], whose inverse-up-to-isomorphism is the arrow $i^{-1}:1 \rightarrow 0$. \end{rmk}

\begin{rmk} The [[2-morphism|2-arrows]] $\iota_{0}$ and $\iota_{1}$ are [[2-isomorphism|2-isomorphisms]]. \end{rmk}

\begin{rmk} The free-standing equivalence is a [[(2,1)-category]], that is, all its 2-morphisms are invertible. \end{rmk}

\begin{defn} Let $F_{\leq 1}$ be as in Definition \ref{DefinitionFreeStandingEquivalence}. Let $F$ be the free strict 2-category on the 2-truncated [[reflexive globular set]] whose 1-truncation is the underlying reflexive directed graph of $F_{\leq 1}$, and which in addition has a 2-arrow $\iota_{1}: i \circ i^{-1} \rightarrow id(1)$, and a 2-arrow $\iota_{1}^{-1}: i \circ i^{-1} \rightarrow id(1)$.

The _free-standing semi-strict equivalence_ is the quotient $\mathcal{E}_{semi}$ of $F$ by the relation on 1-arrows which forces that $i^{-1} \circ i = id(0)$, and by the relation on 2-arrows which forces that $\iota_{1}^{-1} \circ \iota_{1} = id$, and $\iota_{1} \circ \iota_{1}^{-1} = id$.
 \end{defn}

\begin{rmk} The free-standing semi-strict equivalence is the quotient of the free-standing equivalence by the relation on 1-arrows which identifies $i^{-1} \circ i$ and $id(0)$, and which identifies $\iota_{0}$ and $\iota^{-1}$ with $id\left(id(0)\right)$. \end{rmk}

\section{Representing of equivalences}

The 2-category $\mathcal{E}$ is the model for all equivalences in all 2-categories. In other words, any equivalence in a 2-category $\mathcal{A}$ is just a [[2-functor]] from $\mathcal{E}$:

\begin{prpn} Let $\mathcal{A}$ be a 2-category (weak or strict). Let $\mathcal{E}$ denote the free-standing equivalence. Let $f$ be a 1-arrow of $\mathcal{A}$ which is an [[equivalence]], the equivalence being exhibited by a 1-arrow $f^{-1}$, a 2-isomorphism $\phi_{0}: f^{-1} \circ f \rightarrow id$, and a 2-isomorphism $\phi_{1}: f \circ f^{-1} \rightarrow id$. Then there is a unique [[2-functor]] $F: \mathcal{E} \rightarrow \mathcal{A}$ such that the arrow $i:0 \rightarrow 1$ of $\mathcal{I}$ maps under $F$ to $f$, such that the arrow $i^{-1}: 1 \rightarrow 0$ of $\mathcal{I}$ maps under $F$ to $f^{-1}$, such that $\iota_{0}$ maps under $F$ to $\phi_{0}$, and such that $\iota_{1}$ maps under $F$ to $\phi_{1}$. \end{prpn}

\begin{proof} Immediate from the definitions. \end{proof}

\begin{prpn} Let $\mathcal{A}$ be a 2-category (weak or strict). Let $\mathcal{E}_{semi}$ denote the free-standing semi-strict equivalence. Let $f$ be a 1-arrow of $\mathcal{A}$ which is a [[semi-strict equivalence]], the equivalence being exhibited by a 1-arrow $f^{-1}$ and a 2-isomorphism $\phi_{1}: f \circ f^{-1} \rightarrow id$. Then there is a unique [[2-functor]] $F: \mathcal{E}_{semi} \rightarrow \mathcal{A}$ such that the arrow $i:0 \rightarrow 1$ of $\mathcal{I}$ maps under $F$ to $f$, and such that $\iota_{1}$ maps under $F$ to $\phi$. \end{prpn}

\begin{proof} Immediate from the definitions. \end{proof}

\section{Structured interval}

Let $\mathcal{E}_{semi}$ denote the free-standing semi-strict equivalence. We shall view it as an [[interval object]] equipped with all the structures required for Corollary XV.6 and Corollary XV.7 of [Williamson2011](#Williamson2011).

Throughout, we shall denote the category of strict 2-categories by [[2Cat]], and denote the final object of [[2Cat]] by $1$.

\begin{notn} We denote by $i_{0}$ (resp. $i_{1}$) the [[2-functor|functor]] $1 \rightarrow \mathcal{E}_{semi}$ which picks out the object $0$ (resp. $1$) of $\mathcal{E}_{semi}$. \end{notn}

\begin{notn} \label{NotationContractionStructure} We denote by $p$ the canonical functor $\mathcal{E}_{semi} \rightarrow 1$. It is immediate that it defines a _contraction structure_ on $\left(\mathcal{E}_{semi}, i_{0}, i_{1} \right)$ in the sense of VI.6 of [Williamson2011](#Williamson2011). \end{notn}

\begin{notn} We denote by $v$ the functor $\mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi}$ determined by $i \mapsto i^{-1}$, $i^{-1} \mapsto i$, $\iota_{1}, \iota_{1}^{-1} \mapsto id\left(id(0)\right)$. It defines an _involution structure_ on $\left(\mathcal{E}_{semi}, i_{0}, i_{1} \right)$ in the sense of VI.10 of [Williamson2011](#Williamson2011).

Since $1$ is a final object of [[2Cat]], it is immediate that $v$ is compatible with the contraction structure $p$ of Notation \ref{NotationContractionStructure} in the sense of VI.12 of [Williamson2011](Williamson2011).
\end{notn}

\begin{notn} Let

\begin{tikzcd}
1 \ar[r, "i_{0}"] \ar[d, swap, "i_{1}"] & \mathcal{E}_{semi} \ar[d, "r_{0}"] \\
\mathcal{E}_{semi} \ar[r, swap, "r_{1}"] & S
\end{tikzcd}

be a [[pushout|co-cartesian square]] in [[2Cat]].

Explicitly, let $F_{\leq 1}$ be the [[free category]] on the [[directed graph]] with exactly three objects $0$, $1$, and $2$, and with non-identity arrows $r_1: 0 \rightarrow 1$, $r_0: 1 \rightarrow 2$, and $r_1^{-1}: 1 \rightarrow 0$, $r_0^{-1}: 2 \rightarrow 1$. Let $F$ be the free strict 2-category on the 2-truncated [[reflexive globular set]] whose 1-truncation is the underlying reflexive directed graph of $F_{\leq 1}$, and which in addition has 2-arrows $\iota_{1}, \iota_{1}^{-1}: r_1 \circ r_1^{-1} \rightarrow id(1)$ and $\kappa_{2}, \kappa_{2}^{-1}: r_0 \circ r_0^{-1} \rightarrow id(2)$.Then $S$ can be taken to be the quotient of $F$ by the relation on 2-arrows which forces $\iota_{0}$ to be a [[2-isomorphism]] with inverse $\iota_{0}^{-1}$, and similarly for $\kappa_{2}$. The functors $r_0$ and $r_1$ can be taken to be functors picking out the equivalences in $S$ of the same name.

There is a functor $s: \mathcal{E}_{semi} \rightarrow S$ which picks out the semi-strict equivalence in $S$ given by $r_{0} \circ r_{1}$. It is immediately checked that $\left( S, r_0, r_1, s \right)$ defines a _subdivision structure_ with respect to $\left( \mathcal{E}_{semi}, i_0, i_1 \right)$ in the sense of VI.14 of [Williamson2011](Williamson2011). Moreover, since $1$ is a final object, it is immediate that this subdivision structure is compatible with the contraction structure of Notation \ref{NotationContractionStructure} in the sense of VI.18 of [Williamson2011](Williamson2011).

The functor $q_{l}$ of VI.34 in [Williamson2011](Williamson2011) is in this case the functor $S \rightarrow \mathcal{E}_{semi}$ which is determined by $r_{0} \mapsto i$, $r_0^{-1} \mapsto i^{-1}$, $r_1 \mapsto id(0)$, $r^{-1} \mapsto id(0)$, $\iota_1 \mapsto id$, and $\kappa_2 \mapsto \iota_1$. We see then that $\left( \mathcal{E}_{semi}, i_0, i_1, p, S, r_0, r_1, s \right)$ has strictness of left identities in the sense of VI.34 of [Williamson2011](Williamson2011).

It is similarly the case that $\left( \mathcal{E}_{semi}, i_0, i_1, p, S, r_0, r_1, s \right)$ has strictness of right identities in the sense of VI.34 of [Williamson2011](Williamson2011).

\end{notn}

\begin{rmk} Explicitly, $\mathcal{E}_{semi} \times \mathcal{E}_{semi}$ can be described as follows. Let $F_{\leq 1}$ be the [[free category]] on the [[directed graph]] with objects $(0,0)$, $(0,1)$, $(1,0)$, $(1,1)$, and with arrows $(0, i)$, $(0, i^{-1})$, $(i, 0)$, $(i^{-1}, 0)$, $(1, i)$, $(1, i^{-1})$, $(i, 1)$, and $(i^{-1}, 1)$. Let $F$ be the free strict 2-category on the 2-truncated [[reflexive globular set]] whose 1-truncation is the underlying reflexive directed graph of $F_{\leq 1}$, and which in addition has 2-arrows $(0, \iota_{1}): (0, i \circ i^{-1}) \rightarrow id(0, 1)$, $(1, \iota_{1}): (1, i \circ i^{-1}) \rightarrow id(1, 1)$, $(\iota_{1}, 0): (i \circ i^{-1}, 0) \rightarrow id(1, 0)$, and $(\iota_{1}, 1): (i \circ i^{-1}, 1) \rightarrow id(1, 1)$.

Then $\mathcal{E}_{semi} \times \mathcal{E}_{semi}$ is the quotient of $F$ by the relation on 1-arrows which forces $(0, i^{-1} \circ i)$ to be equal to $(0,0)$, and similarly for $(i^{-1} \circ i, 0)$, $(1, i^{-1} \circ i)$, and $(i^{-1} \circ i, 1)$; and on 2-arrows which forces $(0, \iota_{1})$, $(\iota_{1}, 0)$, $(1, \iota_1)$ and $(\iota_1, 1)$ to be 2-isomorphisms.
\end{rmk}

\begin{notn} Let $\Gamma_{ul}$ be the functor $\mathcal{E}_{semi} \times \mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi}$ determined by

$$
\begin{aligned}
(0, i), (i,0) &\mapsto i, \\
(0, i^{-1}), (i^{-1}, 0) &\mapsto i^{-1}, \\
(1, i), (1, i^{-1}), (i, 1), (i^{-1}, 1) &\mapsto id(1), \\
(0, \iota_{1}), (\iota_{1}, 0) &\mapsto \iota_{1}, \\
(1, \iota_{1}), (\iota_{1}, 1) &\mapsto id\left(id(1)\right).
\end{aligned}
$$

Then $\Gamma_{ul}$ defines an _upper left connection structure_ with respect to $\left( \mathcal{E}_{semi}, i_0, i_1, p \right)$ in the sense of VI.22 of [Williamson2011](Williamson2011).  \end{notn}

\begin{notn} Let $\Gamma_{lr}$ be the functor $\mathcal{E}_{semi} \times \mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi}$ determined by

$$
\begin{aligned}
(0, i), (i,0), (0, i^{-1}), (i^{-1}, 0) &\mapsto id(0), \\
(1, i), (i, 1) &\mapsto i, \\
(1, i^{-1}), (i^{-1}, 1) &\mapsto i^{-1}, \\
(0, \iota_{1}), (\iota_{1}, 0) &\mapsto id\left(id(0)\right), \\
(1, \iota_{1}), (\iota_{1}, 1) &\mapsto \iota_{1}.
\end{aligned}
$$

Then $\Gamma_{lr}$ defines an _lower right connection structure_ with respect to $\left( \mathcal{E}_{semi}, i_0, i_1, p \right)$ in the sense of VI.24 of [Williamson2011](Williamson2011).

Since $1$ is a final object of [[2Cat]], it is immediate that $\Gamma_{lr}$ is compatible with $p$ in the sense of VI.26 of [Williamson2011](Williamson2011).

\end{notn}

\begin{notn} Let $\Gamma_{ur}$ be the functor $\mathcal{E}_{semi} \times \mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi}$ determined by

$$
\begin{aligned}
(i,0) &\mapsto i, \\
(i^{-1}, 0) &\mapsto i^{-1}, \\
(0, i), (0, i^{-1}) &\mapsto id(0), \\
(1, i) &\mapsto i^{-1}, \\
(1, i^{-1}) &\mapsto i, \\
(i, 1), (i^{-1}, 1) &\mapsto id(0), \\
(\iota_{1}, 0) &\mapsto \iota_{1}, \\
(0, \iota_{1}) &\mapsto id\left(id(0)\right), \\
(1, \iota_{1}) &\mapsto id\left(id(0)\right), \\
(\iota, 1) &\mapsto id\left(id(0)\right).
\end{aligned}
$$

Then $\Gamma_{ur}$ defines an _upper right connection structure_ with respect to $\left( \mathcal{E}_{semi}, i_0, i_1, p, v \right)$ in the sense of VI.29 of [Williamson2011](Williamson2011).

The functor $x \circ \left( \mathcal{E}_{semi} \times s \right) : \mathcal{E}_{semi} \times \mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi}$ of VI.32 in [Williamson2011](Williamson2011) is determined by

$$
\begin{aligned}
(i, 0), (i,1) &\mapsto i, \\
i^{-1}, 0), (i^{-1}, 1) &\mapsto i^{-1}, \\
(\iota_{1}, 0) &\mapsto \iota_{1}, \\
(\iota_{1}, 1) &\mapsto \iota_{1}, \\
(0, i), (0,i^{-1}) &\mapsto id(0), \\
(1, i), (1, i^{-1}) &\mapsto id(1), \\
(0, \iota_{1}) &\mapsto id\left(id(0)\right), \\
(1, \iota_{1}) &\mapsto id\left(id(1)\right).
\end{aligned}
$$

The key observation here is that $(1, i), (1, i^{-1}) \mapsto id(1)$, which relies on the fact that $i^{-1} \circ i = id(0)$ in $\mathcal{E}_{semi}$.

We deduce that $x \circ \left( \mathcal{E}_{semi} \times s \right) = \mathcal{E}_{semi} \times p$, and thus that $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with $\left(S, r_0, r_1, s \right)$ in the sense of VI.32 of [Williamson2011](Williamson2011).
\end{notn}

\begin{rmk} It follows from the above, Corollary XV.6, and Corollary XV.7 of [Williamson2011](Williamson2011) that there is a model structure on [[2Cat]], the category of strict 2-categories, whose fibrations and cofibrations are 'Hurewicz' fibrations and cofibrations respectively with respect to the [[interval object]] $\left( \mathcal{E}_{semi}, i_0, i_1 \right)$. See [[canonical model structure on 2-categories]] for more. \end{rmk}

\begin{rmk} All of the structures in this section have an analogue for $\mathcal{E}$, the free-standing equivalence, as well. All the required compatibilities hold except for one: $\Gamma_{lr}$ and $\Gamma_{ur}$ are not compatible with $\left(S, r_0, r_1, s \right)$. This is exactly where the semi-strictness of $\mathcal{E}_{semi}$ is needed.  \end{rmk}

\section{References}

* {#Williamson2011} [[Richard Williamson]], _Cylindrical model structures_, DPhil thesis, University of Oxford, 2011. [author's webpage](http://rwilliamson-mathematics.info/cylindrical_model_structures/cylindrical_model_structures.html) [arXiv:1304.0867](https://arxiv.org/abs/1304.0867)

[[!redirects walking equivalences]]
[[!redirects walking semi-strict equivalence]]
[[!redirects walking semi-strict equivalences]]
[[!redirects free-standing equivalence]]
[[!redirects free-standing equivalences]]
[[!redirects free-standing semi-strict equivalence]]
[[!redirects free-standing semi-strict equivalences]]


walking isomorphism

\tableofcontents

\section{Idea}

The  _[[interval object]] in [[groupoids]]_ or _free-standing isomorphism_ or, if you insist, the "[[walking structure|walking]] isomorphism", is the [[groupoid]]:
\[
  I_\sim
  \;\coloneqq\;
  \big\{
    a \overset{\;\; \sim \;\;}{\leftrightarrows} b
  \big\}
\]
with precisely two [[objects]] and (besides their [[identity morphisms]]) one [[isomorphism]] and its [[inverse morphism]] connecting them.

This is such that for $\mathcal{C}$ any [[category]], a [[functor]] of the form

$$
  I_\sim \longrightarrow \mathcal{C}
$$

is precisely a choice of [[isomorphism]] in $\mathcal{C}$.

The interval groupoid can be [[categorified]] in a number of ways: to the [[walking equivalence]], to the [[walking adjoint equivalence]], or to the [[walking 2-isomorphism]]. Related, though not quite a categorification in one of the usual senses, is the [[walking 2-isomorphism with trivial boundary]].

\section{Definition}

\begin{defn}
The _free-standing isomorphism_ is the category generated by two objects $0$ and $1$, one arrow $0 \rightarrow 1$, one arrow $1 \rightarrow 0$, and the equations:
\[
0 \to 1 \to 0 = \mathrm{id}_0, \quad 1 \to 0 \to 1 = \mathrm{id}_1,
\]
which make $0 \to 1$ an [[isomorphism]].
\end{defn}

\begin{rmk} The arrow $0 \rightarrow 1$ is an isomorphism, whose inverse is the arrow $1 \rightarrow 0$. \end{rmk}

\begin{rmk} The free-standing isomorphism is a groupoid. \end{rmk}

\begin{rmk} The free-standing isomorphism can also be described as the [[free groupoid]] on the [[interval category]], that is to say, the walking arrow. Because it is an [[interval object]] for [[Cat]] and [[Grpd]], it is also known as the _interval groupoid_.\end{rmk}

\section{Representing of isomorphisms}

\begin{prpn}
Let $\mathcal{A}$ be a category. Let $\mathcal{I}$ denote the free-standing isomorphism.
Evaluation at the arrow $0\to 1$ establishes a natural bijective
correspondence between functors $\mathcal{I}\to\mathcal{A}$
and isomorphisms in $\mathcal{A}$.
Thus, for any isomorphism $f$ of $\mathcal{A}$ there is a unique [[functor]] $F: \mathcal{I} \rightarrow \mathcal{A}$ such that the arrow $0 \rightarrow 1$ of $\mathcal{I}$ maps under $F$ to $f$.
\end{prpn}

\begin{proof} Immediate from the definitions. \end{proof}

\section{Properties}

\begin{prpn} $\{ 1 \} \subseteq \mathcal{I}$ is the full subcategory classifier of the 1-category $Cat$. That is, for any small category $\mathcal{A}$, there is a bijective correspondence between full subcategories of $\mathcal{A}$ and functors $\mathcal{A} \to \mathcal{I}$, with the reverse direction given by taking the pullback of the inclusion $\{ 1 \} \subseteq \mathcal{I}$.
\end{prpn}
\begin{proof} Functors into $\mathcal{I}$ are uniquely determined by functions on the sets of objects. $\{ 1 \}$ is a full subcategory of $\mathcal{I}$, and any pullback of a full subcategory can be given as a full subcategory.
\end{proof}

In fact, this generalizes. If $X$ is a simplicial set, say that a _full subspace_ of $X$ is a subsimplicial set $S \subseteq X$ with the property there is some subset $S_0 \subseteq X_0$ such that $S$ contains exactly the simplices whose vertices are all contained in $S_0$.

\begin{prpn} The nerve $N(\mathcal{I})$ is the full subspace classifier for $sSet$,
and thus $N(\mathcal{I})$ represents the subpresheaf $FullSub \subseteq Sub$ of full subspaces. \end{prpn}
\begin{proof} This can be determined from the explicit description of $N_n(\mathcal{I}) \cong \{ 0, 1 \}^{n+1}$ given by listing the vertices of a path. However, it's more informative to observe that $N(\mathcal{I}) = indisc(\{ 0, 1 \})$, where $indisc$ is the _indiscrete space_ functor, which is the direct image part of the geometric embedding $Set \subseteq sSet$ whose inverse image is $X \mapsto X_0$.
\end{proof}

\begin{rmk} This also implies $N(\mathcal{I})$ is the full subcategory classifier of $qCat$, the 1-category of [[quasi-category|quasi-categories]], since those are given by full subspaces of simplicial sets. \end{rmk}

## References

A generalization of the notion of the interval groupoid to [[simplicial groupoids]] is considered in

* {#DwyerKan84} [[William Dwyer]], [[Daniel Kan]], §2.8 of: *Homotopy theory and simplicial groupoids*, Indagationes Mathematicae (Proceedings) **87** 4 (1984) 379-385 [doi:10.1016/1385-7258(84)90038-6]

and plays a key role in the discussion of the [[model structure on simplicial groupoids]], see [there](model+structure+on+simplicial+groupoids#SimplicialIntervalGroupoid).

[[!redirects free-standing isomorphism]]
[[!redirects interval groupoid]]


walking structure

# Walking structures
* tables of contents
{: toc}

## Idea

Around the nLab and elsewhere, one occasionally sees an expression "the walking _____" where the blank is some mathematical concept. This is a colloquial way of referring to an archetypal model of the concept or type, and usually refers to a *[[free construction|free]]* or *[[initial object|initial]]* form of such a kind of structure.

Pronunciation is just as in 'John is a walking almanac' or 'Eugene Levy is a walking pair of eyebrows'. The term is [believed](http://golem.ph.utexas.edu/category/2010/01/f_and_the_shibboleth.html#c031066) to have been introduced by [[James Dolan]].

Sometimes, "the free-living _____" or "the free-standing _____" is used instead; this terminology is probably much older.

## Definition

The idea is probably easier to apprehend through examples rather than through a formal definition, but for the record:

If $X$ is a type of [[mathematical structure]] that can be defined in a [[category]], [[higher category]], or category with some [[extra structure]], then the *walking X* refers to the *[[free construction|free]] category (resp. higher category, category with suitable structure) containing an $X$.

More precisely, if $StructCat$ denotes some (higher) category of categories with an appropriate type of structure, then the **walking X** is an object $[X] \in StructCat$ together with a [[natural transformation|natural]] [[equivalence]]
$$
  StructCat([X],C)
  \;\simeq\;
  \big\{Xs \; in \; C\}
$$
between the [[hom-set]]/[[hom-category|category]]/[[hom-space|space]] from $[X]$ to $C$, for any $C\in StructCat$, and the set/category/space of all Xs in $C$.

+-- {: .num_remark}
###### Remark
In other words, the structured category $[X]$ equipped with its canonical type $X$ is [[initial object|initial]] among such structured categories that come equipped with such types $X$. A fancier expression is that $[X]$ 'coclassifies' such types: this is analogous to how a [[classifying space]] $B G$ for a [[topological group]] $G$ classifies $G$-[[bundles]], in that every $G$-bundle $p: E \to X$ over a suitable space $X$ has a classifying map $\chi_p: X \to B G$ (unique up to homotopy) such that pulling back the canonical $G$-bundle type $\pi: E G \to B G$ along $\chi_p$ reproduces the type $p$. Only here we say 'coclassifies' (as for example in this [comment](http://nforum.mathforge.org/discussion/1529/walking-structures/#Item_5)), since here we instead "push forward" the canonical type $X$ of $[X]$ along a structured-category morphism $[X] \to C$ to obtain a given type of $C$.
=--

## Examples

* The [[interval category]] is the *walking morphism*.

* The [[interval groupoid]] is the [[walking isomorphism]].

* [[equivalence|Equivalences]] in a [[2-category]] are represented by the [[walking equivalence]].

* The [[2-category]] [[Adj]] is the [[walking adjunction]].

* The [[walking 2-isomorphism with trivial boundary]] is a 2-groupoid model for the 2-truncation of the [[2-sphere]].

* The augmented/algebraist's [[simplex category]] is "the *walking [[monoid]]*" (in a [[monoidal category]]). That is to say: the simplex category is initial (in a 2-categorical sense) among monoidal categories equipped with a monoid object. Intuitively, it is [[the]] monoidal category that results if all one need be told about it is that it has a monoid object -- all the morphisms of the category are obtainable from the monoid structure by applying the operations of a monoidal category, and they are subject to no further relations beyond those implied by the monoid axioms.

* Similarly, the [[Lawvere theory]] of groups can be described as "the *walking [[group object|group]]*" (qua [[cartesian monoidal categories]]). This gives a good intuitive description: this Lawvere theory can be understood as the category (with finite products) that results if all one need be told about it is that it has a [[group object]]; the rest of the structure of this category can be deduced from this one fact.

The last two examples indicate the need for a little care: the [[doctrine]] or type of structured category in which the '$X$' of "the walking $X$" lives should either be specified or clear from context. For example, if one simply says "the
walking monoid", this means the simplex category if the surrounding context is the doctrine of monoidal categories -- but means something else (the Lawvere theory of monoids) if the ambient context is the doctrine of categories with finite products, and it means the category opposite to that of finitely presentable monoids if we are in the doctrine of [[finitely complete categories]].

If the doctrine is not specified, then a reasonable default is a 'minimal' doctrine in which the concept makes sense; for example, to make sense of monoids, one doesn't need more than monoidal categories. See also [[microcosm principle]].

Thus, more generally,

* The [[syntactic category]] of a [[theory]] $T$ in some [[doctrine]] $D$ is the "walking $T$-model" (in a $D$-category).  In particular, the [[classifying topos]] of a [[geometric theory]] $T$ is "the *walking $T$-model*" *qua* [[Grothendieck toposes]] (where the morphisms are the left-adjoint parts of [[geometric morphisms]]).

## Relation to initial objects

The walking X is, of course, not the same as the [[initial object|initial]] X.

Consider for example the case when X is a pointed monoid (a monoid equipped with an element).  The initial pointed monoid (in $Set$) is the [[natural numbers]] equipped with $1\in \mathbb{N}$.  Whereas the walking pointed monoid (qua category with finite products, say) is a category $C_M$ with finite products containing a [[monoid object]] $M\in C_M$ and an "[[generalized element|element]]" $e:1\to M$.  They have different types and different universal properties: $\mathbb{N}$ has a universal property mapping into other pointed monoids in $Set$, while $C_M$ has a universal property mapping into other categories with products equipped with pointed monoids.

Nevertheless, the first sits inside the second!  Specifically, for any category $C$ with products, the "underlying set" functor $C(1,-):C\to Set$ is product-preserving and hence carries monoid objects to monoid objects, and in the case of the walking pointed monoid we have $C_M(1,M) \cong \mathbb{N}$.

This is true rather generally: *the initial X is the underlying X of the walking X*.  One general theorem of this sort is the following:

+--{: .un_theorem}
###### Theorem
Let $K$ be a [[2-category]] containing an object $S$, and suppose that:

1. The domain projection $K\sslash S \to K$ from the [[lax slice 2-category]] has a section.  Explicitly, for every object $X$ we have a map $s_X : X\to S$ and for every morphism $f:X\to Y$ we have a 2-cell $\sigma_f : s_X \to s_Y f$, such that for every 2-cell $\alpha :f\to g$ we have $\sigma_g . s_Y\alpha = \sigma_f$, and these vary functorially.
2. We have $s_S \cong 1_S$, and for any $X$ the composite $s_X \xrightarrow{\sigma_{s_X}} s_S s_X \cong s_X$ is the identity.

Then for any $X$, the morphism $s_X:X\to S$ is the initial object of the hom-category $K(X,S)$.
=--
+--{: .proof}
###### Proof
We will use the characterization of initial objects via [cones over the identity](/nlab/show/initial+object#ConesOverTheIdentity).  Thus, we must construct a natural transformation from the constant functor $\Delta_{s_X} : K(X,S) \to K(X,S)$ to the identity, which is the identity at $s_X$.  However, given any $f:X\to S$, we have the 2-cell $s_X \xrightarrow{\sigma_f} s_S f \cong f$, and the assumption $\sigma_g . s_Y\alpha = \sigma_f$ makes this a natural transformation; and the final assumption says exactly that this is the identity at $s_X$.
=--

To see how this theorem implies that the initial X is the underlying X of the walking X, consider again the case of pointed monoids.  Let $K$ be the 2-category of categories with products and product-preserving functors, let $S=Set$, and let $s_C : C \to Set$ be $C(1,-)$.  The hypotheses are easy to verify; thus the theorem tells us that $C(1,-)$ is the initial functor $C\to Set$ for any $C$.

Now take $C$ to be the walking pointed monoid $C_M$ above.  Then its universal property tells us that functors $F:C_M\to Set$ are equivalent to pointed monoids $F(M)$ in $Set$; so we see that $C(1,M)$ is the initial pointed monoid in $Set$, i.e. $\mathbb{N}$.

A similar argument applies whenever we have a "$Set$-like" object of a 2-category with "underlying set" morphisms to it.  For instance, in place of categories with finite products we could consider categories with finite limits.  However, in other cases, such as monoidal categories, there is a disconnect: the natural "underlying set" functor for a monoidal category $C$ is $C(I,-)$, where $I$ is the unit object; but in general this is only *lax* monoidal, whereas the universal property of a "walking X qua monoidal category" is relative to *strong* monoidal functors.

It does often happen that this disconnect can be bridged.  For instance, suppose X is something that can be defined in any [[multicategory]] (like a pointed monoid).  Then we can apply the above argument to the 2-category of multicategories, as the walking X qua multicategory has a universal property relative to functors of multicategories, even though such functors correspond to *lax* functors between monoidal categories.  It follows that the underlying set functor $C_M(;-) : C_M\to Set$ of the walking X qua multicategory corresponds to the initial X in $Set$.

Moreover, we can deduce from this that the underlying pointed monoid of the walking X qua monoidal category is also the initial X in $Set$.  This is because the walking X qua monoidal category is the monoidal category freely generated by the walking X qua multicategory, and a multicategory embeds fully-faithfully in the monoidal category that it freely generates.  In terms of the [[type theory]] that generates walking Xs, the latter fact can be seen as a sort of [[canonicity]] for the tensor-product constructor.

Similar arguments apply in other cases.  For instance, in the 2-category of [[double categories]] we can take $S$ to be the double category [[Span]]; this has the same problem as that of monoidal categories, but we can solve it similarly by considering [[virtual double categories]] instead.

On the other hand, there are also cases where this argument does not apply.  For instance, X could be something that can be defined in a monoidal category but *not* in a multicategory, such as a [[Frobenius monoid]].  In this case the claim doesn't even make sense: the functor $C(I,-)$, being only lax monoidal in general, need not preserve Frobenius monoids.  It seems that in most cases where an "underlying set" functor $C(I,-)$ preserves Xs, there is a kind of [[generalized multicategory]] in which Xs can be defined and this argument carried through, but I do not know of a general theory suggesting this.

## References

* A [Café post](http://golem.ph.utexas.edu/category/2010/01/f_and_the_shibboleth.html) essentially about walking objects (among other things), including a [comment](http://golem.ph.utexas.edu/category/2010/01/f_and_the_shibboleth.html#c031066) that explains the terminology.

The terminology "free-living" appears in:

* {#FoltzLairKelly80} [[François Foltz]], [[Christian Lair]], [[G. M. Kelly]], *Algebraic categories with few monoidal biclosed structures or none*, Journal of Pure and Applied Algebra **17** 2 (1980) 171-177 [[pdf](https://core.ac.uk/download/pdf/82322397.pdf), doi:10.1016/0022-4049(80)90082-1]

[[!redirects walking]]
[[!redirects walking structure]]
[[!redirects walking structures]]
[[!redirects walking object]]
[[!redirects walking objects]]
[[!redirects free-living]]


wall crossing

> This entry is about discontinuities in parameter dependence of (often asymptotic) solutions of [[differential equations]] and similar phenomena (notably in the context of [[BPS states]] formalized via [[Bridgeland stability conditions]]) with stability parameters (and their stability slopes) in [[algebraic geometry]] which are often interpreted as crossing the walls of marginal stability in [[physics]]. For the different notions of the same name in [[Morse theory]] see at _[[Cerf wall crossing]]_ and for the (Weyl chamber wall) crossing functors in representation theory see [[wall crossing functor]].

# Contents
* table of contents
{: toc}

## Overview

In the study of [[solitons]], one may try a [[WKB approximation|WKB-style approximation]] to a nonlinear [[wave equation]] (see also _[[eikonal equation]]_, _[[Maslov index]]_, etc.). Stokes has observed that when trying to connect the local solutions, one has discontinuities along certain lines, now called **Stokes lines**. This is called the [[Stokes phenomenon]]. Similar issues appear in study of [[isomonodromic deformation]]s of nonlinear [[ODEs]] in the [[complex plane]], which is also relevant in [[soliton]] theory, and [[integrable systems]], and special functions like [[Painlevé transcendent]]s. This has especially been studied by the Kyoto school (Jimbo, Miwa, Sato, [[Masaki Kashiwara|Kashiwara]] etc.), including the use of [[D-modules]] and [[microlocal analysis]]. The Kyoto school found a connection of isomonodromic theory to what is called [[holonomic quantum field]]s.

The solutions of meromorphic differential equations can be expressed in terms of [[meromorphic connection]]s. Then the slopes related to the solutions can be viewed as features of particular objects in a category of $D$-[[D-module|modules]]. More generally, slope filtrations are structures which appear in many other additive categories, e.g. in [[Hodge theory]], theory of Dieudonné modules and so on. Many of those are related to the stability of the objects, which is important in the construction of [[moduli spaces]].

In [[algebraic geometry]], [[Grothendieck]] has shown how to correctly define and construct some fundamental [[moduli spaces]], like Hilbert schemes and Quot schemes for [[coherent sheaves]]. The work has been continued by [[David Mumford]] who geometrized classical invariant theory into [[geometric invariant theory]]. To keep moduli under control, one needs to impose stability conditions on objects and also look at classes with some fixed data: those involve slopes or equivalently phase factors. This is thus  similar to the phases of eikonal in the case of Stokes phenomenon.
Cf. also Harder-Narasimhan filtration, [[Castelnuovo-Mumford regularity]] (cf. [wikipedia](http://en.wikipedia.org/wiki/Castelnuovo%E2%80%93Mumford_regularity)) etc.

### In supersymmetric field theory

In [[super Yang-Mills theory]] the number of [[BPS states]] is locally constant as a function of the parameters of the theory, but it may jump at certain "walls" in the [[moduli spaces]] of parameters. The precise behaviour of the BPS states as one crosses these walls is studied as "wall crossing phenomena".

Another example are the [[moduli spaces]] of [[Higgs bundles]], studied by [[Carlos Simpson]] and others, which have special cases with interpretations both in geometry and in the gauge theory (instantons). It appears that sometimes they can be linked to the geometric picture. [[Riemann-Hilbert correspondence]], spectral transform and similar correspondences again play a major role.

Surely, one often works at the derived level. An adaptation of the  notion of stability into the setup of [[triangulated categories]] has been introduced by Bridgeland. Bridgeland stability for the derived categories of (boundary conditions of) D-branes (B-model) are relevant for string theory.

## Related concepts

* [[BPS state]], [[D-module]], [[cluster algebra]], [[quiver]], [[representation theory]], [[Donaldson-Thomas invariant]].

[[!include field theory with boundaries and defects - table]]

## References

### Introductions and lectures

* [[Sergio Cecotti]], _Trieste lectures on wall-crossing invariants_ (2010) [[pdf](http://people.sissa.it/~cecotti/ictptext.pdf)&rbrack

* [[Greg Moore]], _PiTP Lectures on BPS states and wall-crossing in $d = 4$, $\mathcal{N} = 2$ theories_ ([pdf](http://www.physics.rutgers.edu/~gmoore/PiTP_July26_2010.pdf))

* {#Dimofte10} [[Tudor Dimofte]], _Refined wall crossing_ ([pdf](http://thesis.library.caltech.edu/5808/4/TD_part1.pdf)), part I of _Refined BPS invariants, Chern-Simons theory, and the quantum dilogarithm_, 2010 ([pdf](http://thesis.library.caltech.edu/5808/1/DimofteTDofficial.pdf), [web](http://thesis.library.caltech.edu/5808/))

### Original articles

#### General

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Stability structures, motivic Donaldson-Thomas invariants and cluster transformations_, [arXiv:0811.2435](http://arxiv.org/abs/0811.2435); _Motivic Donaldson-Thomas invariants: summary of results_, [0910.4315](http://arxiv.org/abs/0910.4315); _Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry_, [arxiv/1303.3253](http://arxiv.org/abs/1303.3253)

* Arend Bayer, [[Yuri Manin|Yuri I. Manin]], _Stability conditions, wall-crossing and weighted Gromov-Witten invariants_, [math.AG/0607580](http://arxiv.org/abs/math.AG/0607580), Mosc. Math. J. __9__ (1), 2009.

* [[Mina Aganagic]], Hirosi Ooguri, [[Cumrun Vafa]], Masahito Yamazaki, _Wall crossing and M-theory_, [arxiv/0908.1194](http://arxiv.org/abs/0908.1194)

* [[Mina Aganagic]], _Wall crossing, quivers and crystals_, [arxiv/1006.2113](http://arxiv.org/abs/1006.2113)

* [[Gregory Moore]], _PiTP lectures on BPS states and wall-crossing in d = 4, N = 2 theories_ [pdf](http://www.physics.rutgers.edu/grad/695/PiTP-LectureNotes.pdf)

* S. Cecotti, [[Cumrun Vafa]], _BPS wall crossing and topological strings_,
[arXiv/0910.2615](http://arxiv.org/abs/0910.2615)

* [[Davide Gaiotto]], [[Greg Moore]], [[Andrew Neitzke]], _Wall-crossing, Hitchin systems, and the WKB approximation, Advances in Mathematics __234__ (2013) 239--403 ([arXiv:0907.3987](https://arxiv.org/abs/0907.3987) [doi](https://doi.org/10.1016/j.aim.2012.09.027))

* E. Diaconescu, [[Greg Moore]], _Crossing the wall: branes vs. bundles_, [arXiv/0706.3193](http://arxiv.org/abs/0706.3193)

* E. Andriyash, [[Frederik Denef]], [[Daniel Jafferis]], [[Greg Moore|G. W. Moore]], _Wall-crossing from supersymmetric galaxies_, [arxiv/1008.0030](http://arxiv.org/abs/1008.0030)

* Tom Bridgeland, [[Valerio Toledano-Laredo]], _Stability conditions and Stokes factors_, [arxiv/0801.3974](http://arxiv.org/abs/0801.3974)

* M. C. N. Cheng, E. P. Verlinde, _Wall crossing, discrete attractor flow and Borcherds algebra_, SIGMA 4 (2008), 068, 33 pages, [pdf](http://www.emis.de/journals/SIGMA/2008/068/sigma08-068.pdf)

* Masahito Yamazaki, _Crystal melting and wall crossing phenomena_, Ph.D. thesis, [arxiv/1002.1709](http://arxiv.org/abs/1002.1709)

* Michele Cirafici, Annamaria Sinkovics, [[Richard Szabo]], _Instanton counting and wall-crossing for orbifold quivers_, [arxiv/1108.3922](http://arxiv.org/abs/1108.3922)

* H.-Y. Chen, N. Dorey, K. Petunin, _Moduli space and wall-crossing formulae in higher-rank gauge theories_, JHEP 11 (2011) 020, doi; _Wall crossing and instantons in compactified gauge theory_, JHEP 06 (2010) 024 [arXiv:1004.0703](ttp://arxiv.org/abs/1004.0703)

### In supergravity

* {#Denef00} [[Frederik Denef]], _Supergravity flows and D-brane stability_, JHEP 0008:050,2000 ([arXiv:hep-th/0005049](http://arxiv.org/abs/hep-th/0005049))

* [[Frederik Denef]], _Quantum Quivers and Hall/Hole Halos_, JHEP 0210:023,2002 ([arXiv:hep-th/0206072](http://arxiv.org/abs/hep-th/0206072))

* [[Daniel Jafferis]], [[Gregory Moore]], _Wall crossing in local Calabi Yau manifolds_ ([arXiv:0810.4909](http://arxiv.org/abs/0810.4909))

### Conferences and seminars

* (past) [[Kontsevich]] in Aarhus, August 2010, [master class on wall crossing](http://qgm.au.dk/events/show/artikel/masterclass-aug-2010/); we will keep a [[wall crossing in Aarhus|nlab page]] on it

* (past) [Focus Week on New Invariants and Wall Crossing](http://member.ipmu.jp/domenico.orlando/FocusInvariants.html), May 18-22, 2009, Kashiwa Campus of the University of Tokyo

* (past) [Wall-crossing in Mathematics and Physics](http://www.math.uiuc.edu/wallcrossing), May 24-28, 2010,
Department of Mathematics, University of Illinois at Urbana-Champaign

* Description of seminar on stability conditions and Stokes factors in Bonn, [pdf](http://www.math.uni-bonn.de/people/compgeo/Hall.pdf)

Also ([Gaiotto-Moore-Witten 15](#GaiottoMooreWitten15)).

### Categorification

A [[categorification]] of wall crossing formulas to an [[(infinity,2)-category]] of sorts is discussed in

* {#GaiottoMooreWitten15} [[Davide Gaiotto]], [[Gregory Moore]], [[Edward Witten]], _An introduction to the web-based formalism_ ([arXiv.1506.04086](http://arxiv.org/abs/1506.04086))

[[!redirects wall crossing]]
[[!redirects wall-crossing]]


wall crossing functor

Various objects in Lie theory have the combinatorics of their representations described in terms of roots and weights; the structure of Weyl chamber walls is crucial for many criteria in their study. Certain important functors in this representation theory cross the chamber walls. We talk about the wall crossing functors in representation theory.

Symmetries related to Weyl groups, Weyl chambers and chamber walls are involved (what is sometimes also in the other notion of [[wall crossing]] in the BPS setup). A priori wall crossing functors in representation theory (introduced in 1970s by the Moscow school, Gelfand, Bernstein etc.) are about certain functors which in take as input an infinite-dimensional representation, tensor it with finite-dimensional and look for certain pieces in their decomposition. In this study, the position with respect to the  chamber walls is crucial.

* A. [[Beilinson]], [[Victor Ginzburg|V. Ginzburg]], _Wall-crossing functors and $D$-modules_, Representation Theory __3__ (electronic), 1--31 (1999) [pdf](http://www.ams.org/ert/1999-003-01/S1088-4165-99-00063-1/S1088-4165-99-00063-1.pdf).

> The composition of the translation functor that sends the category at a regular maximal ideal to the category at a non-regular maximal ideal with the translation functor acting in the opposite direction is called a wall-crossing functor.

* J. Bernstein, S. Gelfand, _Tensor products of finite and infinite dimensional representations of semisimple Lie algebras_, Comp. Math. 41 (1980), 245–285. MR 82c:17003

[[!redirects wall crossing functor]]
[[!redirects wall crossing in representation theory]]


wall crossing in Aarhus 2010

[[!redirects wall crossing in Aarhus]]

[[Maxim Kontsevich]] taught a master class on [[wall crossing]] in Aarhus, August 16-20, 2010

* ([master class webpage](http://qgm.au.dk/events/show/artikel/masterclass-aug-2010/), [video recording](http://qgm.au.dk/video/mc/wall-crossing/))

Kontsevich spoke about two a priori unrelated theories which lead to the same class of wall-crossing formulae.

> The first one is the theory of (motivic, or refined) [[Donaldson-Thomas invariants]]. It applies to
> 1) [[quivers]] (maybe with potentials), including cluster categories and  matrix models,
> 2) moduli of [[coherent sheaves]] on algebraic surfaces, moduli of representations of an arbitrary finitely presented algebra,
> 3) 3-dimensional [[Calabi-Yau categories]].

> One can define in each of these cases appropriate cohomology groups of moduli spaces, carrying a generalization of mixed Hodge structure. The generating series for Serre polynomials splits into an infinite product of quantum dilogarithms, depending on stability structure. Exponents in this expansion are refined Donaldson-Thomas invariants, and they jump in a universal way when we cross a wall in the space of stability structures.

> The second theory is about the behavior of collapsing hyperkahler metrics, and of WKB asymptotics for complex integrable systems. Here one counts holomorphic discs with boundaries on Lagrangian submanifolds.

> Physics partially explains the relation between two theories, via BPS counting and geometry of hypermultiplet moduli spaces.

category: reference


Wall's finiteness obstruction

#Contents#
* table of contents
{:toc}

## Idea

The  [[compact object in an (infinity,1)-category|compact objects]] in [[∞Grpd]] are the [[retracts]] of [[finite homotopy types]] ([[finite CW-complexes]]). Not every such retract is itself a finite homotopy type; the vanishing of _Wall's finiteness [[obstruction]] is a necessary and sufficient condition for this to happen.

## References

* [[C. T. C. Wall]], _Finiteness conditions for CW-complexes I_  [pdf](http://math.uchicago.edu/~shmuel/tom-readings/wall%20finiteness%201.pdf)

* [[C. T. C. Wall]], _Finiteness conditions for CW-complexes II_  [pdf](http://www.maths.ed.ac.uk/~aar/papers/findom2.pdf)

[[!redirects Wall finiteness obstruction]]

[[!redirects Wall finiteness]]

[[!redirects Wall finiteness condition]]
[[!redirects Wall's finiteness condition]]


Wallman compactification

__Wallman compactification__ is a particular compactification of $T_1$-topological spaces introduced in

* Henry Wallman, _Lattices and topological spaces_, Annals of Math. __39__:1 (Jan., 1938), pp. 112-126 [jstor](http://www.jstor.org/stable/1968717)

There are recent applications related to topoi and noncommutative geometry

* [[Olivia Caramello]], _Gelfand spectra and Wallman compactifications_, [arxiv/1204.3244](http://arxiv.org/abs/1204.3244); _Dualité de Gelfand et bases de Wallman_, [seminar talk videos](https://sites.google.com/site/logiquecategorique/autres-seminaires/20130404-caramello-gelfand-wallman)

A __Wallman base__ $B$ for a topological space $X$ is a sublattice of of the [[frame]] $Open(X)$ of open sets of $X$ which is a [[base of a topology|base for the topology]] and satisfies the property that for any $U\in B$ and
$x\in U$ there exists $V\in B$ such that $U\cup V = X$ and $x\notin V$.

Standard references are

* P. T. Johnstone, _[[Stone Spaces]]_, Cambridge Studies in Advanced Math. __3__ (1982).
* [eom]: [Wallman compactification](http://www.encyclopediaofmath.org/index.php?title=Wallman_compactification)
* wikipedia: [Wallman compactification](http://en.wikipedia.org/wiki/Wallman_compactification)

category: topology
[[!redirects Wallman base]]
[[!redirects Wallman basis]]
[[!redirects Wallman bases]]


wallpaper group

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

#Contents#
* table of contents
{:toc}

## Definition

A _wallpaper group_ is a [[crystallographic group]] ([[space group]]) in [[dimension]] 2, hence a [[subgroup]] $G \subset Iso(\mathbb{R}^2)$ of the [[isometry group]] of the [[Euclidean plane]] such that

1. the part of $G$ inside the [[translation group]] is generated by two [[linearly independent subset|linearly independent]] [[vectors]];

1. the [[point group]] is [[finite group|finite]].

{#Tilley2020Fig310} [Fig. 3.10 in Tilley 2020](#Tilley2020):

## Related concepts

* [[flat orbifold]]

* [[finite rotation group]]

* [[tesselation]]

## References

e. g.

* {#Tilley2020} Richard Tilley, Sec. 3.5 in: *Crystals and Crystal Structure*, Wiley (2020) [[ISBN:978-1-119-54838-6](https://www.wiley.com/en-us/Crystals+and+Crystal+Structures,+2nd+Edition-p-9781119548386)]

* [[Patrick Morandi]], _The Classification of Wallpaper Patterns: From Group Cohomology to Escherís Tessellations_ ([pdf](http://sierra.nmsu.edu/morandi/notes/Wallpaper.pdf))

See also

* Wikipedia, _[Wallpaper group](https://en.wikipedia.org/wiki/Wallpaper_group)_

Discussion of [[meromorphic functions]] on the [[complex plane]] invariant under wallpaper groups:

* Richard Chapling, _Invariant Meromorphic Functions on the Wallpaper Groups_ ([arXiv:1608.05677](https://arxiv.org/abs/1608.05677))

[[!redirects wallpaper groups]]


Walter D. Neumann

* [Wikipedia entry](https://en.wikipedia.org/wiki/Walter_Neumann)

## Selected writings

Introducing the notion of [[cutting and pasting of manifolds]]:

* U. Karras, [[Matthias Kreck]], [[Walter D. Neumann]], E. Ossa, *Cutting and Pasting of Manifolds; SK-Groups*, Publish or Perish (1973) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/skbook.pdf), [[Karras-Kreck-Neumann-Ossa_CutNPaste.pdf:file]]]

* [[Walter D. Neumann]], *Manifold cutting and pasting groups*, Topology **14** 3 (1975) 237-244 [doi:10.1016/0040-9383(75)90004-X]

category: people

[[!redirects Walter Neumann]]
[[!redirects Walter David Neumann]]


Walter Feit

* [Wikipedia entry](https://en.wikipedia.org/wiki/Walter_Feit)

## related $n$Lab entries

* [[Feit-Thompson theorem]]

* [[permutation representation]]

category: people


Walter Felscher

* [German Wikipedia entry](https://de.wikipedia.org/wiki/Walter_Felscher)

## Related entries

* [[game semantics]]

category: people


Walter Gordon

German theoretical physicist, 1893-1939.

* [Wikipedia](http://en.wikipedia.org/wiki/Walter_Gordon_%28physicist%29)

## related $n$Lab entries

* [[Klein-Gordon equation]]

category: people


Walter Greiner

* [Wikipedia entry](https://en.wikipedia.org/wiki/Walter_Greiner) ([de](https://de.wikipedia.org/wiki/Walter_Greiner))

## Selected writings

On [[electrodynamics]]:

* [[Walter Greiner]], *Classical Electrodynamics*, Springer (1998) [[doi:10.1007/978-1-4612-0587-6](https://doi.org/10.1007/978-1-4612-0587-6)]

category: people


Walter Gubler

* [webpage](http://www.mathematik.uni-regensburg.de/gubler/)

category: people


Walter Lewis Baily

* [Wikipedia entry](https://en.wikipedia.org/wiki/Walter_Lewis_Baily,_Jr.)

## Selected writings

On [[orbifolds]] in [[complex geometry]]:

* [[Walter Lewis Baily]], _On the quotient of an analytic manifold by a group of analytic homeomorphisms_, PNAS 40 (9) 804-808 (1954) ([doi:10.1073/pnas.40.9.804](https://doi.org/10.1073/pnas.40.9.804))

* [[Walter Lewis Baily]], _The Decomposition Theorem for V-Manifolds_, American Journal of Mathematics Vol. 78, No. 4 (Oct., 1956), pp. 862-888 ([jstor:2372472](https://www.jstor.org/stable/2372472))

category: people


Walter Michaelis

* [Mathematics Genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=18941)

## related $n$Lab entries

* [[coalgebra]]

* [[cofree coalgebra]]

category: people


Walter Neumann

* [webpage](http://www.math.columbia.edu/~neumann/)

## related $n$Lab entries

* [[hyperbolic manifold]]

category: people


Walter Noll

* [Wikipedia entry](http://en.wikipedia.org/wiki/Walter_Noll)

* [website](http://www.math.cmu.edu/~wn0g/)

category: people


Walter Rudin

* [Wikipedia entry](https://en.wikipedia.org/wiki/Walter_Rudin)

## Related $n$Lab entries

* [[analysis]]

category: people


Walter Thirring

* [Wikipedia entry](https://en.wikipedia.org/wiki/Walter_Thirring) ([de](https://de.wikipedia.org/wiki/Walter_Thirring))

## Selected writings

On [[mathematical physics]]:

* [[Walter Thirring]], *A Course in Mathematical Physics* -- 1 *Classical Dynamical Systems* and 2 *Classical Field Theory*, Springer (1988, 1992) [[doi:10.1007/978-1-4684-0517-0](https://doi.org/10.1007/978-1-4684-0517-0)]

category: people


Walter Tholen

* [website](https://tholen.mathstats.yorku.ca/)

## Selected writings

On ([[monadic descent|monadic]]) [[descent]]:

* {#JanelidzeTholen94} [[George Janelidze]], [[Walter Tholen]], *Facets of descent I*, Applied Categorical Structures **2** 3 (1994) 245-281 [[doi:10.1007/BF00878100](https://doi.org/10.1007/BF00878100)]

* {#JanelidzeTholen97} [[George Janelidze]], [[Walter Tholen]], *Facets of descent II*, Applied Categorical Structures **5** 3 (1997) 229-248 [[doi:10.1023/A:1008697013769](https://doi.org/10.1023/A:1008697013769)]

On [[free coproduct completions]]:

* {#HuTholen95} Hongde Hu, [[Walter Tholen]], *Limits in  free  coproduct completions*, Journal of  Pure  and  Applied  Algebra 105 (1995) 277-291 (doi:10.1016/0022-4049(94)00153-7, [pdf](https://core.ac.uk/download/pdf/82604415.pdf))

On a generalization of [[coproducts]] to [[colimits]] over certain [[diagrams]] of the shape of [[skeletal groupoids]]:

* [[Hongde Hu]], [[Walter Tholen]], *Quasi-coproducts and accessible categories with wide pullbacks*, Appl Categor Struct **4** (1996) 387–402 [[doi:10.1007/BF00122686](https://doi.org/10.1007/BF00122686)]

See also:

* [[Reinhard Börger]], [[Walter Tholen]], _Strong regular and dense generators_,  [[Cahiers de Topologie et Géométrie Différentielle Catégoriques]] **32** 3 (1991) 257-276 [[numdam:CTGDC_1991__32_3_257_0](http://www.numdam.org/item?id=CTGDC_1991__32_3_257_0), [MR1158111](http://www.ams.org/mathscinet-getitem?mr=1158111)]

On (co-)[[reflective factorization systems]]:

* {#RosickyTholen08} [[Jiri Rosicky]], [[Walter Tholen]], _Factorization, Fibration and Torsion_, Journal of Homotopy and Related Structures, **2** 2 (2007) 295-314  [[arXiv:0801.0063](http://arxiv.org/abs/0801.0063), [publisher](http://www.emis.de/journals/JHRS/volumes/2007/n2a14/)]

Presenting a [[pretorsion theory]] on [[Cat]] whose torsion(-free) objects are the [[groupoids]] ([[skeletal categories]], respectively), hence whose "trivial objects" are the [[skeletal groupoids]]:

* [[Francis Borceux]], [[Federico Campanini]], [[Marino Gran]], [[Walter Tholen]], *Groupoids and skeletal categories form a pretorsion theory in $Cat$* [[arXiv:2207.08487](https://arxiv.org/abs/2207.08487)]

## Related entries

* [[duality]]

* [[Gelfand duality]]

category: people

[[!redirects W. Tholen]]
[[!redirects Tholen]]


Wanda Szmielew

* [Wikipedia entry](https://de.wikipedia.org/wiki/Wanda_Szmielew)

## related $n$Lab entries

* [[Euclidean geometry]]

category: people


Ward identity

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]] the _quantum master Ward identity_ (def. \ref{OnRegularPolynomialObservablesMasterWardIdentity} below) expresses the relation between the [[quantum field theory|quantum]] (measured by [[Planck's constant]] $\hbar$) [[interacting field theory|interacting]] (measured by the [[coupling constant]] $g$) [[equations of motion]] to the [[classical field theory|classical]] [[free field]] [[equations of motion]] at $\hbar, g\to 0$ (remark \ref{QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations} below). As such it generalizes the [[Schwinger-Dyson equation]], to which it reduces for $g = 0$ (example \ref{QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations} below) as well as the _classical master Ward identity_, which is the case for $\hbar = 0$ (example \ref{MasterWardIdentityClassical} below).

Applied to products of the [[equations of motion]] with any given [[observable]], the master Ward identity becomes a particular _Ward identity_.

This is of interest notably in view of [[Noether's theorem]], which says that every [[infinitesimal symmetry of the Lagrangian]] of, in particular, the given [[free field theory]], corresponds to a [[conserved current]], hence a [[horizontal differential form]] whose [[total spacetime derivative]] vanishes up to a term proportional to the [[equations of motion]]. Under [[transgression of variational differential forms|transgression]] to [[local observables]] this is a relation of the form

$$
  div \mathbf{J} = 0 \phantom{AAA} \text{on-shell}
  \,,
$$

where "on shell" means up to the ideal generated by the [[classical field theory|classical]] [[free field theory|free]] [[equations of motion]]. Hence for the case of [[local observables]] of the form $div \mathbf{J}$, the quantum Ward identity expresses the possible failure of the original [[conserved current]] to actually be conserved, due to both quantum effects ($\hbar$) and interactions ($g$). This is the form in which Ward identities are usually understood (example \ref{NoetherCurrentConservationQuantumCorrection} below).

In terms of [[BV-BRST formalism]], the master Ward identity is equivalent to the _[[quantum master equation]]_ on [[regular polynomial observables]] ([this prop.](quantum+master+equation#QuantumMasterEquation)).

Neither of these equations is guaranteed to hold for any choice of [[renormalization|("re"-)normalization]]. If a Ward identity is violated by the [[renormalization|("re"-)normalized]] [[perturbative QFT]], specifically if there is no possible choice of [[renormalization|("re"-)normalization]] that preserves it, the one speaks of a _[[quantum anomaly]]_. Specifically if the [[conserved current]] corresponding to a [[gauge symmetry]] is _anomalous_ in this way, one speaks of a _[[gauge anomaly]]_.

## Details

### Before renormalization
 {#BeforeRenormalization}

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

Consider a [[free field theory|free]] [[gauge fixing|gauge fixed]] [[Lagrangian field theory]] $(E_{\text{BV-BRST}}, \mathbf{L}')$ ([this def.](A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity)) with global [[BV-differential]] on [[regular polynomial observables]]

$$
  \{-S',(-)\}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

([this def.](A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal)).

Let moreover

$$
  g S_{int}
   \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [ \hbar , g  ] ]
$$

be a [[regular polynomial observable]] (regarded as an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]]) such that the total action $S' + g S_{int}$ satisfies the [[quantum master equation]] ([this prop.](quantum+master+equation#QuantumMasterEquation)); and write

$$
  \mathcal{R}^{-1}(-)
  \;\coloneqq\;
  \mathcal{S}(g S_{int})^{-1}
    \star_H
  (\mathcal{S}(g S_{int}) \star_F (-))
$$

for the corresponding [[quantum Møller operator]] ([this def.](quantum+master+equation#MollerOperatorOnRegularPolynomialObservables)).

Then by [this prop.](quantum+master+equation#QuantumMasterEquation) we have

$$
  \label{OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered}
  \{-S',(-)\} \circ \mathcal{R}^{-1}
  \;=\;
  \mathcal{R}^{-1}
  \left(
    \left\{ -(S' + g S_{int}) \,,\, (-) \right\}_{\mathcal{T}}
    -
    i \hbar   \Delta_{BV}
  \right)
$$

This is the _quantum master Ward identity_ on [[regular polynomial observables]], i.e. before [[renormalization]].

=--

([Rejzner 13, (37)](#Rejzner13))

+-- {: .num_remark #QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations}
###### Remark
**([[quantum master Ward identity]] relates [[quantum field theory|quantum]] [[interacting field theory|interacting field]] [[equation of motion|EOMs]] to [[classical field theory|classical]] [[free field]] [[equation of motion|EOMs]])**

For $A \in PolyObs(E_{\text{BV-BRST}})_{reg}[ [ \hbar, g] ] $ the [[quantum master Ward identity]] on [[regular polynomial observables]] (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered) reads

$$
  \label{RearrangedMasterQuantumWard}
  \mathcal{R}^{-1}
  \left(
    \left\{ -(S' + g S_{int}) \,,\, A \right\}_{\mathcal{T}}
    -
    i \hbar   \Delta_{BV}(A)
  \right)
  \;=\;
  \{-S', \mathcal{R}^{-1}(A) \}
$$

The term on the right is manifestly in the [[image]] of the global [[BV-differential]] $\{-S',-\}$ of the [[free field theory]] ([this def.](A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal)) and hence vanishes when passing to [[on-shell]] observables along the [[isomorphism]] ([this equation](A+first+idea+of+quantum+field+theory#eq:OnShellPolynomialObservablesAsBVCohomology))

$$
  \underset{
    \text{on-shell}
  }{
  \underbrace{
    PolyObs(E_{\text{BV-BRST}}, \mathbf{L}')
  }}
  \;\simeq\;
  \underset{
    \text{off-shell}
  }{
  \underbrace{
    PolyObs(E_{\text{BV-BRST}})_{def(af = 0)}
  }}/im(\{-S',-\})
$$

(by [this example](A+first+idea+of+quantum+field+theory#BVDifferentialGlobal)).

Hence

$$
  \mathcal{R}^{-1}
  \left(
    \left\{ -(S' + g S_{int}) \,,\, A \right\}_{\mathcal{T}}
    -
    i \hbar   \Delta_{BV}(A)
  \right)
  \;=\;
  0
  \phantom{AAA}
  \text{on-shell}
$$

In contrast, the left hand side is the [[interacting field observable]] (via [this def.](S-matrix#MollerOperatorOnRegularPolynomialObservables)) of the sum of the [[time-ordered product|time-ordered]] [[antibracket]] with the [[action functional]] of the [[interacting field theory]] and a quantum correction given by the [[BV-operator]]. If we use the definition of the [[BV-operator]] $\Delta_{BV}$ ([this def.](BV-operator#RearrangedMasterWardWithOnShell)) we may equivalently re-write this as

$$
  \label{RearrangedMasterWardWithOnShell}
  \mathcal{R}^{-1}
  \left(
    \left\{ -S'  \,,\, A \right\}
    +
    \left\{ -g S_{int} \,,\, A  \right\}_{\mathcal{T}}
  \right)
  \;=\;
  0
  \phantom{AAA}
  \text{on-shell}
$$

Hence the [[quantum master Ward identity]] expresses a relation between the ideal spanned by the [[classical field theory|classical]] [[free field theory|free field]] [[equations of motion]] and the [[quantum field theory|quantum]] [[interacting field theory|interacting field]] equations of motion.

=--

+-- {: .num_example #SchwingerDysonReductionOfQuantumMasterWardIdentity}
###### Example
**([[free field]]-limit of [[master Ward identity]] is [[Schwinger-Dyson equation]])**

In the [[free field]]-limit $g \to 0$ (noticing that in this limit $\mathcal{R}^{-1} = id$) the [[quantum master Ward identity]] (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered) reduces to

$$
  \left\{ -S'  \,,\, A \right\}_{\mathcal{T}}
  -
  i \hbar   \Delta_{BV}(A)
  \;=\;
  \{-S', A  \}
$$

which is the defining equation for the [[BV-operator]] ([this equation](BV-operator#eq:BVOperatorDefiningRelation)), hence is isomorphic (under $\mathcal{T}$) to the [[Schwinger-Dyson equation]] ([this prop.](BV-operator#DysonSchwinger))

=--

+-- {: .num_example #MasterWardIdentityClassical}
###### Example
**([[classical limit]] of [[quantum master Ward identity]])**

In the [[classical limit]] $\hbar \to 0$ (noticing that the classical limit of $\{-,-\}_{\mathcal{T}}$ is $\{-,-\}$) the [[quantum master Ward identity]] (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered) reduces to

$$
  \mathcal{R}^{1}
  \left(
    \left\{ -(S' + g S_{int}) \,,\, A \right\}
  \right)
  \;=\;
  \{-S', \mathcal{R}^{-1}(A) \}
$$

This says that the [[interacting field observable]] corresponding to the global [[antibracket]] with the action functional of the [[interacting field theory]] vanishes on-shell, classically.

Applied to an observable which is [[linear map|linear]] in the [[antifields]]

$$
  A
    \;=\;
  \underset{\Sigma}{\int}
    A^a(x)
    \mathbf{\Phi}^\ddagger_a(x)
  \,
  dvol_\Sigma(x)
$$

this yields

$$
  \begin{aligned}
    0
    & =
    \{-S', \mathcal{R}^{-1}(A)\}
    +
    \mathcal{R}^{-1}
    \left(
      \left\{ -(S' + S_{int}) \,,\, A \right\}_{\mathcal{T}}
    \right)
    \\
    & =
    \underset{\Sigma}{\int}
      \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)} \mathcal{R}^{-1}(A^a(x))
    \, dvol_\Sigma(x)
    +
    \mathcal{R}^{-1}
    \left(
      \underset{\Sigma}{\int}
          A^a(x) \frac{\delta (S' + S_{int})}{\delta \mathbf{\Phi}^a(x)}
        \, dvol_\Sigma(x)
    \right)
  \end{aligned}
$$

This is the _classical master Ward identity_ according to ([Dütsch-Fredenhagen 02](#DuetschFredenhagen02), [Brennecke-Dütsch 07, (5.5)](#BrennecketDuetsch07)), following ([Dütsch-Boas 02](#DuetschBoas02)).

=--

+-- {: .num_example #NoetherCurrentConservationQuantumCorrection}
###### Example
**(quantum correction to [[Noether's theorem|Noether current]] [[conserved current|conservation]])**

Let $v \in \Gamma^{ev}_\Sigma(T_\Sigma(E_{\text{BRST}}))$ be an [[evolutionary vector field]], which is an [[infinitesimal symmetry of the Lagrangian]] $\mathbf{L}'$, and let $J_{\hat v} \in \Omega^{p,0}_\Sigma(E_{\text{BV-BRST}})$ the corresponding [[conserved current]], by [[Noether's theorem|Noether's theorem I]] ([this prop.](A+first+idea+of+quantum+field+theory#NoethersFirstTheorem)), so that

$$
  \begin{aligned}
    d J_{\hat v}
    & =
    \iota_{\hat v}  \delta \mathbf{L}'
    \\
    & =
    (v^a dvol_\Sigma) \frac{\delta_{EL} L'}{\delta \phi^a}
    \phantom{AAA}
    \in
    \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})
  \end{aligned}
$$

(by [this equation](A+first+idea+of+quantum+field+theory#eq:CurrentNoetherConservation)), where in the second line we just rewrote the expression in components (using [this equation](A+first+idea+of+quantum+field+theory#eq:EulerLagrangeEquationGeneral))

$$
  v^a
  \,,
  \frac{\delta_{EL} L'}{\delta \phi^a}
  \;\in \Omega^{0,0}_\Sigma(E_{\text{BV-BRST}})
$$

and re-arranged suggestively.

Then for $a_{sw} \in C^\infty_{cp}(\Sigma)$ any choice of [[bump function]], we obtain the [[local observables]]

$$
  \begin{aligned}
    A_{sw}
    & \coloneqq
    \underset{\Sigma}{\int}
      \underset{
        A^a(x)
      }{
      \underbrace{
        a_{sw}(x)
        v^a( \mathbf{\Phi}(x), D\mathbf{\Phi}(x), \cdots )
      }
      }
      \mathbf{\Phi}^\ddagger_a(x)
      \,
      dvol_\Sigma(x)
    \\
    & \coloneqq
    \tau_\Sigma( a_{sw}  v^a \phi^{\ddagger}_a \, dvol_\Sigma)
  \end{aligned}
$$

and

$$
  \begin{aligned}
    (div \mathbf{J})_{sw}
    & \coloneqq
    \underset{\Sigma}{\int}
      \underset{
        A^a(x)
      }{
      \underbrace{
        a_{sw}(x)
        v^a( \mathbf{\Phi}(x), D\mathbf{\Phi}(x), \cdots )
      }
      }
      \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)}
      \,
      dvol_\Sigma(x)
    \\
    & \coloneqq
    \tau_\Sigma
    \left(
      a_{sw}  v^a \frac{\delta_{EL} \mathbf{L}'}{\delta \phi^a}
      \, dvol_\Sigma
    \right)
  \end{aligned}
$$

by [[transgression of variational differential forms]].

This is such that

$$
  \left\{
    -S' , A_{sw}
  \right\}
  =
  (div \mathbf{J})_{sw}
  \,.
$$

Hence applied to this choice of local observable $A$, the quantum master Ward identity (eq:RearrangedMasterWardWithOnShell) now says that

$$
  \mathcal{R}^{-1}
  \left(
    {\, \atop \,}
    (div \mathbf{J})_{sw}
  \right)
  \;=\;
  \mathcal{R}^{-1}
  \left(
      \{g S_{int}, A_{sw} \}_{\mathcal{T}}

    {\, \atop \,}
  \right)
    \phantom{AAA}
  \text{on-shell}
$$

Hence the [[interacting field observable]]-version $\mathcal{R}^{-1}(div\mathbf{J})$ of $div \mathbf{J}$ need not vanish itself on-shell, instead there may be a correction as shown on the right.

=--

## Examples

* [[chiral anomaly]]

* [[conformal blocks]]

## References

Named after _[[John Clive Ward]]_.

Discussion of the [[master Ward identity]] in the rigorous context of [[relativistic field theory|relativistic]] [[perturbative quantum field theory]] formulated via [[causal perturbation theory]]/[[perturbative AQFT]] is in

* {#DuetschBoas02} [[Michael Dütsch]], F.-M. Boas, _The Master Ward Identity_, Rev. Math. Phys 14, (2002) 977-1049 ([pdf](http://cds.cern.ch/record/526377/files/0111101.pdf))

* {#DuetschFredenhagen02} [[Michael Dütsch]], [[Klaus Fredenhagen]], equation (90) in _The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory_, Commun.Math.Phys. 243 (2003) 275-314 ([arXiv:hep-th/0211242](https://arxiv.org/abs/hep-th/0211242))

* {#BrennecketDuetsch07} Ferdinand Brennecke, [[Michael Dütsch]], equation (5.5) in  _Removal of violations of the Master Ward Identity_, in perturbative QFT, Rev.Math.Phys. 20 (2008) 119-172 ([arXiv:https://arxiv.org/abs/0705.3160](https://arxiv.org/abs/0705.3160))

* {#Hollands07} [[Stefan Hollands]], around (322) and (333) and (345) of _Renormalized Quantum Yang-Mills Fields in Curved Spacetime_, Rev. Math. Phys.20:1033-1172, 2008 ([arXiv:0705.3340](https://arxiv.org/abs/0705.3340))

* {#Rejzner11} [[Katarzyna Rejzner]], section 5.3 of _Batalin-Vilkovisky formalism in locally covariant field theory_ ([arXiv:1111.5130](https://arxiv.org/abs/1111.5130))

* {#Rejzner13} [[Katarzyna Rejzner]], equation (37) of _Remarks on local symmetry invariance in perturbative algebraic quantum field theory_ ([arXiv:1301.7037](https://arxiv.org/abs/1301.7037))

* {#Duetsch18} [[Michael Dütsch]], equation (4.2) of _[[From classical field theory to perturbative quantum field theory]]_, 2018

See also

* Wikipedia, _[Ward-Takahashi identity](http://en.wikipedia.org/wiki/Ward%E2%80%93Takahashi_identity)_

[[!redirects Ward identities]]

[[!redirects Ward–Takahashi identity]]
[[!redirects Ward–Takahashi identities]]

[[!redirects classical master Ward identity]]
[[!redirects classical master Ward identities]]

[[!redirects quantum master Ward identity]]
[[!redirects quantum master Ward identities]]

[[!redirects master Ward identity]]
[[!redirects master Ward identities]]

[[!redirects quantum Ward identity]]
[[!redirects quantum Ward identities]]


Warren Dicks

* [personal page](https://mat.uab.cat/~dicks/)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=64122)

## Selected writings

On algebraic presentations of [[mapping class groups]] and [[braid groups]]:

* [[Warren Dicks]], Edward Formanek, *Algebraic Mapping-Class Groups of Orientable Surfaces with Boundaries*, in: *Infinite Groups: Geometric, Combinatorial and Dynamical Aspects*, Progress in Mathematics **248** Birkhäuser (2005) [[doi;10.1007/3-7643-7447-0_4](https://doi.org/10.1007/3-7643-7447-0_4)]

On [[group action|actions]] of [[braid groups]] (such as via [[automorphisms]] of  [[free groups]]):

* [[Lluís Bacardit]], [[Warren Dicks]], *Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue*, Groups Complexity Cryptology **1** (2009) 77-129 [[arXiv:0705.0587](https://arxiv.org/abs/0705.0587), [doi;10.1515/GCC.2009.77](https://doi.org/10.1515/GCC.2009.77)]

category: people


Warren Siegel

* [webpage](http://insti.physics.sunysb.edu/~siegel/plan.html)

* [Wikipedia entry](http://en.wikipedia.org/wiki/Warren_Siegel)

## Selected writings

On [[supersymmetry]] in [[superspace]]:

* {#GatesGrisaruRocekSiegel83} [[Jim Gates]] Jr, [[Marcus Grisaru]], [[Martin Roček]], [[Warren Siegel]], _Superspace, or One thousand and one lessons in supersymmetry_, Front.Phys. 58 (1983) 1-548  (1983) ([arXiv:hep-th/0108200](https://arxiv.org/abs/hep-th/0108200), [spire:195126](inspirehep.net/record/195126))

On [[doubled supergeometry]]:

* Machiko Hatsuda, Kiyoshi Kamimura, [[Warren Siegel]], _Superspace with manifest T-duality from type II superstring_, J. High Energ. Phys. (2014) 2014: 39 ([arXiv:1403.3887](https://arxiv.org/abs/1403.3887))

category: people


Warsaw circle

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

# Contents
* table of contents
{: toc}

## Idea

The **Warsaw circle** is a [[topological space]] that serves to illustrate some of the ideas of [[shape theory]].

A topological space  may have very little separating it from '[[manifold|manifoldness]]', yet a 'singularity' can cause havoc! The simple example, here, is known as the __Warsaw Circle__ as it was studied extensively by K. Borsuk and his Polish collaborators, see the book ([Borsuk 75](#Borsuk75)).

## Definition

The **Warsaw circle** $S_W $ is the [[subset]] of the [[plane]], $\mathbb{R}^2$, specified by

$$\{(x,\sin\left(\frac{1}{x}\right)) \mid -\frac{1}{2\pi} \lt x \leq \frac{1}{2\pi}, x \neq 0\}\cup \{(0,y) | -1 \leq y \leq 1\} \cup C,$$
where $C$ is an arc in $\mathbb{R}^2$ joining $(\frac{1}{2\pi}, 0)$ and $(-\frac{1}{2\pi}, 0)$, disjoint from the other two subsets specified above except at its endpoints.

It looks something like [[warsaw.pdf|this:file]]:

<img src="https://ncatlab.org/nlab/files/warsaw.gif" alt="The Warsaw circle" title="A picture of the Warsaw circle">

**Note** There is a variant version $S_W'$ with no $(x,\sin(\frac{1}{x}))$-bit for the $x\lt 0$ and the curve $C$ joins $(0,0)$ to $(\frac{1}{2\pi}, 0)$.  The discussion adapts very easily to that. For this version, there is a surjective continuous map $\mathbb{R} \to S_W'$. See eg [Wikipedia](https://en.wikipedia.org/wiki/Shape_theory_%28mathematics%29#/media/File:Warsaw_Circle.png) for a picture.

## Properties

The Warsaw circle is a [[compact space|compact]] [[metric space]], but is not [[locally connected space|locally connected]] along the line  corresponding to $\{(0,y) \mid -1 \leq y \leq 1\} $, so is not a [[manifold]], nor for that matter a [[polyhedron]].  It is [[connected space|connected]], but not [[pathwise connected space|pathwise connected]] as no path can get out from the 'line'. (The variant version noted above _is_ pathwise connected.) We note

* $\pi_0(S_W)$ is two points;

* $\pi_1(S_W)$ is trivial at any base point.

There is a simple continuous map from $S^0$, the 0-circle, $\{-1,1\}$, to $S_W$ which is a [[weak homotopy equivalence]]. (For instance define $f(-1) = (0,0)$ and $f(1)$ to be any point in the outer $sin(1/x)$ part of the space, it does not matter which one.)  This is not a [[homotopy equivalence]]. (In fact it is instructive to look at maps from $S_W$ to $S^0$!  It does not take long.)

 A striking thing about the picture is that it 'clearly' divides the plane into two components, an inside and an outside, and has a definite sense of being 'almost' a circle. It has a line of singularities, but otherwise ... .

If we consider, not just $S_W$ as a compact metric space, but as a subspace of the plane, then we can take small [[open neighbourhood|open neighbourhoods]] of $S_W$, to be definite take

$$N_{\frac{1}{n}}(S_W) = \{ \underline{x}\in \mathbb{R}^2 | d(\underline{x},S_W) \lt \frac{1}{n}\}.$$

This looks like an [[annulus]] with a thickenning at one small section. It has the [[homotopy type]] of a [[circle]]. If $N \gt n$, $N_{\frac{1}{N}}(S_W)\subset N_{\frac{1}{n}}(S_W)$, of course, (we will write $i^N_n$ for this map, and this is a homotopy equivalence. The Warsaw circle, $S_W$, is clearly the intersection of all these almost annular neighbourhoods. (Note, also clearly, that the complements  of these neighbourhoods are gradually occupying more and more of the two components of $\mathbb{R}^2- S_W$.)

We have a inverse system ([[pro-object]]) of topological spaces all of which have the [[homotopy type]] of a [[polyhedron]], ... in fact always the same polyhedron, $S^1$.  Note that by our use of a specific [[cofinal diagrams|cofinal]] family of neighbourhoods of $S_W$, indexed by the [[natural number|natural numbers]], we have an _inverse sequence_. That was a choice and we could have chosen differently or not at all. The ability to pick a sequence of neighbourhoods is related to the fact that we are considering a **compact metric space**.

Another point to note is that not only is each of the neighbourhoods [[homotopic]] to $S^1$, but the inclusion maps making up the 'bonds' of the inverse sequence, are homotopy equivalence.  This is a particularity of $S_W$ and other examples, such as the [[solenoid|solenoids]] need not have this 'stability' property.  The Warsaw circle is an example of what is called a [[stable space]].

### A (Borsuk) shape map $f\colon S^1 \to S_W$

There is a sequence of maps, $\{f_n : S^1 \to N_{\frac{1}{n}}(S_W)\mid n\in \mathbb{N}\}$, so that for each pair, $(n,N)$, with $N\gt n$, there is a homotopy $f_n \sim i^N_n f_N$. This makes a (Borsuk) [[shape map]] from the [[circle]] to the Warsaw circle.  Each $f_n$ is in fact a [[homotopy equivalence]] and we can use a choice of homotopy inverses to get another shape map $g : S_W\to S^1$ and these make up a **shape equivalence**.

(A more detailed description of shape maps and shape equivalences in the Borsuk version of shape theory, is given in the entry [[Borsuk shape theory]]. The version given here skates over some points.  It is, in fact, near the ANR-systems approach to shape.)

### From a Čech point of view

To get $\check{C}(S_W,-)$ Čech nerve complex of $S_W$, (see [[Čech methods]]), we can calculate $\check{C}(S_W,\alpha)$ for an arbitrary [[open cover]] $\alpha$ of $S_W$, but we need not do that (in fact that is a silly thing to do!).  We first note that $S_W$ is [[compact space|compact]] so we need only consider finite open covers, as these form a [[cofinal subcategory]] of the category of all open covers. ('Cofinal subcategory' means that its inclusion into the bigger category is a [[cofinal functor]].)

Next we look at any finite open cover and note that it has a refinement in the form of open balls of radius $\frac{1}{n}$, in other words we can restrict to (well chosen) such covers, giving a countable family of open covers that have to be worked with.

For such open covers the nerve will look a bit like [[circle.pdf|this:file]].

There may be fine detail in the rectangle depending on the choice of cover, but that detail will disappear as one passes to finer and finer scales. (New holes may occur, but again going finer those disappear.) Cofinally it looks like  a space obtained by adding in a thin rectangle transverse to a circle at one small segment. For different open coverings, the only difference will be where the region of attachment (marked $*$) will occur and the relative thinness of the rectangle.  The line of singularities given by the interval $[-1,1]$ on the $y$-axis cannot be 'observed', of course.  If one passes to finer and finer covers, most of the curve does not change appreciably. It just gets subdivided,  but the part near $*$ will lengthen, 'spawning' a very large number of new vertices.

There are two important points to note:

* (essentially) each $\check{C}(S_W,\alpha)$ has the homotopy type of a circle

*  the transition maps, $C(S_W,\alpha)\to C(S_W,\beta)$, will be (cofinally) homotopy equivalences.

(With a bit more care in the choice of the covers these can be made exact statements, not just 'essentially' or cofinally true.)

We note that there are obvious maps of pro-objects $\check{C}(S^1,-)\to \check{C}(S_W,-)$, and back again.  These give an isomorphism in $pro-Ho(sSets)$.  This is the Čech homotopy versions of the observations made for Borsuk's shape  above.

## References

* {#Borsuk75} [[K. Borsuk]], _Theory of Shape_, Monografie Matematyczne Tom 59,Warszawa (1975)

[[!redirects Warsaw circle]]
[[!redirects Warsaw Circle]]


Washington Taylor

* [webpage](http://web.mit.edu/physics/people/faculty/taylor_washington.html)

## Selected writings

On [[2d QCD]]:

* [[David Gross]], [[Washington Taylor]], _Two Dimensional QCD is a String Theory_, Nucl. Phys. B400:181-210, 1993

On longitudinal [[M5-branes]] in the [[BFSS matrix model]] and introducing the [[fuzzy 4-sphere]]:

* Judith Castelino, Sangmin Lee, [[Washington Taylor]], _Longitudinal 5-branes as 4-spheres in Matrix theory_, Nucl. Phys. B526:334-350, 1998 ([arXiv:hep-th/9712105](https://arxiv.org/abs/hep-th/9712105))

  (via [[D5-branes]])

On the [[non-abelian DBI action]] for [[intersecting branes]] with non-abelian [[gauge enhancement]] on their worldvolume:

* [[Akikazu Hashimoto]], [[Washington Taylor]], _Fluctuation Spectra of Tilted and Intersecting D-branes from the Born-Infeld Action_, Nucl. Phys. B503: 193-219, 1997 ([arXiv:hep-th/9703217](https://arxiv.org/abs/hep-th/9703217))

* [[Washington Taylor]], [[Mark Van Raamsdonk]], _Multiple D$p$-branes in Weak Background Fields_, Nucl. Phys. B573:703-734, 2000 ([arXiv:hep-th/9910052](https://arxiv.org/abs/hep-th/9910052))

On [[D0-D6 brane bound states]]:

* [[Washington Taylor]], _Adhering 0-branes to 6-branes and 8-branes_, Nucl. Phys. B508: 122-132, 1997 ([arXiv:hep-th/9705116](https://arxiv.org/abs/hep-th/9705116))

On [[F-theory]] [[string phenomenology]]:

* {#TaylorTurner19} [[Washington Taylor]], Andrew P. Turner, _Generic construction of the Standard Model gauge group and matter representations in F-theory_ ([arXiv:1906.11092](https://arxiv.org/abs/1906.11092))

Realization of [[E7|$E_7$]]-[[GUT]] models in [[F-theory]]:

* Shing Yan Li, [[Washington Taylor]], *Towards natural and realistic $E_7$ GUTs in F-theory* [[arXiv:2401.00040](https://arxiv.org/abs/2401.00040)]

category: people


Wasserstein metric

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## Idea

A _Wasserstein metric_ is a certain [[metric]] over a space of [[probability measure]]s on a [[measurable space]] $X$.

By ([JKO](#JKO)) the  [[heat flow]]/[[diffusion equation]] on $X$ is the [[gradient flow]] of the [[entropy|Boltzman-Shannon entropy]] functional with respect to the Wasserstein metric.

The Wasserstein metric does not seem to arise from a [[Riemannian metric|Riemann metric tensor]]. A detailed discussion of the relevant gradient flows in non-smooth [[metric space]]s is in ([AGS](#AGS)).

## References and links

* Related concepts [[information metric]], [[entropy]]

The characterization of heat flow as the gradient flow of Shannon-entropy is due to

* R. Jordan, D. Kinderlehrer, F. Otto, _The variational formulation of the [[Fokker-Planck equation]]_ , SIAM J. Math. Anal. 29 (1998), no. 1, 1-17.([pdf](http://www.imati.cnr.it/~savare/Ravello2010/JKO.pdf))
 {#JKO}

The analog of this for finite probability spaces is discussed in

* Jan Maas, _Gradient flows of the entropy for finite Markov chains_ ([pdf](http://www.janmaas.org/papers/discrete.pdf))

A comprehensive discussion of the corresponding gradient flows is in

* l Ambrosio, N. Gigli, G. Savaré, _Gradient flows in metric spaces and in the space of probability measures_, Second edition. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008. x+334 pp. ([pdf](http://www.imati.cnr.it/~savare/research/optimal_transportation/Introduction.pdf) of toc and introduction)
[MR2009h:49002](http://www.ams.org/mathscinet-getitem?mr=2401600) {#AGS}
* Luigi Ambrosio, Nicola Gigli, _Construction of the parallel transport in the Wasserstein space_, Methods Appl. Anal. 15 (2008), no. 1, 1–29, [MR2010c:49082](http://www.ams.org/mathscinet-getitem?mr=2482206)

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wave

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## Idea

An [[oscillation]] is a *periodic motion of a physical system*, i.e. as a function of time, the position of the system of particles expressed as a coordinate in a configuration space depends as $t\mapsto x(t)$ where $x(t) = x(t+I)$ where $I$ is a constant called the period of the motion. The motion can pertain to a system of particles, to the coordinates determining the position of a rigid body or can express the magnitude of some physical quantity like pressure which evolves in time. In a mechanical system, the point in which the potential energy is minimal is called the equlibrium position; the relative position with respect to the equilibrium position is called the __elongation__. The maximal elongation is usually called the __amplitude of oscillation__.

A __wave__ is the (result of) *propagation of oscillations through an extended medium*.
Examples of extended media are a liquid, the vacuum (where physical fields can propagate, for example the electromagnetic field) or a discrete grid of interacting particles. Many partial differential equations for the mechanics of extended objects have wave solutions. Waves are typically viewed via quantity called __amplitude__ which describes the relative position of each particle in a continuum system from its equilibrium position (this conflicts a bit with the terminology on oscillations, where the amplitude would be the maximnal one). The amplitude depends both on the time and the position in space. Most typical wave propagation is of the form $A(x,t) = f(x-vt)$ where $x$ is the position, $v$ is the speed of propagation (which may depend on the direction of propagation, and for inhomogeneous systems on the coordinate) and $t$ is the time. Partial differential equations (PDE) which have __wave solutions__ are typically parabolic or hyperbolic PDEs. Here wave solutions are recognized by their form (variations of the formula $f(x-vt)$ from above). Wave solutions can add up together linearly or not; in both cases the phenomenon is called __superposition__. Finite superpositions have many typical features (e.g. Lissajous figures).
The superposition can stretch over infinite families of waves, then the sum passes into the integral and many periodic modes add up to something what is not periodic whatsoever, and may be even localized. One then talks about a __wave packet__. A wave packet can from far away look like a point particle. Wave packets can decay (damp) quickly as they propagate and in nonlinear situations can be unstable under collisions with other waves. In simple linear cases, the individual cases in the wave packet correspond to elements of a basis in a Hilbert space of functions, then the expression of a wave packet in terms of basic waves (say plane waves) amount to a case of [[Fourier transform|Fourier mode expansion]].

In some special circumstances, the waves can preserve their features in nonlinear situations even asymptotically after collisions, in that case we talk about [[soliton]]s. Solitons and multisolitons
are especially well-known in gauge theories (including the 4d case localized in time, so called instantons) and in many [[integrable systems]] like the [[sine-Gordon equation]], KdV and the nonlinear Schroedinger equation. The inverse scattering method in those cases can be interpreted as the existence of a certain nonlinear analogue of the Fourier transform (the spectral transform) for the nonlinear wave solutions.

The [[quantum mechanics]], when described mathematically via the solutions of a time-dependent Schroedinger equation, is a wave mechanics where the modulus square of the amplitude is interpreted as the density of probability to find the particle in the corresponding position. Linear superposition is thus one of the basic features of quantum mechanics.

## Related concepts

* [[plane wave]], [[wave vector]]

* [[harmonic analysis]], [[Fourier analysis]]

* [[oscillation]],

* [[semiclassical approximation]]

* [[geometrical optics]]

* [[soliton]]

* [[Stokes phenomenon]]

* [[Berry's phase]].

* [[wave function]]

## References

* [[Howard Georgi]], *The Physics of Waves*, Prentice Hall (1993) $[$[web](https://www.people.fas.harvard.edu/~hgeorgi/new.htm), [pdf](https://www.people.fas.harvard.edu/~hgeorgi/onenew.pdf)$]$

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wave equation

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## Idea

For $(X,g)$ a [[pseudo-Riemannian manifold]] and $\Box : C^\infty(X) \to C^\infty(X)$ its [[Laplace operator]], the **wave equation** on $X$ is the [[linear differential equation|linear]] [[differential equation]]

$$
  \Box_g f = 0
  \,.
$$

where $\Box_g$ denotes the _wave operator_ /[[Laplace operator]], a [[hyperbolic differential operator]].

For $m \in \mathbb{R}$ the inhomogenous equation

$$
  \Box_g f = m^2 f
$$

is called the _[[Klein-Gordon equation]]_.

## Properties

### Fundamental solution

On a [[globally hyperbolic spacetime]] the wave equation/Klein-Gordon equation has unique advanced and retarded [[Green functions]].

Their difference is the [[Peierls bracket]] which gives the [[Poisson bracket]] on the [[covariant phase space]] of the [[free field|free]] [[scalar field]].  This in turn defines the [[Wick algebra]] of the free scalar field, which yields the [[quantization]] of the free scalar field to a [[quantum field theory]].

### Bicharacteristic flow and propagation of singularities

The [[bicharacteristic strips]] of the Klein-Gordon operator are [[cotangent vectors]] along [[lightlike]] [[geodesics]] ([this example](bicharacteristic+flow#BicharachteristicFlowOfKleinGordonOperator)).

## Related concepts

* [[wave]]

* [[plane wave]],

  [[wave vector]], [[wavelength]], [[frequency]]

* [[superposition]]

* [[linear differential equation]]

* [[scalar field]]

* [[Green's function]]

* [[Feynman propagator]]

* [[spectral geometry]], [[hearing the shape of a drum]]

* [[heat equation]]

* [[Klein-Gordon equation]]

* [[sine-Gordon equation]]

* [[Schrödinger equation]]

* [[Dirac equation]]

## References

* F. Friedlander, _The Wave Equation on a Curved Space-Time_, Cambridge: Cambridge University Press, 1975

* [[Howard Georgi]], *The Physics of Waves*, Prentice Hall (1993) $[$[web](https://www.people.fas.harvard.edu/~hgeorgi/new.htm), [pdf](https://www.people.fas.harvard.edu/~hgeorgi/onenew.pdf)$]$

* {#BaerGinouxPfaeffle07} [[Christian Bär]], [[Nicolas Ginoux]], [[Frank Pfäffle]], _Wave Equations on Lorentzian Manifolds and Quantization_, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover ([arXiv:0806.1036](https://arxiv.org/abs/0806.1036))

* {#Ginoux08} [[Nicolas Ginoux]], _Linear wave equations_,  Ch. 3 in [[Christian Bär]], [[Klaus Fredenhagen]], _Quantum Field Theory on Curved Spacetimes: Concepts and Methods_, Lecture Notes in Physics, Vol. 786, Springer, 2009

* [[Sergiu Klainerman]], chapter 4, section 3 of _Lecture notes in analysis_, 2011 ([pdf](https://web.math.princeton.edu/~seri/homepage/courses/Analysis2011.pdf))

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wave function collapse

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## Idea

In the context of [[quantum mechanics]], the **collapse of the wave function**, also known as the **reduction of the wave packet**, is said to occur after [[observation]] or [[measurement]], when a wave function expressed as the sum of [[eigenfunctions]] of the observable is projected randomly onto one of them. Different [[interpretations of quantum mechanics]] understand this process differently.

The perspective associated with the _[[Bayesian interpretation of quantum mechanics]]_ observes (see [below](#RelationToConditionalExpectationValues)) that the apparent collapse is just the mathematical reflection of the formula for [[conditional expectation values]] in [[quantum probability theory]].

## Relation to conditional expectation values
 {#RelationToConditionalExpectationValues}

There is a close relation between wave function collapse and [[conditional expectation values]] in [[quantum probability]] (e.g. [Kuperberg 05, section 1.2](#Kuperberg05), [Yuan 12](#Yuan12)):

Let $(\mathcal{A},\langle -\rangle)$ be a [[quantum probability space]], hence a [[complex numbers|complex]] [[star algebra]] $\mathcal{A}$ of [[quantum observables]], and a [[state on a star-algebra]] $\langle -\rangle \;\colon\; \mathcal{A} \to \mathbb{C}$.

This means that for $A \in \mathcal{A}$ any [[observable]], its _[[expectation value]]_ in the given [[state on a star-algebra|state]] is

$$
  \mathbb{E}(A) \;\coloneqq\;  \langle A \rangle \in \mathbb{C}
  \,.
$$

More generally, if $P \in \mathcal{A}$ is a [[real part|real]] [[idempotent]]/[[projector]]

$$
  \label{RealIdempotent}
  P^\ast = P
  \,,
  \phantom{AAA}
  P P = P
$$

thought of as an event, then for any observable $A \in \mathcal{A}$ the [[conditional expectation value]] of $A$, conditioned on the observation of $P$, is (e.g. [Redei-Summers 06, section 7.3](#RedeiSummers06), see also [Fröhlich-Schubnel 15, (5.49)](#FroehlichSchubnel15), [Fröhlich 19 (45)](#Froehlich16))

$$
  \label{ConditionalExpectation}
  \mathbb{E}(A \vert P)
  \;\coloneqq\;
  \frac{
    \left \langle P A P \right\rangle
  }{
    \left\langle P \right\rangle
  }
  \,.
$$

Now assume a [[star-representation]] $\rho \;\colon\; \mathcal{A} \to End(\mathcal{H})$ of the [[algebra of observables]] by [[linear operators]] on a [[Hilbert space]] $\mathcal{H}$ is given, and that the state $\langle -\rangle$ is a [[pure state]], hence given by a [[vector]] $\psi \in \mathcal{H}$ ("[[wave function]]") via the [[Hilbert space]] [[inner product]] $\langle (-), (-)\rangle \;\colon\; \mathcal{H} \otimes \mathcal{H} \to \mathbb{C}$ as

$$
  \begin{aligned}
    \langle A \rangle
    & \coloneqq
    \left\langle\psi
      \vert A \vert \psi
    \right\rangle
    \\
    & \coloneqq
    \left\langle\psi,
      A \psi
    \right\rangle
  \end{aligned}
  \,.
$$

In this case the expression for the [[conditional expectation value]] (eq:ConditionalExpectation) of an observable $A$ conditioned on an idempotent observable $P$ becomes (notationally suppressing the [[representation]] $\rho$)

$$
  \begin{aligned}
    \mathbb{E}(A\vert P)
    & =
    \frac{
      \left\langle
        \psi \vert P A P\vert \psi
      \right\rangle
    }{
      \left\langle
        \psi \vert P \vert \psi
      \right\rangle
    }
    \\
    & =
    \frac{
      \left\langle
        P \psi \vert A \vert P \psi
      \right\rangle
    }{
     \left\langle
        P \psi \vert P \psi
     \right\rangle
    }
    \,,
  \end{aligned}
$$

where in the last step we used (eq:ConditionalExpectation).

This says that _assuming_ that $P$ has been observed in the [[pure state]] $\vert \psi\rangle$, then the corresponding [[conditional expectation values]] are the same as actual [[expectation values]] but for the new pure state $\vert P \psi \rangle$.

This is the statement of "wave function collapse"

$$
  \vert \psi \rangle \mapsto P \vert \psi \rangle
  \,.
$$

The original [[wave function]] is $\psi \in \mathcal{H}$, and after observing $P$ it "collapses" to $P \psi \in \mathcal{H}$ (up to normalization).

## Related concepts

* [[interpretation of quantum mechanics]]

* [[Bayesian interpretation of quantum mechanics]]

* [[propositions as projections]]

* [[deferred measurement principle]]

[[!include states and observables -- content]]

## References
 {#References}

Early explicit discussion of wavefunction collapse in [[quantum measurements]], including discussion relating to historical experiments:

* {#vonNeumann32} [[John von Neumann]], §III.3 and §VI of:

  *Mathematische Grundlagen der Quantenmechanik*, Springer (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]

  *Mathematical Foundations of Quantum Mechanics* Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), [Wikipedia entry](https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics)]

But the formulation in [von Neumann 1932](#vonNeumann32) postulated  that a [[pure state]] would collapse to a [[mixed state]] when the [[quantum observable]]'s [[eigenvalues]] are degenerate.

This problem was pointed out and the modern formulation of the collapse postulate was formulated by:

* {#Lüders51} [[Gerhart Lüders]]:

  *Über die Zustandsänderung durch den Meßprozeß*, Ann. Phys. **8** (1951) 322–328  [[doi:10.1002/andp.19504430510](https://doi.org/10.1002/andp.19504430510)]

  *Concerning the state-change due to the measurement process*, Ann. Phys. **15** 9 (2006) 663-670 [[pdf](http://myweb.rz.uni-augsburg.de/~eckern/adp/history/historic-papers/2006_518_663-670.pdf), [[Lueders-StateChange.pdf:file]]]

Early review:

* {#Scheibe73} [[Erhard Scheibe]], pp 137 in: _*The logical analysis of quantum mechanics&, Pergamon Press Oxford (1973)

Textbook accounts in [[quantum information theory]]:

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], §2.2.3 of: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

* [[Joseph M. Renes]], eq. (3.2) in: *Quantum Information Theory* (2015) [[pdf](https://edu.itp.phys.ethz.ch/hs15/QIT/renes_lecture_notes14.pdf)], (A.2) in: De Gruyter (2022) [[doi:10.1515/9783110570250](https://doi.org/10.1515/9783110570250)]

Discussion from the point of view of [[quantum probability]]:

* {#Kuperberg05} [[Greg Kuperberg]], _A concise introduction to quantum probability, quantum mechanics, and quantum computation_ (2005) [[pdf](http://www.math.ucdavis.edu/~greg/intro-2005.pdf), [[Kuperberg-ConciseQuantum.pdf:file]]]

* {#RedeiSummers06} [[Miklos Redei]], [[Stephen Summers]], _Quantum Probability Theory_ ([arXiv:quant-ph/0601158](https://arxiv.org/abs/quant-ph/0601158))

* {#Yuan12} [[Qiaochu Yuan]], _[Finite noncommutative probability, the Born rule, and wave function collapse](https://qchu.wordpress.com/2012/09/09/finite-noncommutative-probability-the-born-rule-and-wave-function-collapse/)_, 2012

* {#FroehlichSchubnel15} [[Jürg Fröhlich]],  B. Schubnel, _Quantum Probability Theory and the Foundations of Quantum Mechanics_. In: Blanchard P., Fröhlich J. (eds.) _The Message of Quantum Science_. Lecture Notes in Physics, vol 899. Springer 2015 ([arXiv:1310.1484](https://arxiv.org/abs/1310.1484), [doi:10.1007/978-3-662-46422-9_7](https://doi.org/10.1007/978-3-662-46422-9_7))

* {#Froehlich16} [[Jürg Fröhlich]], _The structure of quantum theory_, Chapter 6 in _The quest for laws and structure_, EMS 2016  ([doi](https://www.researchgate.net/publication/308595814_The_Quest_for_Laws_and_Structure), [doi:10.4171/164-1/8](https://www.ems-ph.org/books/show_abstract.php?proj_nr=207&vol=1&rank=8))

See also

* Wikipedia, _[Wave function collapse](https://en.wikipedia.org/wiki/Wave_function_collapse)_

* Wikipedia, _[Born rule](https://en.wikipedia.org/wiki/Born_rule)_

More speculative discussion of state collapse as caused by stochastic terms added to the [[Schrödinger equation]]:

from the point of view of the [[symplectic formulation of quantum mechanics]]:

* {#Hughston96} [[Lane P. Hughston]], *Geometry of stochastic state vector reduction*, Proceedings of the Royal Society A **452** 1947 (1996) [[doi:10.1098/rspa.1996.0048](https://doi.org/10.1098/rspa.1996.0048), [jstor:52944](https://www.jstor.org/stable/52944)]

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wave packet

## Related concepts

* [[plane wave]]

## References

See also

* Wikipedia, *[Wave packet](https://en.wikipedia.org/wiki/Wave_packet)*

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