set theory - contents

**[[set theory]]**

* fundamentals of set theory
  * [[propositional logic]]
  * [[first-order logic]]
  * [[typed predicate logic]]
  * [[membership relation]]
  * [[propositional equality]]
  * [[set]], [[element]], [[function]], [[relation]]
  * [[universe]], [[small set]], [[large set]]
* [[material set theory]]
  * [[membership relation]], [[propositional equality]], [[axiom of extensionality]]
  * [[pairing structure]], [[axiom of pairing]]
  * [[union structure]], [[axiom of union]]
  * [[powerset structure]], [[axiom of power sets]]
  * [[natural numbers structure]], [[axiom of infinity]]
* presentations of set theory
  * [[first-order set theory]]
  * [[unsorted set theory]]
  * [[simply sorted set theory]]
    * [[one-sorted set theory]]
    * [[two-sorted set theory]]
    * [[three-sorted set theory]]
  * [[dependently sorted set theory]]
  * [[structurally presented set theory]]
* structuralism in set theory
  * [[material set theory]]
    * [[ZFC]]
    * [[ZFA]]
    * [[Mostowski set theory]]
    * [[New Foundations]]
  * [[structural set theory]]
    * [[categorical set theory]]
      * [[ETCS]]
        * [[fully formal ETCS]]
        * [[ETCS with elements]]
        * [[Trimble on ETCS I]]
        * [[Trimble on ETCS II]]
        * [[Trimble on ETCS III]]
      * [[structural ZFC]]
    * [[allegorical set theory]]
      * [[SEAR]]
* [[class-set theory]]
  * [[class]], [[proper class]]
  * [[universal class]], [[universe]]
  * [[category of classes]]
  * [[category with class structure]]
* [[constructive set theory]]
* [[algebraic set theory]]


set theory versus dependent type theory

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## Overview

There are a few major differences between [[set theory]] and [[dependent type theory]].

### Logic

Set theory of all flavors, including [[material set theory]] and [[structural set theory]], have two notions of logic,

* the external [[predicate logic]] with primitive notions of [[proposition]], [[true]], [[false]], [[conjunction]], [[disjunction]], [[implication]], [[existential quantifier]], and [[universal quantifier]]

* the [[internal logic]] built from [[subsingletons]], the [[singleton]], the [[empty set]], binary [[products]] of two subsingletons, the [[support]] of the binary [[disjoint union]] of two subsingletons, [[function sets]] of two subsingletons, the support of the indexed disjoint union of a family of subsingletons, and the indexed product of a [[family of subsets|family of subsingletons]].

Dependent type theory only has one notion of logic, which would correspond to the internal logic of the set theory. Thus, propositions are subsingletons.

### Equality

Set theory of all flavors have only one notion of equality, [[propositional equality]], which is a proposition in the external logic.

Most dependent type theories, like [[Martin-Löf type theory]], [[cubical type theory]], ([[higher observational type theory|higher]]) [[observational type theory]], [[two-level type theory]], [[simplicial type theory]], et cetera, all have two notions of equality, [[judgmental equality]] and [[typal equality]].

The only dependent type theory which only has one notion of equality is [[objective type theory]], and corresponds to typal equality.

In addition, the equality in all set theories are by definition propositions, while only in [[set-level type theories]] such as those with [[UIP]], [[axiom K]], or [[boundary separation]] have subsingleton-valued typal equality. More general dependent type theories have equalities which could be sets, or types representing [[infinity-groupoids]].

### Functions
 {#Functions}

[[category theory|Categorical]]$\;$[[set theory]] has two notions of [[functions]], either as [[elements]] of [[function sets]] $f \in B^A$ or as a primitives $f \colon A \to B$ in the set theory. One could compare functions for [[equality]] in either case.

On the other hand, [[dependent type theory]] has only one notion of function which could be compared for equality: an element of a [[function type]] $f \colon A \to B$. The corresponding primitive notion is the concept of [[family of elements]] $x \colon A \vdash f(x) \colon B$, of which there is no corresponding notion of equality of a family of elements.

## See also

* [[set theory]]
* [[dependent type theory]]
* [[material versus structural set theory]]

[[!redirects set theory and dependent type theory]]


set truncation

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## Idea

Set truncation is an operation in [[type theory]] which turns a [[type]] into an [[h-set]]. The idea is that given a type $T$, the type of [[connected components]] of $T$ is a set.

## Definition

### With propositional truncations and quotient sets

Suppose that the [[dependent type theory]] has [[propositional truncations]] and [[quotient sets]]. Then given a type $T$, the [[type family]] $R(x, y)$ indexed by $x:T$ and $y:T$ defined by the [[propositional truncation]] of the [[identity type]] of $T$, $R(x, y) \equiv [\mathrm{Id}_T(x, y)]_{-1}$ is an [[equivalence relation]]. The **set truncation** of $T$ is the quotient set of $T$ by $R$: $[T]_0 \equiv T / R$.

### With localizations and the circle type

Suppose that [[dependent type theory]] has [[localization of a type|localizations of types]] and the [[circle type]] $S^1$. Then the **set truncation** of a type $T$ is the localization of $T$ at $S^1$: $[T]_0 \equiv L_{S^1}(T)$

### As a higher inductive type

As a higher inductive type, the set truncation of a type $A$ is a type $[A]_0$ inductively generated by

* A function $[-]_0:A \to [A]_0$

* For each point $x:[A]_0$ and path $p:x =_{[A]_0} x$, a 2-path $K_{[A]_0}(x, p):p =_{x =_{[A]_0} x} \mathrm{refl}_A(x)$

Equivalently, it is the higher inductive type generated by

* A function $[-]_0:A \to [A]_0$

* For each function $f:S^1 \to [A]_0$ from the [[circle type]], a 2-path $K_f:\mathrm{ap}_f(\mathrm{loop}) =_{f(\mathrm{base}) =_{[A]_0} f(\mathrm{base})}\mathrm{ap}_f(\mathrm{refl}_{S^1}(\mathrm{base}))$

### Definitional set truncations

There are also definitional or judgmental versions of set truncations, similar to how there are definitional and propositional versions of [[bracket types]] and [[squash types]]:

The **definitional set truncation** or **judgmental set truncation** of a type $A$ is a type $[A]_0$ inductively generated by

* A function $[-]_0:A \to [A]_0$

* For each point $x:[A]_0$ and path $p:x =_{[A]_0} x$, a definitional equality $p \equiv \mathrm{refl}_{[A]_0}(x):x =_{[A]_0} x$

The recursion principle of definitional set truncations says that given a type $B$ with

* A function $f:A \to B$

* For each point $y:B$ and path $q:y =_{B} y$, a definitional equality $q \equiv \mathrm{refl}_B(y):y =_{B} y$

there exists a unique function $u_B:[A]_0 \to B$ such that $u_B([a]_0) = f(a)$ for all $a:A$.

Equivalently, the definitional set truncation of a type $A$ is generated by

* A function $[-]_0:A \to [A]_0$

* For each function $l:S^1 \to [A]_0$ from the [[circle type]], a definitional equality

$$\mathrm{ap}_l(\mathrm{loop}) \equiv \mathrm{ap}_l(\mathrm{refl}_{S^1}(\mathrm{base})):l(\mathrm{base}) =_{[A]_0} l(\mathrm{base})$$

Similarly, the recursion principle of this kind of definitional set truncations says that given a type $B$ with

* A function $f:A \to B$

* For each function $l_f:S^1 \to B$ from the [[circle type]], a definitional equality

$$\mathrm{ap}_{l_f}(\mathrm{loop}) \equiv \mathrm{ap}_{l_f}(\mathrm{refl}_{S^1}(\mathrm{base})):l_f(\mathrm{base}) =_{B} l_f(\mathrm{base})$$

there exists a unique function $u_B:[A]_0 \to B$ such that $u_B([a]_0) = f(a)$ for all $a:A$.

### Inference rules for set truncations

Formation rules for set truncations:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash [A]_0 \; \mathrm{type}}$$

Introduction rules for set truncations:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{settrunc}_A:A \to [A]_0 \; \mathrm{type}}$$

Dependent universal property of set truncations:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:[A]_0 \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{dup}_{[A]_0}^{B}:\left(\prod_{x:[A]_0} \mathrm{isSet}(B(x))\right) \to \mathrm{isEquiv}\left(\lambda f:\prod_{x:[A]_0} B(x).f \circ \mathrm{settrunc}_A\right)}$$

### With a choice of unique representatives

A choice of unique representatives for [[propositional truncation]] consists of a type family $C(x)$ indexed by $x:A$, and an element of type

$$\prod_{x:A} \exists!y:A.C(x) \times [x =_A y]$$

Then the type $\sum_{x:A} C(x)$ is the set truncation of $A$.

Equivalently, one could use functions instead of type families, and say that a choice of unique representatives is a type $C$ with a function $f:C \to A$ and an element of type

$$\prod_{x:A} \exists!y:A.\left(\sum_{z:C} f(z) =_A x\right) \times [x =_A y]$$

Then
$$C \simeq \sum_{x:A} \sum_{z:C} f(z) =_A x$$
is the set truncation of $A$.

## Properties

### Universal property of set truncations

The **universal property of set truncations** says that given a type $A$, a set $B$, and a function $f:A \to B$, [[uniqueness quantifier|there exists a unique]] function $u:[A] \to B$ such that for all $x:A$, $f(x) = u([x]_0)$.

### Power sets

Given a type $A$ and a [[subtype]] $P:A \to \mathrm{Prop}$, by [[propositional extensionality]] and the universal property of set truncations, one can construct a subtype $[P]_0:[A]_0 \to \mathrm{Prop}$ of the set truncation of $A$ such that for all $x:A$, $P(x) = [P]_0([x]_0)$, .

## Related concepts

* [[quotient set]]

* [[cone type]], [[bracket type]], [[set truncation]]

* [[localization of a type]], [[circle type]]

* [[axiom of set truncation]]

* [[n-truncation modality]]

## References

For set truncations in [[dependent type theory]], see section 18.5 of:

* [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

and section 6.9 and 7.3 of:

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

[[!redirects set truncation]]
[[!redirects set truncations]]

[[!redirects set-truncation]]
[[!redirects set-truncations]]

[[!redirects propositional set truncation]]
[[!redirects propositional set truncations]]

[[!redirects propositional set-truncation]]
[[!redirects propositional set-truncations]]

[[!redirects judgmental set truncation]]
[[!redirects judgmental set truncations]]

[[!redirects judgmental set-truncation]]
[[!redirects judgmental set-truncations]]

[[!redirects definitional set truncation]]
[[!redirects definitional set truncations]]

[[!redirects definitional set-truncation]]
[[!redirects definitional set-truncations]]

[[!redirects universal property of set truncations]]
[[!redirects universal properties of set truncations]]

[[!redirects universal property of set-truncations]]
[[!redirects universal properties of set-truncations]]


set-level foundations

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# Set-level foundations
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## Definition

By **set-level foundations** we mean any [[foundation of mathematics]] in which *all* the basic objects are, or behave like, [[sets]].  This includes:

* axiomatic [[set theory]] of all sorts:
  * [[material set theory]], such as [[ZFC]] and its variations
  * [[structural set theory]], such as [[ETCS]] and [[SEAR]].
* [[set-level type theory]], including both
  * definitionally [[extensional type theory]]
  * propositionally/typally extensional type theory, such as [[MLTT]] with [[UIP]], [[Lean]], the default type theory of [[Agda]], [[observational type theory]], and [[XTT]].

By contrast, "set-level foundations" does *not* include foundations of mathematics such as [[homotopy type theory]] in which there are basic objects of higher [[h-level]], which behave like [[groupoids]] or [[infinity-groupoids]] (or even [[categories]] or [[infinity-categories]]).  We call these [[higher-level foundations]].

### Characteristic axiom

Even though all the basic objects in a set-level foundation are sets, it's of course possible to talk about objects of higher [[h-level]] (e.g., via [[Kan complexes]], [[model categories]], [[quasicategories]], etc.). However, as compared to a higher-level foundation, in any set-level foundation, these objects will satisfy the principle that [[sets cover]]. For example, in a set-level foundation, every [[homotopy type]] admits a surjection from a set and every [[(∞,1)-category]] admits a surjection from a set to its [[core]] [[∞-groupoid]].

The axiom of "sets cover" is not equivalent to being set-level, however.  Even if we add this axiom to a higher foundation it remains "higher", since it still has basic objects of higher h-level, even if they are each covered by some set.


set-level type theory

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## Idea

A [[type theory]] where types behave like [[sets]], i.e. in which all types are [[h-sets]].  Thus, it is a form of [[set-level foundations]]. This includes:

* Any (definitionally) [[extensional type theory]], where there is an equality reflection rule for [[identity types]] or [[path types]] which turns their inhabitants into judgmental equalities. This includes [[Martin-Löf type theory]] with equality reflection and [[cubical type theory]] with equality reflection.

* Any [[intensional type theory]] with an axiom like [[UIP]] or [[axiom K]], or rules which prove said axioms such as strong [[pattern matching]] or [[boundary separation]], which makes all [[identity types]] or [[path types]] valued in [[propositions]]/[[subsingletons]]. Examples of the latter include [[MLTT]] with [[UIP]], [[Lean]], the default type theory of [[Agda]], [[observational type theory]], and [[XTT]].

Set-level type theory contrasts with higher-level type theory such as [[homotopy type theory]] or plain [[intensional type theory]], where types behave (at least potentially) like [[n-groupoids]] or even [[infinity-groupoids]].  In the [[negative thinking|other direction]], it contrasts with type theories where types behave like [[propositions]] (type theories of this sort are often called [[logics]]).

## Set-level intensional type theories

In a set-level but intensional type theory, we distinguish [[definitional equality|definitional]] and [[propositional equality]] (unlike in [[extensional type theory]]), but no two terms can be propositionally equal in more than one way (up to propositional equality).  In the language of [[homotopy type theory]], this means that all types are [[h-sets]].  There are a number of equivalent ways to force this to be true by adding [[axioms]] to type theory.

1. We can add as an axiom the "uniqueness of identity proofs" ([[axiom UIP]]) property that any two inhabitants of the same identity type $Id_A(x,y)$ are themselves equal (in the corresponding higher identity type).

1. We can add Streicher's *[[axiom K]]* which says that any inhabitant of a self-equality type $Id_A(x,x)$ is (propositionally) equal to the identity/reflexivity equality $1_x$.  (Axiom K is so named because $K$ comes after $J$, and $J$ usually denotes the [[elimination rule|eliminator]] for [[identity types]].)

1. In the presence of [[dependent sum types]], we can add an axiom saying that if $(a,b_1)$ and $(a,b_2)$ are pairs in a dependent sum $\sum_{x\colon A} B(x)$ with the same first component, and the identity type $Id_{\sum_{x\colon A} B(x)}((a,b_1), (a,b_2))$ is inhabited, then so is $Id_{B(a)}(b_1,b_2)$.

1. Given the [[circle type]] $S^1$, we can add any of the equivalent axioms which state respectively that $S^1$ is a [[set]] $\mathrm{isSet}(S^1)$; $S^1$ is a [[proposition]] $\mathrm{isProp}(S^1)$ or a [[contractible type]] $\mathrm{isContr}(S^1)$, which are the same condition as $S^1$ being a [[set]] because $S^1$ is provably a [[pointed connected type]]; and there is an [[identification]] $K:\mathrm{refl}_{S^1}(\mathrm{base}) =_{S^1} \mathrm{loop}$.

1. We can allow definition of functions by a strong form of *pattern matching*, as in unmodified [[Agda]].  The relevant "extra strength" is to allow output *types* of a pattern match which are only well-defined *after* the pattern has been matched.

1. In [[cubical type theory]] (with [[glue types]] removed, since univalence is incompatible with set-level-ness), we can add a [[boundary separation]] rule, leading to [[XTT]].

## See also

* [[set-level foundations]]

[[!redirects set-level type theories]]


set-like structures in homotopy type theory

[[!redirects sets in homotopy type theory]]
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## Idea

Historically, there have been [[foundations]] which do not take the notion of [[set]] to be the fundamental notion. These include [[preset]] theories over [[first order logic]] such as [[SEAR#eqfree|certain presentations of SEAR]], as well as [[type theories]] like [[MLTT]] and [[cubical type theory]]. In such theories, there have been multiple proposed definitions of what a [[set]] should be. These include:

* An [[h-set]] (as common in [[cubical type theory]] and [[Martin-Löf type theory]])
* A preset/type with an [[equivalence relation]]/[[symmetric relation|symmetric]] [[preorder]] (as common in preset theories)
* An h-set with an [[equivalence relation]]/[[symmetric relation|symmetric]] [[preorder]]
* A [[Bishop set]] (as common in [[Errett Bishop]]'s [[Bishop's constructive mathematics|branch of constructive mathematics]])
* A [[setoid]] (as common in [[MLTT]] without the [[quotient set]] [[higher inductive type]])

In [[homotopy type theory]] (and more generally in any [[intensional type theory]]), all these definitions are valid, but they lead to different structures which behave differently from each other. This article is dedicated to distinguishing between the different set-like structures which have popped up throughout history, in the context of homotopy type theory.

## Definitions

A **[[proposition]]** is a type $A$ such that for all elements $a:A$ and $b:A$ there is an element $p(a, b):a =_A b$. An **[[h-set]]** is a type $A$ where every identity type $a =_A b$ is an proposition for all elements $a:A$ and $b:A$.

We take the approach in [CapriottiKraus17](#CapriottiKraus17) in defining a **[[graph]]** $A \coloneqq (A_0, A_1)$ to be a type $A_0$ with a type family $A_1(a, b)$ indexed by elements $a:A_0$ and $b:A_0$. While not defined in CapriottiKraus17, we could continue to borrow from graph theory terminology and call  the type family $A_1$ the **[[edge]] type family** of a graph.

The edge type family $A_1$ is

* **transitive** if it comes with a family of functions
$$\mathrm{trans}(a, b, c):(A_1(a, b) \times A_1(b, c)) \to A_1(a, c)$$

* **symmetric** if it comes with a family of functions
$$\mathrm{sym}(a, b):A_1(a, b)\to A_1(b, a)$$

* **reflexive** if it comes with a family of elements
$$\mathrm{refl}(a):A_1(a, a)$$

The graph $(A_0, A_1)$ is similarly called

* **transitive** if its edge type family is transitive

* **symmetric** if its edge type family is symmetric

* **reflexive** if its edge type family is reflexive

In analogy with similar concepts in category theory ([[wild category]], [[precategory]], [[strict category]], and [[univalent category]]), we define the following:

A **wild set** $(A_0, A_1,\mathrm{trans}, \mathrm{sym}, \mathrm{refl})$ is a reflexive, symmetric, transitive graph.

A **preset** is a wild set whose hom-types are [[h-propositions]]. This is the same as a type with an [[equivalence relation]]. Note that "preset" used here is different from the notion of [[preset]] in preset theories like [[SEAR]] without equality, which correspond more to bare types in homotopy type theory.

A **strict set** or **strict preset** is a preset whose object type is also an h-set.

A **univalent set** is a preset which satisfies the Rezk completion condition

$$\mathrm{idToArr}:a =_{A_0} b \simeq A_1(a, b)$$

Univalent sets are the same as [[h-sets]], and so they are typically just called **[[sets]]**.

We do the same for the notion of [[setoids]]/[[Bishop set]]:

A **wild setoid** $(A_0, A_1,\mathrm{trans}, \mathrm{sym}, \mathrm{refl})$ is the same as a wild set, a reflexive, symmetric, transitive graph. These are the [[setoids]]/[[Bishop sets]] talked about in general [[intensional type theory]].

A **[[presetoid]]** is a wild setoid whose hom-types are [[h-sets]]. This is the same as a type with a [[pseudo-equivalence relation]].

A **[[strict category|strict]] setoid** is a presetoid whose object type is also an h-set. These are the [[setoids]]/[[Bishop sets]] talked about in [[set-level foundations]].

A **[[univalent setoid]]** is a [[presetoid]] which satisfies the Rezk completion condition

$$\mathrm{idToIso}:a =_{A_0} b \simeq a \cong b$$

where $a \cong b$ are the hom-sets of the [[core]] [[pregroupoid]] of the [[presetoid]].

## Bijections, isomorphisms, and equivalences

We begin with wild sets
$$A \coloneqq (A_0, A_1, \mathrm{trans}_A, \mathrm{refl}_A, \mathrm{sym}_A)$$
$$B \coloneqq (B_0, B_1, \mathrm{trans}_B, \mathrm{refl}_B, \mathrm{sym}_B)$$
$$C \coloneqq (C_0, C_1, \mathrm{trans}_C, \mathrm{refl}_C, \mathrm{sym}_C)$$
An extensional function between two wild sets $f:A \to B$ consists of a function $f_0:A_0 \to B_0$ and for each object $a:A_0$ and $b:B_0$ a function $f_1(a, b):A_1(a, b) \to B_1(a, b)$.

The identity function $id_A:A \to A$ is an extensional function which consists of the identity function $id_{A_0}:A_0 \to A_0$ for all elements $a:A_0$ and $b:A_0$, the identity function $id_\equiv(a, b):(a \equiv_A b) \to (a \equiv_A b))$.

The composition $g \circ f:A \to C$ of extensional functions $f:A \to B$ and $g:B \to C$ is defined as
$$(g \circ f)_0 \coloneqq g_0 \circ f_0$$
and for all objects $a:A_0$ and $b:A_0$,
$$(g \circ f)_\equiv(a, b) \coloneqq g_equiv(f_0(a), f_0(b)) \circ f_equiv(a, b)$$

We naively copy the set-theoretic definitions of [[injection]], [[surjection]], and [[bijection]] over to type theory

An extensional function $f:A \to B$ is a **[[injection]]** if for all elements $b:B_0$ the type of elements $a:A_0$ with a witness $p(a, b):B_1(f_0(a), b)$ is a [[subsingleton]].

$$\mathrm{isInj}(f) \coloneqq \prod_{b:B_0} \mathrm{isProp}\left(\sum_{a:A} B_1(f_0(a), b)\right)$$

An extensional function $f:A \to B$ is a **[[surjection]]** if for all elements $b:B_0$ the type of elements $a:A_0$ with a witness $p(a, b):B_1(f_0(a), b)$ is inhabited.

$$\mathrm{isSurj}(f) \coloneqq \prod_{b:B_0} \left|\sum_{a:A} B_1(f_0(a), b)\right|$$

An extensional function $f:A \to B$ is a **[[bijection]]** if for all elements $b:B_0$ the type of elements $a:A_0$ with a witness $p(a, b):B_1(f_0(a), b)$ is a [[singleton]].

$$\mathrm{isBij}(f) \coloneqq \prod_{b:B_0} \mathrm{isContr}\left(\sum_{a:A} B_1(f_0(a), b)\right)$$

We also naively copy the categorical-theoretic definition of [[isomorphism]] over to type theory

An extensional function $f:A \to B$ is an **[[isomorphism]]** if there exist an extensional function $g:B \to A$ with witnesses
$$\mathrm{ret}_0(f, g):(g \circ f)_0 = id_{A_0}$$
$$\mathrm{sec}_0(f, g):(f \circ g)_0 = id_{B_0}$$
$$\mathrm{ret}_1(f, g)(a, b):(g \circ f)_1(a, b) = id_{A_1}(a, b)$$
$$\mathrm{sec}_1(f, g)(a, b):(f \circ g)_1(a, b) = id_{B_1}(a, b)$$

An extensional function $f:A \to B$ is an **[[equivalence]]** if the function $f_0:A_0 \to B_0$ is an equivalence of types and for all elements $a:A_0$ and $b:A_0$ the function $f_1(a, b):A_1(a, b) \to B_1(f_0(a), f_0(b))$ is an equivalence of types.

The only notion of wild sets for which the set-theoretic notions of [[bijection]], [[isomorphism]], and [[equivalence]] are all the same is an [[h-set]]: a transitive, reflexive, and symmetric graph whose hom-types are propositionally truncated, and which satisfy the [[Rezk completion]] condition

$$\mathrm{idToArr}:a =_{A_0} b \simeq A_1(a, b)$$

Isomorphism and equivalence of wild sets are the same only when both the object type and the hom-types are [[h-sets]]. Bijection and equivalence of wild sets are the same only in the presence of the Rezk completion condition on the structure. Combined with the requirement that the object type is an [[h-set]], this automatically means that the hom-types are [[h-propositions]], which makes the entire structure an [[h-set]].

This is why [[h-sets]] are simply called [[sets]] in [[homotopy type theory]] and more generally in [[intensional type theory]].

## See also

* [[preset]]
* [[set]]
* [[setoid]]
* [[Bishop set]]
* [[h-set]]
* [[equivalence relation]]
* [[graph]]

For a similar phenomenon in defining preorders and posets, see

* [[relation between preorders and (0,1)-categories]]
* [[preorder]]
* [[poset]]
* [[thin category]]
* [[Prop]]-[[enriched category]]
* [[(0,1)-category]]

And for defining a [[category]] in [[homotopy type theory]], see

* [[category]]
* [[precategory]]
* [[univalent category]]
* [[strict category]]
* [[wild category]]

## References

* {#Bishop67} [[Errett Bishop]], _Foundations of Constructive Analysis_. New York: McGraw-Hill (1967)

* {#Palmgren05} [[Erik Palmgren]],  _Bishop's set theory_ (2005) ([pdf](http://www.cse.chalmers.se/research/group/logic/TypesSS05/Extra/palmgren.pdf))

* {#CapriottiKraus17} [[Paolo Capriotti]], [[Nicolai Kraus]], _Univalent Higher Categories via Complete Semi-Segal Types_ ([arXiv:1707.03693](https://arxiv.org/abs/1707.03693))


set-theoretic multiverse

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# The set-theoretic multiverse

* table of contents
{: toc}

## Idea

The **set-theoretic multiverse** is a philosophical perspective on [[set theory]], advocated by [[Joel David Hamkins]], according to which

> there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe.

This is in contrast to the "universe view", which

> asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer.

The set-theoretic multiverse is at least informally analogous to such categorical notions as [[Topos]], the [[2-category]] of [[toposes]], with each topos regarded as a universe of ("variable") sets. See at _[[topos theory]]_ and at _[[categorical logic]]_ for more on this.

## In dependent type theory

In [[dependent type theory]], the set-theoretic multiverse is given by the existence of multiple inequivalent models of [[set theory]] as [[types]] $V$ with [[well-founded relations]] $\in$, which are made into [[Tarski universes]] via the [[dependent sum type]]:

$$x:V \vdash \mathrm{El}(x) \coloneqq \sum_{y:V} y \in x$$

In addition, this extends to any other notion of [[set theory]], such as the categorical models of set theory as [[well-pointed categories]] $\mathcal{E}$, which are made into [[Tarski universes]] by the [[hom-set]]

$$x:V \vdash \mathrm{El}(x) \coloneqq \mathrm{hom}(1, x)$$

where $1:\mathcal{E}$ is the [[terminal object|terminal]] [[separator]] of the category $\mathcal{E}$.

This all coexists with the usual type theoretic notion of universes of sets as [[Tarski universes]] $U$ with universal type family $T$ in which for every type $A:U$ in the universe, $T(A)$ satisfies [[UIP]].

## Related (or unrelated) concepts

* [[multiverse]] (in [[cosmology]])

## References

* {#Hamkins12} [[Joel David Hamkins]], *The set-theoretic multiverse*, Review of Symbolic Logic 5:416-449 (2012), [arxiv](https://arxiv.org/abs/1108.4223)

* [[Claudio Ternullo]], *Maddy on the Multiverse*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019) [[doi:10.1007/978-3-030-15655-8_3](https://doi.org/10.1007/978-3-030-15655-8_3), [pdf](https://www.claudioternullo.com/_files/ugd/8787da_7474d586e9034716b2eea201b3e00191.pdf)]

* [n-Cafe blog discussion](https://golem.ph.utexas.edu/category/2011/08/the_settheoretic_multiverse.html)


Seth Lloyd

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

## Selected writings

On [[topological quantum computation]] with [[abelian anyons]]:

* [[Seth Lloyd]], *Quantum computation with abelian anyons*, Quantum Information Processing **1** 1/2 (2002) ([arXiv:quant-ph/0004010](https://arxiv.org/abs/quant-ph/0004010), [doi:10.1023/A:1019649101654](https://doi.org/10.1023/A:1019649101654))

On [[quantum metrology]]:

* Vittorio Giovannetti, [[Seth Lloyd]], Lorenzo Maccone, *Quantum metrology*,  	Phys. Rev. Lett. **96** (2006) 010401 [[arXiv:quant-ph/0509179](https://arxiv.org/abs/quant-ph/0509179), [doi:10.1103/PhysRevLett.96.010401](https://doi.org/10.1103/PhysRevLett.96.010401)]

On [[adiabatic quantum computation]]:

* [[Dorit Aharonov]], [[Wim van Dam]], [[Julia Kempe]], [[Zeph Landau]], [[Seth Lloyd]], [[Oded Regev]], *Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation*, SIAM Journal of Computing **37** 1 (2007) 166-194 [[arXiv:quant-ph/0405098](https://arxiv.org/abs/quant-ph/0405098), [jstor:20454175](https://www.jstor.org/stable/20454175), [doi:10.1109/FOCS.2004.8](https://doi.org/10.1109/FOCS.2004.8), [doi:10.1137/080734479](https://doi.org/10.1137/080734479)]

Introducing the notion of [[quantum random access memory]]:

* {#GiovanettiyLloydMaccone08} [[Vittorio Giovannetti]], [[Seth Lloyd]], [[Lorenzo Maccone]], *Quantum random access memory*, Phys. Rev. Lett. **100** 160501 (2008) [[doi:10.1103/PhysRevLett.100.160501](https://doi.org/10.1103/PhysRevLett.100.160501), [arXiv:0708.1879](https://arxiv.org/abs/0708.1879)]

* {#GiovanettiyLloydMaccone08b} [[Vittorio Giovannetti]], [[Seth Lloyd]], [[Lorenzo Maccone]], *Architectures for a quantum random access memory*, Phys. Rev. A **78** 052310 (2008) [[doi:10.1103/PhysRevA.78.052310](https://doi.org/10.1103/PhysRevA.78.052310)]

On [[quantum machine learning]]:

* [[Jacob Biamonte]], Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, [[Seth Lloyd]], *Quantum Machine Learning*, Nature **549**  (2017) 195–202 ([doi:10.1038/nature23474](https://doi.org/10.1038/nature23474))

  EurekaAlert, *Quanutum Machine Learning* [14-Sep-2017](https://www.eurekalert.org/pub_releases/2017-09/iiop-qml091417.php)

category: people


setoid

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#### Relations
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#### Constructivism, Realizability, Computability
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#### Graph theory
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# Setoids
* table of contents
{: toc}

## Disambiguation
 {#Disambiguation}

In [[constructive mathematics]], such as in [[intuitionistic type theory]], the notion of *setoids* [[Hofmann (1995)](#Hofmann95)] is a formalization of the idea of *[[Bishop sets]]* (following [Bishop (1967)](#Bishop67)) and serves as a way of "constructively" speaking about [[quotient sets]] without, in a sense, actually constructing these: A setoid is essentially a [[set]] equipped with an [[equivalence relation]] or a similar [[structure]] (such as a *[[pseudo-equivalence relation]]*), and thought of as a stand-in for the would-be [[quotient set]] ([[quotient type]]) by that relation.

{#UsageInTheLiterature} Accounts which take setoids to be...

* ...actual [[equivalence relations]], i.e. $Setoids \coloneqq (X \colon Set) \times (R \colon X \times X \to {\color{purple}Prop}) \times ReflSymTrans(R)$:

  [Hofmann (1995), §5.1.1](#Hofmann95); [Barthe, Capretta & Pons (2003), §2.1](#BartheCaprettaPons03); [Stacks Project](https://stacks.math.columbia.edu/tag/04S9); [hackage.haskell](https://hackage.haskell.org/package/setoid-0.1.0.0/docs/Data-Setoid.html);  [Wikipedia](https://en.wikipedia.org/wiki/Setoid);

* ...[[pseudo-equivalence relations]], i.e. $Setoids \coloneqq (X \colon Set) \times (R \colon X \times X \to {\color{purple}Set}) \times ReflSymTrans(R)$:

  [Wilander (2012), §4](#Wilander12); [Palmgren & Wilander (2014), §2, §6](#PalmgrenWilander14); [Emmenegger & Palmgren (2020), §6](#EmmeneggerPalmgren20);  [Cipriano (2020), Def. 1.1.1](#Cipriano20),

* ... either:

  [van den Berg & Moerdijk (2018)](#vdBergMoerdijk18)

{#PointOfFirstDefinition} Here the definition of setoids in terms of [[equivalence relations]] may be closer to the original intention of [Bishop (1967)](#Bishop67) (though this remains a little vague) and is natural from the point of view of [[category theory]]/[[homotopy theory]] where setoids in this sense, and with equivalence of elements regarded as [[isomorphism]], are exactly those [[groupoids]] which are [[equivalence of categories|equivalent]] to sets (the [[0-truncated]] groupoids).

Another perspective is that setoids in the sense of equivalence relations consitute the [[ex/reg completion]] of [[Sets]], while those in terms of pseudo-equivalence relations constitute the [[ex/lex completion]] of [[Sets]].

In terms of [[type theory]], the definition in terms of [[equivalence relations]] is natural under the *[[propositions as some types]]*-paradigm, while the definition in terms of [[pseudo-equivalence relations]] is natural under the *[[propositions-as-types|types as propositions]]*-paradigm.

{#TerminologyDiffers} In any case, beware that terminology and definitions may differ significantly across authors, cf. [Palmgren (2005)](#Palmgren05) [p. 9](https://ncatlab.org/nlab/files/Palmgren-BishopSetTheory.pdf#page=9).

Setoids are also sometimes used in "impoverished" [[foundations of mathematics]] that lack a primitive notion of [[quotient set]]; see for instance *[[Bishop set]]*.

## Applications

In [[constructive mathematics]], we want the [[real numbers]] to form a [[linearly ordered set|linearly ordered]] [[Heyting field]] $R$ with completeness for located [[Dedekind cuts]].  Using [[power sets]] and a set $N$ of [[natural numbers]], one may form $R$ directly as a [[subset]] of $\mathcal{P}N$ (or something equivalent), but what if you wish to be (at least weakly) [[predicative mathematics|predicative]]? Using [[function sets]], one may form the Cauchy reals as a [[subquotient]] of $N^N$, but these are complete in the desired sense only if a weak form of [[countable choice]] (which follows from either the presentation axiom or [[excluded middle]]) holds. (Essentially, there may not be enough [[sequences]] of natural numbers.) Alternatively, if you have them, you can use [[prefunction]] sets and form $R$ as a subquotient of the set of _presequences_ of natural numbers.

The construction of $R$ above may also be done with entire relations if the [[axiom of fullness]] holds (see also [[real numbers object]]). Conversely, the axiom of fullness follows from the existence of sets of prefunctions; in addition to defining a functional entire prerelation, a prefunction between sets also defines an entire relation, and the set of these satisfies fullness.  (This is related to the idea that prefunctions between sets may be formalised as entire relations.)

See also the discussion at [[net]] about how to force the domain of a net to be [[partially ordered|partial order]], by using either entire relations or prefunctions as nets.

## Fixing inadequate foundations

Sometimes one finds a [[foundation]] of [[predicative mathematics]] in which it appears to be impossible to construct [[quotient sets]], leading to much hand-wringing. (For a simple example, simply remove the axiom of [[power sets]] from [[ZFC]] as normally presented.)

Assuming (as usual) that the original foundation had equality relations on its sets, there will be identity prerelations on the presets, leading to a special class of sets which we may again call the __completely presented sets__.

When you do this, the new kind of set is called a setoid, and then there may be hand-wringing about the need to use setoids instead of sets as one would like.  But if you didn\'t have quotient sets originally, then you shouldn\'t have been talking about 'sets' in the first place; theories of sets without quotient sets are really theories of presets.

Some foundations also adopt an [[axiom of choice]] for functions which do not preserve the equivalence relation that, together with the identity relations, proves the [[presentation axiom]] for general sets, which means that free sets are completely presented sets.

If you are willing to accept the [[presentation axiom]], then you can define a notion of setoid internal to a given theory of sets: as a [[projective set]]. (With the full axiom of choice, therefore, a setoid is simply a set.) Alternatively, you might forgo setoid as such but define a [[prefunction]] between sets to be an entire relation.

A similar result holds for [[SEAR plus epsilon|SEAR+$\epsilon$]].

## Related concepts

* [[Bishop set]]

* [[completely presented set]]

* [[equivalence relation]], [[pseudo-equivalence relation]]

* [[homotopy exact completion]]

## References

The notion of "[[Bishop sets]]" goes to back the definition of sets in [[constructive mathematics]]/[[constructive analysis]] according to:

* {#Bishop67} [[Errett Bishop]], *[[Foundations of Constructive Analysis]]*, Mcgraw-Hill (1967)

* {#BishopBridges85} [[Errett Bishop]], [[Douglas Bridges]], p. 15 of:  *[[Constructive Analysis]]*, Grundlehren der mathematischen Wissenschaften **279**, Springer (1985) [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]

The connection to [[dependent type theory]] and the term *setoid* is due to

* {#Hofmann95} [[Martin Hofmann]], §1.3, §5.1 in: _Extensional concepts in intensional type theory_, Ph.D. thesis, University of Edinburgh (1995), Distinguished Dissertations, Springer (1997) [[ECS-LFCS-95-327](http://www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/), [[HofmannExtensionalIntensionalTypeTheory.pdf:file]], [doi:10.1007/978-1-4471-0963-1](https://doi.org/10.1007/978-1-4471-0963-1)]

Survey of further developments:

* {#BartheCaprettaPons03} [[Gilles Barthe]], Venanzio Capretta, [[Olivier Pons]], *Setoids in type theory*, Journal of Functional Programming **13** 2 (2003) 261-293 [[doi:10.1017/S0956796802004501](https://doi.org/10.1017/S0956796802004501)]

* {#Palmgren05} [[Erik Palmgren]],  _Bishop's set theory_ (2005) [[pdf](http://www.cse.chalmers.se/research/group/logic/TypesSS05/Extra/palmgren.pdf), [[Palmgren-BishopSetTheory.pdf:file]]]

See also:

* {#Wilander12} [[Olov Wilander]], *Constructing a small category of setoids*, Mathematical Structures in Computer Science **22** 1 (2012) 103-121 [[doi:10.1017/S0960129511000478](https://doi.org/10.1017/S0960129511000478), [pdf](http://www.diva-portal.org/smash/get/diva2:399799/FULLTEXT01.pdf)]

* {#PalmgrenWilander14} [[Erik Palmgren]], [[Olov Wilander]], *Constructing categories and setoids of setoids in type theory*, Logical Methods in Computer Science, **10** 3 (2014) lmcs:964 [[arXiv:1408.1364](https://arxiv.org/abs/1408.1364), doi:10.2168/LMCS-10(3:25)2014]

* {#vdBergMoerdijk18} [[Benno van den Berg]], [[Ieke Moerdijk]], *Exact completion of path categories and algebraic set theory: Part I: Exact completion of path categories*, Journal of Pure and Applied Algebra **222** 10 (2018) 3137-3181 [[doi:10.1016/j.jpaa.2017.11.017](https://doi.org/10.1016/j.jpaa.2017.11.017)]

* {#EmmeneggerPalmgren20} [[Jacopo Emmenegger]], [[Erik Palmgren]], *Exact completion and constructive theories of sets*, J. Symb. Log. **85** 2  (2020) [[arXiv:1710.10685](https://arxiv.org/abs/1710.10685), [doi:10.1017/jsl.2020.2](https://doi.org/10.1017/jsl.2020.2)]

* {#Cipriano20} Cioffo Cipriano Junior, Def. 1.1.1 in: *Homotopy setoids and generalized quotient completion* [[pdf](https://air.unimi.it/retrieve/d7ab68cc-fc12-4fe9-8a21-617deb888f36/phd_unimi_R12371.pdf), [[Cipriano-HomotopySetoids.pdf:file]]]

* hackage.haskell, *[Data-Setoid.html](https://hackage.haskell.org/package/setoid-0.1.0.0/docs/Data-Setoid.html)*

* [[Stacks Project]], *Categories fibred in setoids* [[tag:04S9](https://stacks.math.columbia.edu/tag/04S9)]

* Wikipedia, *[Setoid](https://en.wikipedia.org/wiki/Setoid)*

category: disambiguation

[[!redirects setoids]]


setoid > history

< [[setoid]]
[[!redirects setoid -- history]]
[[!redirects setoid->history]]


setoid object

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## Disambiguation

By *setoid objects* one will means [[objects]] equipped with *[[setoid]]*-[[structure]] [[internalization|internal]] to suitable ambient [[categories]]. Setoid objects [[internalization|internal]] to [[Set]] are ordinary [[setoids]].

Similarly to how the plain term *[[setoids]]* may refer to [[sets]] ([[objects]] of the category [[Set]]) equipped with either an [[equivalence relation]] or to sets equipped with a  [[pseudo-equivalence relation]], so the term *setoid object* in a [[finitely complete category]] may refer to either

* an [[object]] quipped with an [[internal equivalence relation]] (i.e. a [[congruence]])

* an [[object]] equipped with an [[internal pseudo-equivalence relation]].

## Related concepts

* [[setoid]]

* [[internalization]]

category:disambiguation

[[!redirects internal setoid]]
[[!redirects internal setoids]]
[[!redirects setoid object]]
[[!redirects setoid objects]]


Sets for Mathematics

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This page collects material related to the book

* [[William Lawvere]], [[Robert Rosebrugh]],

  _Sets for Mathematics_,

  Cambridge UP  2003.

  [doi:10.1017/CBO9780511755460](https://doi.org/10.1017/CBO9780511755460)

  ([book homepage](http://www.mta.ca/~rrosebru/setsformath/), [GoogleBooks](http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics), [pdf](http://patryshev.com/books/Sets%20for%20Mathematics.pdf))

which introduces [[set theory]] and the [[foundations of mathematics]] from a practical and [[categorical logic|category theoretic]] point of view known as _[[structural set theory]]_ ("informal [[ETCS]]").

## Related entries

* [[Paul Taylor]]

  _[[Practical Foundations of Mathematics]]_

  ([web](http://www.paultaylor.eu/~pt/prafm/index.html))

* [[Robert Harper]],

  _[[Practical Foundations for Programming Languages]]_,

  Cambridge University Press (2016), ([webpage](http://www.cambridge.org/us/academic/subjects/computer-science/programming-languages-and-applied-logic/practical-foundations-programming-languages-2nd-edition?format=HB))

category: reference


Sevak Mkrtchyan

* [webpage](https://people.math.rochester.edu/faculty/smkrtchy/)

## Selected writings

On the [[entropy]] of [[Schur-Weyl measures]]:

* [[Sevak Mkrtchyan]], *Entropy of Schur-Weyl Measures*, Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 678-713.  ([arXiv:1107.1541](https://arxiv.org/abs/1107.1541), [numdam:AIHPB_2014__50_2_678_0](http://www.numdam.org/item/AIHPB_2014__50_2_678_0))

category: people


seven trees in one

# Seven trees in one
* table of contents
{: toc}

## Idea

There is a [[natural bijection]] between the set of finite binary [[tree|trees]] and the set of seven-[[tuples]] of such.

This observation, originally made by [[Bill Lawvere]], has since then been expanded to other "polynomial equations" apart from $T \simeq T^7$ and other [[structure type|types]] than [[tree|trees]], revealing connections between [[types and calculus|type theory and calculus]].

## References

* John Baez, this week's finds in mathematical physics, week 202, [web](http://math.ucr.edu/home/baez/week202.html)

* Andreas Blass, seven trees in one, [arXiv:math/9405205v1](http://arxiv.org/abs/math/9405205)

* [[Tom Leinster]], objects of categories as complex numbers, [arXiv:math/0212377v1](http://arxiv.org/abs/math/0212377)

category: type theory, formal calculus


several complex variables

__Several complex variables__ is a well-established common name for [[function theory]]/[[analytic geometry]]/[[potential theory]] in several complex variables, i.e. the study of [[complex manifold]]s of finite even [[dimension]] $d \gt 2$ as well as some generalizations like [[complex space]]s.

The study of several complex variables is qualitatively much different from the theory of functions of one complex variable: for example, natural (inextendable) domains of holomorphy are not at all what one might guess based on the one-variable case ([[Hartogs lemma]]), and similarly for the question of when two domains are [[biholomorphism|biholomorphic]]. More subtle geometric conditions such as [[pseudoconvexity]] come to the fore.

Related $n$lab entries include [[Oka principle]], [[Oka manifold]], [[Weierstrass preparation theorem]].

[[!redirects several complex variables]]
[[!redirects several complex varaibles]]


Severin Bunk

Severin Bunk is working on [[higher structures]] in geometry and topology, field theories and [[mathematical physics]], currently at the University of Oxford.

* [website](http://people.maths.ox.ac.uk/bunk/)

## Selected writings

On [[schreiber:Higher Prequantum Geometry]]:

* _Categorical Structures on Bundle Gerbes and Higher Geometric Prequantisation_ ([arXiv:1709.06174](https://arxiv.org/abs/1709.06174))

On [[smooth ∞-groupoids]] and their [[shape via the cohesive path ∞-groupoid]]:

* {#Bunk20} [[Severin Bunk]], *The $\mathbb{R}$-Local Homotopy Theory of Smooth Spaces* ([arXiv:2007.06039](https://arxiv.org/abs/2007.06039))

On [[principal ∞-bundles]]:

* [[Severin Bunk]], *Principal ∞-Bundles and Smooth String Group Models* ([arXiv:2008.12263](https://arxiv.org/abs/2008.12263))

On variants of [[differential concretification|differentially concretified]] [[infinity-stack|higher]] [[moduli stacks]] of [[ordinary differential cohomology]] (higher [[bundle gerbes with connection]]) with application to [[higher gauge theory]]:

* [[Severin Bunk]], [[Carlos S. Shahbazi]], *Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology* [[arXiv:2304.06633](https://arxiv.org/abs/2304.06633)]

On a smooth open/closed [[functorial field theory]] exhibiting the [[string]]'s [[WZW term]] in a background with [[D-branes]]:

* {#BunkWaldorf21a} [[Severin Bunk]], [[Konrad Waldorf]], *Transgression of D-branes*, Adv. Theor. Math. Phys. **25** 5 (2021) 1095-1198 [[arXiv:1808.04894] (https://arxiv.org/abs/arXiv:1808.04894), [doi:10.4310/ATMP.2021.v25.n5.a1](https://dx.doi.org/10.4310/ATMP.2021.v25.n5.a1)]

* {#BunkWaldorf21b} [[Severin Bunk]], [[Konrad Waldorf]], *Smooth functorial field theories from B-fields and D-branes*, J. Homot. Rel. Struc. **16** 1 (2021) 75-153 [[doi:10.1007/s40062-020-00272-2](https://doi.org/10.1007/s40062-020-00272-2), [arXiv:1911.09990] (https://arxiv.org/abs/arXiv:1911.09990)]

On the relation between [[functorial quantum field theory]] (axiomatizing the [[Schrödinger picture]] of [[quantum field theory]]) and [[algebraic quantum field theory]] (axiomatizing the [[Heisenberg picture]]):

* {#BunkMacManusSchenkel23} [[Severin Bunk]], [[James MacManus]], [[Alexander Schenkel]], *Lorentzian bordisms in algebraic quantum field theory* [[arXiv:2308.01026](https://arxiv.org/abs/2308.01026)]

On [[principal infinity-bundles|principal $\infty$-bundles]]:

* [[Severin Bunk]], *$\infty$-Bundles*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]* [[arXiv:2308.04196](https://arxiv.org/abs/2308.04196)]

category: people


Severino C. Coutinho

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=60032)

## Selected writings

On [[D-modules]]:

* [[Severino C. Coutinho]], Introduction to: *A primer of algebraic D-modules*, London Math. Soc. Stud. Texts **33**, Cambridge University Press (1995) [[doi:10.1017/CBO9780511623653](https://doi.org/10.1017/CBO9780511623653)]

On the history of the [[canonical commutation relations]] ("[[Weyl algebra]]") originating in [[quantum mechanics]]:

* [[Severino C. Coutinho]], *The Many Avatars of a Simple Algebra*, The American Mathematical Monthly **104** 7 (1997) 593-604 [[doi:10.2307/2975052](https://doi.org/10.2307/2975052), [jstor:2975052](https://www.jstor.org/stable/2975052)]

category: people

[[!redirects Severino Coutinho]]
[[!redirects S. C. Coutinho]]


sewing constraint

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#Contents#
* table of contents
{:toc}

## Idea

If one regards a [[quantum field theory]] from the point of view of [[FQFT]] as a [[representation]] of a [[category of cobordisms]] with structure, then it is often helpful to decompose this information into two pieces:

1. the local geometric information: the value of the field theory on [[cobordism]]s of trivial topology (punctured spheres), depending thus only on the geometry (say [[Riemannian geometry]]) of these cobordisms. For instance in the case of [[conformal field theory]] this is encoded in a [[vertex operator algebra]] (as discussed there).

1. the global topological information: to obtain a representation of all cobordisms it must be possible to consistently glue the local data assigned to cobordisms of trivial topology to obtain the data assigned to more complicated cobordisms: it must be possible to "sew" together small cobordisms to large ones. The condition on the local data induced this way is called the **sewing constraint**.

## Examples

* [[Cardy condition]]

* The [[FRS formalism]] is all about solving the sewing constraints for 2-dimensional rational [[CFT]] given the local data in terms of a [[modular tensor category]] of conformal representations.

[[!redirects sewing constraints]]

[[!redirects sewing law]]
[[!redirects sewing laws]]


sextonion

The **sextonions** form a six dimensional algebra, intermediate between the quaternions and octonions.

##References

* J.M. Landsberg, L. Manivel, _The sextonions and $E_{7 \frac{1}{2}}$_, [arXiv:math/0402157](http://arxiv.org/abs/math/0402157)

* [[Bruce Westbury]], _Sextonions and the magic square_, ([pdf](http://homepages.warwick.ac.uk/~masdbn/sex2.pdf))

* Chandrashekar Devchand, _Oxidation of self-duality to 12 dimensions and beyond_, [arXiv:1212.6270](http://arxiv.org/abs/1212.6270)


Sextus Empiricus

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

## Related entries

* [[monad (disambiguation)]]

category: people


sfermion

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The (hypothetical) [[superpartners]] of [[fermions]] are called _sfermions_.

Generally, for ever fermions species, the corresponding superpartner species name is given by pre-fixing an "s".

For instance "[[squarks]]" is the name for the (hypothetical) superpartners of [[quarks]].

## Related concepts

* [[split supersymmetry]]

## References

* Wikipedia, _[Sfermions](https://en.wikipedia.org/wiki/Sfermion)_

[[!redirects sfermions]]


sFinDimVect

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include higher linear algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[category]] of [[super vector spaces]] of [[finite number|finite]] [[dimension]].

## Related conceots

* [[FinDimVect]]

* [[Deligne's theorem on tensor categories]]

[[!redirects sFinVect]]


SGA1

The initials 'SGA I' (or in more modern times 'SGA1') refer to the [[SGA]] seminar notes from [[Grothendieck]]'s seminar (1960-61) at [IHÉS](http://www.ihes.fr), Bures-sur-Yvette (see at *[[SGA]]* for more).

* [[A. Grothendieck]], [[Michèle Raynaud]], _Revêtements étales et groupe fondamental_ (SGA I), Lecture Notes in Mathematics __224__, Springer 1971 [doi:10.1007/BFb0058656](https://link.springer.com/book/10.1007/BFb0058656), retyped as [math.AG/0206203](http://arxiv.org/abs/math/0206203); published  version Documents Mathématiques __3__, Société Mathématique de France, Paris 2003)

These notes contain discussion/introduction of:

* [[Grothendieck fibration]]s, [[Grothendieck construction]]

* [[Grothendieck's Galois theory]]

* [[Galois theory|étale fundamental group]]

* [[Grothendieck's section conjecture]]

* [[descent theory]] (chapter SGA I.6)

[[!redirects SGA I]]
[[!redirects SGA 1]]

category: reference


SGA3

* [SGA 3](https://webusers.imj-prg.fr/~patrick.polo/SGA3/) - Edited by P. Gilles and P. Polo, 2011

* Review of SGA 3 (2011 edition), J.S. Milne ([pdf](http://www.jmilne.org/math/xnotes/SGA3r.pdf))

* [[SGA1]]

* [[SGA4]]

* [[SGA6]]


SGA4

The theory of [[Grothendieck topoi]] and [[etale cohomology]] has been introduced systematically in SGA 4, the fourth volume of [[SGA]].

{#Tome1} (SGA4-1) _Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos_, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4). Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. Lecture Notes in Mathematics __269__, Springer 1972. xix+525 p ([doi:10.1007/BFb0081551](https://link.springer.com/book/10.1007/BFb0081551), [pdf](http://www.cmls.polytechnique.fr/perso/laszlo/sga4/SGA4-1/sga41.pdf))

{#Tome2} (SGA4-2) _Théorie des topos et cohomologie étale des schémas. Tome 2_, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4). Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. Lecture Notes in Mathematics, Vol. 270. Springer-Verlag, Berlin-New York, 1972. iv+418 p ([doi:10.1007/BFb0061319](https://link.springer.com/book/10.1007/BFb0061319))

{#Tome3} (SGA4-3) _Théorie des topos et cohomologie étale des schémas. Tome 3_, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4). Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat. Lecture Notes in Mathematics __305__, Springer 1973. vi+640 p ([doi:10.1007/BFb0070714](https://link.springer.com/book/10.1007/BFb0070714), [pdf](https://www.normalesup.org/~forgogozo/SGA4/tomes/tome3.pdf))

Exposés :

  I. Préfaisceaux, par A. Grothendieck et J.-L. Verdier

  II. Topologies et faisceaux, par J.-L. Verdier

  III. Fonctorialité des catégories de faisceaux, par J.-L. Verdier

  IV.  Topos, par A. Grothendieck

  V. Cohomologie dans les topos, par J.-L. Verdier

  Vbis. Techniques de descente cohomologique, par B. Saint-Donat

  VI. Conditions de finitude. Topos et sites fibrés. Applications aux questions de passage à la limite, par A. Grothendieck et J.-L. Verdier

  VII. Site et topos étales dʼun schéma, par A. Grothendieck

  VIII. Foncteurs fibres, supports, étude cohomologique des morphismes finis, par A. Grothendieck

  IX. Faisceaux constructibles. Cohomologie dʼune courbe algébrique, par M. Artin

  X. Dimension cohomologique : premiers résultats, par M. Artin

  XI. Comparaison avec la cohomologie classique : cas dʼun schéma lisse, par M. Artin

  XII. Théorème de changement de base pour un morphisme propre, par M. Artin

  XIII. Théorème de changement de base pour un morphisme propre : fin de la démonstration, par M. Artin

  XIV. Théorème de finitude pour un morphisme propre ; dimension cohomologique des schémas algébriques affines, par M. Artin

  XV.  Morphismes acycliques, par M. Artin

  XVI. Théorème de changement de base par un morphisme lisse, et applications, par M. Artin

  XVII. Cohomologie à supports propres, par P. Deligne

  XVIII. La formule de dualité globale, par P. Deligne

  XIX. Cohomologie des préschémas excellents dʼégales caractéristiques, par M. Artin

Scans are available from [this page](http://library.msri.org/books/sga/).

* [Re-edition 2015](http://fabrice.orgogozo.perso.math.cnrs.fr/SGA4/index.html)

[[!redirects SGA 4]]
[[!redirects SGA IV]]


SGA6

[[!redirects SGA 6]]
* [[Pierre Berthelot]], [[Alexandre Grothendieck]], [[Luc Illusie]].  _Théorie des intersections et théorème de Riemann-Roch_, _Séminaire de Géométrie Algébrique du Bois Marie - 1966-67_.

## See also

* [[SGA]]

[[!redirects SGA VI]]


Shabnam Beheshti

* [webpage](http://www.maths.qmul.ac.uk/~beheshti/)

## related $n$Lab entries

* [[integrable system]], [[harmonic map]]

category: people


Shahn Majid

__Shahn Majid__ is an English [[mathematical physics|mathematical physicist]], an expert on [[Hopf algebras]] and quantum/[[noncommutative geometry]]. He is a Professor at the Department of Mathematics of the Queen Mary University in London.

* [wikipedia](http://en.wikipedia.org/wiki/Shahn_Majid)

* math genealogy [page](http://www.genealogy.ams.org/id.php?id=63541)

* academic [webpage](http://www.maths.qmul.ac.uk/~majid/Welcome.html)

##Selected publications

* [[T. Brzeziński]] and S. Majid, _Coalgebra bundles_, Commun. Math. Phys, 191, (1998), 467 – 492

On the [[Drinfeld double]]-construction:

* [[Shahn Majid]], _Doubles of quasitriangular Hopf algebras_, Comm. Algebra __19__ 11 (1991) 3061-3073 [[MR92k:16052](http://www.ams.org/mathscinet-getitem?mr=1132774) [doi:10.1080/00927879108824306](http://dx.doi.org/10.1080/00927879108824306)]

* [[Shahn Majid]], _Some remarks on the quantum double_, Czechoslovak J. Phys. __44__ 11-12 (1994) 1059-1071 [[arXiv:hep-th/9409056](https://arxiv.org/abs/hep-th/9409056), [doi:10.1007/BF01690458](https://doi.org/10.1007/BF01690458)]

category: people

[[!redirects S. Majid]]


Shai Ben-David

__Shai Ben-David__ is an Israeli-Canadian theoretical computer scientist who started in [[model theory]] with a thesis under [[Saharon Shelah]].

* [wikipedia](https://en.wikipedia.org/wiki/Shai_Ben-David)
* [webpage](https://cs.uwaterloo.ca/~shai) at University of Waterloo

## Selected writings:

On [[machine learning]]:

* [[Shai Shalev-Shwartz]], Shai Ben-David, _Understanding machine learning: from theory to algorithms_, Cambridge University Press 2014 [doi](https://doi.org/10.1017/CBO9781107298019)

A fundamental no-go theorem in learning theory:

* Shai Ben-David, P. Hrubes, S. Moran, A. Yehudayoff, _Learnability can be undecidable_, Nature Machine intelligence, 1 (2019) 44--48

category: people


Shai Chester

* [webpage](http://www.weizmann.ac.il/physics/shai-chester)

## Selected writings

On the [[conformal bootstrap]]:

* [[Shai Chester]], _Weizmann Lectures on the Numerical Conformal Bootstrap_ ([arXiv:1907.05147](https://arxiv.org/abs/1907.05147))

Discussion of [[large 1/N limit|small N]] effects in [[M-theory]] using the [[conformal bootstrap]]:

* Nathan B. Agmon, [[Shai Chester]], Silviu S. Pufu, _Solving M-theory with the Conformal Bootstrap_, JHEP 06 (2018) 159 ([arXiv:1711.07343](https://arxiv.org/abs/1711.07343))

* [[Shai Chester]], _Bootstrapping M-theory_, 2018 ([[ChesterBootstrappingMTheory.pdf:file]])

Specifically for the [[D=6 N=(2,0) SCFT]] on the [[M5-brane]] via [[AdS7/CFT6]]:

* {#ChesterPerlmutter18} [[Shai Chester]], [[Eric Perlmutter]], _M-Theory Reconstruction from $(2,0)$ CFT and the Chiral Algebra Conjecture_, J. High Energ. Phys. (2018) 2018: 116 ([arXiv:1805.00892](https://arxiv.org/abs/1805.00892))

  from p. 2:

  > On the other hand, given our utter lack of a complete description of M-theory, the bulk is  not  terribly  useful  for  determining  finite aspects of the dual CFT. However, we can turn this problem around using the modern perspective of the conformal bootstrap, which gives an a priori independent formulation of the (local sector of the) CFT. This provides an independent tool for constructing M-theory at the non-perturbative level, a philosophy that we will substantiate in this work.

* [[Luis Alday]], [[Shai Chester]], Himanshu Raj, _6d $(2,0)$ and M-theory at 1-loop_ ([arXiv:2005.07175](https://arxiv.org/abs/2005.07175))

Specifically for the [[D=3 SCFT]] ([[BLG-model]], [[ABJM model]]) on the [[M2-brane]] via [[AdS4/CFT3]]

* Nathan B. Agmon, [[Shai Chester]], Silviu S. Pufu, _The M-theory Archipelago_ ([arXiv:1907.13222](https://arxiv.org/abs/1907.13222))

* Damon J. Binder, [[Shai Chester]], Max Jerdee, Silviu S. Pufu, _The 3d $\mathcal{N}=6$ Bootstrap: From Higher Spins to Strings to Membranes_ ([arXiv:2011.05728](https://arxiv.org/abs/2011.05728))

See also:

* [[Luis F. Alday]], [[Shai M. Chester]], *Pure anti-de Sitter supergravity and the conformal bootstrap* [[arXiv:2207.05085](https://arxiv.org/abs/2207.05085)]

Application of [[conformal bootstrap]] to [[experiment|experimental]] [[superfluid]]-transition:

* [[Shai Chester]], Walter Landry, Junyu Liu, [[David Poland]], [[David Simmons-Duffin]], Ning Su, [[Alessandro Vichi]], _Carving out OPE space and precise $O(2)$ model critical exponents_, JHEP 06 (2020) 142 ([arXiv:1912.03324](https://arxiv.org/abs/1912.03324))

  exposition in:

  [[Slava Rychkov]], _Conformal bootstrap and the $\lambda$-point specific heat experimental anomaly_, Journal Club for Condensed Matter Physics recommendation 2020 ([pdf](https://www.condmatjclub.org/uploads/2020/01/JCCM_January_2020_02.pdf), [doi:10.36471/JCCM_January_2020_02](https://doi.org/10.36471/JCCM_January_2020_02))

On the [[AdS-CFT correspondence]]  for [[higher spin gauge theory]]:

* [[Ofer Aharony]], [[Shai Chester]], [[Erez Urbach]], _A Derivation of AdS/CFT for Vector Models_ ([arXiv:2011.06328](https://arxiv.org/abs/2011.06328))

[[!redirects Shai M. Chester]]

category: people


Shai Shalev-Shwartz

* [GoogleScholar page](https://scholar.google.com/citations?user=uYVc9koAAAAJ&hl=en)

## Selected writings

On [[machine learning]]:

* Shai Shalev-Shwartz, [[Shai Ben-David]], _Understanding machine learning: from theory to algorithms_, Cambridge University Press 2014 [doi](https://doi.org/10.1017/CBO9781107298019)

* Shai Shalev-Shwartz, _Online learning and online convex optimization_, Foundations and trends in machine learning, __4__:2, 107-194 (2011) [doi](http://dx.doi.org/10.1561/2200000018) [pdf](http://www.cs.huji.ac.il/~shais/papers/OLsurvey.pdf)

* A. Brutzkus, A. Globerson, E. Malach, S. Shalev-Shwartz, _SGD learns over-parameterized networks that provably generalize on linearly separable data_, in Proc. Int. Conf. Learn. Represent., 2018, 1--17 [arXiv:1710.10174](https://arxiv.org/abs/1710.10174) [openreview](https://openreview.net/forum?id=rJ33wwxRb) [Globerson's slides](https://simons.berkeley.edu/sites/default/files/docs/9983/simonsjune18.pdf) (presentation)

category: people


Shamil Shakirov

* [InSpire page](https://inspirehep.net/authors/1048151)

## Selected writings

On the [[AGT correspondence]]:

* [[Andrei Mironov]], [[Andrey Morozov]], [[Shamil Shakirov]], *A direct proof of AGT conjecture at $\beta = 1$*, JHEP 1102:067 (2011) [[arXiv:1012.3137](https://arxiv.org/abs/1012.3137), doi:10.1007/JHEP02(2011)067]

category: people

[[!redirects Sh. Shakirov]]


Shamit Kachru

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

## Selected writings

On a [[matrix model]] for the [[3-brane in 6d]] arising as the [[brane intersection|intersection]] of two [[M5-branes]]:

* {#KachruOzYin98} [[Shamit Kachru]], [[Yaron Oz]], [[Zheng Yin]], *Matrix Description of Intersecting M5 Branes* JHEP 9811:004, (1998) ([arXiv:hep-th/9803050](https://arxiv.org/abs/hep-th/9803050))

Systematic construction of [[Ricci tensor|Ricci flat]] [[Riemannian metrics]] on [[K3]] [[orbifolds]]:

* [[Shamit Kachru]], [[Arnav Tripathy]], [[Max Zimet]], _K3 metrics_ ([arXiv:2006.02435](https://arxiv.org/abs/2006.02435))

* [[Arnav Tripathy]], [[Max Zimet]], _A plethora of K3 metrics_ ([arXiv:2010.12581](https://arxiv.org/abs/2010.12581))

category: people


Shane Farnsworth

* [webpage](https://www.perimeterinstitute.ca/people/shane-farnsworth)

## Selected writings

Discussion of [[spectral triples]] over [[Jordan algebras]] in the [[Connes-Lott model]]:

* [[Latham Boyle]], [[Shane Farnsworth]], _The standard model, the Pati-Salam model, and "Jordan geometry"_ ([arxiv:1910.11888](https://arxiv.org/abs/1910.11888))

* [[Shane Farnsworth]], _The geometry of physical observables_ ([arXiv:2003.01708](https://arxiv.org/abs/2003.01708))

* Fabien Besnard, [[Shane Farnsworth]], *Particle models from special Jordan backgrounds and spectral triples* $[$[arXiv:2206.07039](https://arxiv.org/abs/2206.07039)$]$

## Related $n$Lab entries

* [[Connes-Lott-Chamseddine model]]

category: people


Shane Kelly

* [webpage](https://www.ms.u-tokyo.ac.jp/~kelly/)

## Selected writings

On [[motives]] over [[positive characteristic]] bases (developing techniques to avoid [[resolution of singularities]]):

* [[Shane Kelly]], _Voevodsky motives and ldh descent_, Asterisque 391 (2017), [arXiv:1305.5349](https://arxiv.org/abs/1305.5349).

On the proof that [[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]]:

* [[Shane Kelly]], _What is the relationship between Grothendieck pretopologies and Grothendieck topologies?_ ([web] (https://www.ms.u-tokyo.ac.jp/~kelly/pdfs/grothendieckTop.pdf)).

category: people


shape

#Contents#
* table of contents
{:toc}

## Idea

Given a [[space]] of sorts, its _[[shape]]_ is a [[discrete ∞-groupoid|geometrically discrete]] [[homotopy type]] which approximates the spaces as good as possible "from the left".

For instance

* [[shape modality]] (shape of an [[object]] in a [[locally ∞-connected (∞,1)-topos]], see [[geometric realization in a locally ∞-connected (∞,1)-topos]])

* [[shape of an (∞,1)-topos]] (itself)

* [[shape of a topological space]]

## Related concepts

* [[shape theory]]

* [[shape via cohesive path ∞-groupoid]]

[[!include cohesion - table]]

[[!redirects shapes]]


shape fibration

(this is a stub yet)

A shape fibration is an adaptation of the notion of [[Hurewicz fibration]] from the setup of [[homotopy theory]] to [[shape theory]] (of [[locally compact space|locally compact]] [[metrisable spaces]]).

A map between [[metric space|metric]] [[compactum|compacta]] $p : E\to B$ is called a
**shape fibration**, provided it is induced by a level map $p : E\to B$ of ANR-sequences satisfying the [[approximate homotopy lifting property]] with respect to any metric space $X$ and the lifting index, and lifting mesh do
not depend on $X$.

If, more generally, $E$ and $B$ are locally compact metric spaces. A
[[proper map]] $p : E\to B$ is a **shape fibration** if for any compact $C\subset B$ the restriction of $p$ to a map $p^{-1}(C)\to C$ is a shape fibration between metric compacta.

If $E$ and $B$ are locally compact metric ANR's then the notion of shape fibration coincides with a notion of an [[approximate fibration]].

* [[Sibe Mardesic| S. Mardešić]], T. B. Rushing, Shape fibrations. I.  General Topology Appl.  __9__  (1978), no. 3, 193--215.

* S. Mardešić, J. Segal, (1982) Shape Theory, North Holland.

* T. B. Rushing, Cell-like maps, approximate fibrations and shape fibrations


shape map

###Idea

A shape map is a morphism in a shape category, for instance in Borsuk's [[shape theory]].


shape modality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Induction
+-- {: .hide}
[[!include induction - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In a [[locally ∞-connected (∞,1)-topos]] with [[full and faithful (∞,1)-functor|fully faithful]] [[inverse image]] (such as a [[cohesive (∞,1)-topos]]), the extra [[left adjoint]] $\Pi$ to the [[inverse image]] $Disc$ of the [[global sections]] [[geometric morphism]] $\Gamma$ induces a  [[higher modality]] $\esh \coloneqq Disc \circ \Pi$, which sends an [[object]] to something that may be regarded equivalently as its [[geometric realization]] or its [[fundamental ∞-groupoid]] (see at _[[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos]]_ and at _[[shape via cohesive path ∞-groupoid]]_). In either case $\esh X$ may be thought of as the _[[shape]]_ of $X$ and therefore one may call $\esh$ the _shape modality_.  It forms an [[adjoint modality]] with the [[flat modality]] $\flat \coloneqq Disc \circ \Gamma$.

## Definition

### In type theory

We assume a [[dependent type theory]] with crisp term judgments $a::A$ in addition to the usual (cohesive) type and term judgments $A \; \mathrm{type}$ and $a:A$, as well as context judgments $\Xi \vert \Gamma \; \mathrm{ctx}$ where $\Xi$ is a list of crisp term judgments, and $\Gamma$ is a list of cohesive term judgments. A crisp type is a type in the context $\Xi \vert ()$, where $()$ is the empty list of cohesive term judgments. In addition, we also assume the dependent type theory has [[typal equality]] and [[judgmental equality]], as well as the following [[axiom of cohesion]]:

*There is a crisp type $\Xi \vert () \vdash R \; \mathrm{type}$ such that given any crisp type $\Xi \vert () \vdash A \; \mathrm{type}$, $A$ is discrete if and only if the function $\mathrm{const}_{A, R}:A \to (R \to A)$ is an equivalence of types.*

From here, there are two different notions of the shape modality which could be defined in the type theory, the **strict shape modality**, which uses [[judgmental equality]] in the [[computation rule]] and [[uniqueness rule]], and the **weak shape modality**, which uses [[typal equality]] in the [[computation rule]] and [[uniqueness rule]]. When applied to a type they result in **strict shape types** and **weak shape types** respectively. The [[natural deduction]] rules for strict and weak shape types are provided as follows:

* Formation rules for weak and strict shape types:

$$\frac{\Xi \vert \Gamma \vdash A \; \mathrm{type}}{\Xi \vert \Gamma \vdash \esh A \; \mathrm{type}}\esh-\mathrm{form}$$

* Introduction rules for weak and strict shape types:

$$\frac{\Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash \sigma_A(a):\esh A}\esh-\mathrm{intro}1$$

$$\frac{\Xi \vert \Gamma \vdash g:R \to \esh A \quad \Xi, \Gamma \vert () \vdash x:R}{\Xi \vert \Gamma \vdash k_A(g):\esh A}\esh-\mathrm{intro}2$$

$$\frac{\Xi \vert \Gamma \vdash g:R \to \esh A \quad \Xi, \Gamma \vert () \vdash x:R}{\Xi \vert \Gamma \vdash \iota_A(g, x):g(x) =_{\esh A} k_A(g)}\esh-\mathrm{intro}3$$

$$\frac{\Xi \vert \Gamma \vdash g:R \to \esh A \quad \Xi, \Gamma \vert () \vdash x:R}{\Xi \vert \Gamma \vdash k_A'(g):\esh A}\esh-\mathrm{intro}4$$

$$\frac{\Xi \vert \Gamma \vdash x:R}{\Xi \vert \Gamma \vdash \iota_A'(x):x =_{\esh A} k_A'(\mathrm{const}_{\esh A, R}(x))}\esh-\mathrm{intro}5$$

* Elimination rules for weak and strict shape types:

$$\frac{\Xi \vert \Gamma \vdash A \; \mathrm{type} \quad \Xi \vert \Gamma, w:\esh A \vdash Q \; \mathrm{type} \quad \Xi \vert \Gamma \vdash x:A}{\Xi \vert \Gamma \vdash d_{\sigma_A}(x):Q[\sigma_A(x)/w]}\esh-\mathrm{elim}1$$

$$\frac{\Xi \vert \Gamma \vdash A \; \mathrm{type} \quad \Xi \vert \Gamma, w:\esh A \vdash Q \; \mathrm{type} \quad \Xi \vert \Gamma \vdash g:R \to \esh A \quad \Xi \vert \Gamma \vdash x:R \quad \Xi \vert \Gamma \vdash h:Q[g(x)/w]}{\Xi \vert \Gamma \vdash d_{\kappa_A}(g, h):Q[\kappa_A(g)/w]}\esh-\mathrm{elim}2$$

$$\frac{\Xi \vert \Gamma \vdash A \; \mathrm{type} \quad \Xi \vert \Gamma, w:\esh A \vdash Q \; \mathrm{type} \quad \Xi \vert \Gamma \vdash g:R \to \esh A \quad \Xi \vert \Gamma \vdash x:R \quad \Xi \vert \Gamma \vdash h:Q[g(x)/w]}{\Xi \vert \Gamma \vdash d_{\iota_A}(g, x.h):h =_{Q[\kappa_A(g)/w]} d_{\kappa_A}(g, h)}\esh-\mathrm{elim}3$$

$$\frac{\Xi \vert \Gamma \vdash A \; \mathrm{type} \quad \Xi \vert \Gamma, w:\esh A \vdash Q \; \mathrm{type} \quad \Xi \vert \Gamma \vdash g:R \to \esh A \quad \Xi \vert \Gamma \vdash x:R \quad \Xi \vert \Gamma \vdash h:Q[g(x)/w]}{\Xi \vert \Gamma \vdash d_{\kappa_A'}(g, h):Q[\kappa_A'(g)/w]}\esh-\mathrm{elim}4$$

$$\frac{\Xi \vert \Gamma \vdash A \; \mathrm{type} \quad \Xi \vert \Gamma, w:\esh A \vdash Q \; \mathrm{type} \quad \Xi \vert \Gamma \vdash x:\esh A \quad \Xi \vert \Gamma \vdash y:Q[x/w]}{\Xi \vert \Gamma \vdash d_{\iota_A'}(y, x):y =_{Q[\kappa_A'(\mathrm{const}_{\esh A, R}(x))/w]} d_{\kappa_A'}(\mathrm{const}_{\esh A, R}(x), \mathrm{const}_{\esh A, R}(y))}\esh-\mathrm{elim}5$$

* Computation rules for weak and strict shape types:

$$\frac{\Xi \vert \Gamma \vdash A \; \mathrm{type} \quad \Xi \vert \Gamma, w:\esh A \vdash Q \; \mathrm{type} \quad \Xi \vert \Gamma \vdash x:A \quad \Xi \vert \Gamma, y:\esh A \vdash f:Q}{\Xi \vert \Gamma \vdash f[\sigma_A(x)/y] \equiv d_{\sigma_A}(x):Q[\sigma_A(x)/w]}\esh-\mathrm{comp}\mathrm{strict}1$$

$$\frac{\Xi \vert \Gamma \vdash A \; \mathrm{type} \quad \Xi \vert \Gamma, w:\esh A \vdash Q \; \mathrm{type} \quad \Xi \vert \Gamma \vdash x:A \quad \Xi \vert \Gamma, y:\esh A \vdash f:Q}{\Xi \vert \Gamma \vdash \beta_\esh^1(x):f[\sigma_A(x)/y] =_{Q[\sigma_A(x)/w]} d_{\sigma_A}(x)}\esh-\mathrm{comp}\mathrm{weak}1$$

...

The shape modality is an example of a [[higher inductive type]] which states that the [[shape modality]] of a type $A$ $\esh(A)$ is the [[localization of a type|localization]] $L_{R^1}(A)$ of $A$ at the function $\mathrm{const}:A \to (R \to A)$, equivalent to $\mathrm{const}':(1 \to A) \to (R \to A)$. Weak shape modalities are primarily used in cohesive [[objective type theories]], while strict shape modalities are typically used in cohesive type theories with [[judgmental equality]], such as cohesive [[Martin-Löf type theory]] ([[cohesive homotopy type theory]] or cohesive [[higher observational type theory]].

## Properties

### Preservation of some $\infty$-fiber products
 {#PreservationOfSomeHomotopyFiberProducts}

The shape operation is defined to [[preserved limit|preserve]] [[finite products|finite]] [[homotopy products]] and small [[(infinity,1)-colimits|homotopy colimits]] (being a [[left adjoint]]) but it does not preserve general [[(infinity,1)-limits|homotopy limits]], not even all [[homotopy pullbacks]]:

\begin{example}
**(shape does not preserve homotopy fibers of points over positive-dimensional smooth spheres)**
\linebreak
  For $n \in \mathbb{N}_+$ consider the [[n-sphere|$n$-sphere]] in its standard incarnation as a [[smooth manifold]] (or just as a [[D-topological space]]) and as such as a [[0-truncated]] [[smooth infinity-groupoid|smooth $\infty$-groupoid]]
$$
  S^n_{smth} \,\in\, SmthMfd \hookrightarrow SmthGrpd_\infty
  \,.
$$
This is such that the shape is the $n$-sphere as a geometrically [[discrete object|discrete]] higher [[homotopy type]]
$$
  \esh S^n_{smth} \,\simeq\, S^n \,\in\,
  Grpd_\infty
  \xhookrightarrow{Disc}
  SmthGrpd_{\infty}.
$$

Now for $x \colon \ast \to S^n_{smth}$ any point (and using that $\esh \ast \,=\, \ast$ by the condition that $\esh$ preserves [[finite products]]):

1. its [[fiber product]] over $S^n_{smth}$ may be computed in [[SmthMfd]] and hence is that same point:

1. its [[homotopy fiber product]] over $S^n = \esh S^n_{smth}$ is the [[loop space]] $\Omega S^n$, which (under our assumption that $n$ is a [[positive number]]) is far from equivalent to the point (for example: $\Omega S^1 \,\simeq\, \mathbb{Z}$).

Hence

$$
  n \in \mathbb{N}_+
  \;\;\;\;
  \vdash
  \;\;\;\;
  \ast
  \;\simeq\;
  \esh
  \big(
    \ast
     \underset
       {S^n_{smth}}
       {\times}
    \ast
  \big)
  \;\neq\;
  \big(
    (\esh\ast)
     \underset
       {\esh S^n_{smth}}
       {\times}
    (\esh\ast)
  \big)
  \;\simeq\;
  \Omega S^{n}
  \,.
$$
\end{example}

However, the shape modality does preserve some useful classes of homotopy fiber products after all:

#### General

\begin{prop}\label{CohesiveShapePreservesLoopingOfDeloopings}
**(cohesive shape preserves looping and delooping)**
\linebreak
For $\mathbf{H}$ a [[cohesive (infinity,1)-topos|cohesive $\infty$-topos]]
and $\mathbf{B}\mathcal{G} \,\in\, \mathbf{H}$ a pointed connected object, [[May recognition theorem|hence]] the [[delooping]] of an [[infinity-group|$\infty$-group]] $\mathcal{G} \,\in\, Grp(\mathbf{H})$, the cohesive shape modality preserves the [[looping]] homotopy fiber product:
$$
  \esh
  \big(
    \ast
    \underset
     {\mathbf{B}\mathcal{G}}{\times}
    \ast
  \big)
  \;\;
  \simeq
  \;\;
    \ast
    \underset
     {\esh \mathbf{B}\mathcal{G}}{\times}
    \ast
  \,.
$$
\end{prop}
\begin{proof}
  Since [[groupoid objects in an (∞,1)-topos are effective]], the [[delooping]] of an [[infinity-group|$\infty$-group]] is the simplicial [[(infinity,1)-colimit|$\infty$-colimit]] of the [[simplicial object]] $\mathcal{G}^{\times^\bullet}$ of [[finite products]] of $\mathcal{G}$. Since the shape operation [[preserved limit|preserves]] finite products and general colimits, it preserves the delooping:
$$
  \esh
  \mathbf{B}\mathcal{G}
  \;\simeq\;
  \esh
  \underset{\underset{n \in \mathbb{N}}{\longrightarrow}}
  {\lim}
  \mathcal{G}^{\times^n}
  \;\simeq\;
  \underset{\underset{n \in \mathbb{N}}{\longrightarrow}}
  {\lim}
  \big(
    \esh
    \mathcal{G}
  \big)^{\times^n}
  \;\simeq\;
  \mathbf{B} \esh \mathcal{G}
  \,.
$$
Finally, using again that [[groupoid objects in an (∞,1)-topos are effective]], the claim follows:
$$
  \esh
  \big(
    \ast
     \underset{\mathbf{B}\mathcal{G}}{\times}
    \ast
  \big)
  \;\simeq\;
  \esh
   \mathcal{G}
  \;\simeq\;
  \esh
  \big(
    \ast
     \underset{\esh\mathbf{B}\mathcal{G}}{\times}
    \ast
  \big)
  \,.
$$
\end{proof}

\begin{prop}\label{ShapePreservesHomotopyFibersOverDiscreteObjects}
**(cohesive shape preserves homotopy fiber products over discrete objects)**
\linebreak
  If $\mathbf{H}$ is a [[cohesive (infinity,1)-topos|cohesive $\infty$-topos]] [[base (infinity,1)-topos|
over]] [[Infinity-Grpd|$Grpd_\infty$]], then its [[shape modality]] [[preserved limit|preserves]] [[homotopy fiber products]] over [[discrete objects]].
\end{prop}
The following argument follows [[schreiber:differential cohomology in a cohesive topos|dcct, Thm. 3.8.19]], which in turn follows discussion with [[Mike Shulman]]. The diagram is taken from the writeup of the proof given in [[schreiber:Equivariant principal infinity-bundles|SS21, Prop. 3.38]].
\begin{proof}
By a version of the [[(infinity,1)-Grothendieck construction|$\infty$-Grothendieck construction]] ([Beardsley & Péroux 2022, Lem. 3.10](infinity1-Grothendieck+construction#BeardsleyPéroux22)), a [[slice (infinity,1)-topos|slice $\infty$-topos]] over the [[inverse image]] of a plain [[infinity-groupoid|$\infty$-groupoid]] $B$ is [[equivalence of (infinity,1)-categories|equivalent]] to the [[(infinity,1)-category of (infinity,1)-functors|$\infty$-category of $\infty$-functors]] $B \to \mathbf{H}$. This way, the [$B$-sliced adjunction](adjoint+infinity1-functor#OnSlices) $Shp \dashv Dscr$ is seen to be of the following form:
\begin{tikzcd}[column sep=20pt]
    \mathrm{Fnctr}
    \big(
      B
      ,\,
      \mathbf{H}
    \big)
    \ar[
      from=rrrrrr,
      rounded corners,
      to path={
           ([yshift=0pt]\tikztostart.south)
        -- ([yshift=-12pt]\tikztostart.south)
        -- node[yshift=-6.8pt]{ \scalebox{.8}{$
               \mathrm{Fnctr}
                 (B, \mathrm{Dscr})
           $} }
           ([yshift=-12pt]\tikztotarget.south)
        -- ([yshift=-00pt]\tikztotarget.south)
      }
    ]
    \ar[
      rrrrrr,
      rounded corners,
      to path={
           ([yshift=0pt]\tikztostart.north)
        -- ([yshift=+12pt]\tikztostart.north)
        -- node[yshift=+6.8pt]{ \scalebox{.8}{$
             \mathrm{Fnctr}
               (B,\mathrm{Shp})
           $} }
           ([yshift=+12pt]\tikztotarget.north)
        -- ([yshift=-00pt]\tikztotarget.north)
      }
    ]
    &[-24pt]
    \simeq
    &[-24pt]
    \mathbf{H}_{/\mathrm{Dscr}(B)}
    \ar[
      rr,
      shift left=7pt,
      "\mathrm{Shp}_{/B}"
    ]
    \ar[
      from=rr,
      shift left=7pt,
      "{\mathrm{Dscr}_{/B}}"
    ]
    \ar[
      rr,
      "{\scalebox{.8}{$\bot$}}",
      phantom
    ]
    &&
    \big(
      \mathrm{Grpd}_{\infty}
    \big)_{/B}
    &[-24pt]\simeq&[-24pt]
    \mathrm{Fnctr}
    \big(
      S
      ,\,
      \mathrm{Grpd}_\infty
    \big)
    \,.
\end{tikzcd}

Now $Fnctr(B,Shp)$ preserves [[binary products]] as soon as $Shp$ does (because products of presheaves are computed objectwise). But binary products in the equivalent slice category $\mathbf{H}_{/B}$ are the homotopy fiber products in question, and hence the claim follows.
\end{proof}

\begin{prop}\label{ShapePreservesHomotopyFibersOfMapsFromDiscreteToDeloopedObjects}
**(shape preserves homotopy fibers of maps from discrete to delooped objects)**
\linebreak
  For $\mathbf{H}$ is a [[cohesive (infinity,1)-topos|cohesive $\infty$-topos]] [[base (infinity,1)-topos|
over]] [[Infinity-Grpd|$Grpd_\infty$]] the shape modality preserves the [[homotopy fibers]] of morphisms

$$
  f \,\colon\, X \longrightarrow \mathbf{B}\mathcal{G}
$$

* from any [[discrete object]] $X \,\in\, Grpd_\infty \xhookrightarrow{Dscr} \mathbf{H}$

* to any [[delooping|delooped]] object $\mathbf{B}\mathcal{G} \,\in\, \mathbf{H}$

in that

$$
  \esh
  fib(f)
  \;\simeq\;
  fib\big( \esh f \big)
  \,.
$$

\end{prop}
\begin{proof}
Since a discrete $\infty$-groupoid $X$ is the [[coproduct]] of its [[connected components]], and since coproducts are [[preserved colimit|preserved]] by [[homotopy pullback]] (due to [[universal colimits]] in an [[(infinity,1)-topos|$\infty$-topos]]) and by the shape modality (being a [[left adjoint]]), we may assume without restriction of generaliity that also $X$ is connected, hence $X \,\simeq\,B \mathcal{K}$ with $\ast \twoheadrightarrow B\mathcal{K}$ an [[effective epimorphism in an (infinity,1)-category|effective epimorphism]].

First, observe that the [[pasting law for homotopy pullbacks]] gives the pasting diagram of [[homotopy pullbacks]] shown on the left below:

$$
  \array{
    \mathcal{G}
    &\longrightarrow&
    fib(f)
    &\longrightarrow&
    \ast
    \\
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow
    \\
    \ast
    &\twoheadrightarrow&
    B\mathcal{K}
    &\underset{f}{\longrightarrow}&
    \mathbf{B}\mathcal{G}
  }
  \;\;\;\;\;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;\;\;\;\;
  \array{
    \esh \mathcal{G}
    &\longrightarrow&
    \esh fib(f)
    &\longrightarrow&
    \ast
    \\
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow
    \\
    \ast
    &\twoheadrightarrow&
    B\mathcal{K}
    &\underset{\esh f}{\longrightarrow}&
    B\mathcal{G}
    \mathrlap{\,.}
  }
$$

Now, as shown on the right, the shape modality

1. preserves the left pullback square and its bottom effective epimorphism, by Prop. \ref{CohesiveShapePreservesLoopingOfDeloopings};

1. preserves the outer pullback rectangle by Prop. \ref{ShapePreservesHomotopyFibersOverDiscreteObjects}.

Therefore also the square on the far right is homotopy Cartesian, by the "reverse pasting law" ([this Prop.](pasting+law+for+pullbacks#ReversePastingLawForInfinityGroupoids)). But this is equivalently the claim to be shown.
\end{proof}

\linebreak

#### More specific cases
 {#MoreSpecificCases}

The following discussion is about the topological/smooth shape modality commuting with [[looping]] ([[homotopy fiber products]] of points) of [[mapping stacks]] between [[delooping]] stacks of certain [[topological groups]]. This is a technical point of crucial importance in the construction and discussion of [[equivariant classifying spaces]] for [[equivariant principal bundles]] -- though not traditionally presented as being about behaviour under a shape modality. The following perspective is from [[schreiber:Equivariant principal infinity-bundles|SS21]], though the hard results themselves are either classical (essentially [Conner & Floyd 1964, Lemma 31.8](nearby+homomorphisms+from+compact+Lie+groups+are+conjugate#ConnerFloyd64) for the case of compact $\Gamma$) or, in more generality, due to [Uribe & Lück 2014](equivariant+bundle#UribeLück14). (Beware that the following differs from their notation by the exchange "$G$" $\leftrightarrow$ "$\Gamma$"! For us here, $G$ denotes the [[equivariance group]] and $\Gamma$ the [[structure group]].)

\linebreak

\begin{defn}\label{ConditionHOnFamiliesOfGroupHomomorphisms}
**(Condition H on families of topological group homomorphisms)**
\linebreak
  Let $G, \Gamma \,\in\, Grp(kTop)$ be [[topological groups]] and let
$$
  \mathcal{R}
    \,\subset\,
  \underset{
    H \subset G
  }{\coprod}
  Hom(H,\Gamma)
$$
be a [[set]] of [[pairs]] $(H,\rho)$ consisting of a topological [[subgroup]] $H$ of $G$ and a [[continuous map|continuous]] [[homomorphism]] $H \to \Gamma$ such that
([Uribe & Lück 2014, Def. 3.4](equivariant+bundle#UribeLück14)) $\mathcal{R}$ is closed under:

1. [[fiber products]], i.e. under products in $Grp(kTop)_{/\Gamma}$,

1. [[conjugation]] by $\Gamma$ (under postcomposition),

1. [[conjugation]] by $G$ (under precomposition).

We say (with [Uribe & Lück 2014, Def. 6.1](equivariant+bundle#UribeLück14)) that such data satisfies *Condition H* if for all $(H,
\rho) \,\in\, \mathcal{R}$ we have:

1. the [[path component]] of $\rho$ in $HomMap(H,\Gamma)$ is contained in its [[orbit]] under the $\Gamma$-[[conjugation action]]:

   $$
     HomMap(H,\Gamma)_{\rho}
     \;\;
     \subset
     \;\;
     \Gamma(\rho)
     \;\coloneqq\;
     \big\{
       Ad_\gamma \circ \rho
       \;\big\vert\;
       \gamma \in \Gamma
     \big\}
   $$

1. the [[coset space]]-[[coprojections]]

   1. $G \longrightarrow G/H$

   1. $\Gamma \longrightarrow \Gamma/Stab_\Gamma(\rho)$

      (by the the [[stabilizer subgroup]] $Stab_{\Gamma}(\rho) \subset \Gamma$ under the [[conjugation action]])

   [[coset space coprojection admitting local sections|admit local sections]].

1. the map

   $$
     \array{
       \Gamma/Stab_\Gamma(\rho)
       &\longrightarrow&
       HomMap(H,\Gamma)
       \\
       [\gamma] &\mapsto& Ad_\gamma \circ \rho
     }
   $$

   is a [[homeomorphism]] onto its [[image]].

\end{defn}
\begin{example}
 \label{ExamplesOfCollectionsOfTopologicalGroupHomsSatisfyingConditionH}
  Examples of families of topological group homomorphisms satsifying *Condition H* (Def. \ref{ConditionHOnFamiliesOfGroupHomomorphisms}) include the following:

1. $\Gamma$ and $G$ any [[Lie groups]] with $\Gamma$ [[almost connected topological group|almost connected]]

   with $\mathcal{R}$ consisting of all [[compact topological group|compact]] subsgroups $H \subset G$ and all their homomorphisms $H \to \Gamma$

   (established in [Uribe & Lück 2014, Thm. 6.3](equivariant+bundle#UribeLück14))

1. $\Gamma =$ [[PU(ℋ)]] and $G$ any [[finite group]]

   with $\mathcal{R}$ consisting of all subgroups $H \subset G$ and all their homomorphisms $H \to
PU(\mathcal{H})$

   (established in [Uribe & Lück 2014, Sec. 15](equivariant+bundle#UribeLück14))

\end{example}

For the following formulation of the main result of [Uribe & Lück 2014](equivariant+bundle#UribeLück14) in terms of cohesive homotopy theory we use the [[continuous diffeology]]
$$
  kTop
  \xrightarrow{CDfflg}
  DiffSp
  \hookrightarrow
  SmthGrp_\infty
$$
and the fact that this preserves shapes and shapes of [[mapping spaces]] (see the introduction of [[schreiber:Equivariant principal infinity-bundles|SS21]]).

\begin{proposition}
**(shape commutes with looping of mapping stacks between delooping stacks of certain pairs of topological groups)**
\linebreak
For

* $G$ a [[finite group]]

* $\Gamma$ a [[compact Lie group]] or [[PU(ℋ)]],

we have for all [[homomorphisms]] $\rho \,\colon\, H \to \Gamma$ from [[subgroups]] $H \subset G$ that the shape modality commutes with the [[looping]]-operation on the [[mapping stack]] between the [[deloopings]] of $H$ and $\Gamma$ in [[smooth infinity-groupoids|$SmoothGrpd_{\infty}$]]:
$$
  \esh
  \Omega_\rho
  Map
  \big(
    \mathbf{B}H
    ,\,
    \mathbf{B}\Gamma
  \big)
  \;\;
  \simeq
  \;\;
  \Omega_\rho
  \esh
  Map
  \big(
    \mathbf{B}H
    ,\,
    \mathbf{B}\Gamma
  \big)
  \,.
$$
\end{proposition}
\begin{proof}
First notice (e.g. from the discussion at *[[stabilizer subgroup]]*, [here](stabilizer+group#GeneralAbstract)) that
\[
  \label{ConjugationStabilizerSubgroupAsLoopingOfMappingStack}
  Stab_\Gamma(\rho)
  \;\simeq\;
  \Omega_\rho
  Map
  \big(
    \mathbf{B}H
    ,\,
    \mathbf{B}\Gamma
  \big)
\]

From the discussion at *[[equivariant principal bundles]]* we have, under the given assumptions, that
$$
  B_G \Gamma
  \;\;\;
  \colon
  \;\;\;
  G/H
  \;\mapsto\;
  \esh
  \,
  Map
  \big(
    \mathbf{B}H
    ,\,
    \mathbf{B}\Gamma
  \big)
$$
is the $G$-[[equivariant homotopy type]] of the $G$-[[equivariant classifying space]] of $\Gamma$.

(For $\Gamma$ compact Lie this is the [Murayama-Shimakawa result](equivariant+bundle#UniversalEquivariantPrincipalBundles) as reviewed in [Guillou, May & Merling 2017, Thm. 3.1](equivariant+bundle#GuillouMayMerling17), while for $\Gamma = PU(\mathcal{H})$ this is due to [Bárcenas, Espinoza, Joachim & Uribe 2012](equivariant+bundle#BEJU14), with a re-derivation in a more explicitly cohesive context in [[schreiber:Equivariant principal infinity-bundles|SS21]]).

Moreover, by Exp. \ref{ExamplesOfCollectionsOfTopologicalGroupHomsSatisfyingConditionH} both cases satisfy *Condition H* (Def. \ref{ConditionHOnFamiliesOfGroupHomomorphisms}), whence the main result [Uribe & Lück 2014, Sec. 13](equivariant+bundle#UribeLück14) applies, which, using the above identification of the equivariant classifying space, gives a [[weak homotopy equivalence]] of the form

$$
  \underset{
     { [\rho] \in }
     \atop
     { Hom(H,\Gamma)/\Gamma }
  }{\coprod}
  B Stab_\Gamma(\rho)
  \;\;\;
  \simeq
  \;\;\;
  \esh
  Map
  \big(
    \mathbf{B}H
    ,\,
    \mathbf{B}\Gamma
  \big)
  \,.
$$

(For $\Gamma$ a [[Lie group]] this is the classical fact that [[nearby homomorphisms from compact Lie groups are conjugate]].)

Observing that $B Stab_\Gamma(\rho) \,\simeq\, \mathbf{B} \esh Stab_\Gamma(\rho)$, the looping of this equivalence at the basepoint given by $\rho$ yields the claim.
\end{proof}

\linebreak

### Relative shape and factorization system

Generally, given an [[(∞,1)-topos]] $\mathbf{H}$ (or just a 1-[[topos]]) equipped with an [[idempotent monad]] $\esh \colon \mathbf{H} \to \mathbf{H}$ (a [[modal type theory|(higher) modality]]/[[closure operator]]) which preserves [[(∞,1)-pullbacks]] over objects in its [[essential image]], one may call a [[morphism]] $f \colon X \to Y$ in $\mathbf{H}$ _$\esh$-closed_ if the [[unit of an adjunction|unit]]-diagram

$$
  \array{
    X
    &\overset{\eta_X}{\longrightarrow}& \esh(X)
    \\
    \big\downarrow {\mathrlap{{}^f}}
    &&
    \big\downarrow{\mathrlap{{}^{\esh(f)}}}
    \\
    Y
    &
    \overset{\eta_Y}{\longrightarrow}
    &
    \esh(Y)
  }
$$

is an [[(∞,1)-pullback]] diagram. These $\esh$-closed morphisms form the right half of an [[orthogonal factorization system]], the left half being the morphisms that are sent to [[equivalence in an (∞,1)-category|equivalences]] in $\mathbf{H}$.

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

Let $(\Pi\dashv \Disc\dashv \Gamma):H\to\infty\Grpd$ be an [[infinity-connected (infinity,1)-topos]], let $\esh:=\Disc \Pi$ be the geometric path functor / geometric homotopy functor, let $f:X\to Y$ be a $H$-morphism, let $c_{\esh} f$ denote the ∞-pullback

$$
  \array{
    c_{\esh} f
    &\longrightarrow&
    {\esh} X
    \\
    \big\downarrow
    &&
    \big\downarrow^{\mathrlap{{\esh}_f}}
    \\
    Y
    &\xrightarrow{1_{(\Pi\dashv \Disc)}}
    &
    {\esh}Y
  }
$$

$c_{\esh} f$ is called $\esh$**-closure of** $f$.

$f$ is called $\esh$**-closed** if $X\simeq c_{\esh}f$.
=--

If a morphism $f:X\to Y$ factors into $f=g\circ h$ and $h$ is a $\esh$-equivalence then $g$ is $\esh$-closed; this is seen by using that $\esh$ is idempotent.

$\Pi$-closed morphisms are a right class of an [[orthogonal factorization system]] ([[orthogonal factorization system in an (∞,1)-category|in an (∞,1)-category]]) and hence, as discussed there, are closed under [[limits]], [[composition]], [[retracts]] and satisfy the left cancellation property.

\begin{remark}
**(shape-modal maps as open maps)**
\linebreak
A consequence of the previous property is that the class of $\esh$-closed morphisms gives rise to an admissible structure in the sense of [[structured (infinity,1)-topos|structured spaces]] on an (∞,1)-connected (∞,1)-topos, hence they serve as a class of a kind of *[[open maps]]*.
\end{remark}

### Shape via cohesive path ∞-groupoid

See at _[[shape via cohesive path ∞-groupoid]]_.

## Examples

### Internal locally constant $\infty$-stacks
 {#LocallyConstantStacks}

In a [[cohesive (∞,1)-topos]] $\mathbf{H}$ with an [[∞-cohesive site]] of definition, the [[fundamental ∞-groupoid]]-functor $\esh$ satisfies the above assumptions (this is the example gives this entry its name). The $\esh$-closed morphisms into some $X \in \mathbf{H}$ are canonically identified with the [[locally constant ∞-stacks]] over $X$. The correspondence is effectively what is called [[categorical Galois theory]].

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

Let $H$ be a [[cohesive (∞,1)-topos]] possessing a [[∞-cohesive site]] of definition.
Then for $X\in H$ the locally constant ∞-stacks $E\in \L\Const(X)$, regarded as ∞-bundle morphisms $p:E\to X$ are precisely the $\esh$-closed morphisms into $X$

=--

### Formally étale morphisms
 {#EtaleMorphisms}

If a [[differential cohesive (∞,1)-topos]] $\mathbf{H}_{th}$, the [[de Rham space]] functor $\Im$ satisfies the above assumptions. The $\Im$-closed morphisms are precisely the [[formally étale morphisms]].

## Related entries

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

* [[shape via cohesive path ∞-groupoid]]

* [[reflective subuniverse]]

[[!include cohesion - table]]

* [differential cohesion - cotangent bundles](http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#CotangentBundles)

## References

### General

Original discussion:

* [[Urs Schreiber]], [[Zoran Škoda]]: [§7.4.2](https://arxiv.org/pdf/1004.2472.pdf#page=31) of: *Categorified symmetries* [[arXiv:1004.2472](https://arxiv.org/abs/1004.2472)]

* [[Urs Schreiber]], Sec. 3.4.5 in: _[[schreiber:differential cohomology in a cohesive topos]]_ ([arXiv:1310.7930](https://arxiv.org/abs/1310.7930))

* {#Shulman15} [[Mike Shulman]], Sec. 9.7 in: _Brouwer's fixed-point theorem in real-cohesive homotopy type theory_, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

### In Smooth∞Grpd

For the case of [[Smooth∞Grpd]]:

On [[shape via cohesive path ∞-groupoids]]:

* {#Pavlov14} [[Dmitri Pavlov]], _Structured Brown representability via concordance_, 2014 ([pdf](https://dmitripavlov.org/concordance.pdf), [[Pavlov_Concordance.pdf:file]])

* {#BEBP19} [[Daniel Berwick-Evans]], [[Pedro Boavida de Brito]], [[Dmitri Pavlov]], _Classifying spaces of infinity-sheaves_ ([arXiv:1912.10544](https://arxiv.org/abs/1912.10544))

Discussion for [[orbifolds]], [[étale groupoids]] and, generally, [[étale ∞-groupoids]]:

* {#Carchedi15} [[David Carchedi]], _On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces_ ([arXiv:1504.02394](http://arxiv.org/abs/1504.02394))

[[!redirects Pi-closed morphisms]]

[[!redirects cohesive shape]]
[[!redirects cohesive shapes]]

[[!redirects Pi modality]]
[[!redirects Pi-closed morphism]]