structure identity principle

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## Idea
 {#Idea}

By a *stucture identity principle* one means a statement of the form that all "identifications" of [[mathematical structures]] of a given [[type]] are necessarily *structure respecting* identifications (see also *[[structuralism]]*).

The source of the exact term "structure identity principle" seems to be [Aczel (2011, slide 7)](#Aczel11), but the idea will have occurred earlier (see also the *[[principle of equivalence]]*).

The literature on the subject arguably suffers from absence of global conventions on how to name different notions of "identification" (*identity*, *equality*, *isomorphism*, *equivalence*, ...), which can be confusing when speaking about a principle that is all about the fine-print of identifying two nominally different notions of identifications... But the general idea is always the same: Faced with notions of "direct"- and of "structure preserving"-identifications, a structure-identity-principle asserts that they are suitably identical.

Concretely,  [Aczel (2011, p. 9)](#Aczel11) declares that the structure identity principle (SIP) in [[homotopy type theory]] *is* the [[univalence axiom]]: This of course identifies [[terms]] $p \,\colon\, Id_{Type}(A,B)$ of [[identification type]] between [[types]] $A,B \,\colon\, Type$ with [[functions]] that are  [[equivalence in type theory|type-theoretic equivalences]] $f \,\colon\,  A \stackrel{\sim}{\longrightarrow} B$.

The latter is what [Aczel (2011, p. 22)](#Aczel11) calls "[[isomorphisms]]". While this matches the usage of the term "[[isomorphism]]" in abstract [[category theory]] to mean "[[invertible morphism]]", to complete the notion of "structure identity" one may want a further argument to verify that such isomorphisms really are invertible [[homomorphisms|*homo*-morphism]] with respect to some given [[mathematical structure]] (such as [group structure](structure#GroupDataStructure) etc.). Such enhancement of Aczel's notion of SIP is considered in [Coquand & Danielsson (2013)](#CoquandDanielsson13) (who do not use the terminology "structure udentity principle") and in [UFP (2013, §9.8)](#UFP13), [Escardó (2019, §3.33.1)](#Escardó19) (who do).

## Examples
 {#Examples}

To make this more concrete and more manifest, we may notice that:

1. at least a large class of kinds of [[mathematical structures]] on a base type $B \,\colon\, Type$ are expressible as nothing but "[[telescope type|telescopes]]" of [[dependent pairs]] of $B$ with an iteration of further [[dependent pair]]- and [[dependent function]]- and [[identification types]] (for examples, such as [[group object|group structure]], see [here](structure#InDependentTypeTheory));

1. the [[univalence axiom]] implies extensionality principles for all three of these [[type formations]] (so with regards to [UFP13](#UFP13) this is now using §2, instead), namely [[dependent function|dependent]] [[function extensionality]] ([here](function+extensionality#InTypeTheory)) and its analog for dependent pairs ([here](dependent+sum+type#ExtensionalityPrinciple)):

\[
  \label{ExtensionalityPrinciples}
\]

Together this means exactly that

* identifications of two structured types which both are iterated dependent function-, dependent pair- and identity-types of the same form

* are equivalent to giving the corresponding dependent function- and dependent pairing of componenwise identifications...

  ...and this is just what one wants to mean by "preserving" this structure as in the notion of *[[homomorphisms]]*.

\linebreak

For example, with the type of [[group object|group structures]] defined as

then the above extensionality principles (eq:ExtensionalityPrinciples) imply that the [[type of identifications]] of groups is equivalently that of [[bijections]] of the underlying types and equipped with the [[group homomorphism]]-property:

## Related concepts

* [[principle of equivalence]]

* [[structure]]

* [[structuralism]]

* [[homotopy type theory]]

* The idea of building the structure identity principle right into the [[inference rules]] for identifications in a [[type theory]] leads to the notion of ([[higher observational type theory|higher]]) [[observational type theory]].

## References

* {#Aczel11} [[Peter Aczel]], *On Voevodsky’s Univalence Axiom*, talk at  Third European Set Theory Conference (2011) [[pdf](http://www.cs.man.ac.uk/~petera/Recent-Slides/Edinburgh-2011-slides_pap.pdf), [[Aczel-Univalence.pdf:file]]]

* {#CoquandDanielsson13} [[Thierry Coquand]], [[Nils Anders Danielsson]], *Isomorphism is equality*, Indagationes Mathematicae **24** 4 (2013) 1105-1120 [[doi:10.1016/j.indag.2013.09.002](https://doi.org/10.1016/j.indag.2013.09.002)]

* {#UFP13} [[Univalent Foundations Project]], §9.8 but also §2 of: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

* {#Escardó19} [[Martín Escardó]], *[A structure identity principle for a standard notion of structure](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#sns)*, §3.33.1 in: *Introduction to Univalent Foundations of Mathematics with Agda* [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580),  [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html)]

  > (formalized in the [[Agda]] [[proof assistant]])

* [[Benedikt Ahrens]], [[Paige Randall North]], [[Michael Shulman]], [[Dimitris Tsementzis]], *A Higher Structure Identity Principle*, LICS '20 (2020) 53–66 [[arXiv:2004.06572](https://arxiv.org/abs/2004.06572), [doi:10.1145/3373718.3394755](https://doi.org/10.1145/3373718.3394755)]

* [[Benedikt Ahrens]], [[Paige Randall North]], *Univalent foundations and the equivalence principle*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019)  [[arXiv:2202.01892](https://arxiv.org/abs/2202.01892), [doi:10.1007/978-3-030-15655-8](https://doi.org/10.1007/978-3-030-15655-8)]

Discussion in the context of the [[philosophy]] of [[structuralism]]:

* [[Dimitris Tsementzis]], §5.1 in: *Univalent foundations as structuralist foundations*, Synthese **194** 9 (2017) 3583–3617 [[jstor:26748765](https://www.jstor.org/stable/26748765), [doi:10.1007/s11229-016-1109-x](https://doi.org/10.1007/s11229-016-1109-x), [pdf](https://core.ac.uk/download/pdf/157866891.pdf)]


structure in model theory

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#### Model theory
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## Idea

A _[[structure]]_ in [[mathematics]] (also "mathematical structure") is often taken to be a [[set]] equipped with some choice of elements, with some operations and some relations. Such as for instance the "structure of a [[group]]". In [[model theory]] this concept of mathematical structure is formalized by way of [[formal logic]].

Notice however that by far not every concept studied in [[mathematics]] fits as an example of a mathematical structure in the sense of classical [[first-order theory|first order]] model theory, described [below](#Definition). For instance a concept as basic as that of  [[topological spaces]] fails to be a structure in the sense of classical model theory (see [here](#MathSEDisc)).

In [[category theory]] there is a more flexible concept of _[[structure]]_, see there.

## Definition
 {#Definition}

Given a [[first-order language]] $L$, which consists of symbols ([[variable|variable symbols]], constant symbols, [[function symbols]] and [[relation]] symbols including $\epsilon$) and [[quantifiers]]; a __structure__ for $L$, or "$L$-structure", is a [[set]] $M$ with an __interpretation__ for symbols:

* if $R\in L$ is an $n$-ary [[relation]] symbol, then its interpretation $R^M\subset M^n$

* if $f\in L$ is an $n$-ary [[function]] symbol, then $f^M:M^n\to M$ is a function

* if $c\in L$ is a constant symbol, then $c^M\in M$

The underlying set $M$ of the structure is referred to as (universal) __domain__ of the structure (or the universe of the structure).

Interpretation for an $L$-structure inductively defines an interpretation for well-formed formulas in $L$. We say that a sentence $\phi\in L$ is [[true]] in $M$ if $\phi^M$ is true. Given a [[theory]] $(L,T)$, which is a language $L$  together with a given set $T$ of sentences in $L$ ([[axioms]]), the interpretation in a structure $M$ makes those sentences true or false; if all the sentences in $T$ are true in $M$ we say that $M$ is a [[model]] of $(L,T)$.

In [[model theory]], given a language $L$, a structure for $L$ is the same as a [[model]] of $L$ as a [[theory]] with an empty set of axioms. Conversely, a _model_ of a theory is a _structure_ of its underlying language that satisfies the axioms demanded by that theory.

There is a generalization of structure for languages/theories with multiple domains or sorts, called multi-sorted languages/theories.

## Properties

### Elementary classes of structures

A class $K$ of structures of a given signature is an __elementary class__ if there is a [[first-order theory]] $T$ such that $K$ consists precisely of all models of $T$.

There is a vast generalizations for higher-order theories (and more), see at  _[[abstract elementary class]]_ and _[[metric abstract elementary class]]_.

### Categories of structures

Every [[algebraic category]] whose [[forgetful functor]] preserves [[filtered colimits]] is the category of [[models]] for some [[first-order theory]]. The converse is false.

A detailed discussion of characterizations of [[categories]] of structures in the sense of  model theory is in ([Beke-Rosciky 11](#BekeRosciky11)).

### Interpretation in categorical logic
 {#CategoricalInterpretation}

Every [[first-order language]] $L$ gives rise to a [[first-order hyperdoctrine]] with [[equality]] freely generated from $L$. Denoting this by $T(L)$, the base category $C_{T(L)}$ consists of sorts (which are products of basic sorts) and functional terms between sorts; the [[predicates]] are [[equivalence classes]] of [[relations]] definable in the language. The construction of $T(L)$ depends to some extent on the [[logic]] we wish to impose; for example, we could take the free [[Boolean hyperdoctrine]] generated from $L$ if we work in [[classical logic]].

There is also a "tautological" first order hyperdoctrine whose base category is $Set$, and whose predicates are given by the power set functor

$$P \colon Set^{op} \to Bool$$

and then an interpretation of $L$, as described above, amounts to a morphism of hyperdoctrines $T(L) \to Taut(Set)$.

This observation opens the door to a widened interpretation of "interpretation" in [[categorical logic]], where we might for instance generalize [[Set]] to any other [[topos]] $E$, and use instead $Sub \colon E^{op} \to Heyt$ (taking an object of $E$ to its [[Heyting algebra]] of [[subobjects]]) as the receiver of interpretations. This of course is just one of many possibilities.

## Related concepts

* [[model theory]]

* [[first-order theory]]

* [[interpretation]]

* [[diagram of a first-order structure]]

* [[exceptional structure]]

* [[amalgamation property]]

* [[structure]], [[higher structure]]

## References

Standard textbook accounts include

* [[Wilfrid Hodges]], section 1 of _A shorter model theory_, Cambridge University Press (1997)

* Chen Chung Chang, [[H. Jerome Keisler]], _Model Theory. Studies in Logic and the Foundations of Mathematics_. 1973, 1990, Elsevier.

Characterizations of [[categories]] of model-theoretic structures  and [[homomorphisms]] between them (certain accessible categories) is discussed in

* {#BekeRosciky11} [[Tibor Beke]], [[Jiří Rosický]], _Abstract elementary classes and accessible categories_, 2011   ([pdf](http://www.math.muni.cz/~rosicky/papers/elem7.pdf))

Online discussion includes

* {#MathSEDisc} Math.SE _[Topological spaces as model-theoretic structures — definitions?](http://math.stackexchange.com/questions/97856/topological-spaces-as-model-theoretic-structures-definitions)_

category: model theory
[[!redirects structures in model theory]]
[[!redirects first-order structure]]
[[!redirects structure (in model theory)]]
[[!redirects structures (in model theory)]]
[[!redirects first-order structures]]


structure on algebras and their module categories - table

**[[Tannaka duality]] for [[categories of modules]] over [[monoids]]/[[associative algebras]]**

| [[monoid]]/[[associative algebra]]  | [[category of modules]]  |
|--|--|
| $A$ | $Mod_A$ |
| $R$-[[associative algebra|algebra]] | $Mod_R$-[[2-module]] |
| [[sesquialgebra]] | [[2-ring]] = [[monoidal category|monoidal]]  [[presentable category]] with [[colimit]]-preserving [[tensor product]] |
| [[bialgebra]] | strict [[2-ring]]: [[monoidal category]] with [[fiber functor]] |
| [[Hopf algebra]] | [[rigid monoidal category]] with [[fiber functor]] |
| [[hopfish algebra]] (correct version) | [[rigid monoidal category]] (without fiber functor) |
| [[weak Hopf algebra]] | [[fusion category]] with generalized [[fiber functor]] |
| [[quasitriangular bialgebra]] | [[braided monoidal category]] with [[fiber functor]] |
| [[triangular bialgebra]] | [[symmetric monoidal category]] with [[fiber functor]] |
| [[quasitriangular Hopf algebra]] ([[quantum group]]) | [[rigid monoidal category|rigid]] [[braided monoidal category]] with [[fiber functor]] |
| [[triangular Hopf algebra]] | [[rigid monoidal category|rigid]] [[symmetric monoidal category]] with [[fiber functor]] |
|  [[superalgebra|supercommutative]] [[Hopf algebra]] ([[supergroup]]) | [[rigid monoidal category|rigid]] [[symmetric monoidal category]] with [[fiber functor]] and Schur smallness  |
| form [[Drinfeld double]] | form [[Drinfeld center]] |
| [[trialgebra]] | [[Hopf monoidal category]] |

**2-Tannaka duality for [[module categories]] over [[monoidal categories]]**

| [[monoidal category]] | [[2-category of module categories]]  |
|--|--|
| $A$ | $Mod_A$ |
| $R$-[[2-algebra]] | $Mod_R$-[[3-module]] |
|  [[Hopf monoidal category]] | [[monoidal 2-category]] (with some duality and strictness structure) |

**3-Tannaka duality for [[module 2-categories]] over [[monoidal 2-categories]]**

| [[monoidal 2-category]] | [[3-category of module 2-categories]]  |
|--|--|
| $A$ | $Mod_A$ |
| $R$-[[3-algebra]] | $Mod_R$-[[4-module]] |


structure sheaf

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#### Higher geometry
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## Idea

For a [[ringed topos]] $(\mathcal{X}, \mathcal{O})$ the ring object $\mathcal{O} \in \mathcal{X}$ is called the **structure sheaf**.

More generally, for $\mathcal{G}$ a [[geometry (for structured (∞,1)-toposes)]], a [[structured (∞,1)-topos]]

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

is an [[(∞,1)-topos]] equipped with a $\mathcal{G}$-valued **structure sheaf** presented by the finite-limits-preserving and cover-preserving [[(∞,1)-functor]] $\mathcal{O}$.

## Related concepts

* [[sheaf of meromorphic functions]]

* [[sheaf of rational functions]]

## References

* [[James Milne]], section 6 of _[[Lectures on Étale Cohomology]]_

category: sheaf theory, algebraic geometry

[[!redirects structure sheaves]]
[[!redirects structure (∞,1)-sheaf]]
[[!redirects structure (infinity,1)-sheaf]]
[[!redirects structure (∞,1)-sheaves]]
[[!redirects structure (infinity,1)-sheaves]]


structure theorem for finitely generated modules over a principal ideal domain

## Idea

Structure theorem for [[finitely generated object|finitely generated]] [[modules]] over a [[principal ideal domain]]: see [here](/nlab/show/principal+ideal+domain#StructureTheoryOfModules) for a partial account.

special case: [[fundamental theorem of finitely generated abelian groups]]

## References

* Wikipedia, _[Structure theorem for finitely generated modules over a principal ideal domain](https://en.wikipedia.org/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain)_


structure type

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## Idea

A 'structure type' is a type of [[extra structure]] that can placed on [[finite set]]s, e.g. a coloring, or ordering.

## Definition

We define a __structure type__ to be a [[faithful functor]]

$$p: X \to core(FinSet)$$

from some groupoid $X$ to [[core]]([[FinSet]]), which is the [[groupoid]] of [[finite set]]s and [[bijection]]s.  Equivalently, we can think of it as a [[presheaf]] of sets on the groupoid of finite sets and bijections, or in other words a functor

$$F: core(FinSet)^{op} \to Set $$

These two points of view are related by the [[Grothendieck construction]]:

$$F(n) = p^{-1}(n) $$

But since a groupoid is equivalent to its opposite, we can also think of a structure type as a functor

$$core(FinSet) \to Set $$

In this guise, a structure type is more commonly called a __(combinatorial) species of structure__, or [[species]] for short.

A structure type is a special case of a stuff type, so see [[stuff type]] for more information.

## References

* [[John Baez]], ...

[[!redirects structure type]]
[[!redirects structure types]]


structured (infinity,1)-topos

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## Idea

The notion of _structured $(\infty,1)$-topos_ is a generalization of the notion of a [[locally ringed space]] and locally [[ringed topos]] generalized to [[(∞,1)-topos]]es. This is a way to formalize [[higher geometry]]/[[derived geometry]]-structure on [[little topos|little]] [[(∞,1)-topos]]s.

So a structured $(\infty,1)$-topos is an [[(∞,1)-topos]] $\mathcal{X}$ equipped with an [[(∞,1)-sheaf|(∞,1)-]][[structure sheaf]] $\mathcal{O}$ that we think of as the collection of _functions_ on $\mathcal{X}$ that preserve extra geometric structure -- for instance [[topology|continuous structure]] or [[smooth structure]].

Being an $\infty$-function $\infty$-algebra, $\mathcal{O}(\mathcal{U})$ is an algebra over an [[(∞,1)-algebraic theory]] $\mathcal{G}$, called the (pre)[[geometry (for structured (∞,1)-toposes)]], since this encodes the nature of the extra geometric structure on $\mathcal{X}$.

Formally therefore a geometric structure $(\infty,1)$-sheaf of $\mathcal{X}$ is a product/limit-preserving [[(∞,1)-functor]]

$$
  \mathcal{O} : \mathcal{G} \to \mathcal{X}
  \,.
$$

Here we think of $\mathcal{X} = Sh_{(\infty,1)}(C)$ as being the [[(∞,1)-sheaf (∞,1)-topos]] on some [[(∞,1)-site]] $C$ and for any $V \in \mathcal{G}$ we think of

$$
  \mathcal{O}_V \in Sh_{(\infty,1)}(C)
$$

as being the [[(∞,1)-sheaf]] of structure-preserving functions on $C$ with values in $V$.

## Definition

Let $S$ be a [[geometry (for structured (∞,1)-toposes)|geometry]] and let $Sh(S)$ be the [[(∞,1)-topos]] of [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaves]] on $S$.

Notice that if $S = Op(X)$ is the nerve of the [[category of open subsets]] of some [[topological space]] $X$, then $Sh(X) \coloneqq Sh(S)$ is the [[(∞,1)-category of (∞,1)-sheaves]] on $X$, as in the above motivating introduction.

We want to define a _structure sheaf_ on $S$ (for instance on $Op(X)$) of quantities modeled on some $(\infty,1)$-category $V$ to be an [[(∞,1)-functor]]

$$
  O_X : V \to Sh(S)
$$

to be thought of as the assignment to each _value space_ $v \in V$ of an $(\infty,1)$-sheaf  of $v$-valued functions on $S$ (on $X$ if $S = Op(X)$).

But since we are taking care of the sheaf condition on $S$, we also want to allow a similar kind of co-sheaf condition on $V$. In order to do so, $V$ is taken to be equipped with extra structure encoding covers in $V$, and $O_X$ is then required to respect this structure suitably.

### Geometries and admissibility structure

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

An **admissiblility structure** on an $(\infty,1)$-category $V$ is

* a choice of [[sub-quasi-category|sub (∞,1)-category]] $V^{ad} \hookrightarrow V$, whose morphisms are to be called the **admissible morphisms**, such that

  * for every admissible morphism $U \to X$ and any morphism $X' \to X$ there is a diagram

    $$
      \array{
        U' &\to& U
        \\
        \downarrow && \downarrow
        \\
        X' &\to& X
      }
    $$

    with $U' \to X'$ admissible;

  * for every diagram of the form

    $$
      \array{
          && Y
          \\
          & \nearrow && \searrow
         \\
         X &&\to&& Z
      }
    $$

    with $X \to Z$ and $Y \to Z$ admissible, also $X \to Y$ is admissible.

* a [[Grothendieck topology]] on $V$ which has the property that it is generated from a [[coverage]] consisting of admissible morphisms.

=--

This is [[Structured Spaces|StrSh, def 1.2.1]] in view of remark 1.2.4 below that.

+-- {: .num_defn}
###### Definition ([StrSh, def 1.2.5](http://arxiv.org/abs/0905.0459))

An $(\infty,1)$-category $V$ equipped with an admissiblility structure is a **geometry** if it is essentially small, admits finite limits and is [[idempotent complete (infinity,1)-category|idempotent complete]].
=--

The admissible morphisms in an admissibility structure are roughly to be thought of as those morphisms that behave as _open immersions_ ,

### Structure sheaves: local $\infty$-algebras

+-- {: .num_defn}
###### Definition ([StrSh, def 1.2.8](http://arxiv.org/abs/0905.0459))
**(structure sheaf)**

Let $\mathcal{G}$ be a geometry and $\mathcal{X}$ an $(\infty,1)$-topos
An [[(∞,1)-functor]]

$$
  O_{\mathcal{X}} : \mathcal{G} \to \mathcal{X}
$$

is a $\mathcal{G}$-**structure** on $\mathcal{X}$ or
$\mathcal{G}$-**[[structure sheaf]]** on $\mathcal{X}$ if

* it is a left [[exact (∞,1)-functor]];

* it respects gluing in $\mathcal{G}$ in that
  for $\{U_i \to V\}_i$
  a covering [[sieve]] consisting of admissible morphism,
  the induced morphism

  $$
    \coprod_i O_X(U_i) \to O_X(V)
  $$

  is an [[effective epimorphism]] in $\mathcal{X}$.

=--

Write $Str_{\mathcal{G}}(\mathcal{X}) \subset Func(\mathcal{G},\mathcal{X})$ for the full subcategory of such morphisms of the [[(∞,1)-category of (∞,1)-functors]].

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

Without the condition on preservations of covers, the above defined the [[∞-algebra over an (∞,1)-algebraic theory|∞-algebras]] over the [[essentially algebraic (∞,1)-theory]] $\mathcal{G}$. The preservation of covers encodes the **local** $\mathcal{G}$-algebras.

Therefore we shall equivalently write

$$
  \mathcal{G}Alg_{loc}(\mathcal{X}) \simeq Str_{\mathcal{G}}(\mathcal{X})
  \,.
$$

=--

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

$\mathcal{G}Alg_{loc}(\mathcal{X})$ is the $(\infty,1)$-category of algebras over an $(\infty,1)$-[[geometric theory]].

=--

This is discussed [below](#InTermsOfClassifying).

### As algebras over geometric $(\infty,1)$-theories
{#InTermsOfClassifying}

By the [[(∞,1)-Yoneda lemma]], a
cover-preserving functor $\mathcal{O} : \mathcal{G} \to \mathcal{X}$
[[Yoneda extension|Yoneda extends]] equivalently to a [[(∞,1)-colimit]]-preserving [[(∞,1)-functor]]

$$
  \mathcal{O} : Sh_{(\infty,1)}(\mathcal{G}) \to \mathcal{X}
  \,.
$$

By the [[adjoint (∞,1)-functor theorem]] this has a [[right adjoint|right]] [[adjoint (∞,1)-functor]] and if
$\mathcal{O}$ preserves
finite [[(∞,1)-limits]] then so does its extension. Therefore local $\mathcal{G}$-[[∞-algebra over an (∞,1)-algebraic theory|∞-algebras]] in $\mathcal{X}$ are equivalent to [[(∞,1)-geometric morphism]]s

$$
  \mathcal{X} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\to}
  Sh_{(\infty,1)}(\mathcal{G})
  \,.
$$

This means that structure sheaves
$\mathcal{O} : \mathcal{G} \to \mathcal{X}$ are equivalently encoded
in [[geometric morphism]]s to the [[(∞,1)-category of (∞,1)-sheaves]] on the geometry.

Formally we have:

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

For $\mathcal{G}$ a geometry, precomposition of the [[inverse image]] functor with the [[(∞,1)-Yoneda embedding]] $y : \mathcal{G} \to Sh_{(\infty,1)}(\mathcal{G})$ induces an [[equivalence of (∞,1)-categories]]

$$
  Topos(\mathcal{X}, Sh_{(\infty,1)}(\mathcal{G}))
  \stackrel{\simeq}{\to}
  \mathcal{G}Alg_{loc}(\mathcal{X})
$$

between the  [[(∞,1)-category of (∞,1)-functors|(∞,1)-category of (∞,1)-geometric morphisms]] from $\mathcal{X}$ to $Sh_{(\infty,1)}(\mathcal{G})$ and the $(\infty,1)$-category of local $\mathcal{G}$-[[∞-algebra over an (∞,1)-algebraic theory|∞-algebra]]s in $\mathcal{X}$.

=--

This is ([StrSp, prop 1.42](#Lurie)).

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

This follows from the general fact, discussed in the section Local yoneda embedding at [[(∞,1)-Yoneda lemma]] that the essential image of the $(\infty,1)$-functor

$$
  Topos(\mathcal{X}, Sh_{(\infty,1)}(\mathcal{G}))
  \stackrel{L}{\to}
  Topos(\mathcal{X}, PSh_{(\infty,1)}(\mathcal{G}))
  \stackrel{y}{\to}
  Func(\mathcal{G}, \mathcal{X})
$$

is spanned by the left exact and cover preserving functors.

=--

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

We may think of this as saying that $Sh_{(\infty,1)}(\mathcal{G})$ is the $(\infty,1)$-[[classifying topos]] for the $(\infty,1)$-[[geometric theory]] of **local [[∞-algebra over an (∞,1)-algebraic theory|∞-algebras]]** over the [[essentially algebraic (∞,1)-theory]] $\mathcal{G}$.

=--

### Local morphisms between structure sheaves {#LocalMorphisms}

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

The $(\infty,1)$-category $Str_{\mathcal{G}}(\mathcal{X})$
of $\mathcal{G}$-structure sheaves on an $(\infty,1)$-topos $\mathcal{X}$
does _not_ depend on the admissibility structure of $\mathcal{G}$, but only on the [[Grothendieck topology]] induced by it.

=--

(See [[Structured Spaces|StrSp, remark below prop. 1.4.2]]).

The admissibility structure does serve to allow the following definition of _local_ morphisms of structure sheaves.

#### In terms of admissibility structures

+-- {: .num_defn}
###### Definition
**(local morphism of structure sheaves)**

A [[natural transformation]] $\eta : \mathcal{O} \to \mathcal{O}' : \mathcal{G} \to \mathcal{X}$ of structure sheaves is **local** if for every admissible morphism $U \to X$ in $\mathcal{G}$ the naturality diagram

$$
  \array{
    \mathcal{O}(U) &\stackrel{\eta(U)}{\to}& \mathcal{O}'(U)
    \\
    \downarrow && \downarrow
    \\
    \mathcal{O}(X) &\stackrel{\eta(X)}{\to}& \mathcal{O}'(X)
  }
$$

is a [[limit in a quasi-category|pullback square]] in $\mathcal{X}$.

Write

$$
  Str^{loc}_{\mathcal{G}}(\mathcal{X}) \subset Str_{\mathcal{G}}(\mathcal{X})
$$

for the [[sub-quasi-category|sub-(∞,1)-category]] of $\mathcal{G}$-structures on $\mathcal{X}$ spanned by local transformations between them.

=--

#### In terms of classifying $(\infty,1)$-toposes
 {#InTermsOfClassifyingToposes}

Alternatively, the local transformations can be characterized as follows

it turns out the local transformations are the right half of a [[factorization system]] on $Str_{\mathcal{G}}(\mathcal{X})$, and that this factorization system depends functorially on $\mathcal{X}$, in that for every geometric morphism $\mathcal{X} \to \mathcal{Y}$ the induced $Str_{\mathcal{G}}(\mathcal{X}) \to Str_{\mathcal{G}}(\mathcal{Y})$ respects these factorization  systems. (theorem 1.3.1)

This one can turn around, to characterize local transformations (and hence admissibility structures on $\mathcal{G}$) in terms of functorial factorization systems on classifying $(\infty,1)$-toposes (def. 1.4.3):

For $\mathcal{K}$ an $(\infty,1)$-topos, declare that a _geometric structure_ on $\mathcal{K}$ is a choice of factorization systems on $Topos_{geom}(\mathcal{X}, \mathcal{K})^{op}$ that is functorial in $\mathcal{X}$ . Given such we have another way of saying "local transformation": this is the non-full subcategory $Str^{loc}_{\mathcal{K}}(\mathcal{X})$ of $Topos_{geom}(\mathcal{X}, \mathcal{K})^{op}$ on all objects and on the right part of the factorization system.

And this is indeed the same kind of datum as an admissibility structure on a geometry (example 1.4.4): in the case that $\mathcal{K} = Sh(\mathcal{G})$ is the classifying topos for the geometry $\mathcal{G}$, the defining equivalence $Topos_{geom}(\mathcal{X}, Sh(\mathcal{G}))^{op} \stackrel{\simeq}{\to} Str_{\mathcal{G}}(\mathcal{X})$ identifies the two sub-categories of local transformations, $Str^{loc}_{\mathcal{G}}(\mathcal{X})$ and $Str^{loc}_{Sh(\mathcal{G})}(\mathcal{X})$.

### The $(\infty,1)$-category of structured $(\infty,1)$-toposes {#CatOfStructuredToposes}

Let $(\infty,1)Toposes \subset $ [[(∞,1)Cat]]  be the [[sub-quasi-category|sub (∞,1)-category]] of [[(∞,1)-topos]]es: objects are [[(∞,1)-topos]]es, morphisms are [[geometric morphism]]s.

Write $LTop \coloneqq (\infty,1)Toposes^{op}$.

+-- {: .num_defn}
###### Definition
**($(\infty,1)$-category of $\mathcal{G}$-structured $(\infty,1)$-toposes)**

For $\mathcal{G}$ a geometry, the $(\infty,1)$-category of
$\mathcal{G}$-structured $(\infty,1)$-toposes

$$
  LTop(\mathcal{G})
$$

is defined as follows.

It is the [[sub-quasi-category|sub (∞,1)-category]]

$$
  LTop(\mathcal{G})
  \subset
  Func(\mathcal{G}, E LTop) \times_{Func(\mathcal{G}, LTop)} LTop
  \,,
$$

where $E LTop \to LTop$ is the [[coCartesian fibration]] associated by the [[(∞,1)-Grothendieck construction]] to the inclusion functor $LTop \hookrightarrow (\infty,1)Cat$, spanned by the following objects and morphisms:

* objects are $\mathcal{G}$-structures $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ on some $(\infty,1)$-topos $\mathcal{X}$:

  an object in
  $Func(\mathcal{G}, E LTop) \times_{Func(\mathcal{G}, LTop)} LTop$ is an object $\mathcal{X} \on LTop$ together with a functor $\mathcal{G} \to E LTop|_{\mathcal{X}}$ into the fiber of $E Top$ over that object; but that fiber is $\mathcal{X}$ itself, so an object in the fiber product is a functor $\mathcal{G} \to \mathcal{X}$ and this is in $LTop(\mathcal{G})$ if it is a $\mathcal{G}$-structure on $\mathcal{X}$;

* morphisms $\alpha : \mathcal{O} \to \mathcal{O}'$
  are _local_ morphisms of $\mathcal{G}$-structures:

  for $f^* : \mathcal{X} \to \mathcal{Y}$ the image of $\alpha$ in $LTop$,
  $\alpha$ is in $LTop(\mathcal{G})$ precisely if for every admissible morphism $U \to X$ in $\mathcal{G}$ the square

  $$
    \array{
     f^* \mathcal{O}(U) &\to& f^*\mathcal{O}(X)
     \\
     \downarrow^{} && \downarrow^{}
     \\
     \mathcal{O}'(U) &\to& \mathcal{O}'(X)
    }
  $$

  is a pullback square in $\mathcal{Y}$.

=--

This is [[Structured Spaces|StrSp, def 1.4.8]]

### The spectrum construction

For $f : \mathcal{G} \to \mathcal{G}'$ a morphism of geometries, let

$$
  \mathcal{O}_{\mathcal{G}}^{\mathcal{G}'}
  :
  Topos(\mathcal{G}') \to Topos(\mathcal{G})
$$

be the induced functor on categories of structured toposes.

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

This functor is a left [[adjoint (∞,1)-functor]]

$$
  (
    \mathcal{O}_{\mathcal{G}}^{\mathcal{G}'}
      \dashv
    Spec_{\mathcal{G}^{\mathcal{G}'}})
  )
  :
  Topos(\mathcal{G})
  \stackrel{\overset{\mathcal{O}_{\mathcal{G}}^{\mathcal{G}'}
}{\leftarrow}}{\underset{Spec_{\mathcal{G}}^{\mathcal{G}'}
}{\to}}
  Topos(\mathcal{G}')
  \,.
$$

=--

This is ([Lurie, theorem 2.1.1](#Lurie)).

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

For $\mathcal{G}$ a geometry, let $\mathcal{G}_0$ be the corresponding discrete geometry. We have a canonical morphism $\mathcal{G}_0 \to \mathcal{G}$.

Write

$$
  Spec^{\mathcal{G}}
    :
  Pro(\mathcal{G})
   \to
  Topos(\mathcal{G}_0)
   \stackrel{Spec_{\mathcal{G}_0}^{\math}}{\to}
  Topos(\mathcal{G})
$$

for the composite.

=--

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

This fits into an adjunction

$$
  (\mathcal{O} \dashv Spec) :
  Pro \mathcal{G} \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}}
  Top(\mathcal{G})
  \,.
$$

=--

This is ([Lurie, theorem xyz](#Lurie)).

## Examples

### Classes of examples

#### Canonical structure sheaves on objects in the big topos {#CanonicalStructureSheafOnObjectsInBigTopos}

For $\mathcal{G}$ a [[geometry (for structured (infinity,1)-toposes)|geometry]] let

$$
  \mathbf{H} \coloneqq Sh_\infty(\mathcal{G})
$$

be the [[(∞,1)-category of (∞,1)-sheaves]] on $\mathcal{G}$. This is the [[big topos]] of [[higher geometry]] modeled on $\mathcal{G}$. By the [above discussion](#InTermsOfClassifying) it is also the classifying topos of $\mathcal{G}$-[[structure sheaves]] on toposes:

a $\mathcal{G}$-valued structure sheaf $\mathcal{O}_{\mathcal{X}} : \mathcal{G} \to \mathcal{X}$ on an [[(∞,1)-topos]] $\mathcal{X}$ is equivalently an [[(∞,1)-geometric morphism]]

$$
  (p^* \dashv p_*)
   :
  \mathcal{X} \stackrel{\leftarrow}{\to} \mathbf{H}
  \stackrel{j}{\leftarrow}
  \mathcal{G}
$$

in that $\mathcal{O}_{\mathcal{X}} = p^* j$, where $j$ is the [[(∞,1)-Yoneda embedding]].

Notice that for every object $X \in \mathcal{H}$ its [[little topos]]-incarnation is the [[over-(∞,1)-topos]] $\mathbf{H}/X$. This canonically sits over $\mathbf{H}$ by its [[etale geometric morphism]]

$$
  \mathcal{X} \coloneqq \mathbf{H}/X
  \stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}
  \mathbf{H}
  \,.
$$

So we have

+-- {: .num_lemma}
###### Observation

The [[little topos]] $\mathcal{X} \coloneqq \mathbf{H}/X$ of every object $X$ in the big topos $\mathbf{H}$ over $\mathcal{G}$ is canonically equipped with a $\mathcal{G}$-structure sheaf

$$
  \mathcal{O}_X : \mathcal{G} \stackrel{j}{\to} \mathbf{H}
  \stackrel{X^*}{\to}
  \mathbf{H}/X
  \,.
$$

=--

We want to show that for $V \in \mathcal{G}$ the [[(∞,1)-sheaf]] $\mathcal{O}_X(V)$ may indeed be thought of as the "sheaf of $V$-valued functions on $X$".

Notice that for any $V \in \mathbf{H}$ we have that $X^*(F) = (p_2 : V \times X \to X)$.

Now assume first that $X$ is itself representable. Then by the discussion at [[over-(∞,1)-topos]] we have that $\mathbf{H}/X$ is a lucalization of $PSh_\infty(\mathcal{G})/X \simeq PSh_\infty(\mathcal{G}/X)$, where $\mathcal{G}/X$ is the [[big site]] of $X$. Under this equivalence (more details on this at [[over-topos]]) we have that $(V \times X \to X)$ identifies with the presheaf given by

$$
  (U \to X) \mapsto \mathcal{G}(U,V)
  \,.
$$

This is the "sheaf of $V$-valued functions on $X$".

(...)

### Specific examples

#### Structure sheaves of continuous functions {#StrSheafOfcontFuncts}

  Consider $X$ an ordinary [[topological space]] and
   $Sh(X)$ the ordinary [[category of sheaves]] on its
   [[category of open subsets]]. Let $\mathcal{G} = Top$
   be some [[small category|small]] version of [[Top]]
   with its usual [[Grothendieck topology]] with
   admissible [[covering]]
   families being [[open cover]]s. Consider the functor

   $$
     O_X : Top \to Sh(X)
   $$

   that sends a topological space $V$ to the sheaf of
   [[continuous function]]s with values in $V$:

   $$
     O_X(V) : U \mapsto C(U,V) = Hom_{Top}(U,V)
     \,.
   $$

   By general properties of the [[hom-functor]], this respects [[limit]]s.
   The gluing condition says that
   for $V_1, V_2 \subset V$ an open cover of $V$ by two patches,
   the morphism of sheaves

   $$
     O_X(V_1) \coprod O_X(V_2) \to O_X(V)
   $$

   is an [[epimorphism]] of sheaves. This means that
   for each point $x \in X$ the map of [[stalk]]s

   $$
     O_X(V_1)_x \coprod O_X(V_2)_x \to O_X(V)_x
   $$

   is an epimorphism of sets. But this just says that given
   any function $f : U_x \to V$ on a neighbourhood $U_x$ of $x$,
   there is a smaller neighbourhood $W_x \subset U_x$ such that
   the restriction $f|_{W_x}$ factors either through $V_1$ or
   through $V_2$. This is clearly the case by the fact that $V_1,V_2$
   form an open cover. (A neighbourhood  of $f(x) \in V$ exists which
   is contained in $V_1$ or in $V_2$, so take its preimage under
   $f$ as $U_x$).

#### Locally ringed spaces

[StrSh, remark 2.5.11](http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.0459v1.pdf#page=71)

   Let $X$ be a topological space as before, but consider now the
   geometry $\mathcal{G} = CRing^{op}$ to be the opposite category of
   commutative [[ring]]s, where a covering family of
   $Spec R \in CRing^{op}$ is a family of maps of the form
   $R \to R[\frac{1}{r_i}]$ with $\{r_i \in R\}_i$ generating the unit
   ideal in $R$. So we think of $Spec R[\frac{1}{r_i}]$ as
   an the open subset of $Spec R$ on wich the function $r_i$
   does not vanish, and a covering family we think of as an open
   cover by such open subsets, in direct analogy to the above example.

   Now given a sheaf of rings

   $$
     \bar O_X \in Sh(X,CRing)
   $$

   on $X$ (making $X$ a [[ringed space]]), which we
   may regard as the functor

   $$
     O_X : CRing^{op} \to Sh(X)
   $$

   that it [[representable functor|represents]]

   $$
     O_X(Spec R) : U \mapsto
                   Hom_{CRing^{op}}(\bar O_X(U), Spec R)
                           = Hom_{CRing}(R, \bar O_X(U))
   $$

   we check what
   conditions this has to satisfy to qualify as a
   structure sheaf in the above sense.

   The condition that

   $$
     \coprod_i O_X(Spec R[\frac{1}{r_i}])
     \to Spec R
   $$

   is an epimorphism of sheaves again means that it is [[stalk]]wise
   an epimorphism of sets. Now, a ring homomorphism
   $R[\frac{1}{r_i}] \to \bar O_X(U)$ is given by
   a ring homomorphism $f : R \to O_X(U)$ such that $f(r_i)$ is
   invertible in $O_X(U)$. (We think of this as the pullback of
   functions on $Spec R$ to functions on $U$ by a map $U \to Spec R$
   that lands only in the open subset where the functoin $r_i$ is
   non-vanishing).

   So the condition that the above is an epimorphism on small enough
   $U$ says that for every ring homomorphism $\phi : R \to \bar O_X(U)$
   the value of $\phi$ on at least one of the $r_i$ is invertible
   element in $O_X(U)$.

   Thinking of this dually this is just the same kind of statement as
   in the first example, really. But now we can say this more
   algebraically:

   by assumption there is a linear combination of the $r_i$ to the
   identity in $R$

   $$
     1 = \sum_i \alpha_i r_i
   $$

   in $R$ (the partition of unity of functions on $Spec R$)
   and hence $\sum_i \alpha_i \phi(r_1) = 1$ in $(O_X)_x$
   That for this invertible finite sum at least one of the
   summands is invertible is the condition that $(O_X)_x$ is
   a _local ring_ .

   So a [[ringed space]] has a structure sheaf in the above sense
   if it is a [[locally ringed space]].

#### Ordinary ringed spaces

It may be worthwhile to retell the motivating example in the "Idea" introduction above for the maybe more familiar case of ordinary (1-categorical) structure sheaves with values in (unital) rings.

An ordinary [[topological space]] $X$ with its [[category of open subsets]] $Op(X)$ is a [[ringed space]] or $Op(X)$ is a [[ringed site]] if it is equipped with a [[sheaf]] $O_X : Op(X)^{op} \to Rings$ with values in the category of [[rings]]. For $U \subset X$ one thinks of $O_X(U)$ as the ring of allowed functions on $U$.

If for the moment we ignore the technicality that $O_X$ is supposed to be a [[sheaf]] and just regard it as a [[presheaf]], and if we furthermore invoke the idea of [[space and quantity]] and think of a ring $R$ as a generalized quantity in form of a copresheaf, canonically the [[representable functor|co-representable]] co-presheaf

$$
  R : (Ring^fin)^{op} \to \Set
$$

on finitely generated rings, which sends

$$
 R : R' \mapsto Hom(R,R')
$$

then we find that $O_X$ is in fact a presheaf on $Op(X)$ with values in a co-presheaf on $(Ring^{fin})^{op}$

$$
  O_X : Op(X)^{op} \to [(Ring^{fin})^{op}, Set]
$$

or equivalently a generalized quantity on $(Ring^{fin})^{op}$ with values in presheaves on $X$:

$$
  O_X : (Ring^{fin})^{op} \to [Op(X)^{op}, Set]
  \,.
$$

Since rings can be identified with left-exact functors $(Ring^{fin})^{op}\to Set$, we don't need to impose any admissibility structure in order to recover the notion of a sheaf of rings, since left-exactness is part of the definition of a "structure sheaf."  We do, however, need an admissibility structure if we want to recover the notion of a sheaf of *local* rings, as in the previous example above.

#### Derived ringed spaces

Now formulate the previous example according to the above definition:

Let $CRing^{fin}$ be the category of finitely generated commutative rings There is a standard admissibility structure on $(CRing^{fin})^{op}$ that makes it a _geometry_ in the above sense.

Then for $X$ a topological space an $(\infty,1)$-functor $(CRing^{fin})^{op} \to Sh(S)$ to $(infty,1)$-sheaves on $X$  is a sheaf of local commutative rings on $X$. ([StrSh, example 1.2.13](http://arxiv.org/abs/0905.0459))

To generalize this to **derived structure sheaves** we replace the category of rings here with the $(\infty,1)$-category of simplicial rings.

**Definition**  ([StrSh def 4.1.1](http://arxiv.org/abs/0905.0459))

The $(\infty,1)$-category of **simplicial commutative rings** over an ordinary commutative ring $k$ is

$$
  SCR_k \coloneqq PSh_\Sigma(FreeAlg_k)
$$

the $(\infty,1)$-category of [[(∞,1)-presheaves]] on commutative $k$-algebras of the form $k[x_1, \cdots, x_n]$.

Then...

#### Derived smooth manifolds

([StrSh, example 4.5.2](http://arxiv.org/abs/0905.0459))

Every ordinary [[smooth manifold]] $X$ becomes canonically a generalized space with structure sheaf as follows:

Let $V \coloneqq Diff$ be some version of the category of smooth manifolds. This becomes a _pregeometry_ in the above sense by taking admissible morphisms to be inclusions of open submanifolds.

Then for $Sh(X) \coloneqq Sh(Op(X))$ the $(\infty,1)$-topos of $(\infty,1)$-sheaves on $X$, the obvious $(\infty,1)$-functor

$$
  O_X : V \to Sh(X)
$$

which for every co-test manifold $v$ is the sheaf

$$
  O_X(v) : (U \subset X) \mapsto Hom_{Diff}(U,v)
$$

is a $Diff$-structure sheaf. Notice that this is precisely nothing but the structure sheaf of smooth functions from the introduction above.

The point is that there are other, more fancy structure sheaves

$$
  O_X : V \to Sh(X)
$$

possible. They describe **[[derived smooth manifolds]]** as described in [DerSmooth](http://math.berkeley.edu/~dspivak/thesis2.pdf).

## Properties

### Limits and colimits
  {#LimitsAndColimits}

Let $\mathcal{G}$ be a [[geometry for structured (infinity,1)-toposes]]. Write $F : (\infty,1)Topos(\mathcal{G}) \to $ [[(∞,1)Topos]] for the [[forgetful functor|forgetful]] [[(∞,1)-functor]] from $\mathcal{G}$-structured $(\infty,1)$-toposes to their underlying $(\infty,1)$-topos.

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

The $(\infty,1)$-category $(\infty,1)Topos(\mathcal{G})$ has all [[filtered (∞,1)-category|cofiltered]] [[(∞,1)-limit]]s and the forgetful functor
$F : (\infty,1)Topos(\mathcal{G}) \to (\infty,1)Topos$ preserves these.

=--

This appears as ([Lurie, corl 1.5.4](#Lurie)).

### Embedding into the ambient big $(\infty,1)$-topos
 {#EmbeddingIntoTheAmbientBigTopos}

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

For $\mathcal{G}$ a [[geometry (for structured (∞,1)-toposes)]] write

$$
  \hat \mathcal{G} \coloneqq Pro(\mathcal{G})
$$

for its [[(∞,1)-category]] of [[ind-object in an (∞,1)-category|pro-objects]].

Write $\widehat{\infty Grpd}$ for the very large [[(∞,1)-category]] of large [[∞-groupoid]]s and

$$
  \hat Sh(\hat \mathcal{G}, \widehat{\infty Grpd})
$$

for the [[very large (∞,1)-sheaf (∞,1)-topos]].

=--

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

The canonical inclusion

$$
  Scheme(\mathcal{G}) \to
  \hat PSh(\hat \mathcal{G}, \widehat{\infty Grpd})
$$

of [[locally representable structured (∞,1)-topos]]es by

$$
  (X, \mathcal{O}_X) \mapsto Hom(Spec(-), (X, \mathcal{O}_X))
$$

is a [[full and faithful (∞,1)-functor]].

=--

This is ([Lurie, theorem, 2.4.1](#Lurie)).

## Related concepts

* [[ringed space]], [[locally ringed space]]

* [[ringed site]], [[locally ringed site]]

* [[ringed topos]], [[locally ringed topos]]

* [[locally algebra-ed topos]]

* **locally algebra-ed (∞,1)-topos**

* [[geometry (for structured (∞,1)-toposes)]],  **structured $(\infty,1)$-topos** , [[locally representable structured (∞,1)-topos]]

Analogous structures in the axiomatic context of [[differential cohesion]] are discussed in _[differential cohesion --  Structure sheaves](cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructureSheaves)_.

## References

The notion of structured $(\infty,1)$-toposes was introduced in

* [[Jacob Lurie]], _[[Structured Spaces]]_
{#Lurie}

Analogous precursor discussion in 1-category theory, hence for [[ringed toposes]] is in

* [[Monique Hakim]], _Topos annelés et schémas relatifs_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972).

The special case of "smoothly structured spaces" ([[derived smooth manifolds]]) is  discussed in

* [[David Spivak]], _Derived smooth manifolds_ PhD thesis ([pdf](http://www.uoregon.edu/~dspivak/files/thesis1.pdf))

[[!redirects structured (∞,1)-topos]]

[[!redirects structured (∞,1)-toposes]]
[[!redirects structured (∞,1)-topoi]]
[[!redirects structured (infinity,1)-toposes]]
[[!redirects structured (infinity,1)-topoi]]

[[!redirects structured generalized spaces]]
[[!redirects structured generalized space]]
[[!redirects ringed (∞,1)-topos]]
[[!redirects ringed generalized spaces]]
[[!redirects ringed generalized space]]

[[!redirects locally algebra-ed (∞,1)-topos]]
[[!redirects locally algebra-ed (infinity,1)-topos]]
[[!redirects locally algebra-ed (∞,1)-toposes]]
[[!redirects locally algebra-ed (infinity,1)-toposes]]


structured category

Let $C$ be a [[finitely complete category]] and $T = (T,\mu,\eta)$ a [[cartesian monad]] in $C$. A **$T$-structured category** is an [[internal category]] in the [[Eilenberg-Moore category]] $C^T$ of $T$-algebras in $C$.

##Literature##

* [[Albert Burroni]], $T$-catégories (catégories dans un triple). Cahiers de Topologie et Géométrie Différentielle Catégoriques, 12 no. 3 (1971), p. 215-321, [numdam: abstract](http://www.numdam.org/item/?id=CTGDC_1971__12_3_215_0) [djvu](http://archive.numdam.org/article/CTGDC_1971__12_3_215_0.djvu) [pdf](http://archive.numdam.org/article/CTGDC_1971__12_3_215_0.pdf)

* [[Tom Leinster]], Higher operads, higher categories, London Math. Soc. Lec. Notes Series 298, [arXiv](http://arxiv.org/abs/math.CT/0305049)

[[!redirects structured categories]]


structured cospan

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

# Structured cospan
* table of contents
{: toc}

## Idea
 {#Idea}

The notion of a *structured cospan* is a modification of the concept of *[[decorated cospans]]*, introduced to provide an improved definition of [[isomorphism classes]] of decorated cospans.

To better understand this motivation, notice that the decoration of the [[target]] of an [[isomorphism]] of [[decorated cospans]] is completely determined by the decoration of the source and the chosen isomorphism of vertices.

More clearly, let
$$
  \array{
     && c && &&&&&& F(c)\\
     & {}^{i}\nearrow && \nwarrow^{o} &&&&&& {}^d\nearrow\\
     a && \downarrow^f && b &&&& 1 & & \downarrow^{F(f)} \\
     & {}^{i'}\searrow && \swarrow^{o'}  &&&&&& {}^{d'}\searrow \\
     && c' &&&&&&&& F(c')
  }
$$
be an isomorphism of decorated cospans, where $d$ and $d'$ are the decorations of the source and the target, respectively (i.e., the 'top' and the 'bottom' cospan in the left diagram, respectively). Then, since the right diagram sits in $\mathbf{Set}$, it commutes on the nose. Thus the decoration $d'$ is already determined by the data of bijection $c \to c'$, together with the decoration on the source.

Therefore, this definition of isomorphism means we are lacking a degree of freedom, namely the freedom to specify an isomorphism for the decorations as well. For example, when open graphs are treated with decorated cospans, the decoration $d$ of a cospan $a \to c \leftarrow b$ is a [[directed graph]] with $c$ as vertex set. An isomorphism of such decorated cospans simply renames the source and target of each edge in $d$. However, isomorphism of directed graphs can be more than just relabeling of the vertices; hence we end up distinguishing open graphs merely by frivolous details like the specific names we give to edges.

Decorated cospans solve this problem by moving the cospans to the 'decorating category', meaning that the data of an isomorphism of cospans now is an arrow between the decorations. In the example of open graphs, we now have to specify an isomorphism of quivers $d \to d'$ instead of getting this from the (poorer) isomorphism of their vertices.

## Definition
+-- {: .un_def}
Let $\mathbf A$ be a category admitting finite coproducts, $\mathbf X$ a category admitting finite colimits and $L : \mathbf A \to \mathbf X$ a functor preserving finite coproducts. Then the [[symmetric monoidal category|symmetric monoidal]] [[double category]] of **structured cospans** over $L$ is the category $_L\mathrm{Csp}(\mathbf X)$ which has

* objects given by objects of $\mathbf A$,
* vertical 1-morphisms given by morphisms of $\mathbf A$
* horizontal 1-morphisms given by structured cospans:
$$
  \array{
     && x
      \\
      & {}^{i}\nearrow
      && \nwarrow^{o}
     \\
     L(a)
     &&&&
     L(b)
  }
$$
which are composed through the obvious pushout,
* 2-morphisms given by commutative diagrams of the form
$$
  \array{
     L(a) & \leftarrow^i & x & \rightarrow^o & L(b)\\
     \downarrow^{L(f)} & & \downarrow^h & & \downarrow^{L(g)}\\
     L(a') & \leftarrow^{i'} & x' & \rightarrow^{o'} & L(b')\\
  }
$$
which are composed horizontally in the obvious way.
=--

(Assuming the existence and preservations of coproducts are unnecessary simply to get a double category, without monoidal structure.)

A **structured cospan** is then a ($1$-)morphism in such a category, that is, a cospan in $\mathbf X$ with the additional data of the functor $L$ and the two preimages of the feet.

## Examples
* By taking $L : \mathbf{Set} \to \mathbf{Graph}$ to be the *discrete graph functor*, i.e. the functor assigning to a set $V$ the edgeless graph with vertex set $V$, we get a category $_L\mathrm{Csp}(\mathbf{Graph})$ which models open graphs.

## Related concepts
* [[cospan]]
* [[decorated cospan]]
* [[hypergraph category]]

## References ##
Structured cospan categories were invented by [[John C. Baez]], [[Kenny Courser]], and [[Christina Vasilakopoulou]]. An introductory talk was given by Courser at the 4th Symposium on Compositional Structures:

* [[John C. Baez]], [[Kenny Courser]], _Structured cospans_, Theory and Applications of Categories **35** 48 (2020) 1771-1822 $[$[arXiv:1911.04630](https://arxiv.org/abs/1911.04630), [tac:35-48](http://www.tac.mta.ca/tac/volumes/35/48/35-48abs.html)$]$

* [[John C. Baez]], [[Kenny Courser]], *Structured cospans* talk notes (2019) ([pdf](http://math.ucr.edu/home/baez/SYCO4/courser_syco4.pdf))

[[!redirects structured cospans]]


structured set

# Structured sets
* tic
{: toc}

## Idea

A *structured set* is, of course, a [[set]] equipped with [[extra structure|extra]] [[structure]].  It is not the individual structured set that matters so much as the [[concrete category|concrete]] [[category]] of sets with a particular sort of structure.

## Definitions

### Abstract

Very abstractly, we may define a __structured set__ as an [[object]] of any [[concrete category]], that is an object of any category $C$ equipped with a [[faithful functor]] $U\colon C \to Set$ to the [[category of sets]].  (Some authors require that $U$ be [[representable functor|representable]] for a concrete category, but we do not need that here.)  Given two structured sets $X$ and $Y$ (in the same category $C$), a [[function]] $f\colon U(X) \to U(Y)$ between their underlying sets __preserves__ the structure if it lies in the image of $U$, that is if there exists a (necessarily unique since $U$ is faithful) [[morphism]] $\tilde{f}\colon X \to Y$ in $C$ such that $U(\tilde{f}) = f$.

### Concrete
{#Concrete}

More concretely, we may define a __type of structure on sets__ as an operation $T$ that, to any set $A$, assigns a set $T(A)$ of __$T$-structures__ on $A$.  At minimum, we should have $T(A) \cong T(B)$ whenever $A \cong B$ (where $\cong$ is [[isomorphism]] in $Set$), so that the concept is [[structural set theory|structural]].  But for good behaviour, we actually want something more coherent; we want an additional operation that, to any [[bijection]] $f\colon A \to B$, assigns a bijection $T(f)\colon T(A) \to T(B)$, such that:

*  $T(\id_A) = \id_{T(A)}$,
*  $T(f g) = T(f) T(g)$.

In other words, $T\colon Set_\cong \to Set_\cong$ (or equivalently $T\colon Set_\cong \to Set$) is a [[functor]] from the [[underlying groupoid]] of $Set$ to itself (or equivalently to all of $Set$).  In particular, any [[automorphism]] of a single set $A$ defines an automorphism of the $T$-structures on $A$, giving an [[action]] of the [[symmetric group]] $S_A$ on $T(A)$.

(Compare the notion of [[structure type]] from [[combinatorics]], which is a set-valued functor on the groupoid of [[finite sets]].  Every combinatorial structure type can be interpreted as a type of structure, where only finite sets are capable of supporting the structure.)

Given a type $T$ of structure on sets, we define a __$T$-structured set__ to be a set $A$ equipped with an [[element]] of $T(A)$.  Given $T$-structured sets $X = (A,\sigma)$ and $Y = (B,\tau)$, a bijection $f\colon A \to B$ __preserves__ the $T$-structure on $X$ and $Y$ if $T(f)(\sigma) = \tau$.

In general, there is no notion of whether an arbitrary function $f\colon A \to B$ preserves $T$-structure, although such a notion may be defined in many cases.  So to get a concrete construction of a [[concrete category]], we specify whatever morphisms we like, subject to the restriction that they form a category and have the correct [[core]] given above.

[[Bourbaki]]\'s theory of structure, while not described in category-theoretic terms, is essentially the above.

## Conversions

Morally, either of the abstract and concrete versions can be converted into the other.  Technically, there are some restrictions.

### Abstract to concrete

Given a category $C$ and a faithful functor $U\colon C \to Set$, we may define a type $T$ of structure on sets as follows:

For each set $A$, consider the [[essential fiber|essential fibre]] of $U$ over $A$, the collection of pairs $(X,f)$ where $X$ is an object of $C$ and $f\colon U(X) \to A$ is a bijection.  We consider two such pairs $(X,\sigma)$ and $(Y,\tau)$ to be [[equivalent]] if there is an [[isomorphism]] $h\colon X \to Y$ in $C$ such that $U(h) ; \tau = \sigma$.  (Because $U$ is faithful, any such $h$ must be unique.)  Define $T(A)$ to be the [[quotient set]] of the essential fibre modulo this [[equivalence relation]].  That is, a $T$-structure on $A$ is an equivalence class $[(X,\sigma)]$.

Given a bijection $f\colon A \to B$, we must define $T(f)\colon T(A) \to T(B)$.  So given $(X,\sigma)$ as above, let $T(f)$ map $[(X,\sigma)]$ to $[(X,\sigma;f)]$ in $T(B)$.  It\'s easy to check that this is well defined as a function from $T(A)$ to $T(B)$; we can also check that this makes $T$ into a functor and that the abstract and concrete definitions of whether a bijection preserves $T$-structure agree.

Technicality:  If $C$ is a [[large category]], then $T(A)$ might be a [[proper class]] instead of a set.  In this case, we can pass to a larger [[universe]]; it is not essential for $T\colon Set_\cong \to Set$ that both copies of $Set$ be the same size.  But for the above description to make sense as it is, we must require that $U$ have essentially small fibres.

### Concrete to abstract

Given a type $T$ of structure on sets, we cannot quite reconstruct the category $C$, but we can reconstruct its [[core]] $C_\cong$.  That is, we can say what the [[objects]] and [[isomorphisms]] of $C$ are, if not the [[morphisms]] of $C$ in general.

An object of $C$ is simply a pair $(A,\sigma)$ consisting of a set $A$ and an element $\sigma$ of $T(A)$.  Given two such objects, an isomorphism from $(A,\sigma)$ to $(B,\tau)$ is simply a structure-preserving map from $A$ to $B$, that is a bijection $f\colon A \to B$ such that $T(f)(\sigma) = \tau$.  Then it is straightforward to check that this defines a [[groupoid]] $C_\cong$.  This groupoid, the __groupoid of $T$-structured sets__, is naturally equipped with a faithful [[forgetful functor]] $U\colon C_\cong \to Set$, given by $U(A,\sigma) \coloneqq A$.

While defining isomorphisms of structured sets is an exact science, choosing more general morphisms of structured sets is something of an art.  In principle, we may define a (*not* the!) __category of $T$-structured sets__ by picking, for each $(A,\sigma)$ and $(B,\tau)$, a collection $Hom_{\sigma,\tau}(A,B)$ of [[functions]] from $A$ to $B$, such that:

*  whenever $f \in Hom_{\sigma,\tau}(A,B)$ and $g \in Hom_{\tau,\upsilon}(B,C)$, then $f ; g \in Hom_{\sigma,\upsilon}(A,C)$;

*  the [[identity map]] on $A$ belongs to $Hom_{\sigma,\sigma}(A,A)$; and

*  a [[bijection]] $f$ from $A$ to $B$ is an isomorphism (as defined above) if and only if both $f \in Hom_{\sigma,\tau}(A,B)$ and $f^{-1} \in Hom_{\tau,\sigma}(B,A)$.

The last condition states precisely that the [[underlying groupoid]] of any concrete category of $T$-structured sets is the groupoid of $T$-structured sets.

Any category of $T$-structured sets is still (like the groupoid of such sets) a concrete category.

### Back and forth

If we start with a type $T$ of structures on sets, construct from this a groupoid $C_\cong$ and a faithful functor $U\colon C_\cong \to Set$ and then construct from this another type $T'$ of structures, then $T'$ will be equivalent to $T$ in the sense that there is a [[natural isomorphism]] between them as functors from $Set_\cong$ to $Set$.

If we start with a category $C$ and a faithful functor $U\colon C \to Set$, construct from this a type of structures $T\colon Set_\cong \to Set$, and then construct from this a groupoid $C_\cong$ with a faithful functor $U\colon C_\cong \to Set$, then $C_\cong$ will in fact be the [[core]] of $C$, with $U\colon C_\cong \to Set$ to restriction of $U\colon C \to Set$ to this core, up to [[equivalence of categories]].

(Do we need proofs?)

Thus the abstract and concrete approaches to structured sets are equivalent, except that the concrete approach does not include a specification of what are the noninvertible morphisms between structured sets.

## Examples

Almost everything in contemporary [[mathematics]] is an example of a structured set; here we list only a few representative ones (and perhaps also some exceptions).

*  [[sets]] themselves (trivially);
*  [[groups]], [[rings]], and other objects of [[universal algebra]];
*  [[topological spaces]] and other concrete (based on sets of [[points]]) notions of [[spaces]];
*  some other notions of [[spaces]] such as [[locales]], as long as we only consider isomorphisms, but *not* using functions as the other morphisms;
*  *not* some other notions of spaces such as [[homotopy types]] (at least if defined via topological spaces);
*  [[strict categories]] (where the underlying set is the set of [[morphisms]]) but *not* [[categories]] treated up to [[equivalence of categories|equivalence]].

## Structured objects

Given any category $S$ whatsoever, we may define a __type of structure__ on objects of $S$ as a functor $T\colon S_\cong \to Set$ to $Set$ from the underyling groupoid of $S$.  Then any [[faithful functor]] $U\colon C \to S$ whatsoever defines a type of structure on objects of $S$ (at least if its fibres are essentially small), in the same way as a concrete category defines a type of structure on sets.  Indeed, we say that $U$ presents the objects of $C$ as objects of $S$ with __[[extra structure]]__.

## See also

*  [[category of elements]]
*  [[Grothendieck construction]]

[[!redirects structured set]]
[[!redirects structured sets]]


Structured Spaces

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

This entry provides hyperlinks for central keywords of the text

* [[Jacob Lurie]],

  _Structured Spaces_

  ([arXiv:0905.0459](http://arxiv.org/abs/0905.0459), [pdf](http://www-math.mit.edu/~lurie/papers/DAG-V.pdf))

which develops the generalization of the notion of a [[ringed topos]] from [[topos theory]] to [[(∞,1)-topos theory]] and formulates basic notions of [[geometry]] in this context ("[[higher geometry]]").

Precursor in 1-[[category theory]] include

* [[Monique Hakim]], _Topos annelés et schémas relatifs_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972) ([doi:10.1007/978-3-662-59155-0](https://www.springer.com/gp/book/9783540055730))

* [[Cole's theory of spectrum]]

The theory is general, but the focus of the examples and applications is on [[derived algebraic geometry]]/[[E-∞ geometry]].

Apparently a version of this is to eventually appear as:

* [[Jacob Lurie]], part I _Fundamentals of Spectral Algebraic Geometry_, of: _[[Spectral Algebraic Geometry]]_

## Survey ##

A survey is at [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]] in the section [notions of space](http://ncatlab.org/nlab/show/A+Survey+of+Elliptic+Cohomology+-+the+derived+moduli+stack+of+derived+elliptic+curves#notions_of_space_4).

## Contents ##

* [[higher geometry]]

* [[structured (∞,1)-topos]]

  * [[geometry (for structured (∞,1)-toposes)]]

  * [[structure sheaf]]

  * [[generalized scheme]]

    * [[derived algebraic geometry]]

      * [[derived scheme]]

      * [[derived Deligne-Mumford stack]]

    * [[derived smooth manifold]]

category: reference


Structures de Dérivabilité

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###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

This entry is about the text

* [[Bruno Kahn]], [[Georges Maltsiniotis]], _Structures de Dérivabilité_ ([pdf](http://www.math.jussieu.fr/~maltsin/ps/bkgmdef.pdf))

on [[derived functor]]s and [[homotopy Kan extensions]].

category: reference

[[!redirects Structures de Derivabilite]]


Strøm model structure

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#### Model category theory
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#### Homotopy theory
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[[!include homotopy - contents]]
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=--

#Contents#
* table of contents
{:toc}

## Idea

[[Arne Strom|Arne Strøm]] proved that the [[category]] [[Top]] of *all* [[topological spaces]] has a structure of a Quillen [[model category]] where

* fibrations are [[Hurewicz fibrations]],

* cofibrations are *closed* [[Hurewicz cofibrations]]

* and the role of weak equivalences is played by (strong) [[homotopy equivalences]]

  (as opposed to the [[weak homotopy equivalences]] of the "standard" [[Quillen model structure on topological spaces]]).

The theorem might have been a [[folklore]] at the time, but the paper ([Strøm 1972](#Strom72)) has a number of subtleties.

Strøm's proofs are not that well-known today and use techniques better known to the topologists of that time, and there is consequently a slight controversy among topologists now.  One of these is that there are modern reproofs, but these modern techniques essentially use [[compactly generated spaces]], while Strøm's proofs succeeded in avoiding that assumption.

However, for many applications nowadays, one is mainly interested in the analogous model structure on the category of [[k-spaces]], or of [[compactly generated space]]s ([[weak Hausdorff space|weak Hausdorff]] [[k-spaces]]).  Note that any cofibration in the latter category is closed.

## Properties

### General

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

In the Strøm model structure, _every_ object is both a [[fibrant object]] and a [[cofibrant object]].

=--

This is a most rare property for a non-trivial model structure.

### Monoidal structure

The Strøm model structure on the category of [[compactly generated spaces]] is a [[monoidal model category]].  This is proven in section 6.4 of [[A Concise Course in Algebraic Topology]] (without that language) using the fact that a subspace inclusion is a Hurewicz cofibration if and only if it is an [[NDR-pair]].

### Quillen adjunctions
 {#QuillenAdjunctions}

The [[identity functor]] $id \colon Top \to Top$ is [[Quillen adjunction|left Quillen]] from the [[classical model structure on topological spaces]] (or the mixed model structure) to the Strøm model structure, and of course right Quillen in the other direction.

$$
  Top_{Strom}
    \stackrel{\overset{id}{\longleftarrow}}{\underset{id}{\longrightarrow}}
  Top_{Quillen}
  \,.
$$

This is just the observation that any [[Hurewicz fibration]] is a [[Serre fibration]], and any [[homotopy equivalence]] is a [[weak homotopy equivalence]]---or dually, that any [[retract]] of a [[relative cell complex]] inclusion is a Hurewicz cofibration.

It follows, by composition, that the ([[geometric realization]] $\dashv$ [[singular simplicial complex]])-[[adjunction]] $ {\vert-\vert} \colon sSet \leftrightarrows Top \colon Sing$ is a [[Quillen adjunction]] between the [[classical model structure on simplicial sets]] and the Strøm model structure.

$$
  Top_{Strom}
    \stackrel{\overset{id}{\longleftarrow}}{\underset{id}{\longrightarrow}}
  Top_{Quillen}
    \stackrel{\overset{{\vert-\vert}}{\longleftarrow}}{\underset{Sing}{\longrightarrow}}
  sSet_{Quillen}
  \,.
$$

### Simplicial structure

If $Top$ denotes the category of [[compactly generated space]]s, then [[geometric realization]] $ {|-|} \colon sSet \to Top$ preserves finite [[products]], and hence is a [[strong monoidal functor]].  Therefore, in this case the adjunction ${|-|} \dashv Sing$ is a [[strong monoidal Quillen adjunction]], and hence makes the Strøm model structure into a [[simplicial model category]].

### Geometric realization is a Reedy cofibrant replacement

Write $({\vert- \vert} \dashv Sing) : Top\stackrel{\overset{{|-|}}{\leftarrow}}{\underset{Sing}{\to}} $ [[sSet]] for the ordinary [[geometric realization]]/[[singular simplicial complex]] [[adjunction]] (see [[homotopy hypothesis]]).

For $S_{\bullet,\bullet} : \Delta^{op} \times \Delta^{op} \to Set$ a [[bisimplicial set]], write $d S$ for its [[diagonal]] $d X : \Delta^{op} \to \Delta^{op} \times \Delta^{op} \stackrel{S}{\to} Set$.

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

For $X_\bullet$ any [[simplicial topological space]], there is a [[homeomorphism]] between the [[geometric realization of simplicial topological spaces|geometric realization]] of the
[[simplicial topological space]] $[n] \mapsto |Sing(X_n)|$ and the ordinary [[geometric realization]] of the [[simplicial set]] that is the diagonal of the [[bisimplicial set]] $Sing(X_\bullet)_\bullet$

$$
  \left|[n] \mapsto |Sing(X_n)|\right|  \simeq_{iso} | d Sing(X_\bullet)_\bullet |
  \,.
$$

Moreover, the degreewise $(|-| \dashv Sing)$-[[unit of an adjunction|counit]] yields a morphism

$$
 ([n] \mapsto |Sing(X_n)|) \to X_\bullet
$$

and this is a cofibrant [[resolution]] in the [[Reedy model structure]]  $[\Delta^{op}, Top_{Strom}]_{Reedy}$ relative to the Strøm model structure.

=--

See [[geometric realization of simplicial topological spaces]] for more details.

## Related concepts

* [[homotopy hypothesis]]

* [[model structure on topological spaces]]

  * [[classical model structure on topological spaces]]

    * [[classical model structure on pointed topological spaces]]

  * [[Strøm model structure]]

  * [[model structure on Delta-generated topological spaces]]

* [[model structure on simplicial sets]], [[model structure on topological spaces]]

* [[Thomason model structure]]

* [[Hurewicz model structure on chain complexes]]

## References

The model structure was originally established in

* {#Strom72} [[Arne Strøm]], _The homotopy category is a homotopy category_, Archiv der Mathematik 23 (1972) ([pdf](https://www.uio.no/studier/emner/matnat/math/MAT9580/v17/documents/strom-the-homotopy-category-is-a-homotopy-category-1972.pdf), [[Strom_HomotopyCategory.pdf:file]])

using results on [[Hurewicz cofibrations]] from:

* [[Arne Strøm]], _Note on cofibrations_,  Math. Scand.  **19** (1966) 11-14 ([jstor:24490229](https://www.jstor.org/stable/24490229), [dml:165952](https://eudml.org/doc/165952), MR0211403)

* [[Arne Strøm]], _Note on cofibrations II_,  Math. Scand.  **22** (1968) 130--142 (1969) ([jstor:24489730](https://www.jstor.org/stable/24489730), [dml:166037](https://eudml.org/doc/166037), MR0243525)

A new proof using [[algebraic weak factorization systems]], and  its generalization to any [[bicomplete category]] which is [[power|powered]], [[copower|copowered]] and [[enriched category|enriched]] in [[TopSp]] is due to:

* {#BarthelRiehl13} [[Tobias Barthel]], [[Emily Riehl]], _On the construction of functorial factorizations for model categories_, Algebr. Geom. Topol. 13 (2013) 1089-1124 ([arXiv:1204.5427](http://arxiv.org/abs/1204.5427), [doi:10.2140/agt.2013.13.1089](http://dx.doi.org/10.2140/agt.2013.13.1089), [euclid:agt/1513715550](https://projecteuclid.org/euclid.agt/1513715550))

Beware that a proof of the Strøm model structure was also claimed in

* [[Peter May|May]], [[Johann Sigurdsson|Sigurdsson]], *[[Parametrized Homotopy Theory]]*,

but relying on

* {#Cole06} [[Michael Cole]], Prop. 5.3 in *Many homotopy categories are homotopy categories*, Topology and its Applications 153 (2006) 1084–1099 ([doi:10.1016/j.topol.2005.02.006](https://doi.org/10.1016/j.topol.2005.02.006))

* {#MayPonto12} [[Peter May]], [[Kate Ponto]], Lemma 17.1.7 in: *[[More concise algebraic topology]] -- Localization, Completion, and Model Categories*, University of Chicago Press (2012) ([ISBN:9780226511795](https://press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html), [pdf](https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf))

which later was noticed to be false, by [[Richard Williamson]], see [Barthel & Riehl, p. 2 and Rem 5.12 and Sec. 6.1](#BarthelRiehl13) for details.

[[!redirects Strøm's model category]]
[[!redirects Strom's model category]]
[[!redirects Strøm's model category]]
[[!redirects Strom's model category]]
[[!redirects Strøm model category]]
[[!redirects Strom model category]]
[[!redirects Strøm's model structure]]
[[!redirects Strom's model structure]]
[[!redirects Strøm's model structure]]
[[!redirects Strom's model structure]]
[[!redirects Strøm model structure]]
[[!redirects Strom model structure]]
[[!redirects Hurewicz model structure]]
[[!redirects h-model structure]]


Strøm's theorem

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .un_theorem}
###### Strøm's theorem
Let $(X,A)$ and $(Y,B)$ be [[NDR-pairs]]. Their product in the category of topological pairs

$$
(X\times Y,X\times B\cup A\times Y)
$$

is also an NDR-pair.
=--

## References

* [[Arne Strøm]], _Note on cofibrations_,  Math. Scand.  **19** (1966) 11-14 ([jstor:24490229](https://www.jstor.org/stable/24490229), [dml:165952](https://eudml.org/doc/165952), MR0211403)

* [[Arne Strøm]], _Note on cofibrations II_,  Math. Scand.  **22** (1968) 130--142 (1969) ([jstor:24489730](https://www.jstor.org/stable/24489730), [dml:166037](https://eudml.org/doc/166037), MR0243525)

[[!redirects Strom's theorem]]
[[!redirects Strom's theorem]]
[[!redirects Strøm's theorem]]
[[!redirects Strom theorem]]
[[!redirects Strøm theorem]]


StrωGrpd

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=--

$Str \omega Grpd$ denotes the collection of [[strict ∞-groupoid]]s.

This is an [[(∞,1)-category]], presented by the 1-category of the same name equipped with the [[model structure on strict ∞-groupoids]].

category: category


STU relation

[[!redirects STU relations]]

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#### Knot theory
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[[!include knot theory - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In [[knot theory]], by the _STU-relations_, one means the following [[relations]] in the [[linear span]] of [[Jacobi diagrams]]:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

## Properties

### Relation to weight systems and Lie algebra objects
 {#RelationToLieAlgebraObjects}

By definition, the STU-relations are the relations respected by [[weight systems]] on [[Jacobi diagrams]].

When regarded as relations of [[string diagrams]], the STU relations characterize the [[Jacobi identity]] for [[Lie algebra objects]], or more generally the [[Lie action property]] of [[Lie modules]]:

$$
  \begin{aligned}
    \Leftrightarrow
    &
    \;\;\;\;\;
    \rho(f(x,y),z) \;=\; \rho(y,\rho(x,z)) - \rho(x,\rho(y,z))
    \\
    \underset{
      {f = [-,-]}
      \atop
      {Lie\;bracket}
    }{
      \Leftrightarrow
    }
    &
    \;\;\;\;\;
    \underset{
       {Lie\;action\;property}
    }{
    \underbrace{
      \rho([x,y],z) \;=\; \rho(y,\rho(x,z)) - \rho(x,\rho(y,z))
    }
    }
    \\
    \underset{
      {\rho = -[-,-]}
      \atop
      {adjoint\;action}
    }{
      \Leftrightarrow
    }
    &
    \;\;\;\;\;
    \underset{
       {Jacobi\;identity}
    }{
    \underbrace{
      [[x,y],z] \;=\; - [y,[x,z]] + [x,[y,z]]
    }
    }
  \end{aligned}
$$

This is the reason behind the existence of [[Lie algebra weight systems]].

### Relation to 4T-relations
 {#RelationTo4TRelations}

Under the embedding of the set of [[round chord diagrams]] into the set of [[Jacobi diagrams]], the [[STU-relations]] imply the [[4T relations]] on [[round chord diagrams]]:

Using this, one finds that [[chord diagrams modulo 4T are Jacobi diagrams modulo STU]]:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

### Relation to knot-graph complex

That the STU-relations characterize the [[cochain cohomology]] of the [[knot-graph complex]] in the respective degrees is shown in [Koytcheff-Munson-Volic 13, Section 3.4](#KoytcheffMunsonVolic13)

> graphics grabbed from [Koytcheff-Munson-Volic 13](#KoytcheffMunsonVolic13)

## Related concepts

* [[weight system]]

* [[4T relation]]

* [[Vassiliev invariant]]

## References

### General

Original articles:

* {#BarNatan95} [[Dror Bar-Natan]], Def. 1.9 in: _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

Textbook accounts:

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

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

Lecture notes:

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

### In terms of the knot-graph complex

In terms of the [[knot-graph complex]]:

* {#KoytcheffMunsonVolic13} Robin Koytcheff, Brian A. Munson, [[Ismar Volić]], Section 3.4 of: _Configuration space integrals and the cohomology of the space of homotopy string links_, J. Knot Theory Ramif. 22, no. 11, 73 pp. (2013) ([arXiv:1109.0056](https://arxiv.org/abs/1109.0056))

[[!redirects STU relation]]
[[!redirects STU relations]]
[[!redirects STU-relation]]
[[!redirects STU-relations]]


Stuart Hameroff

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

## related entries

* [[quantum biology]], [[Roger Penrose]]

category: people


Stuart Jarvis

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=93794)

## Selected writings

On an identification of the [[moduli space of Yang-Mills monopoles]] with a space of [[rational maps]] ([[holomorphic maps]], really) of "[[scattering]] data" from the [[Riemann sphere]] to a [[flag variety]]:

* [[Stuart Jarvis]], *Euclidian Monopoles and Rational Maps*, Proceedings of the London Mathematical Society **77** 1 (1998) 170-192 ([doi:10.1112/S0024611598000434](https://doi.org/10.1112/S0024611598000434))

* [[Stuart Jarvis]], *Construction of Euclidian Monopoles*, Proceedings of the London Mathematical Society, **77** 1 (1998) ([doi:10.1112/S0024611598000446](https://doi.org/10.1112/S0024611598000446))

* [[Stuart Jarvis]], *A rational map of Euclidean monopoles via radial scattering*, J. Reine angew. Math. **524** (2000) 17-41 ([doi:10.1515/crll.2000.055](https://doi.org/10.1515/crll.2000.055))

category: people


Stuart Presnell

* [webpage](http://www.bristol.ac.uk/school-of-arts/people/stuart-m-presnell/index.html)

## Selected writings

On the understanding of [[transport]] in [[homotopy type theory]] as expressing [[Leibniz]]'s principle of "[[indiscernibility of identicals]]" (aka "substitution of equals" or "substitutivity"), as well as the converse implication of  [[path induction]] from transport (together with the uniqueness principle for id-types):

* {#LadymanPresnell15} [[James Ladyman]], [[Stuart Presnell]], *Identity in Homotopy Type Theory, Part I: The Justification of Path Induction*, Philosophia Mathematica **23** 3 (2015) 386–406 [[doi:10.1093/philmat/nkv014](https://doi.org/10.1093/philmat/nkv014), [pdf](http://philsci-archive.pitt.edu/11079/1/Identity_in_HTT_public.pdf)]

category: people


Stuart Raby

* [webpage](http://www.physics.ohio-state.edu/~raby/)

## Selected writings

On [[supersymmetry]] and the [[strong CP problem]]:

* [[Hans-Peter Nilles]], [[Stuart Raby]], _Supersymmetry and the strong CP problem_, Nuclear Physics B Volume 198, Issue 1, 26 April 1982 (doi:10.1016/0550-3213(82)90547-8)

On [[proton decay]] in [[supersymmetry|supersymmeyric]] [[GUTs]]:

* [[Savas Dimopoulos]] [[Stuart Raby]], [[Frank Wilczek]], _Proton decay in supersymmetric models_, Physics Letters B Volume 112, Issue 2, 6 May 1982, Pages 133-136 (doi:10.1016/0370-2693(82)90313-6)

On the [[flavor problem]] in the [[MSSM]]:

* [[Stuart Raby]], _SUSY Flavor Problem_, In: _Supersymmetric Grand Unified Theories_. Lecture Notes in Physics, vol 939 ([doi:10.1007/978-3-319-55255-2_7](https://doi.org/10.1007/978-3-319-55255-2_7))

## Related $n$Lab entries

* [[gauge coupling unification]]

* [[R-parity]]

category: people


stuff > history

< [[stuff]]


stuff type

#Contents#
* table of contents
{:toc}

## Idea

A 'stuff type' is a type of [[stuff]] that can placed on [[finite sets]], e.g. 'being a 2-colored finite set', or 'being the first of two finite sets'.

## Definition

To make this precise, we define a __stuff type__ to be a [[functor]]

$$p: X \to core(FinSet)$$

from some [[groupoid]] $X$ to the groupoid of finite sets and bijections, which is the [[core]] of the category [[FinSet]].  Equivalently, we can think of a stuff type as a [[2-functor]] (of a suitably weakened sort, namely a [[pseudofunctor]])

$$F: core(FinSet)^{op} \to Gpd $$

The idea is to use the [[Grothendieck construction]] and define

$$F(n) = p^{-1}(n)$$

taking advantage of the fact that we may assume without loss of generality that $p$ is a [[fibration]].

We can think of $F$ as a suitably weakened sort of [[presheaf]] of groupoids on $core(FinSet)$.  But since a groupoid is equivalent to its opposite, we can also think of a stuff type as a functor

$$core(FinSet) \to Gpd $$

If a stuff type

$$p: X \to core(FinSet)$$

is faithful, we call it a [[structure type]].  Structure types are also called [[species]], and they can be thought of as presheaves of _sets_

$$F: core(FinSet)^{op} \to Set $$

A stuff type can also be thought of as a [[categorification|categorified]] [[generating function]].   Whereas a generating function assigns a number to each [[natural number]] (or [[finite set]]), a stuff type assigns a [[groupoid]].  Namely, the stuff type

$$F: core(FinSet)^{op} \to Gpd $$

assigns to the finite set $n$ the groupoid $F(n)$.  We can write $F$ as a power series where the coefficient of $Z^n$ is the groupoid $F(n)$.    In these terms, the structure type 'being a finite set' is
\[E^Z := \frac{1}{\overline{0!}} + \frac{1}{\overline{1!}}Z + \frac{1}{\overline{2!}}Z^2 + \cdots + \frac{1}{\overline{n!}}Z^n + \cdots,\]
where $+$ is disjoint union, $//$ is the weak quotient, $n!$ is the permutation group $S_n$, and $1$ is the one-element set (since there's only one way to be finite).

The structure type 'being a totally ordered even set' is
\[\frac{1}{1-Z^2} := \frac{0!}{\overline{0!}} + 0Z + \frac{2!}{\overline{2!}}Z^2 + 0Z^3 + \cdots,\]
since there are $n!$ ways to order a set with $n$ elements and $0$ ways for an odd set to be even.

## References

One advantage of stuff types over the more familiar structure types (i.e., species) is that they allow one to categorify the theory of [[Feynman diagram|Feynman diagrams]]:

* [[John Baez]] and [[James Dolan]], From finite sets to Feynman diagrams, in _Mathematics Unlimited - 2001 and Beyond_, vol. 1, eds. Björn Engquist and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50.  ([arXiv](http://arxiv.org/abs/math.QA/0004133))

* Jeffrey Morton, Categorified algebra and quantum mechanics, _Theory and Applications of Categories_, **16** (2006), 785--854. ([arXiv](http://arxiv.org/abs/math/0601458))

* [[John Baez]], Fall 2003 to Spring 2004 [seminar notes](http://math.ucr.edu/home/baez/qg-fall2003/).

For more on stuff types see:

* [[John Baez]], [Groupoidification](http://math.ucr.edu/home/baez/groupoidification/)

For the generalization of stuff types using $\infty$-groupoids, see:

* Imma Gálvez-Carrillo, Joachim Kock,
and Andrew Tonks, Homotopy linear algebra, _Proc. Roy. Soc. Edinburgh
Sect. A_, **148(2)** (2018), 293--325.  ([arXiv](https://arxiv.org/abs/1602.05082))

[[!redirects stuff types]]