geometric theory

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### Context
#### Topos Theory
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# Contents
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## Idea

The notion of *geometric theory* has many different incarnations.  A few are:

* A geometric theory is a (possibly infinitary) [[first-order logic|first order]] [[theory]] whose [[model]]s are preserved and reflected by [[geometric morphisms]].

* A geometric theory is a (possibly infinitary) [[first-order logic|first order]] [[theory]] whose axioms can be written as [[sequent|sequents]] in context of formulae constructed from the [[logical connective|connectives]] $\top$ ([[truth]]), $\wedge$ (finite [[logical conjunction|conjunction]]), $\bot$ ([[falsity]]), $\bigvee$ (possibly infinitary [[disjunction]]), and $\exists$ ([[existential quantification]]),
as well as $=$ ([[equality]]).

* A geometric theory is a [[syntax|syntactic]] description of a [[Grothendieck topos]].

The equivalence of these statements involves some serious proofs, including [[Giraud's theorem]] characterizing Grothendieck topoi.

## Logical definition

In logical terms, a geometric theory fits into a hierarchy of theories that can be interpreted in the [[internal logic]] of a hierarchy of types of [[categories]].  Since [[predicates]] in the internal logic are represented by [[subobjects]], in order to interpret any connective or quantifier in the internal logic, one needs a corresponding operation on subobjects to exist in the category in question, and be well-behaved.  For instance:

* Since $\wedge$ and $\top$ are represented by [[intersections]] and identities (top elements), and $=$ is represented by equalizer, these can be interpreted in any [[lex category]].  Theories involving only these are examples of [[cartesian theory|cartesian theories]].

* Since $\exists$ is represented by the [[image]] of a subobject, it can be interpreted in any [[regular category]].  Theories involving only $\wedge$, $\top$, and $\exists$ are [[regular theory|regular theories]].

* Since $\vee$ and $\bot$ are represented by [[union]] and [[bottom elements]], these can be interpreted in any [[coherent category]].  Theories which add these to regular logic are called [[coherent theory|coherent theories]].

* Finally, theories which also involve infinitary $\bigvee$, which is again represented by an infinitary union, can be interpreted in [[infinitary coherent categories]], aka *[[geometric categories]]*.  These are **geometric theories**.

Note that the axioms of one of these theories are actually of the form
$$ \varphi \vdash_{\vec{x}} \psi $$
where $\varphi$ and $\psi$ are formulas involving only the specified connectives and quantifiers, $\vdash$ means entailment, and $\vec{x}$ is a [[context]].  Such an axiom can also be written as
$$ \forall \vec{x}. (\varphi \Rightarrow \psi) $$
so that although $\Rightarrow$ and $\forall$ are not strictly part of any of the above logics, they can be applied "once at top level."  In an axiom of this form for geometric logic, the formulas $\varphi$ and $\psi$ which must be built out of $\top$, $\wedge$, $\bot$, $\bigvee$, and $\exists$ are sometimes called *positive* or *geometric* formulas.

The interpretation of arbitrary uses of $\Rightarrow$ and $\forall$ requires a [[Heyting category]].  In fact, by the [[adjoint functor theorem]] for [[posets]], any geometric category which is [[well-powered category|well-powered]] is automatically a Heyting category, but *geometric functors* are not necessarily Heyting functors.  Likewise, a well-powered geometric category automatically has arbitrary intersections of subobjects as well, so we can interpret infinitary $\bigwedge$ in its internal logic, but again these are not preserved by geometric functors.

By the usual syntactic constructions (see [[internal logic]] and [[context]]), any geometric theory $T$ generates a "free geometric category containing a model of that theory," also known as its *[[syntactic category]]* $G_T$.  This syntactic category $G_T$ has the universal property that for any other geometric category $G'$, geometric functors $G_T \to G'$ are equivalent to $T$-models in $G'$.

## Examples

* Let $\Sigma$ be a signature, then the set of all geometric sequents over $\Sigma$ as well as the empty set are geometric theories called the _inconsistent theory_ $\mathbb{T}^\Sigma_0$ resp. the _empty theory_ $\mathbb{T}^\Sigma_1$ over $\Sigma$. The former admits a more concise axiomatisation as $\{\top\vdash\bot\}$. The most basic examples occur when the signature is empty: $\mathbb{T}^\emptyset_0$ is classified by the initial topos $\mathbf{1}$ and $\mathbb{T}^\emptyset _1$ is classified by the terminal topos $Set$ (see at [[classifying topos]] for details).

* The empty signature occurring in the preceding example is an extreme case of a _propositional signature_ i.e. one lacking sort symbols. These can at most contain 0-ary relation symbols since function symbols and higher order relation symbols require sorts. Whence theories over such signatures can at most contain sequents in the empty context between formulas consisting of quantifier-free nestings of 0-ary relation symbols and $\wedge,\bigvee$ and are therefor called _propositional theories_ - in other words: geometric logic boils down to [[propositional logic]]. Propositional theories are classified by [[localic topos|localic toposes]] (see there for further information).

* Any finitary [[algebraic theory]] is, in particular, a cartesian theory, and hence geometric.  This includes [[monoids]], [[groups]], [[abelian groups]], [[rings]], [[commutative rings]], etc.

* The [[theory of categories|theory $\mathbb{K}$ of categories]] with models the
 ([[strict category|strict]]) [[categories]] is not finitary-algebraic, but it is cartesian, and hence geometric; this generalises to (finitary) [[essentially algebraic theory|essentially algebraic theories]].

* The theory of [[torsion]]-free abelian groups is also cartesian, being obtained from the theory of abelian groups by the addition of the sequents $(n\cdot x = 0) \vdash_{x} (x = 0)$ for all $n\in \mathbb{N}$.

* The theory of [[local rings]] is a coherent theory, being obtained from the theory of commutative rings by adding the sequent $ (0 = 1) \vdash \bot $, for nontriviality, and
  $$\exists z. ((x+y)z = 1) \vdash_{x y} (\exists z.(x z = 1) \vee \exists z.(y z = 1))$$
  saying that if $x+y$ is invertible, then either $x$ or $y$ is so.

* The theory of [[fields]] is also coherent, being obtained from the theory of commutative rings by adding $ (0 = 1) \vdash \bot $ and also
  $$ \top \vdash_{x} (x=0) \vee (\exists y.(x y = 1)) $$
  asserting that every element is either zero or invertible.  In the [[constructive logic]] that holds internal to the categories in question, this is the notion of a "discrete field;" other classically equivalent axiomatizations (called "Heyting fields" or "residue fields" -- see [[field]]) are not coherent.

* The theory of [[torsion]] abelian groups is geometric but not coherent; it can be obtained from the theory of abelian groups by adding the sequent
  $$ \top \vdash_{x} \bigvee_{n\ge 1} (n \cdot x = 0) $$
  asserting that for each $x$, either $x=0$ or $x+x=0$ or $x+x+x=0$ or ....  Similarly, the theory of fields of finite [[characteristic]] is geometric but not coherent.

* The _theory of the standard successor algebra_ is geometric. Its signature has one sort symbol $N$, one function symbol $s:N\to N$ and one constant $0:N$. The theory axiomaticizes [[natural number object|natural number objects]] by taking advantage of the availability of infinitary disjunctions in geometric logic in order to assert in the third axiom that every natural number is _standard_ i.e. either 0 or obtained from 0 by (repeated) application of the successor function:

  - $0 = s(n) \vdash_{n} \bot$
  - $s(n) = s(n') \vdash_{n n'} n=n'$
  - $\top \vdash_{n} \bigvee_{i} n=\underset{i}{\underbrace{s \dots s}}(0)$

  The [[classifying topos]] is $Set$. But due to the trivial subtopos lattice, theories classified by Set admit no consistent quotient theories; thus the *geometric* theory of natural numbers is [[complete theory|complete]]!  Furthermore, since $Set$ is the terminal topos, classifying also the [[empty theory]], one sees that the object/type $\mathbb{N}$ of natural numbers "comes for free" in geometric logic. This observation underlies the ideas for a [[geometric type theory]] (for more on this see there or the [article by Vickers](#Vickers09)).

* The theory of a [[real number]] is geometric.  This is a [[propositional logic|propositional]] theory, having no sorts, and having one relation symbol "$p\lt x\lt q$" for each pair of rational numbers $p\lt q$.  Its axioms are:
  - $(p_1\lt x\lt q_1) \wedge (p_2\lt x\lt q_2) \vdash (\max(p_1,p_2) \lt x \lt \min(q_1,q_2))$ (if $\max(p_1,p_2)\lt \min(q_1,q_2)$), and $(p_1\lt x\lt q_1) \wedge (p_2\lt x\lt q_2) \vdash \bot$ (otherwise).
  - $(p\lt x\lt q) \vdash (p\lt x\lt q') \vee (p' \lt x\lt q)$ (if $p\lt p' \lt q' \lt q$).
  - $(p\lt x\lt q) \vdash \bigvee_{p\lt p'\lt q'\lt q} (p' \lt x\lt q')$.
  - $\top \vdash \bigvee_{p\lt q} (p\lt x\lt q)$.

  The [[classifying topos]] of this theory is the topos of [[sheaves]] on the [[real numbers]].

* The theory of flat diagrams[^flat] over a small category $\mathcal{C}$ is geometric. For each object $i$ of $\mathcal{C}$ there is a sort $\sigma_i$, and for each morphism $u\colon i \to j$ a function symbol $\alpha_u\colon \sigma_i \to \sigma_j$. The axioms are -

  - $\top \vdash_{x\colon\sigma_i} \alpha_{Id_i}(x) = x$

  - $\top \vdash_{x\colon\sigma_i} \alpha_{v}(\alpha_u(x)) = \alpha_{v u}(x)$ ($u\colon i \to j$ and $v\colon j \to k$)

  - $\top \vdash \bigvee_{i \in \mathcal{C}_0} (\exists x\colon \sigma_i) \top$

  - $\top \vdash_{x\colon\sigma_i, y\colon\sigma_j} \bigvee_{u\colon k\to i, v\colon k\to j} (\exists z\colon\sigma_k) (x = \alpha_u(z) \wedge y = \alpha_v(z))$

  - $\alpha_u(x) = \alpha_v(x) \vdash_{x\colon\sigma_i} \bigvee_{w\colon k\to i, u w = v w} (\exists z\colon \sigma_k) x = \alpha_w(z)$ ($u,v\colon i \to j$)

  This theory is classified by the topos of presheaves over $\mathcal{C}$.

  It is commonly more useful to consider the theory of flat _presheaves_ over $\mathcal{C}$, in other words the flat diagrams over $\mathcal{C}^{op}$. ([[Elephant]] calls these _torsors_ over $\mathcal{C}$, generalizing the established terminology for groups.) This is because the representable presheaves are flat, and so Yoneda's lemma transforms objects of $\mathcal{C}$ covariantly into models of the theory. In fact, the models of the theory are the filtered colimits of representables. For example, a finitary algebraic theory is classified by the topos of covariant functors from the category of finitely presented algebras to $Set$.

[^flat]: This is also sometimes called the theory of flat functors on $\mathcal{C}$.

* If $(\mathcal{C},J)$ is a small site, by adding a geometric axiom schema to express J-continuity to the theory of flat diagrams, one obtains the [[theory of flat functors|theory of J-continuous flat diagrams]], which is classified by $Sh(\mathcal{C},J)$. This answers in principle the question, given a site representation what geometric theory is classified by a Grothendieck topos? Of course, in practice one would like to have more concise and less unwieldy axiomatizations.

* For any [[geometric theory]] $\mathbb{T}$ there exists a geometric theory $\mathbb{T}^\mathbf{2}$ called the [[theory of model homomorphisms|theory of $\mathbb{T}$-model homomorphisms]] whose models in a [[Grothendieck topos]] $\mathcal{E}$ are precisely the homomorphisms between $\mathbb{T}$-models in $\mathcal{E}$:
  $$Mod_{\mathbb{T}^\mathbf{2}}(\mathcal{E})=Mod_{\mathbb{T}}(\mathcal{E})^\mathbf{2}=Mod_\mathbb{T}(\mathcal{E}^\mathbf{2})\quad.$$

* A geometric theory whose [[classifying topos]] is a [[presheaf topos]] is called a _[[theory of presheaf type]]_.

## Other characterizations

### In terms of sheaf topoi {#InTermsOfSheafTopoi}

Categories of each "logical" type can also be "completed" with respect to a suitable "exactness" property, without changing their internal logic.  Any regular category has an completion into an [[exact category]] (see [[regular and exact completion]]), any coherent category has a completion into a [[pretopos]], and any geometric category has a completion into an [[infinitary pretopos]].

However, [[Giraud's theorem]] says that an infinitary pretopos having a [[generating set]] is precisely a [[Grothendieck topos]].  Moreover, a functor between Grothendieck topoi is geometric (preserves all the structure of a geometric category) iff it preserves finite [[limits]] and small [[colimits]].  By the [[adjoint functor theorem]], this implies that it is the [[inverse image]] part of a [[geometric morphism]], i.e. an [[adjunction]] $f^* \dashv f_*$ in which $f^*$ (the "inverse image") preserves finite limits.

Thus:

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Grothendieck topoi and inverse-image functors are in some sense the "natural home" for models of geometric theories.

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Note, though, that geometric morphisms are generally considered as pointing in the direction of the *direct* image $f_*$, which is the opposite direction to the geometric functor $f^*$.  (This is because when $E$ and $F$ are the toposes of [[sheaves]] on [[sober space|sober topological spaces]] (or [[locales]]) $X$ and $Y$ respectively, then [[continuous maps]] $X \to Y$ correspond precisely to geometric morphisms $E \to F$.)  Also, of course any Grothendieck topos is an [[elementary topos]] (at least, as long as one works in [[foundations]] for which [[Set]] is an elementary topos), and hence its internal logic also includes "higher-order" constructions such as function-objects and power-objects.  However, these are not preserved by geometric functors, so they (like $\forall$ and $\Rightarrow$) are not part of geometric logic.  (They are, however, preserved by [[logical functors]], a different sort of morphism between topoi.)

In particular, we can apply the "exact completion" operation to the syntactic category $G_T$ of a geometric theory to obtain an infinitary pretopos $Set[T]$.  As long as the theory $T$ was itself small, $Set[T]$ will have a generating set, and therefore will be a Grothendieck topos.  The universal property of the syntactic category, combined with that of the exact completion, implies that for any Grothendieck topos $E$, geometric morphisms $E\to Set[T]$ are equivalent to $T$-models in $E$.  This topos $Set[T]$ is called the **[[classifying topos]]** of $T$.

In the other direction, if $C$ is any small [[site]], we can write down a "geometric theory of cover-preserving [[flat functors]] $C^{op}\to Set$."  By [[Diaconescu's theorem]] classifying geometric morphisms into sheaf topoi, it follows that $Sh(C)$ is a [[classifying topos]] for this theory.  Therefore, not only does every geometric theory have a [[classifying topos]], every Grothendieck topos is the [[classifying topos]] of _some_ theory.  Very different-looking theories can have equivalent classifying topoi; this of course implies that they have all the same models in all Grothendieck topoi (hence a Grothendieck topos is the "extensional essence" of a geometric theory).  We say that two geometric theories with equivalent classifying topoi are [[Morita equivalence|Morita equivalent]].

### Functorial definition {#FunctorialDefinition}

We can approach the same idea by starting instead from the notions of [[Grothendieck topos]] and [[geometric morphism]].  The following approach is described in B4.2 of the [[Elephant]].

Suppose we consider what sorts of "theories" we can define in terms of Grothendieck topoi, that are preserved by inverse image functors.  Any such theory should certainly define a [[2-functor]] $T\colon GrTop^{op}\to Cat$, where $GrTop$ is the 2-category of Grothendieck topoi and geometric morphisms, so for the moment let's call any such $T$ a "theory".  The image $T(E)$ of a functor $E$ is supposed to be "the category of $T$-models," and a *classifying topos* for such a 2-functor will be just a [[representing object]] for it.

Of course, this notion of theory is far too general; we only want to consider theories that are constructed in some reasonable way.  One theory that should certainly be geometric is the [[theory of objects]], $O$.  This 2-functor sends a topos $E$ to itself, considered as a mere category, and an inverse image functor to itself, considered as a mere functor.  The theory $O$ can be shown to have a classifying topos, the [[classifying topos for the theory of objects|object classifier]] $Set[O]$.  Similarly, we have a theory $O_n$ of $n$-tuples of objects that should be geometric.

How can we construct more theories that ought to be geometric?  We should start from some finite collection of objects (i.e. a model of $O_n$), "construct some new objects and morphisms," and then "impose some axioms on them."  For any theory $T$, let's call a transformation $T\to O$ a *geometric construct*.  This is supposed to be "an object constructed out of the axioms of $T$ in a natural way."  More precisely, to every $T$-model in a topos $E$ it assigns an object of $E$, in a way that varies naturally with morphisms of $T$-models and inverse image functors.

Now define a *simple functional extension* of $T$ to be the [[inserter]] of a pair of geometric constructs $T\;\rightrightarrows\; O$.  A model of such a theory will consist of a model of $T$, together with an additional morphism between two objects constructed out of the given $T$-model.  By iteratively applying such constructions, we can add in any number of new morphisms between "constructed objects."

Finally, define a *simple geometric quotient* of $T$ to be the [[inverter]] of a [[modification]] between a pair of geometric constructs $T\;\rightrightarrows\; O$.  That is, we require that a certain naturally defined morphism between objects constructed out of $T$-models must be an isomorphism.  (Applying equalizers, we see that this also includes the ability to set morphisms equal, i.e. to construct [[equifier]]s.)

From this point of view, a **geometric theory** is a theory $GrTop^{op}\to Cat$ obtained from some $O_n$ by a finite sequence of simple functional extensions and simple geometric quotients.  Of course, once we know that each $O_n$ has a classifying topos, it follows immediately that any geometric theory has a classifying topos, since $GrTop$ has inserters and inverters.

### Hybrid definition

The following definition is sort of a "halfway house" between logic and geometry.  Start with a first-order [[signature (in logic)|signature]] $\Sigma$ (this is the logical part).  Then we have a 2-functor $\Sigma Str\colon GrTop^{op}\to Cat$ sending a topos $E$ to the category $\Sigma Str(E)$ of $\Sigma$-structures in $E$.  A **geometric theory** over $\Sigma$ is defined to consist of the following.

* For each Grothendieck topos $E$, we have a [[replete subcategory|replete]] [[full subcategory]] $T(E)$ of $\Sigma Str(E)$, such that
* For each geometric morphism $f\colon E\to F$, if $M\in T(F)$ then $f^*M\in T(E)$ (i.e. $T$ is a subfunctor of $\Sigma Str$), and moreover
* For any set-indexed jointly surjective family $(f_i \colon E_i \to E)_{i\in I}$ of geometric morphisms, and any $M \in \Sigma Str(E)$, if $f_i^* M \in T(E_i)$ for all $i$, then $M\in T(E)$.

The equivalence of this definition with the others can be found in

* Olivia Caramello, "A characterization theorem for geometric logic", [arXiv:0912.1404](http://arxiv.org/abs/0912.1404).

It is not sufficient, in the third condition, to restrict to the case when $I$ is a singleton, but it is sufficient to consider the case when $I$ is a singleton together with all families of coproduct injections $(E_i \to \coprod_i E_i)_{i\in I}$.

## Localization

By framing this notion in the [[internal language]] of a [[topos]] $S$ we can talk of geometric theories over $S$, with models in [[bounded topos|bounded]] $S$-toposes (the [[relative point of view|relative version]] of "Grothendieck topos").  As a simple example, if we have a sheaf $A$ of [[rings]] on a [[topological space]] $X$ we can describe left $A$-[[modules]] as models of a geometric theory over $Sh(X)$, the topos of sheaves on $X$, and this notion is definable in $Sh(X)$-toposes.

Similarly to the $Set$-based case, given a geometric theory $T$ over a topos $S$, we can form the $S$-topos $S[T] \to S$ that classifies $T$, for which the category of $T$-models in a bounded $S$-topos $E$ is naturally equivalent to the category of morphisms of $S$-toposes $E \to S[T]$. In other words

$$T Mod(E) \simeq Top_S(E,S[T])$$

## Morphisms of theories

Since the classifying topos encodes the "extensional essence" of a geometric theory, it makes sense to define a **map of theories** $T \to T'$ to be a morphism of $S$-toposes $h: S[T'] \to S[T]$. Equivalently, of course, this is a $T$-model in $S[T']$.  Composition with $h$ induces a functor, _[[forgetful functor|forget]] along_ $h$, from $T'$-models to $T$-models in any $S$-topos.

## Gros categories of models

Define the [[gros category]] $T Mod$ of $T$-models to have as objects pairs $(E,A)$ where $E$ is an $S$-topos and $A$ is
a $T$-model in $E$. A map $(f,g): (E,A) \to (F,B)$ is given
by a morphism $f: F \to E$ of $S$-toposes and a homomorphism $g: f^*(A) \to B$ of $T$-models in $F$. The
composition of maps should be evident. A map $h: T \to T'$
of geometric theories over $S$ induces a forgetful functor
$T' Mod \to T Mod$ which leaves unchanged the $S$-topos of residence, which has a left adjoint $T Mod \to T' Mod$ which may change the topos.  For if $a: E \to S[T]$ is a $T$-model in $E$, [[base change|pulling]] $a$ back along $h$ yields a $T'$-model, not in $E$ but in the [[pullback]].  This is a consequence of general facts about finite [[2-limits]] of the [[2-category]] of
[[bounded topos|bounded]] $S$-toposes.

The 2-categorical version of $T Mod$ is useful in generalization of the [[spectrum]] construction: See at [[Cole's theory of spectrum]].

## Properties

* [[Barr's theorem]] says, that if a statement in [[geometric logic]] is deducible from a [[geometric theory]] using classical [[logic]] and the [[axiom of choice]], then it is also deducible from it in [[constructive mathematics]].

## Related concepts

* [[geometric type theory]]

* [[geometric homotopy type theory]]

* [[Cole's theory of spectrum]]

* [[classifying topos for the theory of objects]]

* [[theory of objects]]

* [[theory of decidable objects]]

* [[theory of model homomorphisms]]

* [[theory of flat functors|theory of flat diagrams]]

* [[theory of categories]]

## References

Review:

* [[Steven Vickers]], *Continuity and geometric logic*, Journal of Applied Logic Volume 12, Issue 1, March 2014, Pages 14-27 ([doi:10.1016/j.jal.2013.07.004](https://doi.org/10.1016/j.jal.2013.07.004))

Textbook accounts:

* [[Peter Johnstone]], sections B4.2, D1.1 of _[[Sketches of an Elephant]]_

* [[Michael Makkai]], [[Gonzalo E. Reyes]], _First Order Categorical Logic_ , LNM 611, Springer Berlin 1977. ([draft](https://marieetgonzalo.files.wordpress.com/2018/04/makkai-reyes-book.pdf))

A textbook account of (finitary) geometric logic can be found in

* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_

A systematic introduction to topos theory and geometric logic can be found in the following draft by O. Caramello:

* [[Olivia Caramello]], _Topos-theoretic background_ , ms. 2014. ([pdf](http://www.oliviacaramello.com/Unification/ToposTheoreticPreliminariesOliviaCaramello.pdf))

which has become a chapter of the following monograph exploring first-order model theory in the context of Grothendieck toposes:

* [[Olivia Caramello]], _Theories, Sites, Toposes_ , Oxford UP 2018.

For additional background on (finitary) geometric formulas consider:

* Karel Stokkermans, [[Bill Lawvere]], and [[Steve Vickers]]: _Catlist discussion March 1999_ . ([link](http://comments.gmane.org/gmane.science.mathematics.categories/1058))

* H.J. Keisler: _Theory of models with generalized atomic formulas_ , JSL 25 (1960), pp.1-26.

Discussion in the context of [[computer science]] is in

* [[Steve Vickers]], _Geometric logic in computer science_ ([pdf] (http://www.cs.bham.ac.uk/~sjv/GLiCS.pdf))

Discussion with an eye towards [[geometric type theory]] is in

* {#Vickers09}[[Steve Vickers]], _Locales and toposes as spaces_ ([pdf](http://www.cs.bham.ac.uk/~sjv/LocTopSpaces.pdf))

Discussion with an eye towards [[quantum physics]]:

* Guillaume Raynaud, *Fibred Contextual Quantum Physics* ([etheses:1685](https://etheses.bham.ac.uk/id/eprint/1685/), [pdf](https://etheses.bham.ac.uk//id/eprint/1685/1/Raynaud14PhD.pdf))

[[Stone duality]] for geometric theories is discussed in:

* [[Henrik Forssell]], _Topological representation of geometric theories_,  [arxiv/1109.0699](http://arxiv.org/abs/1109.0699)

A nice short exposition together with an unorthodox proposal to expand geometric logic with fix point operators can be found here:

* [[Andreas Blass]], _Topoi and Computation_, Bull. European Assoc.Theoret. Comp.Sci. **36** (1988) pp.57-65. ([draft](http://www.math.lsa.umich.edu/~ablass/eatcs.pdf))

[[!redirects geometric theory]]
[[!redirects geometric theories]]
[[!redirects geometric logic]]
[[!redirects geometric logics]]
[[!redirects geometric formula]]
[[!redirects geometric formulas]]
[[!redirects geometric formulae]]
[[!redirects geometric sequent]]
[[!redirects geometric sequents]]


geometric transformation

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###Context###
#### Topos Theory
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[[!include topos theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _geometric transformation_ is a [[morphism]] between [[geometric morphism]]s between [[topos]]es: a [[2-morphism]] in the [[2-category]] [[Topos]].

## Definition

For

$$
  f = (f^* \dashv f_*) : \mathcal{E} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{F}
$$

and

$$
  g = (g^* \dashv g_*) : \mathcal{E} \stackrel{\overset{g^*}{\leftarrow}}{\underset{g_*}{\to}} \mathcal{F}
$$

two [[geometric morphism]]s, a **geometric transformation**

$$
  \eta : f \Rightarrow g
$$

is a [[natural transformation]] between the [[inverse image]] functors

$$
  f^* \Rightarrow g^*
  \,.
$$

By [[mate]]-calculus, these are in bijection to natural transformations of the [[direct image]] functors

$$
  g_* \Rightarrow f_*
  \,.
$$

## References

Section A4.1 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

[[!redirects geometric transformations]]


geometric type theory

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#### Type theory
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#### Topos Theory
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[[!include topos theory - contents]]
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#### Geometry
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[[!include higher geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

_Geometric type theory_ is a conjectural extension of [[geometric logic]] to  an [[extensional type theory|extensional]] [[dependent type theory|dependent]] [[type theory]].  That is, it is supposed to be a dependent type theory which under the [[relation between category theory and type theory]] corresponds to [[sheaf toposes]], and is preserved by [[geometric morphisms]].

At present it is not clear exactly how such a type theory should be defined.

A *motivating example* is a [[geometric theory]] as follows. Its signature has one sort, $N$, and two function symbols $0\colon 1 \to N$ and $s \colon N \to N$. Its axioms are -

  * $0 = s(n) \vdash_{n} \bot$
  * $s(n) = s(n') \vdash_{n n'} n=n'$
  * $\top \vdash_{n} \bigvee_{i} n=s^{i}(0)$

("$s^{i}(0)$" is not a term in the formal language, but $i$ indexes an inductively defined sequence of formulae "$n=s^{i}(0)$".)

In any model (in a Grothendieck topos) the interpretation of $N$ is uniquely isomorphic to an [[natural numbers object|nno]], so the nno can be characterized uniquely up to isomorphism by geometry theory - something that would be impossible in finitary first-order logic. There is essentially only one model, and this theory is equivalent to the empty theory. What this illustrates is that a sort for the nno can be added "for free" to any theory - and this is consistent with the fact that nno's are preserved by inverse image functors.

List objects ($List X$ is the set of finite sequences of elements from $X$) can similarly be added "for free", and in fact the same goes for free constructions for [[cartesian theory|cartesian theories]]. Thus such constructors can be added as a type theoretic adjunct to geometric logic without altering its expressive power. Since geometric type theory also includes [[quotient types]], the constructions are similar to quotient inductive types.

As an example, the [[geometric theory]] of a real number can be rewritten in a form that more directly describes Dedekind sections of rationals, with sort $Q$ that is geometrically constrained to be the rationals. For further details, see [Vickers 2007](#Vickers07).

All this leads to a conjecture of a "geometric type theory", geometric logic enhanced with the geometrically definable types.

An "arithmetic type theory" has now been formalized [Vickers 2018](#Vickers18) by adjoining such type constructors to coherent logic.
Categorically, it corresponds to replacing [[Grothendieck topos|Grothendieck toposes]] with [[arithmetic universe|arithmetic universes]].
Although it lacks the infinitary disjunctions of geometric logic, they can in many cases be provided by existential quantification over infinite sorts.
[Vickers 2017](#Vickers17) shows how it can be used to prove base-free results for (relative) Grothendieck toposes.

## Related concepts

* [[modal type theory]]

* [[geometric homotopy type theory]]

* [[synthetic geometry]]

* [[geometric mathematics]]

## References

* {#Vickers07}[[Steve Vickers]], _Locales and toposes as spaces_, Handbook of Spatial Logics, Springer 2007 ([pdf](http://www.cs.bham.ac.uk/~sjv/LocTopSpaces.pdf))

* {#Vickers17}[[Steve Vickers]], _Arithmetic universes and classifying toposes_, Cahiers de topologie et géométrie différentielle catégorique **58 (4)** (2017), pp. 213-248 ([pdf](http://arxiv.org/abs/1701.04611))

* {#Vickers18}[[Steve Vickers]], _Sketches for arithmetic universes_, accepted 2018 for Journal of Logic and Analysis ([pdf](http://arxiv.org/abs/1608.01559))

[[!redirects geometric type theories]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called the "geometrization of quantum mechanics" by [Kibble (1979)](#Kibble79) who introduced the idea, or "geometrical formulation of quantum mechanics" [[Ashtekar & Schilling (1999)](#AshtekarSchilling99)] or just "geometric quantum mechanics" [[Brody & Hughston (2001)](#BrodyHughston01)], but which ought to be called the "symplectic formulation", is the observation that, in standard [[quantum mechanics]]:

1. the [[complex projective space]] $P \mathscr{H}$ of the [[Hilbert space|Hilbert]] [[space of quantum states|space of]] [[pure quantum states]] $\mathscr{H}$ is canonically a [[Kähler manifold]] (via the [[Fubini-Study metric]]) and so in particular a [[symplectic manifold]],

1. the [[curves]] in $P \mathscr{H}$ which correspond to solutions of the [[Schrödinger equation]] are [[Hamiltonian flows]] with respect to this symplectic structure.

This is fairly immediate to see from mathematical inspection, but the perspective is somewhat surprising from the point of view of standard accounts of [[classical physics|classical]]/[[quantum physics]], which tend to frame [[symplectic geometry]] and its [[Hamiltonian flows]] as the hallmark of [[classical physics|classical]] [[phase spaces]] and [[classical physics]], and to highlight [[quantization]] as a [[deformation]] of this symplectic structure (whence: "[[deformation quantization]]").

Now, the standard perspective is certainly not wrong, but various authors have inevitably suggested that the "geometrical formulation"  (more descriptive would be: "symplectic formulation") of quantum mechanics may point to some deeper truth, and if only to show some kind of conceptual unity where one is used to amplifying the dichotomy.

## References

The observation is due to:

* {#Kibble79} [[Tom W. B. Kibble]], *Geometrization of quantum mechanics*, Communications in Mathematical Physics **65** (1979) 189–201 [[doi:10.1007/BF01225149](https://doi.org/10.1007/BF01225149)]

The idea was picked up in:

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

  > (with speculations on the implication for [[quantum state collapse]])

* {#AshtekarSchilling99} Abhay Ashtekar, Troy A. Schilling, *Geometrical Formulation of Quantum Mechanics*, in: *On Einstein’s Path* Springer (1999) [[arXiv:gr-qc/9706069](https://arxiv.org/abs/gr-qc/9706069), [doi:10.1007/978-1-4612-1422-9_3](https://doi.org/10.1007/978-1-4612-1422-9_3)]

Further discussion:

* {#BrodyHughston01} [[Dorje C. Brody]], [[Lane P. Hughston]], *Geometric Quantum Mechanics*, J. Geom. Phys. **38** (2001) 19-53 [[arXiv:quant-ph/9906086](https://arxiv.org/abs/quant-ph/9906086), doi:10.1016/S0393-0440(00)00052-8]

* A Benvegnù, N Sansonetto, M Spera, *Remarks on geometric quantum mechanics*, Journal of Geometry and Physics **51** 2 (2004) 229-243 [[doi:10.1016/j.geomphys.2003.10.008](https://doi.org/10.1016/j.geomphys.2003.10.008)]

* J. F. Cariñena, J. Clemente-Gallardo & G. Marmo, *Geometrization of quantum mechanics*, Theoretical and Mathematical Physics **152** (2007) 894–903 [[doi:10.1007/s11232-007-0075-3](https://doi.org/10.1007/s11232-007-0075-3)]

* Secret Blogging Seminar, *[Quantum mechanics and geometry](https://sbseminar.wordpress.com/2009/11/16/quantum-mechanics-and-geometry/)* (2009)

* Hoshang Heydari, *Geometric formulation of quantum mechanics* [[arXiv:1503.00238](https://arxiv.org/abs/1503.00238)]

Discussion of dynamics of [[mixed states]] ([[density matrices]]), now via [[Poisson geometry]]:

* Pritish Sinha, Ankit Yadav, *Poisson Geometric Formulation of Quantum Mechanics* [[arXiv:2312.05615](https://arxiv.org/abs/2312.05615)]

[[!redirects symplectic formulation of quantum mechanics]]


geometrical optics

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

#Contents#
* table of contents
{:toc}

## Idea

What is called **geometrical optics** is an approximate theoretical treatment of [[wave]] [[optics]] in the short wavelength limit. The [[wave equation]] of physical optics is thus replaced by the so called **eikonal equation**.

A formal analogue to this limit in [[quantum mechanics]] is the [[semiclassical approximation]].

Integrals involving exponential function with rapidly oscillating phase have main contributions close to the classical paths which in optics correspond to the light rays of geometric optics. However the geometric optics can tell much more than the crude light ray representation, for example the asymptotic behaviour of light near caustics and phase jumps when passing through focal points.

## References

* [[Victor Guillemin]], [[Shlomo Sternberg]], Chapter III of: *Geometric asymptotics*, Mathematical Surveys and Monographs **14**, AMS (1977) [[ams:surv-14](https://bookstore.ams.org/surv-14)]

See also

* Wikipedia, _[Geometrical optics](https://en.wikipedia.org/wiki/Geometrical_optics)_

[[!redirects geometric optics]]

[[!redirects eikonal equation]]
[[!redirects eikonal equations]]


geometrically admissible action

It is often natural that a symmetry object acting on some space is in the category of spaces itself. For example, algebraic groups acts on varieties, topological groups on topological spaces, and group schemes over $S$ act on $S$-schemes.

Suppose now we want to define an action of a Hopf $k$-algebra $H$ on a noncommutative $k$-scheme, where we view a Hopf algebra as a dual to a noncommutative analogue of an affine group object over $Spec(k)$. In nonaffine situations noncommutative schemes are glued from algebras, but they are not described by a single algebra, but by a system of algebras; in fact a more canonical object is the category of quasicoherent sheaves which localizes to categories of modules over each of the algebras of the cover; these categories of modules are monadic over the category $V = {}_{k}Mod$
which is the category of quasicoherent sheaves over $Spec(k)$. The Hopf algebra, to be considered as a noncommutative space, has to be replace by the *monoidal* category $\mathcal{M} = {}_{H}Mod$ which comes equipped with a canonical action on $V$: tensor and forget the additional structure. The Hopf algebra should be reconstructed from the forgetful functor to $V$; it is itself a comonoid in $M$. What does it mean that $H$ acts on some noncommutative space represented by a category $C$ ? Well it has no meaning unless $C$ has a forgetful functor to $V$ as well (like in the case of possibly noncommutative $k$-schemes). Now the action of $Spec(H)$ corresponds to the action of the monodial category $M$ on $C$. But not any action: the action which is compatible with the action of $M$ on $V$. Such actions are called __geometrically admissible actions__ of monoidal category. Admisibility is a notion relative to the action of the same monoidal category on the base category. In other words admissible actions are the actions of monoidal categories on categories which lift the defining action on the base. In special cases, the liftings correspond to the distributive laws. This is the origin of appearance of coalgebra bundles in noncommutative geometry: [[entwining structure]]s induce lifts to admissible actions; however there are other cases of distributive laws and more general lifts which are not coming from entwinings between coalgebras and algebras.

Another example is given by the action of the monoidal category of sheaves on a topological group $G$ on a topological $G$-space $X$. The action $Sh(G)$ on $Sh(X)$ is given by the pushforward along the action $G\times X\to X$ of the external tensor product of sheaves $Sh(G)\times Sh(X)\to Sh(G\times X)$.

As a general definition, we start with a fixed notion of a symmetry object as a monoidal category $M$ with distinguished action $\lozenge$ on the base category $V$. A geometrically admissible action $\tilde{\lozenge}$ in wider sense is any action of $M$ on some category $C$ over $V$ such that

$$\array{
M\times C&\stackrel{\tilde{\lozenge}}\to& C\\
\downarrow && \downarrow\\
M\times V&\stackrel{\lozenge}\to& V
}$$

commutes.

> Remark: For admissible actions in much more narrow sense one also has a forgetful functor $M\to V$ and one requires that the action below is in the fact factoring through that functor. In other words, the tensor product of $V$ lifts to an action of $M$ in $Cat/V$.

* [[Z. Škoda]], _Some equivariant constructions in noncommutative algebraic geometry_, Georgian Mathematical Journal __16__ (2009), No. 1, 183--202, [arXiv:0811.4770](http://arxiv.org/abs/0811.4770).

[[!redirects geometrically admissible actions]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The term _discrete $\infty$-groupoid_ or _bare $\infty$-groupoid_ or _geometrically [[discrete object|discrete]] [[geometric homotopy type]]_ is essentially synonymous to just _[[∞-groupoid]]_ or just _[[homotopy type]]_. It is used for emphasis in contexts where one considers $\infty$-groupoids with extra [[geometric homotopy type|geometric structure]] (e.g. [[cohesive (∞,1)-topos|cohesive]] [[stuff, structure, property|structure]]) to indicate that this extra structure is being disregarded, or rather that the special case of [[discrete space|discrete]] such structure is considered.

## Definition

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

The [[terminal object in an (∞,1)-category|terminal]] [[(∞,1)-sheaf (∞,1)-topos]] [[∞Grpd]] is trivially a [[cohesive (∞,1)-topos]], where each of the defining four [[(∞,1)-functor]]s $(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \infty Grpd \to \infty Grpd$ is an [[equivalence of (∞,1)-categories]].

=--

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

In the context of [[cohesive (∞,1)-topos]]es we say that [[∞Grpd]] defines **discrete cohesion** and refer to its objects as **discrete $\infty$-groupoids**.

More generally, given any other [[cohesive (∞,1)-topos]]

$$
  (\Pi \dashv Disc \dashv \Gamma \dashv codisc) : \mathbf{H}  \stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}
  \infty Grpd
$$

the [[inverse image]] $Disc$ of the [[global section]] functor is a [[full and faithful (∞,1)-functor]] and hence embeds [[∞Grpd]] as a full [[sub-(∞,1)-category]] of $\mathbf{H}$. A general object in $\mathbf{H}$ is a _cohesive $\infty$-groupoid_ . We say $X \in \mathbf{H}$ is a **discrete $\infty$-groupoid** if it is in the [[image]] of $Disc$.

=--

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

This generalizes the traditional use of the terms [[discrete space]] and [[discrete group]]:

* a [[discrete space]] is equivalently a 0-[[truncated]] discrete $\infty$-groupoid;

* a [[discrete group]] is equivalently a 0-[[truncated]] [[group object in an (∞,1)-category|group object]] in discrete $\infty$-groupoids.

=--

## Structures in $Disc\infty Grpd$

We discuss now some of the general abstract [[cohesive (∞,1)-topos -- structures|structures in a cohesive (∞,1)-topos]] realized in discrete $\infty$-groupoids.

### Geometric homotopy and Galois theory
 {#TopAsCohesiveInfinTopos}

We discuss the general absatract notion of geometric homotopy in cohesive $(\infty,1)$-toposes (see here) in the context of discrete cohesion.

By the [[homotopy hypothesis]]-theorem the [[(∞,1)-topos]]es [[Top]] and [[∞Grpd]] are [[equivalence of (∞,1)-categories|equivalent]], hence indistinguishable by general abstract constructions in [[(∞,1)-topos theory]]. However, in practice it can be useful to distinguish them as two different [[presentable (∞,1)-category|presentations]] for an equivalence class of $(\infty,1)$-toposes.

For that purposes consider the following

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

Define the [[quasi-categories]]

$$
  Top := N(Top_{Quillen})^\circ
$$

and

$$
  \infty Grpd := N(sSet_{Quillen})^\circ
  \,,
$$

where on the right we have the standard [[model structure on topological spaces]] $Top_{Quillen}$ and the standard [[model structure on simplicial sets]] $sSet_{Quillen}$  and $N((-)^\circ)$ denotes the [[homotopy coherent nerve]] of the [[simplicial category]] given by the full [[sSet]]-subcategory of these [[simplicial model categories]] on fibrant-cofibrant objects.

=--

For

$$
  ({|-| \dashv Sing}) : Top_{Quillen} \stackrel{\overset{{|-|}}{\leftarrow}}{\underset{Sing}{\to}} sSet_{Quillen}
$$

the standard [[Quillen equivalence]] of the [[homotopy hypothesis]]-theorem given by the [[singular simplicial complex]]-functor and [[geometric realization]], write

$$
  (\mathbb{L} {|-|} \dashv \mathbb{R}Sing) :
  Top
    \stackrel{\overset{\mathbb{L}{|-|}}{\leftarrow}}{\underset{\mathbb{R}Sing}{\to}}
  \infty Grpd
$$

for the corresponding [[derived functor]]s (the image under the [[homotopy coherent nerve]] of the restriction of ${|-|}$ and $Sing$ to fibrant-cofibrant objects followed by functorial fibrant-cofibrant replacement) that constitute a pair of [[adjoint (∞,1)-functor]]s modeled as morphisms of [[quasi-categories]].

Since this is an [[equivalence of (∞,1)-categories]] either functor serves as the [[left adjoint]] and [[right adjoint]] and so we have

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

[[Top]] is exhibited a [[cohesive (∞,1)-topos]] over [[∞Grpd]] by setting

$$
  (\Pi \dashv Disc \dashv \Gamma \dashv coDisc)
  :
  Top
    \stackrel{\overset{\mathbb{R}Sing}{\to}}{\stackrel{\overset{\mathbb{L}{|-|}}{\leftarrow}}{\stackrel{\overset{\mathbb{R}Sing}{\to}}{\underset{\mathbb{L}{|-|}}{\leftarrow}}}}
  \infty Grpd
  \,.
  \,.
$$

In particular a presentation of the intrinsic [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] is given by the familiar [[singular simplicial complex]] construction

$$
  \Pi(X) \simeq \mathbb{R} Sing X
  \,.
$$

=--

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

While degenerate, it is sometimes useful to make this example of a [[cohesive (∞,1)-topos]] explicit. For instance it allows to think of simplicial models for topological fibrations in terms of topological [[higher parallel transport]]. Some remarks on this are in Flat higher parallel transport in Top.

=--

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

Notice that the [[topology]] that enters the explicit construction of the objects in [[Top]] here does _not_ show up as [[cohesive (∞,1)-topos|cohesive structure]]. A [[topological space]] here is a model for a _discrete_ $\infty$-groupoid, the [[topology]] only serves to allow the construction of $Sing X$.
For discussion of $\infty$-groupoids equipped with genuine _topological cohesion_ see [[Euclidean-topological ∞-groupoid]].

=--

### Cohomology and principal $\infty$-bundles

We discuss the general abstract notion of cohomology and principal $\infty$-bundles a
in cohesive $\infty$-toposes (see here) in the context of  discrete cohesion.

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

  Write $\mathrm{sGrp} = \mathrm{Grp}(\mathrm{sSet})$
 for the category of [[simplicial group]]s.

=--

A classical reference is section 17 of [May](#May).

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

The category $\mathrm{sGrpd}$ inherits a [[model category]] structure [[transferred model structure|transferred]] along the
forgetful functor $F : \mathrm{sGrp} \to \mathrm{sSet}$.

The category $\mathrm{sSet}_0 \hookrightarrow \mathrm{sSet}$ of reduced simplicial sets (simplicial sets with a single vertex) carries a model category structure whose weak equivalences and cofibrations are those of $\mathrm{sSet}_{\mathrm{Quillen}}$.

There is a [[Quillen equivalence]]
$$
  (G \dashv \bar W)
  :
  sGrp
   \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}}
  sSet_{0}
$$

which [[presentable (infinity,1)-category|presents]] the abstract [[looping and delooping]] equivalence of $\infty$-categories

 $$
   (\Omega \dashv \mathbf{B})
   :
   \infty Grpd
    \stackrel{\overset{\Omega}{\leftarrow}}{\underset{B}{\to}}
   \infty Grpd_{connected}
   \,,
$$

=--

The model structures and the Quillen equivalence are classical, discussed in  ([GoerssJardine, section V](#GoerssJardine))

This means on abstract grounds that for $G$ a [[simplicial group]], $\bar W G \in \mathrm{sSet}$ is a model of the
[[classifying space|classifying]] [[delooping]] object $\mathbf{B}G$ for discrte $G$-[[principal ∞-bundles]].
The following statements assert that these principal $\infty$-bundles themselves can  be modeled as ordinary [[simplicial principal bundle]]s

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

 For $G$ a [[simplicial group]] and $\bar W G$  the model for $\mathbf{B}G$ given by the above proposition, write

$$
  W G \to \bar W G
$$

for the simplicial [[decalage]] on $\bar W G$.

=--

This characterization of the object going by the classical name $W G$ is made fairly  explicit in ([Duskin, p. 85](#Duskin)).

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

The morphism $W G \to \bar W G$ is a Kan fibration resolution of the point inclusion ${*} \to \bar W G$.

=--

This follows directly from the characterization of $W G \to \bar W G$ by [[decalage]].
Pieces of this statement appear in ([May](#May)): lemma 18.2 there gives the fibration property, prop. 21.5 the contractibility of $W G$.

+-- {: .num_cor}
###### Corollary
 For $G$ a [[simplicial group]], the sequence of simplicial sets

$$
  G \to W G \to \bar W G
$$

is a presentation of the [[fiber sequence]]

$$
  G \to * \to \mathbf{B}G
  \,.
$$

Hence $W G \to \bar W G$ is a model for the universal $G$-principal discrete $\infty$-bundle (see [[universal principal ∞-bundle]]):

every $G$-principal discrete $\infty$-bundle $P \to X$ in $\infty \mathrm{Grpd}$, which  by definition is a [[homotopy fiber]]

$$
  \array{
    P &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    X &\to& \mathbf{B}G
  }
$$

in [[?Gpd]], is presented in the standard [[model structure on simplicial sets]] by the ordinary [[pullback]]

$$
  \array{
    P &\to& W G
    \\
    \downarrow && \downarrow
    \\
    X &\to& \bar W G
  }
  \,.
$$

=--

The explicit statement that the sequence $G \to W G \to \bar W G$ is a model for the looping fiber sequence appears on p. 239 of _[[Crossed Menagerie]]_ .
The universality of $W G \to \bar W G$ for $G$-principal simplicial bundles is the  topic of section 21 in ([May](#May)), where however it is not made explicit that the "[[twisted cartesian product]]s" considered there are precisely the models for the
pullbacks as above. This is made explicit on page 148 of _[[Crossed Menagerie]]_.

In _[[Euclidean-topological ∞-groupoid]]_ we discuss how this model of discrete principal $\infty$-bundles
by simplicial principal bundles lifts to a model of topological principal $\infty$-bundles
by simplicial topological bundles principal over
[[simplicial topological group]]s.

## Related concepts

* [[discrete space]]

* [[discrete group]]

* [[discrete groupoid]]

[[!include geometries of physics -- table]]

## References

[[simplicial group|Simplicial groups]] and [[simplicial principal bundles]] are discussed in

* [[Peter May]], _Simplicial Objects in Algebraic Topology_ ([djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu)).
 {#May}

and section V of

* [[Paul Goerss]] and [[Rick Jardine]], 1999, _Simplicial Homotopy Theory_, number 174 in Progress in Mathematics, Birkhauser. ([ps](http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html))
 {#GoerssJardine}

The relation of $W G \to \bar W G$ to [[decalage]] is mentioned on p. 85 of

* [[John Duskin]], _Simplicial methods and the interpretation of "triple" cohomology_, number  163 in Mem. Amer. Math. Soc., 3, Amer. Math. Soc. (1975)
 {#Duskin}

Discrete cohesion is the topic of section 3.1 of

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

where much of the above material is taken from.

[[!redirects geometrically discrete infinity-groupoids]]

[[!redirects geometrically discrete ∞-groupoid]]
[[!redirects geometrically discrete ∞-groupoids]]

[[!redirects discrete infinity-groupoid]]

[[!redirects discrete ∞-groupoid]]

[[!redirects discrete ∞-groupoids]]
[[!redirects discrete infinity-groupoids]]

[[!redirects Disc∞Grpd]]

[[!redirects discrete ∞-group]]
[[!redirects discrete ∞-groups]]

[[!redirects discrete infinity-group]]
[[!redirects discrete infinity-groups]]

[[!redirects geometrically discrete homotopy type]]
[[!redirects geometrically discrete homotopy types]]

[[!redirects Discrete∞Groupoid]]
[[!redirects Discrete∞Groupoids]]
[[!redirects Discrete∞Grpd]]


geometries of physics -- table

**[[geometry of physics|geometries of physics]]**

| $\phantom{A}$([[higher geometry|higher]]) [[geometry]]$\phantom{A}$ | $\phantom{A}$[[site]]$\phantom{A}$ | $\phantom{A}$[[sheaf topos]]$\phantom{A}$ | $\phantom{A}$[[(∞,1)-sheaf (∞,1)-topos|∞-sheaf ∞-topos]]$\phantom{A}$ |
|---|--------|-------|-------|
| $\phantom{A}$[[discrete geometry]]$\phantom{A}$ | $\phantom{A}$[[terminal category|Point]]$\phantom{A}$ | $\phantom{A}$[[Set]]$\phantom{A}$ | $\phantom{A}$[[Discrete∞Groupoid|Discrete∞Grpd]]$\phantom{A}$ |
| $\phantom{A}$[[differential geometry]]$\phantom{A}$ | $\phantom{A}$[[CartSp]]$\phantom{A}$ | $\phantom{A}$[[SmoothSet]]$\phantom{A}$ | $\phantom{A}$[[Smooth∞Groupoid|Smooth∞Grpd]]$\phantom{A}$ |
| $\phantom{A}$[[synthetic differential geometry|formal geometry]]$\phantom{A}$ | $\phantom{A}$[[FormalCartSp]]$\phantom{A}$ | $\phantom{A}$[[FormalSmoothSet]]$\phantom{A}$ | $\phantom{A}$[[FormalSmooth∞Groupoid|FormalSmooth∞Grpd]]$\phantom{A}$ |
| $\phantom{A}$[[supergeometry]]$\phantom{A}$ | $\phantom{A}$[[SuperFormalCartSp]]$\phantom{A}$ | $\phantom{A}$[[SuperFormalSmoothSet]]$\phantom{A}$ | $\phantom{A}$[[SuperFormalSmooth∞Groupoid|SuperFormalSmooth∞Grpd]]$\phantom{A}$ |


geometrization conjecture

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Every closed [[3-manifold]] can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure
(...)

The proof of the geometrization conjecture implies that of

* the [[Poincaré conjecture]]

* the [[elliptization conjecture]]

## Related entries

...

## References

The conjecture was stated by [[William Thurston]] in 1982.
The [[Ricci flow]]-strategy for the proof was devised by [[Richard Hamilton]] and the remaining technical problem of controlling the pinching was finally solved by [[Grigori Perelman]].

* Wikipedia, _[Geometrization conjecture](http://en.wikipedia.org/wiki/Geometrization_conjecture#The_conjecture)_

* {#BessiersEtAl} Laurent Bessieres, Gerard Besson, Michel Boileau, Sylvain Maillot, Joan Porti, _Geometrisation of 3-manifolds_ ([pdf](http://www-fourier.ujf-grenoble.fr/~besson/book.pdf))

Discussion for 3-dimensional [[orbifolds]] is in

* Michel Boileau, Bernhard Leeb, Joan Porti, _Geometrization of 3-Dimensional Orbifolds_, Annals of Mathematics Second Series, Vol. 162, No. 1 (2005), pp. 195-290 ([jstor:3597373](https://www.jstor.org/stable/3597373))

* Bruce Kleiner, [[John Lott]], _Geometrization of Three-Dimensional Orbifolds via Ricci Flow_ ([arXiv:1101.3733](https://arxiv.org/abs/1101.3733))

[[!redirects geometrization conjecture]]
[[!redirects geometrisation conjecture]]


geometrodynamics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

> This disappearance and regeneration of space in time and of time in space is motion;– a becoming, which, however, is itself just as much immediately the identically existing unity of both, or matter. ([PdN§261](Science+of+Logic#PN261))

### General

The term _geometrodynamics_ has been coined, or at least promoted, by [[John Wheeler]] as a description for the dynamics of [[gravity]] according to [[general relativity]]. Since the field of [[gravity]] is entirely encoded in the [[spacetime]] [[pseudo-Riemannian metric|metric geometry]], there is some justification for thinking of the dynamics of the gravitational field as being "the dynamics of geometry" itself. Hence the term.

More specifically the term _geometrodynamics_ is associated with Wheeler's speculation that all of physics might fundamentally be described by configurations of gravity coupled to other fields, notably the field of [[electromagnetism]], but without any matter: one can see that certain [[spacetimes]] without any matter content but with certain nontrivial [[topology]] may locally effectively look as if they contained massive and possibly charged bodies.

For instance on a spacetime that is obtained from two copies of [[Minkowski space]] connected by a thin (as measured by the metric) throat -- often called a _wormhole_ -- an electric field configuration whose field lines all converge to the throat's mouth in one of the two Minkowski sheets, pass through the throat and then emerge concentrically in the other Minkowski sheet may have no divergence anywhere, hence according to [[electromagnetism|Maxwell's equations]] have no charge sources anywhere, and still effectively look to an observer constrained to one of the two Minkowski sheets but relatively far away from the throat's mouth as if they were the field lines of a positively or negatively charged point source located where the mouth of the throat is.

These kinds of ideas Wheeler liked to describe by phrases such as _charge without charge_ and *mass without mass*. Later these basic ideas have continued a life notably in the context of attempts to describe gravity by a [[topological quantum field theory]], for instance in approaches to describe [[gravity as a BF-theory]].

### Similarity to KK-compactifications of pure super-gravity

A rather similar perspective arises in [[Kaluza-Klein compactification]] of systems of pure [[gravity]] to [[effective field theories]]. In the case of [[supergravity]] these reductions give rise to [[effective field theories]] which contain not just extra [[force]] field (such as the [[electromagnetic field]] in the original KK-mechanism) but also [[fermion|fermionic]] [[matter]].
For instance models such as the [[G2-MSSM]] consist entirely of pure [[11d supergravity]] [[KK-compactification|KK-compactified]] on [[G2-manifold]] [[fiber bundles]] and the resulting [[effective field theory]] contains all of the [[standard model of particle physics]].

## Geometric description of physical phenomena

### Mass without Mass

...

###Momenta without Momenta

...

### Charge without Charge

...

### Spin without Spin

...

## Related concepts

* [[Wheeler superspace]]

## References

Early speculation of similar nature is due to

* [[William Clifford]], *[[On the Space-Theory of Matter]]*, Proceedings of the Cambridge Philosophical Society **2** (1864-1876 - Printed 1876) 157-158 reprinted in: *The Concepts of Space and Time* Boston Studies in the Philosophy of Science **22** Springer (1976) [[doi:10.1007/978-94-010-1727-5_50](https://doi.org/10.1007/978-94-010-1727-5_50)]

(long before the formulation of [[Einstein gravity]]/[[general relativity]]).

The term "geometrodynamics" is due to:

* [[Charles W. Misner]], [[John A. Wheeler]], *Classical physics as geometry*, Annals of Physics **2** 6 (1957) 525-603 \[doi:10.1016/0003-4916(57)90049-0\]

Review:

* [[Domenico Giulini]], *Matter from Space*, in: *Beyond Einstein* Einstein Studies **14** Birkhäuser (2018) [[doi:10.1007/978-1-4939-7708-6_12](https://doi.org/10.1007/978-1-4939-7708-6_12), [arXiv:0910.2574](https://arxiv.org/abs/0910.2574)]

See also:

* Wikipedia, *[Geometrodynamics](http://en.wikipedia.org/wiki/Geometrodynamics)*

* The Algebraic Rainich Conditions [pdf](http://www.ptep-online.com/index_files/2007/PP-10-08.PDF)

* Return of the Wheeler wormhole [pdf](http://www.gravityresearchfoundation.org/pdf/awarded/1990/vissar.pdf)

* [Spin 1/2 from gravity](http://prl.aps.org/abstract/PRL/v44/i17/p1100_1),


geometry

**geometry** $\leftarrow$ [[Isbell duality]]  $\rightarrow$ [[algebra]]

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

> "Geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring." [[[Philosophiæ naturalis principia mathematica|Newton (1687)]], preface]

In its historical and etymological origins, "geometry" is the science of *measuring the earth* and here specifically of measuring *[[distances]]* and *[[angles]]*, first in the guise of [[Euclidean geometry]] of the ancients and then eventually as the *[[differential geometry of curves and surfaces]]* due to [[Carl Friedrich Gauss|Gauss]] and others.

But in modern [[mathematics|mathematical]] jargon, the term *geometry* is used in much greater generality for the study of *[[spaces]]* equipped with [[extra structure|extra]] "geometrical" [[structure]] of a large variety of sorts, foremost in the guise of *[[Riemannian geometry]]* but subsuming also a wealth of variant notions of "geometries" which the ancient would not have recognized as such, for instance *[[symplectic geometry]]* and other ([[torsion of a G-structure|torsion]]-free) *[[G-structure|$G$-structured]]* [[differentiable manifolds]] (*[[differential geometry|differential]] [[Cartan geometry]]*), or ([[higher topos theory|higher]]) [[topos]]-theoretic notions (cf. "[[geometric logic]]") of ([[higher geometry|higher]]) *[[functorial geometry]]* famously including *[[algebraic geometry]]* or *[[supergeometry]]* but also more exotic variants such as *[[arithmetic geometry]]* or *[[absolute geometry]]*. Finally, taking the [[duality between algebra and geometry]] to the extreme yields notions of *[[noncommutative geometry]]* and/or *[[derived geometry]]* whose underlying "spaces" are only indirectly conceived as whatever it is that given ([[higher algebra|higher]]) [[algebras]] would see (measure!) if they are interpreted as [[algebras of functions]] on a would-be space --- which on the one hand is far remote from what Newton must have imagined geometry to be about, but on the other hand is quite in the spirit of geometry being about (maybe not space as such but) *[[measurement|measuring]]* space.

Beware that bare *[[topology]]* is sometimes regarded as a rudimentary kind of geometry (as reflected for instance in the common terminology *[[geometric realization]]* for an operation that really is *[[topological realization]]*), but more often than not a "geometric space" is meant to be a *[[topological space]]* equipped with [[extra structure|extra]] geometrical [[structure]], of sorts.

Even a plain [[differentiable manifold]] as considered in [[differential geometry]] is often not quite regarded as *geometric* in itself unless equipped with further structure, as reflected for instance in the terminology *[[topological field theory]]* for those [[functorial field theories]] which depend on [[diffeomorphism]]-structure of [[spacetime]] but not on further (notably [[pseudo-Riemannian manifold|pseudo]]-[[Riemannian geometry|Riemannian]]) geometric structure.

## Related entries

* [[geometric type theory]], [[geometric homotopy type theory]]

* [[synthetic geometry]]

* [[differential geometry]], [[differential topology]], [[Diff]], [[cobordism]]

* [[differential form]], [[tangent space]], [[tangent bundle]], [[cotangent bundle]], [[cotangent complex]]

* [[symplectic geometry]], [[symplectic manifold]], [[Poisson manifold]], [[Lagrangian submanifold]]

* [[multisymplectic geometry]], [[n-symplectic manifold]], [[foliation]], [[integrable distribution]], [[G-structure]]
* [[fibre bundle]], [[principal bundle]], [[noncommutative principal bundle]], [[vector bundle]]
* [[connection]] (and links therein), [[connection on a bundle]]
* [[Morse function]], [[Morse lemma]], [[Morse theory]], [[perfect Morse function]]
* [[Casson invariant]], [[Donaldson-Thomas invariant]], [[Kähler manifold]], [[mirror symmetry]]
* [[metric space]], [[convex set]], [[Riemannian manifold]], [[geodesic flow]], [[geodesic convexity]], [[star-shaped]]
* [[differentiable stack]], [[orbifold]], [[stratifold]], [[orientifold]]

* [[algebraic geometry]], [[analytic geometry]],
[[arithmetic geometry]], [[GAGA]], book entry [[EGA]]

* [[scheme]], [[quasicompact]], [[noetherian scheme]], [[reduced scheme]], [[integral scheme]]

* [[open subscheme]], [[closed subscheme]], [[configuration scheme]]

* [[formal scheme]], [[formal group scheme]], [[formal group law]], [[algebraic group]]

* [[spectrum (geometry)]]

* [[algebraic variety]], [[normal variety]], [[hyperplane
line bundle]]

* [[derived algebraic geometry]], [[derived smooth manifold]]
* [[noncommutative geometry]], [[derived noncommutative algebraic geometry]]
* [[noncommutative algebraic geometry]], [[noncommutative scheme]], [[noncommutative thin scheme]]
* [[hyperplane section theorem]], [[BBDG decomposition theorem]]
* [[rational map]], [[rational variety]], [[unirational variety]], [[birational map]], [[birational geometry]], [[image of a rational map]]
* [[algebraic space]], [[algebraic stack]], [[topological stack]]
* [[quasicoherent sheaf]], [[coherent sheaf]], [[relative scheme]]
* [[smooth scheme]], [[smooth morphism of schemes]], [[etale morphism]], [[formally smooth morphism]]
* [[pure motive]], [[Voevodsky motive]]
* [[D-module]], [[local system]], [[regular differential operator]], [[holonomic D-module]]
* [[D-geometry]], [[Grothendieck connection]], [[crystal]], [[diffiety]]
* [[flag variety]], [[geometric quantization]], [[coherent state]], [[orbit]], [[coadjoint orbit]]
* [[Lie group]], [[Lie groupoid]], [[Lie algebroid]], [[Courant algebroid]], [[Atiyah Lie groupoid]]
* [[Lie theory]], [[Lie's three theorems]], [[Erlangen program]]
* [[ringed space]], [[ringed site]], [[ringed topos]], [[lined topos]]
* [[sheaf of ideals]], [[defining sheaf]], [[conormal sheaf]], [[conormal bundle]], [[subscheme of an Abelian category]]
* [[synthetic differential geometry]], [[infinitesimal object]], [[smooth topos]], [[Kock-Lawvere axiom]], [[infinitesimal singular simplicial complex]], [[differential forms in synthetic differential geometry]]

  * book entry [[Models for Smooth Infinitesimal Analysis]]
* [[generalized smooth space]]:

  * [[Froelicher space]], [[diffeological space]], [[smooth space]]
* [[space and quantity]]
* [[Euclidean geometry]], [[trigonometry]]

* [[D-brane geometry]]

There are many entries on [[sheaf]], [[stack]], [[site]], [[locale]] and [[topos]] theory including

* [[germ]], [[etale space]], [[soft sheaf]], [[flabby sheaf]], [[fine sheaf]], [[Grothendieck topos]], [[Grothendieck topology]], [[(infinity,1)-topos]], [[smooth topos]], [[(infinity,1)-category of (infinity,1)-sheaves]], [[cover]], [[hypercover]], [[local object]], [[local isomorphism]], [[coverage]], [[local epimorphism]], [[hypercompletion]], [[sieve]], [[Verdier site]], [[Grothendieck fibration]], [[Pierce spectrum]], [[Stone duality]], [[point of a topos]], [[geometric morphism]]...

and pages on various [[cohomology|cohomologies]], including [[sheaf cohomology]], [[nonabelian cohomology]], [[differential cohomology]], [[Deligne cohomology]], [[etale cohomology]], [[equivariant cohomology]], [[Bredon cohomology]] and their cocycle classes including [[torsors]], [[gerbes]], [[principal 2-bundles]] as well as the related picture of the [[descent]] theory (cf. [[oriental]], [[descent for simplicial presheaves]]...). A modern systematic theory of cohomology and descent can be done using the language of $(\infinity,1)$-categories and abstract [[homotopy theory]], say via Quillen [[model categories]] (e.g. of [[simplicial presheaves]]).

## References

* Wikipedia *[History of geometry](https://en.wikipedia.org/wiki/History_of_geometry)*

[[!include Isbell duality - table]]

[[!redirects geometric]]


+-- {: .rightHandSide}
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###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _geometry_ $\mathcal{G}$ is an [[(∞,1)-category]] equipped in a compatible way with

1. the structure of an [[(∞,1)-site]];

1. the structure of an [[essentially algebraic (∞,1)-theory]].

The [[object]]s of $\mathcal{G}$ are to be thought of as test-[[space]]s with certain [[higher geometry]] structure and the [[morphism]]s as [[homomorphism]]s preserving that geometric structure.

These two structures gives rise to

1. The [[big topos|big]] [[(∞,1)-topos]]  $Sh(\mathcal{G})$ of [[(∞,1)-sheaves]] on $\mathcal{G}$. Its objects are generalized [[space]]s given by rules

   $X : \mathcal{G}^{op} \to $ [[∞Grpd]]

   for how to map test spaces into them.

1. The [[∞-algebra over an (∞,1)-algebraic theory|(∞,1)-algebras]] over $\mathcal{G}$ in some [[little topos]] $\mathcal{X}$, given by rules

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

   that send each obect $U\in \mathcal{G}$ to a $U$-valued [[structure sheaf]].

Using the additional structure of a site on $\mathcal{G}$ allows to identify those structure sheaves $\mathcal{O}$ that are **local** in that they respect coverings. This constitutes a generalized notion of [[locally ringed topos]]es called $\mathcal{G}$-[[structured (∞,1)-topos]]es. Equivalently these local structure sheaves are given by [[(∞,1)-geometric morphism]]s
$ \mathcal{X} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\to} \mathbf{H} = Sh(\mathcal{G}) $ to the [[big topos]] over $\mathcal{G}$.

## Definition

### Geometry {#Geometry}

+-- {: .un_remark}
###### Summary

A **geometry** on an [[(∞,1)-category]] $\mathcal{G}$ is a [[Grothendieck topology]] on $\mathcal{G}$ together with

* the extra structure given the information of which covering morphisms are to be thought of as [[local homeomorphism]]s

* the extra property that it has all finite [[limit]]s.

If only all finite [[product]]s exist we speak of a pre-geometry. Every pregeometry $\mathcal{T}$ extends uniquely $\mathcal{T} \hookrightarrow \mathcal{G}$ to an _enveloping geometry_ $\mathcal{G}$.

When the objects of the geometry $\mathcal{G}$ are thought of as test spaces (affine schemes), the objects of the pregeometry $\mathcal{T} \hookrightarrow \mathcal{G}$ are to be thought of as the [[affine space]]s. This distinction is used to encode [[smooth map|smoothness]] of maps between test spaces: a morphism in $\mathcal{G}$ is smooth if it locally factors through admissible maps between objects in $\mathcal{T}$.

=--

+-- {: .un_defn}
###### Definition (admissibility structure, [[Structured Spaces|StSp]] 1.2.1)

An **admissibility structure** on an [[(∞,1)-category]] $\mathcal{G}$ is a [[Grothendieck topology]] on $\mathcal{G}$ that is generated from its intersection with a subcategory $\mathcal{G}^{ad} \subset \mathcal{G}$ whose morphisms -- called the  **admissible morphisms** have the following properties

* admissible morphisms are stable under [[(∞,1)-pullback]];

* admissible morphisms satisfy "left cancellability", meaning that whenever in

  $$
    \array{
      && Y
      \\
      & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}}
      \\
      X &&\stackrel{h}{\to}&& Z
    }
  $$

  $g$ and $h$ are admissible, then so is $f$.

* admissible morphisms are closed under [[retracts]].

Equivalently, this is a [[Grothendieck topology]] on $\mathcal{G}$ which is _generated_ from admissible morphisms.

=--

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

As will become clear when looking at examples, the notion of
admissible morphisms models the idea of maps between test spaces that behave like _open injections_ or, more generally, as _local homeomorphisms_ .

=--

+-- {: .un_defn}
###### Definition ( [[Structured Spaces]] 1.2.5 )

A **geometry (for $(\infty,1)$-toposes)** is

* An [[(∞,1)-category]] $\mathcal{G}$ that

  * is [[essentially small category|essentially small]];

  * has all finite [[limit]]s

  * is [[idempotent complete category|idempotent complete]].

* equipped with an admissibility structure.

=--

+-- {: .un_defn}
###### Definition (discrete geometry, [[Structured Spaces|StSp]], 1.2.10)

The **discrete geometry** $\mathcal{G}^0$ on $\mathcal{G}$ is given by

* the admissible morphisms in $\mathcal{G}$ are precisely the equivalences

* the [[Grothendieck topology]] on $\mathcal{G}$ is trivial: a [[sieve]] is covering only if it is maximal.

Every [[small (∞,1)-category]] $C$ becomes a
geometry by regarding it as a discrete geometry in the above way.

=--

### Pregeometry {#Pregeometry}

+-- {: .un_defn}
###### Definition (pregeometry, [[Structured Spaces|StSp]], 3.1.1)

A **pregeometry** (for structured [[(∞,1)-topos]]es) is

* an [[(∞,1)-category]] $\mathcal{T}$;

* equipped with an admissibility structure (homotopical topology)

such that

* $\mathcal{T}$ has all [[product]]s.

=--

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

So a geometry differs from a pregeometry in that it is [[idempotent complete category|idempotent complete]] and closed not
only under [[products]] but under all finite [[limit]]s.

Various concepts for geometries have immediate analogues for
pregeometries.

=--

#### Smooth morphisms

+-- {: .un_defn}
###### Definition (smooth morphism, [[Structured Spaces]], 3.1.7)

A morphism $f : X \to S$ in a pregeometry $\mathcal{T}$ is called **smooth**
if it is _locally stably admissible_ in that there exists a  cover $\{u_i : U_i \to X\}$ (meaning: generators of a covering [[sieve]])
of $X$ by admissible morphisms, such that on $U_i$ the morphism $f$
factors admissibly through some $S \times V_i$ in that there is a commuting
diagram

$$
  \array{
    U_i &\stackrel{u_i}{\to}& X
    \\
    \downarrow && \downarrow
    \\
    S \times V_i &\stackrel{p_1}{\to}& S
  }
  \,.
$$

=--

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

To interpret this, recall that we think of
admissible morphisms as injections of open subsets.

=--

+-- {: .un_prop}
###### Proposition ([[Structured Spaces]], 3.1.8)

* Smooth morphisms are stable under [[pullback]].

* pregeometric $\mathcal{T}$-structures $\mathcal{O} : \mathcal{T} \to \mathcal{X}$  preserve pullbacks of smooth morphisms.

=--

### Structured $(\infty,1)$-topos {#StructTop}

+-- {: .un_defn}
###### Definition (geometry structure on an $(\infty,1)$-topos, [[Structured Spaces|StSp]], 1.2.8)

For $\mathcal{G}$ a geometry, and $T \simeq Sh_\infty(S)$ an [[(∞,1)-topos]],
a **$\mathcal{G}$-structure on the $(\infty,1)$-topos $T$** making it a
[[structured (∞,1)-topos]] is a [[(∞,1)-functor]]

$$
  C(-) : \mathcal{G} \to T
$$

such that

* $C(-)$ is left [[exact functor|exact]] (preserves finite [[limit]]s);

* $C(-)$ satisfies [[codescent]] (the dual notion of [[descent]]):
  for $\pi : (V = \coprod_i V_i) \to W$ any
  [[cover]] by admissible morphisms in $G$, the induced morphism
  $$
    C(\pi) : C(V) \to C(W)
  $$
  is an [[effective epimorphism]] in $T$, i.e. its [[?ech nerve]]
  is a [[simplicial resolution]] of $C(W)$:
  $$
    \mathop{Čech}(C(\pi)) \stackrel{\simeq}{\to} C(W)
    \,.
  $$

=--

+-- {: .un_defn}
###### Definition (pregeometry structure on an $(\infty,1)$-topos, [[Structured Spaces|StrSp]] 3.1.4)

Let $\mathcal{T}$ be a pregeometry and $\mathcal{X}$ an [[(∞,1)-topos]].

A **$\mathcal{T}$-structure** on $\mathcal{X}$ is an [[(∞,1)-functor]]
$\mathcal{O} : \mathcal{T} \to \mathcal{X}$ such that

* $\mathcal{O}$ preserves finite [[product]]s.

* $\mathcal{O}$ preserves [[pullback]]s of  admissible morphism
  in that for every [[pullback]] diagram

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

  in $\mathcal{T}$ with $f$ admissible, the image

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

  is again a [[pullback]].

* $\mathcal{O}$ respects covers by admissible morphisms in that for every
  covering sieve $\{f_i : U_i \to X\}$ in $\mathcal{T}$ by admissible $f_i$
  the induced map $\coprod_i \mathcal{O}(U_i) \to \mathcal{O}(X)$
  is an [[effective epimorphism]] in $\mathcal{X}$.

=--

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

The first clause says that $\mathcal{O} : \mathcal{T} \to \mathcal{X}$ is in particular an $\infty$-algebra over the (multi-sorted) [[(∞,1)-algebraic theory]] $\mathcal{T}$.

The other two clauses encode that this $\infty$-algebra $\mathcal{O}$ indeed behaves like a _function algebra_ .

=--

+-- {: .un_defn}
###### Definition (geometric envelope, [[Structured Spaces|StrSp]] 3.4.1)

...the universal geometry extending a pregeometry...

=--

+-- {: .un_prop}
###### Proposition ([[Structured Spaces|StSp]]  3.4.5)

Let $\mathcal{T}$ be a pregeometry and $f : \mathcal{T} \to \mathcal{G}$
a morphism that exhibits the geometry $\mathcal{G}$ as a geometric envelope
of $\mathcal{T}$. Then for every [[(∞,1)-topos]] $\mathcal{X}$ precomposition with $f$ induces an equivalence of
[[(infinity,1)-category|(∞,1)-categories]] of $\mathcal{T}$- and
$\mathcal{G}$-structures on $\mathcal{X}$:

$$
  Str_{\mathcal{G}}(\mathcal{X}) \stackrel{\simeq}{\to}
  Str_{\mathcal{T}}(\mathcal{X})
  \,,\;\;\;\;
  Str_{\mathcal{G}}(\mathcal{X})^{loc} \stackrel{\simeq}{\to}
  Str_{\mathcal{T}}(\mathcal{X})^{loc}
$$

=--

## Examples

### Étale geometry

If we regard the ordinary [[étale site]] as a pregeometry $\mathcal{T}_{et}$, then its geometric envelope $\mathcal{G}_{et}$ is the [[étale (∞,1)-site]]. See [[étale (∞,1)-site#AsDerivedGeometry|derived étale geometry]] for the precise statement.

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

The [[n-localic (∞,1)-topos|1-localic]] $\mathcal{G}_{et}$-[[generalized scheme]]s  are precisely [[Deligne-Mumford stack]]s (without the [[separated (infinity,1)-presheaf|separation axiom]]).

=--

See [[Deligne-Mumford stack]] for details.

### Smooth geometry

There should be a geometry $\mathcal{G}$ such that $\mathcal{G}$-[[generalized scheme]]s are precisely [[derived smooth manifold]]s.

## Related concepts

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 general theory is developed in

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

The definition of a **geometry** $\mathcal{G}$ is def. 1.2.5.

A $\mathcal{G}$-structure on an [[(∞,1)-topos]] is in def. 1.2.8.

The notion of $\mathcal{G}$-spectrum -- which are [[(∞,1)-topos]]es -- is the subject of section 2.1 .

The inclusion

$$
  Spec^{\mathcal{G}} :
  \mathcal{G} \hookrightarrow Str(\mathcal{G})
$$

is definition 2.1.2.

The definition of $\mathcal{G}$-[[generalized scheme]] is definition 2.3.9, page 51.

The inclusion

$$
  Sch(\mathcal{G}) \hookrightarrow Sh_\infty(Ind(\mathcal{G}))
$$

is the topic of section 2.4, theorem 2.4.1

The special case of "smoothly structured spaces" called [[derived smooth manifold]] is  discussed in

* [[David Spivak]], _Derived smooth manifolds_ PhD thesis ([original version pdf](http://www.uoregon.edu/~dspivak/files/thesis1.pdf) [arXiv:0810.5174](http://arxiv.org/abs/0810.5174))

Apart from looking at the special case this article also contains useful introduction
and details on the general case.

In the version of this that is available on the arXiv ([arXiv](http://arxiv.org/abs/0810.5174))
the point of view is more on bi-presheaves, a useful discussion to the relation
to structured morphisms here is in section 10.1 there.

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

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


geometry of 19th century

* [19th century geometry](http://plato.stanford.edu/entries/geometry-19th) at Stanford Encyclopaedia of Philosophy
* [[Felix Klein]], _The development of mathematics in 19th century_
* Jeremy Gray, _World out of nothing. A course in history of geometry in the 19th century_, Springer Undergraduate Mathematics Series (2007)


Geometry of Interaction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What has been called _Geometry of Interaction_ ([Girard 89](#Girard89)) is a kind of [[semantics]] for [[linear logic]]/[[linear type theory]] that is however different in method from the usual [[categorical semantics]] in [[monoidal categories]]. Instead of interpreting a [[proof]] of a linear entailment $A\vdash B$ as a [[morphism]] between [[objects]] $A$ and $B$ in a [[monoidal category]] as in [[categorical semantics]], the _Geometry of Interaction_ interprets it as an [[endomorphism]] on the object $A\multimap B$. This has been named _operational semantics_ to contrast with the traditional _denotational semantics_.

That also the "operational semantics" of GoI has an interpretation in [[category theory]], though, namely in [[compact closed categories]] induced from [[traced monoidal categories]] was first suggested in ([Joyal-Street-Verity 96](#JoyalStreetVerity96)) and then developed in ([Hines 97](#Hines97),[Haghverdi 00](#Haghverdi00), [Abramsky-Haghverdi-Scott 02](#AbramskyHaghverdiScott02), [Haghverdi-Scott 05](#HaghverdiScott05)). See ([Shirahata](#Shirahata)) for a good review.

## Properties

### Relation to superoperators and quantum operations
 {#RelationToQuantumOperations}

We discuss a relation of the GoI to [[superoperators]]/[[quantum operations]] in [[quantum physics]].

According to ([Abramsky-Haghverdi-Scott 02, remark 5.8](#AbramskyHaghverdiScott02), [Haghverdi 00b, section 6](#Haghverdi00b)) all the standard and intended interpretations of the GoI take place in those [[compact closed categories]] $Int(\mathcal{C})$ free on a [[traced monoidal category]] $\mathcal{C}$ (as discussed there). In these references therefore the notation $Int(-)$ from ([Joyal-Street-Verity 96](#JoyalStreetVerity96)) is changed to $\mathcal{G}(-)$, for "Geometry of Interaction construction" (see [Abramsky-Haghverdi-Scott 02, p. 11](#AbramskyHaghverdiScott02)).

The objects of $Int(\mathcal{C})$ are pairs $(A^+, A^-)$ of objects of $\mathcal{C}$, a morphism $(A^+ , A^-) \to (B^+ , B^-)$ in $Int(\mathcal{C})$ is given by a morphism in $\mathcal{C}$ of the form

$$
  \array{
    A^+\otimes B^-
    \\
    \downarrow
    \\
    A^- \otimes B^+
  }
$$

in $\mathcal{C}$, and [[composition]] of two such morphisms $(A^+ , A^-) \to (B^+ , B^-)$ and $(B^+ , B^-) \to (C^+ , C^-)$ is given by [[trace|tracing out]] $B^+$ and $B^-$ in the evident way.

We observe now that subcategories of such $Int(\mathcal{C})$ are famous in [[quantum physics]] as "categories of [[superoperators]]" or "categories of [[quantum operations]]", formalizing [[linear maps]] on spaces of [[operators]] on a [[Hilbert spaces]] that takes [[density matrices]] (hence mixed [[quantum states]]) to density matrices ("[[completely positive maps]]").

First notice that if the [[traced monoidal category]] structure on $\mathcal{C}$ happens to be that induced by a [[compact closed category]] structure, then by compact closure the morphisms $(A^+ , A^-) \to (B^+ , B^-)$ in $Int(\mathcal{C})$ above are equivalently morphisms in $\mathcal{C}$ of the form

$$
  (A^-)^\ast \otimes A^+ \longrightarrow (B^-)^\ast \otimes B^+
$$

and if moreover one concentrates on the case where $A \coloneqq A^- \simeq A^+$ etc. then these are morphisms of the form

$$
  End(A) \longrightarrow End(B)
  \,.
$$

Under this identification the somewhat curious-looking composition in $Int(\mathcal{C})$ given by tracing in $\mathcal{C}$ becomes just plain composition in $\mathcal{C}$.

Subcategories (non-full) of $Int(\mathcal{C})$ consisting of morphisms of this form in some ambient $\mathcal{C}$ have been considered in ([Selinger 05](#Selinger05)), there denoted $CPM(\mathcal{C})$, and shown to formalize the concept of [[quantum operations]] in [[quantum physics]]. For a quick review of Selinger's construction see for instance ([Coecke-Heunen 11, section 2](#CoeckeHeunen11)). Comments related to the relation between GoI and such constructions appear in ([Abramsky-Coecke 02](#AbramskyCoecke02)).

Now the $CPM(-)$ construction requires and assumes that $\mathcal{C}$ has all [[dual objects]] (which in the standard model means to restrict to [[Hilbert spaces]] of [[quantum states]] with are of [[finite]] [[dimension]]). To generalize away from this requirement a more general definition of quantum operations in $\mathcal{C}$ is given in  ([Coecke-Heunen 11, def. 1](#CoeckeHeunen11)), there denoted $CP(\mathcal{C})$. Direct inspection shows that $CP(\mathcal{C})$ is still a (non-full) subcategory of $Int(\mathcal{C})$.

## Related concepts

* [[transcendental syntax]]

## References

The original references are

* [[Jean-Yves Girard]]. Towards a geometry of interaction. In Categories in Computer Science and Logic, pages 69 – 108, Providence, 1989. American Mathematical Society. Proceedings of Symposia in Pure Mathematics n92.
 {#Girard89}

* [[Jean-Yves Girard]]. Geometry of interaction I: interpretation of system F. In Ferro, Bonotto, Valentini, and Zanardo, editors, Logic Colloquium '88, pages 221 – 260, Amsterdam, 1989. North-Holland.

* [[Jean-Yves Girard]]. Geometry of interaction II : deadlock-free algorithms. In Martin-Löf and Mints, editors, Proceedings of COLOG 88, volume 417 of Lecture Notes in Computer Science, pages 76 – 93, Heidelberg, 1990. Springer-Verlag.

* [[Jean-Yves Girard]]. Geometry of interaction III : accommodating the additives. In Girard, Lafont, and Regnier, editors, Advances in Linear Logic, pages 329 – 389, Cambridge, 1995. Cambridge University Press.

* [[Jean-Yves Girard]]. Geometry of interaction IV : the feedback equation. In Stoltenberg-Hansen and Väänänen, editors, Logic Colloquium '03, pages 76 – 117. Association for Symbolic Logic, 2006.

* [[Jean-Yves Girard]]. Geometry of interaction V : logic in the hyperfinite factor, in memoriam Claire Delaleu (1991-2009). Fully revised version (October 2009). ([pdf](http://iml.univ-mrs.fr/~girard/GdI5.1.pdf))

* [[Jean-Yves Girard]]. Geometry of Interaction VI: a blueprint for Transcendental Syntax, January 2013, revised August 28th 2013. ([pdf](http://iml.univ-mrs.fr/~girard/blueprint.pdf))

Reviews include

* [[Jean-Yves Girard]], Part VI of _[[Lectures on Logic]]_

* Masaru Shirahata, _Geometry of Interaction explained_, ([pdf](http://www.kurims.kyoto-u.ac.jp/~hassei/algi-13/kokyuroku/19_shirahata.pdf))
 {#Shirahata}

* [[Samson Abramsky]], E. Haghverdi and P. Scott, "Geometry of Interaction and linear combinatory algebra," Mathematical Structures in Computer Science (2002), vol. 12, pp. 625-665. ([ps](http://www.cs.ox.ac.uk/files/328/ahs.ps))

* [[Samson Abramsky]] and R. Jagadeesan, "New Foundations for the Geometry of Interaction," Proceedings 7th Annual IEEE Symp. on Logic in Computer Science, LICS'92, Santa Cruz, CA, USA, 22–25 June 1992. ([pdf](http://www.cs.ox.ac.uk/files/300/nfgoi.pdf))

* Harry G. Mairson, "From Hilbert Spaces to Dilbert Spaces: Context Semantics Made Simple", Proceedings of FST TCS 2002. ([pdf](http://www.cs.brandeis.edu/~mairson/Papers/fsttcs02.pdf))

* Linear Logic Wiki, _[Geometry of Interaction](http://llwiki.ens-lyon.fr/mediawiki/index.php/Geometry_of_interaction)_

* {#Murfet14} [[Daniel Murfet]], section 5 of _Logic and linear algebra: an introduction_ ([arXiv:1407.2650](http://arxiv.org/abs/1407.2650))

The operational categorical semantics in [[traced monoidal categories]] is due to

* [[André Joyal]], [[Ross Street]], [[Dominic Verity]], _Traced monoidal categories_, Math. Proc. Camb. Phil. Soc. (1996), 119, 447 ([pdf](http://sci-prew.inf.ua/v119/3/S0305004100074338.pdf))
 {#JoyalStreetVerity96}

* [[Peter Hines]],  _The Algebra of Self-Similarity and its Applications,  PhD Thesis, University of Wales, U.K. 1997 ([pdf](https://peterhines.info/papers/phd.pdf))
 {#Hines97}

* [[Esfandir Haghverdi]],  _A Categorical Approach to Linear Logic_, Geometry of Proofs and Full Completeness, PhD Thesis, University of Ottawa, Canada 2000.
 {#Haghverdi00}

* [[Esfandir Haghverdi]], _Unique Decomposition Categories, Geometry of Interaction, and Combinatory Logic, Math. Struc. in Comp. Sci., 10, pp. 205-231.
 {#Haghverdi00b}

* [[Samson Abramsky]], [[Esfandir Haghverdi]],  [[Philip Scott]], _Geometry of Interaction and Linear Combinatory Algebras_. MSCS, vol. 12(5), 2002, 625-665, CUP (2002) ([citeseer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.7818))
  {#AbramskyHaghverdiScott02}

* [[Esfandir Haghverdi]], [[Philip Scott]], _A categorical model for the geometry of interactions_ ,Theoretical Computer Science, 2005 ([pdf](http://xavier.informatics.indiana.edu/~ehaghver/HS-TCS-04.pdf))
 {#HaghverdiScott05}

Discussion of [[quantum operations]]/[[completely positive maps]] mentioned above is in

* [[Peter Selinger]], _Dagger-compact closed categories and completely positive maps_, Electronic Notes in Theoretical Computer
Science (special issue: Proceedings of the 3rd International Workshop on Quantum Programming Languages). 2005 ([[SelingerPositiveMaps.pdf:file]], [ps](http://www.mscs.dal.ca/~selinger/papers/dagger.ps))
 {#Selinger05}

* [[Samson Abramsky]], [[Bob Coecke]], _Physical Traces: Quantum vs. Classical Information Processing_, Electronic Notes in Theoretical Computer Science 69 (2003) ([arXiv:0207057](http://arxiv.org/abs/cs/0207057))
 {#AbramskyCoecke02}

* [[Bob Coecke]], [[Chris Heunen]], _Pictures of complete positivity in arbitrary dimension_, EPTCS 95, 2012, pp. 27-35 ([arXiv:1110.3055](http://arxiv.org/abs/1110.3055))
 {#CoeckeHeunen11}

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