syntactic category

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### Context
#### Topos Theory
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[[!include topos theory - contents]]
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#### Type theory
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[[!include type theory - contents]]
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*Note: This page is about [[categories]] (in the sense of [[category theory]]) built out of [[syntax]], i.e. "term models".  The phrase "syntactic category" is also sometimes used to mean a grouping of syntactic objects, so that e.g. [[terms]] belong to one syntactic category and [[formulas]] to another. For this usage, see also [[categorial grammar]].*

# Contents
* table of contents
{: toc}

## Idea

As described at [[relation between type theory and category theory]], for many kinds of [[logic]] and [[type theory]] there is an [[adjunction]] or [[equivalence]]

$$ Theories \;\rightleftarrows\; Categories $$

where on the left we have a [[category]] or [[2-category]] of [[theories]] of some sort (i.e. in some [[doctrine]]), and on the right we have a category or 2-category of categories with some structure (e.g. finite [[limits]], [[cartesian closed category|cartesian closure]], etc.).

The **syntactic category** construction is the functor from theories to categories, denoted $Syn$ or $Con$.  Given a theory, it generates the [[walking structure|walking]] model of that theory, i.e. a structured category of the appropriate sort which is generated by a model of that theory.  Since the objects of the syntactic category are frequently taken to be the [[contexts]] in the theory, the syntactic category is also called the **category of contexts**.

The functor in the other direction associates to any category its [[internal logic]].

## Definition

Given a [[type theory]] $T$, its **syntactic category** or **category of contexts** $Con(T)$ is defined as follows.

* Its [[objects]] are the [[contexts]] in the type theory;

* Its [[morphisms]] between contexts are **[[substitutions]]**, or **interpretations** of [[variables]].

A morphism from the context $\Gamma$ to the context $\Delta$ consists of a way of fulfilling the assumptions required by $\Delta$ by appropriately interpreting those provided by $\Gamma$, generally by substituting terms available in $\Gamma$ for variables needed in $\Delta$ and proving whatever is necessary from the assumptions at hand.

More precisely, let $\Gamma$ and $\Delta$ be contexts of some [[type theory]] $T$ of the form
$$\Gamma \;=\; x_0:A_0,\; x_1:A_1(x_0),\; x_2:A_2(x_0,x_1),\;\dots x_n:A_n(x_0,\dots,x_{n-1})$$
and
$$\Delta \;=\; y_0:B_0,\; y_1:B_1(y_0),\; y_2:B_2(y_0,y_1),\;\dots y_m:B_m(y_0,\dots,y_{m-1})
  \,.
$$

(Here we allow the possibility that each type in these contexts [[dependent type|depends]] on the variables occurring earlier in the context, but for simplicity we can ignore that. )

Then a [[morphism]] $\Gamma \to \Delta$ in the category of contexts $Con(T)$ consists of a sequence of terms such as the following:
$$
\begin{aligned}
\Gamma &\vdash t_0 :B_0\\
\Gamma &\vdash t_1 : B_1(t_0)\\
\vdots\\
\Gamma &\vdash t_m : B_m(t_0,t_1,\dots, t_{m-1})
\end{aligned}
$$
In other words, to give such a morphism we must give, for each type (or assumption) required by $\Delta$, a way to construct an element of that type (or a proof of that assumption) out of the data and assumptions contained in $\Gamma$.

+-- {: .query}
This might fit better after the motivating examples below; but maybe those examples don\'t make sense to a newcomer.  This is incomplete, however, since it doesn\'t address contexts that include propositional hypotheses.  ---Toby
=--

## Properties

### Structure

Depending on the type-forming operations available in $T$, the category $Con(T)$ will have categorical structure.  Roughly:

* If $T$ has [[function types]], then $Con(T)$ is [[cartesian closed category]].
* If $T$ has [[sum types]], then $Con(T)$ has binary [[coproducts]].
* If $T$ is a [[dependent type theory]] with [[dependent product types]], then $Con(T)$ is a [[locally cartesian closed category]].
* If $T$ is a [[regular theory]], then $Con(T)$ is a [[regular category]].
* If $T$ is a [[coherent theory]], then $Con(T)$ is a [[coherent category]].
* If $T$ is a [[geometric theory]], then $Con(T)$ is a [[geometric category]].
* If $T$ is a finitary [[first-order theory]], then $Con(T)$ is a [[Heyting category]].

And so on.  (If $T$ lacks [[eta-conversion]], then this categorical structure may only be "weak".)  One thing worth noting is that

* $Con(T)$ always has finite [[products]].

This is due to the objects of $Con(T)$ being contexts rather than types.  A way to avoid this is to work instead with a syntactic [[cartesian multicategory]].

In fact, $Con(T)$ is not just a structured category, it is a [[split model of type theory]] in any of the senses described there.  In constructing formally an internal logic that involves dependent types, this is important to keep track of.

### Universal property
 {#UniversalProperty}

The syntactic category $Con(T)$ has the [[universal property]] that for $C$ any suitable category, [[functors]]

$$
  Con(T) \to C
$$

that preserve the relevant structure correspond to [[categorical semantics|interpretations]] of $T$ in $C$.

The construction

$$
  Con\colon TypeTheories \to Categories
$$

is the [[left adjoint]] in an [[adjunction]] that [[relation between type theory and category theory|relates type theories and categories]], whose [[right adjoint]] $Lan : Categories \to TypeTheories$ extracts the [[internal logic]] of a category.

Accordingly, the [[counit of an adjunction|adjunction counit]] evaluated on any category $C$

$$
  Con(Lan(C)) \to C
$$

says that there a canonical interpretation of the internal logic of a category $C$ in $C$ itself, while the [[unit of an adjunction|unit]] evaluated at a theory $T$:

$$ T \to Lan(Con(T)) $$

says that there is a canonical interpretation of $T$ in the internal logic of its syntactic category.

## Examples

### Substitution and introduction of a single term
 {#SubstitutionAndIntroductionOfTerm}

Given a context $\Gamma$ and a [[type]] $X$, there is a new context $[\Gamma, x : X]$.

Given a [[term]] $t : X$ there is a canonical morphism of contexts

$$
  \Gamma \stackrel{[t/x]}{\to} [\Gamma, x : X]
$$

which picks that term in $X$. The [[base change]] functor along this morphism

$$
  [t/x]^* : Con(T)_{/[\Gamma, x : X]} \to Con(T)_{/\Gamma}
$$

is the operation of _[[substitution]] of [[variables]]: it sends a [[dependent type]] $\Gamma, x : X \vdash P(x)$ to $\Gamma \vdash P(t)$.

There is also the canonical morphism of contexts

$$
  \hat x : [\Gamma, x : X] \to \Gamma
$$

which simply forgets the type $X$. The [[base change]] along this morphism is the context extension functor

$$
  \hat x^* : Con(T)_{/\Gamma} \to Con(T)_{/[\Gamma , x : X]}
  \,.
$$

Its [[right adjoint]] is the [[dependent product]] functor $\prod_{x : X}$ giving the [[universal quantifier]] $\forall_{x : X}$, and its [[left adjoint]] is the [[dependent sum]] functor $\sum_{x : X}$ giving the [[existential quantifier]] $\exists_{x :X}$.

### The theory of a group
{#group}

For example, consider these contexts in the theory of a group $G$:
$$ \Gamma \;=\; a\colon G,\; b\colon G $$
$$ \Delta \;=\; a\colon G,\; b\colon G,\; (a b)^2 = a^2 b^2 $$
$$ E \;=\; a\colon G,\; b\colon G,\; a b = b a $$
$$ Z \;=\; x\colon G,\; y\colon G $$

One interpretation of $\Gamma$ in $\Delta$ (that is, a morphism from $\Delta$ to $\Gamma$) is given by the substitution
$$ a \coloneqq a,\; b \coloneqq b .$$
The fact that $(a b)^2 = a^2 b^2$ in $\Delta$ is ignored.  In type-theoretic language, this would consist of the two terms
$$\begin{aligned}
a\colon G,\; b\colon G,\; (a b)^2 = a^2 b^2 &\vdash a:G\\
a\colon G,\; b\colon G,\; (a b)^2 = a^2 b^2 &\vdash b:G
\end{aligned}
$$
Note that in this way of presenting things, the names of the variables in $\Gamma$ do not appear; only the order in which the types appear in $\Gamma$ matters.

A less obvious interpretation of $\Gamma$ in $\Delta$ is the substitution
$$ a \coloneqq b,\; b \coloneqq a .$$
There is no reason to keep variable names the same.  (At this point, compare $\Gamma$ and $Z$; when the definition is complete, it ought to follow that these are isomorphic.)

Another, perhaps even less obvious, morphism $\Delta \to \Gamma$ is
$$ a \coloneqq a^2,\; b \coloneqq a^3 .$$
Not only does this ignore that $(a b)^2 = a^2 b^2$; it also ignores the very existence of $b$ in $\Delta$.  (It also uses the existence of $a$ more than once.  Ignoring and reusing information like this is not always allowed in substructural logics such as [[linear logic]].)

We can interpret $E$ in $\Delta$ without renaming variables because the theory of a group allows us to derive the judgment
$$ a\colon G,\; b\colon G,\; (a b)^2 = a^2 b^2 \;\vdash\; a b = b a. $$
That is, we get a morphism from $\Delta$ to $E$ by performing the substitution
$$ a \coloneqq a,\; b \coloneqq b $$
and then inserting a proof of the above judgment.  As it happens, the argument in such a proof is reversible, so you should expect that $\Delta$ and $E$ are also isomorphic.

Are there any morphisms from $\Gamma$ to $\Delta$?  The obvious substitution does not define a morphism, since the required fact cannot be proved.  However, you get one using the substitution
$$ a \coloneqq a,\; b \coloneqq a $$
and then inserting a proof that
$$ \Gamma \;\vdash\; (a a)^2 = a^2 a^2 .$$
All the same, we would not want to say that $\Gamma$ and $\Delta$ are isomorphic contexts; although there are morphisms in each direction, composing them should never produce identity morphisms on both sides.

The category structure of $Con(T)$ can be seen explicitly as well.  First, given a context $\Gamma$, there is an obvious **[[identity morphism|identity]]** morphism where every variable is substituted for itself and every statement assumed is proved immediately from itself.

Given morphisms $\Gamma \overset{f}\to \Delta \overset{g}\to E$, form the **[[composite]]** as follows:  First, for each variable $X$ required by $\Gamma$, $f$ tells us how to substitute a term $T$ built out of the variables in $\Delta$, while $g$ tells us how to substitute a term from $E$ for each of these variables.  So in the end, $X$ is expressed as a term $T[g]$ involving variables available in $E$.  Also, by combining the proofs provided by $f$ and $g$, we get the proofs required for their composite.

To really complete the definition of a category, I should also describe when two morphisms $\Gamma \to \Delta$ are **[[equality|equal]]**.  There are actually many options here; the most strict is to say that they are equal only if the substitutions and proofs used are syntactically identical, and the most weak is to say that any parallel morphisms are equal.  Neither of these is very useful; for purposes of this example, let us require only that the expressions substituted for each variable $X$ in $\Gamma$ can be proved equal in the context $\Delta$.

Now you should be able to prove that composition satisfies the axioms of a [[category]].

Notice that the exact definition of equality of morphisms can depend heavily on the theory in question and your own purposes.  For example, this definition makes sense only because we have a notion of proving equality of elements of a group.  Also, you can sometimes place interesting conditions on whether two proofs count as equivalent, rather than requiring either syntactic identity or (as we do here) accepting proof irrelevance.

+-- {: .num_example}
###### Exercise
Now that the category of contexts (in one sense) of the theory of a group has been completely defined, describe that category (up to [[equivalence of categories|equivalence]]) in terms familiar to an algebraist.  In particular, compare it to the [[Grp|category of groups]].
=--

+-- {: .proof}
###### Answer
In [rot13](http://www.rot13.com/) (so that you have a chance to think about it yourself without accidentally seeing the answer): gur bccbfvgr bs gur pngrtbel bs svavgryl cerfragrq tebhcf.
=--

The result of this [[exercise]] is true in more generality: it works for any finite-limit theory; see in particular [[Lawvere theory]].  Presumably there are also infinitary generalizations.  There's some general discussion in the [[Elephant]].

## Variations

### Cartesian multicategories

Instead of a syntactic category, for a non-dependent type theory one can construct instead a syntactic [[cartesian multicategory]] (or, in the case of a [[linear logic|linear]] type theory, a plain (symmetric) [[multicategory]]).  This avoids the need to take the objects to be contexts rather than single types.

### The syntactic site

For some doctrines, the syntactic category of any theory is naturally equipped with the structure of a [[site]].  For instance, if $T$ is a regular, coherent, or geometric theory, then $Con(T)$ is a regular, coherent, or geometric category, which comes with a naturally defined topology.  When equipped with this topology, the syntactic category is called the **syntactic site**.

In each of these cases, the [[category of sheaves]] on the syntactic site is the [[classifying topos]] of the theory.  In other words, it has the universal property that for any [[Grothendieck topos]] $\mathcal{E}$, [[geometric morphism]]s $\mathcal{E} \to Sh(Con(T))$ are equivalent to [[model]]s of the [[theory]] $T$ in $\mathcal{E}$.

## Related concepts

* More or less the same concept is that of _[[term model]]_.

* [[free topos]]

## References

Sections D4.1 and D4.4 of

* [[Peter Johnstone]], _[[Sketches of an elephant]]_,
 {#Johnstone}

A description of the construction of the category of contexts is in

* Andrew Pitts, _Categorical logic_, In Handbook of Logic in Computer Science, volume 5, pages 39–128. Oxford University Press (2001)

Another description, via [[sketches]], is in chapter VIII of

* [[Paul Taylor]], _[[Practical Foundations of Mathematics]]_,

A review of this is around [section 2.8](http://www.paultaylor.eu/ASD/foufct/cattype.html) of

* [[Paul Taylor]], _Foundations for computable topology_ ([webpage](http://www.paultaylor.eu/ASD/foufct/abstract.html))
 {#Taylor}

[[!redirects syntactic category]]
[[!redirects syntactic categories]]
[[!redirects syntactic site]]
[[!redirects syntactic sites]]
[[!redirects category of contexts]]
[[!redirects categories of contexts]]


syntactic site

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### Context
#### Topos Theory
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#### Type theory
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#Contents#
* table of contents
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## Idea

For $\mathbb{T}$ a [[theory]], the _syntactic site_ of a [[syntactic category]] $\mathcal{C}_{\mathbb{T}}$  is the structure of a [[site]] on $\mathcal{C}_{\mathbb{T}}$ such that [[geometric morphisms]] $\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})$ into the [[sheaf topos]] over the syntactic site are equivalent to [[models]] for the [[theory]] $\mathbb{T}$ in $\mathcal{E}$, hence such that $Sh(\mathcal{C}_{\mathbb{T}})$ is the [[classifying topos]] for $\mathbb{T}$.

## Definition

For $\mathbb{T}$ a [[theory]] and $\mathcal{C}_{\mathbb{T}}$ its [[syntactic category]], we define [[coverage]]s $J$ on $\mathcal{C}_{\mathbb{T}}$. These depend on which type of theory $\mathbb{T}$ is (or is regarded to be).

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

* For $\mathbb{T}$ a [[cartesian theory]], $J$ is the trivial coverage: the covering families consist of single [[isomorphism]]s.

* For $\mathbb{T}$ a [[regular theory]], $J$ is the [[regular coverage]]: the [[covering]] families consist of single [[regular epimorphism]]s.

* For $\mathbb{T}$ a [[coherent theory]], $J$ is the [[coherent coverage]]: the [[covering]] families consist of morphisms $\{U_i \to U\}$ such that the union $\cup_i U_i \simeq U$ equals $U$.

* For $\mathbb{T}$ a [[geometric theory]], $J$ is the [[geometric coverage]]

=--

## Properties

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

For $\mathcal{T}$ a [[cartesian theory]], [[regular theory]], etc. and $\mathcal{C}_{\mathbb{T}}$ its syntactic site, according to def. \ref{TheCoverage}, we have

* For $\mathbb{T}$ a [[cartesian theory]], left [[exact functor]]s  $\mathcal{C}_{\mathbb{T}} \to \mathcal{E}$ are equivalent to [[geometric morphism]]s $\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})$

  $$
    \mathbb{T}-Model(\mathcal{E})
     \simeq
    Func_\times(\mathcal{C}_{\mathbb{T}}, \mathcal{E})
     \simeq
    Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}}))
    \,.
  $$

* For $\mathbb{T}$ a [[regular theory]],  [[regular functor]]s $\mathcal{C}_{\mathbb{T}} \to \mathcal{E}$ are equivalent to [[geometric morphism]]s $\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})$

  $$
    \mathbb{T}-Model(\mathcal{E})
     \simeq
    RegFunc(\mathcal{C}_{\mathbb{T}}, \mathcal{E})
     \simeq
    Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}}))
    \,.
  $$

* For $\mathbb{T}$ a [[coherent theory]],  [[coherent functor]]s $\mathcal{C}_{\mathbb{T}} \to \mathcal{E}$ are equivalent to [[geometric morphism]]s $\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})$

  $$
    \mathbb{T}-Model(\mathcal{E})
     \simeq
    CohFunc(\mathcal{C}_{\mathbb{T}}, \mathcal{E})
     \simeq
    Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}}))
    \,.
  $$

* For $\mathbb{T}$ a [[geometric theory]],  [[geometric functor]]s $\mathcal{C}_{\mathbb{T}} \to \mathcal{E}$ are equivalent to [[geometric morphism]]s $\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})$

  $$
    \mathbb{T}-Model(\mathcal{E})
     \simeq
    GeomFunc(\mathcal{C}_{\mathbb{T}}, \mathcal{E})
     \simeq
    Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}}))
    \,.
  $$

In each case the [[equivalence of categories]] $Topos(\mathcal{E}, Sh(\mathcal{C}_{\mathbb{T}})) \stackrel{\simeq}{\to} \mathbb{T}-Model(\mathcal{E})$ is given by sending a [[geometric morphism]] $f : \mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})$ to the precomposition of its [[inverse image]] $f^*$ with the [[Yoneda embedding]] $j$ and [[sheafification]] $L$:

$$
  f \;\; \mapsto \;\;
  ( \mathcal{C}_\mathbb{T} \stackrel{j}{\to} PSh(\mathcal{C}_{\mathbb{T}}) \stackrel{L}{\to} Sh(\mathcal{C}_{\mathbb{T}}) \stackrel{f^*}{\to} \mathcal{E} )
  \,.
$$

=--

This appears as ([Johnstone](#Johnstone)), theorem 3.1.1, 3.1.4, 3.1.9, 3.1.12.

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

For cartesian theories this is the statement of [[Diaconescu's theorem]].

The other cases follow from this by using this discussion at [[classifying topos]], which says that geometric morphism $\mathcal{E} \to Sh(\mathcal{C})$ are equivalent to [[site|morphisms of site]]s $\mathcal{C} \to \mathcal{E}$ (for the [[canonical coverage]] on $\mathcal{E}$). This means that in addition to preserving finite limits, as in [[Diaconescu's theorem]], these functors also send [[cover]]s in $\mathcal{C}$ to [[epimorphism]]s in $\mathcal{E}$.

In the cases at hand this last condition means precisely that $\mathcal{C}_{\mathbb{T}} \to \mathcal{E}$ is a [[regular functor]] or [[coherent functor]] etc., respectively.

=--

## References

Section D3.1 of

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

[[!redirects syntactic sites]]


syntax

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#### Type theory
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#### Linguistics
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In [[formal logic]], by *syntax* one means formal specification of the ([[language]] of) a [[theory]] -- this in contrast to *[[semantics]]*.

In [[linguistics]] the meaning of syntax is similar, but a little different and narrower (does not include phonological, morphological and lexical levels of descriptions), see [[syntax (linguistics)]].

## Related concepts

* [[abstract syntax tree]]

* [[syntax - semantics duality]]

* [[relation between type theory and category theory]]

* [[transcendental syntax]]

* [[name binding]]

* [[language]], [[theory]]

* [[word]], [[string (computer science)|string]]

* [[linguistics]], [[syntax (linguistics)]].

## References

See also

* Wikipedia, *Syntax (logic)*

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

[[!redirects syntax]]


syntax (linguistics)

\tableofcontents

## Idea

In [[linguistics]], language expression is described at several levels:

1. sounds (phonetics),

1. groups of equivalent sounds which has a distinctive role (phonology),

1. longer sequences of sounds which carry a meaning or a function (morphems and words studied in morphology and lexicography),

1. rules how these morphems form utterances (sentences, discourse).

Syntax is the part of the description of language expression which in this traditional classification describes the top level of expression, that is how morphems and words form phrases, sentences. It is a part of a [[grammar]].

In addition there is a relation to meaning and usage (semantics, pragmatics) which is often considered completely separate, though it is not in natural language.

Another term is a syntax as a discipline which studies syntax in above sense, often meant with emphasis on a well formed sentences. Syntax as a theoretical discipline also studies how these rules function and are derived (e.g. syntax part of generative and cognitive grammar).

Unlike [[syntax]] in the sense of [[formal logic]] and [[computer science]], syntax in [[linguistics]] is thus not contrasted only to [[semantics]] but also to the description of lower level formations: sounds, lexical units. It usually does not include prosody, which is however also on the level of a phrase or sentence.

The following is from [Tallerman (2020)](#Tallerman98):

> ‘Syntax’ means ‘sentence construction’: how words group together to make phrases and sentences. Some people also use the term grammar to mean the same as syntax, although most linguists follow the more recent practice whereby the grammar of a language includes all of its organizing principles: information about the sound system, about the form of words, how we adjust language according to context, and so on; syntax is only one part of this grammar.

## Related entries

* [[linguistics]]

* [[semantics]], [[syntax]]

* [[semiotics]]

* [[grammar]]

## Literature

Related entries: [[syntax]] (in the senses of logics and computer science), [[grammar]], [[linguistics]]

* wikipedia [syntax](https://en.wikipedia.org/wiki/Syntax)
* Andre Martinet, _A functional view of the language_

* Dubravko Škiljan, _Pogled u lingvistiku_

* [[Noam Chomsky]], _Lectures on government and binding: the Pisa lectures_

* [[Noam Chomsky]], _The Minimalist Program_, MIT Press

* {#Tallerman98} M. Tallerman, _Understanding syntax_ (1sr ed.1998, 5th ed. 2020)


syntax-semantics duality

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#### Duality
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[[!include duality - contents]]
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# Contents
* table of contents
{: toc}

## Idea

There is a [[duality]] between [[syntax]] and [[semantics]].

## Related concepts

* [[Stone duality]]

* [[Gabriel-Ulmer duality]]

* [[Makkai duality]]

* [[relation between category theory and type theory]]

* [[computational trinitarianism]]

* [[initiality conjecture]]

## References

For [[first order logic]]:

* [[Steve Awodey]], [[Henrik Forssell]], _First-order logical duality_, [arxiv/1008.3145](http://arxiv.org/abs/1008.3145)

For [[geometric logic]]:

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

[[!redirects syntax-semantics duality]]
[[!redirects syntax-semantix duality]]
[[!redirects syntax–semantics duality]]
[[!redirects syntax--semantics duality]]
[[!redirects syntax - semantics duality]]
[[!redirects duality of syntax and semantics]]


synthetic algebraic geometry

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

## Idea

[[synthetic mathematics|synthetic]]$\;$ [[algebraic geometry]]

## Related concepts

* [[algebraic geometry]]

* [[synthetic mathematics]]

  * [[synthetic geometry]]

  * [[synthetic topology]]

  * [[synthetic differential geometry]]

  * [[synthetic differential topology]]

  * [[synthetic homotopy theory]]

  * [[synthetic (infinity,1)-category theory|synthetic $(\infty,1)$-category theory]]

  * [[synthetic probability theory]]

## References

*  {#Kock81} [[Anders Kock]], §I.12 in: _Synthetic Differential Geometry_, Cambridge University Press (1981, 2006) [[pdf](http://home.imf.au.dk/kock/sdg99.pdf), [doi:10.1017/CBO9780511550812](https://doi.org/10.1017/CBO9780511550812)]

* [[Ingo Blechschmidt]], *Using the internal language of toposes in algebraic geometry*, PhD thesis (2017) [[pdf](https://rawgit.com/iblech/internal-methods/master/notes.pdf), [[Blechschmidt-InternalLanguage.pdf:file]]]

* [[Felix Cherubini]], [[Thierry Coquand]], [[Matthias Hutzler]], *A Foundation for Synthetic Algebraic Geometry* (2023) [[arXiv:2307.00073](https://arxiv.org/abs/2307.00073)]

* [[Felix Cherubini]], *A Foundation for Synthetic Algebraic Geometry*, talk at *[Homotopy Type Theory Electronic Seminar Talks](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html)* (Oct 2023) [slides:[pdf](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Cherubini-2023-10-18-HoTTEST.pdf), video:[YT](https://www.youtube.com/watch?v=lp4kcmQ0ueY)]

[[!redirects synthetic algebraic geometries]]


synthetic computability theory

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# Contents

* table of contents
{:toc}

## Idea

_Synthetic computability theory_ is the study of [[computability theory]] [[synthetic mathematics|synthetically]] [[axiom|axiomatized]] by [[structures]] present in the [[effective topos]] or other toposes of computable structures, rather than by [[analytic versus synthetic|analytic]] construction such as [[Turing machines]] or [[lambda calculus]].

## Related concepts

* [[synthetic Tait computability]]

* [[synthetic mathematics]]

## References

* [[Andrej Bauer]], _First steps in synthetic computability theory_, ENTCS, Vol 155, 2006, [doi:10.1016/j.entcs.2005.11.049](https://doi.org/10.1016/j.entcs.2005.11.049)

[[!redirects synthetic computability]]


synthetic differential geometry

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#### Formal geometry
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[[!include formal geometry -- contents]]
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=--

# Contents

* table of contents
{:toc}

## Idea

> Der unendlich kleinste Theil des Raumes ist immer ein Raum, etwas, das Continuität hat, nicht aber ein blosser Punct, oder die Grenze zwischen bestimmten Stellen im Raume; ([Fichte 1795, Grundriss, §4.IV](Grundriss+des+Eigenthümlichen+der+Wissenschaftslehre#4IVUnendlichKleinsterTeilDesRaumes))

In _synthetic differential geometry_ one formulates [[differential geometry]] [[axiom|axiomatically]] in [[toposes]] -- called [[smooth toposes]] --  of [[generalized smooth spaces]] by assuming the explicit existence of [[infinitesimal neighbourhoods]] of points.

The main point of the axioms is to ensure that a well defined notion of the [[infinitesimal spaces]] exists in the topos, whose existence concretely and usefully formalizes the wide-spread but often vague intuition about the role of infinitesimals in [[differential geometry]].

In particular, in such toposes $E$ there exists an [[infinitesimal space]] $D$ that behaves like the [[infinitesimal object|infinitesimal interval]] in such a way that for any space $X \in E$ the [[tangent bundle]] of $X$, is, again as an object of the topos, just the [[internal hom]] $T X \;\text{:=}\; X^D$ (using the notation of [[exponential object | exponential objects]] in the [[cartesian closed category]] $E$). So a tangent vector in this context is literally an _infinitesimal path_ in $X$.

This way, in [[smooth topos | smooth toposes]] it is possible to give precise well-defined meaning to many of the familiar computations -- wide-spread in particular in the [[physics]] literature -- that compute with supposedly "infinitesimal" quantities.

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

As quoted by [[Anders Kock]] in his first book ([p. 9 ](http://home.imf.au.dk/kock/sdg99.pdf#page=9)), [[Sophus Lie]] -- one of the founding fathers of [[differential geometry]] and, of course [[Lie theory]] -- once said that he found his main theorems in [[Lie theory]] using "synthetic reasoning", but had to write them up in non-synthetic style (see _[[analytic versus synthetic]]_) just due to lack of a formalized language:

> "The reason why I have postponed for so long these investigations, which are basic to my other work in this field, is essentially the following: I found these theories originally by synthetic considerations. But I soon realized that, as expedient ( _zweckmässig_ ) the synthetic method is for discovery, as difficult it is to give a clear exposition on synthetic investigations, which lead to deal with objects that till now have almost exclusively been considered analytically. After long vacillations, I have decided to use a half synthetic, half analytic form. I hope my work will serve to bring justification to the synthetic method besides the analytical one." ([[Sophus Lie]], _Allgemeine Theorie der partiellen Differentialgleichungen erster Ordnung_, Math. Ann. 9 (1876).)

=--

Synthetic differential geometry provides this formalized language.

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

Another advocate of the use of infinitesimals in the late 19th century was the American philosopher [[Charles Sanders Peirce]]:

> The illumination of the subject by a strict notation for the logic of relatives had shown me clearly and evidently that the idea of an infinitesimal involves no contradiction...As a mathematician, I prefer the method of infinitesimals to that of limits, as far easier and less infested with snares. Charles Sanders Peirce, _The Law of Mind_, The Monist **2** (1892)

According to Bell ([1998, p.5](#Bell98)):

> Peirce was aware, even before Brouwer, that a faithful account of the truly continuous will involve jettisoning the law of excluded(.)

=--

## Axiomatics

The axioms of synthetic differential geometry demand that the [[topos]] $E$ of smooth spaces is

* a [[smooth topos]] (see there for details)

in which in particular

* [[infinitesimal space | infinitesimal spaces]] exist and

* satisfy the [[Kock-Lawvere axiom]].

Depending on applications one imposes further axioms, such as the

* [[integration axiom]].

With that little bit of axiomatics alone, a large amount of [[differential geometry]] may be formulated. This has been carried through quite comprehensively by [[Anders Kock]], see the [reference](#References) below.

In his work he particularly makes use of the fact that as sophisticated as a [[smooth topos]] may be when explicitly constructed (see the section on [models](#models)), being a [[topos]] means that one can reason inside it almost literally as in [[Set]]. Using this Kock's work gives descriptions of synthetic differential geometry which are entirely intuitive and have no esoteric topos-theoretic flavor. All he needs is the assumption that the [[Kock-Lawvere axiom]] is satisfied for "numbers". Here "numbers" is really to be interpreted in the topos, but if one just accepts that they satisfy the KL axiom, one may work with infinitesimals in this context in essentially precisely the naive way, with the topos theory in the background just ensuring that everything makes good sense.

## Models
{#models}

Being axiomatic, reasoning in synthetic differential geometry applies in every **model** for the axioms, i.e. in every concrete choice of [[smooth topos]] $T$.

Models of [[smooth toposes]] tend to be inspired by, but more general than, constructions familiar from [[algebraic geometry]]. In particular the old insight promoted by [[Grothendieck]] in his work, that [[nilpotent ideals]] in [[rings]] are formal duals of spaces with infinitesimal extension is typically used to model [[infinitesimal spaces]] in synthetic differential geometry.

See at _[[synthetic differential geometry applied to algebraic geometry]]_ for more on this.

The main difference between models for [[smooth toposes]]  and  [[algebraic geometry]] from this perspective is that models for [[smooth topos]] tend to employ test spaces that are _richer_ than plain formal duals to commutative [[ring | rings]] or algebras, as in [[algebraic geometry]]: typical models for synthetic differential geometry use test spaces given by formal duals of [[generalized smooth algebras]] that remember "smooth structure" in the usual sense of [[differential geometry]] (and different from, though not entirely unrelated to, the notion of [[smooth scheme]] in algebraic geometry). This is in particular true for the [well adapted models](#wellAdaptedModels).

However, with a a sufficiently general perspective on [[higher geometry]] one finds that algebraic geometry and  synthetic differential geometry are both special cases of a more general notion of theories of generalized spaces. For more on this see [[generalized scheme]].

### Well adapted models
 {#wellAdaptedModels}

A [[topos]] $E$ modelling the axioms of synthetic differential geometry is called **(well) adapted** if the ordinary [[differential geometry]] of [[manifolds]] embeds into it, in particular if there is a [[full and faithful functor]] [[Diff]] $\to E$ from the category of ordinary [[smooth manifold | smooth manifolds]] into $E$.

A standard model for well adapted synthetic toposes is obtained in terms of sheaves on duals of "germ determined" $C^\infty$-rings. This is described in great detail in the textbook _[[Models for Smooth Infinitesimal Analysis]]_.

The conception and discussion of these well adapted toposes goes back to [[Eduardo Dubuc]], who studied them in a long series of articles. He [asks](http://north.ecc.edu/alsani/ct99-00%288-12%29/msg00218.html) people to refer to this topos as the **[[Dubuc topos]]**.

This theory of well-adapted models was later summarized and extended in the standard textbook

* [[Ieke Moerdijk]], [[Gonzalo Reyes]], _[[Models for Smooth Infinitesimal Analysis]]_ (1991)

## Variations

### Higher categorical versions

* Synthetic differential geometry may be thought of as embedded in the general theory of [[derived smooth manifolds]] and, generally, that of [[generalized schemes]].

### Supergeometric versions

The notion of synthetic differential geometry extends to the context of [[supergeometry - contents|supergeometry]]. See

* [[synthetic differential supergeometry]].

## Constructions in synthetic differential geometry {#Constructions}

### Tangent bundle

The [[tangent bundle]] of an object $X$ in a [[smooth topos]] is just the [[exponential object]] $T X := X^D$. The unique inclusion ${*} \to D$ induces a canonical projection $T X \to X$. A [[section]] $X \to T X$ of that projection is a [[tangent vector field]] on $X$. Its [[adjunct]] is a morphism $D \to X^X$ that sends the unique point of $D$ to the identity $Id_X \in X^X$.

### Differential equation

A [[differential equation]] is an extension problem in a [[smooth topos]] along a morphism that includes an [[infinitesimal object]] into another object.

For instance the ordinary first order homogeneous differential equation that asks the derivative of a function $f : X \to A$ along a [[vector field]] $v : D \to X^X$ to be given by a specified map $\alpha: X \to T A$ is given by a diagram of the form

$$
  \array{
    D &\stackrel{v}{\to}& X^X
    \\
    {}^{\mathllap{\alpha}}\downarrow & \swarrow_{\mathrlap{f}}
    \\
    A^X
  }
  \,,
$$

where we have freely identified morphisms with their [[adjunct | adjuncts]]. See [[differential equation]] for details.

### Differential forms

A [[differential form|differential 1-form]] is a morphism $\omega : T X \to R$ that is "fiberwise linear". One elegant way to say this is obtained by considering all higher differential forms at once:

for a sufficiently well behaved object $X$ in a [[smooth topos]], there is the [[simplicial object]] which is the [[infinitesimal singular simplicial complex]] $X^{(\Delta^\bullet_{inf})}$ of $X$. Taking functions on this produces the [[cosimplicial algebra]] $Hom(X^{\Delta^\bullet_{inf}}, R)$. Its [[Moore complex|normalized Moore cochain complex]] is isomorphic to the [[de Rham complex|de Rham dg-algebra]] of differential forms on $X$:

$$
  N^\bullet(Hom(X^{(\Delta^{inf}_\bullet)}),R)
  =
  \Omega^\bullet_{dR}(X)
  \,.
$$

This is discussed at

* Spaces of infinitesimal k-simplices

* [[differential forms in synthetic differential geometry]].

### Flow of a vector field

See at [flow of a vector field](flow+of+a+vector+field#SyntheticDefinition).

## Related concepts

* [[synthetic mathematics]]

  * [[synthetic geometry]]

  * [[synthetic topology]]

  * [[synthetic algebraic geometry]]

  * [[synthetic differential topology]]

  * [[synthetic homotopy theory]]

  * [[synthetic (infinity,1)-category theory|synthetic $(\infty,1)$-category theory]]

  * [[synthetic probability theory]]

* [[differential geometry]]

* [[derived differential geometry]]

* [[D-geometry]]

* [[differential cohesion]]

  * [[synthetic differential infinity-groupoid]]

* [[topos of laws of motion]]

* [[nonstandard analysis]]

* [[synthetic differential topology]]

* [[logical topology]]

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

## References
{#References}

The idea of axiomatizing [[differential geometry]] using ideas inspired by [[topos]] theory originates in

* {#Lawvere67} [[Bill Lawvere]], _[[Categorical dynamics]]_, lecture  (Chicago 1967). ([abstract pdf](http://www.mat.uc.pt/~ct2011/abstracts/lawvere_w.pdf))

They were published as pp.1-28 in

* {#Kocke79} [[Anders Kock]] (ed.), _Topos Theoretic Methods in Geometry_ , Aarhus Univ. Var. Pub. Ser. 30 (1979).

The first model for the axioms presented there served to demonstrate that the theory is non-empty, but was hard to work with. Much of the later work was concerned with refining the model-building. For instance

* [[Eduardo Dubuc]], _Sur les modèles de la géométrie différentielle synthétique_, Cahier Top et Géom. Diff. **XX-3** (1979) pp.231-279. ([pdf](http://archive.numdam.org/article/CTGDC_1979__20_3_231_0.pdf))

These models are constructed in terms of [[sheaf]] [[topos | toposes]] on the category of [[smooth loci]], formal duals to [[generalized smooth algebra|C^∞-rings]]. See there for a detailed list of references.

Transcripts or notes of further talks by Bill Lawvere on the subject are

* [[Bill Lawvere]], _[[Toposes of laws of motion]]_ , transcript of a talk in Montreal, Sept. 1997 ([[LawvereToposesOfLawsOfMotions.pdf:file]])

  (on the description of [[differential equation | differential equations]] in terms of synthetic differential geometry)

* [[Bill Lawvere]], _Outline of synthetic differential geometry_ , lectures in Buffalo (1998). ([[LawvereSDGOutline.pdf:file]])

Two articles that exhibit the link to [[continuum mechanics]]:

* [[Bill Lawvere|F. W. Lawvere]], _Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body_ , Cah. Top. Géom. Diff. Cat. **21** no.4 (1980) pp.377-392. ([pdf](http://archive.numdam.org/article/CTGDC_1980__21_4_377_0.pdf))

* [[Bill Lawvere|F. W. Lawvere]], _Categorical algebra for continuum microphysics_ , JPAA **175** (2002) pp.267-287.

See also

* [[Marta Bunge]], [[Eduardo Dubuc]], _Archimedian local $C^\infty$-rings and models of synthetic differential geometry_ Cahiers de Topologie et Géométrie Différentielle Catégoriques, **XXVII-3** (1986) pp.3-22. ([numdam](http://www.numdam.org/item?id=CTGDC_1986__27_3_3_0)).

For the early French connection see:

* [[André Weil]], _Théorie des points proches sur les variétés différentiables_ , Colloq. Top. et Géom. Diff., Strasbourg (1953) pp.111-117.

* Jean Penon, _De l'infinitésimal au local (Thèse de Doctorat d'État)_ Diagrammes **S13** (1985), pp.1-191. ([pdf](http://archive.numdam.org/article/DIA_1985__S13__1_0.pdf))

Discussion in [[differential cohesion]] is in

* [[Igor Khavkine]], [[Urs Schreiber]], _[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations: I. Jets and comonad structure]]_ ([arXiv:1701.06238](https://arxiv.org/abs/1701.06238))

Discussion in differentially cohesive [[homotopy type theory]] is in

* {#Wellen17} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_, 2017

* {#JazMyers22} [[David Jaz Myers]], *Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory* [[arXiv:2205.15887](https://arxiv.org/abs/2205.15887)]

### Books

A nice elementary introduction which emphasizes calculations and the application as _engineering mathematics_ can be found in

* {#Bell98}[[John Lane Bell]], _A Primer of Infinitesimal Analysis_, Cambridge UP 1998 ([ISBN:9780521887182](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/primer-infinitesimal-analysis-2nd-edition?format=HB&isbn=9780521887182))

The textbooks

*  {#Kock81} [[Anders Kock]], _Synthetic Differential Geometry_, Cambridge University Press (1981, 2006) [[pdf](http://home.imf.au.dk/kock/sdg99.pdf), [doi:10.1017/CBO9780511550812](https://doi.org/10.1017/CBO9780511550812)]

*  {#Kock10} [[Anders Kock]], _Synthetic geometry of manifolds_, Cambridge Tracts in Mathematics **180** (2010) [[pdf](http://home.imf.au.dk/kock/SGM-final.pdf), [doi:10.1017/CBO9780511691690](https://doi.org/10.1017/CBO9780511691690)]

develop in great detail the theory of [[differential geometry]] using the axioms of synthetic differential geometry. The main goal in these books is to demonstrate how these axioms lead to a very elegant, very intuitive and very comprehensive conception of differential geometry. Accordingly, concrete models (whose explicit description is typically much more evolved than the nice axiomatics that holds once they have been constructed) play a minor role in these books.

Somewhat complementary to that, the book

* [[Ieke Moerdijk]], [[Gonzalo Reyes]], _[[Models for Smooth Infinitesimal Analysis]]_ , Springer Heidelberg 1991.

focuses on concrete constructions of well-adapted models using the technology of [[generalized smooth algebra | generalized smooth algebras]] ($C^\infty$-rings).

Another textbook is

* [[René Lavendhomme]], *Basic concepts of synthetic differential geometry*, Kluwer Texts in the Mathematical Sciences **13**,  Springer (1996) [[doi:10.1007/978-1-4757-4588-7](https://doi.org/10.1007/978-1-4757-4588-7)]

### Expositions

Introductory survey includes

* [[Anders Kock]], _Synthetic differential geometry - new methods for old spaces_, talk at _[[New Spaces for Mathematics and Physics]]_, IHP Paris (2015) [[video recording](https://www.youtube.com/watch?v=AXz7xu3WrPE)]

* [[Anders Kock]], _New methods for old spaces: synthetic differential geometry_ [[arxiv/1610.00286](https://arxiv.org/abs/1610.00286)]

Introductory expositions of basic ideas of synthetic differential geometry are

*  [[Mike Shulman]], _Chicago Pizza-Seminar: Synthetic Differential Geometry_ ([pdf](http://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.pdf))

*  [[John Lane Bell]], _An invitation to smooth infinitesimal analysis_ ([pdf](http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf))

* R. P. Kostecki, _Differential Geometry in Toposes_, ms. University of Warsaw (2009) [pdf](http://www.fuw.edu.pl/~kostecki/sdg.pdf)

* [[Urs Schreiber]], _[[geometry of physics -- supergeometry]]_


synthetic differential geometry - contents

**[[synthetic differential geometry|synthetic]] [[differential geometry]]**

**Introductions**

[[Introduction to Topology -- 1|from point-set topology to differentiable manifolds]]

[[geometry of physics]]: [[geometry of physics -- coordinate systems|coordinate systems]], [[geometry of physics -- smooth spaces|smooth spaces]], [[geometry of physics -- manifolds and orbifolds|manifolds]], [[geometry of physics -- smooth homotopy types|smooth homotopy types]], [[geometry of physics -- supergeometry|supergeometry]]

**Differentials**

* [[differentiation]], [[chain rule]]

* [[differentiable function]]

* [[infinitesimal space]], [[infinitesimally thickened point]], [[amazing right adjoint]]

**[[V-manifolds]]**

* [[differentiable manifold]], [[coordinate chart]], [[atlas]]

* [[smooth manifold]], [[smooth structure]], [[exotic smooth structure]]

* [[analytic manifold]], [[complex manifold]]

* [[formal smooth manifold]], [[derived smooth manifold]]

**[[smooth space]]**

* [[diffeological space]], [[Frölicher space]]

* [[manifold structure of mapping spaces]]

**Tangency**

* [[tangent bundle]], [[frame bundle]]

  * [[vector field]], [[multivector field]], [[tangent Lie algebroid]];

  * [[differential forms in synthetic differential geometry|differential forms]], [[de Rham complex]], [[Dolbeault complex]]

    * [[pullback of differential forms]], [[invariant differential form]], [[Maurer-Cartan form]], [[horizontal differential form]],

    * [[cogerm differential form]]

    * [[integration of differential forms]]

* [[local diffeomorphism]], [[formally étale morphism]]

* [[submersion]], [[formally smooth morphism]],

* [[immersion]], [[formally unramified morphism]],

* [[de Rham space]], [[crystal]]

* [[infinitesimal disk bundle]]

**The magic algebraic facts**

* [[embedding of smooth manifolds into formal duals of R-algebras]]

* [[smooth Serre-Swan theorem]]

* [[derivations of smooth functions are vector fields]]

**Theorems**

* [[Hadamard lemma]]

* [[Borel's theorem]]

* [[Boman's theorem]]

* [[Whitney extension theorem]]

* [[Steenrod-Wockel approximation theorem]]

* [[Whitney embedding theorem]]

* [[Poincare lemma]]

* [[Stokes theorem]]

* [[de Rham theorem]]

* [[Hochschild-Kostant-Rosenberg theorem]]

* [[differential cohomology hexagon]]

**Axiomatics**

* [[Kock-Lawvere axiom]]

  * [[smooth topos]], [[super smooth topos]]

  * [[microlinear space]]

  * [[integration axiom]]

[[!include cohesion - table]]

**Models**
{#models_2}

* [[Models for Smooth Infinitesimal Analysis]]

  * [[smooth algebra]] ($C^\infty$-ring)

  * [[smooth locus]]

  * [[Fermat theory]]

* [[Cahiers topos]]

* [[smooth ∞-groupoid]]

* [[formal smooth ∞-groupoid]]

* [[super formal smooth ∞-groupoid]]

**[[Lie theory]], [[∞-Lie theory]]**

* [[Lie algebra]], [[Lie n-algebra]], [[L-∞ algebra]]

* [[Lie group]], [[Lie 2-group]], [[smooth ∞-group]]

**[[differential equations]], [[variational calculus]]**

* [[D-geometry]], [[D-module]]

* [[jet bundle]]

* [[variational bicomplex]], [[Euler-Lagrange complex]]

* [[Euler-Lagrange equation]], [[de Donder-Weyl formalism]],

* [[phase space]]

**[[Chern-Weil theory]], [[∞-Chern-Weil theory]]**

* [[connection on a bundle]], [[connection on an ∞-bundle]]

* [[differential cohomology]]

  * [[ordinary differential cohomology]], [[Deligne complex]]

  * [[differential K-theory]]

  * [[differential cobordism cohomology]]

* [[parallel transport]], [[higher parallel transport]], [[fiber integration in differential cohomology]]

* [[holonomy]], [[higher holonomy]]

* [[gauge theory]], [[higher gauge theory]]

* [[Wilson line]], [[Wilson surface]]

**[[Cartan geometry]] ([[super Cartan geometry|super]], [[higher Cartan geometry|higher]])**

  * [[Klein geometry]], ([[higher Klein geometry|higher]])

  * [[G-structure]], [[torsion of a G-structure]]

  * [[Euclidean geometry]], [[hyperbolic geometry]], [[elliptic geometry]]

  * ([[pseudo-Riemannian geometry|pseudo]]-)[[Riemannian geometry]]

    * [[orthogonal structure]]

    * [[isometry]], [[Killing vector field]], [[Killing spinor]]

    * [[spacetime]], [[super-spacetime]]

  * [[complex geometry]]

  * [[symplectic geometry]]

  * [[conformal geometry]]


synthetic differential geometry applied to algebraic geometry

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

#Contents#
* table of contents
{:toc}

## Introduction

The axioms of [[synthetic differential geometry]] are intended to pin down the minimum [[category theory|general abstract]] [[axioms]] necessary for talking about the _differential_ aspect of [[differential geometry]] using concrete objects that model [[infinitesimal spaces]].

But the typical _models_ for the axioms -- the typical [[smooth toposes]] -- are constructed in close analogy to the general mechanism of [[algebraic geometry]]: [[Models for Smooth Infinitesimal Analysis|well-adapted models]] for [[smooth toposes]] use [[sheaf|sheaves]] on $C^\infty Ring^{op}$ (the [[opposite category]] of [[generalized smooth algebra|smooth algebras]]) where spaces in [[algebraic geometry]] (such as [[scheme]]s) uses [[sheaf|sheaves]] on [[CRing]]${}^{op}$.

In fact, for instance also the [[topos]] of [[presheaf|presheaves]] on $k-Alg^{op}$, which one may think of as being a context in which much of [[algebraic geometry]] over a [[field]] $k$ takes place, happens to satisfy the axioms of a [[smooth topos]] (see the examples there).

This raises the question:

+-- {: .standout}

**Questions**

To which degree do results in [[algebraic geometry]] depend on the choice of [[site]] [[CRing]]${}^{op}$ or similar?

To which degree are these results valid in a much wider context of any [[smooth topos]], or smooth topos with certain extra assumptions?

In the general context of [[structured (∞,1)-topos]]es and [[generalized scheme]]s: how much of the usual lore depends on the choice of the (simplicial)ring-theoretic Zariski or etale [[geometry (for structured (∞,1)-toposes)|(pre)geometry (for structured (∞,1)-toposes)]], how much works more generally?

=--

It is curious that the field of algebraic geometry has induced, first with [[Alexander Grothendieck]] now with [[Jacob Lurie]], so much [[category theory]] and [[higher category theory]], while at the same time it is common practice in this field to effectively disregard one of the major guidelines that practitioners in pure [[category theory]] are fond of adhering to: that of separation of context and implementation. [[Bill Lawvere]]'s famous dichotomy between _theory_ and _model_.

In fact, it seems that [[William Lawvere]] found the axioms of [[synthetic differential geometry]] not without the idea of capturing central structures in [[algebraic geometry]] this way, too. But, possibly due to the very term chosen, [[synthetic differential geometry]] it has apparently always (if at all) attracted more the attention of those interested in ordinary [[differential geometry]] than those interested in [[algebraic geometry]].

But at least in the light of [[Jacob Lurie|Lurie]]'s notion of [[structured (∞,1)-topos]]es and [[generalized scheme]]s, from the point of which ordinary [[algebraic geometry]] as well as [[derived algebraic geometry]] is just one realization of a more general concept of geometry, it seems to be worthwhile to reexamine the wealth of knowledge accumulated in algebraic geometry and see how much of it depends on just general context, how much on concrete implementation.

## Facts

### Functions in the big Zariski topos

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

Let $X$ be a [[scheme]] and $Sh(Sch/X)$ the [[Zariski site|big Zariski topos]] associated to $X$. Denote by $\mathbb{A}^1$ (the [[affine line]]) the [[ring object]] $T \mapsto \Gamma(T,\mathcal{O}_T)$, i.e. the functor represented by the $X$-scheme $\mathbb{A}^1_X \coloneqq X \times Spec(\mathbb{Z}[t])$. Then:

* $\mathbb{A}^1$ is [[internal logic|internally]] a [[local ring]].

* $\mathbb{A}^1$ is internally a [[field]] in the sense that any nonzero element is invertible.

* Internally, any [[function]] $f : \mathbb{A}^1 \to \mathbb{A}^1$ is a [[polynomial]] function, i.e. of the form $f(x) = \sum_i a_i x^i$ for some coefficients $a_i : \mathbb{A}^1$. More precisely,
$$ Sh(Sch/X) \models \forall f : [\mathbb{A}^1,\mathbb{A}^1]. \bigvee_{n \in \mathbb{N}} \exists a_0,\ldots,a_n : \mathbb{A}^1. \forall x : \mathbb{A}^1. f(x) = \sum_i a_i x^i. $$
Furthermore, these coefficients are uniquely determined.

=--

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

See _[[affine line#InternalFormulation]]_.

=--

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

Since the big Zariski topos is [[cocomplete category|cocomplete]] (being a [[Grothendieck topos]]), one can also get rid of the external [[disjunction]] and refer to the object $\mathbb{A}^1[X]$ of internal polynomials: The canonical ring homomorphism $\mathbb{A}^1[X] \to [\mathbb{A}^1,\mathbb{A}^1]$ (given by evaluation) is an [[isomorphism]].

=--

## Questions

> this is old material that needs attention

### Coherent sheaves #

To which degree can the notion of [[quasicoherent sheaf]] generalize from a context modeled on the [[site]] [[CRing]] to a more general context. What is, for instance, a quasicoherent sheaf on a [[derived smooth manifold]]? If at all? What on a general [[generalized scheme]], if at all?

### geometric $\infty$-function theory ##

Closely related to that: [[David Ben-Zvi]] et al have developed a beautiful theory of integral transforms on [[derived stack|derived]] [[∞-stack]], as described at [[geometric ∞-function theory]].

But in their construction it is always assumed that the underlying [[site]] is the (derived) algebraic one, something like simplicial rings.

 How much of their construction actually depends on that assumption? How much of this work carries over to other choices of [[geometry (for structured (∞,1)-toposes)|geometries]]?

For instance, when replacing the category of [[ring]]s /[[affine scheme]] in this setup with that of [[generalized smooth algebra|smooth algebra]] / [[smooth locus|smooth loci]], how much of the theory can be carried over?

It seems that the crucial and maybe only point where they use the concrete form of their underlying [[site]] is the definition of [[quasicoherent sheaf]] on a [[derived stack]] there, which uses essetnially verbatim the usual definition $QC(-) : Spec(A) \mapsto A Mod$.

What is that more generally? What is $A Mod$ for $A$ a [[generalized smooth algebra|smooth algebra]]?

If that doesn't have a good answer, maybe there is a more _intrinsic_ way to say what quasicoherent sheaves on an [[∞-stack]] are, such that it makes sense on more general [[generalized scheme]]s.

## References

Somebody points out the discussion here

* [http://www.mta.ca/~cat-dist/catlist/1999/finite-topos](http://www.mta.ca/~cat-dist/catlist/1999/finite-topos)

on formulating finitely-presented conditions internally in a topos. In particular [[William Lawvere]]'s message (the third one from the top).


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[(∞,1)-topos]] of _synthetic differential super $\infty$-groupoids_ combines the properties of that of

1. [[smooth super ∞-groupoids]]

1. [[synthetic differential ∞-groupoids]].

## Definition

Let [[CartSp]]$_{supersynth}$ be the [[site]] which is the [[full subcategory]] of that of [[Isbell duality|formal duals]] of [[smooth superalgebras]] on those of the form

$$
  \mathbb{R}^p \times D \times \mathbb{R}^{0|q}
  \simeq
  \mathbb{R}^{p|q} \times D
$$

where

* $\mathbb{R}^p$ is the [[Cartesian space]] of [[dimension]] $p$;

* $\mathbb{R}^{p|q}$ is the [[super vector space]] of dimension $(p|q)$ ($\mathbb{R}^{0|1}$ is the [[odd line]]);

* $D$ is an [[infinitesimally thickened point]].

If $D$ here is the formal dual of the [[Artin algebra]] on $k$ commuting nilpotent elements, then such an object is written $\mathbb{R}^{p \oplus k|q}$ in ([Konechny-Schwarz](#KonechnySchwarz)).

Let then

$$
  SynthDiffSuper\infty Grpd
  \coloneqq
  Sh_\infty(CartSp_{supersynth})
$$

be the [[(∞,1)-category of (∞,1)-sheaves]] over this site.

## Related concepts

* [[cohesive (∞,1)-topos]]

  * [[discrete ∞-groupoid]]

  * [[Euclidean-topological ∞-groupoid]]

  * [[smooth ∞-groupoid]]

  * [[synthetic differential ∞-groupoid]]

  * [[super ∞-groupoid]]

  * [[smooth super ∞-groupoid]]

## References

* Anatoly Konechny and [[Albert Schwarz]],

  _On $(k \oplus l|q)$-dimensional supermanifolds_ in _Supersymmetry and Quantum Field Theory_ ([[Dmitry Volkov]] memorial volume) Springer-Verlag (1998) Lecture Notes in Physics, 509 , J. Wess and V. Akulov (editors)([arXiv:hep-th/9706003](http://arxiv.org/abs/hep-th/9706003))

  _Theory of $(k \oplus l|q)$-dimensional supermanifolds_ Sel. math., New ser. 6 (2000) 471 - 486
  {#KonechnySchwarz}

* [[Urs Schreiber]], section 4.5 of: _[[schreiber:differential cohomology in a cohesive topos]]_

* [[Hisham Sati]], [[Urs Schreiber]], Section 3.1.3 of: _[[schreiber:Proper Orbifold Cohomology]]_ ([arXiv:2008.01101](https://arxiv.org/abs/2008.01101))

* [[Urs Schreiber]], _[[schreiber:Introduction to Higher Supergeometry]]_, lecture at _[[Higher Structures in M-Theory 2018]]_, [Durham Symposium](http://www.maths.dur.ac.uk/lms/)

  published in parts as: _Higher Structures in M-Theory_ (with [[Branislav Jurčo]], [[Christian Saemann]], [[Martin Wolf]]), Fortschritte der Physik (2019) ([arXiv:1903.02807](https://arxiv.org/abs/1903.02807), [doi:10.1002/prop.201910001]( https://doi.org/10.1002/prop.201910001))

[[!redirects synthetic differential super ∞-groupoid]]

[[!redirects synthetic differential super infinity-groupoids]]
[[!redirects synthetic differential super ∞-groupoids]]


synthetic differential supergeometry

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## Idea

_Synthetic differential supergeometry_ pairs the concepts of

* [[synthetic differential geometry]]

with those of

* [[supergeometry - contents|supergeometry]].

## Related entries

* [[smooth superalgebra]]

* [[synthetic differential super infinity-groupoid]]

## References

* {#Yetter88} [[David Yetter]], _Models for synthetic supergeometry_, Cahiers, 29, 2 (1988) ([NUMDAM](http://www.numdam.org/item?id=CTGDC_1988__29_2_87_0))

* {#CarchediRoytenberg12} [[David Carchedi]], [[Dmitry Roytenberg]], _On theories of superalgebras of differentiable functions_, Theory and Applications of Categories, Vol. 28, 2013, No. 30, pp 1022-1098. ([arxiv:1211.6134](http://arxiv.org/abs/1211.6134), [TAC](http://www.tac.mta.ca/tac/volumes/28/30/28-30abs.html))

The article

* {#KonechnySchwarz} Anatoly Konechny, [[Albert Schwarz]], _Theory of $(k \oplus l|q)$-dimensional supermanifolds_ Sel. math., New ser. 6 (2000) 471-486

considers manifolds with odd and even nilpotent functions: formal supermanifolds.


synthetic differential topology

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* table of contents
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## Idea

_Synthetic differential topology_ (SDT) is a [[synthetic mathematics|synthetic]] axiomatization of [[differential topology]] in analogy to how [[synthetic differential geometry]] (SDG) is [[synthetic mathematics|synthetic]] axiomatization of [[differential geometry]].

Where in SDG the concept of _[[infinitesimal neighbourhoods]]_ is encoded by the [[axioms]] (the [[Kock-Lawvere axiom]]) in SDT it is _[[germs of spaces]]_ that are being encoded by the axioms.

Notice that the germ of a manifold around a point is in general "larger" than the [[formal neighbourhood]] of that point, reflecting, dually, the fact that there are [[smooth functions]] which are non-vanishing in every [[open neighbourhood]] of that point but all whose [[partial derivatives]] vanish at that point (see also at _[[bump function]]_).

SDT uses the representability of germs by an object defined intrinsically as the points not well-separated from $0 \in R^n$, $\neg \neg \{0\}$ ("[[logical topology]]").

[[!include infinitesimal and local - table]]

## Related concepts

* [[logical topology]]

## References

* [[Marta Bunge]] and [[Eduardo Dubuc]], _Local Concepts in Synthetic Differential Geometry and Germ Representability_, [article](https://www.researchgate.net/publication/230774975_Local_Concepts_in_Synthetic_Differential_Geometry_and_Germ_Representability)

* [[Marta Bunge]], Felipe Gago, _Synthetic aspects of $C^\infty$-mapping II: Mather's theorem for infinitesimally represented germs_, Journal of Pure and Applied Algebra 55 (1988)  213-250 North-Holland ([doi:10.1016/0022-4049(88)90117-X](https://doi.org/10.1016/0022-4049%2888%2990117-X))

* [[Marta Bunge]], Felipe Gago, Ana Maria San Luis, _Synthetic Differential Topology_, 2018, ([CUP](http://www.cambridge.org/gb/academic/subjects/mathematics/logic-categories-and-sets/synthetic-differential-topology)) ([excerpt](http://assets.cambridge.org/97811084/47232/excerpt/9781108447232_excerpt.pdf))

The [[logical topology]] goes back to

* {#Penon81} [[Jacques Penon]], _Infinitésimaux et intuitionnisme_,  Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 22 (1981) no. 1, p. 67-72 ([numdam:CTGDC_1981__22_1_67_0](http://www.numdam.org/item/?id=CTGDC_1981__22_1_67_0))

Review in the context of [[cohesive toposes]], [[modal type theory]] and [[cohesive homotopy type theory]] includes

* [[David Myers]], _Logical Topology and Axiomatic Cohesion_, talk at _[Geometry in Modal Homotopy Type Theory](https://felix-cherubini.de/modal-workshop.html)_ 2019 ([pdf slides](http://www.andrew.cmu.edu/user/fwellen/myers-slides.pdf), [video recording](https://youtu.be/GbzQMsr3Jf4))

[[!redirects SDT]]


synthetic domain theory

#Contents#
* table of contents
{:toc}

## Idea

**Synthetic domain theory**, as its name suggests, is a [[synthetic mathematics|synthetic]] axiomatization of [[domain theory]], typically in [[toposes]]. The slogan for synthetic domain theory is that "domains are sets", or more precisely, "domains are certain intuitionistic sets" in that one works in an [[intuitionistic mathematics|intuitionistic]] [[set theory]] with certain anti-classical [[axioms]] and defines a notion of domain internally. These synthetic domains act like the analytic notions in that domains support arbitrary [[fixed points]] and solutions to non-well-founded recursive domain equations.

## Axiomatics

Specific axioms of SDT vary, but a category for synthetic domain theory is typically a [[topos]] equipped with a [[dominance]] $\Sigma$. From this dominance, a $\Sigma$-[[partial map classifier]] can be constructed called the "lift" [[monad]] $L A$. The lift monad has [[initial algebra of an endofunctor|initial and final algebras]] (as a functor, rather than a monad) called $\omega$ and $\overline\omega$ respectively, with an induced [[monomorphism]] $\omega \to \overline \omega$. Then it is required that the induced function $\Sigma^{\overline \omega} \to \Sigma^\omega$ has an [[inverse]].

An intuition is that maps out of $\omega$ are a synthetic notion of $\omega$-chains (as in [[CPO|$\omega$-CPOs]]) and maps out of $\overline \omega$ are $\omega$-chains with a limit point. Requiring the inverse above then amounts to requiring that every chain in $\Sigma$ has a *unique* limit point.

## Models

There are two main classes of models of synthetic domain theory: [[realizability toposes]] and [[Grothendieck toposes]].

A Grothendieck topos model that is very closely related to the analytic notion of $\omega$-CPO is given by taking sheaves with respect to the [[canonical topology]] on the full subcategory of $\omega$-CPO whose objects are finite products of $\overline \omega$, i.e., the ordinal $\omega + 1$. This model embeds $\omega$-CPO and the embedding preserves (among other things) limits, coproducts and the [[natural numbers object]]. The $\omega$-CPOs can be characterized as certain $\neg\neg$-separated sheaves.

## Related pages

* [[synthetic guarded domain theory]]
* [[synthetic topology]]
* [[synthetic computability theory]]

##References

* [[Martin Hyland]], _First Steps in Synthetic Domain Theory_, Category Theory. Lecture Notes in Mathematics, vol 1488 ([pdf](https://www.dpmms.cam.ac.uk/~martin/Research/Oldpapers/synthetic91.pdf))

* [[Marcelo Fiore]] and [[Gordon Plotkin]], _An extension of models of Axiomatic Domain Theory to models of Synthetic Domain Theory_, CSL 1996 ([doi](https://doi.org/10.1007/3-540-63172-0_36))

* [[Marcelo Fiore]] and [[Giuseppe Rosolini]], _The Category of Cpos From a Synthetic Viewpoint_, ENTCS Vol 6, 1997, ([doi](https://doi.org/10.1016/S1571-0661(05)80165-3))

* [[Marcelo Fiore]] and [[Giuseppe Rosolini]], _Two models of synthetic domain theory_, Journal of Pure and Applied Algebra, Volume 116, Issues 1–3, 28 March 1997, ([doi](https://doi.org/10.1016/S0022-4049(96)00164-8))


synthetic geometry

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## Idea

> To speak of points of space, as if they constituted the positive element of space, is inadmissible ([PdN§254b](Science+of+Logic#PN254b))

> That the line does not consist of points, nor the plane of lines, follows from their concepts ([PdN§256b](Science+of+Logic#PN256b))

_Synthetic geometry_ is about formalization of [[geometry]] by [[axioms]] that directly speak about the fundamental concepts of geometry -- such as points and lines -- instead of about a backdrop for such objects -- such as [[Cartesian spaces]].

In the 19th century, after [[noneuclidean geometries]] were found (Lobachevski, Bolya, Gauss), mathematicians reexamined the [[foundations]] of [[geometry]]. Some schools concentrated on the [[axiom|axiomatic]] construction of geometry,  independent from the tools which were offered by the [[models]] with concrete [[coordinate]] [[algebra of functions|algebras]]. Synthetic geometry in this sense referred to doing geometry without recourse to [[algebras of functions]] and analytic computations. This was especially successful in [[projective geometry]], see _[[synthetic projective geometry]]_.

The main part of synthetic geometry is the study of incidence structures in geometry, sometimes also called [[incidence geometry]]. A modern point of view on incidence geometry is through the theory of [[buildings]].

## Comparison to synthetic differential geometry
 {#ComparisonToSDG}

This perspective of synthetic geometry is somewhat different from the more modern subject called _[[synthetic differential geometry]]_, although the terminology has the same origin. Indeed, in view of the fact that in the axiomatic approach the emphasis is on the relationship between the elements of geometry ([[points]], [[lines]], [[curves]]) and that in modern geometry these are viewed as subvarieties, hence [[subobjects]], in his original axiomatization of _[[synthetic differential geometry]]_ [[William Lawvere]] has reinterpreted the meaning of synthetic geometry as being concerned with the categorical relationship between geometrical objects. However, he was not concerned with working without analytic and algebraic calculations (which do appear in his axioms for [[infinitesimally thickened points]], the _[[Kock-Lawvere axioms]]_); he just wanted to introduce the foundations internally. (But this is different for his later development of [[cohesion|cohesive geometry]] which includes the original synthetic differential geometry via _[[differential cohesion]]_. Here no analysis enters the [[axioms]].)

Lawvere also notes the  failure (for physical applications) of the correspondence between evolution paths and corresponding maps in differential geometry, and wanted to work in [[closed monoidal categories]] (having the [[exponential law]] for [[mapping spaces]]). In [[synthetic differential geometry]], which he created with Kock, Dubuc and others, this enabled also intuitive, pre-Cauchy reasoning with [[differentials]], what also reminded him of 19th century mathematics what probably also influenced on choosing the name. But still, the subject of synthetic differential geometry is non-synthetic in the sense of traditional
synthetic geometry. A genuinely synthetic axiomatization of differential geometry is [[cohesion]].

## Related concepts

| [[synthetic geometry]]  |
|-------------------------|
| [[Euclidean geometry]]  |
| [[hyperbolic geometry]] |
| [[elliptic geometry]]   |

* [[parallel postulate]]

* [[synthetic mathematics]]

* [[geometric homotopy type]]

* [[cohesive homotopy type]]

## References

* Wikipedia, _[Synthetic geometry](http://en.wikipedia.org/wiki/Synthetic_geometry)_

* {#Tarski} [[Alfred Tarski]], _What is elementary geometry?_, in _The axiomatic method. With special reference to geometry and physics_ Proceedings of an International Symposium held at the Univ. of Calif., Berkeley, Dec. 26, 1957-Jan. 4, 1958 (ed. L. Henkin, P. Suppes, and A. Tarski), Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland (1959), pp. 16–29. ([pdf](http://www.geographicknowledge.de/SeminarLFG/Tarski_Whatiselementarygeometry.pdf))

Textbook accounts of the axiomatization of [[Euclidean geometry]] includes these:

* [[Wolfram Schwabhäuser]], [[Wanda Szmielew]], [[Alfred Tarski]], _Mathematische Methoden in der Geometrie_, Springer 1983

* {#Petrunin13} [[Anton Petrunin]], _Euclidean Plane and its Relatives; a minimalistic introduction_, ([arXiv:1302.1630](https://arxiv.org/abs/1302.1630))

Full formalization of the first part of this book in [[Coq]] (as synthetic geometry but following Tarski's work) is discussed at

*  [[Gabriel Brau]], [[Pierre Boutry]], [[Julien Narboux]], _[GeoCoq](http://geocoq.github.io/GeoCoq/)_, _[La géométrie de Tarski en Coq](http://gabrielbraun.free.fr/Geometry/Tarski/)_


synthetic guarded domain theory

#Contents#
* table of contents
{:toc}

## Idea

**Synthetic guarded domain theory** (SGDT), is a field of [[synthetic mathematics]] that provides an alternative to [[synthetic domain theory]], where all [[guarded recursion|guarded recursive]] definitions have (guarded) [[fixed points]].

More generally, the aim of [[synthetic domain theory]] is roughly that computability should be built in the logic. As a result, constructions on domains would be set-theoretic with no extra structure and proofs of continuity are for free.
However, such a theory would yield an unrestricted fixed-point combinator at all types which would render the theory itself inconsistent when viewed as a logical system.

SGDT solves this problem by introducing a notion of "time". For a type $A$, $\triangleright A$ is the type of computations available one step from now.
The $\triangleright$ modality (pronounced "later") yields a restricted fixed-point operator

$$\text{fix} : (\triangleright A \to A) \to A$$

at all types $A$, which is consistent when viewing this theory as a logical framework. In particular, it corresponds to [[Löb's theorem]].

## A model of SGDT

The original model of (single-clock) SGDT is the category of  [[presheaves]] over $\omega$, the first infinite [[ordinal]]. The objects of this category, also known as the [[topos of trees]], here named $\mathcal{S}$, are contravariant set-valued functors $X : \omega^{\text{op}} \to \text{Set}$ which have  the following shape

$$X(0) \xleftarrow{r_0} X(1)\xleftarrow{r_1}  \dots \xleftarrow{r_{n-1}} X(n)\xleftarrow{r_{n}} \dots $$

where the maps $r_i$, for $i \in \omega^{\text{op}}$, are the restriction maps induced by the functor $X$. The arrows of this category are natural transformations between two functors (satisfying the naturality condition).

The later modality $\triangleright$ is an endofunctor in this category defined as $\triangleright A (0) = 1$ and $\triangleright A (n+1) = A(n)$, for an object $A$. Furthermore, for every object $X$ there exists an obvious map $\text{next} : X\to \triangleright X$ witnessing the fact that everything that is available now is also available later.

Inspired by terminology used in the metric spaces (see below), every map $X \to X$ in $\text{Set}^{\omega^\text{op}}$ is called *non-expansive* while a map $f : X \to X$ is *contractive* if it factors through the $\text{next}$ map, i.e. there exists a map $g : \triangleright X \to X$ such that $f = g \circ \text{next}$.

### Fixed-point construction
A $\triangleright$-algebra $(X, f)$ yields a natural transformation with components $f_1 : 1 \to X(0)$ and $f_i : X(i) \to X(i+1)$ commuting with the restriction maps of the object $X$.
It can therefore be thought of as an $\omega$-chain as follows

$$1 \xrightarrow{f_0} X(0) \xrightarrow{f_1} X(1)  \dots \to X(n) \dots $$

The morphism $\text{fix}_f : 1 \to X$ is itself a natural transformation whose components are $f_0 \circ f_1 \circ \dots f_{i-1} \circ f_{i} : 1 \to X(i)$, for all $i \in \omega^{\text{op}}$ satisfying

$$\text{fix}_f = f \circ \text{next} \circ (\text{fix}_f)$$

The idea here is that the recursive call is **guarded** by the $\text{next}$ map.

### Relation with metric spaces
The category of [[complete space|complete]] bisected [[metric space|ultrametric spaces]] (BiCUlt) is a [[coreflective subcategory]] of the topos of trees ([Proposition 5.1](#BMSS2012)).

$$
  \text{BiCUlt}
    \underoverset
      {\underset{}{\leftarrow}}{\overset{\iota}{\hookrightarrow}}{\bot }
  \mathcal{S}
$$

Moreover, the inclusion functor restricts to an [[equivalence of categories]] when considering objects in $\mathcal{S}$ whose restriction maps are surjective, which are sometimes called the total objects or [[flabby sheaves]].

First, recall that every point in a ball $\mathcal{B}$ of an ultrametric space is at its centre while intersecting balls are contained into each other which  means that closed $2^{-n}$-balls partition a space.
Thus, we can partition a metric space $(X, d)$ by defining a family of equivalence relation

$$x =_n y \Leftrightarrow d(x,y) \le 2^{-n}$$

for all $n \in \mathbb{N}$. This family is also known as Complete Ordered Family of Equivalences  (c.o.f.e) [(Di Gianantonio and Miculan, 2003)](#GianantonioM03).
Intuitively, $x$ and $y$ are equal at $n$ if they belong to the same $2^{-n}$-ball.

The inclusion functor $\iota: \text{BiCUlt} \hookrightarrow \mathcal{S} $ takes a metric space $(X,d)$ in CBUlt into the presheaf of its space decompositions

$$(X/=_1) \leftarrow (X/=_2) \leftarrow  \dots \leftarrow (X/=_n) \leftarrow \dots $$

where each restriction map $(X/=_{(n+1)}) \to (X/=_{n})$ takes a $2^{-(n+1)}$-ball to the $2^{-n}$-ball that contains it.

The inclusion functor has a right adjoint $\mathcal{S} \to \text{BiCUlt}$ which maps a presheaf $X$ to the homset $\mathcal{S}(1, X)$ (the global elements of $X$). This is also the limit of $X$ when seen as a diagram in **Set**.
The metric on $\mathcal{S}(1, X)$ is defined as

$$d(x,y) =  \bigsqcap \{ 2^{-n} \mid \forall i \lt n . x_j(\star)  = y_j(\star)  \}$$

for natural transformations $x, y : 1 \to X$. Intuitively, this metric is the greatest lower bound on which the elements of $X$ agree.

A summary of these results can be found in Bizjak's Ph.D. thesis ([Bizjak 16, Section 1.2](#Bizjak16)).

## Axiomatics
[Palombi and Sterling](#PalombiSterling22) provide the following simple axiomatization of a model of (single-clock) synthetic guarded domain theory:

1. An [[elementary topos]] $E$ with [[natural numbers object]].

2. with a [[left exact]] [[endofunctor]] $\triangleright$

3. with a [[natural transformation]] $\text{next}
\;\colon\; 1 \to \triangleright$ satisfying $\triangleright\text{next} = \text{next}\triangleright$

4. supporting [[Löb's theorem|Löb-induction]], i.e., $\phi : \Omega|\triangleright \phi \Rightarrow \phi \vdash \phi$ in the [[internal logic]] of $E$.

Then "multi-clock" SGDT is simply a topos $E$ with an object of clocks $K \colon E$ such that the [[slice topos|slice]] $E/K$ is a [[model]] of single-clock SGDT.

## Type Theory

Mathematics in synthetic guarded domain theory can be formalized in the [[internal language]] of SGDT models. Several presentations have been proposed such as:

* *guarded dependent type theory* ([BGCMB2016](#BGCMB2016)) which is an [[extensional type theory]]

* *clocked cubical type theory* ([KMV2022](#KMV2022)), an [[intensional type theory]] extending [[cubical type theory]].

[[Agda]] includes an implementation of some portions of clocked cubical type theory under the name *[Guarded Cubical Agda](#guardedcubical)*.

## Related pages

* [[synthetic domain theory]]
* [[domain theory]]
* [[guarded recursion]]

## References

* {#BMSS2012} [[Lars Birkedal]], Rasmus Ejlers Møgelberg, Jan Schwinghammer, Kristian Støvring, _First steps in synthetic guarded domain theory: step-indexing in the topos of trees_.  Logical Methods in Computer Science **8** 4 (2012)   [doi:10.2168/LMCS-8(4:1)2012]

* {#BGCMB2016} Aleš Bizjak, Hans Bugge Grathwohl, Ranald Clouston, Rasmus E. Møgelberg, Lars Birkedal. _Guarded Dependent Type Theory with Coinductive Types_. FoSSaCS 2016 ([doi](https://doi.org/10.1007/978-3-662-49630-5_2))

* {#KMV2022} Magnus Baunsgaard Kristensen, Rasmus Ejlers Møgelberg, Andrea Vezzosi _Greatest HITs: Higher inductive types in coinductive definitions via induction under clocks_. LICS 2022 ([preprint](https://arxiv.org/abs/2102.01969))

* {#PalombiSterling22} Daniele Palombi and [[Jonathan Sterling]], _Classifying topoi in synthetic guarded domain theory: The universal property of multi-clock guarded recursion_. Mathematical Foundations of Program Semantics 2022 ([doi](https://doi.org/10.48550/arXiv.2210.04636))

* {#Paviotti16} Marco Paviotti, _Denotational semantics in Synthetic Guarded Domain Theory_. PhD thesis, IT University of Copenhagen ([pdf](https://mpaviotti.github.io/assets/papers/paviotti-phdthesis.pdf))

* {#Bizjak16} Aleš Bizjak, _On Semantics and Applications of Guarded Recursion_. PhD thesis, Aarhus University ([pdf](https://abizjak.github.io/documents/thesis/semantics-applications-gr.pdf))

* {#GianantonioM03} Pietro Di Gianantonio, Marino Miculan, _A Unifying Approach to Recursive and Co-recursive Definitions. In Geuvers, Wiedijk, editors, Proceedings of TYPES’02. LNCS 2646, 2003.

* {#guardedcubical} [Guarded Cubical Agda ](https://agda.readthedocs.io/en/latest/language/guarded-cubical.html)

* [[Jonathan Sterling]], [Bibliography of Guarded Domain Theory](https://www.jonmsterling.com/jms-005S.xml)


synthetic homotopy theory

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#### Type theory
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#### Homotopy
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#### Homotopy type theory
+-- {: .hide}
[[!include homotopy type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[Homotopy type theory]] provides a [[synthetic mathematics|synthetic]]  formalization of [[homotopy theory]] in its modern and most powerful incarnation in the guise of [[(∞,1)-toposes]].

Just like plain [[dependent type theory]] is the [[internal logic]] of
[[locally cartesian closed categories]], so homotopy type theory is the
internal logic of [[locally cartesian closed (∞,1)-categories]], and
homotopy type theory with [[univalence|univalent]] type [[universes]] is the internal
logic of [[elementary (∞,1)-toposes]]. This means that homotopy type
theory provides a "structural" foundation of the kind that William
Lawvere had found in [[topos theory]], but refined to homotopy theory
in the refined guise of (∞,1)-topos theory.

Homotopy type theory natively knows about the formalization of simplicial homotopy theory and higher topos theory without the need to first formalize several textbooks worth of material starting from bare set theory. For instance, the formal proof of the [[Blakers-Massey theorem]] ([FFLL](#FFLL)) does not need to begin by first formalizing what a simplicial set is,
what a Kan fibrancy condition is, what the infinite tower of homotopy
groups is, what weak homotopy equivalences are, what homotopy pushouts
are, how they reflect in long exact sequences of homotopy groups;
because all this is native to homotopy type theory. Accordingly, the
proof, on top of being a formal proof, is elegantly transparent
and of actual practical interest. Moreover, since the formal HoTT
proof generalizes the traditional statement to more general
(∞,1)-toposes, it is actually a genuine new mathematical result of
genuine interest in modern homotopy theory, to practicing
mathematicians not concerned about foundations.

## Cubical synthetic homotopy theory

[Mörtberg & Pujet (2020)](#MörtbergPujet2020) make the case for the use of [[cubical type theory|cubical]] versions of [[HoTT]] in synthetic homotopy theory since [[univalence]] and [[HITs]] are natively supported there, rather than axiomatically added as in HoTT. The path algebra in HoTT is made complicated by the fact that many equalities do not hold definitionally, even in the proof of simple results such as that the torus is equivalent to the product of two circles. The proof of this result is trivial in cubical type theory.

## Related concepts

* [[synthetic topology]]

* [[synthetic differential topology]]

* [[synthetic (infinity,1)-category theory|synthetic $(\infty,1)$-category theory]]

## References

The above text of the Idea section follows [Schreiber 14](https://cs.nyu.edu/pipermail/fom/2014-October/018251.html).

The idea of regarding [[homotopy type theory]] as synthetic homotopy theory is essentially [[Awodey's proposal]]:

* {#Awodey10} [[Steve Awodey]], §3 of: *Type theory and homotopy*,  in: Dybjer P., Lindström S., Palmgren E., Sundholm G. (eds.),  _Epistemology versus Ontology_, Springer (2012) 183-201 $[$[arXiv:1010.1810](http://arxiv.org/abs/1010.1810), [doi:10.1007/978-94-007-4435-6_9](https://doi.org/10.1007/978-94-007-4435-6_9)$]$

Exposition:

* {#BrunerieLicataLumsdaine13} [[Guillaume Brunerie]],  [[Dan Licata]], [[Peter LeFanu Lumsdaine]]: _Homotopy theory in type theory_ (2013) $[$[pdf slides](http://dlicata.web.wesleyan.edu/pubs/bll13homotopy/bll13homotopy.pdf), [[Licata-HomotopyInTypeTheory.pdf:file]], [blog entry 1](https://homotopytypetheory.org/2013/03/08/homotopy-theory-in-homotopy-type-theory-introduction), [blog entry 2](https://homotopytypetheory.org/2013/05/20/homotopy-theory-in-type-theory-progress-report/)$]$

* [[Mike Shulman]], slides 37 onwards in _Homotopical trinitarianism: A perspective on homotopy type theory_, 2018 ([pdf](https://home.sandiego.edu/~shulman/papers/trinity.pdf))

Further discussion in [[homotopy type theory]]:

* {#Rijke19} [[Egbert Rijke]], _Classifying Types_ ([arXiv:1906.09435](https://arxiv.org/abs/1906.09435))

* {#FFLL} [[Favonia|Kuen-Bang Hou (Favonia)]], [[Eric Finster]], [[Dan Licata]], [[Peter LeFanu Lumsdaine]], _A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory_, ([arXiv:1605.03227](https://arxiv.org/abs/1605.03227))

Therefore, most of the articles listed at

* [[mathematics presented in homotopy type theory]]

count as synthetic homotopy theory.

The use of [[cubical type theory]] for synthetic homotopy theory is discussed in:

* {#MörtbergPujet2020} [[Anders Mörtberg]], [[Loïc Pujet]], *Cubical synthetic homotopy theory*,  CPP 2020: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (Jan 2020) 158–171 [[doi:10.1145/3372885.3373825](https://doi.org/10.1145/3372885.3373825), [pdf](https://staff.math.su.se/anders.mortberg/papers/cubicalsynthetic.pdf)]


synthetic mathematics

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## Idea

In "[[analytic versus synthetic|synthetic]]" approaches to the formulation of [[theories]] in [[mathematics]] the emphasis is on [[axioms]] that directly capture the core aspects of the intended [[structures]], in contrast to more traditional "analytic" approaches where axioms are used to encode some basic substrate out of which everything else is then built [[analytic versus synthetic|analytically]].

| analytic | synthetic |
|-----------|---------|
| [[structures]] built out of "point-[[set]]"-substrate |  [[axioms]] imposed on the available [[type theory|types]] |

Often the "synthetic approach" is just referred to as "axiomatic". For instance [[model categories]] were introduced as "axiomatic homotopy theory" and indeed they may be regarded as providing a synthetic axiomatization of [[homotopy theory]], which is not based on (but does subsume) the traditional "point-set model" provided by [[topological spaces]].

## Examples

* **[[synthetic geometry]]** axiomatizes the nature of objects in a [[plane]] or [[hyperplane]] without speaking of the [[plane]] or [[hyperplane]] itself as a collection of points and without speaking of these figures as being [[subsets]] of points.

* **[[synthetic differential geometry]]** is  refinement of this to contemporary research-level mathematics.

  Here instead of defining the concept of _[[smooth manifold]]_ analytically by "point-set methodology" as is traditional (a [[set]] of points equipped with a [[topology]] and a [[smooth structure]], etc.) one imposes one single [[axiom]] scheme in the ambient [[topos]] (the "[[Kock-Lawvere axiom]]" in this case) which essentially ensures that for all [[objects]]/[[types]] $X$ there exists also an object/type $T X$ which behaves as the [[tangent bundle]] of $X$ should. These axioms then turn out to have [[models]] in [[smooth spaces]] (which include [[smooth manifolds]]) but also in other, such as [[algebraic geometry]] and [[supergeometry]].

  Alternatively one may set up synthetic differential geometry via axioms for [[differential cohesion]].

* **[[synthetic homotopy theory]]** -- With the advent of [[homotopy type theory]], which may be regarded to some extent as a further abstraction of axioms similar to those of model categories, it became more common to speak of this as "synthetic homotopy theory", as, for example, in the [[Homotopy Type Theory -- Univalent Foundations of Mathematics|HoTT book]]. For a list of work carried out in homotopy type theory, see at *[[mathematics presented in homotopy type theory]]*

* [[synthetic topology]]

* [[synthetic differential topology]]

* [[synthetic domain theory]]

  * [[synthetic guarded domain theory]]

* [[synthetic algebraic geometry]]

* [[synthetic probability theory]]

* [[synthetic computability theory]]

* {#SyntheticTaitComputatibility} [[synthetic Tait computability]]

* [[synthetic (infinity,1)-category theory|synthetic $(\infty,1)$-category theory]]

## Relation to constructivism

Synthetic approaches are naturally compatible with [[constructive mathematics]]/[[intuitionistic mathematics]], but [[synthetic mathematics]] is about a different issue than [[constructivism]]. For instance even most constructive [[proof assistants]] in existence have in their libraries defined basic mathematics concepts, such as [[topological spaces]], in the traditional analytic way.

## Relation to computer science

There is at least some similarity between synthetic mathematics and [[domain specific embedded programming languages]], see for instance ([Hudak 98, section 3.2](http://ncatlab.org/nlab/show/domain+specific+embedded+programming+language#Hudak98)). In ([Hudak 98, figure 2](http://ncatlab.org/nlab/show/domain+specific+embedded+programming+language#Hudak98)) this shows aspects of a real-world DSL for "geometric region analysis" embedded in [[Haskell]] which under the [[relation between type theory and category theory]]/[[computational trinitarianism]] one immediately recognizes as a fragment of [[synthetic geometry]].

## Related concepts

* [[analytic versus synthetic]]

## References

### General

See also

* [[Andrej Bauer]], _Synthetic Mathematics with an Excursion Into Computability Theory_, 2021 ([slides and video](http://math.andrej.com/2021/02/03/synthetic-mathematics-with-excursion-to-computability/))

* Wikipedia, _[Analytic-synthetic distinction](https://en.wikipedia.org/wiki/Analytic%E2%80%93synthetic_distinction)_

### Synthetic homotopy theory
 {#ReferencesSyntheticHomotopyTheory}

Discussion of _synthetic [[homotopy theory]]_ (typically understood as [[homotopy type theory]]):

* [[Ulrik Buchholtz]], Sec. 3.1 of _Higher Structures in Homotopy Type Theory_ ([arXiv:1807.02177](https://arxiv.org/abs/1807.02177))

* [[Mike Shulman]], slides 37 onwards in _Homotopical trinitarianism:A perspective on homotopy type theory_, 2018 ([pdf](https://home.sandiego.edu/~shulman/papers/trinity.pdf))

* {#Rijke19} [[Egbert Rijke]], _Classifying Types_ ([arXiv:1906.09435](https://arxiv.org/abs/1906.09435))

Via [[cubical type theory]]:

* [[Anders Mörtberg]], [[Loïc Pujet]], *Cubical synthetic homotopy theory*,  CPP 2020: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (Jan 2020) 158–171 [[doi:10.1145/3372885.3373825](https://doi.org/10.1145/3372885.3373825), [pdf](https://staff.math.su.se/anders.mortberg/papers/cubicalsynthetic.pdf)]


synthetic probability theory

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## Idea

The application of [[synthetic mathematics]] to [[probability theory]]. This is related to [[measurable locales]].

## References

* [[Alex Simpson]], _Synthetic probability theory, [slides](http://mysite.science.uottawa.ca/phofstra/Simpson.pdf)


synthetic projective geometry

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## Idea

__Synthetic projective geometry__ is an [[axiom|axiomatic]] approach to  [[projective geometry]] (usually of projective spaces) without use of (algebraic or analytic) [[coordinate]] calculations (unlike the wider, modern study of projective and [[quasiprojective algebraic varieties]]).
In particular, the method does not require that [[projective space]] be defined over an [[algebraically closed field|algebraically closed ground field]], or even a [[field]] at all; for example the noncommutative ground [[division rings]] are possible as well as combinatorial descriptions of finite projective geometries.
In dimension different from $2$ only the projective spaces over division rings are possible, while in dimension $2$ other cases are possible, in particular non-Desarguesean planes (e.g. planes coming from Hall [[quasifield]]s).

## Related concepts

* [[projective plane]]
* [[synthetic geometry]]

## References

* U. Schoenwaelder, [Literatur zur synthetischen projektiven Geometrie](http://www.math.rwth-aachen.de/%7EUlrich.Schoenwaelder/Lit/Geom/litgsprojn.pdf)
* Karl Georg Christian von Staudt, _Geometrie der Lage_, 1857 (projective geometry over reals)
* O. Veblen, J. W. Young, _Projective geometry_, 2 Vols. Blaisdell Publishing Company, 1966
* H. S. M. Coxeter, _Reelle projektive Geometrie der Ebene_,  Mathematische Einzelschriften Band 3. Oldenbourg, 1955; _The Real projective plane_, 1949, 1992; _Projective Geometry_, Springer–Verlag, 1994
* D. R. Hughes, F. C. Piper, _Projective planes_, GTM 6. Springer–Verlag, 1973
* H. Lenz, _Vorlesungen uber projektive Geometrie_, Akademische
Verlagsgesellschaft, Leipzig, 1965
* G. Pickert, _Projektive Ebenen_, Die Grundlehren der math. Wissenschaften 80. Springer–Verlag, 1955, 1975
* A. Seidenberg, _Lectures in projective geometry_, The University Series in Undergraduate Mathematics. D. van Nostrand Company, 1962
* A. Seidenberg, _Pappus implies Desargues_, Amer. Math. Monthly, 83(3):190–192, 1976
* O. Veblen, J.H.M. Wedderburn, _Non-Desarguesian and non-Pascalian geometries_, Trans AMS __8__, 379–388 (1907) [pdf](http://www.ams.org/tran/1907-008-03/S0002-9947-1907-1500792-1/S0002-9947-1907-1500792-1.pdf)
* Marshall Hall Jr., _Projective Planes_, Transactions of the American Mathematical Society __54__: 229–277 (1943) [pdf](http://www.ams.org/journals/tran/1943-054-02/S0002-9947-1943-0008892-4/S0002-9947-1943-0008892-4.pdf)
* [[Charles Weibel]], _Survey of Non-Desarguesian Planes_, Notices of the American Mathematical Society __54__ (10): 1294–1303 (2007) [pdf](http://www.ams.org/notices/200710/tx071001294p.pdf)
* wikipedia en: [category: projective geometry](http://en.wikipedia.org/wiki/Category:Projective_geometry), [Hall plane](http://en.wikipedia.org/wiki/Hall_plane), de: [Projektive Geometrie](http://de.wikipedia.org/wiki/Projektive_Geometrie), [Satz von Pappos](http://de.wikipedia.org/wiki/Satz_von_Pappos)
* Wolfram MathWorld [Projective geometry](http://mathworld.wolfram.com/ProjectiveGeometry.html) (many references!)

The following treatment is not axiomatic but still concentrates mainly on the classical subject

* Lars Kadison, Matthias T. Kromann, _Projective geometry and modern algebra_, Birkhäuser Boston Inc., Boston, MA, 1996
* R. Hartshorne, _Foundations of projective geometry_, Lecture notes Harvard University, W. A. Benjamin, New York 1967

category: geometry


synthetic Tait computability

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## Idea

_Synthetic Tait computability_ is an approach to constructing [[logical relations]] (also known as Tait's method of computability) using the [[internal language]] of [[toposes]] constructed using [[Artin gluing]].

## Terminology

Note that despite the name, synthetic Tait computability is not a subfield of [[synthetic computability theory]], but rather a [[synthetic mathematics|synthetic]] approach to [[logical relations|Tait's method of computability]]. This suggests that synthetic Tait computability is one technique in a larger field of synthetic mathematics internal to a glued topos.

## Related concepts

* [[synthetic computability theory]]

* [[synthetic mathematics]]

* [[cubical type theory]]

## References

* [[Jonathan Sterling]], _First Steps in Synthetic Tait Computability: The Objective Metatheory of Cubical Type Theory_, ([PhD thesis](https://www.jonmsterling.com/pdfs/sterling:2021:thesis.pdf))

* [[Jonathan Sterling]], _Naïve logical relations in synthetic Tait computability_, ([pdf](https://www.jonmsterling.com/papers/sterling-2022-naive.pdf))

Further references are documented on [Jon Sterling's personal website](https://www.jonmsterling.com/jms-005T.xml).

[[!redirects synthetic Tait computability theory]]