successor

# Contents
* automatic table of contents goes here
{: toc}

## Idea

The _successor_ of something is the thing one step after it.

## Successors of natural numbers

Given a [[natural number]] $n$, the __successor__ $n^+$ of $n$ is simply $n + 1$.  In a topos, a [[natural numbers object]] $\mathbb{N}$ is equipped with a successor morphism $\mathbb{N}\to\mathbb{N}$, which, together with its zero element $1\to \mathbb{N}$, is used to characterize its abstract universal property of [[recursion]].

The formula $n^+ \coloneqq n + 1$ may be usefully generalized to other types of numbers; for example, in the [[real numbers]], the [[golden ratio]] $\Phi \approx 0.618$ is the positive number whose successor equals its [[reciprocal]] (or more commonly, the golden ratio $\phi \approx 1.618$ is the common successor and reciprocal of $\Phi$).

## Successors in well-orderings and cardinals

Generalizing successors in $\mathbb{N}$, in any [[well-ordered set]] $S$, the **successor** $w^+$ of an element $w$ is the least element of $S$ which is (strictly) greater than $w$ (if such an element exists).  If $S$ has no [[maximal element]], then the successor map $w \mapsto w^+$ is always defined; it is sometimes used to make recursive definitions.  We say that an element of a well-ordered set *is a successor* if it is the successor of something.

This notion is sometimes also used for some well-ordered proper classes, for example for the classes $\mathbf{Ord}$ of [[ordinal number]]s and $Card$ of [[cardinal numbers]].  (The latter is only well-ordered if we assume the [[axiom of choice]].)  Thus the **successor ordinal** of an ordinal $\alpha$ is the least ordinal greater than $\alpha$, which, if we use the von Neumann definition of ordinals, is $\alpha \cup \{\alpha\}$.  Similarly, the **successor cardinal** of a cardinal $\kappa$ is the least cardinal greater than $\kappa$.  We speak of "a successor ordinal" or "a successor cardinal" to mean an ordinal or cardinal which is the successor of some other ordinal or cardinal, respectively.

Note that if we identify a cardinal number with the least ordinal having that cardinality, as is common in [[material set theory]], then the *successor ordinal* and the *successor cardinal* of that cardinal are different.  For instance, the successor ordinal of $\omega = \aleph_0$ is $\omega+1$ (which is not a cardinal), whereas its successor cardinal is $\aleph_1$ (which is an ordinal but not a successor ordinal).

For definitions by [[transfinite recursion]], one usually specifies the value at $0$, the rule for recursion along the successor map, and a separate rule of recursion for the limiting ordinals (infinite ordinals which are not successors). (For example, the [[von Neumann hierarchy]] of well-founded [[pure set]]s is defined in that way.)  One can (and in [[constructive mathematics]] must) also handle all three cases at once, and the successor function is used there as well.

In [[constructive mathematics]], we take the successor of an ordinal to be $\alpha^+ \coloneqq \alpha \cup \{\alpha\}$ again, but it\'s not clear in what sense this is the smallest ordinal strictly greater than $\alpha$.  (On the one hand, given any [[truth value]] $P$, $\alpha_P \coloneqq \alpha \cup \{\alpha \mid P\}$ satisfies $\alpha \leq \alpha_P \leq \alpha^+$, but we cannot prove that $\alpha_P$ equals either $\alpha$ or $\alpha^+$; on the other hand, $(\alpha_P)^+$ is provably strictly greater than $\alpha$ in a sense, but we cannot prove that $\alpha^+ \leq (\alpha_P)^+$.)

## Successors of material sets

In [[material set theory]], the formula $\alpha^+ \coloneqq \alpha \cup \{\alpha\}$ is generalized from von Neumann ordinals to arbitrary [[pure sets]] (or material sets with [[urelements]]).  The successor $X^+ \coloneqq X \cup \{X\}$ of a material set $X$ is the smallest material set that contains $X$ as both a [[subset]] and a [[member]].

## Related concepts

* [[zero]]

* [[natural numbers object]]

* [[numeral]]

* [[Hartogs number]]

* [[Lindenbaum number]]

[[!redirects successor]]
[[!redirects successors]]

[[!redirects successor ordinal]]
[[!redirects successor cardinal]]
[[!redirects successor ordinals]]
[[!redirects successor cardinals]]

[[!redirects predecessor]]
[[!redirects predecessors]]


successor monad

# The Successor Monad

The successor monad is an example of a [[monad]] on [[Set]]. For $X$ a [[set]], we define $S X$ as a [[disjoint union]] of $S$ and a [[singleton set]] $1$.  Given a [[function]] $f\colon X \to Y$, we have $S f\colon S X \to S Y$ as
$$ \begin {matrix} S f(x) \coloneqq f(x) & x\colon X ;\\
                   S f({*}) \coloneqq * & *\colon 1 \end {matrix} $$

In [[material set theory]] (with the [[axiom of foundation]]), it is handy to define this as $S X \coloneq X \cup \{X\}$; then for $f\colon X \to Y$,
$$ S f(x) \coloneqq \left\{ \begin{matrix} f(x) & x \in X \\ Y & x = X \end{matrix} \right .$$
The structure map $\eta_X$ is the [[inclusion function|inclusion]] of $X$ in $S X$; the map $\mu_X\colon S S X \to S X$ [[restriction|restricts]] to $S X$ as the [[identity function|identity]], and futhermore has as [[image]] $\mu_X(S X) = X$.

The [[hom-set]] of morphisms in the [[Kleisli category]] of $(S,\eta,\mu)$ from $X$ to $Y$ is canonically equivalent (using [[excluded middle]]) to the set of [[partial functions]] from $X$ to $Y$; its [[Eilenberg–Moore category]] is equivalent to the category of [[pointed sets]] and [[pointed function]]s, i.e. functions of the form $1 \to X$ and [[commuting squares]] between such functions.

The successor monad as defined here is also interesting in that it stabilizes the finite [[von Neumann ordinals]] and *monotone* maps between them, and that $\eta$ and $\mu$ are also monotone.  Thus the successor monad restricts as a monad to the augmented [[simplex category]] (this is the opposite of the [[decalage#DecalageComonad|Décalage comonad]]).  Furthermore, every monotone map of finite ordinals can be written as a composite of arrows of the form $S^k \mu_l$ and $S^m \eta_n$.  Indeed, the [[monoidal category]] $\Delta_a$ is generated by the [[monoid object]] $0 \overset{\iota_0}\rightarrow 1 \overset{\mu_0}\leftarrow 2$.

+-- {: .query}
[[JCMcKeown]]: I want to say something like $(S,\eta,\mu)$ _generates_ the (skeletal augmented) simplex category; there is surely a right way to say that, but what is it?

_Toby_:  I don\'t think that it\'s quite true that $S$ generates the simplex category, because you need an object to start applying it to.  But I\'d agree that $(S,\eta,\mu)$ generates it starting from $0$.  I don\'t know any slick way to say that.

I\'ve put in a suggestion by Finn above.  How\'s that?
=--


Sudip Ghosh

* [webpage](https://www.kent.ac.uk/physical-sciences/people/1458/ghosh-sudeep)

## Selected writings

On breakdown of the [[locality]] due to ill-behaved [[perturbative quantum field theory|perturbation series]] of [[perturbative quantum gravity]] and [[perturbative string theory]] for [[n-point functions]] in the limit of large $n$:

* [[Sudip Ghosh]], [[Suvrat Raju]], *Loss of locality in gravitational correlators with a large number of insertions*, Phys. Rev. D 96, 066033 (2017) ([arXiv:1706.07424](https://arxiv.org/abs/1706.07424))

* [[Sudip Ghosh]], [[Suvrat Raju]], *The Breakdown of String Perturbation Theory for Many External Particles*, Phys. Rev. Lett. 118, 131602 (2017) ([arXiv:1611.08003](https://arxiv.org/abs/1611.08003))

category: people


sufficiently cohesive topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Cohesion
+-- {: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A **sufficiently cohesive topos** is a cohesive [[topos]] that has enough connected objects in the sense that every object embeds into a [[connected object]].

This can be viewed as a strong form of [[cohesive topos|cohesiveness]] in the context of [[William Lawvere|Lawvere's]] axiomatic approach to [[gros toposes]]. In fact, in Lawvere ([1986](#Law86)) a big topos of spaces was  defined as (one equivalent to) a sufficiently cohesive topos.

## Terminology

Sufficient cohesion is a relative concept and requires minimally the presence of an [[essential geometric morphism]] $p:\mathcal{E}\to\mathcal{S}$. Here we state it for an [[adjoint quadruple]] between toposes

$$p_!\dashv p^* \dashv p_* \dashv p^!\;\colon\;\mathcal{S}\to\mathcal{E}$$

such that $p^!$ (and hence $p^\ast$) is [[fully faithful]] and $p_!$ preserves [[finite products]].

This (and $\mathcal{E}$ in particular) is called a 'cohesive topos' (over $\mathcal{S}$) at _[[cohesive topos]]_, and will be referred to as a _'weakly cohesive topos'_ in the present entry - a sufficiently cohesive topos in this context corresponds to the three axioms (0-2) for a 'gros topos' in Lawvere ([1986](#Law86)) where the concept of sufficient cohesion was considered for the first time.

A weakly cohesive topos is called _pre-cohesive_ if $p$ furthermore satisfies the [[Nullstellensatz]] i.e. the canonical map $\theta:p_\ast\to p_!$ is an epimorphism. This is the situation explored in Menni ([2014a](#Menni14a), [2014b](#Menni14b)).

A pre-cohesive topos that moreover satisfies the _continuity principle_ that $p_!(X^{p^\ast(Y)})\simeq p_!(X)^{Y}$ natural in $X\in\mathcal{E}$, $Y\in\mathcal{S}$, is called _'cohesive'_ in Lawvere ([2007](#Law07)) where the term 'sufficently cohesive' occurs for the first time although the notion is defined in a more restricted environment than the earlier papers (cf. Lawvere [1986](#Law86), [1992](#Law92), [1999](#Law99)).

## Definitions

+-- {: .num_defn #Contractible_object}
###### Definition
An object $X$ in a weakly cohesive topos $p:\mathcal{E}\to\mathcal{S}$ is called _[[contractible space|contractible]]_ if for every object $Y\in\mathcal{E}$: $p_!(X^Y)=1$.
=--

+-- {: .num_remark}
###### Remark
In particular, a contractible object is connected: $p_!(X)=p_!(X^1)=1$. Since exponentiation $(-)^Y$ preserves $1$ in any Cartesian closed category, the terminal object is contractible in any weakly cohesive topos.
=--

+-- {: .num_defn #Sufficient_Cohesion}
###### Definition
A weakly cohesive topos $p:\mathcal{E}\to\mathcal{S}$ is called _sufficiently cohesive_ if the [[subobject classifier]] $\Omega\in\mathcal{E}$ is contractible i.e. $p_!(\Omega^X)=1$ for every object $X\in\mathcal{E}$ or, in other words, if all [[power object|power objects]] are connected.
=--

+-- {: .num_remark}
###### Remark
Since in general, every object $X$ embeds into its power object via the singleton map $\{\}:X\rightarrowtail\Omega^X$ it follows that in a sufficiently cohesive topos $\mathcal{E}$ every object embeds into a [[connected object]] i.e. $\mathcal{E}$ has _enough connected objects_.

Furthermore, since in a [[Cartesian closed category]]
$$(X^Y)^Z\simeq X^{(Y\times Z)}\simeq X^{(Z\times Y)}\simeq (X^Z)^Y$$
one sees that in a sufficiently cohesive topos power objects $\Omega^X$ are not only connected but even contractible:
$$p_!((\Omega^X)^Z)=p_!(\Omega^{X\times Z})=1$$
and hence it follows that in a sufficiently cohesive topos _every object embeds into a contractible object_.

Conversely, if every power object $\Omega^X$ embeds into a connected object then the power objects $\Omega^X$ will be connected themselves by proposition \ref{connected_retract} below since power objects are injective in general. Whence a (weakly) cohesive topos is sufficiently cohesive iff every object embeds into a connected object iff every object embeds into a contractible object. The last formulation is taken as the definition of sufficient cohesion in Lawvere ([2007](#Law07)).
=--

+-- {: .num_remark}
###### Remark
The definition of sufficient cohesion by 'having enough contractible objects' (Lawvere [2007](#Law07)) has of course the advantage that it even works in the wider context of Cartesian closed [[extensive category|extensive categories]] that lack a subobject classifier. The definition by 'having enough connected objects' would be feasible for merely [[distributive category|distributive categories]] that are not necessarily Cartesian closed (cf. Lawvere [1991](#Law91a), [1992](#Law92)) but Lawvere ([1991](#Law91a), p.4) suggests that in this case the definition should be strenghtened to demand that every object is the equaliser of a pair of maps between two connected objects.
=--

## Properties

Let us first record some easy but useful facts concerning the interplay between connectedness and constancy.

Recall that in a category with a terminal object a morphism $c:X\to Y$ is called _constant_ if $c$ factors through the [[terminal object]] $1$:
$$X\overset{c}{\to} Y=X\overset{!_X}{\to}1\overset{c_\ast}{\to} Y\quad .$$

+-- {: .num_prop #constant_identity}
###### Observation
An object $X$ is a terminal object iff $id_X$ is constant.
=--

**Proof**. "$\Leftarrow$": By assumption $id_X=(id_X)_\ast\circ !_X$. Since $X$ has a point $(id_X)_\ast$ there exists for every object $Z$ at least one map $Z\to X$. Suppose then that $f$ is some map $Z\to X$:

$$f=id_X\circ f=(id_X)_\ast\circ !_X\circ f=(id_X)_\ast\circ !_Z\quad .$$

But the righthand side does not depend on $f$ hence it is the only map $Z\to X$. $\qed$

+-- {: .num_prop #constant_homotopy}
###### Observation
Let $X$ be an object in a weakly cohesive topos and $c:X\to X$ be a constant endomap such that $p_!(c)=p_!(id_X)$. Then $X$ is connected i.e. $p_!(X)=1$.
=--

**Proof**. Since by assumption $p_!(1)=1$, $p_!(c)$ and, accordingly, $p_!(id_X)=id_{p_!(X)}$ are constants. $\qed$

+-- {: .num_prop #connected_retract}
###### Observation
In a weakly cohesive topos, retracts of connected objects are connected themselves.
=--

**Proof**. Let $X$ be a retract of $Y$ with $p_!(Y)=1$. Then $id_X$ factors as $X\rightarrowtail Y\to X$ and applying $p_!$ shows that $id_{p_!(X)}$ factors through the terminal object $p_!(Y)$. $\qed$

Since in general, injective objects $I$ are retracts of the objects $X$ that they embed into because such inclusions $I\rightarrowtail X$ factor through $id_I$ by injectivity, it follows that in a weakly cohesive topos _injective objects that embed into a connected object are connected_ themselves.

In particular, _all_ injective objects are connected in a sufficiently cohesive topos. Since power objects are injective in general one gets the converse as well:

+-- {: .num_prop #sufficient_injective}
###### Proposition
A weakly cohesive topos is sufficiently cohesive iff all injective objects are connected. $\qed$
=--

We can replace 'connected' with 'contractible' in this proposition as the following propositions show.

+-- {: .num_prop #contractinjective}
###### Proposition
Let $\mathcal{E}$ be a topos. Then all injective objects are connected iff all injective objects are contractible.
=--

**Proof**. "$\Rightarrow$": by the following lemma. $\qed$

+-- {: .num_prop #exponential_injective}
###### Lemma
Let $I$ be an injective object in a topos $\mathcal{E}$. Then $I^X$ is injective for any object $X$ , or in other words, exponentials $(_-)^X$ preserve injective objects.
=--

**Proof**. (cf. at [[injective object#Adjuncts_Injectives|injective object]]) We have to show that for every mono $m:S\rightarrowtail B$ the induced function $m^\ast:Hom(B,I^X)\to Hom(S,I^X)$ is onto. From the exponential adjunction one has the following commutative diagram

$$
\array{
Hom(B,I^X) &\overset{m^\ast}{\rightarrow} & Hom(S,I^X)
\\
\simeq \parallel& &\parallel
 \simeq
\\
Hom(X\times B,I) &\underset{(m\times id_X)^\ast}{\rightarrow} &Hom(X\times S,I)
}
$$

whence it suffices to show that $m\times id_X:S\times X\to B\times X$ is a mono since $I$ is injective. But this is easy to see:

$$
\begin{aligned}
(m\times id_X)\circ \langle f_1,f_2\rangle &= (m\times id_X)\circ\langle g_1,g_2\rangle
\\
\langle m\circ f_1, f_2\rangle &=\langle m\circ g_1,g_2\rangle
\end{aligned}
$$
which is equivalent to $\m\circ f_1=m\circ g_1$ and $f_2=g_2$. $\qed$

Summing up we get:

+-- {: .num_prop #sufficient_contractinjective}
###### Proposition
A weakly cohesive topos $\mathcal{E}$ is sufficiently cohesive iff all injective objects are contractible. $\qed$
=--

## Cohesively connected truth

In a sufficiently cohesive topos the subobject classifier is obviously connected since $\Omega=\Omega^1$. It is the aim of this section to prove that in a weakly cohesive topos the converse holds as well: $\Omega$ is connected iff $\Omega$ is contractible.

Before embarking on a proof let us consider two non-examples that display varied behaviors in respect to the  _connectedness_ of the subobject classifier and the _exactness_ properties of the components functor $p_!$:

+-- {: .num_example}
###### Example
The [[Sierpinski topos]] $Set^\to$ is weakly cohesive over $Set$ since there exists a string of adjoint functors $L\dashv\Pi\dashv\Delta\dashv\Gamma\dashv B: Set\to Set^\to$ with

* $L(Z)=\emptyset \to Z$

* $\Pi(X\to Y) = Y$

* $\Delta(Z)=Z\overset{id}{\to} Z$

* $\Gamma (X\to Y) = X$

* $B(Z)=Z\to 1$.

The Nullstellensatz fails as does the continuity principle. As a right adjoint $\Pi$ preserves all limits and the terminal object in particular whence $1$ is connected in $Set^\to$. Since the underlying category $\to$ satisfies the [[Ore condition]] trivially, it follows then from a general result[^bouquet] of Lawvere that $\Omega$ is not connected[^bouquet] and, accordingly, that the _Sierpinski topos is not sufficiently cohesive!_
=--

[^bouquet]: (Theorem 12.2.3 in La Palme Reyes et al. ([2004](#RRZ04), p.221)). Of course, this can also easily be proved directly or read off the concrete objects and properties worked out in La Palme Reyes et al. (2004) where the Sierpinski topos is called the category of bouquets.

+-- {: .num_example}
###### Example

On the other hand, consider the [[Quiv|topos of directed graphs]] $Set^\rightrightarrows$: it lacks the right adjoint $p^!$ to the section functor $p_*$ but has a connected components functor $p_!$ and the [[Quiv#Omega_graph|subobject classifier for directed graphs]] $\Omega$ is connected.

Although $p_!$ does not preserve finite products in general, it nevertheless preserves finite products with the subobject classifier $\Omega$ since $\Omega$ has arcs between all nodes in both directions. This implies that the proof of proposition \ref{homotopy_components} below still goes through and since this is the only argument that hinges on exactness properties of $p_!$ and nothing hinges on (the existence of) the right adjoint $p^!$ at all, the conclusions from \ref{connected_truth} hold as well: _the subobject classifier of $Set^\rightrightarrows$ is not only connected but contractible!_ In other words, despite not being sufficiently cohesive $Set^\rightrightarrows$ nevertheless has enough contractible objects.
=--

Recall that in any topos, the subobject classifier $\Omega$ has two points $\mathsf{true},\mathsf{false}$ fitting into the following pullback diagram (which is an equaliser diagram as well) due to the classifying property of $\Omega$ for the monomorphism $0\to 1$:

$$
\array{
0 &\to & 1
\\
\downarrow & &\downarrow\mathsf{true}
\\
1 &\underset{\mathsf{false}}{\to} &\Omega
}
$$

In a sufficiently cohesive topos $\Omega$ is furthermore connected whence together with its [[Heyting algebra|Heyting algebra structure]] we can view it as a generalized (non linear) "interval" object:

+-- {: .num_defn #connector}
###### Definition
An object $T$ in a weakly cohesive topos is called a _(cohesive) connector_ if $p_!(T)=1$ and $T$ has two points $t_0,t_1:1\to T$ with empty [[equaliser]]: $0\overset{e}{\to} 1\overset{t_0}{\underset{t_1}{\rightrightarrows}} T$.
=--

In a topos with a connected subobject classifier $\Omega$ itself is a connector. Conversely the existence of a connector implies the connectedness of $\Omega$:

+-- {: .num_prop #connecting_truth}
###### Proposition
Let $\mathcal{E}$ be weakly cohesive topos. $\mathcal{E}$ has a connector $T$ iff $p_!(\Omega)=1$.
=--

**Proof**.
"$\Rightarrow$": Let $1\overset{t_0}{\underset{t_1}{\rightrightarrows}} T$ be a connector. Then $t_1:1\to T$ is a subobject with characteristic map $\chi_1:T\to\Omega$. Consider the two composites $\chi_1\circ t_i$ , $i=0,1$:

For $i=1$ this simply yields $\mathsf{true}$ by the definition of $\chi_1$.

For $i=0$ we claim that $\chi_1\circ t_0=\mathsf{false}$ since we have the following diagram:

$$
\array{
0 &\to & 1 &\to & 1
\\
\downarrow & &t_1\downarrow& & \downarrow \mathsf{true}
\\
1&\underset{t_0}{\to} &T&\underset{\chi_1}{\to}&\Omega
}
$$

Here the left square is a pullback since $t_o,t_1$ have empty equaliser by assumption. The right square is a pullback since it classifies $t_1$ whence the outer square is a pullback too. Therefore $\chi_1\circ t_0$ classifies $0\to 1$ which is exactly the definition of $\mathsf{false}$.

Since $p_!(T)$ is terminal $p_!(t_0)=p_!(t_1)$. Whence
$$p_!(\mathsf{true})=p_!(\chi_1\circ t_1)=p_!(\chi_1)\circ p_!(t_1)=p_!(\chi_1)\circ p_!(t_0)=p_!(\chi_1\circ t_0)=p_!(\mathsf{false})\quad$$

This says that $\mathsf{true}$ and $\mathsf{false}$ are in the same connected component but a lattice whose top and bottom elements are in the same component is necessarily connected. $\qed$

One can use connectors to define a (generalized) homotopy relation between maps that behaves well under taking connected components.

+-- {: .num_defn #I-homotopy}
###### Definition
Let $I$ be a connector. Two parallel maps $f,g:A\to B$ are called _I-homotopic_, in signs: $f\sim_I g$, if there exists a map $h:A\times I\to B$ with the property that
$$f=h\circ\langle id_A, t_0\circ !_A\rangle\quad and\quad g=h\circ\langle id_A, t_1\circ !_A\rangle\quad.$$
(In this case, $h$ is also called an ($I$-)homotopy between $f$ and $g$.)
=--

The following result brings together two ingredients for the equivalence between contractability and connectedness of $\Omega$, namely, the preservation of finite products by $p_!$ and $p_!(\Omega)=1$.

+-- {: .num_prop #homotopy_components}
###### Proposition
Let $f=h\circ\langle i, k_1\rangle$ and $g=h\circ\langle i, k_2\rangle$ be a pair of parallel maps in a weakly cohesive topos with the property that $p_!(k_1)=p_!(k_2)$. Then $p_!(f)=p_!(g)$. In particular, $f\sim_I g$ implies $p_!(f)=p_!(g)$.
=--

**Proof**. Since $p_!$ preserves finite products it maps product diagrams to product diagrams whence
$$p_!(h\circ\langle i, k_j\rangle)=p_!(h)\circ p_!(\langle i, k_j\rangle)=p_!(h)\circ \langle p_!( i), p_!(k_j)\rangle\quad,\quad j\in\{1,2\}$$
but these two maps coincide since $p_!(k_1)=p_!(k_2)$ by assumption.

Since for an I-homotopy $k_j=t_j\circ !_A:A\to I$ and, $p_!(I)=1$ by assumption, $p_!(k_j):p_!(A)\to 1$, $j\in\{1,2\}$, and these maps necessarily coincide since $1$ is terminal whence $p_!(t_0\circ !_A)=p_!(t_1\circ !_A)$ whence $p_!(f)=p_!(g)$ as claimed. $\qed$

For the following the monoid structure of $\Omega$ will become important. So let us briefly review the basics:

In a general topos, the Heyting algebra structure endows the subobject classifier with the structure of an [[internal monoid]]: The multiplication is given by conjunction

$$\Omega\times\Omega\overset{\wedge}{\to}\Omega\quad ,
\text{ and the unit by }\quad 1\overset{\mathsf{true}}{\to}\Omega\quad .
$$
The conjunction $\wedge$ is defined as the characteristic map of $1\xrightarrow{\langle\mathsf{true},\mathsf{true}\rangle}\Omega\times\Omega$.

Importantly, the other truth value $1\xrightarrow{\mathsf{false}}\Omega$ plays the role of a (multiplicative) _zero_ with respect to this multiplication.

For the following we need

+-- {: .num_prop #omega_homotopy}
###### Proposition
Let $\mathcal{E}$ be a weakly cohesive topos whose subobject classifier is a connector i.e. $p_!(\Omega)=1$. Then the conjunction $\wedge :\Omega\times\Omega{\to}\Omega$ is an $\Omega$-homotopy from $id_{\Omega}$ to the constant map $\mathsf{false}\circ !_\Omega$.
=--

**Proof**.
We have to show that $\wedge\circ\langle id_\Omega, \mathsf{true}\circ !_\Omega\rangle=id_\Omega$ and $\wedge\circ\langle id_\Omega, \mathsf{false}\circ !_\Omega\rangle=\mathsf{false}\circ !_\Omega$. This is more or less clear from the propositional structure of $\Omega$ but let us spell out the details diagrammatically:

For the first equation, consider the commutative diagram:
$$
\array{
1 &\to & 1
\\
{}_\mathsf{true}\downarrow & &\downarrow_{\langle\mathsf{true},\mathsf{true}\rangle}
\\
\Omega &\xrightarrow{\langle id_\Omega,\mathsf{true}\circ !_\Omega\rangle}&\Omega\times\Omega
}
$$
This pullback pasted to the classifying pullback diagram for $\langle\mathsf{true},\mathsf{true}\rangle$ displays $\wedge\circ\langle id_\Omega, \mathsf{true}\circ !_\Omega\rangle$ as the characteristic map of $\mathsf{true}$ which of course is none other than $id_\Omega$.

For the second, consider the following pullback:
$$
\array{
X &\xrightarrow{!_X} & 1
\\
{}_{\mathsf{true}\circ !_X}\downarrow & &\downarrow_{\langle\mathsf{true},\mathsf{true}\rangle}
\\
\Omega &\xrightarrow{\langle id_\Omega,\mathsf{false}\circ !_\Omega\rangle }&\Omega\times\Omega
}
$$
Chasing the arrows around in the second component yields the equation
$$\mathsf{true}\circ !_X=\mathsf{false}\circ !_\Omega\circ\mathsf{true}\circ !_X=\mathsf{false}\circ !_X$$
but this implies $X=0$ since it corresponds to  the pullback of $\mathsf{true}$ and $\mathsf{false}$. Hence the pasted

$$
\array{
0 &\xrightarrow{!_0} & 1 &\to & 1
\\
\downarrow & &\downarrow_{\langle\mathsf{true},\mathsf{true}\rangle}& &\downarrow _\mathsf{true}
\\
\Omega &\xrightarrow{\langle id_\Omega,\mathsf{false}\circ !_\Omega\rangle }&\Omega\times\Omega&\xrightarrow{\wedge}&\Omega
}
$$

is the classifying pullback of $0\rightarrowtail\Omega$. Since it is easily seen that $\mathsf{false}\circ !_\Omega$ is also the characteristic map of $0\rightarrowtail\Omega$ the claim follows. $\qed$

In order to show that the connectedness of $\Omega$ implies its contractibility we will now lift this $\Omega$-homotopy between $id_\Omega$ and a constant map $\Omega\to\Omega$ to a $\Omega$-homotopy between $id_{\Omega^X}$ and a constant map $\Omega^X\to\Omega^X$.

+-- {: .num_prop #homotopy_lifting}
###### Proposition
Let $\mathcal{E}$ be a weakly cohesive topos with connector $1\overset{t_0}{\underset{t_1}{\rightrightarrows}} T$. Suppose that $m:X\times T\to X$ is a $T$-homotopy between $id_X$ and a constant map $c:X\to X\,$.Then there exists for every object $Y$ a $T$-homotopy $\mu:X^Y\times T\to X^Y$ from $id_{X^Y}$ to a constant endomap $X^Y\to X^Y$.
=--
**Proof**.
 We get $\mu$ from $X\times T \overset{m}{\to}  X$ as follows
$$
\begin{aligned}
X& \to X^{T}\quad\text{by transposal}
\\
X^Y&\to (X^{T})^{Y}\quad\text{by application of the endofunctor (-)}^Y
\\
X^Y &\to (X^Y)^{T}\quad\text{by using rules for powers}
\\
X^Y\times T&\overset{\mu}{\to}X^Y\quad \text{by reversing transposal .}
\end{aligned}
$$

The corresponding terms in the [[internal language]] are
$$
\array{
m=\lambda (x,t).m(x,t)
\\
\lambda x\lambda t.m(x,t)
\\
\lambda f\lambda y\lambda t. m(f(y),t)
\\
\lambda f\lambda t \lambda y. m(f(y),t)
\\
\mu =\lambda (f,t)\lambda y. m(f(y),t)
}
$$
respectively. The last term evaluates to $\lambda f\lambda y.f(y)$ i.e. to $id_{X^Y}$ at $t_0$ and at $t_1$ evaluates to $\lambda f\lambda y.c$ , i.e. the function mapping f to $c^Y$ , the function constantly $c$ on $Y$,  whence $\mu$ is a $T$-homotopy from $id_{X^Y}$ to the constantly  $c^Y$ map. $\qed$

By prop. \ref{omega_homotopy} and the preceding the next is immediate:

+-- {: .num_prop #omega_action}
###### Corollary
Let $\mathcal{E}$ be a weakly cohesive topos whose subobject classifier is a connector. The $\Omega$-homotopy $\wedge :\Omega\times\Omega{\to}\Omega$ lifts to a $\Omega$-homotopy between $id_{\Omega^X}$ and a constant map $\Omega^X\to\Omega^X$. $\qed$
=--

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

In classical [[topology]], a space $X$ with the property that $id_X$ is homotopic to a constant map is called a [[contractible space]]. Hence  proposition \ref{homotopy_lifting} can be viewed as a synthetic internal avatar of the classical fact, that any two parallel maps into a contractible space $X$ are homotopic.

Pursuing this analogy, the preceding result shows that in a weakly cohesive topos with connected subobject classifier, the power object $\Omega^X$ of an object $X$ is akin to the [[cone]] $\mathbf{C}X$ of a topological space $X$ in providing a contractible object to embed into. In this perspective, one can think of a sufficiently cohesive topos as being equipped with a generalized cone construction.
=--

+-- {: .num_prop #connected_truth}
###### Theorem
Let $\mathcal{E}$ be a weakly cohesive topos. Then the subobject classifier $\Omega$ is connected iff $\Omega$ is contractible. In other words,  $\mathcal{E}$ is sufficiently cohesive iff $p_!(\Omega)=1$.
=--

**Proof**.
"$\Rightarrow$": The propositions \ref{omega_action} and \ref{homotopy_components} imply that for all $X$ $p_!(id_{\Omega^X})=id_{p_!(\Omega^X)}$ is a constant map. By observation \ref{constant_homotopy} it then follows that $p_!(\Omega^X)=1$ for all $X$ which is just the definition of $\Omega$ being contractible. $\qed$

For convenience and summary let us collect all the equivalent formulations of sufficient cohesion in one place:

+-- {: .num_prop #sufficient_cohesion_equivalences}
###### Theorem (Lawvere)
A weakly cohesive topos $\mathcal{E}$ is sufficiently cohesive iff $\mathcal{E}$ satisfies the following equivalent conditions:

* The [[subobject classifier]] $\Omega\in\mathcal{E}$ is contractible i.e. $p_!(\Omega^X)=1$ for every object $X\in\mathcal{E}$. 'power objects are connected' or '_truth is contractible_'

* The [[subobject classifier]] $\Omega\in\mathcal{E}$ is connected i.e. $p_!(\Omega)=1$. '_truth is connected_'

* The [[subobject classifier]] $\Omega\in\mathcal{E}$ is a connector. '_truth is a connector_'

* $\mathcal{E}$ has a connector.

* Every object $X\in\mathcal{E}$ embeds into a contractible object. '$\mathcal{E}$ _has enough contractible objects_'

* Every object $X\in\mathcal{E}$ embeds into a connected object. '$\mathcal{E}$ _has enough connected objects_'

* All injective objects are connected.

* All injective objects are contractible. $\qed$
=--

## Related entries

* [[cohesive topos]]

* [[Nullstellensatz]]

* [[quality type]]

* [[connected object]]

* [[injective object]]

## References

* {#JS96}[[Peter Johnstone|P. Johnstone]], _Remarks on Quintessential and Persistent Localizations_ , TAC **2** no.8 (1996) pp.90-99. ([pdf](http://www.tac.mta.ca/tac/volumes/1996/n8/n8.pdf))

* {#J02} [[Peter Johnstone|P. Johnstone]], _Sketches of an Elephant vol. 1_ , Cambridge UP 2002.

* {#JS11} [[Peter Johnstone|P. Johnstone]], _Remarks on Punctual Local Connectedness_ , TAC **25** no.3 (2011) pp.51-63. ([pdf](http://www.tac.mta.ca/tac/volumes/25/3/25-03abs.html))

* {#RRZ04} M. La Palme Reyes, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _Generic Figures and their Glueings_ , Polimetrica Milano 2004.

* {#Law86} [[F. W. Lawvere]], _Categories of Spaces may not be Generalized Spaces as Exemplified by Directed Graphs_, Revista Colombiana de Matemáticas **XX** (1986) pp.179-186. Reprinted with commentary as TAC Reprint **9** (2005) pp.1-7. ([pdf](ftp://ftp.tac.mta.ca/tac/html/tac/reprints/articles/9/tr9.pdf))

* {#Law91a} [[F. W. Lawvere]], _Some Thoughts on the Future of Category Theory_ , pp.1-13 in Springer LNM **1488** (1991).

* {#Law92} [[F. W. Lawvere]], _Categories of Space and Quantity_, pp.14-30 in: J. Echeverria et al. (eds.), _The Space of Mathematics_ , de Gruyter Berlin 1992.

* {#Law99} [[F. W. Lawvere]], _Kinship and Mathematical Categories_ , pp.411-425 in: R. Jackendoff, P. Bloom, K. Wynn (eds.), _Language, Logic, and Concepts - Essays in Memory of John Macnamara_, MIT Press 1999.

* {#Law07} [[F. W. Lawvere]], _Axiomatic cohesion_ , TAC **19** no.3 (2007) pp.41–49. ([pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf))

* {#LawvereMenni15} [[F. W. Lawvere]], [[Matías Menni|M. Menni]], _Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness_, TAC **30** no. 26 (2015) pp.909-932. ([abstract](http://www.tac.mta.ca/tac/volumes/30/26/30-26abs.html))

* {#LawRose03} [[F. W. Lawvere]], [[Robert Rosebrugh|R. Rosebrugh]], _Sets for Mathematics_ , Cambridge UP 2003.

* {#MarmoMenni16}F. Marmolejo, [[Matías Menni|M. Menni]], _On the relation between continuous and combinatorial_ , arXiv:1602.02826 (2016). ([abstract](http://arxiv.org/abs/1602.02826))

* {#Menni14a} [[Matías Menni|M. Menni]], _Sufficient Cohesion over Atomic Toposes_ , Cah. Top. Géom. Diff. Cat. **LV** (2014). ([preprint](https://sites.google.com/site/matiasmenni/SufCohesion12.pdf?attredirects=0))

* {#Menni14b} [[Matías Menni|M. Menni]], _Continuous Cohesion over Sets_ , TAC **29** no.20 (2014) pp.542-568. ([pdf](http://www.tac.mta.ca/tac/volumes/29/20/29-20.pdf))

* {#Menni17} [[Matías Menni|M. Menni]], _Every sufficiently cohesive topos is infinitesimally generated_ , ms. 2017. (to appear in Cah. Top. Géom. Diff. Cat.; [preprint](https://drive.google.com/file/d/0B5-jAxlI_G5xdzdoUllNbDB6Rkk/view))

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

[[!redirects sufficiently cohesive topos]]
[[!redirects Sufficiently cohesive topos]]
[[!redirects sufficiently cohesive toposes]]
[[!redirects sufficiently cohesive topoi]]
[[!redirects sufficiently cohesive]]
[[!redirects sufficient cohesion]]
[[!redirects Sufficient Cohesion]]


Sufian Husseini

* [obituary](https://www.math.wisc.edu/node/695)

* [obituary](https://lists.lehigh.edu/pipermail/algtop-l/2017q4/002964.html)

## Selected writings

On [[configuration spaces of points]]:

* [[Edward Fadell]], [[Sufian Husseini]], _Geometry and topology of configuration spaces_, Springer Monographs in Mathematics (2001), [MR2002k:55038](http://www.ams.org/mathscinet-getitem?mr=2002k:55038), xvi+313

On [[loop spaces]] of [[configuration spaces of points]]:

* [[Edward Fadell]], [[Sufian Husseini]], _The space of loops on configuration spaces and the Majer-Terracini index_, Topol. Methods Nonlinear Anal. Volume 11, Number 2 (1998), 249-271 ([euclid:tmna/1476842829](https://projecteuclid.org/euclid.tmna/1476842829))

category: people


sugaring

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

#Contents#
* table of contents
{:toc}

##Idea

In [[computer science]] and to some extent also in [[formal logic]], **sugaring** refers to the modification of formal notation ([[syntax]]) into a form which is more readable for humans, thereby "sweetening" it for consumption. Syntactic sugar does not add to the functionality or expressivity of the language.

[[Peter Landin]] first used the expression 'syntactic sugaring' in the context of interpretations of terms in the [[λ-calculus]] ([Landin64](#Landin64)).

The reverse process to expressions in the more formal language is sometimes known as 'desugaring'.

## Examples

+-- {: .num_example}
###### Example

In a [[set theory]] where [[classes]] are not "first-class objects" (for example [[ZFC]], but not [[NBG]]), classes are only syntactic sugar and have no independent existence. For instance, if one defines a class $M \coloneqq \{ X | \varphi(X) \}$ by [[comprehension]], then "$X \in M$" is syntactic sugar for "$\varphi(X)$".

=--

+-- {: .num_example}
###### Example

In [[formal linguistics]], natural language versions of formal expressions are described as 'sugarings' (Ranta 1994). Often there are many ways to sugar a formal expression into natural language.

Consider the [[type theory|type theoretic]] expression $\sum_{x: man} \sum_{y: donkey} (x\;owns\; y)$. Ranta (1994, p. 68) gives nine natural language sugarings for it by applying three different operations twice:

* there is a man and there is a donkey and he owns it,
* there is a man and he owns a donkey,
* there is a man and there is a donkey that he owns,
* there is a donkey and a man owns it,
* a man owns a donkey,
* there is a donkey that a man owns,
* there is a man such that there is a donkey and he owns it,
* there is a man who owns a donkey,
* there is a man such that there is a donkey that he owns.

=--

## References

* {#Landin64} [[Peter Landin]], _The mechanical evaluation of expressions_ [[pdf](http://www.cs.cmu.edu/~crary/819-f09/Landin64.pdf)]

* {#Ranta93} [[Aarne Ranta]], _Type theory and the informal language of mathematics_, TYPES '93: Proceedings of the international workshop on Types for proofs and programs, pp. 352–365, ([pdf](/nlab/files/InfLang.pdf))

* {#Ranta94} [[Aarne Ranta]], §1.6, §1.7, §9 in: *Type-theoretical grammar*, Oxford University Press (1994) [[ISBN:9780198538578](https://global.oup.com/academic/product/type-theoretical-grammar-9780198538578)]

See also:

* Wikipedia, *[Syntactic sugar](https://en.wikipedia.org/wiki/Syntactic_sugar)*

[[!redirects syntactic sugar]]


Sugimoto string

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
A [[Green-Schwarz mechanism|non-anomalous]] [[tachyon|tachyon-free]] [[superstring|non-supersymmetric]] string theory obtained from an [[orientifold]] projection of the [[type II string theory|Type IIB string]].

First described in [Sugimoto (1999)](#Sug99), this string theory can be obtained from introducing an [[orientifold plane]] $O9^+$ along with 32 [[D9-brane|$\overline{D9}$]] [[anti D-brane|anti D-branes]]. The resulting theory has the $USp(32)$ [[compact symplectic group]] as a [[gauge group]]. This is in contrast with the [[type I string theory|Type I string]], which can be obtained from a $O9^-$ plane and 32 $D9$ branes, with a gauge group $SO(32)$.

## References

The orientifold projection leading to this string was first described in:

* {#Sug99} [[Shigeki Sugimoto]]. *Anomaly cancellations in type I D-9 - anti-D-9 system and the USp(32) string theory*. Prog. Theor. Phys. **102** (1999) 685-699 \[[hep-th/9905159](https://arxiv.org/abs/hep-th/9905159), [doi:10.1143/PTP.102.685](https://doi.org/10.1143/PTP.102.685)\]


suitable monad

# Suitable monads
* table of contents
{: toc}

## Idea

Given a [[cartesian monad]] $T$ on a [[locally cartesian category]] $C$ one can define a generalized graph and a [[generalized multicategory]] of type $T$ ($T$-multicategory). We want to impose a "suitability" condition on $C$ and $T$ which would ensure that one can talk about a free $T$-multicategory. In other words, the forgetful functor
$$
U : (C,T)Multicat \to (C,T)Graph
$$
from $T$-multicategories to $T$-graphs has a left adjoint $F$; this adjunction induces hence a monad $T^+ = (UF, U\epsilon F,\eta)$ on $C^+ := (C,T)Graph$. Moreover one wants to be able to build new cartesian monads iteratively, namely to ensure the same suitability conditions on the $T^+$, $T^{++}$ and so on. This is used for example in some of the possible inductive definitions of [[opetope]]s.

## Definition

A *category* $C$ is **suitable** (in the sense of [[Tom Leinster]]) if

* $C$ is cartesian
* $C$ has disjoint finite coproducts, which are also stable under pullback
* $C$ has colimits of all nested sequences; they commute with pullback and have monic injections (coprojections)

A *monad* $T = (T,\mu,\eta)$ is **suitable** if

* $(T,\mu,\eta)$ is cartesian
* its underlying endofunctor $T$ preserves colimits of nested sequences

## Properties

Any [[presheaf topos]] is suitable. Any [[finitary monad|finitary]] [[cartesian monad]] is suitable.

The forgetful functor from the category of $T$-multicategories to the category of $T$-graphs has a left adjoint and the adjunction is monadic; the category of $T$-graphs is then suitable and the induced monad is suitable.

## Literature

Section 6.5 and appendix D in

* [[Tom Leinster]], _Higher operads, higher categories_, London Math. Soc. Lec. Note Series __298__, [math.CT/0305049](http://arxiv.org/abs/math.CT/0305049)
* [[Tom Leinster]], _Generalized enrichement for categories and multicategories_,  [math.CT/9901139](http://arxiv.org/abs/math.CT/9901139)

[[!redirects suitable monad]]
[[!redirects suitable monads]]


Sullivan chord diagram

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

#Contents#
* table of contents
{:toc}

## Idea

A _Sullivan chord diagram_ ([Cohen-Godin 03, Def. 1](#CohenGodin03), following [Chas-Sullivan 02](#ChasSullivan02)) is a finite undirected [[graph]] given by attaching the external [[vertices]] of [[trees]], equipped with [[cyclic ordering]] around their [[vertices]], to a number of embedded [[orientation|oriented]] [[circles]], e.g.

such that the embedded circles are [[boundary]] [[connected component|components]] of the induced [[ribbon graph]] [[surface]]:

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

This way Sullivan chord diagrams $D$ induce 2-dimensional [[cobordisms]], namely  [[cospans]] of [[topological spaces]],

$$
  \array{
    && \Sigma_D
    \\
    & \nearrow && \nwarrow
    \\
    \mathllap{ \big( \underset{q}{\sqcup}  S^1  \big) \simeq \; }
    \partial_{in}\Sigma_D
    && &&
    \partial_{out} \Sigma_D
    \mathrlap{ \; \simeq \big( \underset{p}{\sqcup} S^1 \big) }
  }
$$

with the induced [[ribbon graph]] [[surface]] $\Sigma_D$ at the tip, the $p$ [[boundary]] [[connected component|components]] corresponding to the embedded circles (shown above in grey) as one leg, and the remaining $q$ boundary components (shown in blue) as the other.

Applying to this [[cospan]] the [[space of maps]]-[[functor]] $Maps(-,X)$ into any [[topological space]] $X$, yields a [[span]]/[[correspondence]]

$$
  \array{
    && Maps(\Sigma_D, X)
    \\
    & \swarrow && \searrow
    \\
    \big(\mathcal{L}X\big)^q
    && &&
    \big(\mathcal{L}X\big)^p
  }
$$

between [[product topological spaces|products]] of the [[free loop space]] of $X$ with itself. The induced [[integral transform]] operations on the [[ordinary homology]] of the [[free loop spaces]] are generalized [[string topology]]-operation ([Cohen-Godin 03, Section 2](#CohenGodin03)).

(It is for this [[integral transform]] construction that the [[complement]] of the embedded circles is required to be a [[tree]], for this allows to [[contractible topological space|contract]] the [[spaces of maps]] out of these components, up to [[homotopy]].)

Sullivan chord diagrams also naturally [[action|act]] on [[Hochschild homology]]/[[cyclic homology]] ([Tradler-Zeinalian 04](#TradlerZeinalian04), [Tradler-Zeinalian 06](#TradlerZeinalian06)).

## Properties

### Relation to round chord diagrams

Every [[round chord diagram]] is also a Sullivan chord diagram.

### Relation to Jacobi diagrams

Sullivan chord diagrams are much like [[Jacobi diagrams]], but more general in that they may have more than one (disjointly) embedded circle and in that they are not required to be trivalent, and less general in that the complement of the embedded circles is required to be a disjoint union of [[trees]]. The last condition need not be satisfied by [[Jacobi diagrams]], and hence not every Jacobi diagram is a Sullivan diagram.

For example, the following [[Jacobi diagram]] is *NOT* a Sullivan chord diagram:

### Relation to horizontal chord diagrams
 {#RelationToHorizontalChordDiagrams}

One obtains Sullivan chord diagrams with $p$ disjoint embedded circles from [[horizontal chord diagrams]] by closing up strands after acting with a [[permutation]] with $p$ cycles ($p$ [[orbits]])

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

## Related concepts

[[!include chord diagrams and weight systems -- table]]

## Applications

[[!include chord diagrams as multi-trace observables in BMN matrix model -- example]]

## References

In [[string topology]]:

The definition is due to

* {#CohenGodin03} [[Ralph Cohen]], [[Veronique Godin]], _[[A Polarized View of String Topology]]_, In: [[Graeme Segal]], [[Ulrike Tillmann]] (eds.) _Topology, Geometry and Quantum Field Theory_, LMS, Lecture Notes Series 308, 2004 ([arXiv:math/0303003](https://arxiv.org/abs/math/0303003), [doi:10.1017/CBO9780511526398.008](https://doi.org/10.1017/CBO9780511526398.008))

following

* {#ChasSullivan02} [[Moira Chas]], [[Dennis Sullivan]], _Closed string operators in topology leading to Lie bialgebras and higher string algebra_,  In: O.A. Laudal , R. Piene (eds.) _The Legacy of Niels Henrik Abel_, Springer 2004 ([arXiv:math/0212358](https://arxiv.org/abs/math/0212358), [doi:10.1007/978-3-642-18908-1_27](https://doi.org/10.1007/978-3-642-18908-1_27))

Also:

* [[Ralph Kaufmann]], Def. 1.4 of: _Moduli space actions on the Hochschild Co-Chains of a Frobenius algebra I: Cell Operads_ ([arXiv:math/0606064](https://arxiv.org/abs/math/0606064))

Review:

* [[Ralph Cohen]], [[Alexander Voronov]], around Def. 3.2.2 in: _Notes on string topology_ ([arXiv:math/0503625](https://arxiv.org/abs/math/0503625))

* [[Ralph Kaufmann]], Section 4.6 of: _Graphs, Strings, and Actions_, In: Tschinkel Y., Zarhin Y. (eds.) _Algebra, Arithmetic, and Geometry_, Progress in Mathematics, vol 270. Birkhäuser 2009 ([doi:10.1007/978-0-8176-4747-6_5](https://doi.org/10.1007/978-0-8176-4747-6_5))

Action on [[Hochschild homology]]:

* {#TradlerZeinalian04} [[Thomas Tradler]], [[Mahmoud Zeinalian]], _On the Cyclic Deligne Conjecture_, Journal of Pure and Applied Algebra, Volume 204, Issue 2, 1 February 2006, Pages 280-299 ([arXiv:math/0404218](https://arxiv.org/abs/math/0404218), [doi:10.1016/j.jpaa.2005.04.009](https://doi.org/10.1016/j.jpaa.2005.04.009))

* {#TradlerZeinalian06} [[Thomas Tradler]], [[Mahmoud Zeinalian]], _Algebraic String Operations_, K-Theory v. 38, n. 1 (November, 2007): 59-82 ([arXiv:math/0605770](https://arxiv.org/abs/math/0605770), [doi:10.1007/s10977-007-9005-2](http://dx.doi.org/10.1007/s10977-007-9005-2))

[[!redirects Sullivan chord diagrams]]


Sullivan conjecture

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Sullivan conjecture_ (due to [[Dennis Sullivan]], now a [[theorem]] due to [Miller 84](#Miller84)) states that -- under certain conditions and after suitable [[p-adic integers|p-adic]] [[completion]] -- the canonical map for a [[G-space]] from its ordinary [[fixed points]] to its [[homotopy fixed points]] is a [[weak homotopy equivalence]].

A [[proof]] was given in [Carlsson 91](#Carlsson91), using the [[Segal-Carlsson theorem]].

## References

* Wikipedia, _[Sullivan conjecture](https://en.wikipedia.org/wiki/Sullivan_conjecture)_

## Related entries

* [[equivariant cohomology]]

* [[equivariant stable cohomotopy]]

## References

* {#Miller84} [[Haynes Miller]], _The Sullivan conjecture on maps from classifying spaces_, Annals of Mathematics Second Series, Vol. 120, No. 1 (Jul., 1984), pp. 39-87 ([jstor:2007071](http://www.jstor.org/stable/2007071))

* {#Carlsson91} [[Gunnar Carlsson]], _Equivariant stable homotopy and Sullivan's conjecture_. Invent. Math. 103: 497–525, 1991 ([dml:143867](https://eudml.org/doc/143867), [pdf](http://www.digizeitschriften.de/download/PPN356556735_0103/PPN356556735_0103___log38.pdf))

See also

* Wikipedia, _[Sullivan conjecture](https://en.wikipedia.org/wiki/Sullivan_conjecture)_


Sullivan construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The **Sullivan construction** constructs a ([[rational topological space|rational]]) [[topological space]] from a [[dg-algebra]] (a graded commutative cochain dg-algebra in positive degree).

This is the special case of the construction of [[differential forms on presheaves]] from the definition of polynomial [[differential forms on simplices]].

It is one part of an [[equivalence of categories]] between the [[homotopy category]] of (connected, simply connected) [[dg-algebra]]s and that of ([[simply connected]]) [[rational topological space]]s.
As such it is a central tool in [[rational homotopy theory]].

## Definition

Let $\Delta_{Diff} : \Delta \to $ [[Diff]] be the standard smooth [[simplex]]es, and write $\Omega^\bullet_{\mathbb{Q}}(\Delta_{Diff}^n)$ for the [[dg-algebra]] of (polynomial, rational, whatevber) [[differential form]]s on $\Delta^n_{Diff}$.

For $A \in dgAlg_{\mathbb{Q}}$ a [[dg-algebra]], consider the [[simplicial set]]

$$
  |A| : [n] \mapsto Hom_{dgAlg^{op}}( \Omega^\bullet(\Delta^n_{Diff}), A)
  \,.
$$

This, or rather its [[geometric realization]] to a [[rational topological space]], is the Sullivan construction.

## Related concepts

* [[Lie integration]]

## References

See the references at [[rational homotopy theory]].


Sullivan model

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **Sullivan model** of a [[rational space]] $X$ is a particularly well-behaved commutative [[dg-algebra]] [[quasi-isomorphism|quasi-isomorphic]] to the dg-algebra of Sullivan forms on $X$.
These _Sullivan algebras_ are precisely the cofibrant objects in the standard [[model structure on dg-algebras]].

Sullivan models are a central tool in [[rational homotopy theory]].

## Definition

Sullivan models are particularly well-behaved [[differential graded-commutative algebras]] that are equivalent to the dg-algebras of [piecewise polynomial differential forms on topological spaces](differential+forms+on+simplices#PiecewisePolynomialDifferentialForms). Conversely, every rational space can be obtained from a dg-algebra and the _minimal_ Sullivan algebras provide convenient representatives that correspond bijectively to rational homotopy types under this correspondence.

Abstractly, (relative) Sullivan models are the (relative) [[cell complexes]] in the standard [[model structure on dg-algebras]].

We now describe this in detail. First some notation and preliminaries:

+-- {: .num_defn #FiniteType}
###### Definition
**([[of finite type]])**

* A [[graded vector space]] $V$ is _[[of finite type]]_ if in each degree it is finite dimensional. In this case we write $V^*$ for its degreewise dual.

* A [[Grassmann algebra]] is *[[of finite type]]* if it is the Grassmann algebra $\wedge^\bullet V^*$ on a graded vector space of finite type

  (the dualization here is just convention, that will help make some of the following constructions come out nicely).

* A [[CW-complex]] is of finite type if it is built out of finitely many cells in each degree.

=--

For $V$ a $\mathbb{N}$-[[graded vector space]] write
$\wedge^\bullet V$ for the [[Grassmann algebra]] over it. Equipped with the trivial differential $d = 0$ this is a [[semifree dgc-algebra]] $(\wedge^\bullet V, d=0)$.

With $k$ our ground [[field]] we write $(k,0)$ for the corresponding [[dg-algebra]], the tensor unit for the standard [[monoidal category|monoidal structure]] on $dgAlg$. This is the [[Grassmann algebra]] on the 0-vector space
$(k,0) = (\wedge^\bullet 0, 0)$.

+-- {: .num_defn #SullivanAlgebra}
###### Definition
**(Sullivan algebras)**

A **relatived Sullivan algebra** is a [[homomorphism]] of [[differential graded-commutative algebras]] that is an inclusion of the form

$$
  (A,d) \hookrightarrow (A \otimes_k \wedge^\bullet V, d')
$$

for $(A,d)$ any [[dgc-algebra]] and for $V$ some [[graded vector space]], such that

1. there is a [[well ordered set]] $J$ indexing a [[linear basis]] $\{v_\alpha \in V| \alpha \in J\}$ of $V$;

1. writing $V_{\lt \beta} \coloneqq span(v_\alpha | \alpha \lt \beta)$ then for all basis elements $v_\beta$ we have that

  $$
    d' v_\beta \in A \otimes \wedge^\bullet V_{\lt \beta}
    \,.
  $$

This is called a **minimal** relative Sullivan algebra if in addition the condition

$$
  (\alpha \lt \beta) \Rightarrow (deg v_\alpha  \leq deg v_\beta)
$$

holds. For a Sullivan algebra $(k,0) \to (\wedge^\bullet V, d)$ relative to the tensor unit we call the [[semifree dgc-algebra]] $(\wedge^\bullet V,d)$ simply a **Sullivan algebra**, and we call it a **minimal Sullivan algebra** if $(k,0) \to (\wedge^\bullet V, d)$ is a minimal relative Sullivan algebra.

=--

(e.g. [Hess 06, def. 1.10, remark 1.11](#Hess06))

See also the section [Sullivan algebras](model+structure+on+dg-algebras#SullivanAlgebras) at [[model structure on dg-algebras]].

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

The special condition on the ordering in the relative Sullivan algebra says that these morphisms are composites of [[pushouts]] of the [[cofibrantly generated model category|generating cofibrations]] for the [[model structure on dg-algebras]], which are the inclusions

$$
  S(n) \hookrightarrow D(n)
  \,,
$$

where

$$
  S(n) = (\wedge^\bullet \langle c \rangle, d = 0)
$$

is the dg-algebra on a single generator in degree $n$ with vanishing differential, and where

$$
  D(n) = (\wedge^\bullet (\langle b \rangle \oplus \langle c \rangle), d b = c, d c = 0)
$$

with $b$ an additional generator in degree $n-1$.

Therefore for $A \in dgcAlg$, a pushout

$$
  \array{
     S(n) &\stackrel{\phi}{\to}& A
     \\
     \downarrow && \downarrow
     \\
     D(n) &\to& (A \otimes \wedge^ \bullet \langle b \rangle, d b = \phi)
  }
$$

is precisely a choice $\phi \in A$ of a $d_A$-closed element in degree $n$
and results in adjoining to $A$ the element $b$ whose differential is
$d b = \phi$. This gives the condition in the above definition: the differential of any new element has to be a sum of wedge products of the old elements.

Notice that it follows in particular that the cofibrations in $dgAlg_{proj}$ are precisely all the [[retracts]] of relative Sullivan algebra inclusions.

=--

+-- {: .num_remark }
###### Remark
**($L_\infty$-algebras)**

Because they are [[semifree dga]]s, Sullivan dg-algebras $(\wedge^\bullet V,d)$ are (at least for degreewise finite dimensional $V$) [[Chevalley-Eilenberg algebra]]s of [[L-∞-algebra]]s.

The co-commutative differential co-algebra encoding the corresponding [[L-∞-algebra]] is the free cocommutative algebra $\vee^\bullet V^*$ on the degreewise dual of $V$ with differential $D = d^*$, i.e. the one given by the formula

$$
  \omega(D(v_1 \vee v_2 \vee \cdots v_n)) =
  - (d \omega) (v_1, v_2, \cdots,  v_n)
$$

for all $\omega \in V$ and all $v_i \in V^*$.

=--

+-- {: .num_defn }
###### Definition
**(Sullivan models)**

For $X$ a [[simply connected]] [[topological space]] $X$, a **Sullivan (minimal) model** for $X$ is a Sullivan (minimal) algebra $(\wedge^\bullet V^\ast, d_V)$ equipped with a [[quasi-isomorphism]]

  $$
    (\wedge^\bullet V^*, d_V)
      \stackrel{\simeq}{\longrightarrow}
    \Omega^\bullet_{pwpoly}(X)
  $$

  to the dg-algebra of [[piecewise polynomial differential forms]].

=--

## Properties

### As cofibrations

+-- {: .num_prop }
###### Proposition
**(cofibrations are relative Sullivan algebras)**

The cofibrations in the [[projective model structure on differential graded-commutative algebras]] $(dgcAlg_{\mathbb{N}})_{proj}$ are precisely the [[retracts]] of relative  Sullivan algebra inclusions (def. \ref{SullivanAlgebra}).

Accordingly, the cofibrant objects in $(dgcAlg_{\mathbb{N}})_proj{}$ are precisely the [[retracts]] of Sullivan algebras.

=--

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

Minimal Sullivan models are unique up to [[isomorphism]].

=--

e.g [Hess 06, prop 1.18](#Hess06).

### Rationalization

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

Consider the [[derived adjunction]]
$$
  Ho(Top)
   \simeq
  Ho(sSet)
   \underoverset
     {\underset{\mathbb{R} \Omega^\bullet_{poly}}{\longrightarrow}}
     {\overset{\mathbb{L} K_{poly} }{\longleftarrow}}
     {\bot}
  Ho( (dgcAlg_{\mathbb{Q}, \geq 0})^{op} )
$$

induced from the of the [[PL de Rham complex]]-[[Quillen adjunction]]

$$
  (dgcAlg_{\mathbb{Q}, \geq 0}_{proj})^{op}
    \underoverset
      {\underset{K_{poly}}{\longrightarrow}}
      {\overset{\Omega^\bullet_{poly}}{\longleftarrow}}
      {\bot}
  sSet_{Quillen}
$$

([this theorem](rational+homotopy+theory#SullivanRationalizationAdjunction)).

Then: On the [[full subcategory]] $Ho(Top_{\mathbb{Q}, \geq 1}^{fin})$ of [[nilpotent topological space|nilpotent]] [[rational topological spaces]] of [[finite type]] this adjunction restricts to an [[equivalence of categories]]

$$
  Ho(Top_{\mathbb{Q}, \gt 1}^{fin})
   \underoverset
     {\underset{\mathbb{R} \Omega^\bullet_{poly}}{\longrightarrow}}
     {\overset{\mathbb{L} K_{poly} }{\longleftarrow}}
     {\simeq}
  Ho( (dgcAlg_{\mathbb{Q}, \gt 1}^{fin})^{op} )
  \,.
$$

In particular the [[derived adjunction unit]]

$$
  X \longrightarrow K_{poly}(\Omega^\bullet_{pwpoly}(X))
$$

exhibits the [[rationalization]] of $X$.

=--

This is the [[fundamental theorem of dgc-algebraic rational homotopy theory]], see there for more.

It follows that the [[cochain cohomology]] of the cochain complex of [[piecewise polynomial differential forms]] on any topological, hence equivalently that of any of its [[Sullivan models]], coincides with its [[ordinary cohomology]] with coefficients in the [[rational numbers]]:

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

Let $(\wedge^\bullet V^*, d_V)$ be a minimal Sullivan model of a simply connected rational topological space $X$. Then there is an [[isomorphism]]

$$
  \pi_\bullet(X)
    \simeq
  V
$$

between the [[homotopy groups]] of $X$ and the generators of the minimal Sullivan model.

=--

e.g. [Hess 06, theorem 1.24](#Hess06).

### Relation to nilpotent $L_\infty$-algebras

Under the [[formal duality]] between [[L-∞ algebras|$L_\infty$-algebras]] and their [[Chevalley-Eilenberg algebra|Chevalley-Eilenberg]] [[dgc-algebras]],
the connected Sullivan models
correspond [[bijection|bijectively]]
to
[[connective]]
[[nilpotent L-infinity algebra|nilpotent $L_\infty$-algebras]]
([Berglund 2015, Thm. 2.3](#Berglund15)).

### Relation to Whitehead products

See at _[[the co-binary Sullivan differential is the dual Whitehead product]]_.

\linebreak

## Examples

[[!include Sullivan models -- examples]]

## Related concepts

* [[formal dg-algebra]]

* [[minimal dg-module]]

* [[minimal fibration]]

* [[equivariant Sullivan model]]

## References

Original articles:

* [[Dennis Sullivan]], *Differential forms and the topology of manifolds*, in Proc. International  Conf.: *Manifolds Tokyo (1973)*, Univ. Tokyo Press (1975) 37-49 [[pdf](https://www.math.stonybrook.edu/~dennis/publications/PDF/DS-pub-0017.pdf), [[Sullivan-DiffFormsManifolds.pdf:file]]]

* [[Alan J. Deschner]], *Sullivan's theory of minimal models*, MSc thesis, Univ. British Columbia (1976) [[doi:10.14288/1.0080132](http://hdl.handle.net/2429/20052), [pdf](https://open.library.ubc.ca/media/stream/pdf/831/1.0080132/2)]

* {#Sullivan77} [[Dennis Sullivan]], _Infinitesimal computations in topology_, Publications Mathématiques de l'IHÉS, 47 (1977), p. 269-331 ([numdam:PMIHES_1977__47__269_0](http://www.numdam.org/item/PMIHES_1977__47__269_0/))

* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]], _[[On PL deRham theory and rational homotopy type]]_, Memoirs of the AMS, vol. 179 (1976) ([ams:memo-8-179](https://bookstore.ams.org/memo-8-179))

* {#GriffithMorgan13} [[Phillip Griffiths]], [[John Morgan]], _Rational Homotopy Theory and Differential Forms_,  Progress in Mathematics Volume 16, Birkhauser (2013) ([doi:10.1007/978-1-4614-8468-4](https://doi.org/10.1007/978-1-4614-8468-4))

Review and application:

* {#Halperin83} [[Steve Halperin]], _Lectures on minimal models_, Mem. Soc. Math. Franc. no 9/10 (1983) ([doi:10.24033/msmf.294](https://doi.org/10.24033/msmf.294))

* {#FelixHalperinThomas00} [[Yves Félix]], [[Stephen Halperin]], [[Jean-Claude Thomas]], Chapter II of: _Rational Homotopy Theory_, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 ([doi:10.1007/978-1-4613-0105-9](https://link.springer.com/book/10.1007/978-1-4613-0105-9))

* {#Hess06} [[Kathryn Hess]], around def 1.10  of _Rational homotopy theory: a brief introduction_ ([arXiv:math.AT/0604626](http://arxiv.org/abs/math.AT/0604626))

* {#Menichi13} [[Luc Menichi]], *Rational homotopy -- Sullivan models*, in: *Free Loop Spaces in Geometry and Topology*, IRMA Lect. Math. Theor. Phys., EMS (2015) [[arXiv:1308.6685](https://arxiv.org/abs/1308.6685), [doi:10.4171/153](https://doi.org/10.4171/153)]

* {#FelixHalperin17} [[Yves Félix]], [[Steve Halperin]], _Rational homotopy theory via Sullivan models: a survey_,  Notices of the International Congress of Chinese Mathematicians Volume 5 (2017) Number 2 ([arXiv:1708.05245](https://arxiv.org/abs/1708.05245), [doi:10.4310/ICCM.2017.v5.n2.a3](https://dx.doi.org/10.4310/ICCM.2017.v5.n2.a3))

Dual interpretation  as
[[nilpotent L-infinity algebra|nilpotent $L_\infty$-algebras]]:

* {#Berglund15} [[Alexander Berglund]], Def. 2.1 in: _Rational homotopy theory of mapping spaces via Lie theory for $L_\infty$ algebras_,  Homology, Homotopy and Applications, Volume 17 (2015) Number 2 ([arXiv:1110.6145](https://arxiv.org/abs/1110.6145), [doi:10.4310/HHA.2015.v17.n2.a16]( http://dx.doi.org/10.4310/HHA.2015.v17.n2.a16))

[[!redirects Sullivan models]]
[[!redirects Sullivan algebra]]
[[!redirects Sullivan algebras]]
[[!redirects Sullivan minimal algebra]]
[[!redirects Sullivan minimal algebras]]

[[!redirects relative Sullivan model]]
[[!redirects relative Sullivan models]]

[[!redirects relative Sullivan algebra]]
[[!redirects relative Sullivan algebras]]

[[!redirects minimal Sullivan algebra]]
[[!redirects minimal Sullivan algebras]]

[[!redirects minimal Sullivan model]]
[[!redirects minimal Sullivan models]]

[[!redirects Sullivan minimal model]]
[[!redirects Sullivan minimal models]]


Sullivan model of a classifying space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

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

Let $G$ be a [[finite number|finite]] [[dimension|dimensional]] [[simply connected topological space|simply connected]] [[compact Lie group]], with [[Lie algebra]] denoted $\mathfrak{g}$

Then the [[Sullivan minimal model]] of the [[classifying space]] $B G$ is the [[graded algebra]]

$$
  inv^\bullet(\mathfrak{g})
  \; \coloneqq \;
  Sym
  \big(
    \mathfrak{g}^\ast[2]
  \big)^G
$$

of [[invariant polynomials]] on $\mathfrak{g}$, equipped with vanishing [[differential]]:

$$
  \big(
    inv^\bullet(\mathfrak{g}), d = 0
  \big)
  \overset{\simeq}{\longrightarrow}
  CE
  \big(
    \mathfrak{l}
    B G
  \big)
  \,.
$$

=--

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

That the [[rational cohomology]] of $B G$ is given by $inv(\mathfrak{g})$

$$
  inv^\bullet(\mathfrak{g})
  \;\simeq\;
  H^\bullet
  \big(
    B G,
    k
  \big)
$$

is due to [Chern 1950 (11)](#Chern50); [Bott 1973, p. 239 (5 of 15)](#Bott73).

(Here $k$ is the [[ground field]] of [[characteristic zero]]).

That the [[rational cohomology]] algebra of $B G$, with trivial differential, is the [[minimal Sullivan model]] for $B G$ is discussed for instance in [Félix, Oprea & Tanré 20 08](#FelixOpreaTanre08).

=--

## Related concepts

* [[Chern-Weil homomorphism]]

[[!include Sullivan models -- examples]]

## References

* {#Chern50} [[Shiing-shen Chern]], _Differential geometry of fiber bundles_, in: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., (August-September 1950), vol. 2, pages 397-411,  Amer. Math. Soc., Providence, R. I. (1952) ([[Chern-DifferentialGeometryOfFiberBundles.pdf:file]], [full proceedings vol 2 pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1950.2/ICM1950.2.ocr.pdf))

* {#Bott73} [[Raoul Bott]], _On the Chern-Weil homomorphism and the continuous cohomology of Lie-groups_, Advances in Mathematics Volume 11, Issue 3, December 1973, Pages 289-303 (doi:10.1016/0001-8708(73)90012-1)

* {#FelixOpreaTanre08} [[Yves Félix]], [[John Oprea]], [[Daniel Tanré]], _Algebraic models in geometry_, Oxford University Press 2008 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/tanre.pdf), [ISBN:9780199206520](https://global.oup.com/academic/product/algebraic-models-in-geometry-9780199206520))

[[!redirects Sullivan models of classifying spaces]]

[[!redirects Sullivan model of classifying space]]


Sullivan model of a finite G-quotient

#Contents#
* table of contents
{:toc}

## Idea

(...) [[Sullivan model]] of [[quotient spaces]] for [[group]] [[actions]] by [[finite groups]] (...)

(e.g. [Félix-Tanré 10, Section 1](#FelixTanre10))

## Examples

(...) Sullivan models for unordered [[configuration spaces of points]] as [[symmetric group]]-quotients of those for ordered configuration spaces

(e.g. [Félix-Tanré 10](#FelixTanre10))

## Related concepts

[[!include Sullivan models -- examples]]

## References

* {#FelixTanre10} [[Yves Félix]], [[Daniel Tanré]], _Rational homotopy of symmetric products and Spaces offinite subsets_, In: [[Yves Félix]], Gregory Lupton, Samuel B. Smith, _Homotopy theory of function spaces and related topics_, AMS Contemporary Mathematics Volume 51 (2010) ([pdf](http://tanre.org/Pro/Articles_files/SpnFiniteDef.pdf), [doi:10.1090/conm/519](http://dx.doi.org/10.1090/conm/519))

[[!redirects Sullivan models of finite G-quotients]]

[[!redirects Sullivan model of finite G-quotient space]]
[[!redirects Sullivan models of finite G-quotient spaces]]

[[!redirects Sullivan model of finite quotient]]
[[!redirects Sullivan models of finite quotients]]
[[!redirects Sullivan model of finite quotient space]]
[[!redirects Sullivan models of finite quotient spaces]]


Sullivan model of a spherical fibration

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[minimal model]] for a [[spherical fibration]] in [[rational homotopy theory]].

## Properties

### Rational Euler- and Pontryagin-class
 {#RationalEulerAndPontryaginClasses}

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

Let $n \in \mathbb{N}$ be a [[natural number]], $n \geq 1$, let

\[
  \label{SphericalFibration}
  \array{
    S^n &\longrightarrow& E
    \\
    && \big\downarrow
    \\
    && X
  }
\]

be a [[spherical fibration]] of [[topological spaces]] such that $X$ admits a [[Sullivan model]] $A_X \in dgcAlg$. Then a [[Sullivan model]] $A_E$ for the total space $E$ is of the following form:

**$n$ odd**

If $n = 2k+1$ is an [[odd number]], then

\[
  \label{SullivanModelForOddDimensionalSphericalFibration}
  A_E
  \;=\;
  A_X
    \otimes
  \mathbb{Q}\big[
    \omega_{2k+1}
  \big]
  /
  \big(
    d \omega_{2k+1} = c_{2k+2}
  \big)
\]

for some

\[
  \label{RationalEulerClass}
  c_{2k+2} \in A_X
\]

being the rational [[Euler class]] of the [[spherical fibration]].

In particular, if $E = S(V)$ is the [[unit sphere bundle]] of a [[real vector bundle]] $V \to X$, then

$$
  [c_{2k}] = \chi
$$

is the [[Euler class]] of that vector bundle and $\omega_{2k+1}$ is a [[cochain]] on the [[unit sphere bundle]] $S(E)$ which on the [[fundamental class]] of any [[n-sphere|(2k+1)-sphere]] [[fiber]] evaluates to minus unity:

\[
  \label{FiberIntegrationOfOddGenerator}
  \left\langle
    \omega_{2k+1},
    \left[
      S^{2k+1}
    \right]
  \right\rangle
  \;=\;
  -1
  \,.
\]

\linebreak

**$n$ even**

If $n = 2k$ is an [[even number]], then [[generalized the|the]] [[Sullivan model]] $A_E$ for a rank-$2k$ [[spherical fibration]] over some $X$ with [[Sullivan model]] $A_X$ is

\[
  \label{FibS2kModel}
  A_E
  \;=\;
  A_X
    \otimes
  \mathbb{Q}
  \Big[
    \omega_{2k} , \omega_{4k-1}
  \Big]
  /
  \left(
    \array{
      d
      \,
      \omega_{2k}
      &=&
      0
      \\
      d \omega_{4k-1}
      & =&
      -
      \omega_{2k} \wedge \omega_{2k}
      +
      c_{4k}
    }
  \right)
\]

where

1. the new generator $\omega_{2k}$ evaluates to unity on the [[fundamental classes]] of the [[n-sphere|2k-sphere]] [[fibers]] $S^{2k} \simeq E_x \hookrightarrow E$ over each point $x \in X$:

   $$
     \big\langle  \omega_{2k}, [S^{2k}] \big\rangle
     \;=\;
     1
   $$

1. $c_{4k} \in A_X$ is some element in the base algebra, which by (eq:FibS2kModel) is closed and represents the rational [[cohomology class]] of the [[cup product|cup]] square of the class of $\omega_{2k}$:

   $$
     \big[
       c_{4k}
     \big]
     \;=\;
     \big[
       \omega_{2k}
     \big]^2
     \;\in\;
     H^{4k}
     \big(
      X, \mathbb{Q}
     \big)
   $$

   and this class classifies the spherical fibration, rationally.

Moreover, if the [[spherical fibration]] $E \to X$ happens to be the [[unit sphere bundle]] $E = S(V)$ of a [[real vector bundle]] $V \to X$, then

1. the class of $\omega_{2k}$ is $1/2$ the rationalized [[Euler class]] $\chi(\widehat V)$ of the corresponding (...) rank [[reduction of the structure group|reduction]] $\widehat V$ of $V$:

   $$
     \big[ \omega_{2k} \big]
     \;=\;
     \tfrac{1}{2}\chi\big( \widehat V \big)
     \;\in\;
     H^{2k}\big( X, \mathbb{Q} \big)
   $$

1. the class of $c_{4k}$ is $1/4$th the rationalized $k$th [[Pontryagin class]] $p_k(V)$ of $V$:

   $$
     \big[
       c_{4k}
     \big]
     \;=\;
     \tfrac{1}{4}
     p_k(V)
     \;\in\;
     H^{4k}\big( X, \mathbb{Q}\big)
     \,.
   $$

=--

This may be found as [Félix-Halperin-Thomas 00, 15, Example 4, p. 202](#FelixHalperinThomas00), see also [Félix-Oprea-Tanré 16, Prop. 2.3](#FelixOpreaTanre16). The fiber integral (eq:FiberIntegrationOfOddGenerator) follows by [this Prop.](Euler+class#TrivializationOfEulerFormOnUnitSphereBundle).

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

Beware that the Sullivan models for spherical fibrations in Prop. \ref{SullivanModelForSphericalFibration} are not in general _minimal_ Sullivan models.

For example over the [[classifying space]] $B SO(8)$ of [[SO(8)]] with indecomposable [[Euler class]] generator $\chi_8$ the equation $d \omega_7 = \chi_8$ (eq:SullivanModelForOddDimensionalSphericalFibration) for the univeral 7-sperical fibration $S^7 \sslash SO(8) \to B SO(8)$ violates the Sullivan minimality condition (which requires that the right hand side is at least a binary wedge product of generators, or equivalently that the degree of the new generator $\omega_7$ is greater than that of any previous generators).

But the Sullivan models in Prop. \ref{SullivanModelForSphericalFibration} are _relative_ minimal models, relative to the Sullivan model for the base.

This means in particular that the new generators of these models reflect non-[[torsion subgroup|torsion]] [[relative homotopy groups]], but not in general non-torsion absolute homotopy groups.

=--

### Relation to rational mapping space of spheres

By general facts (see at [[∞-action]]) a spherical fibration as in (eq:SphericalFibration) is classified by a map to the [[classifying space]] $B Aut(S^n)$ of the [[automorphism ∞-group]] $Aut(S^n) \hookrightarrow Maps(S^n, S^n)$ inside the [[mapping space]] from $S^n$ to itself, which is those [[connected components]] corresponding to [[degree of a continuous function|degree]] $\pm 1$

$$
  Aut(S^n) \;=\; Maps_{\pm 1}\big( S^n, S^n\big)
  \,.
$$

Hence the spherical fibration is given by the [[homotopy pullback]]

\[
  \label{SphericalFibration}
  \array{
    S^n
      &\longrightarrow&
    E
      &\longrightarrow&
    S^{n}\sslash Aut(S^n)
    \\
    &&
    \big\downarrow
    &{}_{(pb)}&
    \big\downarrow
    \\
    &&
    X
    &\underset{c}{\longrightarrow}&
    B Aut(S^n)
  }
\]

of the universal spherical fibration along a classifying map $c$.

The rational homotopy type of these [[connected components]] of the [[mapping space]] are given by [[Sullivan models of mapping spaces]]:

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

Let $n \in \mathbb{N}$ be a [[natural number]] and $f \colon S^n \to S^n$ a [[continuous function]] from the [[n-sphere]] to itself. Then the [[connected component]] $Maps_f\big( S^n, S^n\big)$ of the [[mapping space]] which contains this map has the following [[rational homotopy theory|rational]] [[homotopy type]]:

\[
  \label{RationalHomotopyTypeOfMappingSpacesSnToSn}
  Maps_f\big( S^n, S^n\big)
  \;\simeq_{\mathbb{Q}}\;
  \left\{
    \array{
      S^n \times S^{n-1} &\vert& n\,\text{even}\,, deg(f) = 0
      \\
      S^{2n-1} &\vert& n \, \text{even}\,, deg(f) \neq 0
      \\
      S^n &\vert& n\, \text{odd}
    }
  \right.
\]

where $deg(f)$ is the [[degree of a continuous function|degree]] of $f$.

=--

([Møller-Raussen 85, Example 2.5](#MollerRaussen85), [Cohen-Voronov 05, Lemma 5.3.5](#CohenVoronov05))

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

Here Prop. \ref{RationalTypeOfMappingSpaceSnToSn} and Prop. \ref{SullivanModelForSphericalFibration} are two aspects of the same situation:

For $n = 2k+1$ an [[odd number]] the rational [[Euler class]] (eq:RationalEulerClass) of the spherical fibration is the class of the rational classifying map to the shift of $S^{2k+1}$ in (eq:RationalHomotopyTypeOfMappingSpacesSnToSn);

for $n = 2k$ an [[even number]] the rational [[Pontryagin class]] (eq:RationalPontryaginClass) of the spherical fibration is the class of the rational classifying map to the shift of $S^{4k-1}$ in (eq:RationalHomotopyTypeOfMappingSpacesSnToSn).

=--

## Related concepts

[[!include Sullivan models -- examples]]

## References

* {#FelixHalperinThomas00} [[Yves Félix]], [[Steve Halperin]], J.C. Thomas, _Rational Homotopy Theory_, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000.

* {#MollerRaussen85} [[Jesper Møller]], [[Martin Raussen]], _Rational Homotopy of Spaces of Maps Into Spheres and Complex Projective Spaces_, Transactions of the American Mathematical Society Vol. 292, No. 2 (Dec., 1985), pp. 721-732 ([jstor:2000242 ](https://www.jstor.org/stable/2000242))

* {#CohenVoronov05} [[Ralph Cohen]], [[Alexander Voronov]], _Notes on string topology_ ([arXiv:math/0503625](https://arxiv.org/abs/math/0503625))

* {#FelixOpreaTanre16} [[Yves Félix]], John Oprea, [[Daniel Tanré]], Prop. 2.3 in _Lie-model for Thom spaces of tangent bundles_, Proc. Amer. Math. Soc. 144 (2016), 1829-1840 ([pdf](http://www.ams.org/journals/proc/2016-144-04/S0002-9939-2015-12829-8/S0002-9939-2015-12829-8.pdf), [doi:10.1090/proc/12829](https://doi.org/10.1090/proc/12829))

[[!redirects Sullivan mode of spherical fibrations]]

[[!redirects Sullivan models of spherical fibrations]]


Sullivan model of a suspension

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## Idea

A [[Sullivan model]] for the ([[reduced suspension|reduced]]) [[suspension]] of a ([[pointed topological space|pointed]]) [[topological space]] $X$ is obtained from a [[Sullivan model]] for $X$ itself essentially by shifting up all generators in degree by one and forcing all their wedge products to vanish.

## Details

See e.g. [Félix-Murillo-Tanré 10, Prop. 5.1](#FelixMurilloTanre10)

## Related concepts

* [[suspension object]]

  * [[suspension]], [[reduced suspension]]

    * [[Freudenthal suspension theorem]]

    * [[Spanier-Whitehead category]]

    * [[suspension isomorphism]]

  * [[suspension of a chain complex]]

* [[rational parametrized stable homotopy theory]]

[[!include Sullivan models -- examples]]

## References

* {#FelixMurilloTanre10} [[Yves Félix]], Aniceto Murillo [[Daniel Tanré]], _Fibrewise stable rational homotopy_, Journal of Topology, Volume 3, Issue 4, 2010, Pages 743–758 ([doi:10.1112/jtopol/jtq023](https://doi.org/10.1112/jtopol/jtq023))

[[!redirects Sullivan model of suspensions]]

[[!redirects Sullivan models for suspensions]]
[[!redirects Sullivan models of suspensions]]

[[!redirects rational model for a suspension]]


Sullivan model of complex projective space

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## Statement

The [[Sullivan model]] of [[complex projective space]] $\mathbb{C}P^n$ is

\[
  \label{TheSullivanModel}
  \mathbb{Q}
  [
    f_2, h_{2n+1}
  ]
  \big/
  \left(
  \begin{aligned}
    d\,f_2 & = 0
    \\
    d\,h_{2n+1} & = (f_2)^{n+1}
  \end{aligned}
  \right)
\]

(e.g. [Félix-Halperin-Thomas 00, p. 203](#FelixHalperinThomas00), [Menichi 13, 5.3](#Menichi13))

## Related concepts

[[!include Sullivan models -- examples]]

## References

* [[Alan J. Deschner]], Exp. 3.5 (b) in: *Sullivan's theory of minimal models*, MSc thesis, Univ. British Columbia (1976) [[doi:10.14288/1.0080132](http://hdl.handle.net/2429/20052), [pdf](https://open.library.ubc.ca/media/stream/pdf/831/1.0080132/2)]

* {#FelixHalperinThomas00} [[Yves Félix]], [[Stephen Halperin]], [[Jean-Claude Thomas]], _Rational Homotopy Theory_, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 ([doi:10.1007/978-1-4613-0105-9](https://link.springer.com/book/10.1007/978-1-4613-0105-9))

* {#Menichi13} [[Luc Menichi]], _Rational homotopy -- Sullivan models_, IRMA Lect. Math. Theor. Phys., EMS ([arXiv:1308.6685](https://arxiv.org/abs/1308.6685))