topological localization

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#### $(\infty,1)$-Topos Theory
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## Idea

A **topological localization** is a left exact [[localization of an (∞,1)-category]] -- in the sense of passing to a [[reflective sub-(∞,1)-category]] -- at a collection of morphisms that are [[monomorphism in an (∞,1)-category|monomorphisms]].

A topological localization of an [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(C)$ is precisely a localization at [[Cech cover]]s for a given [[Grothendieck topology]] on $C$, yielding the corresponding [[(∞,1)-topos]] [[(∞,1)-category of (∞,1)-sheaves|of (∞,1)-sheaves]].

$$
  Sh_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C)
$$

and in fact equivalence classes of such topological localizations are in bijection with [[Grothendieck topology|Grothendieck topologies]] on $C$.

Notice that in general a topological localization is not a [[hypercomplete (∞,1)-topos]]. That in general requires localization further at [[hypercover]]s.

## Definition

Recall that a [[reflective sub-(∞,1)-category]]
$D \stackrel{\stackrel{L}{\leftarrow}}{\hookrightarrow} C$
is obtained by localizing at a collection $S$ of morphisms of $C$.

The class $\bar S$ of all morphisms of $C$ that the left adjoint $L : C \to D$ sends to equivalences is the _strongly saturated class_ of morphisms generated by $S$. By the recognition principle for exact localizations, the functor $L$ is exact if and only if $\bar S$ is stable under the formation of pullbacks.

We now define such localizations where the collection $S$ consists of _[[monomorphism in an (∞,1)-category|monomorphisms]]_ .

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

Call a morphism $f : X \to Y$ in an [[(∞,1)-category]] $C$ a **[[monomorphism in an (∞,1)-category|monomorphism]]** if it is a [[n-truncated object of an (infinity,1)-category|(-1)-truncated object]] in the [[overcategory]] $X_{/Y}$.

Equivalently: if for every object $A \in C$ the induced morphism in the [[homotopy category]] of [[∞-groupoid]]s

$$
   C(A,f) : C(A,X) \to C(A,Y)
$$

exhibits $C(A,X)$ as a [[direct sum|direct summand]] of $C(A,Y)$.

Equivalence classes of monomorphisms into an object $X$ form a [[poset]] $Sub(X)$ of [[subobject]]s of $X$.

=--

This is [[Higher Topos Theory|HTT, p. 460]]

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

The standard example to keep in mind is that of a [[Cech nerve]]. In fact, as the propositions below will imply, this is for the purposes of localizations of an [[(∞,1)-category of (∞,1)-presheaves]] the _only_ kind of example.

Let [[Diff]] be the category of smooth manifolds and $PSh_{(\infty,1)}(Diff)$ the [[(∞,1)-category of (∞,1)-presheaves]] on $Diff$, which may be modeled by the global [[model structure on simplicial presheaves]] on $Diff$.

For $X \in Diff$ a [[manifold]], let $\{U_i \hookrightarrow X\}$ be an [[open cover]]. Let $C(\{U_i\})$ be the [[Cech nerve]] of this cover, the [[simplicial object]] of presheaves

$$
  C(\{U_i\}) =
  \left(
     \cdots
     \coprod_{i j} U_i \cap U_j  \stackrel{\to}{\to}\coprod_{i} U_i
  \right)
  \,.
$$

which we may regard as a [[simplicial presheaf]] and hence as an object of $PSh_{(\infty,1)}(Diff)$.

Then for $V$ any other [[manifold]], we have that

$$
  PSh_{(\infty,1)}(V, C(\{U_i\}))
$$

is the [[∞-groupoid]] whose

* [[object]]s are maps $V \to X$ that factor through one of the $U_i$;

* there is a unique [[morphism]] between two such maps precisely if they factor through a double intersection $U_{i} \cap U_j$;

* and so on.

In the [[homotopy category of an (∞,1)-category|homtopy category]] of [[∞-groupoid]]s, this is equivalent to the [[0-groupoid]]/[[set]] of those maps $V \to X$ that factor through one of the $U_i$. Notice that this constitutes the [[sieve]] generated by the [[covering]] family $\{U_i \to X\}$. This is a subset of the [[0-groupoid]]/[[set]] $PSh_{(\infty)}(V,X) = Hom_{Diff}(V,X)$, hence a [[direct sum]]mand.

=--

+-- {: .num_defn }
###### Definition
**topological localization**

Let $C$ be a [[presentable (∞,1)-category]].

A strongly saturated class $\bar S \subset Mor(C)$ of morphisms is called **topological** if

* there is a subclass $S \subset \bar S$ of _[[monomorphism in an (infinity,1)-category|monomorphisms]]_ that generates $\bar S$;

* under [[pullback]] in $C$ elements in $\bar S$ pull back to elements in $\bar S$.

A [[reflective sub-(∞,1)-category]]

$$
  D \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} C
$$

is called a **topological localization** if the class of morphisms $\bar S := L^{-1}(equiv)$ that $L$ sends to equivalences is topological.

=--

This is [[Higher Topos Theory|HTT, def. 6.2.1.4]]

+-- {: .num_defn }
###### Definition
**$(\infty,1)$-sheaves**

Let $C$ be an [[(∞,1)-site]].

Let $S$ be the collection of all [[monomorphism in an (∞,1)-category|monomorphisms]] $U \to c$ to objects $c \in Y$ (under [[(infinity,1)-Yoneda lemma|Yoneda embedding]]) that correspond to [[covering]] [[sieve]]s in $C$. Say an object $c \in PSh_{(\infty,1)}(C)$ in the [[(∞,1)-category of (∞,1)-presheaves]] on $C$ is an **[[(∞,1)-sheaf]]** if it is an $S$-[[local object]] (i.e. if it satisfies [[descent]] along all morphisms $U \to c$ coming from covering sieves).

Write

$$
  Sh_{(\infty,1)}(C)  \hookrightarrow PSh_{(\infty,1)}(X)
$$

for the [[reflective sub-(∞,1)-category]] on these $(\infty,1)$-sheaves.

=--

This is [[Higher Topos Theory|HTT, def. 6.2.2.6]]

**Warning:** A topological localization is, by definition, a *left exact* localization at a set of monomorphisms; left exactness is part of the definition.  A general localization at a set of monomorphisms need not be left exact.  For instance, the localization at one of the inclusions $1\to 1+1$ is the $(-1)$-truncation, which is not left exact.

## Properties

Let throughout $C$ be a [[locally presentable (∞,1)-category]].

### General

+-- {: .num_cor }
###### Corollary
**(topological localizations are exact)**

Every topological localization is an _exact localization_ in that the reflector $L : C \to D$ preserves [[finite limit]]s.

=--

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

At  Properties of exact localizations it is shown that a reflective localization is exact precisely if the class of morphisms that it inverts is stable under pullback. This is the case for topological localizations by definition.

=--

+-- {: .num_prop }
###### Proposition
**(generation from a small set of morphisms)**

For every topological localization of $C$ at a strongly saturated class $\bar S$ there exists a [[small set]] of [[monomorphism in an (∞,1)-category|monomorphisms]] that generates $\bar S$.

=--

This is [[Higher Topos Theory|HTT, prop. 6.2.1.5]].

+-- {: .un_corollary }
###### Corollary

Every topological localization of $C$ is necessarily [[accessible (∞,1)-category|accessible]] and exact.

=--

This is [[Higher Topos Theory|HTT, cor. 6.2.1.6]]

The following proposition asserts that for the construction of [[(n,1)-topos]]es the notion of topological localization is empty: if colimits commute with products, then already every localization is topological. Accordingly, also the notion of [[hypercompletion]] is relevant only for [[(∞,1)-topos]]es.

+-- {: .un_prop }
###### Proposition
**(localizations of presentable $n$-categories are topological)**

Let $C$ be a [[locally presentable (∞,1)-category|locally presentable]] [[(n,1)-category]] for $n \in \mathbb{N}$ finite with [[universal colimits]]. Then every left exact [[localization of an (∞,1)-category|localization of]] $C$ is a topological localization

=--

This is [[Higher Topos Theory|HTT, prop. 6.4.3.9]].

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

...

=--

This means that every [[(n,1)-topos]] of $n$-sheaves is a localization at [[Cech nerve]]s of covers.

**Remark** Notice in this context the statement found for instance in

* [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], _Hypercovers and simplicial presheaves_ ([web](http://www.math.uiuc.edu/K-theory/0563/))

that a [[simplicial presheaf]] that satisfies [[descent]] on all [[Cech cover]]s already satisfies descent for all _bounded_ hypercovers. If the simplicial presheaf is $n$-truncated for some $n$, then it won't "see" $k$-bounded hypercovers for large enough $k$ anyway, and hence it follows that truncated simplicial presheaves that satisfy Cech descent already satisfy hyperdescent.

This is in line with the above statement that for $n$-toposes with finite $n$ there is no distinction between Cech descent and hyperdescent. The distinction becomes visible only for untruncated $\infty$-presheaves.

### For $(\infty,1)$-presheaf $(\infty,1)$-categories

Let throughout $C$ be a [[small (∞,1)-category]] and write $PSh_{(\infty,1)}(C)$ for the [[(∞,1)-category of (∞,1)-presheaves]] on $C$.

+-- {: .un_prop }
###### Proposition
**sheaves form a topological localization**

If $C$ is endowed with a [[(∞,1)-site|Grothendieck topology]], the inclusion

$$
  Sh_{(\infty,1)}(C)  \hookrightarrow PSh_{(\infty,1)}(C)
$$

is a topological localization.

=--

This is [[Higher Topos Theory|HTT, Prop. 6.2.2.7]].

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

See (∞,1)-category of (∞,1)-sheaves.

=--

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

All topological localizations of $PSh_{(\infty,1)}(C)$ arise this way:

There is a bijection between [[(∞,1)-site|Grothendieck topologies]] on $C$ and equivalence classes of topological localizations of $PSh_{(\infty,1)}(C)$.

=--

This is [[Higher Topos Theory|HTT, prop. 6.2.2.17]].

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

...

=--

### Model category presentation

See at _[[?ech model structure on simplicial sheaves]]_.

## References

Topological localizations are the topic of section 6.2, from def. 6.2.1.5 on, in

* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .

[[!redirects topological localizations]]


topological M-theory

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## Idea

Many aspects of the theory of "[[topological strings]]" (the [[A-model]] and the [[B-model]]) proceed in close analogy (just simpler) to the "physical" [[string theory]]. Accordingly, as the latter can usefully be organized as the dimensional reduction of a conjectured [[UV-completion]] of [[11-dimensional supergravity]] -- [[M-theory]] -- there seems to be an analogous higher dimensional organizational principle for topological strings, hence termed _topological M-theory_.

One way to understand it is as a [[TQFT]]-analog of the [[M2-brane]] [[sigma-model]], the _[[topological membrane]]_.

## Related concepts

* [[bosonic M-theory]]

* [[Hitchin functional]]

* [[M-theory on G2-manifolds]]

* [[7d Chern-Simons theory]]

## References

The notion has been introduced in

* {#NeitzkeVafa04} [[Andrew Neitzke]], [[Cumrun Vafa]], section 7 of: *Topological strings and their physical applications_*, [hep-th/0410178](http://xxx.lanl.gov/abs/hep-th/0410178)

* [[Robbert Dijkgraaf]], [[Sergei Gukov]], [[Andrew Neitzke]], [[Cumrun Vafa]], *Topological M-theory as Unification of Form Theories of Gravity*, Adv. Theor. Math. Phys. **9** (2005) 603-665 [[arXiv:hep-th/0411073](http://arxiv.org/abs/hep-th/0411073), [doi:10.4310/ATMP.2005.v9.n4.a5](https://dx.doi.org/10.4310/ATMP.2005.v9.n4.a5)]

Under the term "[[Z-theory]]" aspects were discussed in

* [[Nikita Nekrasov]], _Z Theory_, ([arXiv:hep-th/0412021](http://arxiv.org/abs/hep-th/0412021))

Overview:

* {#Neitzke2005} [[Andrew Neitzke]], *Elements of topological M-theory* (2005) [[[Neitzke-TopologicalMTheory.pdf:file]]]

The _[[topological membrane]]_ is made explicit in

* Ling Bao, Viktor Bengtsson, [[Martin Cederwall]], [[Bengt Nilsson]], _Membranes for Topological M-Theory_ ([arXiv:hep-th/0507077](http://arxiv.org/abs/hep-th/0507077))
 {#BaoBengtssonCederwallNillson05}

* Giulio Bonelli, Alessandro Tanzini, [[Maxim Zabzine]], _On topological M-theory_ ([arXiv:hep-th/0509175](http://arxiv.org/abs/hep-th/0509175))

Relation to [[topological string theory]]:

* [[Jan de Boer]], [[Paul de Medeiros]], Sheer El-Showk, Annamaria Sinkovics, _Open $G_2$ Strings_ ([arXiv:hep-th/0611080](http://arxiv.org/abs/hep-th/0611080))

Relation to [[BTZ black holes]]:

* Javier Chagoya, Graciela Reyes-Ahumada, M. Sabido, _BTZ entropy from topological M-theory_ ([arXiv:2011.01094](https://arxiv.org/abs/2011.01094))


topological M5-brane

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## Idea

A [[membrane]] [[sigma-model]] [[topological quantum field theory]]. Roughly related to [[topological M-theory]] as the [[M5-brane]] is related to [[M-theory]] and to the [[topological M2-brane]]  as the [[M5-brane]] is related to the [[M2-brane]].

The [[target space]] of the topological membrane is a [[G2-manifold]], the [[action functional]] is governed by the [[higher holonomy]] of the compatible [[C6-field]] over the membrane [[worldvolume]].

## Related concepts

[[!include table of branes]]

## References

* Ling Bao, [[Martin Cederwall]], Bengt E.W. Nilsson, _A Note on Topological M5-branes and String-Fivebrane Duality_ ([arXiv:hep-th/0603120](http://arxiv.org/abs/hep-th/0603120))

[[!redirects topological M5-branes]]


topological machine learning

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#### Constructivism, Realizability, Computability
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## Idea

In *topological machine learning* one tries to combine methods from both [[topological data analysis]] and [[machine learning]] for application to [[data analysis]].

A popular approach (originating around [RHBK15](#RHBK15), [KHNL15](#KHNL15)) is to use [[persistent homology]] as a pre-processing stage and train [[neural networks]] not on full data sets, but on their [[persistence diagram]]/[[barcode]] (which may  still be a large amount of data), by making these amenable to [[kernel methods]].

## References

On feeding [[persistence diagrams]] into [[machine learning]]-algorithms by equipping them with [[kernel method|kernels]]:

* {#RHBK15} Jan Reininghaus, [[Stefan Huber]], Ulrich Bauer, Roland Kwitt, *A Stable Multi-Scale Kernel for Topological Machine Learning*, NIPS'15: Proceedings of the 28th International Conference on Neural Information Processing System, **2** (2015 3070–3078  $[$[arXiv:1412.6821](https://arxiv.org/abs/1412.6821)$]$

* {#KHNL15} Roland Kwitt, [[Stefan Huber]], Marc Niethammer, Weili Lin, [[Ulrich Bauer]], *Statistical Topological Data Analysis -- A Kernel Perspective*, in: *Advances in Neural Information Processing Systems*  (NIPS 2015) $[$[ISBN:9781510825024](https://proceedings.neurips.cc/paper/2015), [doi:10.5555/2969442.2969582](https://dl.acm.org/doi/10.5555/2969442.2969582)$]$

* {#RSL} [[Bastian Rieck]], Filip Sadlo, Heike Leitte, *Topological Machine Learning with Persistence Indicator Functions*,  In: *Topological Methods in Data Analysis and Visualization V* TopoInVis (2017) 87-101 Mathematics and Visualization. Springer, $[$[arXiv:1907.13496](https://arxiv.org/abs/1907.13496), [doi:10.1007/978-3-030-43036-8_6](https://doi.org/10.1007/978-3-030-43036-8_6)$]$

* Raphael Reinauer, Matteo Caorsi, Nicolas Berkouk, *Persformer: A Transformer Architecture for Topological Machine Learning* $[$[arXiv:2112.15210](https://arxiv.org/abs/2112.15210)$]$

Review:

* Felix Hensel, Michael Moor, [[Bastian Rieck]], *A Survey of Topological Machine Learning Methods*, Front. Artif. Intell., **26** (2021) $[$[doi:10.3389/frai.2021.681108](https://doi.org/10.3389/frai.2021.681108), [[HenselMoorRieck-TopologicalMachineLearning.pdf:file]]$]$

See also:

* [[Stefan Huber]], *Topological machine learning* ([webpage](https://www.sthu.org/research/topmachinelearning/))


topological manifold

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#### Manifolds and cobordisms
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## Idea

A _topological manifold_ is a [[topological space]] (usually required to be [[Hausdorff topological space|Hausdorff]] and [[paracompact topological space|paracompact]]) which is _locally_ [[homeomorphism|homeomorphic]] to a [[Euclidean space]] $\mathbb{R}^n$ equipped with its [[metric topology]].

Often one is interested in extra structure on topological manifolds, that make them for instance into [[differentiable manifolds]] or [[smooth manifolds]] or [[analytic manifolds]] or [[complex manifolds]], etc. See at _[[manifold]]_ for more on the general concept.

Topological manifolds form a category [[TopMfd]].

## Definition

### Locally Euclidean topological spaces
 {#LocallyEuclideanTopologicalSpace}

+-- {: .num_defn #LocallyEuclideanSpace}
###### Definition
**([[locally Euclidean topological space]])**

A [[topological space]] $X$ is _[[locally Euclidean topological space|locally Euclidean]]_ if every point $x \in X$ has an [[open neighbourhood]] $U_x \supset \{x\}$ which is [[homeomorphism|homeomorphic]] to the [[Euclidean space]] $\mathbb{R}^n$ with its [[metric topology]]:

$$
  \mathbb{R}^n
    \overset{\phantom{AA} \simeq \phantom{AA}}{\longrightarrow}
  U_x
    \subset
  X
  \,.
$$

=--

The "local" [[topological properties]] of Euclidean space are inherited by locally Euclidean spaces:

+-- {: .num_prop #LocalPropertiesOfLocallyEuclideanSpace}
###### Proposition
**([[locally Euclidean spaces]] are [[T1|$T_1$]]-[[separation axiom|separated]], [[sober topological space|sober]], [[locally connected topological space|locally connected]], [[locally compact topological space|locally compact]])**

Let $X$ be a [[locally Euclidean space]] (def. \ref{LocallyEuclideanSpace}). Then

1. $X$ satisfies the $T_1$ [[separation axiom]];

1. $X$ is [[sober topological space|sober]];

1. $X$ is [[locally connected topological space|locally connected]];

1. $X$ is [[locally compact topological space|locally compact]] in the sense that every open neighbourhood of a point contains a [[compact topological space|compact]] neighbourhood.

=--

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

Regarding the first statement:

Let $x \neq y$ be two distinct points in the locally Euclidean space. We need to show that there is an open neighbourhood $U_x$ around $x$ that does not contain $y$.

By definition, there is a Euclidean open neighbourhood $\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} U_x \subset X$ around $x$. If $U_x$ does not contain $y$, then it already is an open neighbourhood as required. If $U_x$ does contain $y$, then $\phi^{-1}(x) \neq \phi^{-1}(y)$ are equivalently two distinct points in $\mathbb{R}^n$. But Euclidean space, as every [[metric space]], is $T_1$, and hence we may find an open neighbourhood $V_{\phi^{-1}(x)} \subset \mathbb{R}^n $ not containing $\phi^{-1}(y)$. By the nature of the [[subspace topology]], $\phi(V_{\phi^{-1}(x)}) \subset X$ is an open neighbourhood as required.

Regarding the second statement:

We need to show that the map

$$
  Cl(\{-\}) \;\colon\; X \to IrrClSub(X)
$$

that sends points to the [[topological closure]] of their [[singleton sets]] is a [[bijection]] with the set of [[irreducible closed subsets]]. By the first statement above the map is [[injective function|injective]] (via [this lemma](separation+axioms#T1InTermsOfClosureOfPoints)).
Hence it remains to see that every irreducible closed subset is the topological closure of a singleton. We will show something stronger: every irreducible closed subset is a singleton.

So let $P \subset X$ be an open proper subset such that if there are two open subsets $U_1, U_2 \subset X$ with $U_1 \cap U_2 \subset P$ then $U_1 \subset P$ or $U_2 \subset P$. By [this prop.](irreducible+closed+subspace#OpenSubsetVersionOfClosedIrreducible) we need to show that there exists a point $x \in X$ such that $P = X \setminus \{x\}$ it its [[complement]].

Now since $P \subset X$ is a proper subset, and since the locally Euclidean space $X$ is covered by Euclidean neighbourhoods, there exists a Euclidean neighbourhood $\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} U \subset X$ such that $P \cap U \subset U$ is a proper subset. In fact this still satisfies the condition that for $U_1, U_2 \underset{\text{open}}{\subset} U$ then $U_1 \cap U_2 \subset P \cap U$ implies $U_1 \subset P \cap U$ or $U_2 \subset P \cap U$. Accordingly, by [that prop.](irreducible+closed+subspace#OpenSubsetVersionOfClosedIrreducible) it follows that $\mathbb{R}^n \setminus \phi^{-1}(P \cap U)$ is an irreducible closed subset of [[Euclidean space]]. Sine [[metric spaces]] are [[sober topological space]] as well as $T_1$-[[separation axiom|separated]], this means that there exists $x \in \mathbb{R}^n$ such that $\phi^{-1}(P \cap U) = \mathbb{R}^n \setminus \{x\}$.

In conclusion this means that the restriction of an irreducible closed subset in $X$ to any Euclidean chart is either empty or a singleton set. This means that the irreducible closed subset must be a disjoint union of singletons that are separated by Euclidean neighbourhoods. But by irreducibiliy, this union has to consist of just one point.

Regarding the third statement:

Let $x \in X$ be a point and $U_x \supset \{x\}$ a neighbourhood. We need to find a [[connected topological space|connected]] open neighbourhood $Cn_x \subset U_x$.

By local Euclideanness, there is also a Euclidean neighboruhood $\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} V_x \subset X$. Since $\phi$ is a [[homeomorphism]], and since $U_x \cap V_x$ is open, also $\phi^{-1}(U_x \cap V_x) \subset \mathbb{R}^n$ is open. This means that there exists an [[open ball]] $B_{\phi^{-1}(x)}^\circ(\epsilon) \subset \phi^{-1}(U_x \cap V_x)$. This is open and connected, and hence so is its homeomorphic image $\phi(B^\circ_{\phi^{-1}(x)}(\epsilon)) \subset X$. This is a connected open neighbourhood of $x$ as required.

Regarding the fourth statement:

Let $x \in X$ be a point and let $U_x \supset \{x\}$ be an open neighbourhood. We need to find a compact neighbourhood $K_x \subset U_x$.

By assumption there exists a Euclidean open neighbourhood $\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} V_x \subset X$. By definition of the [[subspace topology]] the intersection $U_x \cap V_x$ is still open as a subspace of $V_x$ and hence $\phi^{-1}(U_x \cap V_x)$ is an open neighbourhood of $\phi^{-1}(x)  \in \mathbb{R}^n$.

Since Euclidean spaces are locally compact, there exists a compact neighbourhood $K_{\phi^{-1}(x)} \subset \mathbb{R}^n$ (for instance a sufficiently small [[closed ball]] around $x$, which is compact by the [[Heine-Borel theorem]]). Now since [[continuous images of compact spaces are compact]], it follows that also $\phi(K) \subset X$ is a compact neighbourhood.

=--

But the "global" topological properties of Euclidean space are not generally inherited by locally Euclidean spaces. This sounds obvious, but notice that also Hausdorff-ness is a "global property":

+-- {: .num_remark}
###### Remark
**(locally Euclidean spaces are not necessarily $T_2$)**

It might superficially seem that every locally Euclidean space (def. \ref{LocallyEuclideanSpace}) is necessarily a [[Hausdorff topological space]], since [[Euclidean space]], like any [[metric space]], is Hausdorff, and since by definition the neighbourhood of every point in a locally Euclidean spaces looks like Euclidean space.

But this is not so, Hausdorffness is a "non-local condition".

=--

\begin{example}\label{NonHausdorffManifolds}
counter-example: **([[non-Hausdorff locally Euclidean spaces]])**
\linebreak
An example of a [[locally Euclidean space]] (def. \ref{LocallyEuclideanSpace}) which is a [[non-Hausdorff topological space]], is the [[line with two origins]].
\end{example}

+-- {: .num_lemma #PathConnectedFromConnectedLocallyEuclideanSpace}
###### Lemma
**(connected locally Euclidean spaces are path-connected)**

A locally Euclidean space which is [[connected topological space|connected]] is also [[path-connected topological space|path-connected]].

=--

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

Fix any $x \in X$. Write $PConn_x(X) \subset X$ for the subset of all those points of $x$ which are connected to $x$ by a path, hence

$$
  PConn_x(X)
  \;\colon\;
  \left\{
    y \in X
   \;\vert\;
   \underset{[0,1] \underoverset{cts}{\gamma}{\to} X }{\exists}
   \left(
      \left(\gamma(0) = x\right)
      \phantom{A} \text{and} \phantom{a}
      \left(
        \gamma(1) = y
      \right)
   \right)
  \right\}
  \,.
$$

Observe now that both $PConn_x(X) \subset X$ as well as its [[complement]] are [[open subsets]]:

To see this it is sufficient to find for every point $y \on PConn_x(X)$ an [[open neighbourhood]] $U_y \supset \{y\}$ such that $U_y \subset PConn_x(X)$, and similarly for the complement.

Now by assumption every point $y \in X$ has a Euclidean neighbourhood $\mathbb{R}^n \overset{\simeq}{\to} U_y \subset X$. Since Euclidean space is path connected, there is for every $z \in U_y$ a path $\tilde \gamma \colon [0,1] \to X$ connecting $y$ with $z$, i.e. with $\tilde \gamma(0) = y$ and $\tilde \gamma(1) = z$. Accordingly the composite path

$$
  \array{
     [0,1] &\overset{\tilde \gamma\cdot\gamma}{\longrightarrow}& X
     \\
     t &\overset{\phantom{AAA}}{\mapsto}&
    \left\{
      \array{
         \gamma(2t) &\vert& t \leq 1/2
         \\
         \tilde(2t-1/2) &\vert& t \geq 1/2
      }
    \right.
  }
$$

connects $x$ with $z \in U_y$. Hence $U_y \subset PConn_x(X)$.

Similarly, if $y$ is not connected to $x$ by a path, then also all point in $U_y$ cannot be connected to $x$ by a path, for if they were, then the analogous concatenation of paths would give a path from $x$ to $y$, contrary to the assumption.

It follows that

$$
  X = PConn_x(C) \sqcup (X \setminus PConn_x(X))
$$

is a decomposition of $X$ as the [[disjoint union]] of two open subsets. By the assumption that $X$ is connected, exactly one of these open subsets is empty. Since $PConn_x(X)$ is not empty, as it contains $x$, it follows that its compement is empty, hence that $PConn_x(X) = X$, hence that $(X,\tau)$ is path connected.

=--

+-- {: .num_prop #RegularityConditionsForTopologicalManifoldsComparison}
###### Proposition
**(equivalence of regularity conditions for Hausdorff  locally Euclidean spaces)**

Let $X$ be a [[locally Euclidean space]] (def. \ref{LocallyEuclideanSpace}) which is [[Hausdorff topological space|Hausdorff]].

Then the following are equivalent:

1. $X$ is [[sigma-compact topological space|sigma-compact]],

1. $X$ is [[second-countable topological space|second-countable]],

1. $X$ is [[paracompact topological space|paracompact]] and has a [[countable set]] of [[connected components]],

=--

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

Generally, observe that $X$ is [[locally compact]]:
By prop. \ref{LocalPropertiesOfLocallyEuclideanSpace} every locally Euclidean space is locally compact in the sense that every point has a [[neighbourhood base]] of compact neighbourhoods, and since $X$ is assumed to be Hausdorff, this implies all the other variants of definition of local compactness, by [this prop.](locally+compact+topological+space#InHausdorffSpacesDefinitionsOfLocalCompactnessAgree).

**1) $\Rightarrow$ 2)**

Let $X$ be sigma-compact. We show that then $X$ is [[second-countable topological space|second-countable]]:

By sigma-compactness there exists a [[countable set]] $\{K_i \subset X\}_{i \in I}$ of compact subspaces. By $X$ being locally Euclidean, each admits an [[open cover]] by restrictions of [[Euclidean spaces]]. By their compactness, each of these has a subcover $\{ \mathbb{R}^n \overset{\phi_{i,j}}{\to} X \}_{j \in J_i}$ with $J_i$ a finite set. Since [[countable unions of countable sets are countable]], we have obtained  a countable cover by Euclidean spaces $\{ \mathbb{R}^n  \overset{\phi_{i,j}}{\to} X\}_{i \in I, j \in J_i}$. Now Euclidean space itself is second countable (by [this example](second-countable+space#SecondCountableEuclideanSpace)), hence admits a countable set $\beta_{\mathbb{R}^n}$ of base open sets. As a result the union $\underset{{i \in I} \atop {j \in J_i}}{\cup} \phi_{i,j}(\beta_{\mathbb{R}^n})$ is a base of opens for $X$. But this is a countable union of countable sets, and since [[countable unions of countable sets are countable]] we have obtained a countable base for the topology of $X$. This means that $X$ is second-countable.

**1) $\Rightarrow$ 3)**

Let $X$ be sigma-compact. We show that then $X$ is paracompact with a countable set of connected components:

Since [[locally compact and sigma-compact spaces are paracompact]], it follows that $X$ is paracompact. By [[locally connected topological space|local connectivity]] (prop. \ref{LocalPropertiesOfLocallyEuclideanSpace}) $X$ is the [[disjoint union space]] of its [[connected components]] ([this prop.](locally+connected+topological+space#AlternativeCharacterizationsOfLocalConnectivity)). Since, by the previous statement, $X$ is also second-countable it cannot have an uncountable set of connected components.

**2)$\Rightarrow$ 1)** Let $X$ be second-countable, we need to show that it is sigma-compact.

This follows since [[locally compact and second-countable spaces are sigma-compact]].

**3) $\Rightarrow$ 1)**

Now let $X$ be paracompact with countably many connected components. We show that $X$ is sigma-compact.

Since $X$ is locally compact, there exists a cover $\{K_i = Cl(U_i) \subset X\}_{i \in I}$ by [[compact topological space|compact]] [[subspaces]].
By paracompactness there is a locally finite refinement of this cover. Since [[paracompact Hausdorff spaces are normal]], the [[shrinking lemma]] applies to this refinement and yields a locally finite open cover

$$
  \mathcal{V} \coloneqq \{V_j \subset X \}_{j \in J}
$$

as well as a locally finite cover $\{Cl(V_j) \subset X\}_{j \in J}$ by closed subsets. Since this is a refinement of the orignal cover, all the $Cl(V_j)$ are contained in one of the compact subspaces $K_i$. Since [[subsets are closed in a closed subspace precisely if they are closed in the ambient space]], the $Cl(V_j)$ are also closed as subsets of the $K_i$. Since [[closed subsets of compact spaces are compact]] it follows that the $Cl(V_j)$ are themselves compact and hence form a locally finite cover by compact subspaces.

Now fix any $j_0 \in J$.

We claim that for every $j \in J$ there is a finite sequence of indices $(j_0, j_1, \cdots, j_n = j)$ with the property
that $V_{j_k} \cap V_{j_{k+1}} \neq \emptyset$.

To see this, first observe that it is sufficient to show sigma-compactness for the case that $X$ is [[connected topological space|connected]]. From this the general statement follows since [[countable unions of countable sets are countable]]. Hence assume that $X$ is connected.
It follows from lemma \ref{PathConnectedFromConnectedLocallyEuclideanSpace} that  $X$ is [[path-connected topological space|path-connected]].

Hence for any $x \in V_{j_0}$ and $y \in V_{j}$ there is a path $\gamma \colon [0,1] \to X$ connecting $x$ with $y$. Since the [[closed interval]] is compact and since [[continuous images of compact spaces are compact]], it follows that there is a finite subset of the $V_i$ that covers the image of this path. This proves the claim.

It follows that there is a function

$$
  f \;\colon\; \mathcal{V} \longrightarrow \mathbb{N}
$$

which sends each $V_j$ to the [[minimum]] natural number as above.

We claim now that for all $n \in \mathbb{N}$ the [[preimage]] of $\{0,1, \cdots, n\}$ under this function is a [[finite set]]. Since [[countable unions of countable sets are countable]] this implies that $\{ Cl(V_j) \subset X\}_{j \in J}$ is a countable cover of $X$ by compact subspaces, hence that $X$ is sigma-compact.

We prove this last claim by [[induction]]. It is true for $n = 0$ by construction. Assume it is true for some $n \in \mathbb{N}$, hence that $f^{-1}(\{0,1, \cdots, n\})$ is a finite set. Since finite unions of compact subspaces are again compact ([this prop.](compact+space#UnionsAndIntersectionOfCompactSubspaces)) it follows that

$$
  K_n
    \coloneqq
  \underset{V \in f^{-1}(\{0,\cdots, n\})}{\cup} V
$$

is compact. By local finiteness of the $\{V_j\}_{j \in J}$, every point $x \in K_n$ has an open neighbourhood $W_x$ that intersects only a finite set of the $V_j$. By compactness of $K_n$, the cover $\{W_x \subset X\}_{x \in K_n}$ has a finite subcover. In conclusion this implies that only a finite number of the $V_j$ intersect $K_n$.

Now by definition $f^{-1}(\{0,1,\cdots, n+1\})$ is a subset of those $V_j$ which intersect $K_n$, and hence itself finite.

=--

### Topological manifold

+-- {: .num_defn #TopologicalManifold}
###### Definition
**([[topological manifold]])**

A _topological manifold is a [[topological space]] which is

1. [[locally Euclidean topological space|locally Euclidean]] (def. \ref{LocallyEuclideanSpace}),

1. [[paracompact Hausdorff topological space|paracompact Hausdorff]].

If the local [[Euclidean spaces|Euclidean]] neighbourhoods $\mathbb{R}^n \overset{\simeq}{\to} U \subset X$ are all of [[dimension]] $n$
for a fixed $n \in \mathbb{N}$, then the topological manifold is said to be a _$n$-dimensional manifold_ or _$n$-fold_. This is usually assumed to be the case.

=--

+-- {: .num_remark}
###### Remark
**(varying terminology)**

Often a topological manifold (def. \ref{TopologicalManifold}) is required to be [[sigma-compact]]. But by prop. \ref{RegularityConditionsForTopologicalManifoldsComparison} this is not an extra condition as long as there is a [[countable set]] of [[connected components]].

=--

### Differentiable manifolds

+-- {: .num_defn #Charts}
###### Definition
**([[local chart]] and [[atlas]] and [[gluing function]])

Given an $n$-dimensional topological manifold $X$ (def. \ref{TopologicalManifold}), then

1. an [[open subset]] $U \subset X$ and a [[homeomorphism]] $\phi \colon \mathbb{R}^n \overset{\phantom{A}\simeq\phantom{A}}{\to} U$ is also called a _[[local coordinate chart]]_ of $X$.

1. an [[open cover]] of $X$ by local charts $\left\{ \mathbb{R}^n \overset{\phi_i}{\to} U \subset X \right\}_{i \in I}$ is called an _[[atlas]]_ of the topological manifold.

1. denoting for each $i,j \in I$ the [[intersection]] of the $i$th chart with the $j$th chart in such an atlas by

   $$
     U_{i j} \coloneqq U_i \cap U_j
   $$

   then the induced homeomorphism

   $$
     \mathbb{R}^n
       \supset
        \phantom{AA}
     \phi_i^{-1}(U_{i j})
       \overset{\phantom{A}\phi_i\phantom{A}}{\longrightarrow}
     U_{i j}
       \overset{\phantom{A}\phi_j^{-1}\phantom{A}}{\longrightarrow}
     \phi_j^{-1}(U_{i j})
       \phantom{AA}
       \subset
     \mathbb{R}^n
   $$

   is called the _[[gluing function]]_ from chart $i$ to chart $j$.

> graphics grabbed from [[The Geometry of Physics - An Introduction|Frankel]]

=--

+-- {: .num_defn #Differentiable manifold}
###### Definition
**([[differentiable manifold|differentiable]] and [[smooth manifolds]])**

For $p \in \mathbb{N} \cup \{\infty\}$ then a $p$-fold _[[differentiable manifold]]_ is

1. a [[topological manifold]] $X$ (def. \ref{TopologicalManifold});

1. an [[atlas]] $\{\mathbb{R}^n \overset{\phi_i}{\to} X\}$ (def. \ref{Charts}) all whose [[gluing functions]]  are $p$ times continuously [[differentiable function|differentiable]].

A $p$-fold [[differentiable function]] between $p$-fold differentiable manifolds

$$
  (X, \{\mathbb{R}^{n} \overset{\phi_i}{\to} U_i \subset X\}_{i \in I})
   \overset{\phantom{AA}f\phantom{AA}}{\longrightarrow}
  (Y, \{\mathbb{R}^{n'} \overset{\psi_j}{\to} V_j \subset Y\}_{j \in J})
$$

is

* a [[continuous function]] $f \colon X \to Y$

such that

* for all $i \in I$ and $j \in J$ then

  $$
     \mathbb{R}^n
     \supset
      \phantom{AA}
     (f\circ \phi_i)^{-1}(V_j)
       \overset{\phi_i}{\longrightarrow}
     f^{-1}(V_j)
      \overset{f}{\longrightarrow}
     V_j
       \overset{\psi_j^{-1}}{\longrightarrow}
     \mathbb{R}^{n'}
  $$

  is a $p$-fold [[differentiable function]] between open subsets of [[Euclidean space]].

=--

Notice that this in in general  a non-trivial condition even if $X = Y$ and $f$ is the identity function. In this case the above exhibits a passage to a different, but equivalent, differentiable atlas.

## Properties

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

Let $X$ be a $k$-fold differentiable manifold and let $S \subset X$
be an [[open subset]] of the underlying [[topological space]] $(X,\tau)$.

Then $S$ carries the structure of a $k$-fold differentiable manifold such that the
inclusion map $S \hookrightarrow X$ is an [[open embedding|open]]
[[embedding of differentiable manifolds]].

=--

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

Since the underlying [[topological space]] of $X$ is [[locally connected topological space|locally connected]] ([this prop.](topological+manifold#LocalPropertiesOfLocallyEuclideanSpace)) it is the [[disjoint union space]] of its [[connected components]] ([this prop.](locally+connected+topological+space#AlternativeCharacterizationsOfLocalConnectivity)).

Therefore we are reduced to showing the statement for the case that $X$ has a single [[connected component]]. By [this prop](topological+manifold#RegularityConditionsForTopologicalManifoldsComparison) this implies that $X$ is [[second-countable topological space]].

Now a [[subspace]] of a second-countable Hausdorff space is clearly itself second countable and Hausdorff.

Similarly it is immediate that $S$ is still [[locally Euclidean space|locally Euclidean]]: since $X$ is locally Euclidean every point $x \in S \subset X$ has a Euclidean neighbourhood in $X$ and since $S$ is open there exists an open ball in that (itself [[homeomorphism|homeomorphic]] to Euclidean space) which is a Euclidean neighbourhood of $x$ contained in $S$.

For the differentiable structure we pick these Euclidean neighbourhoods from the given atlas.
Then the [[gluing functions]] for the Euclidean charts on $S$ are $k$-fold differentiable follows since these are restrictions of the gluing functions for the atlas of $X$.

=--

## Examples

See the examples at _[[differentiable manifold]]_.

## Related concepts

* [[triangulation theorem]]

## References

Historical articles:

* [[Hellmuth Kneser]], *Die Topologie der Mannigfaltigkeiten*, Jahresbericht der Deutschen Mathematiker-Vereinigung (1926), Volume: 34, page 1-13 ([eudml:145701](https://eudml.org/doc/145701))

* J. W. Cannon, *The recognition problem: What is a topological manifold?*, Bull. Amer. Math. Soc. 84 (1978), 832-866 ([doi:10.1090/S0002-9904-1978-14527-3](https://doi.org/10.1090/S0002-9904-1978-14527-3))

Textbook accounts:

* [[John M. Lee]], _Introduction to topological manifolds_. Graduate Texts in Mathematics 202 (2000), Springer.  ISBN: 0-387-98759-2, 0-387-95026-5.
Second edition: Springer, 2011.  ISBN: 978-1-4419-7939-1 ([doi:10.1007/978-1-4419-7940-7](https://doi.org/10.1007/978-1-4419-7940-7), errata [pdf](https://sites.math.washington.edu/~lee/Books/ITM/errata.pdf))

See also:

* Wikipedia, _[Topological manifold](https://en.wikipedia.org/wiki/Topological_manifold)_

[[!redirects topological manifolds]]

[[!redirects locally Euclidean topological space]]
[[!redirects locally Euclidean topological spaces]]

[[!redirects locally Euclidean space]]
[[!redirects locally Euclidean spaces]]


topological map

#Contents#
* table of contents
{:toc}

## Definition

A **topological map** (or **embedded graph**) $M$ is a [[graph]] $G = (V,E,d)$ (loops and multiple edges allowed) equipped with an embedding $\theta$ of $G$ into a [[surface]] $X$ in such a way that:

1. vertices $x \in V$ are represented as distinct points $\theta(x) \in X$;

2. edges $e \in E$ are represented as curves $\theta(e)$ on the surface $X$ that intersect only at the vertices;

3. the complement $X \setminus \theta(G)$ of the graph on the surface is a disjoint union of connected components, called _faces_, each [[homeomorphic]] to an open [[disk]] (in other words, each face is [[simply connected]]).

These conditions can be summarized by saying that $\theta$ is a _cellular embedding_ (or "2-cell embedding").

A morphism of topological maps $(G,X,\theta) \to (G',X',\theta')$ is a continuous function between the underlying spaces $f : X \to X'$ whose restriction to $\theta(G)$ defines a homomorphism of graphs $f : G \to G'$.
Sometimes one may place additional conditions on the underlying surfaces and limit the morphisms between them accordingly.  For instance, typically one might assume that $X$ is [[compact space|compact]], [[connected space|connected]], [[oriented]] and without [[boundary]], and consider a map on $X$ as defined up to orientation-preserving homeomorphism.

Topological maps inherit various properties of their underlying surfaces: for instance, a graph embedded in a connected surface is itself necessarily a connected graph.  One also speaks of the _genus_ of a map $(G,X,\theta)$ as the [[genus of a surface|genus]] of the underlying surface $X$.

## Embedded graphs versus abstract graphs

It is important to distinguish topological maps, which are graphs equipped with an embedding, from graphs which may happen to have the property of being embeddable in different surfaces.  For example, a **planar map** is a graph equipped with a cellular embedding into the [[sphere]], while a **planar graph** is a graph which is _graph isomorphic_ to the underlying graph of a planar map.  Two planar graphs are considered equivalent if they are isomorphic as graphs $G_1 \cong G_2$, but two planar maps are only considered equivalent if this isomorphism of graphs can be realized on their embeddings $\theta_1(G_1) \cong \theta_2(G_2)$ by a homeomorphism of the sphere.  In particular, one planar graph might have multiple, inequivalent embeddings into the sphere.

To emphasize the distinction between embedded graphs and graphs which may happen to be embeddable, one sometimes refers to the latter as **abstract graphs**.[^Bartlett]

[^Bartlett]: For more on the distinction between embedded graphs and abstract graphs, see [[Bruce Bartlett]]'s n-Category Café post on  [Feynman's Fabulous Formula](https://golem.ph.utexas.edu/category/2015/06/feynmans_fabulous_formula.html).

## Permutation representations

One of the remarkable properties of topological maps is that they can be given a purely algebraic representation as a collection of permutations satisfying a few properties.  For now, see the articles ([[cartographic group]]) and ([[combinatorial map]]).

## Related concepts

* [[combinatorial map]]

* [[ribbon graph]]

* [[child's drawing]]

* [[cartographic group]]

* [[branched cover of the Riemann sphere]]

## References

A formal treatment of topological maps (including the relationship to [[algebraic maps]] and the definition of several associated categories) is in

* [[Gareth A. Jones]] and [[David Singerman]], Theory of Maps on Orientable Surfaces. Proceedings of the London Mathematical Society, 37:273-307, 1978.

Fairly elementary sources for background material include:

* Arthur T. White: 1984 _Graphs, groups and surfaces_
  {#White1984}

This gives the Edmonds algorithm which given a graph and some permutations outputs an embedding. The original source for that is:

* J. Edmonds (1960), A combinatorial representation for polyhedral surfaces, Notices Amer. Math. Soc. 7, 646.

More recent treatments can be found in:

* Sergei K. Lando and Alexander K. Zvonkin, _Graphs on Surfaces and Their Applications_, Springer, 2004.

* Roman Nedela, _Maps, Hypermaps and Related Topics_, 2007. ([pdf](http://www.savbb.sk/~nedela/CMbook.pdf))

Wikipedia:

* [Graph embedding](https://en.wikipedia.org/wiki/Graph_embedding)

[[!redirects topological maps]]
[[!redirects embedded graph]]
[[!redirects embedded graphs]]


topological membrane

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

#Contents#
* table of contents
{:toc}

## Idea

A [[membrane]] [[sigma-model]] [[topological quantum field theory]] that is roughly related to [[topological M-theory]] as the [[M2-brane]] is related to [[M-theory]] and to the [[topological string]] ([[A-model]]/[[B-model]]) as the [[M2-brane]] is related to the [[string]] and to the [[topological M5-brane]] as the [[M2-brane]] is related to the [[M5-brane]].

The [[target space]] of the topological membrane is a [[G2-manifold]], the [[action functional]] is governed by the [[higher holonomy]] of the compatible [[supergravity C-field]] over the membrane [[worldvolume]].

## Definition

According to ([Bao-Bengtsson-Cederwall-Nillson 05, equation (2.14)](#BaoBengtssonCederwallNillson05)) the topological $(p=2)$-brane is the super 2-brane which exists in $D = 7$ according to the [[brane scan]], which says that the [[super Poincare Lie algebra]] in $D = 7$ carries an exceptional [[Lie algebra cocycle]] of degree $(2+2)$, hence admits the [[Green-Schwarz action functional]] for a super 2-brane.

## Related concepts

[[!include brane scan]]

[[!include table of branes]]

## References

* Ling Bao, Viktor Bengtsson, [[Martin Cederwall]], [[Bengt Nilsson]], _Membranes for Topological M-Theory_ ([arXiv:hep-th/0507077](http://arxiv.org/abs/hep-th/0507077))
 {#BaoBengtssonCederwallNillson05}

* Giulio Bonelli, Alessandro Tanzini, [[Maxim Zabzine]], _On topological M-theory_ ([arXiv:hep-th/0509175](http://arxiv.org/abs/hep-th/0509175))

* Ling Bao, _Topological membranes_, Nucl. Phys. Proc. Suppl. 171 (2007) 259-260 ([inspire](http://inspirehep.net/record/767705?ln=en))

In the context of [[exceptional generalized geometry]]:

* [[Noriaki Ikeda]], Tatsuya Tokunaga, _Topological Membranes with 3-Form H Flux on Generalized Geometries_,  	Adv.Theor.Math.Phys.12:1259-1281,2008 ([arXiv:hep-th/0609098](http://arxiv.org/abs/hep-th/0609098))

[[!redirects topological membranes]]
[[!redirects topological M2-brane]]
[[!redirects topological M2-branes]]


topological modular form

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

On manifolds with rational [[string structure]] the [[Witten genus]] takes values in [[modular forms]]. On manifolds with actual [[string structure]] this refines further to a ring of "topological modular forms". This ring is at the same time the ring of [[homotopy groups]] of an [[E-∞ ring]] spectrum, called _[[tmf]]_.

## Properties

### Relation to modular forms

Write $\overline{\mathcal{M}}$ for the [[Deligne-Mumford compactification]] of the [[moduli stack of elliptic curves]] regarded as a [[derived scheme]], such that [[tmf]] is defined as the [[global sections]] of the derived [[structure sheaf]]

$$
  tmf = \mathcal{O}(\overline{\mathcal{M}})
  \,.
$$

Write $\omega$ for the standard [[line bundle]] on $\overline{\mathcal{M}}$ such that the [[sections]] of $\omega^{\otimes k}$ are the ordinary [[modular forms]] of weight $k$ (as discussed there).

Then there is the [[descent spectral sequence]]

$$
  H^s(\overline{M}, \omega^{\otimes t}) \Rightarrow tmf_{2t-s}
$$

and since the ordinary modular forms embed on the left as

$$
  H^0(\overline{\mathcal{M}}, \omega^{\otimes t}) \hookrightarrow
   H^s(\overline{M}, \omega^{\otimes t})
$$

this induces an [[edge morphism]]

$$
  tmf_{2 \bullet} \longrightarrow MF_\bullet
$$

from topological modular forms to ordinary [[modular forms]].

The [[kernel]] and [[cokernel]] of this map are 2-[[torsion]] and 3-[[torsion]] and hence "away from 6" this map is an [[isomorphism]].

## Related concepts

* [[modular form]]

* [[Jacobi form]]

##  References

See also the references at _[[tmf]]_.

An introductory exposition is in

* {#MazelGee13} [[Aaron Mazel-Gee]], _You could've invented $tmf$_, April 2013 ([pdf slides](http://math.berkeley.edu/~aaron/writing/ustars-tmf-beamer.pdf))

A collection of resources is in

* [[Nora Ganter]], _[Topological modular forms literature list](http://www.ms.unimelb.edu.au/~nganter/talbot/index.html)_

The original identification of topological modular forms as the coefficient ring of the [[tmf]] [[E-∞ ring]] and the refinement of the [[Witten genus]] to a morphism of [[E-∞ rings]], hence to the [[string orientation of tmf]] is due to

* [[Michael Hopkins]], _Topological modular forms, the Witten Genus, and the theorem of the cube_, Proceedings of the International Congress of Mathematics, Zürich 1994 ([pdf](http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf))

* [[Michael Hopkins]], _Algebraic topology and modular forms_, Proceedings of the ICM, Beijing 2002, vol. 1, 283--309 ([arXiv:math/0212397](http://arxiv.org/abs/math/0212397))

* [[Matthew Ando]], [[Michael Hopkins]], [[Charles Rezk]], _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))
  {#AndoHopkinsRezk}

see also remark 1.4 of

* [[Paul Goerss]], _Topological modular forms (after Hopkins, Miller and Lurie)_ ([pdf](http://arxiv.org/PS_cache/arxiv/pdf/0910/0910.5130v1.pdf)).

and for more on the [[sigma-orientation]] see

* [[Matthew Ando]], _The sigma orientation for analytic circle-equivariant elliptic cohomology_, Geom. Topol. 7 (2003) 91-153 ([arXiv:math/0201092](http://arxiv.org/abs/math/0201092))

Equivariant topological modular forms are discussed in

* [[David Gepner]], [[Lennart Meier]], _On equivariant topological modular forms_, ([arXiv:2004.10254](https://arxiv.org/abs/2004.10254))

[[!redirects topological modular forms]]


topological monoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A _topological monoid_ is a [[monoid object]] [[internalization|internal]] to [[topological spaces]].

This is a slightly weaker form of the more common definition of a _[[topological group]]_.

## Related concepts

* [[James construction]]

* [[topological ring]]

* [[topological field]]

* [[H-space]]

* [[group completion]]

[[!redirects topological monoids]]


topological notions of Frölicher spaces

{:myproof: .proof style="margin-left:2em;"}
{:mynumdef: .num_defn style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;"}
{:goal: .un_remark style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

* tic
{: toc}

# Idea #

Although the definition of a [[Frölicher space]] does not use a topology, it is topological in flavour and there are many topological concepts that can be defined for Frölicher spaces.  In other pages, we have used the notion of a Hausdorff Frölicher space and have used a vague notion of a topology on the functions on a Frölicher space.  In this page, we shall investigate the connections between the theory of Frölicher spaces and topological spaces a little more consistently.

There are two different ways of thinking of topological notions on Frölicher spaces.
One says that there is a functor (actually two functors) from the category of Frölicher space to the category of topological spaces so we can say that a Frölicher space has topological property $P$ if the corresponding topological space has it.
The other approach says that we can directly define a property for Frölicher spaces that is _analogous_ to a topological property.
We then might hope for a theorem saying that a Frölicher space with property $P$ defines a topological space with the corresponding property $P$.
However, this would definitely be a theorem.

The first approach involves defining a functor (or more than one) from the category of Frölicher spaces to that of topological spaces and looking at the image.  The second approach involves comparing the two categories and trying to transfer ideas from the more well-known category of topological spaces to that of Frölicher spaces.
We shall study both.

# Functors #

Let us start by defining the two functors to topological spaces.

+-- {: mynumdef #FroTop}
###### Definition
The _curvaceous_ topology on a Frölicher space is the strongest topology for which the smooth curves are continuous.

The _functional_ topology on a Frölicher space is the weakest topology for which the smooth functions are continuous.
=--

It is clear that these assignments are functorial, and that the curvaceous topology is always at least as strong as the functional topology.

As it is an inductive topology, the curvaceous topology has the following characterisation.

+-- {: .num_lemma #LemCurvTop}
###### Lemma
Let $(X,C_X,F_X)$ be a Frölicher space.
A subset $U \subseteq X$ is open in the curvaceous topology if and only if $c^{-1}(U)$ is open in $\mathbb{R}$ for all $c \in C_X$.

Thus a subset $A \subseteq X$ is not open if and only if there is some $c \in C_X$ such that $c^{-1}(A)$ is not open in $\mathbb{R}$.
=--

+-- {: myproof}
###### Proof
We merely need to observe that the family $\{U : c^{-1}(U) \;\text{open}\; \forall c \in C_X\}$ defines a topology.  This is elementary.
=--

The functional topology is a projective topology and so it is _generated_ by sets of the form $f^{-1}(a,b)$.  That is, these sets form a [[subbasis]] for the topology.  Using the structure of the functionals, we can strengthen that.

+-- {: .num_lemma #LemFunTop}
###### Lemma
Let $(X,C_X,F_X)$ be a Frölicher space.
A subset $U \subseteq X$ is open in the functional topology if and only if for each $p \in U$ there is a function $f \in F_X$ such that $f(p) = 1$ and $f(q) = 0$ for $q \notin U$.
=--

+-- {: myproof}
###### Proof
The "if" is obvious so we prove the "only if".  Thus let $U \subseteq X$ be open and $p \in U$.  Then there are $f_1, \dots, f_n$ and $a_i, b_i \in \mathbb{R}$ such that

$$
p \in \bigcap_i f_i^{-1}(a_i,b_i) \subseteq U\;.
$$

We combine the functions $f_i$ into a single function $f \coloneqq (f_1, \dots, f_n) \colon X \to \mathbb{R}^n$.  Then

$$
\bigcap_i f_i^{-1}(a_i,b_i) = f^{-1} (a_1,b_1) \times \dots \times (a_n,b_n)
$$

Now we use the fact that the topology on $\mathbb{R}^n$ is smoothly regular to find a smooth function $g \colon \mathbb{R}^n \to \mathbb{R}$ such that $g(f(p)) = 1$ and $g(y) = 0$ outside the specified rectangle.  Then the composition $g \circ f$ has the required properties.
=--

# Basic Properties #

Let us start with some very simple definitions.

+-- {: mynumdef #DiscIndisc}
###### Definition
A Frölicher space is said to be _indiscrete_ if all curves are smooth.

A Frölicher space is said to be _discrete_ if all functions are smooth.
=--

Let us observe that there is no need for functional or curvaceous versions of these definitions.

+-- {: .num_lemma #LemDiscIndisc}
###### Lemma
A Frölicher space is indiscrete if and only if the only smooth functions are the constant ones.

A Frölicher space is discrete if and only if the only smooth curves are the constant ones.
=--

+-- {: myproof}
###### Proof
If all curves are smooth then for any points $x, y \in X$ the curve
\[
\alpha(t) = \begin{cases}
x & t \leq 0 \\
y & t \ge 0
\end{cases}
\]
is smooth.
Composition with $\phi \in F$ yields the function
\[
t \mapsto \begin{cases}
\phi(x) & t \leq 0 \\
\phi(y) & t \ge 0
\end{cases}
\]
For this to be a smooth function in $C^\infty(\mathbb{R}, \mathbb{R})$ we must have $\phi(x) = \phi(y)$, hence $\phi$ is constant.

For the converse, if the only smooth functions are constant then any curve $\alpha : \mathbb{R} \to X$ satisfies the condition that $\phi \circ \alpha \in C^\infty(\mathbb{R}, \mathbb{R})$ for all $\phi \in F$ so is in $C$.
Hence all curves are smooth.

The discrete case is similar.
=--

# Separation Axioms #

In other sections, namely [[examples of Frölicher spaces]] and [[Frölicher spaces and Isbell envelopes]]  we used the notion of a _Hausdorff_ Frölicher space.
Technically, we ought to have called that _functionally Hausdorff_ as it used the smooth functions in its definition.

+-- {: mynumdef #DefFroHausdorff}
###### Definition
A Frölicher space is said to be _functionally Hausdorff_ if the smooth functions separate points.

A Frölicher space is said to be _curvaceously Hausdorff_ if the only smooth curves with finite image are constant.
=--

However, the distinction is not important as the following lemma shows.

+-- {: .num_lemma #LemFroHausdorff}
###### Lemma
The notions of functional Hausdorff and curvaceously Hausdorff coincide and are equivalent to the underlying topological spaces being Hausdorff.
=--

+-- {: myproof}
###### Proof
Suppose that $(X,C,F)$ is not functionally Hausdorff.
Then there are $x \ne y \in X$ such that $\phi(x) = \phi(y)$ for all $\phi \in X$.
Let $\alpha : \mathbb{R} \to X$ be the function taking the value $x$ for $t \leq 0$ and $y$ for $t \ge 0$.
Then $\phi \circ \alpha$ is constant for all $\phi \in F$ so $\alpha \in C$.
However, $\alpha$ has finite image but is not constant.
Thus $(X,C,F)$ is not curvaceously Hausdorff.

Conversely, suppose that $(X,C,F)$ is not curvaceously Hausdorff.
Then there is some $\alpha \in C$ with finite image which is not constant.
Let $\phi \in F$.
Then $\phi \circ \alpha$ has finite image in $\mathbb{R}$ and hence is constant.
Thus for $x, y \in \im \alpha$, $\phi(x) = \phi(y)$ for all $\phi \in F$.
As $\alpha$ is not constant, there are thus $x \ne y \in X$ such that $\phi(x) = \phi(y)$ for all $\phi \in F$ and so $(X,C,F)$ is not functionally Hausdorff.

If a Frölicher space is Hausdorff then smooth functions separate points.
Thus for $x \ne y \in X$, there is a smooth function $\phi \in F$ with $\phi(x) = -1$ and $\phi(y) = 1$.
Then the sets $\phi^{-1}(-\infty,0)$ and $\phi^{-1}(0,\infty)$ are sufficient to show that $X$ with the functional topology is Hausdorff.
As the curvaceous topology is stronger than the functional one, it is thus also Hausdorff.

Suppose that $X$ with the curvaceous topology is Hausdorff.
Then any finite subset is discrete and so there are no non-constant continuous maps $\mathbb{R} \to X$ with finite image.
In particular, there are no non-constant smooth maps and so the original Frölicher space was Hausdorff.
=--

In light of this, we shall refer to just _Hausdorff_ Frölicher spaces.

Just as with topological spaces, there is a "Hausdorffification" functor.
Unlike topological spaces, this functor is split.

+-- {: .num_lemma #LemHausFunct}
###### Lemma
Let $(X,C,F)$ be a Frölicher space.
Let $Y$ be the quotient of $X$ by the relation $x \sim y$ if $\phi(x) = \phi(y)$ for all $\phi \in F$.
Then $Y$ inherits a Frölicher space structure from $X$ with respect to which it is Hausdorff.
The natural map $X \to Y$ is a quotient mapping in the category of Frölicher spaces.
It is split, but not canonically so.
However, any two splittings are related by a diffeomorphism on $X$.

The assignment $X \mapsto Y$ is left adjoint to the inclusion of the category of Hausdorff Frölicher spaces in the category of all Frölicher spaces.
=--

+-- {: myproof}
###### Proof
The Frölicher structure on $Y$ is defined by setting $F_Y$ to be the set of functions $\phi : Y \to \mathbb{R}$ such that the composition $X \to Y \xrightarrow{\phi} \mathbb{R}$ is in $F_X$.
The smooth curves are then defined by the saturation condition.
It is automatic from this definition that any smooth curve in $X$ projects down to a smooth curve in $Y$ which explains why this family of functions on $Y$ is also saturated and hence we have a Frölicher space structure on $Y$.

To show that $Y$ is Hausdorff, we merely observe that by slight abuse of notation, $F_X = F_Y$ so if $x,y \in X$ are such that $\phi(\overline{x}) = \phi(\overline{y})$ for all $\phi \in F_Y$ then $\phi(x) = \phi(y)$ for all $\phi \in F_X$, whence $\overline{x} = \overline{y}$ in $Y$.

That this is a quotient is straightforward.
Any smooth map $g : X \to Z$ which factors through $Y$ _as a set_ must also do so as a Frölicher space.
In particular, if $g : X \to Z$ is a smooth map with $Z$ Hausdorff then this must factor through $Y$ as a set, whence as a Frölicher space.
This also establishes the necessary adjunction.

Finally, let us look at the splitting.
For each point in $Y$ choose a representative of the equivalence class.
This choice defines a map on the underlying sets $Y \to X$.
This is also smooth since the sets of functions $F_X$ and $F_Y$ are identified by the quotient mapping $X \to Y$.

Given two such splittings, say $i, j : Y \to X$, define a bijection $X \to X$ which interchanges $i(y)$ and $j(y)$.
As $i$ and $j$ are splits of the quotient mapping $X \to Y$ this is well-defined.
It is also clearly a diffeomorphism since $F_X$ cannot detect the difference between $i$ and $j$.
=--

This shows, incidentally, that every smooth curve in the Hausdorffification of $X$ lifts to a smooth curve in $X$.
This sort of behaviour does not _usually_ happen with quotients in the category of Frölicher spaces.

The fibres of the Hausdorffification are straightforward to identify.

+-- {: .num_lemma #FibHaus}
###### Lemma
The fibres of the Hausdorffification of a Frölicher space correspond precisely to the maximal subsets which inherit an indiscrete structure from the ambient space.
=--

+-- {: myproof}
###### Proof
Let $X$ be a Frölicher space, $Y$ its Hausdorffification.
For $y \in Y$, let $X_y$ be the corresponding fibre.
Then $X_y$ inherits a Frölicher space structure from its inclusion in $X$.
The smooth curves in $X_y$ are those that are smooth when considered as curves in $X$.
Let $\alpha : \mathbb{R} \to X_y$ be an arbitrary curve.
Then for $\phi \in F_X$, as $\im \alpha \subseteq X_y$, $\phi \circ \alpha$ is constant.
Hence $\alpha$ is smooth as a curve in $X$, and thus in $X_y$.
Thus $X_y$ is indiscrete.

Conversely, let $Z \subseteq X$ be a subset that inherits an indiscrete structure from $X$.
Let $x,y \in Z$.
Then there is a smooth curve $\alpha$ in $Z$ with $\im \alpha = \{x,y\}$.
This is then smooth in $X$ so for all $\phi \in F_X$, $\phi(x) = \phi(y)$.
Hence $Z$ is contained in a (unique) fibre of the quotient map $X \to Y$.
=--

Thus when we pass to the Hausdorffification we lose almost no information at all and one could certainly say that we lose no _interesting_ information.

Having dealt with Hausdorff Frölicher spaces, the obvious next thing to do is to consider the other separation properties.
Our next definition may be a little surprising at first.

+-- {: mynumdef #FroReg}
###### Definition
A Frölicher space is said to be _regular_ if the curvaceous and functional topologies agree.
=--

The point of this definition is that for the underlying topological space of a Frölicher space what one really wants to know is not whether or not it is regular but whether or not it is _smoothly_ regular.
This is automatic for the functional topology so the only reasonable question is whether or not it happens for the curvaceous topology.
However, a topology is smoothly regular if and only if the smooth functions generate the topology which means that the curvaceous topology is smoothly regular if and only if it agrees with the functional topology.
Hence the definition.

# Compactness #

It is straightforward to see what one version of [[compact space|compactness]] should be.

+-- {: mynumdef #FunComp}
###### Definition
A Frölicher space is _functionally compact_ if every smooth function has bounded image.
=--

The images are automatically compact as a smooth function with non-compact image can be converted to a smooth function with unbounded image by suitable composition.

+-- {: .num_lemma #LemFunComp}
###### Lemma
A Frölicher space is functionally compact if and only if the functional topology is compact.
=--

+-- {: myproof}
###### Proof
One way is obvious: if the functional topology is compact then as the smooth functions are continuous, they have compact image, hence bounded.

For the converse, assume that the functional topology is not compact.
Then we can find a countable family of points $(x_n)$ with no accumulation points.
As the functional topology is (smoothly) regular, we can find smooth functions $(\phi_n)$ such that $\phi_n(x_m) = \delta_{n m}$.
We claim that it is possible to modify these to have disjoint support.
This is done recursively using postcomposition by suitably chosen functions.
Once this is done, we can define a new smooth function by $\sum n \widehat{\phi_n}$.
This is smooth, as the components have disjoint support, and is unbounded.
Hence the Frölicher space is not functionally compact.
=--

+-- {: .num_lemma #CurvComp}
###### Lemma
The curvaceous topology of a Frölicher space is compact if and only if the Frölicher space is functionally compact and regular.
=--

+-- {: myproof}
###### Proof
Since the topologies on a Frölicher space are the pull-backs of the topologies on the Hausdorffification, it is sufficient to prove this for a Hausdorff Frölicher space.

As the curvaceous topology is stronger than the functional, if the curvaceous topology is compact then so is the functional.
Moreover, as both are Hausdorff spaces, the identity map is a continuous bijection from a compact space to a Hausdorff space and hence a homeomorphism.
Thus the Frölicher space is regular.

Conversely, if the Frölicher space is regular its topologies agree and thus if it is functionally compact then its curvaceous topology is compact.
=--

In [[manifolds of mapping spaces]], there is an important issue as to compactness of the actual topological space, rather than compactness notions of Frölicher spaces.  The property needed is about open sets in the product $\mathbb{R} \times X$.  For Frölicher spaces, this property is equivalent to sequential compactness.

+-- {: .num_proposition #ProdSeqCpt}
###### Proposition
Let $X$ be a Frölicher space.  The curvaceous topology on $X$ is sequentially compact if and only if $\mathbb{R} \times X$ has the following property.

A subset $U \subseteq \mathbb{R} \times X$ containing $\{0\} \times X$ is a neighbourhood of $\{0\} \times X$ if and only if it contains a subset of the form $(-\epsilon,\epsilon) \times X$ for some $\epsilon \gt 0$.
=--

+-- {: myproof}
###### Proof
Suppose that $X$ is sequentially compact.  Let $U \subseteq \mathbb{R} \times X$ be a subset containing $\{0\} \times X$.  Suppose that $U$ does not contain a subset of the form $(-\epsilon,\epsilon) \times X$.  Then for each $n \in \mathbb{N}$ we can find some $(t_n, x_n) \in U$ such that $\abs{t_n} \le \frac1n$.  The sequence $(x_n)$ in $X$ has a convergent subsequence, say $(x_{n_k}) \to x$.  Then $(t_{n_k}, x_{n_k})$ converges in $\mathbb{R} \times X$, but $(t_{n_k}, x_{n_k}) \notin U$ for all $k$ but its limit, $(0,x)$ is in $U$.  Hence $U$ is not open.

Conversely, assume that neighbourhoods of $\{0\} \times X$ contain slices as claimed.  Then let $(x_n)$ be a sequence in $X$.  Consider the set $U$ formed by taking $\mathbb{R} \times X$ and removing $(\frac1n, x_n)$.  This does not contain a subset of the form $(-\epsilon,\epsilon) \times X$ and so cannot be open in $\mathbb{R} \times X$.

As the set $U$ is not open, there must be a curve $c \colon \mathbb{R} \to U$ which detects the fact that it is not open.  That is, $c^{-1}(U)$ is not open in $\mathbb{R}$.  Now we can find a sequence $(s_k) \to s$ in $\mathbb{R}$ such that $(s_k) \notin c^{-1}(U)$ but $s \in U$.  Then as $c(s_k) \notin U$, we must have $c(s_k) = (t_{n_k}, x_{n_k})$ for some $n_k$.  By passing to a subsequence if necessary, we can assume that the $n_k$s are strictly increasing.  Then as $(s_k) \to s$, $c(s_k) \to c(s)$ and so $x_{n_k} \to p_X c(s)$.  Hence $X$ is sequentially compact.
=--

# Connectedness #

Another obvious topological property is connectedness.
Here it is obvious what the two definitions should be.

+-- {: mynumdef #FroConnect}
###### Definition
A Frölicher space is _functionally connected_ if the only idempotents in its algebra of functions are the trivial ones.

More generally, the _functional connected components_ of a Frölicher space $(X,C,F)$ are the equivalence classes of the relation $x \sim y$ if whenever $\phi \in F$ is idempotent then $\phi(x) = \phi(y)$.

A Frölicher space is _curvaceously connected_ if every pair of points lie on a curve.

More generally, the _curvaceous connected components_ of a Frölicher space $(X,C,F)$ are the equivalence classes of the relation $x \sim y$ if there is a smooth curve $\alpha \in C$ with $\alpha(0) = x$ and $\alpha(1) = y$.
=--

That the second relation is an equivalence relation follows from the fact that piecewise smooth curves can be reparametrised to smooth curves.

+-- {: .num_lemma #LemConnect}
###### Lemma
The notions of functionally connected and curvaceously connected coincide.
=--

+-- {: myproof}
###### Proof
Let $(X,C,F)$ be a Frölicher space.
It is clear that if $x, y \in X$ are such that there is a smooth curve connecting them then $\phi(x) = \phi(y)$ for any idempotent $\phi \in F$.
Thus we need to show the reverse implication.
To do this, let $X' \subseteq X$ be a curvaceously connected component of $X$.
Let $\phi \colon X \to \mathbb{R}$ be the characteristic function of $X'$.
Then for any $\alpha \in C$, either $\im \alpha \subseteq X'$ or $\im \alpha \cap X' = \emptyset$.
Thus $\phi \circ \alpha$ is a constant function.
Hence $\phi \in F$.
Thus if $x$ and $y$ are in different curvaceously connected components there is an idempotent element of $F$ separating them.
Hence the two notions are the same.
=--

+-- {: .num_corollary #CorConnect}
###### Corollary
The functional and curvaceous topologies have the same connected components, and these are the same as the path-connected components.
=--

# More on Hausdorff Frölicher Spaces # {#hausdorff}

There is not a great deal of difference between a Hausdorff Frölicher space and a generic one.
Much less than the case with topological spaces.
To pass from all Frölicher spaces to Hausdorff Frölicher spaces involves only collapsing everything that is _indiscrete_.
This clears out a considerable amount of junk from the category but does remove two properties: it no longer has a weak subobject classifier and it is no longer topological over $\Set$.

On the other hand, the relationship between the category of Hausdorff Frölicher spaces and that of all Frölicher spaces is very good.
Not only is it a [[reflective subcategory]] with all that that implies, but the morphisms from the [[unit|unit natural transformation]] are [[retract|split epimorphisms]] (though not naturally split).

The category of Hausdorff Frölicher spaces is thus complete and co-complete.
It is also cartesian closed since the product and exponential objects of Hausdorff Frölicher spaces are again Hausdorff.

+-- {: .query}
Do I need to prove this, or is it automatic?
(I can prove it if necessary)

[[Mike Shulman|Mike]]: Completeness and cocompleteness are of course automatic.  It is not automatic that a reflective subcategory inherits cartesian closure; the closest thing I can think of is A4.3.1 in the [[Elephant]] which says that a reflective subcategory is an [[exponential ideal]] iff its reflector preserves finite products.

[[Andrew Stacey|Andrew]]: Is it true, then, that all I need to do is to prove that the exponential of one Hausdorff object by another Hausdorff object is again Hausdorff?

[[Mike Shulman|Mike]]: Yes, that would certainly suffice to show that the category of Hausdorff objects is cartesian closed.

[[Andrew Stacey|Andrew]]: Ah, and now I see from [[exponential ideal]] that I could just show that the Hausdorffification of a product is the product of the Hausdorffifications.  Both seem quite simple, not sure which is the simplest.
=--

In considering [[Isbell duality]] in the context of Frölicher spaces we saw one good reason to restrict to Hausdorff Frölicher spaces.
Another reason comes from the inclusion of the category of manifolds in that of Frölicher spaces.
This factors through Hausdorff Frölicher spaces and this inclusion has some very pleasant properties.

+-- {: .num_theorem #Manifolds}
###### Theorem
The inclusion functor from the category of Manifolds to that of Hausdorff Frölicher spaces preserves limits and colimits.
=--

+-- {: myproof}
Let us write $\mathcal{M}$ for the category of manifolds, $\mathcal{H}$ for the category of Hausdorff Frölicher spaces, and $\mathcal{F}$ for the category of all Frölicher spaces.
We shall not give the inclusion functors special symbols but trust to context to distinguish.
Let $\mathfrak{F} : I \to \mathcal{M}$ be a functor where $I$ is a small category.

Let us assume first that $\mathfrak{F}$ has a limit in $\mathcal{M}$, say $M_0$ with maps $\alpha_i : M_0 \to \mathfrak{F}(i)$.
Let us write $X_0$ for the limit of $\mathfrak{F}$ viewed as a functor into $\mathcal{H}$, with maps $\beta_i : X_0 \to \mathfrak{F}(i)$.
As $\mathcal{H}$ is a [[reflective subcategory]] of $\mathcal{F}$, $X_0$ is the same as the limit of $\mathfrak{F}$ in $\mathcal{F}$.

Since $M_0$, as a Frölicher space, is a source of $\mathfrak{F}$, there is a unique map, say $\gamma : M_0 \to X_0$, such that $\beta_i \gamma = \alpha_i$.

As a Frölicher space, $X_0$ is completely determined by its underlying set and its smooth curves.
Its underlying set is (naturally isomorphic to) $\mathcal{F}(*,X_0)$.
Let $x \in |X_0|$.
Composing with the $\beta_i$ defines maps $\beta_i x : * \to \mathfrak{F}$.
Since $*$ is a manifold and $M_0$ is the limit of $\mathfrak{F}$ in $\mathcal{M}$, there is a unique map $\widehat{x} : * \to M_0$ such that $\alpha_i \widehat{x} = \beta_i x$ for all $i \in I$.
Using the uniqueness of the factorisations, we see that $\gamma \widehat{x} = x$ and thus $\gamma$ induces a bijection $|M_0| \to |X_0|$.
Hence the underlying sets of $M_0$ and $X_0$ are the same.

The smooth curves of $X_0$ are the morphisms $\mathbb{R} \to X_0$.
Since $\mathbb{R}$ is a manifold, the same argument shows that $\gamma$ induces a bijection from the set of smooth curves in $M_0$ to that in $X_0$.
Hence $\gamma$ is an isomorphism of Frölicher spaces and so the inclusion functor $\mathcal{M} \to \mathcal{H}$ preserves limits.

Now let us assume that $\mathfrak{F}$ has a colimit in $\mathcal{M}$, say $M_1$ with maps $\lambda_i : \mathfrak{F}(i) \to M_1$.
Let us write $X_1$ for the colimit of $\mathfrak{F}$ viewed as a functor into $\mathcal{F}$, with maps $\mu_i \colon \mathfral{F}(i) \to X_1$.
Note that this is in $\mathcal{F}$ not $\mathcal{H}$.
To obtain the colimit in $\mathcal{H}$ we apply the reflector functor (Hausdorffification) to $X_1$.

Since $M_1$,  as a Hausdorff Frölicher space, is a sink of $\mathfrak{F}$ there is a unique morphism, say $\nu : X_1 \to M_1$, such that $\nu \mu_i = \lambda_i$.
This morphism factors uniquely through the Hausdorffification of $X_1$.

For the same argument as with the limits, the smooth functions on $X_1$ factor through those of $M_1$.
However, the underlying set functor is not represented by morphisms _into_ a smooth manifold so we have to be a little more careful to see that the $\nu$ induces an isomorphism from the Hausdorffification of $X_1$ to $M_1$.

Firstly, let us show that $\nu$ is surjective on underlying sets.
To see this, suppose for a contradiction that it is not.
Let $x \in |M_1|$ be a point not in the image of $\nu$.
Let $N = M_1 \smallsetminus \{x\}$.
Then $N$ is an open submanifold of $M_1$ and $\nu$ factors through the inclusion $N \to M_1$.
As $X_1$ is the colimit of $\mathfrak{F}$, the morphism $X_1 \to N$ establishes $N$ as a sink for $\mathfrak{F}$.
As $N$ is a manifold, there is thus a unique morphism $M_1 \to N$ factoring the morphisms from $\mathfrak{F}$ to $N$.
That is to say, the morphism $X_1 \to N$ uniquely factors through $\nu : X_1 \to M_1$.
This gives a factorisation of $\nu$ as
$$
X_1 \longrightarrow^{\nu} M_1 \to N \to M_1
$$
where the last morphism is the inclusion of $N$ in $M_1$.
However, the properties of $\nu$ imply that the morphism $M_1 \to M_1$ in the above diagram is the identity, contradicting the non-surjectivity of $N \to M_1$ and thus the non-surjectivity of $X_1 \to M_1$.

Hence $X_1 \to M_1$ is surjective.
We also have that the smooth functions on $X_1$ factor through $M_1$.
This is not enough to prove that $X_1$ and $M_1$ are isomorphic, but is enough to prove that the Hausdorffification of $X_1$ is isomorphic to $M_1$ (note that $M_1$, being a manifold, is already Hausdorff as a Frölicher space).
To see this, observe that with what we already have, all that remains is to show that $\nu$ induces an injective map from the underlying set of the Hausdorffification of $X_1$ to the underlying set of $M_1$.
Thus let $x, y$ be distinct points in the Hausdorffification of $X_1$.
There is thus a smooth function on $X_1$ which distinguishes them.
As this smooth function factors through $M_1$, we must have $\nu(x) \ne \nu(y)$ and hence $\nu$ is injective on the required underlying sets.

Thus the inclusion functor $\mathcal{M} \to \mathcal{H}$ preserves colimits.
=--

The inclusion of the category of Manifolds in that of all Frölicher spaces preserves limits (by the same proof as above) but not colimits.
However, it is thus only the issue of being Hausdorff that prevents it preserving colimits.
The simplest example is the classic non-Hausdorff manifold: consider the [[coequalizer|coequaliser]] of $\mathbb{R} \smallsetminus \{0\}$ included in each piece of $\mathbb{R} \coprod \mathbb{R}$.
The colimit in the category of [[manifold|Manifolds]] is simply $\mathbb{R}$.
The colimit of this in the category of Frölicher spaces is the real line with a double point at $\{0\}$, but upon Hausdorffification this becomes the real line.

+--{: .query}
[[Mike Shulman|Mike]]: What if we re-define "manifold" to remove the Hausdorff axiom?  Does the inclusion into Frölicher spaces then preserve colimits?

[[Andrew Stacey|Andrew]]: I think so, but the inclusion from manifolds to Frölicher spaces is then not full.  Let $X$ be the real line with a double point at the origin.  Take a curve $\mathbb{R} \to X$ which oscillates between the two points.  This is a morphism into the Frölicher space, but not into the manifold.

I think that to make it work, you have to redefine "Euclidean space" to include anything that becomes a Euclidean space upon Hausdorffification.
=--

[[!redirects curvaceous topology]]
[[!redirects functional topology]]
[[!redirects Froelicher topology]]


topological operad

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

#Contents#
* table of contents
{:toc}

## Definition

A **topological operad** is an [[operad]] over [[Top]]. It has for each $k \in \mathbb{N}$ a [[topological space]] of $k$-ary operations.

## Model structure

There is a [[model category]] structure on the [[category]] of topological operads that makes them [[presentable (∞,1)-category|present]] [[(∞,1)-operad]]s. See [[model structure on operads]].

## Examples

* Every operad over [[Set]] may be regarded as a topological operad, via the [[discrete space]] embedding $Disc : Set \to Top$.

* the [[little k-cubes operad]] is naturally a topological operad

## Related concepts

* **topological operad**, [[simplicial operad]]

* [[dg-operad]]

[[!redirects topological operads]]