singular cohomology

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### Context
#### Topology
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[[!include topology - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

The **singular cohomology** (also **Betti cohomology**) of a [[topological space]] $X$ is the [[cohomology]] in [[∞Grpd]] of its [[fundamental ∞-groupoid]] $\Pi(X)$:

for $\mathcal{B}^n \mathbb{Z} \in \infty Grpd$ the [[Eilenberg-MacLane object]] with the group $\mathbb{Z}$ in degree $n$, the degree $n$-singular cohomology of $X$ is

$$
  H^n(X,\mathbb{Z}) := \pi_0 \infty Grpd(\Pi(X), \mathcal{B}^n \mathbb{Z})
  \,.
$$

With $\infty Grpd$ [[presentable (∞,1)-category|presented]] by the category [[sSet]] of [[simplicial set]]s, the fundamental $\infty$-groupoid $\Pi(X)$ is modeled by the [[Kan complex]]

$$
  \Pi(X) = Sing X = Hom_{Top}(\Delta^\bullet_{Top}, X)
  \,,
$$

the [[singular simplicial complex]] of $X$.

The object $\mathcal{B}^n \mathbb{Z}$ is usefully modeled by the simplicial set

$$
  \mathcal{B}^n \mathbb{Z} = U (\Xi \mathbb{Z}[n])
$$

which is

* the underlying simplicial set under the [[forgetful functor]]

  $$
    (F \dashv U)
    sAb \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
    sSet
  $$

  from abelian [[simplicial group]]s to [[simplicial sets]];

* of the abelian [[simplicial group]] $\Xi \mathbb{Z}[n]$ which is the image under the [[Dold-Kan correspondence]]

  $$
    sAb \stackrel{\overset{\Xi}{\leftarrow}}{\underset{}{\to}}
    Ch^+
  $$

* of the [[chain complex]]

  $$
    \mathbb{Z}[n] =
    (\cdots \to \mathbb{Z} \to 0 \to 0 \to \cdots \to 0)
  $$

  concentrated in degree $n$.

So in this model we have

$$
  H^n(X,\mathbb{Z}) = \pi_0 sSet(Sing X, U(\Xi \mathbb{Z}[n]))
  \,.
$$

A [[cocycle]] in this cohomology theory is a [[cochain on a simplicial set]], on the singular complex $Sing X$.

Using the [[adjunction]] $(F \dashv U)$ this is [[isomorphism|isomorphic]] to

$$
  \cdots \simeq \pi_0 sAb( Ch_n(X), \Xi \mathbb{Z}[n] )
  \,,
$$

where

$$
  F(Sing X) = \mathbb{Z}[Sing X]
$$

is the [[free functor|free]] abelian simplicial group on the simplicial set $Sing X$: this is the simplicial abelian group of **singular chains** of $X$. Its elements are formal sums of continuous maps $\Delta^n_{Top} \to X$. In this form

$$
  \cdots \simeq \pi_0 sAb( \mathbb{Z}[Sing X], \Xi \mathbb{Z}[n] )
  \,.
$$

Using next the [[Dold-Kan correspondence|Dold-Kan adjunction]] this is

$$
  \cdots \simeq H_0 Ch( Ch_\bullet(X), \mathbb{Z}[n] )
  \,,
$$

where

$$
  Ch_\bullet(X) := N^\bullet(\mathbb{Z}(Sing X))
$$

is the [[Moore complex]] of normalized chains of $\mathbb{Z}[Sing X]$: this is the complex of **singular chains**, formal sums over $\mathbb{Z}$ of simplices in $X$.

This way singular cohomology is the abelian dual of [[singular homology]].

...

## Related concepts

* [[cochain on a simplicial set]]

* [[cup product]]

* [[de Rham theorem]]

## Comparison to sheaf cohomology

If the topological space $X$ is semi-locally contractible
(meaning: any open subset $U\subset X$ has an open cover $W$ by open subsets $W_i\subset U$ that are contractible in $U$),
then the [[sheaf cohomology]] of $X$ is isomorphic
to the singular cohomology of $X$ for any abelian
group of coefficients.

This was proved in ([Sella 16](#Sella16)).

## Discussion

A previous version of this entry led to the following discussion, which later led to extensive discussion by email. Partly as a result of this and similar discussions, there is now more information on how Kan complexes _are_ $\infty$-groupoids at

* [[Kan complex]]

* [[∞-groupoid]].

## References

[[!include cohomology -- early references]]

### General

Relation to [[sheaf cohomology]]:

* {#Sella16} [[Yehonatan Sella]], _Comparison of sheaf cohomology and singular cohomology_ ([arXiv:1602.06674](http://arxiv.org/abs/1602.06674))

A simplified proof using [[hypercovers]] can be found in

* {#Petersen21} [[Dan Petersen]], _A remark on singular cohomology and sheaf cohomology_ ([arXiv:2102.06927](https://arxiv.org/abs/2102.06927))

[[!redirects Betti cohomology]]

[[!redirects singular chain]]
[[!redirects singular chains]]
[[!redirects singular chain complex]]
[[!redirects singular chain complexes]]
[[!redirects singular cochain]]
[[!redirects singular cochains]]
[[!redirects singular cochain complex]]
[[!redirects singular cochain complexes]]


singular foliation

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###Context###
#### Topology
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[[!include topology - contents]]
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#### Higher Lie theory
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[[!include infinity-Lie theory - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}

## Idea

A _singular foliation_ is a kind of _[[foliation]]_ in which the [[dimension]] of the [[leaves]] is allowed not to be constant.

## References

* [[Iakovos Androulidakis]], [[Georges Skandalis]], _The holonomy groupoid of a singular foliation_ ([arXiv:math/0612370](http://arxiv.org/abs/math/0612370))

[[!redirects singular foliations]]


singular homology

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###Context###
#### Algebraic topology
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[[!include algebraic topology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _singular homology_ of a [[topological space]] $X$ is the [[simplicial homology]] of its [[singular simplicial complex]]:

a **singular $n$-[[chain]]** on $X$ is a [[formal linear combination]] of [[singular simplices]] $\sigma : \Delta^n \to X$, and a singular $n$-[[cycle]] is such a chain such that its oriented [[boundary]] in $X$ vanishes. Two singular chains are **homologous** if they differ by a boundary. The **singular homology** of $X$ in degree $n$ is the group of $n$-cycles modulo those that are boundaries.

Singular homology of a topological space conincide with its [[ordinary homology]] as defined more abstractly (see at [[generalized homology theory]]).

(Here "singular" refers to the contrast with [[cellular homology]], referring to the fact that a [[simplex]] $\Delta_{top} \to X$ in the [[singular simplicial complex]] is not required to be a [[topological embedding]], but may be a "singular map", such as for instance a [[constant function]].)

## Definition
 {#Definition}

Let $X \in $ [[Top]] be [[topological space]].
Write $Sing X \in $ [[sSet]] for its [[singular simplicial complex]].

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

For $n \in \mathbb{N}$, a **singular $n$-chain** on $X$ is
an element in the [[free abelian group]] $\mathbb{Z}[(Sing X)_n]$:

a [[formal linear combinations]] of [[singular simplices]] in $X$.

=--

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

These are the [[chains on a simplicial set]] on $Sing X$.

=--

The groups of singular chains combine to the [[simplicial abelian group]]
$\mathbb{Z}[Sing X] \in Ab^{\Delta^{op}}$.

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

The [[alternating face map complex]]

$$
  C_\bullet(X)
    \coloneqq
  C_\bullet(\mathbb{Z}[Sing X]) \in Ch_\bullet
$$

is the **singular complex** of $X$.

Its [[chain homology]] is the ordinary **singular homology** of $X$.

=--

One usually writes $H_n(X, \mathbb{Z})$ or just $H_n(X)$ for the singular homology of $X$ in degree $n$. See also at _[[ordinary homology]]_.

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

So we have

$$
  C_\bullet(X)
  =
  [
    \cdots
     \stackrel{\partial_2}{\to}
    \mathbb{Z}[(Sing X)_2]
     \stackrel{\partial_1}{\to}
    \mathbb{Z}[(Sing X)_1]
     \stackrel{\partial_0}{\to}
    \mathbb{Z}[(Sing X)_0]
  ]
$$

where the differentials are defined on [[basis]] elements
$\sigma \in (Sing X)_n$ by

$$
  \partial_n \sigma
    =
  -
  \sum_{i = 0}^n
    (-1)
    d_i \sigma
$$

(with $d_i$ the $i$ [[simplicial set|simplicial face map]]) and then extended linearly.

(One may change the global signs and obtain a [[quasi-isomorphism|quasi-isomorphic]] complex, in particular with the same homology groups.)

=--

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

This means that a [[singular chain]] is a [[cycle]] if the formal linear combination of the oriented [[boundaries]] of all its constituent [[singular simplices]] sums to 0. See the  _[basic examples](#BasicExamples)_ below

=--

More generally, for $R$ any unital [[ring]] one can form the degreewise [[free module]] $R[Sing X]$ over $R$. The corresponding homology is the _singular homology with coefficients in $R$, denoted $H_n(X,R)$.

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

Given a [[continuous map]] $f : X \to Y$ between topological spaces,
and given $n \in \mathbb{N}$, every singular $n$-simplex $\sigma : \Delta^n \to X$ in $X$ is sent to a singular $n$-simplex

$$
  f_* \sigma : \Delta^n \stackrel{\sigma}{\to} X \stackrel{f}{\to} Y
$$

in $Y$. This is called the **push-forward** of $\sigma$ along $f$. Accordingly there is a push-forward map on groups of singular chains

$$
  (f_*)_n : C_n(X) \to C_n(Y)
  \,.
$$

=--

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

These push-forward maps make all diagrams of the form

$$
  \array{
     C_{n+1}(X) &\stackrel{(f_*)_{n+1}}{\to}& C_{n+1}(Y)
     \\
     \downarrow^{\mathrlap{\partial^X_n}} && \downarrow^{\mathrlap{\partial^Y_n}}
     \\
     C_n(X) &\stackrel{(f_*)_n}{\to}& C_n(Y)
  }
$$

commute. In other words, push-forward along $f$ constitutes a [[chain map]]

$$
  f_* : C_\bullet(X) \to C_\bullet(Y)
  \,.
$$

=--

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

It is in fact evident that push-forward yields a functor of [[singular simplicial complexes]]

$$
 f_* : Sing X \to Sing Y
 \,.
$$

From this the statement follows since $\mathbb{Z}[-] : sSet \to sAb$ is a functor.

=--

Accordingly we have:

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

Sending a topological space to its singular chain complex $C_\bullet(X)$, def. \ref{SingularComplex}, and a continuous map to its push-forward chain map,
prop. \ref{PushForwardChainMap}, constitutes a [[functor]]

$$
  C_\bullet(-,R) : Top \to Ch_\bullet(R Mod)
$$

from the category [[Top]] to the [[category of chain complexes]].

In particular for each $n \in \mathbb{N}$ singular homology extends to a [[functor]]

$$
  H_n(-,R) : Top \to R Mod
  \,.
$$

=--

## Examples

### Basic examples
 {#BasicExamples}

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

Let $X$ be a topological space. Let $\sigma^1 : \Delta^1 \to X$
be a singular 1-simplex, regarded as a 1-chain

$$
  \sigma^1 \in C_1(X)
  \,.
$$

Then its [[boundary]] $\partial \sigma \in H_0(X)$ is

$$
  \partial \sigma^1 = \sigma(0)  -\sigma(1)
$$

or graphically (using notation as for [[orientals]])

$$
  \partial
  \left(
    \sigma(0) \stackrel{\sigma}{\to} \sigma(1)
  \right)
  =
  (\sigma(0)) - (\sigma(1))
  \,.
$$

Let $\sigma^2 : \Delta^2 \to X$ be a singular 2-chain. The boundary is

$$
  \partial
  \left(
    \array{
       && \sigma(1)
       \\
       & {}^{\mathllap{\sigma(0,1)}}\nearrow
       & \Downarrow^{\mathrlap{\sigma}}&
       \searrow^{\mathrlap{\sigma^{1,2}}}
       \\
       \sigma(0) &&\underset{\sigma(0,2)}{\to}&& \sigma(2)
    }
  \right)
  =
  \left(
    \array{
       && \sigma(1)
       \\
       & {}^{\mathllap{\sigma(0,1)}}\nearrow
       & &
       \\
       \sigma(0)
    }
  \right)
  -
  \left(
    \array{
       &&
       \\
       &
       & &
       \\
       \sigma(0) &\underset{\sigma(0,2)}{\to}& \sigma(2)
    }
  \right)
  +
  \left(
    \array{
       && \sigma(1)
       \\
       &
       & &
       \searrow^{\mathrlap{\sigma^{1,2}}}
       \\
       && && \sigma(2)
    }
  \right)
  \,.
$$

Hence the boundary of the boundary is

$$
  \begin{aligned}
  \partial \partial \sigma
  &=
  \partial
  \left(
  \left(
    \array{
       && \sigma(1)
       \\
       & {}^{\mathllap{\sigma(0,1)}}\nearrow
       & &
       \\
       \sigma(0)
    }
  \right)
  -
  \left(
    \array{
       &&
       \\
       &
       & &
       \\
       \sigma(0) &\underset{\sigma(0,2)}{\to}& \sigma(2)
    }
  \right)
  +
  \left(
    \array{
       && \sigma(1)
       \\
       &
       & &
       \searrow^{\mathrlap{\sigma^{1,2}}}
       \\
       && && \sigma(2)
    }
  \right)  \right)
  \\
  & =
  \left(
    \array{
       &&
       \\
       &
       & &
       \\
       \sigma(0)
    }
  \right)
  -
  \left(
    \array{
       && \sigma(1)
       \\
       &
       & &
       \\

    }
  \right)
  -
  \left(
    \array{
       &&
       \\
       &
       & &
       \\
       \sigma(0) &&
    }
  \right)
  +
  \left(
    \array{
       &&
       \\
       &
       & &
       \\
       && \sigma(2)
    }
  \right)
  +
  \left(
    \array{
       && \sigma(1)
       \\
       &
       & &
       \\
       && &&
    }
  \right)
  -
  \left(
    \array{
       &&
       \\
       &
       & &
       \\
       && && \sigma(2)
    }
  \right)
  \\
  & = 0
  \end{aligned}
$$

=--

For more illustrations see for instance ([Ghrist, (4.5)](#Ghrist)).

### Homology of cells: disks and spheres
 {#HomologyOfDisksAndSpheres}

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

For all $n \in \mathbb{N}$ the [[reduced singular homology]] of the $n$-[[sphere]] $S^n$ is

$$
  \tilde H_k(S^n)
  =
  \left\{
    \array{
      \mathbb{Z} & if\; k = n
      \\
      0 & otherwise
    }
  \right.
  \,.
$$

=--

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

The $n$-sphere may be realized as the [[pushout]]

$$
  S^n \simeq D^n/S^{n-1} \coloneqq D^{n} \coprod_{S^{n-1}} *
$$

which is the $n$-[[ball]] with its [[boundary]] $(n-1)$-sphere identified with the [[point]]. The inclusion $S^{n-1} \hookrightarrow D^n$ is a "good pair" in the sense of def. \ref{GoodPair}, and so the [[long exact sequence]] from prop. \ref{RelativeHomologyLongExactSequence} yields a long exact sequence

$$
  \cdots \to \tilde H_{k+1}(S^n) \to \tilde H_k(S^{n-1}) \to \tilde H_k(D^n) \to \tilde H_k(S^n) \to \tilde H_{k-1}(S^{n-1}) \to \cdots
  \,.
$$

Since the [[disks]] are all [[contractible topological spaces]] we have $H_k(D^n) \simeq 0$ for all $k,n$ by [this example](reduced+homology#ReducedHomologyOfPoints) at _[[reduced homology]]_. This means that in the above long exact sequence all the morphisms

$$
  \tilde H_{k+1}(S^{n+1}) \to \tilde H_k(S^n)
$$

are [[isomorphisms]], for all $k \in \mathbb{N}$. Since

$$
  \tilde H_n(S^0) \simeq
  \left\{
    \array{
       \mathbb{Z} & if \; n = 0
       \\
       0 & otherwise
    }
  \right.
$$

(by [this example](reduced+homology#ReducedHomologyOf0Sphere) at _[[reduced homology]]_) the statement follows by [[induction]] on $n$.

=--

## Properties

### Homotopy invariance
 {#HomotopyInvariant}

Singular homology is _homotopy invariant_:

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

If $f : X \to Y$ is a [[continuous map]] between [[topological spaces]] which is a [[homotopy equivalence]], then the induced morphism on singular homology groups

$$
  H_n(f) : H_n(X) \to H_n(Y)
$$

is an [[isomorphism]].

In other words: the singular chain functor of prop. \ref{SingularHomologyAsAFunctor}
sends [[weak homotopy equivalences]] to [[quasi-isomorphisms]].

=--

A proof (via [[CW approximations]]) is spelled out for instance in ([Hatcher, prop. 4.21](#Hatcher)).

### Relation to homotopy groups
 {#RelationToHomotopyGroups}

The singular [[homology groups]] of a topologial space serve to some extent as an approximation to the [[homotopy groups]] of that space.

+-- {: .num_defn #HurewiczHomomorphism}
###### Definition
**(Hurewicz homomorphism)**

For $(X,x)$ a [[pointed object|pointed]] [[topological space]], the **Hurewicz homomorphism** is the [[function]]

$$
  \Phi : \pi_k(X,x) \to H_k(X)
$$

from the $k$th [[homotopy group]] of $(X,x)$ to the $k$th [[singular homology]] group defined by sending

$$
  \Phi : (f : S^k \to X)_{\sim} \mapsto f_*[S_k]
$$

a representative singular $k$-sphere $f$ in $X$ to the push-forward along $f$ of the [[fundamental class]] $[S_k] \in H_k(S^k) \simeq \mathbb{Z}$.

=--

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

For $X$ a topological space the Hurewicz homomorphism in degree 0 exhibits an [[isomorphism]] between the [[free abelian group]] $\mathbb{Z}[\pi_0(X)]$ on the set of [[connected components]] of $X$ and the degree-0 singular homlogy:

$$
  \mathbb{Z}[\pi_0(X)]
   \simeq
  H_0(X)
  \,.
$$

=--

Since a [[homotopy group]] in positive degree depends on the [[homotopy type]] of the [[connected component]] of the base point, while the singular homology does not depend on a basepoint, it is interesting to compare these groups only for the case that $X$ is connected.

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

For $X$ a [[connected topological space]] the [[Hurewicz homomorphism]]
in degree 1

$$
  \Phi : \pi_1(X,x) \to H_1(X)
$$

is [[surjection|surjective]]. Its [[kernel]] is the [[commutator subgroup]]
of $\pi_1(X,x)$. Therefore it induces an [[isomorphism]] from the [[abelianization]] $\pi_1(X,x)^{ab} \coloneqq \pi_1(X,x)/[\pi_1,\pi_1]$:

$$
  \pi_1(X,x)^{ab}
  \stackrel{\simeq}{\to}
  H_1(X)
  \,.
$$

=--

For higher connected $X$ we have the

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

If $X$ is [[n-connected object in an (infinity,1)-topos|(n-1)-connected]] for $n \geq 2$ then

$$
  \Phi : \pi_n(X,x) \to H_n(X)
$$

is an [[isomorphism]].

=--

This is known as the _[[Hurewicz theorem]]_.

### Relation to relative homology
 {#RelationToRelativeHomology}

For the present purpose one makes the following definition.

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

A [[topological subspace]] inclusion $A \hookrightarrow X$ in [[Top]] is called a **good pair** if

1. $A$ is [[inhabited set|inhabited]] and [[closed subset|closed]] in $X$;

1. $A$ has a [[neighbourhood]] in $X$ of which it is a [[deformation retract]].

=--

Write $X/A$ for the [[cokernel]] of the inclusion, hence for the [[pushout]]

$$
  \array{
    A &\hookrightarrow& X
    \\
    \downarrow && \downarrow
    \\
    * &\to& X/A
  }
$$

in [[Top]].

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

If $A \hookrightarrow X$ is a good pair, def. \ref{GoodPair}, then the singular homology of $X/A$ coincides with the [[relative homology]] of $X$ relative to $A$.  In particular, therefore, it fits into a [[long exact sequence]] of the form

$$
  \cdots
  \to
  \tilde H_n(A)
  \to
  \tilde H_n(X)
  \to
  \tilde H_n(X/A)
  \to
  \tilde H_{n-1}(A)
  \to
  \tilde H_{n-1}(X)
  \to
  \tilde H_{n-1}(X/A)
  \to
  \cdots
  \,.
$$

=--

For instance ([Hatcher, theorem 2.13](#Hatcher)).

### Relation to generalized homology

Singular homology computes the [[generalized homology]] with coefficients in the [[Eilenberg-MacLane spectrum]] $H \mathbb{Z}$ or $H R$.

## Related concepts

* [[Čech homology]]

* The [[duality|dual]] notion is that of _[[singular cohomology]]_.

* The analogous notion in [[algebraic geometry]] is given by [[Chow groups]].

* [[Kan-Thurston theorem]]

* [[Borel-Moore homology]]

## References

### General

Original references on [[chain homology]]/[[cochain cohomology]] and [[singular homology]]/[[singular cohomology]]:

* [[Henri Poincaré]], *[[Analysis Situs]]*, Journal de l'École Polytechnique. (2). 1: 1–123  (1895) (Orig.: [gallica:12148/bpt6k4337198/f7](https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7), Engl.: [[John Stillwell]]: [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf), [[Stillwell_AnalysisSitus.pdf:file]])

* [[Andrei Kolmogoroff]], _Über die Dualität im Aufbau der kombinatorischen Topologie_, Recueil Mathématique 1(43) (1936), 97–102.  ([mathnet](http://mi.mathnet.ru/msb5361))

* [[Andrei Kolmogoroff]], _Homologiering des Komplexes und des lokal-bicompakten Raumes_, Recueil Mathématique 1(43) (1936), 701–705.  [mathnet](http://mi.mathnet.ru/msb5475).

* [[J. W. Alexander]], _On the chains of a complex and their duals_, Proc. Nat. Acad. Sei. USA, 21(1935), 509–511 ([doi:10.1073/pnas.21.8.50](https://doi.org/10.1073/pnas.21.8.509))

* [[J. W. Alexander]], _On the ring of a compact metric space_, Proc. Nat. Acad. Sci. USA, 21(1935), 511–512 ([doi:10.1073/pnas.21.8.511](https://doi.org/10.1073/pnas.21.8.511))

* [[J. W. Alexander]], _On the connectivity ring of an abstract space_, Ann. of Math., 37 (1936), 698–708 ([doi:10.2307/1968484](https://doi.org/10.2307/1968484), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/alexcon.pdf))

Lecture notes:

* Rob Thompson, Len Evens , _[Topology notes](http://math.hunter.cuny.edu/~rthompso/topology_notes/)_ _Chapter 6, Singular homology_ ([pdf](http://math.hunter.cuny.edu/~rthompso/topology_notes/chapter%20six.pdf))

* [[David H. Fremlin]], *Singular homology for amateurs* (2016) $[$[pdf](https://www1.essex.ac.uk/maths/people/fremlin/n16703.pdf), [[FremlinSIngularHomology.pdf:file]]$]$

Textbook discussion in the context of [[homological algebra]] is around Application 1.1.4 of

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_
 {#Weibel}

and in the context of [[algebraic topology]] in chapter 2.1 of

* {#Hatcher} [[Allen Hatcher]], _Algebraic Topology_ ([web](http://www.math.cornell.edu/~hatcher/AT/ATpage.html))

and [chapter 4](http://www.math.upenn.edu/~ghrist/EAT/EATchapter4.pdf) of

* [[Robert Ghrist]], _Elementary applied topology_ ([web](http://www.math.upenn.edu/~ghrist/notes.html))
 {#Ghrist}

Discussion in the context of computing [[homotopy groups]] is in

* [[Michael Hutchings]], _Introduction to higher homotopy groups and obstruction theory_ ([pdf](http://math.berkeley.edu/~hutching/teach/215b-2011/homotopy.pdf))
 {#Hutchings}

Lecture notes include

* [[Marco Gualtieri]], _Homology_ ([pdf](http://www.math.toronto.edu/mgualt/MAT1300/Week%206-9%20Term%202.pdf))

See also

* Wikipedia, _[Singular homology](http://en.wikipedia.org/wiki/Singular_homology)_

### Examples and applications

* Michael  Barratt, [[John Milnor]], _An example of anomalous singular homology_, Proceedings of the American Mathematical Society
Vol. 13, No. 2 (Apr., 1962), pp. 293-297 ([JSTOR]( http://www.jstor.org/stable/2034486))

[[!redirects singular chain]]
[[!redirects singular chains]]

[[!redirects singular simplicial chain]]
[[!redirects singular simplicial chains]]

[[!redirects chain complex of singular simplices]]
[[!redirects chain complexes of singular simplices]]

[[!redirects singular chain homology]]


singular knot

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###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

## Idea

A **singular knot** is a [[smooth map]] (or [[isotopy]] class of smooth maps) $S^1 \to \mathbb{R}^3$ which is not an [[embedding]].  That is, it is not a [[knot]].  Singular knots come in various flavours according to the ways in which it is possible to fail to be an embedding.  One of the simplest type of singular knot is where the only failures allowed are _simple double points_.  That is, places where the curve intersects itself transversally, and at the intersection only two threads intersect.

Formally, $f \colon S^1 \to \mathbb{R}^3$ has a **(simple) double point** at $x \in \mathbb{R}^3$ if $f^{-1}(x) = \{t_1, t_2\}$ with $t_1 \ne t_2$ and $\{f'(t_1), f'(t_2)\}$ are linearly independent.

Singular knots are an important piece of the theory of [[Vassiliev finite type invariants]] of knots.

category: knot theory

[[!redirects singular knots]]
[[!redirects Singular knot]]
[[!redirects Singular knots]]


singular learning theory

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### Context
#### Probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[machine learning]], *singular learning theory* applies results from [[algebraic geometry]] to [[statistical learning theory]]. In the case of learning algorithms, such as [[deep neural networks]], where there are multiple parameter values corresponding to the same [[statistical distribution]], the [[preimage]] of the target distribution may take the form of a [[singular point|singular]] subspace of the parameter space. Techniques from algebraic geometry may then be applied to study learning with such devices.

## References

Historically, it has been understood that the neural networks are singuar statistical models in

* [[Shun-ichi Amari]], T. Ozeki, H. Park, _Learning and inference in hierarchical models with singularities_, Syst. Comput. Japan __34__:7 (2003) 34--42
* [[Sumio Watanabe]], _Almost all learning machines are singular_, Proc. IEEE Symp. Found. Comput. Intell., Apr. 2007, 383--388.

Textbook treatments:

* [[Sumio Watanabe]], _Algebraic geometry and statistical learning theory_, CRC Press (2009) [[doi:10.1017/CBO9780511800474]( https://doi.org/10.1017/CBO9780511800474)]

* [[Sumio Watanabe]], _Mathematical theory of Bayesian statistics_, Cambridge University Press (2018) [[ISBN:9780367734817](https://www.routledge.com/Mathematical-Theory-of-Bayesian-Statistics/Watanabe/p/book/9780367734817), [pdf](http://196.189.45.87/bitstream/123456789/36917/1/Sumio%20Watanabe_2018.pdf)]

For an informal discussion:

* {#Hoogland} Jesse Hoogland, _Neural networks generalize because of this one weird trick_ ([blog post](https://www.jessehoogland.com/article/neural-networks-generalize-because-of-this-one-weird-trick)); Jesse Hoogland, Filip Sondej, _Spooky action at a distance in the loss landscape_ ([blog post](https://www.lesswrong.com/posts/2N7eEKDuL5sHQou3N/spooky-action-at-a-distance-in-the-loss-landscape)).

For a series of talks and further texts:

* Singular Learning Theory seminar, ([webpage](https://metauni.org/slt/))

* S. Wei, [[Daniel Murfet]], M. Gong, H. Li , J. Gell-Redman, T. Quella, _Deep learning is singular, and that’s good_, IEEE Transactions on neural networks and learning systems [pdf](https://www.suswei.com/publication/wei-2022-singular/wei-2022-singular.pdf)


singular point of a curve

## Examples

* [[nodal singularity]]

* [[cusp]]

## Related concepts

* [[curve]]

* [[singularity]]

* [[Deligne-Mumford compactification]]

* [[augmented Teichmüller space]]

## References

* Wikipedia, _[Singular point of a curve](https://en.wikipedia.org/wiki/Singular_point_of_a_curve)_

[[!redirects singular points of a curve]]
[[!redirects singular points of a curves]]


singular point of an algebraic variety

#Contents#
* table of contents
{:toc}

## Idea

An [[algebraic variety]] $X \hookrightarrow \mathbb{A}^n$ defined as the [[zero locus]] of [[polynomials]] $\{f^i \in R[x_1, \cdots, x_n]\}$ is _singular_ at a point $p$, if the [[Jacobian matrix]] of first [[derivatives]] at this point has [[rank]] lower than at other points.

A variety is called _singular_ if it has at least one singular point and _non-singular_ if it has none.

## Examples

* [[cusp]]

* [[nodal curve]]

## Related concepts

* [[geometric point]]

## References

* Wikipedia, _[Singular point of an algebraic variety](http://en.wikipedia.org/wiki/Singular_point_of_an_algebraic_variety)_

[[!redirects singular points of an algebraic variety]]

[[!redirects singular point]]
[[!redirects singular points]]

[[!redirects singular algebraic variety]]
[[!redirects singular algebraic varieties]]

[[!redirects non-singular algebraic variety]]
[[!redirects non-singular algebraic varieties]]

[[!redirects nonsingular algebraic variety]]
[[!redirects nonsingular algebraic varieties]]


singular simplicial complex

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

#Contents#
* table of contents
{:toc}

##Idea

This construction 'probes' a space $X$ by mapping geometric simplices into it. It is one of the classical approaches to determining invariants of the homotopy type of the space.

## Definition

The _singular simplicial complex_ $S_\bullet(X)$ of a [[topological space]] $X$ is the [[nerve]] of $X$ with respect to the standard [[cosimplicial object|cosimplicial]] topological space $\Delta_{Top} : \Delta \to Top$.  It is thus the [[simplicial set]], $S_\bullet(X)$, having
$$
  S_n(X) = Hom_{Top}(\Delta_{Top}^n, X)
  \,.
$$
as its set of $n$-simplices, and fairly obvious faces and degeneracy mappings obtains by restriction along the structural maps of $\Delta_{Top} : \Delta \to Top$.  This is always a [[Kan complex]] and as such has the interpretation of the [[fundamental ∞-groupoid]] $\Pi(X)$ of $X$.

The $n$-simplices of this are just _[[singular simplex|singular n-simplices]]_ generalising paths in $X$. (The term _singular_ is used because there is no restriction that the continuous function used should be an embedding, as would be the case in, for instance, a [[triangulation]] where a simplex in the underlying [[simplicial complex]] corresponds to an embedding of a simplex.)

## Properties

### Preservation of model structure

The singular complex functor preserves all five classes
of maps in a [[model category]]: [[weak equivalences]], [[cofibrations]], [[acyclic cofibrations]], [[fibrations]], and [[acyclic fibrations]].

* [[weak equivalences|Weak equivalences]] are preserved; the reason depends on how the model structures are constructed.  The [[classical model structure on topological spaces]] can be defined to be [[transferred model structure|transferred]] from [[simplicial sets]], in which case the singular complex preserves weak equivalences by definition.  More traditionally, the weak equivalences in the [[classical model structure on simplicial sets]] are defined to be those whose geometric realization is a (weak) [[homotopy equivalence]], and the preservation then follows from the 2-out-of-3 property and the fact that the geometric realization of the total singular complex is a [[CW replacement]] that is [[weak homotopy equivalence|weakly homotopy equivalent]] to the original space.

* [[Cofibrations]] are preserved because all cofibrations of topological spaces are [[injective functions]], injective maps are preserved, and all injective maps of simplicial sets are cofibrations.

* [[fibrations|Fibrations]] and [[acyclic fibrations]] are preserved because the singular complex functor is a [[right Quillen functor]].  More explicitly, this is because geometric realization takes the generating cofibrations and acyclic cofibrations of simplicial sets to those of topological spaces.

### Relation to geometric realization

Together with its [[adjoint functor|adjoint]]—[[geometric realization]] $|-| : sSet \to Top$—the functor $Sing : Top \to sSet$ is part of the [[Quillen equivalence]] between the [[model structure on topological spaces]] and the [[model structure on simplicial sets]] that is sometimes called the [[homotopy hypothesis]]-theorem.

### Relation to ordinary (co)homology

Forming from the singular simplicial complex $Sing(X)$ first the free [[simplicial abelian group]] $\mathbb{Z}[Sing(X)]$ and then under the [[Dold-Kan correspondence]] the corresponding [[normalized chain complex]] yields the [[chain complex]] of (normalized!) singular chains, which computes the [[singular homology]] of $X$.

## Related concepts

* [[singular simplex]]

* [[fundamental infinity-groupoid]]

* [[singular homology]]

[[!redirects singular complex]]

[[!redirects singular simplicial complexes]]

[[!redirects singular simplicial set]]
[[!redirects singular simplicial sets]]


singular-smooth ∞-groupoid

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###Context###
#### Cohesive $\infty$-Toposes
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[[!include cohesive infinity-toposes - contents]]
=--

#### Representation theory
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[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In generalization of how a [[smooth ∞-groupoid]] is an [[∞-groupoid]] equipped with [[generalized smooth structure]] modeled on [[Cartesian spaces]] with [[smooth functions]] between them (hence: on [[smooth manifolds]]), a singular-smooth $\infty$-groupoid carries geometric structure which, in addition to being smooth almost everywhere, may have [[orbi-singularities]], in that it is locally modeled on [[Cartesian space]] regarded possibly as $G$-[[fixed loci]] for any [[finite group]] $G$.

## Definition

Write

* $CartesionSpaces \xhookrightarrow{\;} SmthMfd$ for the [[category]] of [[Cartesian spaces]] with [[smooth functions]] between them, regarded as a [[site]] via the [[coverage]] of [[differentiably good open covers]],

* $Singularities \xhookrightarrow{\;} Grpd_\infty$ for the [[(2,1)-category]] of [[groupoids]] which are [[deloopings]] of [[finite groups]], regarded as an [[(infinity,1)-site|$\infty$-site]] via the trivial topology.

Then singular-smooth $\infty$-groupoids are the [[objects]] in the [[hypercomplete (infinity,1)-topos|hypercomplete]] [[(infinity,1)-sheaf|$\infty$-sheaf]] [[(infinity,1)-topos|$\infty$-topos]] over the [[product category|product]] of these sites:

$$
  SingSmoothGrpd_\infty
  \;\coloneqq\;
  Sh_\infty
  \big(
    CartesianSpaces \times Singularities
  \big)
  \,.
$$

## Properties

In generalization of how [[smooth infinity-groupoids|smooth $\infty$-groupoids]] form a [[cohesive (infinity,1)-topos|cohesive $\infty$-topos]] over [[Infinity-Grpd|$Grpd_\infty$]],  so singular-smooth $\infty$-groupoids form a [[singular-cohesive (infinity,1)-topos|singular-cohesive $\infty$-topos]].

Where the [[cohesive (infinity,1)-topos|cohesive $\infty$-topos]] [[smooth infinity-groupoid|$SmthGrpd_\infty$]] is the natural home of [[smooth manifolds]] and [[diffeological spaces]], reflecting their [[differential cohomology]], so $SingSmoothGrpd_\infty$ is the natural home of [[orbifolds]] reflecting their proper [[orbifold cohomology]] ([SaSc 2020](#SaSc20)).

## Related concepts

* [[orbifold]], [[orbifold cohomology]]

* [[singular-cohesive (infinity)-topos|singular-smooth $\infty$-topos]]

* [[cohesion of global- over G-equivariant homotopy theory]]

## References

* {#SaSc20} [[Hisham Sati]], [[Urs Schreiber]], Ex. 3.56 in: _[[schreiber:Proper Orbifold Cohomology]]_ ([arXiv:2008.01101](https://arxiv.org/abs/2008.01101))

[[!redirects singular-smooth ∞-groupoids]]

[[!redirects singular-smooth infinity-groupoid]]
[[!redirects singular-smooth infinity-groupoids]]


singularity

## Related concepts

* [[pole]]

* [[angular deficit]]

* [[singular point of a curve]]

* [[differential form with logarithmic singularities]]

* [[Picard-Lefschetz theory]]

* [[orbifold]]

* [[conical singularity]]

* [[ADE singularity]]

* [[resolution of singularities]]

* [[triangulated category of singularities]]

## References

* Wikipedia, _[Mathematical singularity](http://en.wikipedia.org/wiki/Mathematical_singularity)_

* A. O. Remizov, _A brief introduction to singularity theory_, ([pdf](http://www.sissa.it/fa/download/publications/remizov.pdf))

[[!redirects singularities]]

[[!redirects orbifold singularity]]
[[!redirects orbifold singularities]]

[[!redirects orbi-singularity]]
[[!redirects orbi-singularities]]

[[!redirects singularity theory]]


Sini Karppinen

## Selected writings

On the [[slice theorem]] for [[topological G-spaces]]:

* [[Sini Karppinen]], _The existence of slices in $G$-spaces, when $G$ is a Lie group_, Helsinki 2016 ([hdl:10138/190707](http://hdl.handle.net/10138/190707))

category: people


sink

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###Context###
#### Category theory
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[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given an [[object]] $Y$ of a [[category]] $C$, a __sink__ or __wide cospan__ to $Y$ in $C$ is a [[family]] of [[morphisms]] of $C$ whose [[targets]] (codomains) are all $Y$, or equivalently, a family of objects in the [[over category]] $C / Y$:

$$ \array {
   X_1 \\
       & \searrow^{f_1} \\
   X_2 & \overset{f_2}\to & Y \\
       & \nearrow_{f_3} \\
   X_3
} $$

We do not, in general, require that this family be [[small category|small]]; if it is so we would call it a "small sink".

The [[duality|dual]] concept is a family of morphisms of $C$ whose [[sources]] (domains) are all $Y$, or equivalently, a family of objects in the [[under category]] $Y / C$:

$$ \array {
   & & X_1 \\
       & {}^{f_1}\nearrow \\
   Y & \overset{f_2}\to & X_2 \\
       & {}_{f_3}\searrow \\
   & & X_3
} $$

Confusingly, this dual concept is called a __source__ from $Y$ in $C$, even though the term 'source' has another meaning, one which we just used in the definition!  One can of course say 'domain' instead of 'source' for this other meaning, but that leads to [[domain|other confusions]].  Or one can say 'cosink' or __wide span__ for a source in the sense dual to a sink or wide cospan, since a source from $Y$ in $C$ is the same as a sink to $Y$ in the [[opposite category]] $C^{\mathrm{op}}$.

### Structured sinks

If $U\colon C\to D$ is a functor, then a **$U$-structured sink** is a collection of objects $X_i\in C$ together with a sink in $D$ of the form $\{U(X_i) \to Y\}$.  This notion figures in the definition of a [[final lift]].

## Examples

* Any [[cocone]] under a [[diagram]] is a sink; indeed a cocone is precisely a sink indexed by the objects of the domain of the diagram together with a commutativity condition for the arrows in the diagram.

* A source with two morphisms is a [[span]], a sink with two morphisms is a [[cospan]].

* A [[terminal object|terminal source]] is a [[dependent product]], while an [[initial object|initial sink]] is a [[dependent sum]].

## Related pages

* [[final lift]], [[topological concrete category]], [[solid functor]]

* [[factorization structure for sinks]]

* [[wide pullback]], [[wide pushout]]

* [[dependent product type]], [[dependent sum type]]

[[!redirects sink]]
[[!redirects cosink]]

[[!redirects sinks]]
[[!redirects cosinks]]

[[!redirects wide span]]
[[!redirects wide cospan]]

[[!redirects wide spans]]
[[!redirects wide cospans]]


site

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### Context
#### Topos Theory
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[[!include topos theory - contents]]
=--
=--
=--

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

## Idea

A _site_  is a [[locally presentable category|presentation]] of a [[sheaf topos]] as a structure [[free cocompletion|freely generated under colimits]] from a category, subject to the relation that certain [[covering]] colimits are preserved.

As such, sites generalise [[topological spaces]] and [[locales]], which present [[localic topos|localic]] sheaf toposes.  More precisely, sites generalise and [[categorify]] [[posites]], which present localic toposes but also present locales themselves in a decategorified manner.

In technical terms, a _site_ is a [[small category]] equipped with a [[coverage]] or [[Grothendieck topology]]. The [[category of sheaves]] over a site is a [[sheaf topos]] and the site is a _site of definition_ for this topos.

## Definition

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

A _site_ $(C,J)$ is a [[category]] $C$ equipped with a [[coverage]] $J$.

For $\mathcal{E}$ a topos equipped with an [[equivalence of categories]]

$$
  \mathcal{E} \simeq Sh(C,J)
$$

to the [[sheaf topos]] over a site, one says that $(C,J)$ is a **site of definition** for $\mathcal{E}$.

=--

Some classes of sites have their special names

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

A site is called

* a [[small site]], [[large site]], _essentially small site_ if the underlying category is a [[small category]], [[large category]], [[essentially small category]], respectively;

* a **[[cartesian site]]** if the underlying category is [[finitely complete category|finitely complete]] (which the [[Elephant]] calls a [[cartesian category]]);

* a **[[standard site]]** if it is a cartesian site equipped with a [[subcanonical coverage]].

=--

The term [[standard site]] appears in ([Johnstone, example A2.1.11](#Johnstone)).

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

Often a site is required to be a [[small category]]. But also [[large site]]s play a role.

=--

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

Every [[coverage]] induces a [[Grothendieck topology]]. Often sites are defined to be categories equipped with a Grothendieck topology. Some constructions and properties are more elegantly handled with coverages, some with Grothendieck topologies.

Notice that there are many equivalent ways to define a [[Grothendieck topology]], for instance in terms of a system of [[local isomorphisms]], or in terms of a system of [[dense monomorphisms]] in the [[category of presheaves]] $PSh(S)$.

=--

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

For $(C,J)$ a site, we write $Sh_J(C)$ for the
[[category of sheaves]] on $C$ with respect to the [[coverage]] $J$.

=--

## Properties

### Morita equivalent sites

Many inequivalent sites may have equivalent [[sheaf topos]]es.

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

Every [[sheaf topos]] has a [standard site](#TypesOfSites) of definition.

=--

This appears as ([Johnstone, theorem C2.2.8 (iii)](#Johnstone)).

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

By [this corollary](classifying+topos#SheafToposesAreClassifyingForTheirTheoryOfLocalAlgegras) at _[[classifying topos]]_ this means that every sheaf topos is the [[classifying topos]] for some [[theory]] of [[local algebras]].

=--

### Subcanonical sites

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

For $\mathcal{E}$ a [[sheaf topos]], the [[essentially small category|essentially small]] sites of definition $(\mathcal{C}, J)$ of $\mathcal{E}$ such that $J$ is a [[subcanonical coverage]] are precisely the [[full subcategories]] on [[generating families]] of objects equipped with the coverages induced from the [[canonical coverage]] of $\mathcal{E}$.

=--

This appears as ([Johnstone, prop. C2.2.16](#Johnstone)).

## Examples

### Classes of sites

* Every [[frame]] is canonically a site, where $U$ is covered by $\{U_i\}$ precisely if it is their [[join]].

  A subclass of examples is the [[category of open subsets]] of a [[topological space]].

  This are examples of [[posite]]s/[[(0,1)-site]].

* Various categories come with canonical structures of sites on them:

  * For every category $C$ there is its [[canonical coverage]].

  * On every [[regular category]] there is its [[regular coverage]].

  * On every [[coherent category]] there is its [[coherent coverage]].

  * Generalizing the previous two examples, on an [[∞-ary regular category]] there is a $\kappa$-canonical coverage.

  If the category in question is the [[syntactic category]] of a [[theory]], the corresponding site is the [[syntactic site]].

* For every site there is the corresponding [[double negation topology]] that forces the sheaf topos to a [[Boolean topos]].

Other classes of sites are listed in the following.

* [[big site]]

* [[dense sub-site]]

* [[large site]], [[essentially small site]]

* [[concrete site]]

* [[locally connected site]] / [[locally ∞-connected site]]

  * [[connected site]] / [[∞-connected site]]

  * [[strongly connected site]] / [[strongly ∞-connected site]]

  * [[totally connected site]] / [[totally ∞-connected site]]

* [[local site]] / [[∞-local site]]

* [[cohesive site]], [[∞-cohesive site]]

* [[atomic site]]

### Specific sites

* Sites for [[big topos]]es defining notions of [[higher geometry|geometry]] are:

  * The sites that define the [[higher geometry|geometry]] called [[differential geometry]] are [[CartSp]]${}_{smooth}$, [[SmoothMfd]], etc, equipped with the [[open cover]] [[coverage]]. Or more generally [[smooth loci]] etc.

  * The sites that induce [[topological geometry]] are small versions of [[Top]] equipped with the [[open cover]] [[coverage]].

  * The sites that induce the [[higher geometry]] modeled on [[Euclidean topology]] are the large site of [[paracompact manifold]]s and its [[dense sub-site]] [[CartSp]]${}_{top}$.

* The sites that define the [[higher geometry|geometry]] called [[algebraic geometry]] are site structures on categories of formal duals of [[commutative ring]]s or commutative [[associative algebra]]s

  * [[fpqc-site]] $\to$ [[fppf-site]] $\to$ [[syntomic site]] $\to$ [[étale site]] $\to$ [[Nisnevich site]] $\to$ [[Zariski site]]

    [[crystalline site]]

## Related concepts

* [[posite]]

* **site**

  * [[internal site]]
  * [[∞-ary site]]

* [[2-site]], [[(2,1)-site]]

* [[(∞,1)-site]]

  * [[model site]], [[simplicial site]]

* [[morphism of sites]], [[covering lifting property]]

## References

In

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

sites are discussed in section C2.1.

[[!redirects sites]]

[[!redirects 1-site]]
[[!redirects (1,1)-site]]
[[!redirects 1-sites]]
[[!redirects (1,1)-sites]]


sitting instant

#Contents#
* table of contents
{:toc}

# Sitting instants
* table of contents
{: toc}

## Idea

A [[function]] or [[differential form]] on a [[manifold with boundary]] to a is said to have _sitting instants_ at the boundary if there is an [[open neighbourhood]] $\partial X \subset U \subset X$ of the boundary such that the [[restriction]] $f|_U$ of $f$ to that neighbourhood is [[locally constant map|locally constant]] in the direction perpendicular to the boundary.

## Related concepts

* [[Lie integration]]

## References

* {#FSS} [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], Def. 4.2.1 in_[[schreiber:Cech Cocycles for Differential characteristic Classes]]_, Adv. in Theor. and Math. Phys., Volume 16 Issue 1 (2012), pages 149-250
   ([arXiv:1011.4735](http://arxiv.org/abs/1011.4735), [Euclid](http://projecteuclid.org/euclid.atmp/1358950853))

* {#BraunackMayer18} [[Vincent Braunack-Mayer]], Def. 3.1.3 _[[schreiber:thesis Braunack-Mayer|Rational parameterized stable homotopy theory]]_, Zurich 2018 ([pdf](https://ncatlab.org/schreiber/show/thesis+Braunack-Mayer#pdf))

[[!redirects sitting instant]]
[[!redirects sitting instants]]


situs

\tableofcontents

\section{Idea of a situs }

A _situs_ is a notion of generalised topological space, and
 defined as a simplicial object of
the [[category of filters|category of filters on sets]]
or, equivalently, the category of finitely additive measures
taking values 0 and 1 only, i.e.
a simplicial set equipped, for each $n\geq 0$,
with a filter, equiv. such (not quite) a measure,
on the set of $n$-simplices such that
under any face or degeneration map the preimage of a large set is large.
We denote the category of situses by sዋ.
Most of what we say below would also apply to the category $sM$ of _simplicial sets with finitely additive measures_ obtained by
dropping the restriction that the measure takes only two values.
With appropriate definitions, sዋ is a full subcategory of $sM$.

Intuitively, these filters (measures) are viewed
as additional structure of topological nature on a simplicial set
("the situs structure on a simplicial set")
giving a precise meaning to the phrase "a simplex is sufficiently small":
by definition, we say "a property holds for all small enough $n$-simplices"
iff it holds "almost everywhere" according to the filter (measure),
i.e. on a set in the filter, or, resp., on a set of full measure.
We refer to sets
in the filter, resp., sets of full measure,
as _neighbourhoods_.
In particular,
the situs structure "on a set" (i.e. on the simplicial set represented by the set) allows one to talk about an n-tuple of points
being "sufficiently near each other" for $n\ge 2$.
This extends to simplicial language the standard intuition of topology
available for  pairs of points (n=2):
given a topological structure on a set, the precise meaning of the phrase
"a property $P_{x_0}(x)$ holds for all points $x$ sufficiently close to a given point $x_0$"
is that the property holds on a neighbourhood of $x_0$;
intuitively, the pair $(x_0,x)$ with $x$ near $x_0$
is thought of as  small (a small simplex) because $x$ approximates $x_0$ up to a small error.

Situses  generalise metric and topological spaces, filters, and simplicial sets,
and the concept is designed to be flexible enough to formulate
categorically
a number of standard basic elementary definitions
in various fields, e.g. in analysis,
limit, (uniform) continuity and convergence,
equicontinuity of sequences of functions;
in algebraic topology, being locally trivial and geometric realisation; in
geometry, quasi-isomorphism;
in model theory, stability and simplicity and several Shelah's dividing lines, e.g. NIP, NOP, NSOP, $NSOP_i$, $NTP_i$, $NATP$, $NFCP$, of a theory.

No homotopy theory for situses has been developed,
although the naive definition of an [interval object](#IntervalSitus)
(namely, the simplicial set represented by the linear order $[0,1]$
equipped
with some situs structure based on the metric/topology)
leads to a [[directed topological space|directed (not symmetric) notion of homotopy]], and in sዋ there is a diagram chasing reformulation of local triviality [(becoming trivial after a certain base-change/pullback)](#locallytrivial).

\section{Examples}

We now give a number of examples demonstrating the expressive power
of the category of situses.

\subsection{Simplicial sets as situses}

A simplicial set can be equipped with discrete or indiscrete situs structure:
%The (in)discrete situs structure on a simplicial set
for each $n$, equip $X_n$ with the discrete or indiscrete filter, respectively.

\subsection{Metric spaces: uniformly continuous maps and quasi-isometries}

Let $M$ be a metric space. View $M$ as a simplicial set represented by the set of points of $M$, and equip each $M^n$ with the filter
generated by uniform neighbourhoods of the diagonal,
i.e. subsets containing all tuples of small enough diameter.
With this situs structure,
a map  $f:M\to N$ is uniformly continuous iff
it induces a morphism $f_\bullet:M_\bullet\to N_\bullet$ of situses.
In fact this defines a fully faithful embedding of the category
of metric spaces with uniformly continuous maps
into the category of situses.

We can also consider a different situs structure
capturing the notion of quasi-isomorphism in large scale geometry.
Equip each $M^n$ with the filter
such that a subset of $M^n$ is large iff it contains
all $n$-tuples such that the distance between
distinct points is at least $D$, for some $D\geq 0$.
With this situs structure,
for quasi-geodesic metric spaces,
a map $f_\bullet:M_\bullet\to N_\bullet$ is an isomorphism of situses
iff $f:M\to N$ is an quasi-isometry.

\subsection{A filter as a situs}

Given a filter $\mathfrak{F}$ on a set $X$,
there is a coarsest situs structure on
$X$ viewed as a simplicial set (i.e.
the simplicial set $|X|_\bullet$ represented by $X$)
such that its filter on the set $X$ of $0$-simplices is
finer than $\mathfrak{F}$.
Dually, there is a finest  situs structure on $|X|_\bullet$
such that its filter on the set of $0$-simplices is coarser than $\mathfrak{F}$.
{#DiagCartDefn} We denote these situses by
 $|X^\mathfrak{F}|_\bullet^{\operatorname{cart}}$
and  $|X^\mathfrak{F}|_\bullet^{\operatorname{diag}}$, respectively.

 In fact this gives two fully faithful embeddings of the category of filters into the category of situses
$$ |-^\mathfrak{F}|_\bullet^{\operatorname{cart}}, |-^\mathfrak{F}|_\bullet^{\operatorname{diag}} : ዋ  \to sዋ $$

{#PreordersAsSituses} In a similar way one can define two fully faithful embeddings of the category of filters on preorders
and continuous monotone maps. We denote these by
$X^{\leq\mathfrak{F}}_\bullet^{\operatorname{cart}}$
and  $X^{\leq\mathfrak{F}}_\bullet^{\operatorname{diag}}$, respectively.

\subsection{Topological and uniform spaces as situses }

{#DiagCartDefnNdim}
More generally, given an arbitrary
simplicial set $X_\bullet$ and a filter $\mathfrak{F}$ on the set of $n$-simplices $X_n$,
there is a coarsest/finest situs structure on $X_\bullet$
such that its filter on the set of $n$-simplices is finer/coarser
than $\mathfrak{F}$. Taking $n=0$ and the filter $\mathfrak{F}$ always indiscrete gives two fully faithful embeddings of the category of simplicial sets into the category of situses.

We use this to define situses corresponding to uniform and topological spaces.

\subsubsection{Uniform spaces as situses }

Take a set $X$ and view it as a simplicial set $X_\bullet$ (represented by $X$). Recall that a [[uniform space|uniform structure]] on $X$ is a filter on  $X\times X$;
take the coarsest situs structure with this filter on
the set $X\times X$ of $1$-simplices. This is the situs
associated with the uniform structure on $X$.
Intuitively, we defined the situs  structure "on a set"
such that two points $x_1,x_2\in X$ of points are
"sufficiently close to each other" in the uniform structure
iff the 1-simplex $(x_1,x_2)\in X\times X$ is "sufficiently small".

{#SymmetricSitus}
In fact, it is easy to define uniform spaces in terms of situses.
A filter on $X\times X$ is a uniform structure iff it is symmetric
(i.e. the endomorphism of $X\times X$, $(x,y)\mapsto (y,x)$ permuting the coordinates is continuous)
and this construction produces a situs such that the filter on $X\times X\times X$
is the coarsest filter such that the two maps $X^3\to X\times X$, $(x_1,x_2,x_3)\mapsto (x_i,x_{i+1}),i=1,2$
are continuous. This can be used to characterise situses arising
from uniform structures as those symmetric situses such that
the filter of $X\times X\times X$ has this property.
We say that a situs is symmetric iff
it factors though the category of non-empty finite sets.

\subsubsection{Topological spaces as situses}

The situs associated to a topological structure on $X$
is defined in the same way starting from
the filter of non-uniform neighbourhoods of the diagonal on $X\times X$
defined as consisting of the subsets of
form $\bigcup_{x\in X} \{x\}\times U_x$ where
$U_x\ni x$ is a not necessarily open neighbourhood of $x\in X$.

A trivial verification shows these constructions
define fully faithful embeddings
of the categories of topological and of uniform spaces
into the category of situses, and in fact
there are corresponding forgetful functors to these categories
such that the following compositions are the identity:
$$Top \to sዋ \to Top$$
$$UniformSpaces \to sዋ \to UniformSpaces .$$

\subsection{Cauchy sequences and equicontinuity }

A filter $\mathfrak{F}$ on a metric space $M$ is _Cauchy_ iff
$\mathfrak{F}^{\operatorname{cart}}_\bullet \to M_\bullet$ is continuous.

A sequence of functions $f_i:L \to M$, $i\in \mathbb{N}$ of metric spaces
is _uniformly equicontinuous_ iff the map $(\mathbb{ N}^{cofinite})^{\operatorname{diag}}_\bullet\times L_\bullet \to M_\bullet$ is continuous.
This gives a precise meaning to the phrase
"if $i\in\mathbb{N}$, $x\in L$ and $j\in \mathbb{N}$, $y\in L$ are sufficiently close to each other, so are $f_i(x)$ and $f_j(y)$".
A different choice of the situs strucuture gives a different precise meaning:

A sequence of functions $f_i:X \to M$, $i\in \mathbb {N}$ from a topological space $X$ to a metric space $M$
is _equicontinuous_ iff the map $(\mathbb{ N}^{cofinite})^{\operatorname{diag}}_\bullet\times X_\bullet \to M_\bullet$ is continuous.

\subsection{Limits, compactness, and completeness }

An endomorphism of the category $\Delta$ of finite linear orders gives rise to an endomorphism of the category of situses.
Of particular interest is the shift
endomorphism $\Delta\to\Delta$ adding a new least element ([[decalage]] considers the endomorphism adding a new greatest element rather than least)
on objects, $n\mapsto 1+n$, and on morphisms,
$f:n\to m$ goes to $f[+1]:1+n\to 1+m$, $f[+1](0)=0$, $f[+1](1+i)=1+f(i)$, $0\leq i\leq n$.
The object $X_\bullet\circ [+1]$ and morphism $X_\bullet\circ [+1]\to X$
allows one to talk about local properties of $X_\bullet$,
e.g. limits and local triviality.

\subsubsection{Limits via shift endomorphism}

For example, taking a _limit_ of a filter $\mathfrak{F}$ on
a topological or metric space $X$ corresponds to taking the factorization
$$\mathfrak{F}^{\operatorname{diag}}_\bullet\to X_\bullet\circ [+1]\to X_\bullet.$$
Indeed, the underlying simplicial set of $\mathfrak{F}^{\operatorname{diag}}_\bullet$
is connected and thus maps to a single connected component
of  $X_\bullet\circ [+1]=\sqcup_{x\in X} \{x\}\times X_\bullet$
(here we consider the equality of the underlying simplicial sets);
continuity of the map $\mathfrak{F} \to  X\times X$, $x\mapsto (x_0,x)$
means exactly that the first coordinate $x_0$ is a limit point
of $\mathfrak{F}$ on $X$.

\subsubsection{Compactness and completeness as lifting properties}
Diagram chasing reformulation of the notions above allows to define compactness and completeness as [lifting properties](en.wikipedia.org/wiki/Lifting_property).

Let $\mathbb{N}^{cofinite}$ and $\mathbb{N}^{\leq cofinite}$
denote the set, resp. the linear order $ \mathbb{N}^\leq$, equipped
with the filter of cofinite subsets.

A metric space $M$ is _complete_ iff either of the following equivalent conditions holds:

i. $\bot\to (\mathbb{N}^{\leq cofinite})^{\operatorname{cart}}_\bullet \rightthreetimes M_\bullet\circ [+1]\to M_\bullet$.

ii. $\bot\to (\mathbb{N}^{cofinite})^{\operatorname{cart}}_\bullet \rightthreetimes M_\bullet\circ [+1]\to M_\bullet$.

iii. $(\mathbb{N}^{\leq cofinite})^{\operatorname{cart}}_\bullet\to
(\mathbb{N}^{\leq cofinite}\cup\{\infty\})^{\operatorname{cart}}_\bullet  \rightthreetimes M_\bullet\to\top$.

Such a reformulation raises the question whether the notion of completeness may be
defined with help of the archetypal counterexample: is a metric space $M$ complete
iff
$$ M_\bullet\circ [+1]\to M_\bullet \in (\mathbb {R}_\bullet\circ [+1]\to \mathbb{R}_\bullet)^{\rightthreetimes lr} ? $$
It also allows to define the completion of a metric space
in terms of a weak factorisation system
$$M_\bullet \xrightarrow{((\mathbb{N}^{\leq cofinite})^{\operatorname{cart}}_\bullet\to
(\mathbb{N}^{\leq cofinite}\cup\{\infty\})^{\operatorname{cart}}_\bullet)^{rl}} \hat M_\bullet \xrightarrow{((\mathbb{N}^{\leq cofinite})^{\operatorname{cart}}_\bullet\to
(\mathbb{N}^{\leq cofinite}\cup\{\infty\})^{\operatorname{cart}}_\bullet)^r} \top
$$

A topological space $X$ is _compact_ iff for each ultrafilter $\mathfrak{U}$ either of the following equivalent conditions holds:

i. $\bot\to \mathfrak{U}^{\operatorname{diag}}_\bullet \rightthreetimes X_\bullet\circ [+1]\to X_\bullet$.

ii. $\mathfrak{U}^{\operatorname{diag}}_\bullet
\to (\mathfrak{U}\cup\{\infty\})^{\operatorname{diag}}_\bullet \rightthreetimes X_\bullet\to \top$.

A topological space $K$ is _compact_ iff
$$K_\bullet \circ [+1]\to K_\bullet \in
(\{\{o\},\{o,1\}\}^{\operatorname{cart}}_\bullet \cup  \{\{1\},\{o,1\}\}^{\operatorname{cart}}_\bullet\to  \{\{o,1\}\}^{\operatorname{cart}}_\bullet)^{\rightthreetimes lr}.$$
Here
$\{\{o\},\{o,1\}\}$, $\{\{1\},\{o,1\}\}$, and $\{\{o,1\}\}$
are viewed as filters on the set $\{o,1\}$. (needs verification)

\subsection{Local triviality }

{#locallytrivial}
A map $f:X\to Y$ of topological or metric spaces is locally trivial with fibre $F$
iff in sዋ becomes a direct product with $F_\bullet$ ("globally trivial")
after base-change
$Y_\bullet\circ [+1]\to Y_\bullet$. That is,
$f_\bullet:(Y_\bullet\circ [+1])\times_{Y_\bullet} X_\bullet \to  Y_\bullet\circ [+1]$
is of form $ (Y_\bullet\circ [+1])\times F_\bullet \to  Y_\bullet\circ [+1]$.

\subsection{Geometric realisation}

The notion of [[geometric realization]] involves
topological spaces and simplicial sets,
which both are situses. This allows one to interpret
the Besser-Drinfeld-Grayson construction
of geometric realisation in sዋ, as follows.

View the standard geometric simplex consisting of sequences $0\leq x_1\leq ... \leq x_n\leq 1$ in $\mathbb{R}^n$
as the space of monotone maps
$[0,1]^\leq \to (n+1)^\leq$ with
Skorokhod-type metric
$dist(f,g):= sup_{x} inf_{y} \{ |x-y| : f(x)=g(y) \}$.
Recall that Skorokhod metric is used in probality theory
to express the intution that two random variables are close
if one can be obtained from another by a small perturbation
of both time and space (values).
The category of situses allows us to view both linear orders as situses:
the situs structure on $[0,1]_\bullet$ "remembers" the metric,
and the situs structure on $\Delta_n=Hom(-,(n+1)^\leq)$
"remembers" the equality $f(x)=g(x)$, i.e. is the finest situs
structure such that the filter on the set of 0-simplices is indiscrete.
Then one may define a situs structure on the
[inner hom](https://ncatlab.org/nlab/show/internal+hom#in_simplicial_sets)
$$HHom([0,1]_\bullet, -) :sዋ\to sዋ$$
of the underlying simplicial sets
motivated by the definition of Skorokhod metric.

This gives the construction of geometric realisation due to
Besser, Drinfeld, and Grayson.
See details at section 3.2 of [[geometric realization]].

\subsection{The interval object and homotopy theory }
{#IntervalSitus} No homotopy theory for situses has been developed.
The naive definition of an [[interval object]] used to define
in sዋ geometric realisation
 eqipped with some situs structure "remembering" the topology,
does not appear very useful, particularly
for dealing with situses arising in model theory
and the [lifting properties](en.wikipedia.org/wiki/Lifting_property)
defining stability and simplicity (defined below).
However, note that the naive interval object
 $[0,1]^\leq_\bullet$ reminds one of [[directed topological space|directed topological spaces]]:
it is directed by definition
and so would be
any naive notion of homotopy associated with it:
a homotopy from $A$ to $B$ cannot in general
be reversed to get a homotopy from $B$ to $A$.

\subsection{Stability and simplicity in model theory}

Let us now describe how situses can be used to reformulate
two notions of model theory: stability and simplicity of a first-order theory. An alternative and more general approach is given in the next section on the Shelah's dividing lines which completely covers simplicity(NTP).

\subsubsection{Stability. }

Consider a model $M$ in a language $\mathcal {L}$, and a linear order $I$.
For an $r$-ary $\mathcal {L}$-formula $\phi(x_1,...,x_r)$,
we say that a sequence $(a_i)_{i\in I}$ of elements of $M$
is $\phi$-indiscernible (with repetitions)
iff for either all or none of the
subsequences $(a_{i_1},...,a_{i_r}), i_1\le ... \le i_r$ (of distinct elements)
the formula $\phi(a_{i_1},...,a_{i_r})$ holds in $M$.

Equip $M^n$ with the filter generated by
the sets of all $n$-tuples which are $\phi$-indiscernible with repetitions,
where $\phi$ varies through all $\mathcal{L}$-formulas.
The situs so obtained is called {the generalised pre-Stone space
of $M$ in sዋ} because the forgetful functor $sዋ\to Top$ takes it
to the set of elements of $M$ equipped with the preimage
of [[Stone topology]];
by this we mean the topology on $M$ generated by
sets of elements realising unary $\mathcal{L}$-formulas.
There are many variants of this definition, notably instead of
being $\phi$-indiscernible
one may require being a part of an {infinite} $\phi$-indiscernible sequence,
and instead of $M^n$ consider the set of $n$-types  $S_n(\emptyset)$
or $S_n(M)$.

We shall reformulate the following characterisation of
stable theories as a lifting property in sዋ.

A first-order theory
is stable iff in a saturated enough model it holds that each
 $\phi$-indiscernible sequence of $n$-tuples is in fact a $\phi$-indiscernible set,
for each $n\ge 0$ and each formula $\phi$ of the language of the theory.

For $n=1$ this can be reformulated as a lifting property in sዋ as follows.

Fix a linear order $I$.
Let $I^\leq_\bullet:=(I^\leq)^{\operatorname{cart}}_\bullet$ be the situs associated with the preorder $I^\leq$ with the indiscrete filter.
[Recall](#PreordersAsSituses) that this is the simplicial set
$n^\leq \mapsto Hom_{preorders} (n^\leq, I^\leq)$ represented by $I^\leq$ as a linear order,
equipped with indiscrete filters.
Let  $(I^{\leq tails})^{\operatorname{cart}}_\bullet$ denote the  situs associated with the preorder $I^\leq$ with the  {filter of tails}
generated by the subsets containing all elements large enough.

Let  $(|I|^{tails})^{\operatorname{cart}}_\bullet$ denote the situs associated with the filter of tails on the set of elements of $I$.

An indiscernible sequence indexed by a linear order $I$ is an injective continuous map $(I^{\leq})^{\operatorname{cart}}_\bullet \to M_\bullet$.

An indiscernible set indexed by $I$ is an injective continuous map $|I|^{\operatorname{cart}}_\bullet \to M_\bullet$.

An eventually indiscernible sequence indexed on a linear order $I$ is an injective continuous map
$(I^{\leq tails})^{ cart}_\bullet \to M_\bullet$.

\begin{proposition}
Let $M$ be a model. The following are equivalent:

i. each $\phi$-indiscernible sequence of elements is in fact a $\phi$-indiscernible set.

ii. the situs $M_\bullet$ is [symmetric](#SymmetricSitus)

iii. the following lifting property holds in sዋ:
$$ (I^{\leq tails})^{\operatorname{cart}}_\bullet\to (|I|^{tails})^{\operatorname{cart}}_\bullet
\rightthreetimes M_\bullet\to \top .$$
\end{proposition}
\begin{proof} iii. says that each continuous map $ (I^{\leq tails})^{\operatorname{cart}}_\bullet\to M_\bullet$ factors
as $(I^{\leq tails})^{\operatorname{cart}}_\bullet\to (|I|^{tails})^{\operatorname{cart}}_\bullet
\to M_\bullet$. If these maps are injective, remarks above
say it is equivalent to i. If not injective, then
in some end segment each element occurs infinitely often,
and in that case being indiscernible with repetitions
means being set indiscernible.
\end{proof}

([Simon,2021](#Simon2021)) implies that a theory is stable iff the lifting property iii. holds
for the situs associated with $M\times M$ considered in the language with arbitrary parameters,
for a saturated enough model $M$ of the theory.

\subsubsection{Simplicity. }

The definition of simplicity is not as simple combinatorially.
A more general approach to simplicity is discussed in the next section on Shelah's dividing lines.

First let us introduce the situs associated with a model
for this purpose; this situs structure is defined to talk
about consistency of instances of a formula. Fix a formula $\phi$.
As usual, the situs is based on the simplicial set represented by the
set of elements of $M$.
The filter on $M^n$ is generated by a single set
of those tuples $(a_1,...,a_n)$ such that
$M\models \exists x \bigwedge_{1\leq i\leq n}  \phi(x,a_i)$.
Let us denote this situs by $M_\bullet^{\exists\phi}$ and call it the _$\phi$-characteristic_ situs of model $M$. Note that
$\phi$-characteristic situs captures the same structure as
the characteristic sequence of a first order formula in
([M.Malliaris.
The characteristic sequence of a first-order formula. 2010](#Mallaris2010jsl)), see also ([M.Malliaris.
"Edge distribution and density in the characteristic sequence](#Mallaris2010apal)).

##### Finite Cover Property

Recall that a formula φ(x;y) has the _finite cover property_ if for arbitrarily large $n \le \omega$ there exist $ a_0,...a_n$ such that $\{\phi(x;a_0),...\phi(x;a_n)\}$ is n-consistent but (n+1)-inconsistent.

Recall that a simplicial object $M_\bullet$, e.g. a situs, is said to have finite dimension if
for some $k$ for each $n$
 $M_\bullet(n)$ is the pullback of
all the simplicial maps $M_\bullet(n)\to M_\bullet(l)$, $l\leq k$.
Let $P_n$ denote the formula $\exists x \wedge_{i\leq n} \phi(x,y_i)$.

\begin{theorem}

1. $\phi$ has no finite cover property

2. the characteristic sequence $\lt P_n\gt $ has finite support (in terminology of [Malliaris, Def.2.6](#Mallaris2010jsl))

3. the situs $M^{\exists\phi}_\bullet$ has finite dimension, i.e. explicitly, for some $k$ for each $n$
the filter on $M^{\exists\phi}_\bullet(n)=M^n$ is the coarsest filter such that
all the simplicial maps $M^{\exists\phi}_\bullet(n)=M^n\to M^{\exists\phi}_\bullet(k)=M^k$ are continuous

4. there is $k$ such that for each $n$ $\exists x \wedge_{i\leq n} \phi(x,y_i)$ holds iff  $\exists x \wedge_{i\leq n} \phi(x,y_i)$ for any $k$-element subset $y_{i_1},...,y_{i_k}$

5. there is $k$ such that for each $n$ $\exists x \wedge_{i\leq n} \phi(x,y_i)\leftrightarrow \wedge_{1\leq i_1\le ...\le i_k\le n}\exists x \wedge_{1\leq l\leq k} \phi(x,y_{i_l})$

\end{theorem}
\begin{proof} Items 1 and 2 are  Remark 2.7 of [Malliaris](#Mallaris2010jsl), item 4 and 5 are  both item 2 and item 3 written explicitly.
\end{proof}

##### No tree property NTP

The reformulation in terms of situses uses
the definition of a simple first-order theory which
says
that each formula of the theory has
NTP ("not the tree property")
see [Tent-Ziegler, Def.7.2.1](#TentZiegler), or [3,\S9](#3)
NTP is defined as a lifting property with respect to a morphism
involving the following combinatorial structures.

We recall the definition of NTP and a simple theory.

\begin{definition}[Tent-Ziegler,7.2.1] 1. A formula $\varphi(x, y)$ has the tree property
(with respect to k) if there is a tree of parameters $(a_s\,\,|\,\,\emptyset \neq s \in {}^{\lt\omega}\omega )$ such that:

 a) For all $s\in {}^{\lt\omega}\omega $, $(\varphi(x, a_{si} )\,\,|\,\,i \lt\omega )$ is $k$-inconsistent.

 b) For all $\sigma\in {}^{\omega}\omega $ $ \{\varphi(x, a_s )\,\,|\,\,\emptyset \neq s\subseteq \sigma  \}$ is consistent.

2. A theory T is simple if there is no formula $\varphi(x, y)$ with the tree property.

\end{definition}

Let $T^\leq$ be an infinitely branching tree of infinite depth,
viewed as preorder, and equipped with the indiscrete filter.
We may take $T^\leq$ to be ${}^{\lt\omega}\omega$.
Let $T^\leq_\bullet:=(T^\leq)^{cart}_\bullet$
denote the corresponding situs.
[Recall](#PreordersAsSituses) that by definition $T^\leq_\bullet(n)$
is the set of ordered (weakly increasing) $n$-tuples
of vertices of $T$, and there is  only one large subset,
namely the whole set.

Note that to give a morphism $T^\leq_\bullet \to M_\bullet^{\exists\phi}$ of the underlying simplicial sets
is the same as to give a a tree of parameters $(a_s\,\,|\,\,\emptyset \neq s \in {}^{\lt\omega}\omega )$. This morphism is continious
iff these parameters satisfy item b,
in notation of the definition: indeed,
continuity means that the preigame of the large (by definition)
set of $\phi$-consistent tuples is large, i.e. the whole
set $T^\leq_\bullet(n)$ of ordered tuples, for each $n$.

Let $|T|_\bullet$ be the simplicial set represented by the set
$|T|$ of vertices of $T$, namely
$|T|_\bullet(n^\leq)=|T|^n$.

\begin{defn} Let $|T|^{TP}_\bullet$ denote the simplicial set $|T|_\bullet$
equipped with the _$TP$-tautological_ filter on $|T|_\bullet(n^\leq)$ defined as follows:
a subset is not small iff it either contains

1) some tuple in weakly increasing order, or

2) all   the lexicographically ordered tuples
   required to be inconsistent by the tree property
   with respect to a subtree-counterexample to the tree property.

In more detail, a subset $\epsilon$ is large iff

1') it contains the subset of tuples in weakly increasing order

2') for each isomorphic copy
of $T'={}^{\lt\omega}\omega$ in $T^\leq$ there is a vertex $v\in T'$
and its immediate (in $T'$) descendants
$v_1\leq_{lex}...\leq_{lex} v_k $ such that $(v_1,..,v_k) \in \epsilon \cap T'$.
%A verification shows that this indeed defines a filter.
\end{defn}

Note that by item (i) the map $T^\leq_\bullet \to |T|_\bullet^{TP}$ is continuous.
Also note that no tuple of increasing elements is required to be $\phi$-inconsistent by the tree property.

\begin{proposition}
The following are equivalent:

i. the formula $\phi$ has NTP with respect $k$ in the model $M$

ii. in sዋ there is no morphism $\tau:T^\leq_\bullet\to M_\bullet^{\exists\phi}$ such that
    for each tuple $k$-tuple $v_1,..,v_k$, for each $k$,
     of immediate descendants of the same vertex,
     $M \models\neg \exists x (\phi(x,\tau(v_1))\wedge ... \wedge \phi(x,\tau(v_n))$

iii. In sዋ the following lifting property holds:
$T^\leq_\bullet \to |T|^{TP}_\bullet
\rightthreetimes M_\bullet^{\exists\phi}\to \top $
\end{proposition}
\begin{proof} ii. is exactly the definition of NTP for formula $\phi$
as stated in ([Tent-Ziegler,  Def.7.2.1](#TentZiegler)), cf. [3,\S9](#3).
In iii., one only needs to check that the unique lifting is
continuous, namely that the set of tuples
$(v_1,...,v_k)$ such that  $M \models \exists x (\phi(x,\tau(v_1))\wedge ... \wedge \phi(x,\tau(v_k))$ is large.
By the definition of the filter, this set is large iff there is
an infinitely branching subtree of infinite depth satisfying ii.
This implies that ii. and iii. are equivalent.

Finally, let us prove our $TP$-tautological filters are well-defined.
We need only to show that the union of
any two small sets $X\cup Y$ is small.
Assume it is not small.
Label each vertex of the tree with the largest $n\lt\omega$
such that the first small subset contains
above the vertex all tuples required to be inconsisent
in some copy of ${}^{\lt n}\omega$.
Above each vertex in $X$ there are at most finitely many vertices in $X$ labelled
by the same or greater number. Removing them
leaves $X\cup Y$ not small.
But then we get that vertices of $X$ are
labelled by numbers strictly decreasing along
any branch, hence $X$ is of finite depth.
Now pick a vertex labelled $0$. This means
that below that vertex
there is no infinite set of siblings that
each lexicographically ordered tuple is in $X$,
hence among any infinite set of siblings by Ramsey theorem there is
an infinite set of siblings not in $X$, i.e. in $Y$.
Hence, $Y$ is not small.
\end{proof}

\begin{remark} One can similarly define
$TP_i$-tautological situs of a tree $T$, for $i=1$,
and see that $TP_i$ is defined by a lifting property.
The same argument gives lifting properties
related  to cdt, [inp](modeltheory.fandom.com/wiki/INP_pattern),
and sct patterns
in classification theory.
\end{remark}

\begin{remark} This raises the question whether $NTP=NTP_1\&NTP_2$
holds in the category of situses. It seems the standard proof would go through if one defines the corresponding lifting properties carefully enough. In particular, to reflect the use of Ramsey theorem,
it may be necessary to replace $|M|_\bullet$
by the simplicial set of types $S^M_\bullet$ where $S^M(n)$ is the set of $n$-types, and also do the same for $T^\leq_\bullet$ and $|T|^{TP}_\bullet$ for quantifier-free types in an appropriate language.

In fact, it would seem that the standard proof of $NTP=NTP_1\&NTP_2$
gives that $T^\leq_\bullet\to |T|_\bullet^{TP}$ is
the pushout of $T^\leq_\bullet\to |T|_\bullet^{TP_1}$
and $T^\leq_\bullet\to |T|_\bullet^{TP_2}$,
just as diagram chasing considerations show would
be sufficient for the corresponding relation between the lifting properties.
Though, possibly one needs to modify
the definitions of the filters appropriately modified
to reflect the need to use Ramsey theorem and
consider the tree properties with respect all the finite conjunctions
$\&_i\phi(x,y_i)$ of $\phi(x,y)$ at the same time.
\end{remark}

\begin{remark}
The notion of a $\phi$-characteristic situs captures the same structure
as
the characteristic sequence of a first order formula in
([M.Malliaris.
The characteristic sequence of a first-order formula. 2010](#Mallaris2010jsl)), see also ([M.Malliaris.
"Edge distribution and density in the characteristic sequence](#Mallaris2010apal)). Moreover, it appears that several properties
of characteristic sequences can be defined as lifting properties
in the category of situses.
\end{remark}

\subsection{Shelah's dividing lines NOP, NSOP, NSOPi, NTP, NTPi, NATP, and tautological filters}

We show that in the category of situses
a number of the Shelah's divining lines,
namely  $NOP, NSOP, NSOP_i, NTP, NTP_i$, and $NATP$
are expressed as Quillen lifting properties of form
$$A_\bullet \to B_\bullet \rightthreetimes M_\bullet\to\top$$
where $\top$ is the terminal object, and
$M$ is a situs associated with a model and a formula,
and $A$ and $B$ are objects of combinatorial nature.

\subsection{Informal explanation}

Recall the common pattern of definitions of
$NOP, NSOP, NSOP_i, NTP, NTP_i, NATP$.
As usually stated, such a property wrt a formula $\phi$ and a model $M$
require
that there is no combinatorial structure ("counterexample" or "witness")
formed by elements of the model $M$ satisfying certain "positive" requirement
that certain collections of formulas are consistent, and "negative" requirement
that certain collections of formulas are inconsistent.
The combinatorics is coded by the maps of underlying simplicial sets
$|A_\bullet|\to |M_\bullet|$ and $|B_\bullet|\to |M_\bullet|$.
The positive requirements are coded by continuity of the morphism $A_\bullet\to M_\bullet$.
The failure of negative requirement are coded by continuity of the diagonal morphism
$B_\bullet \to M_\bullet$.

The underlying simplicial sets of $A_\bullet, B_\bullet$, and $M_\bullet$ are representable,
and are so chosen that both morphisms  $A_\bullet\to M_\bullet$ and $B_\bullet \to M_\bullet$
both correspond to maps of sets $a_\bullet :|I|\to |M|$ and $a'_\bullet:|I'|\to |M|$.
The $A_n$'s and $B_n$'s represent the indices of tuples mentioned
in positive, resp. negative, requirements.
The filters on $A_n$ are taken indiscrete; thus continuity of $f:A_\bullet\to M_\bullet$ means that for each $n\in \mathbb {N}$,
each tuple $(a_i)_{0\leq i\le n} \in a(A_n)\subset |M|^n$ satisfies a certain formula.
Namely, if $M_\bullet$ is the situs of $\phi$-indiscernible sequences,
the continuity says that each such sequence is $\phi$-indiscernile;
if $M$ is the situs of consisent finite $\phi$-types, the continuity says
that each finite type $\phi(x,a_i)$ is consistent.

To code by continuity the failure of negative requirements,
we need to define non-trivial filters on $B_n$. The first (incorrect) attempt
would be to say that a subset of $B_n$ is large (a neighbourhood)
iff it contains a tuple representing one of the negative requirements;
unfortunately, this is not a filter. But if it were, then continuity of $B_n\to M_\bullet(n)$
would mean exactly that one of negative requirements fails, and hence
the map $A_\bullet\to M_\bullet$ does not represent a counterexample/witness.
To fix this definition, define a subset of $B_n$ to be large (a neighbourhood)
iff it contains a tuple representing a negative requirement with respect to each
_substructure_ of $A$ of the same shape as required by the property.
We call the filters on $B_n$'s _tautological_ because of the tautological argument
showing equivalence  of the continuity and the failure of the negative requirements.

Below we do the reformulations for non-order properties $NOP$, $NSOP$, and $NSOP_i$, $i\geq 3$
which involve the situs of $\phi$-indiscernible sequences. These reformulations involve
reformutading the standard definitions in terms of $\phi$-indiscernible sequences.

The no-tree-properties were formulated above and they involve the situs of consistent $\phi$-types.

\begin{remark} Note our considerations allow to formulate a precise conjecture corresponding to
the equality $NATP = NSOP_1+ NTP_2$ of [On the Antichain Tree Property](https://arxiv.org/pdf/2106.03779.pdf).
Indeed, all classes occurring in the formula are defined by lifting properties wrt the same morphism,
and the equalitity should represented a relationship of the "combinatorial" morphisms on the left.
Though doing so should perhaps require a rather more careful reformulations taking care
of formulas of arbitrary arity.
\end{remark}

####Rewriting NOP (no order property)

Recall a formula $\phi(-,-)$ has NOP (no order property) iff  there no sequence $(a_i)_{i\in\omega}$ such that
$ \phi(a_i,a_j) \Leftrightarrow i\leq j$.

Because NOP cares about {\em consistency of instances of a formula on tuples of ordered variables},
equip $M_\bullet(n^\leqslant):=M^n$ with the filter of subsets containing
$$\left\{
(a_1,..,a_n)\in M^n\,:\, i\le j \,\&\, a_i\neq a_j\implies \varphi(a_i,a_j) \right\}$$

Then rewrite the standard definition of NOP
in a form with two items representing positve and negative requirements.

\begin{definition}
 A formula $\varphi(x, y)$ has {\em NOP} iff
 there is no sequence $(a_i)_{i\in\omega}$ such that
\begin{itemize}
\item[a)] for $i\le j$, $\varphi(a_j,a_i)$ is false. %each subsequence $(a_i,a_j,a_i)$ is not $\varphi$-indiscernible with repeated elments
\item[b)] for $i\le j$, $\varphi(a_i,a_j)$ holds.
\end{itemize}
\end{definition}
In analogy with NTP,  think of the sequence as  a tree $T:=\omega^\leqslant$ with a single branch,
and equip
$\omega^\leqslant_\bullet(n^\leqslant) := \Hom_{\text{preorders}} ( n^\leqslant, \omega^\leqslant ), n\ge 0.
$ with indiscrete filters.
Let
$|\omega|^\text{NOP}_\bullet(n^\leqslant) := \Hom_{\text{Sets}} ( n^\leqslant, |\omega|)=|\omega|^n, n\ge 0 $.
where
a  subset $U$ of $|\omega|^\text{NOP}_\bullet(n^\leqslant) =|\omega|^n$ is
{\em large} iff it is closed under simplicial maps and
\begin{center}\color{blue}
each infinite subsequence $(a_{i_j})_j$ of $\omega$  %there are
has a subsequence \\
$(a_{i_{j_1}},..,a_{i_{j_n}})\in U$ where ${j_1}\le ...\le {j_n}$
\end{center}
With these definitions, $\omega_\bullet^\leqslant\to |\omega|^\text{NOP}_\bullet \,\rtt\, M_\bullet\to \top$ defines NOP.

#### NSOP$_\ell$ for $\ell\geq 3$ (no strict order property)
\begin{definition}
		For $\ell \geqslant 3$, a formula $\varphi(x,y)$ has SOP$_\ell$ (%the
		$\ell$-strong order property) iff
\begin{itemize}
\item[a)]  the set
				$\{\varphi(x_1,x_2),\ldots,\varphi(x_{n-1},x_n),\varphi(x_\ell,x_1)\}$ is inconsistent.

\item[b)] 		there are $(a_i)_{i\lt\omega}$ such that $\models\varphi(a_i,a_j)$ for all $ i\lt j $

\end{itemize}				A theory is NSOP$_\ell$ if no formula has SOP$_\ell$.
\end{definition}
Equivalently,  NSOP$_\ell$   means that ``item a) implies  NOP''.

Let $|\ell|_\bullet(n^\leqslant):=\Hom_{\textrm{Sets}}(n^\leqslant,|\ell|)=|\ell|^n$, $n\ge 0$
denote the simplicial set represented by the set $\{1,2,...\ell\}$ with $\ell$ elements which we denote by $|\ell|$.
Equip $|\ell|_\bullet(2^\leqslant)=|\ell|^2$ with the filter of subsets containing
{$\{(1,2),(2,3),...,(\ell-1,\ell),(\ell,1), (1,1),..,(\ell,\ell)\}$,} and for each $n\ge 0$	 equip
$|\ell|_\bullet(n^\leqslant)=|\ell|^n$ with the coarsest filter such that
all simplicial maps $|\ell|_\bullet(n^\leqslant)\to |\ell|_\bullet(2^\leqslant)$ are continuous.

Then a map $|\ell|^{\ell\text{-cycle}}_\bullet\longrightarrow M_\bullet, i\mapsto a_i$ is  continuous iff either $a_1=...=a_\ell$ or
	$M\models\{\varphi(a_{j_1},a_{j_2}),\ldots,\varphi(a_{j_{\ell'-1}},a_{\ell'}),\varphi(x_{a_\ell'},a_{j_1})\}$ holds
	for the subsequence of $(a_{j_k})_k$ of distinct elements.
Hence, item a) holds iff $|\ell|^{\ell\text{-cycle}}_\bullet \longrightarrow \top\,\rtt\, M_\bullet\longrightarrow \top$.
Then\\  NSOP$_\ell$ can be stated as:
\begin{center}
$|\ell|^{\ell\text{-cycle}}_\bullet \longrightarrow \top \,\rtt\, M_\bullet\longrightarrow \top$ implies
 $\omega_\bullet^\leqslant\to |\omega|^\text{NOP}_\bullet \,\rtt\, M_\bullet\to \top$
\end{center}

####NFCP (no finite cover property)

We also note that in the category $\sPhi$  of generalised topological spaces
NFCP (no finite cover property) means
that $M_\bullet$ is of infinite dimension. Recall that ``$M_\bullet$ is of infinite dimension'' (as a simplicial object)
means that the whole of $M_\bullet$ is fully determined by a finite piece of $M_\bullet$,
namely  there is $N\ge 0$ such that for each $n\ge N$ $M_\bullet(n^\leqslant)$
is the pullback of the diagram consisting of simplicial maps $M_\bullet(n^\leqslant)\to M_\bullet(m^\leqslant)$
and $M_\bullet(m_1^\leqslant)\to M_\bullet(m_2^\leqslant)$ where $m,m_1,m_2\le N$.
In terminology of ([M.Malliaris.
"Edge distribution and density in the characteristic sequence](#Mallaris2010apal)) this is expressed by saying that
``the characteristic sequence $\lt P_n\gt$ of $\varphi$  has finite support''.

####{Stability and NIP}
Both stability and NIP can be expressed in terms of indiscernible sequences.
Hence,  we modify the filters on $|M|_\bullet$  to reflect that:
the filter on   $|M|_\bullet(n^\leqslant)=M^n$
is generated by sets of  {\em $\varphi$-indiscernible sequences with repeations},
i.e.~sequences $(a_i)_i$ such that
$\varphi(a_{i_1},...,a_{i_r})\Leftrightarrow \varphi(a_{j_1},...,a_{j_r})$ whenever
$i_1\le ...\le i_r$, $j_1\le ...\le j_r$, all the $a_{i_1},..,a_{i_r}$ are distinct, and all
the $a_{j_1},...,a_{j_r}$ are distinct.  %any subsequence of distinct elements $\varphi$-indiscernible.
(If you feel it is more natural, you may require each subsequence with distinct elements
to be part of an {\em infinite}
$\varphi$-indiscernible sequence.)

This construction of $M_\bullet$ generalises to formulas of arbitrary arity,
and, moreover, for arbitrary collections of formulas.

There is a forgetful functor  $\sPhi\longrightarrow \mathrm{Top}$  taking a generalised topological space
into a topological space (possibly empty).
For a unary formula $\varphi(-)$, it takes $M_\bullet$ defined above into
the (usual) Stone space of $\varphi$-types.

#### Each indiscernible sequence is a set.
Let
{
$\omega^{\leqslant,\text{cof}}_\bullet(n^\leqslant) := \Hom_{\text{preorders}} ( n^\leqslant, \omega^\leqslant ), n\ge 0,
$\\
$|\omega|^\text{cof}_\bullet(n^\leqslant) := \Hom_{\text{sets}} ( n^\leqslant, |\omega| )=|\omega|^n, n\ge 0$}
equip $|\omega|=\omega^{\leqslant,\text{cof}}_\bullet(1^\leqslant) =|\omega|^{\text{cof}}_\bullet(1^\leqslant)$
with the filter of cofinite subsets, and equip each $\omega^{\leqslant,\text{cof}}_\bullet(n^\leqslant)$,
resp.~$|\omega|^{\text{cof}}_\bullet(n^\leqslant)$,
 with the coarsest filter such that
all simplicial maps
$\omega^{\leqslant,,\text{cof}}_\bullet(n^\leqslant)\to \omega^{\leqslant,\text{cof}}_\bullet(2^\leqslant)$,
resp.~$|\omega|^\text{cof}_\bullet(n^\leqslant)\to  |\omega|^\text{cof}_\bullet(2^\leqslant)$,  are continuous.
Then
{each (infinite) $\varphi$-indiscernible sequence is a $\varphi$-indiscernible set iff \\
$ \omega^{\leqslant,\text{cof}}\longrightarrow |\omega|^\text{cof}_\bullet \,\rtt\, M_\bullet \longrightarrow \top $ }

#### Each indiscernible sequence is eventually indiscernible over any parameter
Let
{
$M[+1]_\bullet(n^\leqslant) := \Hom_{\text{sets}} ( (n+1)^\leqslant, |\omega| )=|M|\times|M|^n, n\ge 0$,}
and
equip $|M|^{n+1}=M[+1]_\bullet(n^\leqslant)$
with the filter generated by subsets %, for each formula $\varphi$
{$\{(a_0,a_1,...,a_n): \text{ the seq.  }(a_1,...,a_n)\text{ is }\varphi\text{-indiscernible over }a_0\}$, $\varphi$ a formula}
One can check that the map $M[+1]_\bullet\to M_\bullet, (a_0,a_1,...,a_n)\mapsto (a_1,..,a_n)$ forgetting the first coordinate,
is continuous.

Each indiscernible sequence in $M$ is eventually indiscernible over any parameter iff
{$\{0\}_\bullet \to \omega^{\leqslant,\text{cof}} \,\rtt\, M[+1]_\bullet\to M_\bullet$}
where $\{0\}_\bullet(n^\leqslant)=\{0\}$ is equipped with the filter containing the empty set, and
$\{0\}_\bullet \to \omega^{\leqslant,\text{cof}}$
is the map $(0,..,0)\mapsto (0,..,0)$. Note that the map $\{0\}_\bullet \to  M[+1]_\bullet$
picks an arbitrary element of $M$.

#### Speculations: limits and contractible models ?

In $\sPhi$ the notions of homotopy and limit are closely related: the same construction in $\sPhi$
describes both
picking a homotopy contracting a topological space (via singular complexes) and taking
a limit of a filter on a topological space, and applies to an arbitrary morphism in $\sPhi$.
One may perhaps hope that this construction can somehow be used to define a useful notion
of a {\em contractible} (generalised topological space associated with a) {\em model}, %$M_\bullet$,
or that the notion of a  {\em limit of a combinatorial structure on a model}, say a tree as in NTP, is useful
(i.e.~taking a limit of the morphism associated with a combinatorial structure on a model, such as those arising
in the lifting properties describing NTP).

Below we sketch this construction; see [6,\S3] for more details.
We warn the reader that our considerations here are unusually preliminary,
and apologise for including them;  our excuse is that we are likely unable
to pursue them.

\begin{definition}[(Limit in a generalised space)]\label{def:limit}
Let $\mathfrak{F}_\bullet : F_\bullet\longrightarrow X_\bullet$ be a morphism in $ \sPhi$.
A morphism $x_\bullet$ is said to be a {\em limit morphism} (or simply {\em a limit})
of $\mathfrak{F}_\bullet$ iff
$F_\bullet\longrightarrow X_\bullet$ factors as
$$F_\bullet\longrightarrow X_\bullet\circ[+1] \xrightarrow{x_\bullet} X_\bullet$$
where $[+1]:\Delta\longrightarrow\Delta$ is the shift
$$n\mapsto n+1, \ \ f\!:\!n\to m \,\longmapsto\, f'\!:\!n+1\to m+1, f'(0):=0; f'(i+1):=f(i) \text{ for }i\ge 0,$$
and $X_\bullet\circ[+1]\longrightarrow X_\bullet$ is the expected map ``forgetting the first coordinate''.
 \end{definition}

To recover the Bourbaki definition of a limit of a filter $\mathfrak{F}$ on a topological space $X$,
associate with $\mathfrak{F}\in \Ob \PPhi$ the simplical set represented by $X$
$${F}_\bullet(n^\leqslant):=\Hom(n^\leqslant,X)$$
equip $X_\bullet(1^\leqslant)=X$ with $\mathfrak{F}$, and equip
each ${F}_\bullet(n^\leqslant):=\Hom(n^\leqslant,X)$ %is equipped
with the finest filter
such that the diagonal map  ${F}_\bullet(1^\leqslant)\longrightarrow  {F}_\bullet(n^\leqslant)$
is continuous. A verification shows that  (possibly discontinuous) liftings correspond to points of $X$
(indeed, as simplicial sets, $ F_\bullet$ is connected, and
$X_\bullet[+1]$ is the disjoint union of connected component which are copies of $X_\bullet$ parametrised by $X$),
and the continuity requirement means precisely that they are limit points. %  (see Fig.~2).

\begin{example}[Limit of the tree in NTP] Let $T^\leqslant_\bullet$,
$|T|^\text{NTP}_\bullet$, and $M_\bullet^{\phi\text{-NTP}}$ denote the objects
corresponding to a tree $T$ and a model $M$ appearing in the lifting property for NTP  of formula $\phi$.
A verification shows the following.

A limit of $T^\leqslant_\bullet \longrightarrow M_\bullet^{\phi\text{-NTP}}$
is a point $a\in M$ such that the map $T^\leqslant_\bullet \longrightarrow M_\bullet^{\phi'\text{-NTP}}$
is continuous where $\phi'(x,y):=\phi(x,a) \,\&\, \phi(x,y)$.
\end{example}

To see this, first note that, as simplicial sets (i.e.~if we ignore the filters),
$M_\bullet^{\phi\text{-NTP}}[+1]$ is the disjoint union of connected component which are copies of
$M^{\phi\text{-NTP}}_\bullet$ parametrised by elements of $M$,
that both $T^\leqslant_\bullet$,
$|T|_\bullet$ are connected. Thus to give a map of simplicial sets
 $T^\leqslant_\bullet \longrightarrow M_\bullet^{\phi\text{-NTP}}$  or  $|T|_\bullet \longrightarrow M_\bullet^{\phi\text{-NTP}}$
 is the same as to pick a point $a\in M$.
The continuity requirement on $T^\leqslant_\bullet \longrightarrow M_\bullet^{\phi\text{-NTP}}$
means precisely that  $\phi(x,a_1),...,\phi(x,a_n)$ is consistent whenever $a_1,...,a_n$ lie on the same branch of $T$.
The continuity requirement on $T^\leqslant_\bullet \longrightarrow M_\bullet^{\phi\text{-NTP}}[+1]$
means precisely
that $\phi(x,a),\phi(x,a_1),...,\phi(x,a_n)$ is consistent whenever $a_1,...,a_n$ lie on the same branch of $T$.

The case of $|T|^\text{NTP}_\bullet \longrightarrow  M_\bullet^{\phi\text{-NTP}}$ is similar.
\def\sing{\operatorname{sing}}
Now
let $F$ and $X$ denote topological spaces.
\begin{remark}
A homotopy
contracting $F$ in $X$ to  a point (i.e.~a map
\begin{center}$h:F\times [0,1]/F\times \{1\} \longrightarrow X$\end{center}
from the cone of $F$ to $X$), gives rise to a map $$\sing F_\bullet \longrightarrow \sing X_\bullet[+1]$$
of singular complexes lifting the map $\sing F_\bullet \longrightarrow \sing X_\bullet$:
\begin{center}
$\sing F_\bullet((n+1)^\leqslant):=\Hom_\text{Top}( \Delta^n, F)$,\\
$\sing X_\bullet((n+1)^\leqslant):=\Hom_\text{Top}( \Delta^n, X)$, \\
$\sing X_\bullet\circ[+1]((n+1)^\leqslant)=\Hom_\text{Top}( \Delta^n\times [0,1]/{\Delta^n\times\{1\}}, X)$\\
\end{center}
Take $\delta: \Delta^n \to F$ to $h_*(\delta):\Delta^n\times [0,1]/\Delta^n\times\{1\}\to X$ defined by
\begin{center}$h_*(\delta)( x,t ):= h(\delta(x),t)$.\end{center}
Here  $n\geqslant 0$ and  $ \Delta^n\times [0,1]/{\Delta^n\times\{1\}}$ is the cone of $n$-simplex $\Delta^n$.
\end{remark}
\begin{remark}
A map
$h:F\times [0,1]/F\times \{1\} \longrightarrow X$
continuous in a neighbourhood of  ``the top of the cone'' point $F\times \{1\} $
is the same as a map in $\sPhi$
$$ F_\bullet\times ([0,1]_1)_\bullet  \longrightarrow X_\bullet[+1]$$
$$(x_1,..,x_n,t_1,..,t_n) \mapsto (h(x_1,1),h(x_1,t_1),h(x_2,t_2),...,h(x_n,t_n))$$
where $[0,1]_1$ denotes the interval $[0,1]$ equipped with the filter of neighbourhoods of point $1$,
and $F_\bullet, X_\bullet$ denote the generalised topological spaces corresponding to $F$ and $X$.
\end{remark}

\subsection{References}

Some of these constructions are sketched in the drafts below.

Topology and analysis:

* {#1} [1] Misha Gavrilovich. The category of simplicial sets with a notion of smallness.
 [(pdf)](http://mishap.sdf.org/6a6ywke/6a6ywke.pdf)

Geometric realisation:

* {#2} [2] Misha Gavrilovich, Konstantin Pimenov.  Geometric realisation as the Skorokhod semi-continuous path space endofunctor. [(pdf)](http://mishap.sdf.org/SkorokhodGeometricRealisation.pdf)1

Stability and simplicity:

* {#TentZiegler} [TentZiegler] K.Tent, M.Ziegler.
A Course in Model Theory. CUP. 2012.

* {#Mallaris2010jsl} Maryanthe Malliaris.
The characteristic sequence of a first-order formula. J Symb Logic 75, 4 (2010) 1415-1440. [(pdf)](https://math.uchicago.edu/~mem/mm-char-seq-revised.pdf)

* {#Mallaris2010apal} Maryanthe Malliaris.
"Edge distribution and density in the characteristic sequence," Ann Pure Appl Logic 162, 1 (2010) 1-19. [pdf](https://math.uchicago.edu/~mem/apal-malliaris-revised-version.pdf)

* {#Scow2012} [Scow2012] Lynn Scow.
Characterization of nip theories by ordered graph-indiscernibles. Annals of
Pure and Applied Logic, 163(11):1624 – 1641, 2012. [(pdf)](https://arxiv.org/abs/1106.5153)

* {#Simon2021} [Simon2021] Pierre Simon. A note on stability and NIP in one variable.
[(pdf)](https://arxiv.org/abs/2103.15799)

* {#AhnKimLee2021} [AhnKimLee2021] JinHoo Ahn, Joonhee Kim, and Junguk Lee. On the Antichain Tree Property. [pdf](https://arxiv.org/pdf/2106.03779.pdf)

* {#3} [3] Misha Gavrilovich. Remarks on Shelah's classification theory and Quillen's negation.
 [(pdf)](http://mishap.sdf.org/yetanothernotanobfuscatedstudy.pdf)