# Contents

## Idea of a situs

A situs is a notion of generalised topological space, and defined as a simplicial object of the 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 (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 (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).

## Examples

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

### 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.

### 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.

### 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 $\mathfrak{F}$. 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

$|-^\mathfrak{F}|_\bullet^{\operatorname{cart}}, |-^\mathfrak{F}|_\bullet^{\operatorname{diag}} : ዋ \to sዋ$

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.

### Topological and uniform spaces as situses

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.

#### Uniform spaces as situses

Take a set $X$ and view it as a simplicial set $X_\bullet$ (represented by $X$). Recall that a 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”.

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.

#### 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:

$Top \to sዋ \to Top$
$UniformSpaces \to sዋ \to UniformSpaces .$

### 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.

### 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.

#### 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 $X$.

#### Compactness and completeness as lifting properties

Diagram chasing reformulation of the notions above allows to define compactness and completeness as lifting properties.

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)

### Local triviality

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 $(Y_\bullet\circ [+1])\times F_\bullet \to Y_\bullet\circ [+1]$.

### 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

$HHom([0,1]_\bullet, -) :sዋ\to sዋ$

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.

### The interval object and homotopy theory

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 defining stability and simplicity (defined below). However, note that the naive interval object $[0,1]^\leq_\bullet$ reminds one of 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 $A$.

### 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).

#### 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 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 $(I^{\leq tails})^{ cart}_\bullet \to M_\bullet$.

###### 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

iii. the following lifting property holds in sዋ:

$(I^{\leq tails})^{\operatorname{cart}}_\bullet\to (|I|^{tails})^{\operatorname{cart}}_\bullet \rightthreetimes M_\bullet\to \top .$

###### 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.

(Simon,2021) 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.

#### 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), see also (M.Malliaris. “Edge distribution and density in the characteristic sequence).

##### 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 $\exists x \wedge_{i\leq n} \phi(x,y_i)$.

###### 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)

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 $n$ $\exists x \wedge_{i\leq n} \phi(x,y_i)\leftrightarrow \wedge_{1\leq i_1\le ...\le i_k\le n}\exists x \wedge_{1\leq l\leq k} \phi(x,y_{i_l})$

###### Proof

Items 1 and 2 are Remark 2.7 of Malliaris, item 4 and 5 are both item 2 and item 3 written explicitly.

##### 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, or 3,\S9 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.

###### 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.

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

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 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 $|T|_\bullet(n^\leq)=|T|^n$.

###### Definition

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.

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.

###### 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: $T^\leq_\bullet \to |T|^{TP}_\bullet \rightthreetimes M_\bullet^{\exists\phi}\to \top$

###### Proof

ii. is exactly the definition of NTP for formula $\phi$ as stated in (Tent-Ziegler, Def.7.2.1), cf. 3,\S9. 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.

###### 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, and sct patterns in classification theory.

###### 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.

###### 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), see also (M.Malliaris. “Edge distribution and density in the characteristic sequence). Moreover, it appears that several properties of characteristic sequences can be defined as lifting properties in the category of situses.

### 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.

### 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.

###### 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. 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.

#### No order property NOP

Let $\phi(-,-)$ be a binary formula, and let $M^{\{\phi\}}_\bullet$ be the situs on simplicial set $|M|_\bullet$ represented by the set of elements of $M$ where $|M|^n$ is equipped with the filter generated by the set of all $\phi$-indiscernible sequences

$\{(x_1,...,x_n): \forall 1\leq i\le j\leq n, 1\leq k \le l\leq n \phi(x_i,x_j)\leftrightarrow \phi(x_k,x_l)\}\subset |M|^n.$

We call it the situs of $\phi$-indiscernible sequences.

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

$\phi(a_i,a_j) \leftrightarrow i\leq j$

Let $I^{\leq tails}_\bullet$ be the situs associated with the filter of final segments on the linear order $I$.

###### Definition

Let $|I|^{NOP}_\bullet$ be the simplicial set $|I|_\bullet$ represented by the set $|I|$ of elements of $I$, equipped with NOP-tautological filters defined as follows.

1. A subset $X\subset |I|^3$ is large in the NOP-tautological filter on $|I|^n$ iff

a. it contains each triple $(a_i,a_j,a_k)$ for $i\leq j\leq k$

b. for each infinite increasing subsequence $a_i\in I, i\in\omega$ there is a pair $i\lt j$ such that $(a_i,a_j,a_i)\in X$

1. for $n\neq 3$ the filter on $|I|^n$ is the coarsest filter such that all the simplcial maps $I^n\to I^3$ are continuous.

We call this filter tautological because by definition any large subset contains a “witness” $\phi(a_i,a_j)\leftrightarrow \phi(a_j,a_i)$ of failure of the order property. More precisely, by very definition both maps $a_\bullet: I^{\leq tails}_\bullet\to M^{\{\phi\}}$ and $a_\bullet: |I|^{NOP}_\bullet\to M^{\{\phi\}}$ are continuous iff (a) $a_i$ is a sequence such that $\phi(a_i,a_j)$ whenever $i\leq j$, and (b) each (infinite) subsequence of $(a_i)_{i\in I}$ does not have the order property for $\phi$. A little argument using Ramsey theorem shows that the NOP-tautological filter is indeed a filter, as follows. Assume that the intersection $X\cap Y$ of two large subsets is not large, i.e. there is an infinite subsequence $a_{i_l}$ such that for each $k\lt l$ either $(a_{i_k},a_{i_l},a_{i_k})\notin X$ or $(a_{i_k},a_{i_l},a_{i_k})\notin Y$. This gives a colouring of pairs $i\lt j$ in two colours, and by Ramsey theory there is an infinite clique of the same colour, which by definition means that either $X$ or $Y$ is not large.

The following theorem summarises the considerations above.

###### Theorem

1. formula $\phi$ has NOP

2. there no sequence $(a_i)_{i\in\omega}$ such that $\phi(a_i,a_j) \leftrightarrow i\leq j$

3. the following lifting property holds in sዋ:

$I^\leq_\bullet \to |I|^{NOP}_\bullet \rightthreetimes M^{\{\phi\}}_\bullet\to\top$

###### Proof

Each map $I^\leq_\bullet \to M^{\{\phi\}}_\bullet$ corresponds to a sequence $(a_i)_{i\in I}$ such that $i\leq j\implies \phi(a_i,a_j)$.

If NOP fails, take $(a_i)_{i\in I}$ to be a witness of this. Then evidently the induced map $|I|^{NOP}_\bullet \to M^{\{\phi\}}_\bullet$ is not continuous as there are no witnesses for (b).

Assume NOP. Then for each infinite subsequence there is a witness of NOP, i.e. a tuple as in (b) contained in the preimage of the set of $\phi$-consistent tuples in $M$. Hence, this preimage is large and the induced diagonal map is continuous.

The trick behind the definition of tautological filter works for some other dividing lines such as NSOP and tree properties. We discuss the tree properties in the next section.

#### No strict order property NSOP

To define NSOP-tautological filters we need to define a notion of a witness to failure of NSOP formulated in terms of $\phi$-indiscernible sequences.
The following reformulation is convenient for us: we say that a formula $\phi(-,-)$ has NSOP’ iff there is an infinite sequence $(a_i)_I$ such that

(a) $\phi(a_i,a_j)$ iff $i\leq j$

(b) $\phi(M,a_i)\subset \phi(M,a_j)$ for $i\leq j$.

Failure of this is witnessed by

(a’) a sequence $(a_i,a_j,a_i)$ being $\phi$-indiscernible for $i\lt j$

(b’) both sequences $(x,a_i,a_j)$ and $(x,a_l,a_k)$ being $\phi$-indiscernible for $i\lt j, k\lt l$, and $j\lt l$.

More precisely:

###### Statement

1. If items (a) and (b) hold for $\phi$, then items (a’) and (b’) never hold

1. If items (a’) and (b’) never hold for formula $\phi$ and a long enough sequence $(a_i)_i$, then (a) and (b) holds for subsequence $(a_i)_{0\lt i\lt \omega}$ and formula $\phi'(x,y)$ or $\neg\phi'(x,y)$ where
$\phi'(x,y):=\phi(x,y) \wedge \neg \phi(x,a_0)\wedge \phi(x,a_\omega)$

###### Proof

Indeed, assuming (a), item (b’) means that $\phi(x,a_i),\phi(x,a_j)$, $\neg \phi(x,a_k)$, and $\neg \phi(x,a_l)$, for some $i\lt j, k\lt l$, and $j\lt l$, which does contradict (b).

Item (a’) means that for each $i\lt j$ $\phi(a_i,a_j)\leftrightarrow \neg\phi(a_j,a_i)$, hence by Ramsey theory (a) holds for an infinite subsequence, possibly replacing $\phi$ by $\neg\phi$. So without loss of generality we may assume (a) holds for $\phi$.

Now let us prove that (a) and (b’) imply SOP’ holds for some infinite subsequence $(a_i)_{0\lt i\lt \omega}$ and

$\phi'(x,y):=\phi(x,y) \wedge \neg \phi(x,a_0)\wedge \phi(x,a_\omega)$

We only need to prove that (a) and (b’) implies that (b’’) $\phi'(M,a_i)\subset \phi'(M,a_j)$ for $i\leq j$. Indeed, let $i\lt j$ and $x$ be a counterexample, i.e.
$\phi(x,a_i) \wedge \neg \phi(x,a_0)\wedge \phi(x,a_\omega)$ and $\neg (\phi(x,a_j) \wedge \neg \phi(x,a_0)\wedge \phi(x,a_\omega))$. These formulae imply that $\neg\phi(x,a_j)$ and thus both $(x,a_i,a_\omega)$ and $(x,a_j,a_0)$ are $\phi$-indiscernible.

Hence we define NSOP-tautological filters on $|I\cup I^4|_\bullet$ as:

###### Definition

1. A subset $U\subset |I\cup I^4|^3$ is large in the {\em $NSOP$-tautological filter} on $|I\cup I^4|^3$ iff each infinite subsequence $I'\subset I$ contains a witness of (a’) and (b’), i.e.

i. there is a pair $i\lt j,i,j\in I'$ such that $(a_i,a_j,a_i)\in U$

ii. there are $i\lt j, k\lt l$, and $j\lt l$, $i,j,k,l\in I'$ such that $((i,j,k,l),i,j)\in U$ and $((i,j,k,l),a_l,a_k)\in U$

1. for $n\neq 3$ the filter on $|I\cup I^4|^n$ is the coarsest filter such that all the simplcial maps $|I\cup I^4|^n\to |I\cup I^4|^3$ are continuous.

###### Theorem

1. theory $T$ has NSOP

2. there no formula $\phi$ and an infinite sequence $(a_i)_{i\in\omega}\in M$ satisfying (a’) and (b’) above, for some saturated model $M$

3. for each linear order $I$, model $M$ and formula $\phi$ it holds

$|I|^\leq_\bullet \to |I\cup I^4|^{NSOP}_\bullet \rightthreetimes M^{\{\phi\}}_\bullet\to\top$

###### Proof

Let $(a_i)_{i\in I}$ be a witness for SOP. Take the corresponding map $|I|^\leq_\bullet \to M^{\{\phi\}}_\bullet$. By SOP there are no witnesses of (b’) in $M$, thus wherever I^4 is sent to, the preimage of the subset of $\phi$-consistent tuples will not contain a witness of ii., hence by definition is large in the NTP-tautological filter. Therefore the lifting property fails.

Now assume NOP and let us show the lifting property holds. The map $|I|^\leq_\bullet \to M^{\{\phi\}}_\bullet$ corresponds to a sequence $(a_i)_{i\in I}$ witnessing (a). If this sequence has only finitely many distinct elements, then (b) is witnessed by all tuples where $i,j,k,l$ belong to an infinite constant subsequence, and therefore the preimage of the subset of $\phi$-consistent tuples is large.

Thus we may assume that all $a_i$‘s are distinct. Take the diagonal map sending each $(i,j,k,l)\in I^4$ into a witness of (b’) whenever it exists. Each infinite subsequence also fails SOP, hence there is a witness of (b’) for this. Hence, the preimage of the subset of $\phi$-consistent tuples is large.

#### $NSOP_n$ for $n\gt 3$

Recall that a formula $\phi$ has $SOP_n$ iff there is an infinite sequence $(a_i)_{i\in \omega}$ such that

(a) $\phi(a_i,a_j)$ iff $i\leq j$

($b_n$) there are $(a_i)_{0\leq i \leq n-1}$ such that $\phi((a_i,a_{(i+1)mod\,n})$ for any $0\leq i\leq n$

Say a formula $\phi$ has $SOP'_n$ iff there is an infinite sequence $(a_i)_{i\in \omega}$ such that

(a) $\phi(a_i,a_j)$ iff $i\leq j$

($b'_n$) there are $(a_i)_{0\leq i \leq n-1}$ such that for any $0\leq i\leq n$ the sequence $(a_i,a_{(i+1)mod\,n},a_{(i+2)mod\,n})$ is $\phi$-indiscernible.

It is easy to see that for $n\gt 3$ $SOP'_n(\phi)\leftrightarrow SOP_n(\phi)\vee SOP_n(\neg\phi)$.

Indeed, either there is a 3-cycle $\phi(a_i,a_{(i+1)mod\,n}), \phi(a_{(i+1)mod\,n},a_{(i+2)mod\,n}), \phi(a_{(i+2)mod\,n},a_{i})$ or for each $0\leq i\less n$ $\neg\phi(a_{(i+2)mod\,n},a_{i})$, hence there is an $n$-cycle $n-1,..,(n-1-2k)mod\,n,..(n-1-2n)mod\,n$ for $\neg\phi$.

Let us reformulate this as a lifting property.

Let $|\{0,...,n-1\}|^{cycle}_\bullet$ be the situs associated with the filter on $|\{0,...,n-1\}|^3$ generated by the set of triples $(a_i,a_{(i+1)mod\,n},a_{(i+2)mod\,n})$, $0\leq i\leq n-1$.

###### Theorem

A formula $\phi$ has $NSOP'_n$ iff $\phi$ has NOP and the following lifting property holds:

$|\{0,...,n-1\}|^{diag}_\bullet \to |\{0,...,n-1\}|^{cycle}_\bullet \rightthreetimes M^{\{\phi\}}\to \top$

###### Proof

The proof is straightforward.

### References

Some of these constructions are sketched in the drafts below.

Topology and analysis:

•  Misha Gavrilovich. The category of simplicial sets with a notion of smallness. (pdf)

Geometric realisation:

•  Misha Gavrilovich, Konstantin Pimenov. Geometric realisation as the Skorokhod semi-continuous path space endofunctor. (pdf)1

Stability and simplicity:

• [TentZiegler] K.Tent, M.Ziegler. A Course in Model Theory. CUP. 2012.

• Maryanthe Malliaris. The characteristic sequence of a first-order formula. J Symb Logic 75, 4 (2010) 1415-1440. (pdf)

• Maryanthe Malliaris. “Edge distribution and density in the characteristic sequence,” Ann Pure Appl Logic 162, 1 (2010) 1-19. pdf

• [Scow2012] Lynn Scow. Characterization of nip theories by ordered graph-indiscernibles. Annals of Pure and Applied Logic, 163(11):1624 – 1641, 2012. (pdf)

• [Simon2021] Pierre Simon. A note on stability and NIP in one variable. (pdf)

• [AhnKimLee2021] JinHoo Ahn, Joonhee Kim, and Junguk Lee. On the Antichain Tree Property. pdf

•  Misha Gavrilovich. Remarks on Shelah’s classification theory and Quillen’s negation. (pdf)

Last revised on August 25, 2021 at 10:34:20. See the history of this page for a list of all contributions to it.