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).
We now give a number of examples demonstrating the expressive power of the category of 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.
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.
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
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.
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.
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.
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:
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.
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.
For example, taking a limit of a filter $\mathfrak{F}$ on a topological or metric space $X$ corresponds to taking the factorization
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 $X$.
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
It also allows to define the completion of a metric space in terms of a weak factorisation system
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
Here $\{\{o\},\{o,1\}\}$, $\{\{1\},\{o,1\}\}$, and $\{\{o,1\}\}$ are viewed as filters on the set $\{o,1\}$. (needs verification)
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 $(Y_\bullet\circ [+1])\times F_\bullet \to Y_\bullet\circ [+1]$.
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
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.
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).
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 $(I^{\leq tails})^{ cart}_\bullet \to M_\bullet$.
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:
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.
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).
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 $\exists x \wedge_{i\leq n} \phi(x,y_i)$.
$\phi$ has no finite cover property
the characteristic sequence $\lt P_n\gt$ has finite support (in terminology of Malliaris, Def.2.6)
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
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 $y_{i_1},...,y_{i_k}$
there is $k$ such that for each $n$ $\exists x \wedge_{i\leq n} \phi(x,y_i)\leftrightarrow \wedge_{1\leq i_1\le ...\le i_k\le n}\exists x \wedge_{1\leq l\leq k} \phi(x,y_{i_l})$
Items 1 and 2 are Remark 2.7 of Malliaris, item 4 and 5 are both item 2 and item 3 written explicitly.
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.
[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.
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 $|T|_\bullet(n^\leq)=|T|^n$.
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.
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: $T^\leq_\bullet \to |T|^{TP}_\bullet \rightthreetimes M_\bullet^{\exists\phi}\to \top$
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.
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.
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.
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.
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
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.
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.
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.
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
Then rewrite the standard definition of NOP in a form with two items representing positve and negative requirements.
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}
With these definitions, $\omega_\bullet^\leqslant\to |\omega|^\text{NOP}_\bullet \,\rtt\, M_\bullet\to \top$ defines NOP.
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$_\ell$.
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:
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 “\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’’.
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.
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 \ $\omega^{\leqslant,\text{cof}}\longrightarrow |\omega|^\text{cof}_\bullet \,\rtt\, M_\bullet \longrightarrow \top$ }
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 $M$.
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.
[(Limit in a generalised space)] 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
where $[+1]:\Delta\longrightarrow\Delta$ is the shift
and $X_\bullet\circ[+1]\longrightarrow X_\bullet$ is the expected map “rgetting the first coordinate’’.
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 $X$
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).
[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 $\phi'(x,y):=\phi(x,a) \,\&\, \phi(x,y)$.
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.
A homotopy contracting $F$ in $X$ to a point (i.e.~a map
from the cone of $F$ to $X$), gives rise to a map $\sing F_\bullet((n+1)^\leqslant):=\Hom_\text{Top}( \Delta^n, F)$ar complexes lifting the map $\sing F_\bullet \longrightarrow \sing X_\bullet$:
Take $\delta: \Delta^n \to F$ to $h_*(\delta):\Delta^n\times [0,1]/\Delta^n\times\{1\}\to X$ defined by
Here $n\geqslant 0$ and $\Delta^n\times [0,1]/{\Delta^n\times\{1\}}$ is the cone of $n$-simplex $\Delta^n$.
A map
$h:F\times [0,1]/F\times \{1\} \longrightarrow X$ continuous in a neighbourhood of “e top of the cone’‘ point $F\times \{1\}$ is the same as a map in $\sPhi$
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 $X$.
Some of these constructions are sketched in the drafts below.
Topology and analysis:
Geometric realisation:
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
[3] Misha Gavrilovich. Remarks on Shelah’s classification theory and Quillen’s negation. (pdf)
Last revised on July 14, 2022 at 17:05:39. See the history of this page for a list of all contributions to it.