nLab
situs

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, i.e. a simplicial set equipped, for each n0n\geq 0, with a filter on the set of nn-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.

Intuitively, these filters are viewed as additional structure of topological nature on a simplicial set (“the situs structure on a simplicial set”) giving a meaning to the phrase “a simplex is sufficiently small”: by definition, we say “a property holds for all small enough nn-simplices” iff it holds on a set in the filter, and we refer to sets in the filter as neighbourhoods. This extends to simplicial language the standard intuition of topology: given a topological structure on a set, the precise meaning of the phrase “a property P x 0(x)P_{x_0}(x) holds for all points xx sufficiently close to a given point x 0x_0” is that the property holds on a neighbourhood of x 0x_0; intuitively, xx near x 0x_0 is thought of as small because xx approximates x 0x_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, NSTOP, NSOP iNSOP_i, NTP iNTP_i, 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][0,1] equipped with some situs structure based on the metric/topology) leads to a directed (not symmetric) notion of homotopy.

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 nn, equip X nX_n with the discrete or indiscrete filter, respectively.

Metric spaces: uniformly continuous maps and quasi-isometries

Let MM be a metric space. View MM as a simplicial set represented by the set of points of MM, and equip each M nM^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:MNf:M\to N is uniformly continuous iff it induces a morphism f :M N 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 nM^n with the filter such that a subset of M nM^n is large iff it contains all nn-tuples such that the distance between distinct points is at least DD, for some D0D\geq 0. With this situs structure, for quasi-geodesic metric spaces, a map f :M N f_\bullet:M_\bullet\to N_\bullet is an isomorphism of situses iff f:MNf:M\to N is an quasi-isometry.

A filter as a situs

Given a filter 𝔉\mathfrak{F} on a set XX, there is a coarsest situs structure on XX viewed as a simplicial set (i.e. the simplicial set |X| |X|_\bullet represented by XX) such that its filter on the set XX of 00-simplices is finer than 𝔉\mathfrak{F}. Dually, there is a finest situs structure on |X| |X|_\bullet such that its filter on the set of 00-simplices is coarser than 𝔉\mathfrak{F}. We denote these situses by |X 𝔉| cart|X^\mathfrak{F}|_\bullet^{\operatorname{cart}} and |X 𝔉| diag|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

| 𝔉| cart,| 𝔉| diag:s |-^\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 𝔉cartX^{\leq\mathfrak{F}}_\bullet^{\operatorname{cart}} and X 𝔉diagX^{\leq\mathfrak{F}}_\bullet^{\operatorname{diag}}, respectively.

Topological and uniform spaces as situses

More generally, given an arbitrary simplicial set X X_\bullet and a filter 𝔉\mathfrak{F} on the set of nn-simplices X nX_n, there is a coarsest/finest situs structure on X X_\bullet such that its filter on the set of nn-simplices is finer/coarser than 𝔉\mathfrak{F}. Taking n=0n=0 and the filter 𝔉\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 XX and view it as a simplicial set X X_\bullet (represented by XX). A uniform structure on XX is a filter on X×XX\times X; take the coarsest situs structure with this filter on the set X×XX\times X of 11-simplices. This is the situs associated with the uniform structure on XX.

In fact, it is easy to define uniform spaces in terms of situses. A filter on X×XX\times X is a uniform structure iff it is symmetric (i.e. the endomorphism of X×XX\times X, (x,y)(y,x)(x,y)\mapsto (y,x) permuting the coordinates is continuous) and this construction produces a situs such that the filter on X×X×XX\times X\times X is the coarsest filter such that the two maps X 3X×XX^3\to X\times X, (x 1,x 2,x 3)(x i,x i+1),i=1,2(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×X×XX\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 XX is defined in the same way starting from the filter of non-uniform neighbourhoods of the diagonal on X×XX\times X defined as consisting of the subsets of form xX{x}×U x\bigcup_{x\in X} \{x\}\times U_x where U xxU_x\ni x is a not necessarily open neighbourhood of xXx\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:

TopsTopTop \to sዋ \to Top
UniformSpacessUniformSpaces.UniformSpaces \to sዋ \to UniformSpaces .

Cauchy sequences and equicontinuity

A filter 𝔉\mathfrak{F} on a metric space MM is Cauchy iff 𝔉 cartM \mathfrak{F}^{\operatorname{cart}}_\bullet \to M_\bullet is continuous.

A sequence of functions f i:LMf_i:L \to M, ii\in \mathbb{N} of metric spaces is uniformly equicontinuous iff the map ( cofinite) diag×L M (\mathbb{ N}^{cofinite})^{\operatorname{diag}}_\bullet\times L_\bullet \to M_\bullet is continuous.

A sequence of functions f i:XMf_i:X \to M, ii\in \mathbb {N} from a topological space XX to a metric space MM is equicontinuous iff the map ( cofinite) diag×X M (\mathbb{ N}^{cofinite})^{\operatorname{diag}}_\bullet\times X_\bullet \to M_\bullet is continuous.

Limits, compactness, and completeness

An endomorphism of the category Δ\Delta of finite linear orders gives rise to an endomorphism of the category of situses. Of particular interest is the shift endomorphism ΔΔ\Delta\to\Delta adding a new least element (decalage considers the endomorphism adding a new greatest element rather than least) on objects, n1+nn\mapsto 1+n, and on morphisms, f:nmf:n\to m goes to f[+1]:1+n1+mf[+1]:1+n\to 1+m, f(0)=0f(0)=0, f[+1](1+i)=1+f(i)f[+1](1+i)=1+f(i), 0in0\leq i\leq n. The object X [+1]X_\bullet\circ [+1] and morphism X [+1]XX_\bullet\circ [+1]\to X allows one to talk about local properties of X X_\bullet, e.g. limits and local triviality.

Limits via shift endomorphism

For example, taking a limit of a filter 𝔉\mathfrak{F} on a topological or metric space XX corresponds to taking the factorization

𝔉 diagX [+1]X .\mathfrak{F}^{\operatorname{diag}}_\bullet\to X_\bullet\circ [+1]\to X_\bullet.

Indeed, the underlying simplicial set of 𝔉 diag\mathfrak{F}^{\operatorname{diag}}_\bullet is connected and thus maps to a single connected component of X [+1]= xX{x}×X 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 𝔉X×X\mathfrak{F} \to X\times X, x(x 0,x)x\mapsto (x_0,x) means exactly that the first coordinate x 0x_0 is a limit point of 𝔉\mathfrak{F} on XX.

Compactness and completeness as lifting properties

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

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

A metric space MM is complete iff either of the following equivalent conditions holds:

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

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

iii. ( cofinite) cart( cofinite{}) cartM (\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 MM complete iff

M [+1]M ( [+1] ) lr? 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 (( cofinite) cart( cofinite{}) cart) rlM^ (( cofinite) cart( cofinite{}) cart) rM_\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 XX is compact iff for each ultrafilter 𝔘\mathfrak{U} either of the following equivalent conditions holds:

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

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

A topological space KK is compact iff

K [+1]K ({{o},{o,1}} cart{{1},{o,1}} cart{{o,1}} cart) lr.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}}\{\{o\},\{o,1\}\}, {{1},{o,1}}\{\{1\},\{o,1\}\}, and {{o,1}}\{\{o,1\}\} are viewed as filters on the set {o,1}\{o,1\}. (needs verification)

Local triviality

A map f:XYf:X\to Y of topological or metric spaces is locally trivial with fibre FF iff in sዋ becomes a direct product with F F_\bullet (“globally trivial”) after base-change Y [+1]Y Y_\bullet\circ [+1]\to Y_\bullet. That is,
f :(Y [+1])× Y X Y [+1]f_\bullet:(Y_\bullet\circ [+1])\times_{Y_\bullet} X_\bullet \to Y_\bullet\circ [+1] is of form (Y [+1])×F Y [+1] (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 in n\mathbb{R}^n as the space of monotone maps [0,1] (n+1) [0,1]^\leq \to (n+1)^\leq with Skorokhod-type metric dist(f,g):=sup xinf y{|xy|:f(x)=g(y)}dist(f,g):= sup_{x} inf_{y} \{ |x-y| : f(x)=g(y) \}. The category of situses allows us to view both linear orders as situses: the situs structure on [0,1] [0,1]_\bullet “remembers” the metric, and the situs structure on Δ n=Hom(,(n+1) )\Delta_n=Hom(-,(n+1)^\leq) “remembers” the equality f(x)=g(x)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 Skorokhod([0,1] ,):ssHHom_{Skorokhod}([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] [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 AA to BB cannot in general be reversed to get a homotopy from BB to AA.

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.

Stability.

Consider a model MM in a language \mathcal {L}, and a linear order II. For an rr-ary \mathcal {L}-formula ϕ(x 1,...,x r)\phi(x_1,...,x_r), we say that a sequence (a i) iI(a_i)_{i\in I} of elements of MM is ϕ\phi-indiscernible (with repetitions) iff for either all or none of the subsequences (a i 1,...,a i r),i 1...i r(a_{i_1},...,a_{i_r}), i_1\le ... \le i_r (of distinct elements) the formula ϕ(a i 1,...,a i r)\phi(a_{i_1},...,a_{i_r}) holds in MM.

Equip M nM^n with the filter generated by the sets of all nn-tuples which are ϕ\phi-indiscernible with repetitions, where ϕ\phi varies through all \mathcal{L}-formulas. The situs so obtained is called {the generalised pre-Stone space of MM in sዋ} because the forgetful functor sTopsዋ\to Top takes it to the set of elements of MM equipped with the preimage of Stone topology?; by this we mean the topology on MM generated by sets of elements realising unary \mathcal{L}-formulas. There are many variants of this definition, notably instead of being ϕ\phi-indiscernible one may require being a part of an {infinite} ϕ\phi-indiscernible sequence, and instead of M nM^n consider the set of nn-types S n()S_n(\emptyset) or S n(M)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 ϕ\phi-indiscernible sequence of nn-tuples is in fact a ϕ\phi-indiscernible set, for each n0n\ge 0 and each formula ϕ\phi of the language of the theory.

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

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

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

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

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

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

Proposition

Let MM be a model. The following are equivalent:

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

ii. the situs M M_\bullet is symmetric

iii. the following lifting property holds in sዋ:

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

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

Simplicity.

The definition of simplicity is not as simple combinatorially.

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 ϕ\phi. As usual, the situs is based on the simplicial set represented by the set of elements of MM. The filter on M nM^n is generated by a single set of those tuples (a 1,...,a n)(a_1,...,a_n) such that
Mx 1inϕ(x,a i)M\models \exists x \bigwedge_{1\leq i\leq n} \phi(x,a_i). Let us denote this situs by M ϕM_\bullet^{\exists\phi} and call it the ϕ\phi-characteristic situs of model MM. Note that ϕ\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

Let P nP_n denote the formula x inϕ(x,y i)\exists x \wedge_{i\leq n} \phi(x,y_i).

Theorem

  1. ϕ\phi has no finite cover property

  2. the characteristic sequence <P n>\lt P_n\gt has finite support (in terminology of Malliaris, Def.2.5(2))

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

  4. there is kk such that for each nn x inϕ(x,y i)\exists x \wedge_{i\leq n} \phi(x,y_i) holds iff x inϕ(x,y i)\exists x \wedge_{i\leq n} \phi(x,y_i) for any kk-element subset y i 1,...,y i ky_{i_1},...,y_{i_k}

  5. there is kk such that for each nn x inϕ(x,y i) 1i 1...i knx 1lkϕ(x,y i l)\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 2.6 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 φ(x,y)\varphi(x, y) has the tree property (with respect to k) if there is a tree of parameters (a s|s <ωω)(a_s\,\,|\,\,\emptyset \neq s \in {}^{\lt\omega}\omega ) such that:

a) For all s <ωωs\in {}^{\lt\omega}\omega , (φ(x,a si)|i<ω)(\varphi(x, a_{si} )\,\,|\,\,i \lt\omega ) is kk-inconsistent.

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

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

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

Note that to give a morphism T M ϕT^\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|s <ωω)(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 ϕ\phi-consistent tuples is large, i.e. the whole set T (n)T^\leq_\bullet(n) of ordered tuples, for each nn.

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

Definition

Let |T| TP|T|^{TP}_\bullet denote the simplicial set |T| |T|_\bullet equipped with the TPTP-tautological filter on |T| (n )|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 ϵ\epsilon is large iff

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

2’) for each isomorphic copy of T= <ωωT'={}^{\lt\omega}\omega in T T^\leq there is a vertex vTv\in T' and its immediate (in TT') descendants v 1 lex... lexv kv_1\leq_{lex}...\leq_{lex} v_k such that (v 1,..,v k)ϵT(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 |T| TPT^\leq_\bullet \to |T|_\bullet^{TP} is continuous. Also note that no tuple of increasing elements is required to be ϕ\phi-inconsistent by the tree property.

Proposition

The following are equivalent:

i. the formula ϕ\phi has NTP with respect kk in the model MM

ii. in sዋ there is no morphism τ:T M ϕ\tau:T^\leq_\bullet\to M_\bullet^{\exists\phi} such that for each tuple kk-tuple v 1,..,v kv_1,..,v_k, for each kk, of immediate descendants of the same vertex, M¬x(ϕ(x,τ(v 1))...ϕ(x,τ(v n))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 |T| TPM ϕT^\leq_\bullet \to |T|^{TP}_\bullet \rightthreetimes M_\bullet^{\exists\phi}\to \top

Proof

ii. is exactly the definition of NTP for formula ϕ\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)(v_1,...,v_k) such that Mx(ϕ(x,τ(v 1))...ϕ(x,τ(v k))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 TPTP-tautological filters are well-defined. We need only to show that the union of any two small sets XYX\cup Y is small. Assume it is not small. Label each vertex of the tree with the largest n<ωn\lt\omega such that the first small subset contains above the vertex all tuples required to be inconsisent in some copy of <nω{}^{\lt n}\omega. Above each vertex in XX there are at most finitely many vertices in XX labelled by the same or greater number. Removing them leaves XYX\cup Y not small. But then we get that vertices of XX are labelled by numbers strictly decreasing along any branch, hence XX is of finite depth. Now pick a vertex labelled 00. This means that below that vertex there is no infinite set of siblings that each lexicographically ordered tuple is in XX, hence among any infinite set of siblings by Ramsey theorem there is an infinite set of siblings not in XX, i.e. in YY. Hence, YY is not small.

Remark

One can similarly define TP iTP_i-tautological situs of a tree TT, for i=1i=1, and see that TP iTP_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 2NTP=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| |M|_\bullet by the simplicial set of types S MS^M_\bullet where S M(n)S^M(n) is the set of nn-types, and also do the same for T T^\leq_\bullet and |T| TP|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 2NTP=NTP_1\&NTP_2 gives that T |T| TPT^\leq_\bullet\to |T|_\bullet^{TP} is the pushout of T |T| TP 1T^\leq_\bullet\to |T|_\bullet^{TP_1} and T |T| TP 2T^\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 & iϕ(x,y i)\&_i\phi(x,y_i) of ϕ(x,y)\phi(x,y) at the same time.

Remark

The notion of a ϕ\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, FCP, etc, and tautological filters

We show that several dividing lines of Shelah can also be defined by lifting properties in . Namely, we show that properties NOP, NSOP, and NSOP nNSOP_n, n>3n\gt 3, can be defined by lifting properties using tautological filters.

This subsection is preliminary.

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

{(x 1,...,x n):1ijn,1klnϕ(x i,x j)ϕ(x k,x l)}|M| n.\{(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.

No order property NOP

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

ϕ(a i,a j)ij \phi(a_i,a_j) \leftrightarrow i\leq j

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

Definition

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

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

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

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

  1. for n3n\neq 3 the filter on |I| n|I|^n is the coarsest filter such that all the simplcial maps I nI 3I^n\to I^3 are continuous.

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

The following theorem summarises the considerations above.

Theorem

  1. formula ϕ\phi has NOP

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

  3. the following lifting property holds in sዋ:

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

Proof

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

If NOP fails, take (a i) iI(a_i)_{i\in I} to be a witness of this. Then evidently the induced map |I| NOPM {ϕ}|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 ϕ\phi-consistent tuples in MM. 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 ϕ\phi-indiscernible sequences.
The following reformulation is convenient for us: we say that a formula ϕ(,)\phi(-,-) has NSOP’ iff there is an infinite sequence (a i) I(a_i)_I such that

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

(b) ϕ(M,a i)ϕ(M,a j)\phi(M,a_i)\subset \phi(M,a_j) for iji\leq j.

Failure of this is witnessed by

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

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

More precisely:

Statement

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

  1. If items (a’) and (b’) never hold for formula ϕ\phi and a long enough sequence (a i) i(a_i)_i, then (a) and (b) holds for subsequence (a i) 0<i<ω(a_i)_{0\lt i\lt \omega} and formula ϕ(x,y)\phi'(x,y) or ¬ϕ(x,y)\neg\phi'(x,y) where
    ϕ(x,y):=ϕ(x,y)¬ϕ(x,a 0)ϕ(x,a ω)\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 ϕ(x,a i),ϕ(x,a j)\phi(x,a_i),\phi(x,a_j), ¬ϕ(x,a k)\neg \phi(x,a_k), and ¬ϕ(x,a l)\neg \phi(x,a_l), for some i<j,k<li\lt j, k\lt l, and j<lj\lt l, which does contradict (b).

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

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

ϕ(x,y):=ϕ(x,y)¬ϕ(x,a 0)ϕ(x,a ω)\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’’) ϕ(M,a i)ϕ(M,a j)\phi'(M,a_i)\subset \phi'(M,a_j) for iji\leq j. Indeed, let i<ji\lt j and xx be a counterexample, i.e.
ϕ(x,a i)¬ϕ(x,a 0)ϕ(x,a ω)\phi(x,a_i) \wedge \neg \phi(x,a_0)\wedge \phi(x,a_\omega) and ¬(ϕ(x,a j)¬ϕ(x,a 0)ϕ(x,a ω))\neg (\phi(x,a_j) \wedge \neg \phi(x,a_0)\wedge \phi(x,a_\omega)). These formulae imply that ¬ϕ(x,a j)\neg\phi(x,a_j) and thus both (x,a 0,a i)(x,a_0,a_i) and (x,a ω,a j)(x,a_\omega,a_j) are ϕ\phi-indiscernible.

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

Definition

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

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

ii. there are i<j,k<li\lt j, k\lt l, and j<lj\lt l, i,j,k,lIi,j,k,l\in I' such that ((i,j,k,l),i,j)U((i,j,k,l),i,j)\in U and ((i,j,k,l),a l,a k)U((i,j,k,l),a_l,a_k)\in U

  1. for n3n\neq 3 the filter on |II 4| n|I\cup I^4|^n is the coarsest filter such that all the simplcial maps |II 4| n|II 4| 3|I\cup I^4|^n\to |I\cup I^4|^3 are continuous.

Theorem

  1. theory TT has NSOP

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

  3. for each linear order II, model MM and formula ϕ\phi it holds

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

Proof

Let (a i) iI(a_i)_{i\in I} be a witness for SOP. Take the corresponding map |I| M {ϕ}|I|^\leq_\bullet \to M^{\{\phi\}}_\bullet. By SOP there are no witnesses of (b’) in MM, thus wherever I^4 is sent to, the preimage of the subset of ϕ\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| M {ϕ}|I|^\leq_\bullet \to M^{\{\phi\}}_\bullet corresponds to a sequence (a i) iI(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,li,j,k,l belong to an infinite constant subsequence, and therefore the preimage of the subset of ϕ\phi-consistent tuples is large.

Thus we may assume that all a ia_i‘s are distinct. Take the diagonal map sending each (i,j,k,l)I 4(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 ϕ\phi-consistent tuples is large.

NSOP nNSOP_n for n>3n\gt 3

Recall that a formula ϕ\phi has SOP nSOP_n iff there is an infinite sequence (a i) iω(a_i)_{i\in \omega} such that

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

(b nb_n) there are (a i) 0in1(a_i)_{0\leq i \leq n-1} such that ϕ((a i,a (i+1)modn)\phi((a_i,a_{(i+1)mod\,n}) for any 0in0\leq i\leq n

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

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

(b nb'_n) there are (a i) 0in1(a_i)_{0\leq i \leq n-1} such that for any 0in0\leq i\leq n the sequence (a i,a (i+1)modn,a (i+2)modn)(a_i,a_{(i+1)mod\,n},a_{(i+2)mod\,n}) is ϕ\phi-indiscernible.

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

Indeed, either there is a 3-cycle ϕ(a i,a (i+1)modn),ϕ(a (i+1)modn,a (i+2)modn),ϕ(a (i+2)modn,a i)\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 0ilessn0\leq i\less n ¬ϕ(a (i+2)modn,a i)\neg\phi(a_{(i+2)mod\,n},a_{i}), hence there is an nn-cycle n1,..,(n12k)modn,..(n12n)modnn-1,..,(n-1-2k)mod\,n,..(n-1-2n)mod\,n for ¬ϕ\neg\phi.

Let us reformulate this as a lifting property.

Let |{0,...,n1}| cycle|\{0,...,n-1\}|^{cycle}_\bullet be the situs associated with the filter on |{0,...,n1}| 3|\{0,...,n-1\}|^3 generated by the set of triples (a i,a (i+1)modn,a (i+2)modn)(a_i,a_{(i+1)mod\,n},a_{(i+2)mod\,n}), 0in10\leq i\leq n-1.

Theorem

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

|{0,...,n1}| diag|{0,...,n1}| cycleM {ϕ}|\{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:

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

Geometric realisation:

  • [2] 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)

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

Last revised on May 8, 2021 at 07:58:54. See the history of this page for a list of all contributions to it.