nLab situs


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 n0n\geq 0, with a filter, equiv. such (not quite) a measure, 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. Most of what we say below would also apply to the category sMsM 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 sMsM.

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 nn-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 n2n\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)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, the pair (x 0,x)(x_0,x) with xx near x 0x_0 is thought of as small (a small simplex) 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, NSOP, NSOP iNSOP_i, NTP iNTP_i, NATPNATP, NFCPNFCP, 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, 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.

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). Recall that 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. Intuitively, we defined the situs structure “on a set” such that two points x 1,x 2Xx_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)X×X(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×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. This gives a precise meaning to the phrase “if ii\in\mathbb{N}, xLx\in L and jj\in \mathbb{N}, yLy\in L are sufficiently close to each other, so are f i(x)f_i(x) and f j(y)f_j(y)”. A different choice of the situs strucuture gives a different precise meaning:

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[+1](0)=0f[+1](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 notions 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 consisting of sequences 0x 1...x n10\leq x_1\leq ... \leq x_n\leq 1 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) \}. 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] [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([0,1] ,):ssHHom([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. 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 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 :=(I ) cartI^\leq_\bullet:=(I^\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.


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 .


iii. says that each continuous map (I tails) cartM (I^{\leq tails})^{\operatorname{cart}}_\bullet\to M_\bullet factors as (I tails) cart(|I| tails) cartM (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×MM\times M considered in the language with arbitrary parameters, for a saturated enough model MM 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 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

Recall that a formula φ(x;y) has the finite cover property if for arbitrarily large nωn \le \omega there exist a 0,...a n a_0,...a_n such that {ϕ(x;a 0),...ϕ(x;a n)}\{\phi(x;a_0),...\phi(x;a_n)\} is n-consistent but (n+1)-inconsistent.

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


  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.2.6)

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


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.


[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.


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.


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


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.


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.


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.


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, 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 iNOP, NSOP, NSOP_i, NTP, NTP_i, and NATPNATP are expressed as Quillen lifting properties of form

A B M A_\bullet \to B_\bullet \rightthreetimes M_\bullet\to\top

where \top is the terminal object, and MM is a situs associated with a model and a formula, and AA and BB are objects of combinatorial nature.

Informal explanation

Recall the common pattern of definitions of NOP,NSOP,NSOP i,NTP,NTP i,NATPNOP, NSOP, NSOP_i, NTP, NTP_i, NATP. As usually stated, such a property wrt a formula ϕ\phi and a model MM require that there is no combinatorial structure (“counterexample” or “witness”) formed by elements of the model MM 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 ||M ||A_\bullet|\to |M_\bullet| and |B ||M ||B_\bullet|\to |M_\bullet|. The positive requirements are coded by continuity of the morphism A M A_\bullet\to M_\bullet. The failure of negative requirement are coded by continuity of the diagonal morphism B M B_\bullet \to M_\bullet.

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

To code by continuity the failure of negative requirements, we need to define non-trivial filters on B nB_n. The first (incorrect) attempt would be to say that a subset of B nB_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 nM (n)B_n\to M_\bullet(n) would mean exactly that one of negative requirements fails, and hence the map A M A_\bullet\to M_\bullet does not represent a counterexample/witness. To fix this definition, define a subset of B nB_n to be large (a neighbourhood) iff it contains a tuple representing a negative requirement with respect to each substructure of AA of the same shape as required by the property.
We call the filters on B nB_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 NOPNOP, NSOPNSOP, and NSOP iNSOP_i, i3i\geq 3 which involve the situs of ϕ\phi-indiscernible sequences. These reformulations involve reformutading the standard definitions in terms of ϕ\phi-indiscernible sequences.

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


Note our considerations allow to formulate a precise conjecture corresponding to the equality NATP=NSOP 1+NTP 2NATP = 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.

Rewriting NOP (no order property)

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.

Because NOP cares about {\em consistency of instances of a formula on tuples of ordered variables}, equip M (n ):=M nM_\bullet(n^\leqslant):=M^n with the filter of subsets containing

{(a 1,..,a n)M n:ij&a ia jφ(a i,a j)}\left\{ (a_1,..,a_n)\in M^n\,:\, i\le j \,\&\, a_i\neq a_j\implies \varphi(a_i,a_j) \right\}

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


A formula φ(x,y)\varphi(x, y) has {\em NOP} iff there is no sequence (a i) iω(a_i)_{i\in\omega} such that \begin{itemize} \item[a)] for iji\le j, φ(a j,a i)\varphi(a_j,a_i) is false. %each subsequence (a i,a j,a i)(a_i,a_j,a_i) is not φ\varphi-indiscernible with repeated elments \item[b)] for iji\le j, φ(a i,a j)\varphi(a_i,a_j) holds. \end{itemize}

In analogy with NTP, think of the sequence as a tree T:=ω T:=\omega^\leqslant with a single branch, and equip ω (n ):=Hom preorders(n ,ω ),n0.\omega^\leqslant_\bullet(n^\leqslant) := \Hom_{\text{preorders}} ( n^\leqslant, \omega^\leqslant ), n\ge 0. with indiscrete filters. Let |ω| NOP(n ):=Hom Sets(n ,|ω|)=|ω| n,n0|\omega|^\text{NOP}_\bullet(n^\leqslant) := \Hom_{\text{Sets}} ( n^\leqslant, |\omega|)=|\omega|^n, n\ge 0 . where a subset UU of |ω| NOP(n )=|ω| n|\omega|^\text{NOP}_\bullet(n^\leqslant) =|\omega|^n is {\em large} iff it is closed under simplicial maps and

\color{blue} each infinite subsequence $(a_{i_j})_j$ of $\omega$ %there are has a subsequence \\ $(a_{i_{j_1}},..,a_{i_{j_n}})\in U$ where ${j_1}\le ...\le {j_n}$

With these definitions, ω |ω| NOPrttM \omega_\bullet^\leqslant\to |\omega|^\text{NOP}_\bullet \,\rtt\, M_\bullet\to \top defines NOP.

NSOP_\ell for 3\ell\geq 3 (no strict order property)


For 3\ell \geqslant 3, a formula φ(x,y)\varphi(x,y) has SOP_\ell (%the \ell-strong order property) iff \begin{itemize} \item[a)] the set {φ(x 1,x 2),,φ(x n1,x n),φ(x ,x 1)}\{\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<ω(a_i)_{i\lt\omega} such that φ(a i,a j)\models\varphi(a_i,a_j) for all i<j i\lt j

\end{itemize} A theory is NSOP_\ell if no formula has SOP_\ell.

Equivalently, NSOP_\ell means that “em a) implies NOP’’.

Let || (n ):=Hom textrmSets(n ,||)=|| n|\ell|_\bullet(n^\leqslant):=\Hom_{\textrm{Sets}}(n^\leqslant,|\ell|)=|\ell|^n, n0n\ge 0 denote the simplicial set represented by the set {1,2,...}\{1,2,...\ell\} with \ell elements which we denote by |||\ell|. Equip || (2 )=|| 2|\ell|_\bullet(2^\leqslant)=|\ell|^2 with the filter of subsets containing {{(1,2),(2,3),...,(1,),(,1),(1,1),..,(,)}\{(1,2),(2,3),...,(\ell-1,\ell),(\ell,1), (1,1),..,(\ell,\ell)\},} and for each n0n\ge 0 equip || (n )=|| n|\ell|_\bullet(n^\leqslant)=|\ell|^n with the coarsest filter such that all simplicial maps || (n )|| (2 )|\ell|_\bullet(n^\leqslant)\to |\ell|_\bullet(2^\leqslant) are continuous.

Then a map || -cycleM ,ia i|\ell|^{\ell\text{-cycle}}_\bullet\longrightarrow M_\bullet, i\mapsto a_i is continuous iff either a 1=...=a a_1=...=a_\ell or M{φ(a j 1,a j 2),,φ(a j 1,a ),φ(x a ,a j 1)}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(a_{j_k})_k of distinct elements. Hence, item a) holds iff || -cyclerttM |\ell|^{\ell\text{-cycle}}_\bullet \longrightarrow \top\,\rtt\, M_\bullet\longrightarrow \top. Then\ NSOP_\ell can be stated as:

$|\ell|^{\ell\text{-cycle}}_\bullet \longrightarrow \top \,\rtt\, M_\bullet\longrightarrow \top$ implies $\omega_\bullet^\leqslant\to |\omega|^\text{NOP}_\bullet \,\rtt\, M_\bullet\to \top$

NFCP (no finite cover property)

We also note that in the category sPhi\sPhi of generalised topological spaces NFCP (no finite cover property) means that M M_\bullet is of infinite dimension. Recall that “\bulletisofinfinitedimension(asasimplicialobject)meansthatthewholeof is of infinite dimension'' (as a simplicial object) means that the whole of M\bulletisfullydeterminedbyafinitepieceof is fully determined by a finite piece of M_\bullet,namelythereis, namely there is N\ge 0suchthatforeach such that for each n\ge N M_\bullet(n^\leqslant)isthepullbackofthediagramconsistingofsimplicialmaps is the pullback of the diagram consisting of simplicial maps M_\bullet(n^\leqslant)\to M_\bullet(m^\leqslant)and and M_\bullet(m_1^\leqslant)\to M_\bullet(m_2^\leqslant)where where m,m_1,m_2\le N.Interminologyof([M.Malliaris."Edgedistributionanddensityinthecharacteristicsequence](#Mallaris2010apal))thisisexpressedbysayingthat``thecharacteristicsequence. 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\gtof of \varphi$ has finite support’’.

{Stability and NIP}

Both stability and NIP can be expressed in terms of indiscernible sequences. Hence, we modify the filters on |M| |M|_\bullet to reflect that: the filter on |M| (n )=M n|M|_\bullet(n^\leqslant)=M^n is generated by sets of {\em φ\varphi-indiscernible sequences with repeations}, i.e.~sequences (a i) i(a_i)_i such that φ(a i 1,...,a i r)φ(a j 1,...,a j r)\varphi(a_{i_1},...,a_{i_r})\Leftrightarrow \varphi(a_{j_1},...,a_{j_r}) whenever i 1...i ri_1\le ...\le i_r, j 1...j rj_1\le ...\le j_r, all the a i 1,..,a i ra_{i_1},..,a_{i_r} are distinct, and all the a j 1,...,a j ra_{j_1},...,a_{j_r} are distinct. %any subsequence of distinct elements φ\varphi-indiscernible. (If you feel it is more natural, you may require each subsequence with distinct elements to be part of an {\em infinite} φ\varphi-indiscernible sequence.)

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

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

Each indiscernible sequence is a set.

Let { ω ,cof(n ):=Hom preorders(n ,ω ),n0,\omega^{\leqslant,\text{cof}}_\bullet(n^\leqslant) := \Hom_{\text{preorders}} ( n^\leqslant, \omega^\leqslant ), n\ge 0, \ |ω| cof(n ):=Hom sets(n ,|ω|)=|ω| n,n0|\omega|^\text{cof}_\bullet(n^\leqslant) := \Hom_{\text{sets}} ( n^\leqslant, |\omega| )=|\omega|^n, n\ge 0} equip |ω|=ω ,cof(1 )=|ω| cof(1 )|\omega|=\omega^{\leqslant,\text{cof}}_\bullet(1^\leqslant) =|\omega|^{\text{cof}}_\bullet(1^\leqslant) with the filter of cofinite subsets, and equip each ω ,cof(n )\omega^{\leqslant,\text{cof}}_\bullet(n^\leqslant), resp.~|ω| cof(n )|\omega|^{\text{cof}}_\bullet(n^\leqslant), with the coarsest filter such that all simplicial maps ω ,,cof(n )ω ,cof(2 )\omega^{\leqslant,,\text{cof}}_\bullet(n^\leqslant)\to \omega^{\leqslant,\text{cof}}_\bullet(2^\leqslant), resp.~|ω| cof(n )|ω| cof(2 )|\omega|^\text{cof}_\bullet(n^\leqslant)\to |\omega|^\text{cof}_\bullet(2^\leqslant), are continuous. Then {each (infinite) φ\varphi-indiscernible sequence is a φ\varphi-indiscernible set iff \ ω ,cof|ω| cofrttM \omega^{\leqslant,\text{cof}}\longrightarrow |\omega|^\text{cof}_\bullet \,\rtt\, M_\bullet \longrightarrow \top }

Each indiscernible sequence is eventually indiscernible over any parameter

Let { M[+1] (n ):=Hom sets((n+1) ,|ω|)=|M|×|M| n,n0M[+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] (n )|M|^{n+1}=M[+1]_\bullet(n^\leqslant) with the filter generated by subsets %, for each formula φ\varphi {{(a 0,a 1,...,a n): the seq. (a 1,...,a n) is φ-indiscernible over a 0}\{(a_0,a_1,...,a_n): \text{ the seq. }(a_1,...,a_n)\text{ is }\varphi\text{-indiscernible over }a_0\}, φ\varphi a formula} One can check that the map M[+1] M ,(a 0,a 1,...,a n)(a 1,..,a n)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 MM is eventually indiscernible over any parameter iff {{0} ω ,cofrttM[+1] M \{0\}_\bullet \to \omega^{\leqslant,\text{cof}} \,\rtt\, M[+1]_\bullet\to M_\bullet} where {0} (n )={0}\{0\}_\bullet(n^\leqslant)=\{0\} is equipped with the filter containing the empty set, and {0} ω ,cof\{0\}_\bullet \to \omega^{\leqslant,\text{cof}} is the map (0,..,0)(0,..,0)(0,..,0)\mapsto (0,..,0). Note that the map {0} M[+1] \{0\}_\bullet \to M[+1]_\bullet picks an arbitrary element of MM.

Speculations: limits and contractible models ?

In sPhi\sPhi the notions of homotopy and limit are closely related: the same construction in sPhi\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\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 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 𝔉 :F X \mathfrak{F}_\bullet : F_\bullet\longrightarrow X_\bullet be a morphism in sPhi \sPhi. A morphism x x_\bullet is said to be a {\em limit morphism} (or simply {\em a limit})
of 𝔉 \mathfrak{F}_\bullet iff F X F_\bullet\longrightarrow X_\bullet factors as

F X [+1]x X F_\bullet\longrightarrow X_\bullet\circ[+1] \xrightarrow{x_\bullet} X_\bullet

where [+1]:ΔΔ[+1]:\Delta\longrightarrow\Delta is the shift

nn+1,f:nmlongmapstof:n+1m+1,f(0):=0;f(i+1):=f(i) for i0,n\mapsto n+1, \ \ f\!:\!n\to m \,\longmapsto\, f'\!:\!n+1\to m+1, f'(0):=0; f'(i+1):=f(i) \text{ for }i\ge 0,

and X [+1]X 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 𝔉\mathfrak{F} on a topological space XX, associate with 𝔉ObPPhi\mathfrak{F}\in \Ob \PPhi the simplical set represented by XX

F (n ):=Hom(n ,X){F}_\bullet(n^\leqslant):=\Hom(n^\leqslant,X)

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


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

A limit of T M ϕ-NTPT^\leqslant_\bullet \longrightarrow M_\bullet^{\phi\text{-NTP}} is a point aMa\in M such that the map T M ϕ-NTPT^\leqslant_\bullet \longrightarrow M_\bullet^{\phi'\text{-NTP}} is continuous where ϕ(x,y):=ϕ(x,a)&ϕ(x,y)\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 ϕ-NTP[+1]M_\bullet^{\phi\text{-NTP}}[+1] is the disjoint union of connected component which are copies of M ϕ-NTPM^{\phi\text{-NTP}}_\bullet parametrised by elements of MM, that both T T^\leqslant_\bullet, |T| |T|_\bullet are connected. Thus to give a map of simplicial sets T M ϕ-NTPT^\leqslant_\bullet \longrightarrow M_\bullet^{\phi\text{-NTP}} or |T| M ϕ-NTP|T|_\bullet \longrightarrow M_\bullet^{\phi\text{-NTP}} is the same as to pick a point aMa\in M. The continuity requirement on T M ϕ-NTPT^\leqslant_\bullet \longrightarrow M_\bullet^{\phi\text{-NTP}} means precisely that ϕ(x,a 1),...,ϕ(x,a n)\phi(x,a_1),...,\phi(x,a_n) is consistent whenever a 1,...,a na_1,...,a_n lie on the same branch of TT. The continuity requirement on T M ϕ-NTP[+1]T^\leqslant_\bullet \longrightarrow M_\bullet^{\phi\text{-NTP}}[+1] means precisely that ϕ(x,a),ϕ(x,a 1),...,ϕ(x,a n)\phi(x,a),\phi(x,a_1),...,\phi(x,a_n) is consistent whenever a 1,...,a na_1,...,a_n lie on the same branch of TT.

The case of |T| NTPM ϕ-NTP|T|^\text{NTP}_\bullet \longrightarrow M_\bullet^{\phi\text{-NTP}} is similar. \def\sing{\operatorname{sing}} Now let FF and XX denote topological spaces.


A homotopy contracting FF in XX to a point (i.e.~a map

$h:F\times [0,1]/F\times \{1\} \longrightarrow X$

from the cone of FF to XX), gives rise to a map singF ((n+1) ):=Hom Top(Δ n,F)\sing F_\bullet((n+1)^\leqslant):=\Hom_\text{Top}( \Delta^n, F)ar complexes lifting the map singF singX \sing F_\bullet \longrightarrow \sing X_\bullet:

$\sing F_\bullet((n+1)^\leqslant):=\Hom_\text{Top}( \Delta^n, F)$,\\ $\sing X_\bullet((n+1)^\leqslant):=\Hom_\text{Top}( \Delta^n, X)$, \\ $\sing X_\bullet\circ[+1]((n+1)^\leqslant)=\Hom_\text{Top}( \Delta^n\times [0,1]/{\Delta^n\times\{1\}}, X)$\\

Take δ:Δ nF\delta: \Delta^n \to F to h *(δ):Δ n×[0,1]/Δ n×{1}Xh_*(\delta):\Delta^n\times [0,1]/\Delta^n\times\{1\}\to X defined by

$h_*(\delta)( x,t ):= h(\delta(x),t)$.

Here n0n\geqslant 0 and Δ n×[0,1]/Δ n×{1} \Delta^n\times [0,1]/{\Delta^n\times\{1\}} is the cone of nn-simplex Δ n\Delta^n.


A map
h:F×[0,1]/F×{1}Xh:F\times [0,1]/F\times \{1\} \longrightarrow X continuous in a neighbourhood of “e top of the cone’‘ point F×{1}F\times \{1\} is the same as a map in sPhi\sPhi

F ×([0,1] 1) X [+1] F_\bullet\times ([0,1]_1)_\bullet \longrightarrow X_\bullet[+1]
(x 1,..,x n,t 1,..,t n)(h(x 1,1),h(x 1,t 1),h(x 2,t 2),...,h(x n,t n))(x_1,..,x_n,t_1,..,t_n) \mapsto (h(x_1,1),h(x_1,t_1),h(x_2,t_2),...,h(x_n,t_n))

where [0,1] 1[0,1]_1 denotes the interval [0,1][0,1] equipped with the filter of neighbourhoods of point 11, and F ,X F_\bullet, X_\bullet denote the generalised topological spaces corresponding to FF and XX.


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)

  • [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.