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Misha Gavrilovich

Russia

IPRERAS

SCS Slide Set Lifting property as a category-theoretic negation: a source of educational examples of elementary uses of category theoretic language.

SCS Poster A category-theoretic reformulation of Shelah’s dividing lines in model theory.

Motivation (joint with K.Pimenov )

Playing with the lifting property, we observed that the lifting property and other categorical constructions from algebraic topology has the power to express concisely and uniformly textbook definitions across disparate domains, including topology, analysis, group theory, model theory, in terms of simple(st?) or archetypal (counter)examples, and that an apparently straightforward attempt to read line by line the text of these definitions and rephrase it in categorical language leads to this observation.

We were startled that this never appeared in print, and our presentations attempt to write up some of it.

How these examples can be used in teaching, is discussed in my slides.

Slightly more complicated examples in topology---compact, contractible, and trivial fibration---are discussed in the slides by K.Pimenov.

Analysis of NTP₁ and other properties

The poster by M.Gavrilovich reads line by line the definition of NTP (no tree property) and recovers its reformulation as a lifting property in the category of generalised topological spaces. Let us here sketch how the same analysis applies to the other dividing lines. We do this for NTP₁.

Definition of NTP₁

A formula φ(x,y) has the tree property 1 (TP₁) if there are (a_η;:η &in; ω^<ω) such that

a) for each incomparable η,μ &in; ω^<ω, {φ(x,a_μ), φ(x,a_η)} is inconsistent.
b) for each σ&in; ω^ω branch {φ(x,a_σ|n):n <ω} is consistent.

A theory is NTP₁ if no formula has TP₁.

Note that the only difference from the definition of NTP is in item a). Therefore, the only the only difference from the reformulaton of the definition of NTP is in the definition of filters on |T|_•.

As in the case of NTP, we want the map |T|_•→ M_• in sዋ to be continuous iff there is a witness of failure of the negative requirement. In case of NTP₁, it means that we define the filter on |T|_• by:

a subset of |T|_•(n^<) is big iff it contains a pair of incomparable vertices from each subtree of T isomorphic to ω^<ω.

The case of NTP_i,NATP, NSOP₁ is similar: their definitions also only differ in item a). We also note that in the category sዋ of generalised topological spaces NFCP (no finite cover property) means that M_• is of infinite dimension.

Definition of NOP, NSOP_i(i>2)

The analysis of No-Order-Properties is similar but different mathematical structures arise. Instead of an infinitely branching tree T_•^&lt we consider the tree consisting of a single branch, i.e. a sequence. The order properties care about sequences indiscernible with respect to binary formulas, hence we modify the filters on |M|_• to reflect that: the filter on |M|_•(n^<)=Mⁿ is generated by indiscernible sequences with repeated elements, i.e. sequences such that any subsequence of distinct elements is indiscernible. Finally, a little care is needed to reformulate the standard definitions of no-order-properties so that witnesses of failure are defined in terms of indiscernible sequences.