Misha Gavrilovich: Publications, Preprints and other Expositions.

Contacts: text chat, former online Oxford logic seminar tea room (upon request), email familyname at ihes.fr.

  • A first-order theory is stable iff its type space is simplicially contractible. A definable type of a first-order theory is the same as a section (retraction) of the simplicial path space (decalage) of its space of types viewed as a simplicial topological space; as is well-known, in the category of simplicial sets such sections correspond to homotopies contracting each connected component. Without the simplicial language this is stated in Exercise 8.3.3 in the model theory textbook [Tent-Ziegler], which defines a bijection between the set of all 1-types definable over a parameter set B and the set of all "coherent" families of continuous sections πn:Sn(B)→Sn+1(B) where Sn(B) is the Stone space of types with n variables of the theory T with parameters in B. Thus the definition of stability ``each type is definable'' says that {a first order theory is stable iff its space of types is simplicially contractible}, in the precise sense that the simplicial type space functor S(B):Δop→Top, n⟼Sn+1(B) fits into a certain well-known simplicial diagram in the category of simplicial topological spaces which does define contractibility for fibrant simplicial sets. In this note we spell out this and similar diagrams representing notions in model theory such as a parameter set and a type, a type being invariant, definable, and product of invariant types, and give pointers to the same diagrams in homotopy theory.
  • Convergence and homotopical triviality are defined by the same simplicial formula . (a version for readers less familiar with simplicial notation) We observe that convergence can be defined by a well-known simplicial construction which also defines homotopical triviality, in an appropriate simplicial category of generalised topological spaces, and also spell out how to obtain this observation by transcribing into simplicial language the Bourbaki definition of a limit point of a filter on a topological space. In particular, a metric space is complete iff all maps from 0-coskeletons to the associated generalised topological space are homotopically trivial.
  • The unreasonable expressive power of the lifting property and simplicial constructions in elementary mathematics. We rephrase a number of textbook definitions in terms of the lifting property and in simplicial language. In particular, we observe that in simplicial terms a limit of a filter on a topological space (as defined in Bourbaki, General Topology) is a contracting homotopy.
  • A category-theoretic reformulation of Shelah dividing lines in model theory, a poster with complete definitions. We reformulate NTP,NOP,NSOP,NIP,NFCP in terms of a certain simplicial category of generalised topological spaces, and wonder if this "homotopical" point of view may be used in model theory.
  • A suggestion towards a finitist's realisation of topology. We observe that the notion of a trivial Serre fibration, a Serre fibration, and being contractible, for finite CW complexes, can be defined in terms of the Quillen lifting property with respect to a single map M→Λ of finite topological spaces (preorders) of size 5 and 3, one of the simplest examples of a map contracting something (namely, the V in M), and of a trivial Serre fibration. In particular, we observe that the double Quillen orthogonal {M→Λ}lr is precisely the class of trivial Serre fibrations if calculated in a certain category of nice topological spaces. This suggests a question whether there is a finitistic/combinatorial definition of a model structure on the category of topological spaces entirely in terms of by the single morphism M→Λ, apparently related to the Michael continuous selection theory.
  • Extremally disconnected as {{u->a,b<-v}-->{u->a=b<-v}}^l, and being proper as ({{o}-->{o->c}}^r_<4)^lr. We rewrite a couple of standard definitions in general topology in terms of lifting properties and finite topological spaces, i.e. finite preorders. This reveals preorders implicit in these notions, e.g. we see that for Gleason theorem that extremally disconnected spaces are projective in the category of topological spaces with proper maps, it is important the map of preorders implicit in the definition of extremally disconnected, is surjective and proper.
  • A (hopefully well-written) overview of the current understanding of the notion of generalised topological spaces based on simplicial filters An overview of the current understanding of a notion of generalised topological spaces based onthe category of simplicial objects in the category of filters and continuous maps, or, equivalently, finitely additive 0-1 valued measures and maps such that the preimage of a big set is big. It includes sketches of reformulations in this category of the notions of limit, uniform convergence; local triviality; geometric realisation of a simplicial set; Shelah dividing lines in model theory including various no-tree-properties and no-order-properties (stability/NOP; simplicity/NTP; NSOP, NSOP_i, NTP_i, NATP, NFCP)
  • Geometric realisation as the Skorokhod semi-continuous path space endofunctor. same with Haskell-type notation for Hom sets

    (preliminary notes, with K.Pimenov) We interpret a construction of geometric realisation by [Besser], [Grayson], and [Drinfeld] as constructing a space of maps from the interval to a simplicial set, in a certain formal sense, reminiscent of the Skorokhod space of semi-continuous functions; in particular, we show the geometric realisation functor factors through an endofunctor of a certain category. Our interpretation clarifies the explanation of [Drinfeld] "why geometric realization commutes with Cartesian products and why the geometric realization of a simplicial set [...] %(resp.cyclic set) is equipped with an action of the group of orientation preserving homeomorphisms of the segment $[0, 1]$".

  • Formulating basic notions of finite group theory via the lifting property. As a tex file. We reformulate several basic notions in finite group theory in terms of iterations of the lifting property (orthogonality) with respect to particular morphisms. Our examples include the notions of a finite group being nilpotent, solvable, perfect, torsion-free; $p$-groups, and prime-to-$p$ groups, perfect core, Fitting subgroup, and $p$-core. We also reformulate in similar terms the conjecture that a localisation of a nilpotent group is nilpotent.
  • Standard conjectures in model theory, and categoricity of comparison isomorphisms. As a tex or djvu file. an older arxiv version We formulate two conjectures about etale cohomology and fundamental groups motivated by categoricity conjectures in model theory.
  • Exercises de style: A homotopy theory for set theory. Part I and II., joint with Assaf Hasson, 51 page, a shortened exposition of the two preprints below

  • Exercises de style: A homotopy theory for set theory. Part I., same in djvu, joint with Assaf Hasson, 36 pages, a paper explaining the main construction but not set theoretic applications.

  • Exercises de style: A homotopy theory for set theory. Part II., same in djvu, joint with Assaf Hasson, 36 pages, a paper about set theoretic applications

  • My DPhil thesis Model Theory of the Universal Covering Spaces of Complex Algebraic Varieties. Please note parts of it are superseded by later work of Martin Bays, esp. Categoricity results for exponential maps of 1-dimensional algebraic groups & Schanuel Conjectures for Powers and the CIT. Please note there is an innaccuracy in the definition of the topology in section II.1.2, which is corrected in the preprint Covers of Abelian varieties as analytic Zariski structures, and an error in chapter IV which is corrected in the paper A remark on transitivity of Galois action on the set of uniquely divisible abelian extensions of E(bar Q) by Z2.

    Out of date (not the best expositions).

  • Category theoretic formulations of some notions in classification theory, and their topological intuition. Slides of a talk at Oxford Logic Advanced Class. We suggest a category-theoretic approach to stability theory by rewriting several standard definitions in terms of diagram chasing in a category of generalised spaces. In particular, stability and simplicity are expressed as Quillen lifting properties/negations, and this allows us to concisely define various classes of stable or simple models.
  • Remarks on Shelah's classification theory and Quillen's negation. Versions of notation: 1. standard alphabet, Haskell arrowy notation for Hom as {X=>Y}. 2. standard alphabet, standard notation for Hom sets

    We give category-theoretic reformulations of stability, NIP, NOP, and non-dividing by observing that their characterisations in terms of indiscernible sequences are naturally expressed as Quillen lifting properties %(negation) of certain morphisms associated with linear orders, in a certain category extending the category of topological spaces and that of simplicial sets. This suggests an approach to a homotopy theory for model theory.

  • The category of simplicial sets with a notion of smallness. (usual notation for Hom sets) same with Haskell-type notation for Hom sets

    We consider simplicial sets equipped with a notion of smallness, and observe that this slight ``topological'' extension of the ``algebraic'' simplicial language allows a concise reformulation of a number of classical notions in topology, e.g.~continuity, limit of a map or a sequence along a filter, various notions of equicontinuity and uniform convergence of a sequence of functions; completeness and compactness; in algebraic topology, locally trivial bundles as a direct product after base-change and geometric realisation as a space of discontinuous paths.

    In model theory, we observe that indiscernible sequences in a model form a simplicial set with a notion of smallness which can be seen as an analogue of the Stone space of types.

    These reformulations are presented as a series of exercises, to emphasise their elementary nature and that they indeed can be used as exercises to make a student familiar with computations in basic simplicial and topological language. (Formally, we consider the category of simplicial objects in the category of filters in the sense of Bourbaki.)

    This work is unfinished and is likely to remain such for a while, hence we release it as is, in the small hope that our reformulations may provide interesting examples of computations in basic simplicial and topological language on material familiar to a student in a first course of topology or category theory.

  • A naive diagram-chasing approach to formalisation of tame topology. As a tex or djvu file. A draft of a research proposal. We translate excerpts of (Bourbaki, General Topology) into diagram chasing arguments, and speculate it might lead to a naive diagram-chasing approach to the formalisation and foundations of tame topology.

  • Topological and metric spaces as full subcategories of the category of simplicial objects of the category of filters. As a tex file. A draft of a research proposal at a very early stage reflecting current work (stalled). Not proofread yet.

  • A diagram chasing formalisation of elementary topological properties A shortened exposition of the results of the earlier draft, with a focus on formalisation. As a tex file.

  • The unreasonable power of the lifting property in elementary mathematics. A draft of a research proposal. As a text file (slightly outdated) or a tex file.

  • Tame topology: a naive elementary approach via finite topological spaces. an unproofread draft of a research proposal.

  • Expressing the statement of the Feit-Thompson theorem with diagrams in the category of finite groups. same in djvu,

  • Elementary general topology as diagram chasing calculations with finite categories. A draft of a research proposal. same in djvu,

  • Separation axioms as Quillen lifting properties, a modified wikipedia page. as an article in in pdf,, in tex

  • An example of a lifting property, slides

  • Point set topology as diagram chasing computations. Lifting properties as instances of negation. same in djvu, as published in De Morgan Gazette. 8p. An updated version, 14pp.

  • Point-set topology as diagram chasing computations. same in djvu, 21 page. A draft of a research proposal.

  • A decidable equational fragment of category theory without automorphisms. 6 pages, joint with Alexandre Luzgarev and Vladimir Sosnilo. A preliminary draft (a beta-version; read at your own risk!). The first public version (pre-July 2014)

    Other preprints

    Papers and preprints with abstracts

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    Preprints on the homotopy theory of set theory

    Drafts, out-of-date

  • Exercises de style: A homotopy theory for set theory. Part B. , same in djvu, notes by Misha Gavrilovich on joint with Assaf Hasson, 20 pages, an unfinished write-up briefly sketching the main construction and its set theoretic application.

  • A homotopy approach to set theory, same in djvu, 15 pages, a brief annnouncement of current results, open questions and motivations
  • A construction of a model category, same in djvu, 42 pages, an unfinished write-up of the proofs, motivations and basic ideas containing full proofs. The style is intentionally unorthodox, and the author would appreciate comments whether readers find the exposition conductive to mathematical reasoning.