+--[ RSA 2048]----+ iiii Misha Gavrilovich

Misha Gavrilovich: Publications, Preprints and other Expositions.

  • 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 $[0, 1]$".

  • 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.

  • 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.

  • 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.