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Misha Gavrilovich

Publications and source records attributed to Misha Gavrilovich.

At least 19 recordsLinked to original sources
arXiv2025

Stable first order theories as simplicial profinite sets

We rewrite simplicially the standard definitions of a complete first order theory, a model of it, and various characterisations of stability of a complete first order theory. In our reformulations the simplicial language replaces the standard definitions based on syntax, making them formally unnecessary. We view a complete first-order theory as a symmetric simplicial object in the category of profinite sets and open continuous maps, defined by the functor sending a finite set of variables into the Stone space of complete types in those variables. A model of a complete first-order theory is then a morphism from a representable simplicial set satisfying certain lifting properties reminiscent of, but weaker then, those in the definition of a fibration. The class of simplicial profinite sets corresponding to complete first order theories is characterised by the same lifting properties required of the map from the simplicial covering space (decalage) forgetting the extra degeneracy.

math.CT↗
arXiv2024

Finite combinatorics implicit in the basic definitions of topology

We explain how to see finite combinatorics of preorders implicit in the {text} of basic topological definitions or arguments in (Bourbaki, General topology, Ch.I), and define a concise combinatorial notation such that complete definitions of connectedness, compactness, contractibility, having a generic point, subspace, closed subspace, fit into $2$ or $4$ bytes. This notation is homotopy theoretic in nature, and is based on the following observation: A number of basic properties of continuous maps and topological spaces are defined using a single category-theoretic operation, taking left or right orthogonal complement with respect to the Quillen lifting property, repeatedly applied to a simple example illustrating the definition or its failure. Moreover, for most of these definitions this example can be chosen to be a map of finite topological spaces (=preorders) of size at most $5$. This includes the properties of a space being connected, compact, contractible, discrete, having a generic point, and a map having dense image, being the inclusion of an (open or closed) subspace, or of a component into a disjoint union, and others. Our reformulations illustrate the generative power of the lifting property as a means of defining basic mathematical properties starting from their simplest or typical example. The exposition is accessible to a student.

math.CT↗
arXiv2024

The Quillen negation monoid of a category, and Schreier graphs of its action on classes of morphisms

The free monoid with two generators acts on classes (=properties) of morphisms of a category by taking the left or right orthogonal complement with respect to the lifting property, and we define the Quillen negation monoid of the category to be its largest quotient which acts faithfully. We consider the category of topological spaces and show that a number of natural properties of continuous maps are obtained by applying this action to a single example. Namely, for the category of topological spaces we show finiteness of the orbit of the simplest class of morphisms { \emptyset \to {*} }, and we calculate its Schreier graph. The orbit consists of 21 classes of morphisms, and most of these classes are explicitly defined by standard terminology from a typical first year course of topology: a map having a section or dense image; quotient and induced topology; surjective, injective; (maps representing) subsets, closed subsets; disjoint union, disjoint union with a discrete space; each fibre satisfying separation axiom T0 or T1 . Also, the notions of being connected, having a generic point, and being a complete lattice, can be defined in terms of the classes in the orbit. In particular, calculating parts of this orbit can be used in an introductory course as exercises connecting basic definitions in topology and category theory.

math.CT↗
arXiv2023

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 $π_n:S^T_n(B)\to S^T_{n+1}(B)$ where $S^T_n(B)$ 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 $\mathbb{S}^T_\bullet(B):Δ^{op}\to {\rm Top}$, $n\longmapsto \mathbb{S}_{n+1}(B)$ 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.

math.CT↗
arXiv2020

Remarks on Shelah's classification theory and Quillen's negation

We give category-theoretic reformulations of stability, NIP, NTP, 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 categories of topological spaces and of simplicial sets. This suggests an approach to a homotopy theory for model theory.

math.LO↗
arXiv2020

Geometric realisation as the Skorokhod semi-continuous path space endofunctor

We interpret a construction of geometric realisation by [Besser], [Grayson], and [Drinfeld] of a simplicial set 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 [...] is equipped with an action of the group of orientation preserving homeomorphisms of the segment [0,1]".

math.AT↗
arXiv2019

Formulating basic notions of finite group theory via the lifting property

We reformulate several basic notions of notions in finite group theory in terms of iterations of the lifting property (orthogonality) with respect to particular morphisms. Our examples include the notions being nilpotent, solvable, perfect, torsion-free; p-groups and prime-to-p-groups; Fitting subgroup, perfect core, p-core, and prime-to-p core. We also reformulate as in similar terms the conjecture that a localisation of a (transfinitely) nilpotent group is (transfinitely) nilpotent.

math.GR↗
arXiv2018

Standard conjectures in model theory, and categoricity of comparison isomorphisms

We formulate two conjectures about etale cohomology and fundamental groups motivated by categoricity conjectures in model theory. One conjecture says that there is a unique Z-form of the etale cohomology of complex algebraic varieties, up to Aut(C)-action on the source category; put differently, each comparison isomorphism between Betti and etale cohomology comes from a choice of a topology on C. Another conjecture says that each functor to groupoids from the category of complex algebraic varieties which is similar to the topological fundamental groupoid functor, in fact factors through it, up to a field automorphism of the complex numbers acting on the category of complex algebraic varieties. We also try to present some evidence towards these conjectures, and show that some special cases seem related to Grothendieck standard conjectures and conjectures about motivic Galois group.

math.AG↗
arXiv2018

A naive diagram-chasing approach to formalisation of tame topology

We rewrite classical topological definitions using the category-theoretic notation of arrows and are led to concise reformulations in terms of simplicial categories and orthogonality of morphisms, which we hope might be of use in the formalisation of topology and in developing the tame topology of Grothendieck. Namely, we observe that topological and uniform spaces are simplicial objects in the same category, a category of filters, and that a number of elementary properties can be obtained by repeatedly passing to the left or right orthogonal (in the sense of Quillen model categories) starting from a simple class of morphisms, often a single typical (counter)example appearing implicitly in the definition. Examples include the notions of: compact, discrete, connected, and totally disconnected spaces, dense image, induced topology, and separation axioms, and, outside of topology, finite groups being nilpotent, solvable, torsion-free, p-groups, and prime-to-p groups; injective and projective modules; injective and surjective (homo)morphisms.

math.CT↗
arXiv2018

Topological and metric spaces are full subcategories of the category of simplicial objects of the category of filters

We observe that the category of topological space, uniform spaces, and simplicial sets are all, in a natural way, full subcategories of the same larger category, namely the simplicial category of filters; this is, moreover, implicit in the definitions of a topological and uniform space. We use these embeddings to rewrite the notions of completeness, precompactness, compactness, Cauchy sequence, and equicontinuity in the language of category theory, which we hope might be of use in formalisation of mathematics and tame topology. We formulate some arising open questions.

math.CT↗
arXiv2017

The unreasonable power of the lifting property in elementary mathematics

We illustrate the generative power of the lifting property (orthogonality of morphisms in a category) as means of defining natural elementary mathematical concepts by giving a number of examples in various categories, in particular showing that many standard elementary notions of abstract topology can be defined by applying the lifting property to simple morphisms of finite topological spaces. Examples in topology include the notions of: compact, discrete, connected, and totally disconnected spaces, dense image, induced topology, and separation axioms. Examples in algebra include: finite groups being nilpotent, solvable, torsion-free, p-groups, and prime-to-p groups; injective and projective modules; injective, surjective, and split homomorphisms. We include some speculations on the wider significance of this.

math.CT↗
arXiv2017

Separation axioms as lifting properties

We observe that many of the separation axioms of topology (including $T_0-T_4$) can be expressed concisely and uniformly in terms of category theory as lifting properties (in the sense of Quillen model categories) with respect to (usually open) continuous maps of finite spaces (involving up to 4 points) and the real line.

math.GN↗
arXiv2014

Point-set topology as diagram chasing computations: Lifting property as negation

We observe that some natural mathematical definitions are lifting properties relative to simplest counterexamples, namely the definitions of surjectivity and injectivity of maps, as well as of being connected, separation axioms $T_0$ and $T_1$ in topology, having dense image, induced (pullback) topology, and every real-valued function being bounded (on a connected domain). We also offer a couple of brief speculations on cognitive and AI aspects of this observation, particularly that in point-set topology some arguments read as diagram chasing computations with finite preorders.

math.HO↗
arXiv2013

Some Definability Results in Abstract Kummer Theory

Let $S$ be a semiabelian variety over an algebraically closed field, and let $X$ be an irreducible subvariety not contained in a coset of a proper algebraic subgroup of $S$. We show that the number of irreducible components of $[n]^{-1}(X)$ is bounded uniformly in $n$, and moreover that the bound is uniform in families $X_t$. We prove this by purely Galois-theoretic methods. This proof applies in the more general context of divisible abelian groups of finite Morley rank. In this latter context, we deduce a definability result under the assumption of the Definable Multiplicity Property (DMP). We give sufficient conditions for finite Morley rank groups to have the DMP, and hence give examples where our definability result holds.

math.LO↗
arXiv2012

Exercices de style: A homotopy theory for set theory II

This is the second part of a work initiated in \cite{GaHa}, where we constructed a model category, $\Qt$, for set theory. In the present paper we use this model category to introduce homotopy-theoretic intuitions to set theory. Our main observation is that the homotopy invariant version of cardinality is the covering number of Shelah's PCF theory, and that other combinatorial objects, such as Shelah's revised power function - the cardinal function featuring in Shelah's revised GCH theorem - can be obtained using similar tools. We include a small "dictionary" for set theory in $\QtNaamen$, hoping it will help in finding more meaningful homotopy-theoretic intuitions in set theory.

math.CT↗
arXiv2012

Exercices de style: a homotopy theory for set theory, I

We construct a model category (in the sense of Quillen) for set theory, starting from two arbitrary, but natural, conventions. It is the simplest category satisfying our conventions and modelling the notions of finiteness, countability and infinite equi-cardinality. In a subsequent paper \cite{GaHa1} we give a homotopy theoretic dictionary of set theoretic concepts, most notably Shelah's covering number $\cov(λ, \aleph_1,\aleph_1,2)$, recovered from this model category. We argue that from the homotopy theoretic point of view our construction is essentially automatic following basic existing methods, and so is (almost all) the verification that the construction works.

math.LO↗
arXiv2011

The univalence axiom in posetal model categories

In this note we interpret Voevodsky's Univalence Axiom in the language of (abstract) model categories. We then show that any posetal locally Cartesian closed model category $Qt$ in which the mapping $Hom^{(w)}(Z\times B,C):Qt\longrightarrow Sets$ is functorial in $Z$ and represented in $Qt$ satisfies our homotopy version of the Univalence Axiom, albeit in a rather trivial way. This work was motivated by a question reported in [Ob], asking for a model of the Univalence Axiom not equivalent to the standard one.

math.CT↗