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

Covers of Abelian varieties as analytic Zariski structures

We use tools of mathematical logic to analyse the notion of a path on an complex algebraic variety, and are led to formulate a "rigidity" property of fundamental groups specific to algebraic varieties, as well as to define a bona fide topology closely related to etale topology. These appear as criteria for uncountable categoricity, or rather stability and homogeneity, of the formal countable language we propose to describe homotopy classes of paths on a variety, or equivalently, its universal covering space. We also show that, with this topology, the universal covering space of the variety is an analytic Zariski structure. Technically, we present a countable $L_{omega_1\omega}$-sentence axiomatising a class of analytic Zariski structures containing the universal covering space of an algebraic variety over a number field, under some assumptions on the variety

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.Comment: up-to-date version at http://mishap.sdf.org/by:gavrilovich/mints_simplicial_filters.pd

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

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

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.Comment: 8p

A homotopy approach to set theory

We observe that the notion of two sets being equal up to finitely many elements is a homotopy equivalence relation in a model category, and suggest a homotopy-invariant variant of Generalised Continuum Hypothesis about which more can be proven within ZFC and which first appeared in PCF theory. The formalism allows to draw analogies between notions of set theory and those of homotopy theory, and we indeed observe a similarity between homotopy theory ideology/yoga and that of PCF theory. We also briefly discuss conjectural connections with model theory and arithmetics and geometry.Comment: 15 pages, work-in-progress, 4 figure

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.Comment: arXiv admin note: substantial text overlap with arXiv:1608.0592

A remark on transitivity of Galois action on the set of uniquely divisible abelian extensions of the group of algebraic points of an elliptic curve, by Z2Z^2

We study Galois action on \Ext^1(E(\bar \Q),\Z^2) and interpret our results as partially showing that the notion of a path on a complex elliptic curve $E$ can be characterised algebraically. The proofs show that our results are just concise reformulations of Kummer theory for $E$ as well as the description of Galois action on the Tate module. Namely, we prove (a),(b) below by showing they are equivalent to (c) which is well-known: (a) Absolute Galois group acts transitively on the set of uniquely divisible abelian \EndE-module extensions of E(\bar\Q) of algebraic points of an elliptic curve, by $\Lambda\cong\Z2$, (b) natural algebraic properties characterise uniquely the Poincare's fundamental groupoid of a complex elliptic curve, restricted to the algebraic points, (c) (Kummer theory) up to finite index, the image of the Galois action on the sequences $(P_i)_{i>0},jP_{ij}=P_i,i,j>0$ of points P_i\in E^k(\bar\Q) is as large as possible with respect to linear relations between the coordinates of the points $P_i$'s. Our original motivations come from model theory; this paper presents results from the author's thesis.Comment: 22 pages, part of the author's thesis, available at http://misha.uploads.net.r

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]"