Our reformulations illustrate the generative power of the lifting property as a means of defining natural basic mathematical concepts starting from their simplest or archetypal (counter)example. We also offer a couple of brief speculations on cognitive and AI aspects of this observation, in particular in relation to the ergo-logic of Gromov.
The exposition is accessible to a student.
Namely, for the category of topological spaces we show finiteness of the orbit of the simplest class of morphisms {∅ Ð→ {★}}, 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.
(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 $[0, 1]$".
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 Please note parts of it are superseded by later work of Martin Bays, esp.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.
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.
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)
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
The univalence axiom for posetal model categories. A shortened version. joint with Assaf Hasson and Itay Kaplan, 12 pages, a note where we observe that the set-theoretic model category constructed in "Exercises de style" delivers an example of the Univalence Axiom, albeit in a rather trivial way.
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.
Standard conjectures in model theory, and categoricity of comparison isomorphisms. As a tex or djvu file. 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 pi_{1}^{top}, 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.
Topological and metric spaces as full subcategories of the category of simplicial objects of the category of filters. As a tex file. This is a research proposal at a very early stage reflecting current work (stalled). Not proofread. We define fully faithful embeddings of the category of topological spaces and that of uniform spaces into the category of simplicial objects of the category of filters, and, based on this, use these two functors to reformulate several elementary notions including that of being compact, complete, a Cauchy sequence, and equicontinuity. We also define of quasi-geodesic metric spaces with large scale Lipschitz maps, or rather coarse spaces, into the category of simplicial objects of a category of filters with another kind of morphisms.
A diagram chasing formalisation of elementary topological properties As a tex file. A shortened exposition of the results of the earlier draft, with a focus on formalisation. Abstract: We introduce a formal syntax %and semantics and use it to rewrite in a concise, uniform and intuitive way several %a number of standard definitions in topology which are usually expressed in words. The definitions include compact, discrete, connected, and totally disconnected spaces, dense image, induced topology, closed subsets, and some of the separation axioms. We hope our reformulations may be use in formalisation of elementary topology and in teaching.
The unreasonable power of the lifting property in elementary mathematics. We illustrate the generative power of the lifting property as a 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, particularly in formalisation of mathematics. A draft. As a text file (more convenient for some readers).
Tame topology: a naive elementary approach via finite topological spaces. An unproofread draft of a research proposal. We also observe that several elementary notions in topology can be expressed in a very concise and uniform manner using the Quillen lifting property and finite topological spaces, and, further, that parts of several standard elementary arguments in topology correspond to manipulations (calculations) with arrows and labels, of category-theory style. We suggest it is worthwhile to try to develop a set of rules for manipulating arrows and labels which allow to represent standard elementary arguments in topology as calculations, both human readable and computer verifiable. Should this be possible, we suggest it is then worthwhile to think whether these rules lead to an approach to the tame topology of Grothendieck, i.e. a foundation of topology "without false problems" and "wild phenomena" "at the very beginning". It appears that our observations suggest it is worthwhile to try to develop the abstract theory of the Quillen lifting property and examples in specific categories including the categories of finite groups, topological spaces, and metric spaces.
Expressing the statement of the Feit-Thompson theorem with diagrams in the category of finite groups. same in djvu, We reformulate the statement of the Feit-Thompson theorem in terms of diagrams in the category of finite groups, namely iterations of the Quillen lifting property with respect to particular morphisms.
Elementary general topology as diagram chasing calculations with finite categories. same in djvu, We observe that elementary general topology arguments can be seen as diagram chasing computations with finite categories, via a straightforward, naive translation which often involves the Quillen lifting property playing the role of negation.
This paper is a draft of a research proposal: we argue it is worthwhile to to try to see whether these observations lead to another axiomatisation of topology and a proof system amenable to automatic theorem proving. Help in proofreading much appreciated.
An example of a lifting property, slides. We explain how an example of a proof in (Ganesalingam, Gowers, {\em A fully automatic problem solver with human-style output} is a sequence of applications of a single diagram chasing rule, the lifting property. Their example is: a closed subset of a complete metric space is complete.
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
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.
An updated version with new examples in group theory--abelian, perfect, p-groups, groups of order prime to p, soluble (in finite groups), and Feit-Thompson theorem reformulated in terms of lifting properties.
We give several examples of properties in point-set topology, such as being Hausdorff or that equalizier {x:f(x)=g(x)} is being closed for maps to a Hausdorff space, that translate to diagram chasing rules, and where the lifting property is a "category theory negation". We suggest this observation can be implemented in a problem solver of a kind suggested by M. Ganesalingam and T. Gowers. This is a preliminary publication; perhaps a little more like a research proposal that a proper paper.
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!)
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
The univalence axiom for posetal model categories. A shortened version. joint with Assaf Hasson and Itay Kaplan, 12 pages, a note where we observe that the set-theoretic model category constructed in "Exercises de style" delivers an example of the Univalence Axiom, albeit in a rather trivial way.
A little place to discuss QtNaamen (typos, corrections, questions...)
A description/summary of the preprints below:We construct a model category and use it 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 the model category QtNaamen, hoping it will help in finding more meaningful homotopy-theoretic intuitions in set theory.
- Yuri Manin gives a nice description of some background relevant to the works above:
3.4. Quillen’s homotopical algebra and univalent foundations project. In his influential book [Qu] Quillen developed the idea that the natural language for homotopy theory should appeal not to the initial intuition of continuous deformation itself, but rather to a codified list of properties of category of topological spaces stressing those that are relevant for studying homotopy.
Quillen defined a closed model category as a category endowed with three special classes of morphisms: fibrations, cofibrations, and weak equivalences. The list of axioms to which these three classes of morphisms must satisfy is not long but structurally quite sophisticated. They can be easily defined in the category of topological spaces using homotopy intuition but remarkably admit translation into many other situations. An interesting new preprint [GaHa] even suggests the definition of these classes in appropriate categories of discrete sets, contributing new insights to old Cantorian problems of the scale of infinities.
Closed model categories become in particular a language of preference for many contexts in which objects of study are quotients of “large” objects by “large” equivalence relations, such as homotopy.
It is thus only natural that the most recent Foundation/Superstructure, Vo- evodsky’s Univalent Foundations Project (cf. [Vo] and [Aw]) is based on direct axiomatization of the world of homotopy types.