You may depend upon my bare word, reader, without any further security, that I could wish this offspring of my brain were as ingenious, sprightly, and accomplished as yourself could desire; but the mischief of it is, nature will have to take its course: every production must resemble its author, and my barren and unpolished understanding can produce nothing but what is very dull, very impertinent, and extravagant beyond imagination.You may suppose it the child of disturbance, engendered in some dismal prison, where wretchedness keeps its residence, and every dismal sound its habitation. Some parents are so blinded by a fatherly fondness, that they mistake the very imperfections of their children for so many beauties; and the folly and impertinence of the brave boy must pass upon their friends and acquaintance for wit and sense. But I, who am only a stepfather, disavow the authority of this modern and prevalent custom; nor will I earnestly beseech you, with tears in my eyes, which is many a poor author's case, dear reader, to pardon or dissemble my child's faults; for what favour can I expect from you, who are neither his friend nor relation?
You have a soul of your own, and the privilege of free-will, whoever you be, as well as the proudest he that struts in a gaudy outside; you are a king by your own fireside, as much as any monarch on his throne; you have liberty and property, which set you above favour or affection, and may therefore freely like or dislike this history, according to your humour.
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.
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. To be updated in a week.
Drafts, earlier versions:
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.
Old:
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.