Playing with the lifting property, we observed that the lifting property and other categorical constructions from algebraic topology has the power to express concisely and uniformly textbook definitions across disparate domains, including topology, analysis, group theory, model theory, in terms of simple(st?) or archetypal? (counter)examples, and that an apparently straightforward attempt to read line by line the text of these definitions and rephrase it in categorical language leads to this observation. We were startled that this never appeared in print, and our presentations attempt to write up some of it.
Examples in topology include the notions of: compact, contractible, discrete, connected, and extremally disconnected spaces, dense image, induced topology, subset, closed or open subsets, quotient, Lebesgue dimension, and separation axioms. Examples in algebra include: finite groups being nilpotent, solvable, p-groups, and prime-to-p groups; injective and projective modules; injective, surjective, and split homomorphisms. Each of these can be defined by iteratively applying the lifting property (i.e. taking either left or right orthogonal in a category) from simple and concrete examples of maps. Moreover, in topology such an example often is a map of finite spaces which is the simplest (counter)example to the property being defined, and this leads to a concise notation for basic topological notions in terms of maps of finite preorders (=finite topological spaces).
How these examples can be used in teaching, is discussed in the slides by M.Gavrilovich
Slightly more complicated examples in topology---compact, contractible, and trivial fibration---are discussed in the slides by K.Pimenov
Rephrasing in simplicial language the definitions of topological and uniform spaces, in (Bourbaki, General Topology) led us to define a category of generalised topological spaces flexible enough to formulate categorically a number of standard basic elementary definitions in various fields, e.g. in analysis, limit, (uniform) continuity and convergence, equicontinuity of sequences of functions; in algebraic topology, being locally trivial and geometric realisation; in geometry, quasi-isomorphism; in model theory, stability and simplicity and several Shelah’s dividing lines, e.g. NIP, NOP, NSOP, NSOPi, NTP, NTPi,NATP, NFCP, of a theory.
The poster by M.Gavrilovich reads line by line the definition of NTP (no tree property) and recovers its reformulation as a lifting property in the category of generalised topological spaces.
We'd like to have similar formulae defining
a model structure (notions of fibrations, cofibrations, and homotopy equivalence fitting well together)
spheres Sn, disks Dn, and other spaces of interest (via something like Bing and Kline sphere characterisation theorems)
soft maps, and other clasess of maps arising in Michael continuous selection theory.
and, in general, a proof system or a computer algebra system manipulating formulae defining topological properties via finite preorders.
Elena Volk, Konstantin Pimenov, M.Gavrilovich. Russian trace in family history and work of Alexandre Grothendieck.
Journal of Mathematical Sciences 252(2), 2021.
M.Gavrilovich, K.Pimenov. A suggestion towards a finitist's realisation of the intuition of topology.
A category of generalised topological spaces.