Misha Gabrilovich: Infinity and category theory

(Joint work with Assaf Hasson [HG].)

Take a computer scientist perspective on the notion of a model category, something fashionable some ten years ago. By definition, a model category is a category with three distinguished classes of morphisms satisfying diagram chasing axioms.

What is a category ? Well, it is a directed  =  oriented graph, with a class of distinguished subgraphs called commutative diagrams such that every string of consequitive arrows  →  → ...  → fits into a unique distinguished triangle subgraph

\nearrow_{{}_{\!\!\!---------------------------->}}^{\bullet---->\bullet-->....------>\bullet}\!\!\!\!\searrow

where the bottom arrow is called the composition of the string.

. "Three distinguished classes of morphisms" -- its edges = arrows are labelled by combinations of three labels (c), (w) and (f).

The axioms? Rules for diagram chasing: adding an arrow or a label to the part of graph you already constructed (as a computer scientist, you can not really have the whole of an infinite graph, can you?). For example, axiom M2 of model categories says that any arrow is equal to a composition

\bullet_1---(c)-->\bullet_2---(wf)--->\bullet_3

of a (c)-labelled arrow 

\bullet_1--(c)-->\bullet_2

and a (wf)-labelled arrow

\bullet_2--(wf)-->\bullet_3.

For a computer scientist, that's a diagram chasing rule: given an arrow

\bullet_1--->\bullet_3,

 in the model category graph, add the arrow

\bullet_1---(c)-->\bullet_2---(wf)--->\bullet_3

 and a commutative diagram of these arrows.

What a computer scientist would do now? Use these rules for a script generating a dungeon for a roguelijk: rooms are objects, doors are arrows/morphisms and labels, random seed is a labelled commutative diagram in a model category.

Keeping track of commutative diagrams is a mess -- the morphisms sets between even moderate objects are usually unlistable [Gr]. So s/he assumes all diagrams commute, in v0 of the script\dots

But are these rules consistent? Is there a model category where all the diagrams commute---in other words, is there a partial order admitting structure of a model category and say there is an arrow with no labels?

A set theorist quickly finds some: there is one for every regular cardinal \kappa including 0. As they say, a model category is a tool to prove results about its homotopy category, and let's describe the homotopy categories of such model categories.

Objects of Ht_0 are clasess of countable sets; for \kappa>0 objects of Ht_\kappa's are (arbitrary) classes of sets of size \kappa.

X\longrightarrow Y in Ht_0 iff every x\in X exists y\in Y such that |y \setminus x| is finite.

X\longrightarrow Y in Ht_\kappa iff for every x\in X exists x_S\subseteq Yof |x_S|<\kappa such that |x\setminus \bigcup x_S |<\kappa

A homotopy theorist tells that the ((cwf)-labelled) model category wrt its homotopy category  is akin to a category wrt its isomorphism classes of objects: for purposes of a good ("meaningful") question, isomorphic, resp. homotopic, objects are aways equivalent, but to prove anything you need to pick representatives and work in the category, resp. the category with its cwf-labels.  A typical task is to calculate some homotopy invariants, e.g. some properties of homotopy categories or of canonical functors from the homotopy categories.

functor L:Ht_\kappa \longrightarrow C is another word for an order preserving (or order reversing); it is canonical iff "it depends on the category itself only", says the homotopy theorist. S/he means that for any automorphism s:Ht_\kappa\longrightarrow Ht_\kappa of the partial order ("category"), F\circ s=F are "equivalent"; a logician (almost correctly) understands it to say "definable in the language of a category in an strong enough logic".

The easiest canonical functor to think of is perhaps

\Bbb L_c^\kappa {\rm card}(X)=\min \{|X|: X\longrightarrow Y \}

 cardinality  made homotopy invariant.

And then the revisited continuum hypothesis theorem of Shelah:

\Bbb L_c^\kappa {\rm card}(\{A: |A|=\kappa,\;\;A\subseteq \lambda\})=\lambda

for many \kappa<\lambda---is a homotopy invariant continuum hypothesis. The equality

\lambda^{\leq\kappa}=2^{\kappa}+\Bbb L_c^\kappa {\rm card}(\{A:|A|=\kappa,A\subseteq \lambda\})

shows this homotopy invariant helps in a classical question: the cardinality of the set of all subsets of size \kappa splits into the homotopy invariant part and non-homotopy invariant part. [Sh:460,p.4]

But a set theorist may ask a simpler question first: are homotopy categories Ht_\kappa dense (as partial orders)? Ht_0 is dense; Ht_\omega is not; Ht_\kappa is dense iff \kappa is not measurable.

Proof : I\leq_\kappa \kappa means that I is a <\kappa-closed ideal on \kappa; I is maximal such (in Ht_\kappa) iff there is nothing strictly <_\kappa-between I and \kappa. Finally, note that there is nothing between X\leq\kappa Y implies that for every y\in Y, there is nothing in between \{x\cap y:x\in X\}\leq_\kappa{ y} and X<_\kappa Y implies that at least one of these inequalities is strict.

To summarise: Amazingly, this language of a labelled category has "sufficient generality to cover in a uniform way the different homotopy theories encountered" if "properly, often non-obviously, developed" to express "a large number of arguments that [are] formally similar to well-known ones in algebraic topology", and that these "homotopy theories" are the (non-obvious) structure the homotopy category does inherit. [Qu] An optimist may hope that thereby the homotopy theory shall "contribute new insights to old Cantorian problems of the scale of infinities" [Ma], e.g. one indeed finds in set theory  "a large number of arguments that [are] formally similar to well-known ones in algebraic topology".

As always in mathematics, there's an interplay between "too much structure, too much information" and "too little information, too little to say".  A model category is a means of  forgetting some of the "too much information" while (hopefully) not getting  lost in the "too little to say". A key fact in  this construction is that it collapses the power function, probably, the greatest creator of chaos in set theory. But, maybe because this is the whole point of model categories, it also creates some replacement (in the form, e.g., (c)-labelled arrows --(c)--> and (wf)-arrows ---(wf)--> which turn out to be closely related to Shelah's covering families and numbers), which turn out to be of interest.

[HG] M. Gavrilovich and A. Hasson. Exercises de style: A homotopy theory for set theory.Part I and II.

[Gr] M. Gromov, Ergobrain. \S2.5 On Categories and Functors, p.69.

[Ma] Yu. Manin, Foundations as Superstructure. (Reflections of a practicing mathematician).

[Qu] D. Quillen, Homotopical Algebra. Available online.

[Sh:460] S. Shelah, The Generalized Continuum  Hypothesis revisited.