(Joint work withÂ Assaf HassonÂ [HG].)

Take a computer scientist perspective on the notion of aÂ *modelÂ **category*, somethingÂ fashionable some ten years ago. By definition, aÂ model categoryÂ is a category with three distinguished classes ofÂ morphismsÂ satisfyingÂ diagram chasingÂ axioms.

What is a category ? Well, it is a directed Â = Â oriented graph, with aÂ class of distinguishedÂ subgraphs calledÂ *commutative diagrams *such that every string of consequitive arrowsÂ â†’ Â â†’ ... Â â†’ fits into a unique distinguished triangle subgraph

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

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

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

of a (c)-labelled arrowÂ

and a (wf)-labelledÂ arrow

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

Â in the model category graph, add the arrow

Â and a commutative diagram of these arrows.

What a computer scientist would do now? Use these rules for a script generatingÂ a dungeon for a roguelijk: rooms are objects, doors are arrows/morphisms and labels,Â 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 areÂ usually unlistable [Gr]. So s/he assumes all diagrams commute, in of the script

But are these rules consistent? Is there a model category where allÂ the diagrams commute---in otherÂ words, is there aÂ partial orderÂ admitting structure of a model categoryÂ and say there is an arrow with no labels?

A set theorist quickly finds some:Â there is one for every regular cardinal including .Â As they say, aÂ model category is a tool to prove resultsÂ about itsÂ homotopy category, and let's describe the homotopyÂ categories of such model categories.

Objects of are clasess of countable sets; forÂ objects of 's are (arbitrary) classes of sets of size .

in iff every exists such thatÂ is finite.

in iff for every exists of Â such that

A homotopy theorist tells that the ((cwf)-labelled) model category wrt its homotopy categoryÂ 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.Â A typical task is to calculate some homotopy invariants, e.g. some properties of homotopy categories or of canonical functorsÂ from the homotopy categories.

AÂ *functor*Â Â is another word for an order preserving (or order reversing); it isÂ *canonical*Â iff "it depends on the category itself only", saysÂ the homotopy theorist. S/he means that for any automorphism Â of the partial order ("category"), 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

Â cardinality Â madeÂ homotopy invariant.

And then the revisited continuum hypothesis theoremÂ of Shelah:

for many ---isÂ a homotopy invariant continuum hypothesis. The equality

shows this homotopy invariant helps in a classical question: the cardinality of the set of all subsets of size 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 Â dense (as partial orders)? is dense; is not;Â is dense iff is not measurable.

Proof : means that I is a -closed ideal on ; is maximal such (in ) iffÂ there is nothing strictly -between and . Finally, note that there is nothing betweenÂ implies that for every , there is nothing in between and Â implies that at least one of these inequalities is strict.

**To summarise:**Â *Amazingly*, this language of a labelled category has "sufficient generality to coverÂ in a uniform way the different homotopy theories encountered" if "properly, oftenÂ non-obviously, developed" to express "a large number of arguments that [are] formally similar to well-known ones in algebraic topology", and that these "homotopyÂ 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 ofÂ infinities" [Ma], e.g. one indeed finds in set theoryÂ "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".Â A model category is a means ofÂ forgetting some of the "too much information" while (hopefully) not gettingÂ lost in the "too little to say". A key fact inÂ this construction is that it collapses the power function, probably, the greatest creator of chaos in set theory.Â But, maybe because this is the whole point of model categories, it also creates some replacement (in the form, e.g., (c)-labelled arrows and (wf)-arrows 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 IÂ andÂ II.

[Gr] M. Gromov,Â Ergobrain. Â On Categories and Functors, p.69.

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

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

[Sh:460] S. Shelah,Â TheÂ Generalized Continuum Â HypothesisÂ revisited.