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Preprint Number 245

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245. Misha Gavrilovich
A logical construction of a model category

Submission date: 3 May 2010.

Abstract:

(Notes of a talk given in Oxford, March 2010).
In 1967 Quillen introduced model categories to “cover in a uniform wa” “a large number of arguments [in the different homotopy theories encountered] that were formally similar to well-known ones in algebraic topology”. We show the same homotopy-theory of model categories formalism “covers in a uniform way” a number of arguments in (naive) set theory, e.g. it draws an analogy between a fibre bundle and an infinite union of an increasing chain. We argue that the formalism is curious as it suggests to look at a homotopy-invariant variant of Generalised Continuum Hypothesis which has less independence of ZFC, and first appeared in PCF theory independently but with a similar motivation.

Technically, we show how naive and straightforward homotopy theory style diagramme chasing leads to define a Quillen's model category of set-theoretic nature and show that some invariants in set theory, the covering numbers of PCF theory, are (minor variations) of left derived functors of cofibrantly replaced cardinality, in the Quillen's formalism of model categories. We suggest a homotopy-invariant version of Generalised Continuum Hypothesis, and some similarities between PCF's and homotopy theory's ideologies.

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