Spectra As Locally Finite Z-Groupoids

Lessard, Paul Roy

Graduate Thesis Or Dissertation

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Citeable URL: https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/st74cr650

Abstract

Since 1983, Grothendieck’s suggestion that:

“...the study of homotopical n-types should be essentially equivalent to the study of so-called n-groupoids...”

has gone from suggestion in [26], to conjecture, to theorem in [33], to counter-example in [46], and finally to abiding definition. Through a remarkable instance of Lakatos’, “method of proofs and refutations,” weak ω-groupoid, is now taken as synonymous with spaces by many.

As for analytic models of ω-groupoids perhaps the most intuitive, although certainly not the most widely known, is made possible by the category Θ. If 4 is the category of composition data for compositions of morphisms in a 1-category, then Θ is the category of composition data for compositions of morphisms in an ω-category¹ . There is a Cisinski model category structure on Θb equivalence to space, first constructed in [9] and then developed by alternative techniques in [19].

In [31] where the category Θ is first suggested, and indeed first defined as dual to the category of combinatorial disks, it is noted that the dimensional shift on Θ suggests an elegant presentation of the unreduced suspension on cellular sets. In this thesis we follow that thread.

We discover that stabilizing Θ at this dimensional shift provides a category on which may be written a sketch of an essentially algebraic theory for strict Z-categories. This natural notion is analogous to strict ω-categories but in place of the objects and N_≥1-sorts of morphisms of an ω-category, a Z-category has only Z-sorts of morphisms with every (z + 1)-morphism being a morphism between some z-morphisms.

Finally, we prove that the category of pointed, locally finite, weak Z-groupoids admit a model structure Quillen equivalent to the Hovey structure on Sp^N ( Θ^{^}•, Σ_J); we provide a new naive weak stable homotopy hypothesis.

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