Dissertation/Thesis Abstract

Spectra as Locally Finite Z-Groupoids
by Lessard, Paul Roy, Ph.D., University of Colorado at Boulder, 2019, 169; 22589944
Abstract (Summary)

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 Δ 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-category. There is a Cisinski model category structure on Θ 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 SpNJ); we provide a new naive weak stable homotopy hypothesis.

Indexing (document details)
Advisor: Wise, Jonathan
Commitee: Beaudry, Agnes, Grochow, Joshua, Kearnes, Keith, Pflaum, Markus
School: University of Colorado at Boulder
Department: Mathematics
School Location: United States -- Colorado
Source: DAI-A 81/3(E), Dissertation Abstracts International
Subjects: Mathematics, Logic, Computer science
Keywords: Higher category theory, Homotopy theory, Stable homotopy theory
Publication Number: 22589944
ISBN: 9781088374450
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