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 , to conjecture, to theorem in , to counter-example in , 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  and then developed by alternative techniques in .
In  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 SpN(Θ⋅,ΣJ); we provide a new naive weak stable homotopy hypothesis.
|Commitee:||Beaudry, Agnes, Grochow, Joshua, Kearnes, Keith, Pflaum, Markus|
|School:||University of Colorado at Boulder|
|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|
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