Spectra as locally finite Z-groupoids

Paul Lessard – 2 October 2019

At least since Baez-Dolan '95, we've understood that any correct definition for the notion of symmetric monoidal n-category is equivalent to the one which defines symmetric monoidal n-categories as infinite loop categories. We will synthesize this position with a purely combinatorial treatment of spectra found in Kan '63. More explicitly, we will extend the proof that the theta categories of Joyal, together with the sets of (higher) spine inclusions provide essentially algebraic presentations of strict categories to provide a definition of the abiding notion of strict Z-categories. We'll then describe a natural subcategory of the category of pointed presheaves on this essentially algebraic theory for Z-categories we'll call the locally finite subcategory and we'll then use Kan's observation to put a model structure on this subcategory which: -) corresponds intuitively to a Z-graded groupoidal composition law; and -) presents the category of spectra.