## Spectra as locally finite Z-groupoids (part 2)

### Paul Lessard – 6 November 2019

In the previous talk we provided a brief development of the objects of stable homotopy theory. In particular, we began with the Freudenthal suspension theorem and argued that homotopy theoretic phenomena split naturally into a low dimensional and a dimension invariant part. We then introduced two Quillen equivalent models for the objects of this dimension invariant part, spectrum objects and Kan's combinatorial spectra.
In this talk, we'll:
-) develop a just so story for the now disproved Cisinski-Joyal conjecture for (\infty,n)-categories
-) attempt to convince the audience that the reason the conjecture does not hold is that it was premised on specifically low dimensional intuition.
-) we'll then explain a grand scheme, inspired by Kan's model, in which higher category theory may be split into a low dimensional and a dimension invariant part; we will introduce a definition for the abiding notion of Z-categories
-) lastly, we'll show that in this grand scheme, spectra are Quillen equivalent to locally finite Z-groupoids and we'll make some few conjecture about future directions.

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