## Spectra as locally finite Z-groupoids (part 5)

### Paul Lessard – 25 March 2020

In this fifth talk in the series, we'll leverage:

-) our new intuition for Cisinski model categories; and
-) our study of Berger's wreath product of categories and its relationship to the Segal sketch for 1-categories;

to re-imagine a result from the heroic age of combinatorial topology.

We'll begin with the development of strict \Z and \Z-{\leq n} categories; we give a simple finite limit sketch definition for these abiding notions.

We'll then describe how the homotopification of those strict notions, gotten by way of our understanding of Cisinski's machine, fits into a grand scheme. More, in that grand scheme we will locate both Grothendieck's homotopy hypothesis and the Baez-Dolan Stabilization hypothesis.

Lastly, we'll describe, in some detail, a re-imagining of Kan's combinatorial description of spectra as a proof that spectra are equivalent to locally finite pointed \Z-groupoids. This result which we call the naive stable homotopy hypothesis, is an analogue of the weak form of the G.H.H.

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