## Spectra as locally finite Z-groupoids (part 3)

### Paul Lessard – 19 February 2020

In this third talk in the series, we'll begin by pulling at a loose end from the previous talk; we'll explore the interpretation of fibrant objects for an accessible Cisinski localizer as weak/homotopy models for a sketch.

We'll see how Cisinski's Lambda^{\infty} construction informs us that fibrant objects are homotopy coherent models in the sense that:

-) they interpret all partial operations in the regulus;
-) any functional relation is homotopy/substitution invariant; and
-) witnesses to these facts are essentially unique/coherent.

What's more, Ara has proved that in many cases of interest, this notion of homotopy coherent models coincides with the natural generalization of (complete) Segal spaces to an arbitrary sketch.

We'll then describe the theory of strict Z-categories and strict Z_{\leq n}-categories, and lay out how weak versions, defined by way of the earlier discussion, fit into a grander scheme synthesizing Grothendieck's homotopy hypothesis and the Baez-Dolan stabilization hypothesis.

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