Research Scientist

Department of Mathematics

MIT

Email: dspivak--math/mit/edu

### Our research:

The following joint research with
Professor Dan Dugger was undertaken while I was a postdoc at the
University of Oregon (2007-2010).

Rigidification of
quasi-categories. Here we prove that the mapping space between two objects in the rigidification of a
quasi-category
is isomorphic to the nerve of the category of necklaces connecting those
two objects. On the arXiv.

Mapping spaces in
quasi-categories. Here we prove that the mapping spaces in a quasi-category are as one who is familiar
with the homotopy mapping complexes of Dwyer-Kan might expect. We prove that various formulations of these
mapping spaces found in Lurie's book
HTT can all be obtained in this way. Finally we give a self-contained proof of Lurie's result that the
Joyal model structure on simplicial sets is Quillen equivalent to
Bergner's model structure on simplicial categories. On the arXiv.

### Some related research:

Jacob Lurie Higher Topos
Theory.

Emily Riehl On the structure of simplicial
categories associated to quasi-categories.

Josh Nichols-Barrer Combinatorial Quasi-Categories
(Thesis Chapters 1 and 2).

This work by David I. Spivak is licensed under a Creative Commons
Attribution-Share Alike 3.0 Unported License.