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YTM 2015: Infinity category theory from scratch, 1st lecture

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Emily Riehl, Harvard University july 07, 2015 1h5mn32s
We use the terms infinity-categories and infinity-functors to mean the objects and morphisms in an infinity-cosmos: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of "fibrant objects." Quasi-categories, Segal categories, complete Segal spaces, iterated complete Segal spaces, and fibered versions of each of these are all all infinity-categories in this sense. In joint work with Dominic Verity, we show that the basic category theory of infinity-categories and infinity-functors can be developed only in reference to the axioms of an infinity-cosmos; indeed, most of the work is internal to a strict 2-category of infinity-categories, infinity-functors, and natural transformations. In the infinity-cosmos of quasi-categories, we recapture precisely the same category theory developed by Joyal and Lurie, although in most cases our definitions, which are 2-categorical rather than combinatorial in nature, present a new incarnation of the classical concept. In the first lecture, we define an infinity-cosmos and introduce its homotopy 2-category, the strict 2-category mentioned above. We illustrate the use of formal category theory to develop the basic theory of equivalences of and adjunctions between infinity-categories. Lecture notes available at http://www.math.harvard.edu/~eriehl/
From:Nicolas Peslerbe
Event type:Congress&Conferences
Video type:iPod Mpeg4 video
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Video number:2803
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