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Keywords:
$\infty$-categories; adjunctions; limits; colimits; fibrations; modules; Kan extensions
$\infty$-categories; adjunctions; limits; colimits; fibrations; modules; Kan extensions
Summary:
These lecture notes were written to accompany a mini course given at the 2015 Young Topologists’ Meeting at École Polytechnique Fédérale de Lausanne, videos of which can be found at http://hessbellwald-lab.epfl.ch/ytm2015. We use the terms $\infty-categories$ and $\infty-functors$ to mean the objects and morphisms in an $\infty-cosmos$: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of “fibrant objects.” Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal spaces, $\theta_n$-spaces, and fibered versions of each of these are all $\infty$-categories in this sense. We show that the basic category theory of $\infty$-categories and $\infty$-functors can be developed from the axioms of an $\infty$-cosmos; indeed, most of the work is internal to a strict 2-category of $\infty$-categories, $\infty$-functors, and natural transformations. In the $\infty$-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 standard concepts. In the first lecture, we define an $\infty$-cosmos and introduce its {\it 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 $\infty$-categories. In the second lecture, we study limits and colimits of diagrams taking values in an $\infty$-category and relate these concepts to adjunctions between $\infty$-categories. In the third lecture, we define comma $\infty$-categories, which satisfy a particular weak 2-dimensional universal property in the homotopy 2-category. We illustrate the use of comma $\infty$-categories to encode the universal properties of (co)limits and adjointness. Because comma ∞-categories are preserved by all cosmological functors and created by certain cosmological biequivalences, these characterizations form the foundations for “model independence” results. In the fourth lecture, we introduce (co)cartesian fibrations, a certain class of $\infty$-functors, and also consider the special case with groupoidal fibers. We then describe the calculus of {\it modules} between $\infty$-categories — comma $\infty$-categories being the prototypical example — and use this framework to introduce the Yoneda lemma and develop the theory of pointwise Kan extensions of $\infty$-functors.
References:
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