In these talks we use the nickn ame "&infin\;-category" to refer to either a quasi -category\, a complete Segal space\, a Segal categ ory\, or 1-complicial set (aka a naturally marked quasi-category) - these terms referring to Quillen equivalent models of (&infin\;\,1)-categories\, t hese being weak infinite-dimensional categories wi th all morphisms above dimension 1 weakly invertib le. Each of these models has accompanying notions of &infin\;-functor\, and &infin\;-natural transfo rmation and these assemble into a strict 2-categor y like that of (strict 1-)categories\, functors\, and natural transformations.

In the first t alk\, we'\;ll use standard 2-categorical techni ques to define adjunctions and equivalences betwee n &infin\;-categories and limits and colimits insi de an &infin\;-category and prove that these notio ns relate in the expected ways: eg that right adjo ints preserve limits. All of this is done in the a forementioned 2-category of &infin\;-categories\, &infin\;-functors\, and &infin\;-natural transform ations. In the 2-category of quasi-categories our definitions recover the standard ones of Joyal/Lur ie though they are given here in a "synthetic" rat her than their usual "analytic" form.

In th e second talk\, we'\;ll justify the framework i ntroduced in the first talk by giving an explicit construction of these 2-categories. This makes use of an axiomatization of the properties common to the Joyal\, Rezk\, Bergner/Pellissier\, and Verity /Lurie model structures as something we call an &i nfin\;-cosmos.

In the third talk\, we'\; ll encode the universal properties of adjunction a nd of limits and colimits as equivalences of comma &infin\;-categories. We also introduce co/cartesi an fibrations in both one-sided and two-sided vari ants\, the latter of which are used to define "mod ules" between &infin\;-categories\, of which comma &infin\;-categories are the prototypical example.

In the fourth talk\, we'\;ll prove that theory being developed isn&rsquo\;t just "model-a gnostic&rdquo\; (in the sense of applying equally to the four models mentioned above) but invariant under change-of-model functors. As we explain\, it follows that even the "analytically-proven" theor ems that exploit the combinatorics of one particul ar model remain valid in the other biequivalent mo dels.

Related Links

- http://www.math.jhu.edu/~eriehl/ scratch.pdf \;- lecture notes from a simil ar series of four talks delivered at EPFL
- http://www.math. jhu.edu/~eriehl/elements.pdf \;- book in p rogress on the subject of these lectures

LOCATION:Seminar Room 1\, Newton Institute CONTACT:info@newton.ac.uk END:VEVENT END:VCALENDAR