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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:The model-independent theory of (&\;infin\;\,1)
-categories (4) - Emily Riehl (Johns Hopkins Univ
ersity)
DTSTART;TZID=Europe/London:20180705T100000
DTEND;TZID=Europe/London:20180705T110000
UID:TALK107773AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/107773
DESCRIPTION:Co-author: Dominic Verity (Macquarie Univers
ity)
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.
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