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The modelindependent theory of (∞,1)categories (4)
http://sms.cam.ac.uk/media/2783276
Riehl, E
Thursday 5th July 2018  10:00 to 11:00
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The modelindependent theory of (∞,1)categories (4)
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The modelindependent theory of (∞,1)categories (4)
ucs_sms_2781281_2783276
http://sms.cam.ac.uk/media/2783276
The modelindependent theory of (∞,1)categories (4)
Riehl, E
Thursday 5th July 2018  10:00 to 11:00
Fri, 06 Jul 2018 14:16:30 +0100
Isaac Newton Institute
Riehl, E
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Riehl, E
Thursday 5th July 2018  10:00 to 11:00
Riehl, E
Thursday 5th July 2018  10:00 to 11:00
Cambridge University
4020
http://sms.cam.ac.uk/media/2783276
The modelindependent theory of (∞,1)categories (4)
Riehl, E
Thursday 5th July 2018  10:00 to 11:00
Coauthor: Dominic Verity (Macquarie University)
In these talks we use the nickname "∞category" to refer to either a quasicategory, a complete Segal space, a Segal category, or 1complicial set (aka a naturally marked quasicategory)  these terms referring to Quillen equivalent models of (∞,1)categories, these being weak infinitedimensional categories with all morphisms above dimension 1 weakly invertible. Each of these models has accompanying notions of ∞functor, and ∞natural transformation and these assemble into a strict 2category like that of (strict 1)categories, functors, and natural transformations.
In the first talk, we'll use standard 2categorical techniques to define adjunctions and equivalences between ∞categories and limits and colimits inside an ∞category and prove that these notions relate in the expected ways: eg that right adjoints preserve limits. All of this is done in the aforementioned 2category of ∞categories, ∞functors, and ∞natural transformations. In the 2category of quasicategories our definitions recover the standard ones of Joyal/Lurie though they are given here in a "synthetic" rather than their usual "analytic" form.
In the second talk, we'll justify the framework introduced in the first talk by giving an explicit construction of these 2categories. 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 ∞cosmos.
In the third talk, we'll encode the universal properties of adjunction and of limits and colimits as equivalences of comma ∞categories. We also introduce co/cartesian fibrations in both onesided and twosided variants, the latter of which are used to define "modules" between ∞categories, of which comma ∞categories are the prototypical example.
In the fourth talk, we'll prove that theory being developed isn’t just "modelagnostic” (in the sense of applying equally to the four models mentioned above) but invariant under changeofmodel functors. As we explain, it follows that even the "analyticallyproven" theorems that exploit the combinatorics of one particular model remain valid in the other biequivalent models.
Related Links
http://www.math.jhu.edu/~eriehl/scratch.pdf  lecture notes from a similar series of four talks delivered at EPFL
http://www.math.jhu.edu/~eriehl/elements.pdf  book in progress on the subject of these lectures
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