Spring 2022
Announcement
Hello everyone!
This term our seminar continues horsing around (this time in English). As was promised we shall cover higher categories and homotopical geometry. Note that although this is the second term of a two-term seminar, this term will be relatively independent of the first one.The syllabus looks something like this:
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§0 (Early February 2022) Monoidal model categories (based on chapter 4 in [Ho])
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§1 Homotopical categories
§1.1 (February–March 2022) Homotopy colimits revisited [Ri, Part I].
§1.1.1 Kan extensions.
§1.1.2 Derived functors.
§1.1.3 Enriched category theory.
§1.1.4 (Co-)Bar construction.
§1.1.5 Homotopy (co-)limits — new perspective.
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§1.2 (April 2022) Enriched homotopy theory [Ri, Part II].
§1.2.1 Weighted (co-)limits.
§1.2.2 Application to computations of homotopy (co-)limits.
§1.2.3 Weighted homotopy (co-)limits.
§1.2.4 Derived enrichment
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§1.3 (May 2022) Quasi-categories [Ri, Part IV], [Kerodon, Part I], [1, Part 3].
§1.3.1 Models for ∞-categories.
§1.3.2 Simplicial categories and homotopy coherence.
§1.3.3 Isomorphisms in quasi-categories.
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§2 Topos theory
§2.1 (February 2022) Categorical preliminaries [MM, Chapter I].
§2.1.1 Subobject classifiers.
§2.1.2 Subfunctors and Sieves.
§2.2.2 Heyting algebras.
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§2.2 (February 2022) Sheaves in the classical sense. Review [MM, Chapter II].
§2.3 (March–April 2022) Grothendieck topologies on model categories [MM, Chapter III], [TV], [Jo]
§2.3.1 Grothendieck topologies on a category. (Model) Sites.
§2.3.2 Sheaves of sets on a site.
§2.3.3 Prestacks on model sites
§2.3.4 Local model structure. Stacks on model sites.
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§2.4 (April–May 2022) Topoi [MM, Chapter IV], [TV], [Jo].
§2.4.1 Classical (lower) topoi.
§2.4.2 Geometric morphisms of topoi.
§2.4.3 Model topoi.
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References:
[Ci] Cisinski, Denis-Charles. Higher categories and homotopical algebra. Vol. 180.
Cambridge University Press, 2019.
[Ho] Hovey, Mark. Model categories. No. 63. American Mathematical Soc., 2007.
[MM] MacLane, Saunders, and Ieke Moerdijk. Sheaves in geometry and logic: A first
introduction to topos theory. Springer Science & Business Media, 2012.
[Ri] Riehl, Emily. Categorical homotopy theory. Vol. 24. Cambridge University Press,
2014.
[TV] Toën, Bertrand, and Gabriele Vezzosi. Homotopical algebraic geometry I: Topos
theory. Advances in mathematics 193.2 (2005): 257-372.
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Scientific advisor of the seminar prof. Dmitry Kaledin
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If you want to give a talk e-mail
Grisha Taroyan at tgv628@yahoo.com
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