Seminar on ∞-categories - information about this branch of the MSTR Elective Seminar (NMAG475), for the program Mathematical Structures.

Basic information

The seminar is currently suspended due to anti-covid restrictions.

The seminar is running as a branch of the MSTR Elective Seminar. In the summer semester 2019/2020, it is scheduled on Monday at 2.00-3.30pm in K9.

The main aim is to learn more about the phenomenon of ∞-categories; see e.g. Wikipedia, nLab or a short article in the Notices of the AMS. This is an extension of classical category theory which serves as a modern language and a theoretical framework for several aspects of homotopy theory and homological algebra. The concept has attracted a lot of interest in the last years, certainly also thanks to the tremendous work by Jacob Lurie, who was awarded the Breakthrough Prize in Mathematics in 2015 for this achievement.

An important aspect is that homomorphisms in ∞-categories do not form mere sets, but rather suitably encoded homotopy types of topological spaces. Another important fact is that the composition of morphisms is not defined uniquely, but only up to homotopy. This in fact places ∞-categories in the realm of higher categories. More precisely, ∞-categories realize the idea of (∞,1)-categories, i.e. categories which have homomorphisms of arbitrary dimensions, but all homomorphisms of dimension greater than one are invertible in a suitable sense.


A brief overview of what has been told in the seminar will be updated below.

October 11, 2018
Groupoids - definition and a fundamental groupoid of a topological space. A fundamental 2-groupoid of a space, strict 2-categories and bicategories. The concept of (n,r)-category, a summary of low-dimensional representatives and the purported place for ∞-groupoids and ∞-categories in this classification. Basics about simplicial sets.
October 18, 2018
Simplicial sets continued - the geometric realization functor (an elementary construction and as a left Kan extension). Kan complexes. The nerve of an ordinary category and the category of small categories as a full subcategory of simplicial sets. The definition of an ∞-category.
October 25, 2018
Strict n-groupoids and their homotopy groups.
November 1, 2018
Strict 3-groupoids are not sufficient to model the 3-type of the 2-dimensional sphere (after [Simp]).
November 9, 2018
The fundamental category functor τ: sSet → Cat and properties of the adjunction (τ, N). The essential image of the nerve functor via the Grothendieck-Segal conditions and the unique inner horn extensions. The homotopy category of an ∞-category (directly and via τ) and a glimpse of mapping spaces in ∞-categories.
November 22, 2018
Categories enriched over groupoids, categories enriched over (compactly generated Hausdorff) topological spaces, first steps towards simplicial categories (with the plan to explain the equivalence of two approaches to (∞,1)-categories: via ∞-categories and via simplicial categories with Dwyer-Kan equivalences).
November 29, 2018
Weak equivalences between topologically enriched categories. An explicit construction of the localization functor θ: CG → H, where CG is the category of compactly generated Hausdorff topological spaces and H is the homotopy category of CW-complexes.
December 6, 2018
Lax monoidal functors and change of the enrichment, examples for the monoidal functors θ: CG → H, Sing: CG → sSet and |-|: sSet → CG.
December 20, 2018
Comparison of CG-enriched and simplicially enriched categories and the fact that the adjunction (|-|, Sing) changes the corresponding enriched categories only up to weak equivalence. First steps to a comparison of simplicial categories to ∞-categories. A description of the "simplicially thickened version" C[Δn] ∈ CatsSet of the ordinal [n] ∈ Cat.
January 10, 2019
The adjunction between the rectification and simplicial nerve functors. The simplicial nerve of a category enriched over Kan complexes in an ∞-category. The topological nerve (of a topological category) and the topological rectification functors. The equivalence between the theories of CG-enriched categories and ∞-categories.
February 25, 2019
Basic constructions in ∞-categories: The opposite ∞-category and the mapping spaces (right, left and self-dual versions, as in [Lur1, Sec. 1.2.2]).
March 11, 2019
Equivalences in higher categories (2-categories, topological categories and ∞-categories), the fact that ∞-groupoids (= Kan complexes) are precisely the ∞-categories whose homotopy categories are groupoids, [Lur1, Sec. 1.2.3 - 1.2.5].
March 18, 2019
More on ∞-groupoids, homotopy coherent diagrams, functors and mapping spaces of ∞-categories, [Lur1, Sec. 1.2.5 - 1.2.7].
March 25, 2019
Joins of ∞-categoires, over- and undercategories of an object or a functor, faithful and essentially surjective functors, subcategories of ∞-categories, [Lur1, Sec. 1.2.8 - 1.2.11].
April 1, 2019
Subcategories of ∞-categories, initial and final objects, limits and colimits, [Lur1, Sec. 1.2.11 - 1.2.13].
May 13, 2019
Constructing model structures from exact cylinders and anodyne extensions, with the goal of constructing the Joyal model structure on simplicial sets, [Cis, Sec. 2.1 - 2.4].
October 7, 2019
Recollections on final objects, cones and (co)limits in ∞-categories, their existence and their characterization via the homotopy 2-category of ∞-categories (after [RV1]).
October 14, 2019
Crash course on ∞-categories for new members of the seminar.
October 21, 2019
The "initial" cocomplete ∞-category of spaces obtained by applying the simplicial nerve functor to the simplicial category of Kan complexes.
November 4, 2019
An overview of stable ∞-categories and stabilization (after [Har]).
November 11, 2019
Stabilization of locally presentable ∞-categories, excisive functors (after [Har]).
November 25, 2019
Left and right fibrations and the relation to functors of ∞-categories with values in spaces (after [BS]).
December 4, 2019
Cartesian and cocartesian fibrations and the relation to functors with values in ∞-categories (after [BS]).
February 24, 2020
Introduction to prestable ∞-categories and relation to t-structures (after [Lur3, Appendix C]).
March 2, 2020
More details on general prestable ∞-categories and the meaning of the defining properties, [Lur3, Appendix C, §C.1.2].
March 9, 2020
More details on general prestable ∞-categories and the meaning of the defining properties, [Lur3, Appendix C, §C.1.2].


The following sources are specifically devoted to the theory of ∞-categories (the list is alphabetically ordered):

[Cis] D.-C. Cisinski, Higher categories and homotopical algebra, a book project. [PDF]
[Gro] M. Groth, A short course on ∞-categories, a preprint. [arXiv]
[Hin] V. Hinich, Lectures on infinity categories, a preprint. [arXiv]
[Lur1] J. Lurie, Higher Topos Theory, Annals of Mathematics Studies 170, Princeton, 2009. [PDF]
[Lur2] J. Lurie, Higher Algebra, a book project. [PDF]
[Lur3] J. Lurie, Spectral Algebraic Geometry, a book project. [PDF]
[Rezk] C. Rezk, Stuff about quasi-categories, lecture notes (draft version). [PDF]

There is also a new on-line textbook project which is supposed to become a "Stacks project for homotopy coherent mathematics":

[Lur4] Kerodon, an online resource for homotopy-coherent mathematics. [link]

A crash course of stable ∞-categories, which frequently occur in algebra (derived categories) and homotopy theory (spectra as objects which represent (co)homological theories), can be found on the nLab:

[Har] Y. Harpaz, Introduction to stable ∞-categories, an article on nLab. [PDF]

A concise bestiary of various kinds of fibrations which appear in the theory of ∞-categories (and which generalize various classical notions of category theory, such as categories of elements or Grothendieck (op)fibrations) can be found in:

[BS] C. Barwick, J. Shah, Fibrations in ∞-category theory, 2016 MATRIX annals, 17–42, MATRIX Book Ser. 1, Springer, Cham, 2018. [arXiv]

A model independent approach (see [Berg] below for a discussion of models for (∞,1)-categories), which systematically employs the homotopy 2-category of ∞-categories and, thus, reduces a lot of work to this 2-category, is being developed by Riehl and Verity:

[RV1] E. Riehl, D. Verity, Infinity category theory from scratch, to appear in Higher Structures. [arXiv]
[RV2] E. Riehl, D. Verity, Elements of ∞-Category Theory, a book project. [PDF]

A crash course on ∞-categories can be as well found in Section 2 of the paper

[BSW] B. Jurčo, Ch. Sämann, M. Wolf, Higher Groupoid Bundles, Higher Spaces, and Self-Dual Tensor Field Equations, Fortschr. Phys. 64, 674-717 (2016). [arXiv]

An introduction to simplicial sets and an explanation of the geometric intuition behing them is given in:

[Fr] G. Friedman, An elementary illustrated introduction to simplicial sets, Rocky Mountain J. Math. 42 (2012), no. 2, 353-423. [arXiv]

As the intuition behind ∞-categories relies to a large extent on simplicial techniques in homotopy theory and also model categories are used in the expositions above, some textbooks on these topics are also included (alphabetically ordered):

[GZ] P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory, Springer-Verlag New York, Inc., New York, 1967.
[GJ] P. Goerss, J. F. Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser Verlag, 1999.
[Hov] M. Hovey, Model categories, Mathematical Surveys and Monographs 63, AMS, 1999.
[Hir] P. S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, AMS, 2003.
[May] J. P. May, A Concise Course in Algebraic Topology. Chicago Lectures in Mathematics, Chicago, 1999. [PDF]

The topological aspects of homotopy theory are usually discussed in the framework of compactly generated (weakly) Hausdorff spaces. A concise introduction to these can be found here:

[Str] N. P. Strickland, The category of CGWH spaces, a preprint. [PDF]

On of the reasons why we need the compactly generated Hausdorff spaces is that otherwise the geometric realization functor really might not preserve finite products. An example on that, going back to [Dow], can be also found on pages 524-525 of the Appendix of [Hat].

[Dow] C. H. Dowker, Topology of metric complexes, Amer. J. Math. 74, (1952). 555-577.
[Hat] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. [PDF]

Some references to the standard category theory may be also useful for the seminar (again alphabetically ordered).

[Bor] F. Borceux, Handbook of categorical algebra (3 volumes), Encyclopedia of Mathematics and its Applications 50-52, Cambridge, 1994.
[ML] S. Mac Lane, Categories for the working mathematician, Second edition. Graduate Texts in Mathematics 5, Springer-Verlag, 1998.

The ∞-category theory is compicated in that the composition is defined only up to homotopy, which is unique only up to a higher homotopy, and so on. The reason for this level of complication is explained by a result in [Simp], which among other says that the fundamental 3-groupoid of the 2-dimensional shpere cannot be equivalent to a strict 3-groupoid. Historically, this was a source of mistakes (notably in [KV]; see [Voe] for more information).

[KV] M. M. Kapranov, V. Voevodsky, ∞-groupoids and homotopy types, International Category Theory Meeting (Bangor, 1989 and Cambridge, 1990), Cahiers Topologie Géom. Différentielle Catég. 32 (1991), 29-46.
[Simp] C. Simpson, Homotopy types of strict 3-groupoids, a preprint. [arXiv]
[Voe] V. Voevodsky, The Origins and Motivations of Univalent Foundations, an article for the IAS, Princeton, 2014. [link]

The general (and somewhat vague) concept of (∞,1)-categories is put on a solid footing by the mathematically rigorous definition of an ∞-category. There are, however, other possibilities how to define (∞,1)-categories precisely. Various definitions of these kind lead to notions which are in some sense as good as ∞-categories in that they give the same "homotopy theory of homotopy theories". This is discussed in the following paper:

[Berg] J. Bergner, A survey of (∞,1)-categories, Towards higher categories, 69-83, IMA Vol. Math. Appl., 152, Springer, 2010. [arXiv]