| Title:
|
Infinity category theory from scratch (English) |
| Author:
|
Riehl, Emily |
| Author:
|
Verity, Dominic |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
4 |
| Issue:
|
1 |
| Year:
|
2020 |
| Pages:
|
115-167 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
These lecture notes were written to accompany a mini course given at the 2015 Young Topologists’ Meeting at École Polytechnique Fédérale de Lausanne, videos of which can be found at http://hessbellwald-lab.epfl.ch/ytm2015. We use the terms $\infty-categories$ and $\infty-functors$ to mean the objects and morphisms in an $\infty-cosmos$: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of “fibrant objects.” Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal spaces, $\theta_n$-spaces, and fibered versions of each of these are all $\infty$-categories in this sense. We show that the basic category theory of $\infty$-categories and $\infty$-functors can be developed from the axioms of an $\infty$-cosmos; indeed, most of the work is internal to a strict 2-category of $\infty$-categories, $\infty$-functors, and natural transformations. In the $\infty$-cosmos of quasi-categories, we recapture precisely the same category theory developed by Joyal and Lurie, although in most cases our definitions, which are 2-categorical rather than combinatorial in nature, present a new incarnation of the standard concepts. In the first lecture, we define an $\infty$-cosmos and introduce its {\it homotopy 2-category}, the strict 2-category mentioned above. We illustrate the use of formal category theory to develop the basic theory of equivalences of and adjunctions between $\infty$-categories. In the second lecture, we study limits and colimits of diagrams taking values in an $\infty$-category and relate these concepts to adjunctions between $\infty$-categories. In the third lecture, we define comma $\infty$-categories, which satisfy a particular weak 2-dimensional universal property in the homotopy 2-category. We illustrate the use of comma $\infty$-categories to encode the universal properties of (co)limits and adjointness. Because comma ∞-categories are preserved by all cosmological functors and created by certain cosmological biequivalences, these characterizations form the foundations for “model independence” results. In the fourth lecture, we introduce (co)cartesian fibrations, a certain class of $\infty$-functors, and also consider the special case with groupoidal fibers. We then describe the calculus of {\it modules} between $\infty$-categories — comma $\infty$-categories being the prototypical example — and use this framework to introduce the Yoneda lemma and develop the theory of pointwise Kan extensions of $\infty$-functors. (English) |
| Keyword:
|
$\infty$-categories |
| Keyword:
|
adjunctions |
| Keyword:
|
limits |
| Keyword:
|
colimits |
| Keyword:
|
fibrations |
| Keyword:
|
modules |
| Keyword:
|
Kan extensions |
| MSC:
|
18G55 |
| MSC:
|
55U35 |
| idZBL:
|
Zbl 1451.18042 |
| idMR:
|
MR4074275 |
| DOI:
|
10.21136/HS.2020.04 |
| . |
| Date available:
|
2026-03-11T20:13:32Z |
| Last updated:
|
2026-03-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153419 |
| . |
| Reference:
|
[1] Barwick, Clark, Schommer-Pries, Christopher: On the unicity of the homotopy theory of higher categories.arxiv:1112.0040 http://arxiv.org/pdf/1112.0040 |
| Reference:
|
[2] Cruttwell, G. S. H., Shulman, Michael A.: A unified framework for generalized multicategories.Theory Appl. Categ. 24, no. 21, 580–655 |
| Reference:
|
[3] Heller, A.: Homotopy theories.Memoirs of the American Mathematical Society 71, no. 383 |
| Reference:
|
[4] Joyal, A.: Quasi-categories and Kan complexes.Journal of Pure and Applied Algebra 175, 207–222 10.1016/S0022-4049(02)00135-4 |
| Reference:
|
[5] Joyal, A., Tierney, M.: Quasi-categories vs Segal spaces.Categories in Algebra, Geometry and Mathematical Physics (StreetFest) (A. Davydov et al, ed.), Contemporary Mathematics, vol. 431, American Mathematical Society, pp. 277–325 |
| Reference:
|
[6] Lurie, J.: Higher topos theory.volume 170 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ |
| Reference:
|
[7] Riehl, E., Verity, D.: The 2-category theory of quasi-categories.Adv. Math. 280, 549–642, arxiv:1306.5144 http://arxiv.org/pdf/1306.5144 |
| Reference:
|
[8] Riehl, E., Verity, D.: Homotopy coherent adjunctions and the formal theory of monads.Adv. Math 286, 802–888. arxiv:1310.8279 http://arxiv.org/pdf/1310.8279 |
| Reference:
|
[9] Riehl, E., Verity, D.: Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions.Homol. Homotopy Appl. 17, no. 1, 1–33. arxiv:1401.6247 http://arxiv.org/pdf/1401.6247 |
| Reference:
|
[10] Riehl, E., Verity, D.: Fibrations and Yoneda’s lemma in an ∞-cosmos.J. Pure Appl. Algebra 221, no. 3, 499–564, arxiv:1506.05500 http://arxiv.org/pdf/1506.05500 |
| Reference:
|
[11] Riehl, E., Verity, D.: Kan extensions and the calculus of modules for ∞-categories.Algebr. Geom. Topol. 17, no. 1, 189–271, arxiv:1507.01460 http://arxiv.org/pdf/1507.01460 |
| Reference:
|
[12] Toën, Bertrand: Vers une axiomatisation de la théorie des catégories supérieures.K-Theory 34, no. 3, 233–263 |
| Reference:
|
[13] Wood, Richard J.: Abstract proarrows I..Cahiers de Topologie et Géom. Diff. XXIII-3, 279–290 |
| . |
