# Lecture notes, videos and other learning materials about $\infty$-category theory

I've heard several times (and realized myself) that Lurie's tomes (extraordinary as they are) are not so ideal for self study.

I think it's a good idea to have some kind of compiled list of learning material about infinity categories (could be articles as well if they are considered to have a pedagogical style). Preferably they would have emphasis on intuition, and application rather than formality and rigor.

Lecture notes, Videos and any other relevant form of studying material would help a lot in travelling this technical terrain.

• Charles Rezk is currently writing his "Stuff about quasicategories" ( math.uiuc.edu/~rezk/595-fal16/quasicats.pdf ). Jan 5, 2017 at 19:37
• Re "videos and other learning materials": Gunnar Carlsson's nice introductory lecture on homotopy limits and homotopy colimits should be in this thread, because homotopy limits are an important tool in contemporary $\infty$-category theory. A basic thing to mention: if $\mathfrak{C}$ is a higher category, then the homotopy limit of a diagram in $\mathfrak{C}$ seems not to be the same as the limit of the same diagram taken w.r.t. the homotopy category of $\mathfrak{C}$. (The 'usually' I cannot precisely justify.) Aug 23, 2017 at 11:45
• @Peter Heinig: The justification is that diagrams in the homotopy category often do not even have limits! Aug 6, 2021 at 11:11

I found A Whirlwind Tour of the World of (∞,1)-categories by Omar Antolín Camarena (a student of Jacob Lurie) to be quite insightful for quasicategories.

Here are some extra references. (Maybe this is easier to read: link)

├── Complete Segal Spaces
│   ├── [BC, 10 Pages] Equivariant Complete Segal Spaces.pdf
│   ├── [Brito, 26 Pages] Segal Objects and the Grothendieck Construction.pdf
│   ├── [Osborne, 25 Pages] Decomposing the Classifying Diagram in Terms of Classifying Spaces of Groups.pdf
│   ├── [Rasekh, 17 Pages] A Model for the Higher Category of Higher Categories.pdf
│   ├── [Rasekh, 44 Pages] Complete Segal Objects.pdf
│   ├── [Rasekh, 478 Pages] A Theory of Elementary Higher Toposes.pdf
│   ├── [Rasekh, 61 Pages] Introduction to Complete Segal Spaces.pdf
│   ├── [Rasekh, 75 Pages] Yoneda Lemma for Simplicial Spaces.pdf
│   ├── [Rasekh, 81 Pages] Cartesian Fibrations and Representability.pdf
│   ├── [Stenzel, 21 Pages] Univalence and Completeness of Segal Objects.pdf
│   └── [Stenzel, 26 Pages] Bousfield-Segal Spaces.pdf
├── Elementary ∞-Topoi
│   ├── [Rasekh, 10 Pages] Yoneda Lemma for Elementary Higher Toposes.pdf
│   ├── [Rasekh, 30 Pages] Filter Quotients and Non-Presentable (∞,1)-Toposes.pdf
│   ├── [Rasekh, 40 Pages] A Theory of Elementary Higher Toposes.pdf
│   ├── [Rasekh, 51 Pages] Every Elementary Higher Topos Has a Natural Number Object.pdf
│   └── [Rasekh, 84 Pages] An Elementary Approach to Truncations.pdf
├── Enriched (∞,1)-Categories
│   ├── [AMR, 68 Pages] Factorization Homology of Enriched ∞-Categories.pdf
│   ├── [GH, 100 Pages] Enriched ∞-Categories via Non-Symmetric ∞-Operads.pdf
│   ├── [Haugseng, 29 Pages] Bimodules and Natural Transformations for Enriched ∞-Categories.pdf
│   └── [Haugseng, 52 Pages] Rectification of Enriched Infinity-Categories.pdf
├── General References
│   ├── [AL, 26 Pages] Exponentiable Higher Toposes.pdf
│   ├── [Arctaedius, 38 Pages] Grothendieck's Homotopy Hypothesis and the Homotopy Theory of Homotopy Theories.pdf
│   ├── [Bergner, 13 Pages] A Survey of (∞, 1)-Categories.pdf
│   ├── [Bergner, 287 Pages] The Homotopy Theory of (∞,1)-Categories.pdf
│   ├── [Bergner, 29 Pages] A Survey of Models for (∞,n)-Categories.pdf
│   ├── [Bergner, 39 Pages] Workshop on the Homotopy Theory of Homotopy Theories.pdf
│   ├── [Camarena, 45 Pages] A Whirlwind Tour of the World of (∞,1)-Categories.pdf
│   ├── [Clough, 35 Pages] An Outline of the Theory of (∞,1)-Categories.pdf
│   ├── [Dorn, 99 Pages] Basic concepts in homotopy theory.pdf
│   ├── [HF, 148 Pages] The Homotopy Theory of (∞,1)-Categories.pdf
│   ├── [Mazel-Gee, 26 Pages] The Zen of ∞-Categories.pdf
│   ├── [Porter, 37 Pages] 𝒮-categories, 𝒮-groupoids, Segal categories and quasicategories.pdf
│   ├── [Porter, 54 Pages] Spaces as ∞-groupoids.pdf
│   ├── [Porter, 759 Pages] The Crossed Menagerie.pdf
│   ├── [Schommer-Pries, 65 Pages] Dualizability in Low-Dimensional Higher Category Theory.pdf
│   └── [Simpson, 653 Pages] Homotopy Theory of Higher Categories.pdf
├── Model (∞,1)-Categories
│   ├── [LM, 21 Pages] From Fractions to Complete Segal Spaces.pdf
│   ├── [Mazel-Gee, 16 Pages] A User's Guide to Co⧸Cartesian Fibrations.pdf
│   ├── [Mazel-Gee, 26 Pages] The Universality of the Rezk Nerve.pdf
│   ├── [Mazel-Gee, 29 Pages] Model ∞-Categories II: Quillen Adjunctions.pdf
│   ├── [Mazel-Gee, 34 Pages] Model ∞-Categories III: The Fundamental Theorem.pdf
│   ├── [Mazel-Gee, 41 Pages] All About the Grothendieck Construction.pdf
│   ├── [Mazel-Gee, 43 Pages] Hammocks and Fractions in Relative ∞-Categories.pdf
│   ├── [Mazel-Gee, 528 Slides] Goerss–Hopkins obstruction theory for ∞-Categories.pdf
│   ├── [Mazel-Gee, 545 Pages] Goerss–Hopkins obstruction theory via model ∞-categories.pdf
│   ├── [Mazel-Gee, 54 Pages] Goerss–Hopkins Obstruction Theory for ∞-Categories.pdf
│   ├── [Mazel-Gee, 66 Pages] Model ∞-Categories I: Some Pleasant Properties of the ∞-Category of Simplicial Spaces.pdf
│   └── [Mazel-Gee, 6 Pages] 𝔼_∞ Automorphisms of Motivic Morava E-Theories.pdf
├── Other Models
│   ├── Batanin ∞-Categories
│   │   └── [Ara, 168 Pages] Sur les ∞-groupoïdes de Grothendieck et une variante ∞-catégorique.pdf
│   ├── Grothendieck–Maltsiniotis ∞-Categories
│   │   ├── [AL, 65 Pages] The Folk Model Category Structure on Strict ω-Categories Is Monoidal.pdf
│   │   ├── [AM, 123 Pages] A Quillen's Theorem A for Strict ∞-Categories II: The ∞-Categorical Proof.pdf
│   │   ├── [AM, 227 Pages] Join and Slices for Strict ∞-Categories.pdf
│   │   ├── [AM, 25 Pages] The Brown–Golasinski Model Structure on Strict ∞-Groupoids Revisited.pdf
│   │   ├── [AM, 42 Pages] Comparison of the n-Categorical Nerves.pdf
│   │   ├── [AM, 51 Pages] A Quillen's Theorem A for Strict ∞-Categories I: The Simplicial Proof.pdf
│   │   ├── [AM, 68 Pages] The Homotopy Type of the ∞-Category Associated to a Simplicial Complex.pdf
│   │   ├── [AM, 92 Pages] Towards a Thomason Model Structure on the Category of Strict n-Categories.pdf
│   │   ├── [Ara, 22 Pages] On Homotopy Types Modelized by Strict ∞-Groupoids.pdf
│   │   ├── [Ara, 22 Pages] Strict ∞-Groupoids Are Grothendieck ∞-Groupoids.pdf
│   │   ├── [Ara, 33 Pages] A Quillen Theorem B for Strict ∞-Categories.pdf
│   │   ├── [Ara, 41 Pages] The Groupoidal Analogue Θ to Joyal's Category Θ Is a Test Category.pdf
│   │   └── [Ara, 58 Pages] On the Homotopy Theory of Grothendieck ∞-Groupoids.pdf
│   ├── Miscellany
│   │   ├── [CL, 61 Pages] Weak ∞-Categories via Terminal Coalgebras.pdf
│   │   ├── [Harpaz, 69 Pages] Quasi-Unital ∞-Categories.pdf
│   │   ├── [LM, 68 Pages] Linear Quasi-Categories as Templicial Modules.pdf
│   │   └── [Nikolaus, 26 Pages] Algebraic Models for Higher Categories.pdf
│   ├── Relative Categories
│   │   ├── [BK, 19 Pages] n-Relative Categories.pdf
│   │   ├── [BK, 27 Pages] Relative Categories: Another Model for the Homotopy Theory of Homotopy Theories.pdf
│   │   ├── [BK, 3 Pages] A Thomason-Like Quillen Equivalence Between Quasi-Categories and Relative Categories.pdf
│   │   ├── [BK, 5 Pages] In the Category of Relative Categories the Rezk Equivalences Are Exactly the DK-equivalences.pdf
│   │   └── [Meier, 21 Pages] Fibration Categories Are Fibrant Relative Categories.pdf
│   └── Topological Categories
│       ├── [Amrani, 22 Pages] A Model Structure on the Category of Topological Categories.pdf
├── Parametrised (∞,1)-Categories
│   ├── [BDGNS, 11 Pages] Parametrized Higher Category Theory and Higher Algebra: A General Introduction.pdf
│   ├── [BDGNS, 23 Pages] Parametrized Higher Category Theory and Higher Algebra: Exposé I -- Elements of Parametrized Higher Category Theory.pdf
│   ├── [Nardin, 21 Pages] Parametrized Higher Category Theory and Higher Algebra: Exposé IV -- Stability With Respect to an Orbital ∞-Category.pdf
│   ├── [Shah, 81 Pages] Parametrized Higher Category Theory and Higher Algebra: Exposé II - Indexed Homotopy Limits and Colimits.pdf
│   └── [Shah, 86 Slides] Parametrized Higher Category Theory.pdf
├── Quasicategories
│   ├── [AL, 26 Pages] Exponentiable Higher Toposes.pdf
│   ├── [Beardsley, 78 Pages] Coalgebraic Structure and Intermediate Hopf–Galois Extensions of Thom Spectra in Quasicategories.pdf
│   ├── [Berman, 13 Pages] On Lax Limits in ∞-Categories.pdf
│   ├── [BG, 11 Pages] On the Fibrewise Effective Burnside ∞-Category.pdf
│   ├── [BM, 30 Pages] Spectral Sequences in (∞,1)-Categories.pdf
│   ├── [BS, 18 Pages] Fibrations in ∞-Category Theory.pdf
│   ├── [BV, 267 Pages] Homotopy Invariant Algebraic Structures On Topological Spaces.djvu
│   ├── [Campbell, 3 Pages] A Counterexample in Quasi-Category Theory.pdf
│   ├── [Cisinski, 204 Pages] Algèbre Homotopique et Catégories Supérieures.pdf
│   ├── [Cisinski, 446 Pages] Higher Categories and Homotopical Algebra.pdf
│   ├── [Cisisnki, 57 Pages] Catégories Supérieures et Théorie des Topos.pdf
│   ├── [DS, 46 Pages] Mapping Spaces in Quasi-Categories.pdf
│   ├── [Fiore, 24 Pages] Quasicategorical Adjunctions.pdf
│   ├── [GR, 11 Pages] Simplified HTT 4.3.2.15.pdf
│   ├── [GR, 85 Pages] Some Higher Algebra.pdf
│   ├── [Gregoric, 21 Pages] Gregoric Blitzkrieg.pdf
│   ├── [Groth, 77 Pages] A Short Course on ∞-Categories.pdf
│   ├── [Harpaz, 116 Pages] Little Cube Algebras and Factorisation Homology.pdf
│   ├── [Harpaz, 7 Pages] Limits, colimits and adjunctions in ∞-categories.pdf
│   ├── [Haugseng, 12 Pages] On (Co)ends in ∞-Categories.pdf
│   ├── [Haugseng, 28 Pages] A Fibrational Mate Correspondence for ∞-Categories.pdf
│   ├── [Haugseng, 87 Pages] Introduction to ∞-Categories.pdf
│   ├── [HH, 684 Pages] Higher Categories I & II
│   │   ├── 0    [Hebestreit, 13 Pages] A Fairytale.pdf
│   │   ├── 10  [Hebestreit, 11 Pages] Fat and Thin Slices.pdf
│   │   ├── 11  [Hebestreit, 43 Pages] Cartesian Fibrations.pdf
│   │   ├── 12  [Hebestreit, 121 Pages] Straightening and Unstraightening.pdf
│   │   ├── 12' [HH, 53 Pages] Straightening and Unstraightening (Heuts's Notes).pdf
│   │   ├── 13  [Hebestreit, 14 Pages] Homotopy Colimits.pdf
│   │   ├── 14  [Hebestreit, 10 Pages] Simplicial Model Categories.pdf
│   │   ├── 15  [Hebestreit, 35 Pages] Yoneda's Lemma, Adjunctions and (Co)Limits.pdf
│   │   ├── 1    [Hebestreit, 37 Pages] Categories.pdf
│   │   ├── 2    [Hebestreit, 25 Pages] Simplicial Sets.pdf
│   │   ├── 3    [Hebestreit, 10 Pages] Quasicategories.pdf
│   │   ├── 4    [Hebestreit, 53 Pages] Simplicial Categories.pdf
│   │   ├── 5    [Hebestreit, 38 Pages] Simplicial Homotopy Theory.pdf
│   │   ├── 6    [Hebestreit, 57 Pages] Quasicategories and Anima.pdf
│   │   ├── 7    [Hebestreit, 67 Pages] Equivalences, Equivalences, Equivalences.pdf
│   │   ├── 8    [Hebestreit, 9 Pages] A Fairytale.pdf
│   │   └── 9    [Hebestreit, 88 Pages] Localisations and Model Categories.pdf
│   ├── [Hinich, 111 Pages] Lectures on ∞-Categories.pdf
│   ├── [Joyal, 244 Pages] Notes on Quasi-Categories.pdf
│   ├── [Joyal, 350 Pages] The Theory of Quasi-Categories and its Applications II.pdf
│   ├── [Joyal, 479 Pages] The Theory of Quasi-Categories.pdf
│   ├── [Lejay, 139 Pages] Algèbres à Factorisation et Topos Supérieurs Exponentiables.pdf
│   ├── [Lurie, 60 Pages] On ∞-Topoi.pdf
│   ├── [Lurie, 841 Pages] Kerodon.pdf
│   ├── [Lurie, 949 Pages] Higher Topos Theory.pdf
│   ├── [Lysenko, 208 Pages] Lysenko's comments to Gaitsgory–Rozenblyum.pdf
│   ├── [Morel, 118 Pages] ∞-Categories.pdf
│   ├── [Nguyen, 94 Pages] Theorems in Higher Category Theory and Applications.pdf
│   ├── [NRS, 21 Pages] Adjoint Functor Theorems for ∞-Categories.pdf
│   ├── [Porta, 100 Pages] Derived formal moduli problems.pdf
│   ├── [Rezk, 11 Pages] Degenerate Edges of Cartesian Fibrations are Cartesian Edges.pdf
│   ├── [Rezk, 175 Pages] Stuff About Quasicategories.pdf
│   ├── [Rezk, 50 Pages] Toposes and homotopy toposes.pdf
│   ├── [Riehl, 20 Pages] Quasi-Categories as (∞,1)-Categories.pdf
│   ├── [Riehl, 292 Pages] Categorical homotopy theory.pdf
│   ├── [Riehl, 9 Pages] Associativity Data in an (∞,1)-Category.pdf
│   ├── [Rovelli, 38 Pages] Weighted Limits in an (∞,1)-Category.pdf
│   ├── [RS, 151 Pages] Notes on Higher Categories.pdf
│   ├── [RV, 33 Pages] Completeness Results for Quasi-Categories of Algebras, Homotopy Limits, and Related General Constructions.pdf
│   ├── [Stevenson, 12 Pages] Stability for Inner Fibrations Revisited.pdf
│   ├── [Tanaka, 14 Pages] Functors (Between ∞-Categories) That Aren't Strictly Unital.pdf
│   ├── [Thanh, 46 Pages] Quasicategories.pdf
│   ├── [Wong, 83 Pages] The Grothendieck construction in enriched, internal and ∞-Category Theory.pdf
│   ├── [Zsámboki, 31 Pages] A summary of higher topos theory.pdf
│   └── [無, 5 Pages] HTT ToC.pdf
├── Simplicial Categories
│   ├── [Bergner, 16 Pages] A model category structure on the category of simplicial categories.pdf
│   ├── [Bergner, 22 Pages] Complete Segal Spaces Arising From Simplicial Categories.pdf
│   ├── [Bergner, 40 Pages] Three models for the homotopy theory of homotopy theories.pdf
│   ├── [Cordier, 21 Pages] Sur la notion de diagramme homotopiquement cohérent.pdf
│   ├── [CP, 26 Pages] Vogt's Theorem on Categories of Homotopy Coherent Diagrams.pdf
│   ├── [CP, 54 Pages] Homotopy Coherent Category Theory.pdf
│   ├── [DS, 29 Pages] Rigidification of Quasi-Categories.pdf
│   ├── [DS, 64 Pages] Mapping Spaces in Quasi-Categories.pdf
│   ├── [Hinich, 23 Pages] Homotopy Coherent Nerve in Deformation Theory.pdf
│   ├── [HK, 14 Pages] Mapping Spaces in Homotopy Coherent Nerves.pdf
│   ├── [Joyal, 66 Pages] Quasi-Categories vs Simplicial Categories.pdf
│   ├── [Riehl, 16 Pages] On the Structure of Simplicial Categories Associated to Quasi-Categories.pdf
│   ├── [Riehl, 26 Pages] Homotopy Coherent Structures.pdf
│   ├── [Riehl, 292 Pages] Categorical Homotopy Theory.pdf
│   └── [Riehl, 7 Pages] Understanding the Homotopy Coherent Nerve.pdf
├── Unicity
│   ├── [Bergner, 16 Pages] Equivalence of Models for Equivariant (∞,1)-Categories.pdf
│   ├── [BS, 47 Pages] On the Unicity of the Homotopy Theory of Higher Categories.pdf
│   ├── [JT, 49 Pages] Quasi-Categories vs Segal Spaces.pdf
│   ├── [Riehl, 58 Pages] Seminar Notes on the Barwick–Schommer–Pries Unicity Theorem.pdf
│   └── [Toën, 32 Pages] Vers une axiomatisation de la théorie des catégories supérieures.pdf
└── Un⧸Straightening
├── [AF, 89 Pages] Fibrations of ∞-Categories.pdf
├── [BGN, 19 Pages] Dualizing Cartesian and Cocartesian Fibrations.pdf
├── [BS, 18 Pages] Fibrations in ∞-Category Theory.pdf
├── [Campbell, 31 Slides] A modular proof of the straightening theorem.pdf
├── [GHN, 42 Pages] Lax Colimits and Free Fibrations in ∞-Categories.pdf
├── [GR, 85 Pages] Some Higher Algebra.pdf
├── [Hebestreit, 121 Pages] Straightening and Unstraightening.pdf
├── [HH, 53 Pages] Straightening and Unstraightening (Heuts's Notes).pdf
├── [HM, 17 Pages] Left Fibrations and Homotopy Colimits II.pdf
├── [HM, 27 Pages] Left Fibrations and Homotopy Colimits.pdf
├── [Mazel-Gee, 16 Pages] A User's Guide to (Co)Cartesian Fibrations.pdf
├── [Mazel-Gee, 41 Pages] All About the Grothendieck Construction.pdf
├── [Noel, 1 Page] Cartesian Model Structure.pdf
├── [PK, 43 Pages] Straightening and Unstraightening.pdf
├── [Richardson, 14 Pages] Mapping Spaces and Straightening-Unstraightening.pdf
├── [Ruit, 71 Pages] Grothendieck constructions in higher category theory.pdf
├── [Stevenson, 41 Pages] Model Structures for Correspondences and Bifibrations.pdf
├── [Stevenson, 49 Pages] Covariant Model Structures and Simplicial Localization.pdf
└── [Wong, 83 Pages] The Grothendieck Constructionin Enriched, Internal and ∞-Category Theory.pdf


It depends on from what direction you are coming and in what direction you are wanting to go!

As one of the developers of the basic theory from way back, I like to approach things via homotopy coherence as that is where the quasi-category approach comes from, and it also more or less guarantees some nice diagrams and situations that are slightly more geometric or visual. Have a look at my n-Lab page: https://ncatlab.org/timporter/show/simplicial+foundations+for+homotopy+coherence

Other material that includes intuitive approaches include my book with Heiner Kamps: Abstract Homotopy and Simple Homotopy Theory, World Scientific, 462pp (ISBN 981-02-1602-5)

and the Cubo notes: Abstract Homotopy Theory, the interaction of category theory and homotopy theory, (survey article, updated version of lecture notes from summer school course at Bressanone), Cubo, 5 (2003)115-165, 2003)(and here: https://ncatlab.org/nlab/files/Abstract-Homotopy.pdf).

Those do NOT go very far but concentrate on the intuition and basic structure. They may be too elementary for you but are intended to be readable. You can also access a version of the crossed menagerie: https://ncatlab.org/timporter/show/crossed+menagerie which discusses a lot of the relationship with non-abelian cohomology, if that is what is of interest to you.

I hope this helps.

Moritz Groth put up some excellent lecture notes: https://arxiv.org/abs/1007.2925v2

If you are more categorically minded, Emily Riehl's book has a lot about quasi-categories: http://www.math.jhu.edu/~eriehl/cathtpy.pdf and her website has a lot of exposition about cosmoi, an approach to $$\infty$$-categories via 2-category theory, and not requiring one to work in the setting of quasi-categories: http://www.math.jhu.edu/~eriehl/

• The link to Riehl's pdf if broken. May 26, 2018 at 8:14
• Current link: math.jhu.edu/~eriehl/cathtpy.pdf Jan 22, 2019 at 22:20

There are now more resources since the question was raised. Markus Land's Introduction to infinity-categories grew out from lecture notes, and seemingly much better fitted for pedagogy.

On the other hands, due (or thanks?) to pandemic, there are more video resources. For example, there is a series of lectures by Achim Krause and Thomas Nikolaus recorded and uploaded on the YouTube channel.