Textbooks on higher category theory

What textbooks on higher category theory are there? What books do you recommend? I am looking for self-contained introductions, no research reports. There are lots of informal summaries and arXiv papers, but I am really only asking for textbooks here.

I know of Lurie's Higher Topos Theory, which "only" treats $(\infty,1)$-categories. I am looking for books which treat $\infty$-categories in general. Then I know of Leinster's Higher Operads, Higher Categories, which is from 2004. Is it still up to date? Is Leinster's book the best introduction to the subject? What do you think of Higher-Dimensional Categories: an illustrated guide book by Cheng and Lauda, which is also from 2004 and still a draft? Is it too informal when one really wants to work with the concepts?

Bonus question: Meanwhile, is there some "preferred" definition of an $\infty$-category among the dozen definitions which have been studied?

Solution 1:

1. First of all, make sure to have a few references on category theory available. Good ones include:

• Basic Category Theory (Tom Leinster);
• Category Theory in Context (Emily Riehl);
• The nLab.

2. It also pays of to learn about the insights leading to $\infty$-categories before learning about their theory proper. A good reference here is John Baez's An Introduction to $n$-Categories. Another one is Section 1.2 of Hellstrøm-Finnsen's thesis.

3. $\infty$-Categories require two fundamental prerequisites: model category theory and simplicial sets.

Simplicial Sets. Friedman's An elementary illustrated introduction to simplicial sets is a marvelous introduction for beginners. For more in-depth references, there are May's Simplicial Objects in Algebraic Topology, and Simplicial Homotopy Theory by Goerss–Jardine.

Model Categories. Good references for model category theory include:

• Sections 2.1–2.3 of Higher Categories and Homotopical Algebra (Cisinski);
• Introduction to Homotopy Theory (nLab);
• Categorical Homotopy Theory (Riehl);
• Model Categories (Hovey);
• Part 4 of More Concise Algebraic Topology (May);

4. ($\infty$-Categories, finally) It is hard to capture in a precise way the idea of an $\infty$-category as a set of objects, together with a set of morphisms, a set of $2$-morphisms, and so on. There are two ways of approaching this difficulty, one traditional, the other very recent.

Via Quasicategories. The traditional one is to use models for $\infty$-categories. One such model is given by a special kind of simplicial set called a quasicategory. This is the approach developed by Joyal and Lurie. For learning the theory of quasicategories, there are:

• A Short Course on $\infty$-Categories (Groth);
• Kerodon (Lurie; a (WIP) textbook);
• Higher Categories and Homotopical Algebra (Cisinski);
• Higher Topos Theory (Lurie; this is best used as a reference, not as a textbook).

Via $\infty$-Cosmoi. The second one is the model-independent approach of Riehl and Verity (which is currently being developed). Instead of axiomatizing what $\infty$-categories are via models, Riehl–Verity axiomatize the mathematical object in which $\infty$-categories live, and call it an $\infty$-cosmos.

When working with an specific model for $\infty$-categories, one is often lead to complicated arguments involving its combinatorics. On the other hand, in Riehl–Verity's framework, it is possible to prove statements about $\infty$-categories in a much simpler, model-independent, way

Riehl and Verity are currently compiling their work in a textbook, called Elements of $\infty$-Category Theory.

Extra References

$\infty$-Categories

• Notes on $\infty$-categories (Hinich);
• Notes on $\infty$-categories (Morel, in French);
• Homotopy Theory of Higher Categories: From Segal Categories to $n$-Categories and Beyond (Simpson);
• The Homotopy Theory of $(\infty,1)$-Categories (Bergner).

Background on model categories and simplicial sets

• Notes on Homotopical Algebra (Low);
• Homotopy Limit Functors on Model Categories and Homotopical Categories (Dwyer et al.);
• Model Categories and Their Localizations (Hirschhorn);

Edit: Here are some extra (extra) references, in the (way far from optimal) format of a directory tree.

(P.S. Please take the above attempt at a guide with a big grain of salt; there's a lot missing from it!)

├── Complete Segal Spaces
│   ├── [BC, 10 Pages] Equivariant Complete Segal Spaces.pdf
│   ├── [Brito, 26 Pages] Segal Objects and the Grothendieck Construction.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, 20 Pages] Quillen Adjunctions Induce Adjunctions of Quasicategories.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
│       └── [Lindberg, 70 Pages] Equivariant Sheaves on 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
    ├── [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