\infty-Categories — Mathematics Knowledge System

Overview

An \infty-category (or (\infty,1)-category) is a higher-categorical structure where there are morphisms at all levels, but morphisms above level 1 are all invertible (i.e., they encode homotopies, homotopies between homotopies, etc.). The most tractable model is that of quasi-categories (Joyal, Lurie): simplicial sets satisfying a weak inner horn-filling condition. \infty-categories provide the correct framework for homotopy-coherent mathematics: derived algebraic geometry, topological field theories, and stable homotopy theory.

Intuition

In a 1-category, morphisms compose strictly: f \circ g is a well-defined morphism. In an \infty-category, composition is only defined up to a contractible space of choices — but all these choices are coherently equivalent. The data of an \infty-category is a simplicial set where inner horns (like the space of composable pairs) can be filled, but the filling is not unique (only homotopically unique). This encodes all the homotopy-coherent associativity and unitality data without imposing strict equations.

Formal Definition

Definition

A quasi-category (or \infty-category in Lurie's sense) is a simplicial set C (a functor \Delta^{\mathrm{op}} \to \mathbf{Set}) satisfying the inner horn extension property: every inner horn \Lambda^n_k \hookrightarrow \Delta^n (for 0 < k < n) admits a lift to C.

\Lambda^n_k \hookrightarrow \Delta^n \quad (0 < k < n)

Inner horn: \Delta^n with the k-th face removed (k strictly between 0 and n)

inner-horn

C_n = C(\Delta^n) = \mathrm{Hom}_{\mathbf{sSet}}(\Delta^n, C)

n-simplices of C: the set of n-simplices generalizes objects (n=0), morphisms (n=1), homotopies (n=2)

n-simplices

\mathrm{Hom}_C(x, y) = \{\sigma \in C_1 : d_1(\sigma) = x,\, d_0(\sigma) = y\}

The mapping space (a Kan complex) between objects x, y in a quasi-category C

mapping-space

h(C) = \tau_1(C)

The homotopy category h(C) of an \infty-category C: objects are 0-simplices, morphisms are equivalence classes of 1-simplices under homotopy

homotopy-category

Notation

Notation	Meaning
$N(\mathcal{C})$	Nerve of an ordinary category C, a quasi-category
$\Delta^n$	Standard n-simplex
$\Lambda^n_k$	k-th horn of the n-simplex
$\mathrm{Map}_C(x, y)$	Mapping space (a Kan complex) in \infty-category C
$\mathrm{Fun}(C, D)$	\infty-category of functors between \infty-categories C and D
$\mathbf{Sp}$	Stable \infty-category of spectra
$\mathbf{Cat}_\infty$	\infty-category of \infty-categories

Theorems

Theorem 1: Joyal's Inner Horn Extension Theorem

A simplicial set C is a quasi-category if and only if every inner horn \Lambda^n_k \to C (0 < k < n, n \geq 2) has a (non-unique) extension to \Delta^n \to C.

Theorem 2: Equivalence of Models

The following models of (\infty,1)-categories are all equivalent (as Quillen equivalent model categories): quasi-categories, complete Segal spaces, Segal categories, simplicially enriched categories (Bergner), and naturally marked simplicial sets.

Theorem 3: \infty-Categorical Yoneda Lemma

For an \infty-category C and an object x \in C, the Yoneda embedding j: C \to \mathrm{Fun}(C^{\mathrm{op}}, \mathcal{S}) (where \mathcal{S} is the \infty-category of spaces) is fully faithful: \mathrm{Map}_{\mathrm{Fun}(C^{\mathrm{op}},\mathcal{S})}(j(x), F) \simeq F(x) for any functor F.

Worked Examples

The nerve N(C) has n-simplices = strings of n composable morphisms in C: (f_1, ..., f_n) with cod(f_i) = dom(f_{i+1}).
$N(C)_n = \{(f_1, \ldots, f_n) : f_i \in \mathrm{Mor}(C), \, \mathrm{cod}(f_i) = \mathrm{dom}(f_{i+1})\}$
An inner horn \Lambda^n_k \to N(C) for 0 < k < n is a compatible collection of (n-1)-simplices missing the k-th face. This is a composable string with one composition 'missing'. By associativity in C, the missing composite g_k \circ g_{k+1} (or rather the combined morphism) exists uniquely.
For n=2, k=1: a horn \Lambda^2_1 gives two morphisms f: A -> B and g: B -> C. The unique filler is the 2-simplex with third edge g \circ f. In N(C), the filling is unique because composition in C is strict.

Answer: N(C) is a quasi-category because inner horns correspond to composable morphisms with one gap, which is filled by the composition in C. Moreover, N(C) is actually a 1-coskeletal simplicial set — all inner horns have unique fillings — encoding that C is a strict (1,0)-category.

Practice Problems

Difficulty 9/10

What is the homotopy category h(C) of a quasi-category C, and when is the \infty-category determined by its homotopy category?

Difficulty 10/10

Show that limits and colimits in a quasi-category C are unique up to a contractible space of equivalences.

Difficulty 10/10

Explain the role of the Grothendieck construction in \infty-category theory (the \infty-categorical straightening/unstraightening equivalence).

Common Mistakes

Common Mistake

Thinking that an \infty-category is just a category enriched over simplicial sets (a simplicial category).

Simplicial categories are one model for \infty-categories, but quasi-categories (the dominant model in modern literature) are simplicial sets with a horn-filling property, not categories enriched over simplicial sets. The two models are Quillen equivalent but technically distinct.

Common Mistake

Assuming that limits in a quasi-category are unique (as in 1-categories).

Limits exist up to contractible choice: the space of limit cones is contractible, not a single object. Two limit cones are equivalent by a unique (up to homotopy) equivalence, but there is a whole space of such equivalences forming a contractible Kan complex.

Historical Background

The quest for a good model of \infty-categories began with Boardman-Vogt's weak Kan complexes (1973) and was developed by Joyal in the 1990s under the name quasi-categories. Lurie's monumental Higher Topos Theory (2009) built a complete theory of \infty-toposes using quasi-categories, followed by Higher Algebra (2017) for stable \infty-categories and \infty-operads. Parallel models (complete Segal spaces, Segal categories, simplicial categories) were shown to be equivalent by Bergner and others.

1973
Boardman-Vogt introduce weak Kan complexes (quasi-categories)
Michael Boardman, Rainer Vogt
1997
Joyal develops the theory of quasi-categories
André Joyal
2006
Bergner proves equivalence of all models of (\infty,1)-categories
Julia Bergner
2009
Lurie publishes Higher Topos Theory
Jacob Lurie
2017
Lurie publishes Higher Algebra
Jacob Lurie

Summary

A quasi-category is a simplicial set C satisfying inner horn extension: every \Lambda^n_k -> C (0 < k < n) extends to \Delta^n -> C (not necessarily uniquely).
Objects are 0-simplices, morphisms are 1-simplices, 2-simplices witness composition (homotopies), and higher simplices encode all coherence data.
The homotopy category h(C) is the 1-category of equivalence classes of morphisms; most \infty-categories are not determined by h(C).
The Yoneda lemma, limits/colimits, adjunctions, and Kan extensions all generalize from ordinary category theory to \infty-categories.
Stable \infty-categories (e.g., spectra, derived categories) are pointed \infty-categories where \Sigma \simeq \Omega; they model homotopy-coherent homological algebra.

References

BookLurie, J. Higher Topos Theory. Princeton University Press, 2009. arXiv:math/0608040.
BookLurie, J. Higher Algebra. 2017. Available at math.ias.edu/~lurie/.
BookCisinski, D.-C. Higher Categories and Homotopical Algebra. Cambridge University Press, 2019.
PaperJoyal, A. Quasi-categories and Kan complexes. Journal of Pure and Applied Algebra, 175 (2002), 207-222.