Higher Categories and the Topos Bridge

Overview

Higher category theory keeps relations between morphisms as mathematical data, then keeps relations between those relations. This overview concentrates on $(\infty,1)$-categories: morphisms above dimension one are invertible in the appropriate homotopical sense. Other higher-category theories allow more noninvertible dimensions and require different definitions.

Prerequisites

Ordinary categories, functors, limits, and adjunctions provide the first layer. A technical reading also requires simplicial sets, horns, and homotopy theory. To follow the topos connection, understand presheaves and localization; the formulation below explicitly names the additional accessibility and exactness requirements.

Definition and conventions

In the quasi-category model, an $(\infty,1)$-category is a simplicial set $K$ such that every map $\Lambda_i^n\to K$, for $0<i<n$, extends to $\Delta^n\to K$. These are inner horn fillers. Fillers need not be unique. Lurie calls these objects infinity-categories; the terminology identifies this model of higher categories rather than every possible higher-category structure.

For a small infinity-category $\mathcal C$, let $\mathcal P(\mathcal C)=\operatorname{Fun}(\mathcal C^{\mathrm{op}},\mathcal S)$, where $\mathcal S$ is the infinity-category of spaces. An infinity-topos is an infinity-category obtained as an accessible left exact localization of such a presheaf category. Left exact means preserving finite limits. Accessibility is a technical size condition, not an optional adjective.

Intuition

Ordinary uniqueness of factorization becomes a contractible space of coherent choices. Recording only homotopy classes of maps can hide the homotopies needed to assemble a construction. Space-valued presheaves retain those choices, while suitable localizations impose descent conditions without losing finite-limit compatibility.

Examples and non-examples

The nerve of an ordinary category is a quasi-category. Its inner horn fillers are uniquely determined by composition. A Kan complex is another example and models a space, with all morphisms invertible homotopically.

Consider the inner horn $\Lambda_1^2$ itself as a simplicial set. It has composable edges $0\to1$ and $1\to2$, but no required two-simplex completing their composite. The identity map from this horn cannot extend across $\Delta^2$, so it is a concrete non-example of a quasi-category.

The presheaf infinity-category on the terminal category is $\mathcal S$ itself, an infinity-topos via the identity localization. Merely possessing higher morphisms does not make an arbitrary infinity-category a topos.

Important properties

The ordinary nerve example explains the weakened uniqueness: retaining fillers while allowing them to vary admits coherent homotopy information. In a genuine infinity-category, mapping objects are spaces, and appropriate universal properties use these spaces rather than just sets of arrows.

Topos recognition is a separate structural theorem. The localization requirement controls how presheaf constructions interact with finite limits; not every reflective subcategory has the required exactness. This overview does not supply a proof of the higher Giraud theorem.

Connections and further reading

The bridge from set-valued to space-valued presheaves connects categorical foundations with descent, homotopy theory, and derived geometry. Read Lurie, Definition 1.1.2.4 and Examples 1.1.2.5–1.1.2.6; Definition 6.1.0.4. The Chinese name is independently evidenced in Xu Fei’s university research introduction.