Enrichment
Recursive Hom-universes.
- (∞,0)-Cat = spaces
- (∞,1)-Cat = categories enriched in spaces
- (∞,n)-Cat = categories enriched in (∞,n−1)-Cat
The one-sentence map
A higher category lets processes be compared, those comparisons be compared, and all higher comparisons be organized coherently.
Category: objects + arrows 2-category: arrows between arrows ∞-category: categories with Hom-spaces (∞,n)-category: n directed layers + infinite coherent bookkeeping
The central move is to stop demanding literal equality and instead track equivalence with witnesses.
The coherence problem
Associativity looks harmless until you refuse brittle equality.
strict: f(g h) = (f g)h weak: f(g h) ──associator──▶ (f g)h higher: different associator paths need higher witnesses
Topology becomes the compression codec for this infinite tower of coherence.
Walk the seven rooms
Click a room to update the lesson card.
Rooms 1–2: arrows and coherence
The first jump is from an arrow to an arrow between arrows. The second jump is from strict equality to coherent equivalence.
f, g : A → B α : f ⇒ g Γ : α ⇛ β
When there are multiple ways to compose, a weak higher category stores comparison data rather than pretending all routes are literally equal.
Rooms 3–5: topology as coherence storage
Points, paths, homotopies, and higher homotopies are the native shape of higher morphisms.
Simplicial sets turn this geometry into combinatorics: vertices, edges, triangles, tetrahedra, and higher cells.
A quasicategory asks only that inner horns can be filled. Composition exists, but it is not forced to be unique on the nose.
ordinary category: unique inner horn fillers quasicategory: inner horn fillers exist Kan complex: all horn fillers exist
Three interfaces for (∞,n)-categories
Θₙ-spaces
A higher-dimensional shape library.
- Θₙ indexes composable higher diagrams
- Segal conditions say diagrams glue correctly
- Completeness fixes equivalence of objects
n-fold Segal spaces
A multidirectional grid.
- Use Δⁿ instead of custom shapes
- Segal conditions in multiple directions
- Constancy/completeness conditions recover (∞,n)-categories
The tower rule
The parameter n measures how many layers are genuinely directional.
Sets: objects only (∞,0): all morphisms invertible = spaces / ∞-groupoids (∞,1): 1-morphisms may be non-invertible (∞,2): 1- and 2-morphisms may be non-invertible (∞,n): 1 through n may be non-invertible; above n invertible
Everything above level n is reversible coherence data.
Definition dependency graph
This graph shows what should be learned before quasicategories, Segal spaces, and (∞,n)-categories. Click any node for a definition. Use filters to highlight routes.
Click a node
The shortest path is: equality → equivalence → homotopy → simplicial encoding → models of ∞-categories → models of (∞,n)-categories.
Legend
Quasicategory route
Segal-space route
(∞,n) route
Core heuristic: quasicategories are the horn-filling interface; Segal spaces are the composable-string interface; (∞,n)-categories add controlled non-invertibility through dimension n.
Generated visual plates
Arrow Hall + Coherence Engine.
Homotopy Ocean + Simplicial Workshop + Quasicategory Forge.
(∞,n)-Category Tower + three models.
Minimal learning route
1. Ordinary categories 2. Natural transformations / 2-morphisms 3. Weak vs strict composition 4. Homotopy and homotopy types 5. Simplicial sets and horns 6. Quasicategories or Segal spaces 7. Enrichment / Θₙ / n-fold Segal spaces
One-line compression
An ∞-category is a category whose composition is controlled by homotopy-coherent spaces of choices.
An (∞,n)-category is an ∞-category-like object with exactly n layers of non-invertible process.