Quasi-category¶
A simplicial set satisfying every inner horn-filling condition, modeling an infinity-category with composition coherent up to higher homotopy.
Core Idea¶
Vertices act as objects, edges as morphisms and higher simplices as compositional homotopies; inner horn fillers exist but need not be unique, replacing strict associativity with coherent higher structure. Compatible partial simplex data specify an inner horn, a filler supplies a composite and its witness, and iterated higher fillers organize all choices into coherent homotopies. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Quasi-category belongs to higher category theory and is useful where the analyst can specify the typed higher category theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the simplicial set, face and degeneracy maps, inner horns and dimensions, filler-existence condition, objects and morphisms, homotopy category and equivalence convention are explicit. The scope is broad within that domain but bounded by the need for the simplicial set, face and degeneracy maps, inner horns and dimensions, filler-existence condition, objects and morphisms, homotopy category and equivalence convention are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the simplicial set, face and degeneracy maps, inner horns and dimensions, filler-existence condition, objects and morphisms, homotopy category and equivalence convention are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Quasi-category can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Quasi-category. Quasi-category compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed higher category theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the simplicial set, face and degeneracy maps, inner horns and dimensions, filler-existence condition, objects and morphisms, homotopy category and equivalence convention are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of higher category theory because they reuse the typed higher category theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, Compatible partial simplex data specify an inner horn, a filler supplies a composite and its witness, and iterated higher fillers organize all choices into coherent homotopies., and type the carrier, state every parameter and convention in the definition, test that the simplicial set, face and degeneracy maps, inner horns and dimensions, filler-existence condition, objects and morphisms, homotopy category and equivalence convention are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Quasi-category Domain-specific
Parents (1) — more general patterns this builds on
-
Quasi-category is a kind of Composition Prime
The proposed strict upward parent is
prime:composition.
Hierarchy path (1) — routes to 1 parentless root
- Quasi-category → Composition → Gestalt Principles → Holism
Neighborhood in Abstraction Space¶
Quasi-category sits in a crowded region of the domain-specific corpus (10th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Category-Theoretic Structures (79 abstractions)
Nearest neighbors
- Simplicially enriched category — 0.93
- Accessible quasi-category — 0.92
- 2-group — 0.92
- Join (simplicial sets) — 0.92
- Tower of objects — 0.92
Computed from structural-signature embeddings · 2026-09-08