Accessible quasi-category¶

An infinity-category equivalent to the closure of a small infinity-category under kappa-filtered colimits for some regular cardinal kappa.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3185
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Higher Category Theory → Mathematics

Core Idea¶

Accessibility is relative to some regular cardinal, accessible infinity-categories are typically large, small accessible cases are highly restricted and presentability adds existence of all small colimits. Kappa-compact objects form an essentially small generating subcategory and every object is reconstructed as a kappa-filtered colimit of those generators through an Ind-completion. 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.

How would you explain it like I'm…

No faithful explanation at this level. All three generators judged this rung infeasible: any child-level picture of a huge collection built from small pieces drops the higher, homotopy-coherent structure (arrows between arrows at every level) and collapses the idea into an ordinary accessible category or plain building-block assembly.

Endless Arrows, Small Blocks

In some very advanced math, there are objects, arrows between objects, arrows between those arrows, and so on forever. These are called infinity-categories, and quasi-categories are one way to describe them. An accessible quasi-category is a huge one of these where there is a manageable collection of 'small' objects, and every object can be built as an orderly, ever-growing combination of those small ones. How 'small' counts as small is set by a chosen infinite size limit.

Accessible Infinity-Categories

An accessible quasi-category is the higher-category version of an accessible category. A quasi-category is a model of an infinity-category, where objects are linked by morphisms, morphisms between morphisms, and so on. Accessibility is measured against a regular cardinal kappa: the small, 'kappa-compact' objects form an essentially small generating subcategory, and every object is rebuilt as a kappa-filtered colimit of them, which is to say the whole thing is the Ind-completion of a small infinity-category. Accessible infinity-categories are usually large; the small ones that qualify are very restricted. Presentable infinity-categories are the accessible ones that also have all small colimits.

An accessible quasi-category, or accessible ∞-category, is an ∞-category C that is κ-accessible for some regular cardinal κ. This means C has κ-filtered colimits, its κ-compact objects form an essentially small subcategory, and every object of C is a κ-filtered colimit of κ-compact objects. Equivalently, C is equivalent to Ind_κ of some small ∞-category, the free completion under κ-filtered colimits. Accessibility is always relative to some regular cardinal, and 'accessible' means κ-accessible for some κ, which should be kept distinct from κ-accessibility for a specific κ. Accessible ∞-categories are typically large; small accessible ones are highly restricted. Presentability is the stronger condition that adds existence of all small colimits. The notion also supports accessible functors, and it should be related carefully to ordinary accessible categories and to presentable ∞-categories rather than conflated with them.

Scope of Application¶

Accessible quasi-category belongs to higher category theory and is useful where the analyst can specify the typed higher category theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the infinity-category or quasi-category C, regular cardinal kappa, kappa-filtered colimits, kappa-compact objects, essentially small generating subcategory, equivalence with Ind-kappa of a small infinity-category, accessibility versus kappa-accessibility, accessible functors and relation to presentable and ordinary accessible categories are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the infinity-category or quasi-category C, regular cardinal kappa, kappa-filtered colimits, kappa-compact objects, essentially small generating subcategory, equivalence with Ind-kappa of a small infinity-category, accessibility versus kappa-accessibility, accessible functors and relation to presentable and ordinary accessible categories 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.

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 Accessible quasi-category. Accessible 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 objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the infinity-category or quasi-category C, regular cardinal kappa, kappa-filtered colimits, kappa-compact objects, essentially small generating subcategory, equivalence with Ind-kappa of a small infinity-category, accessibility versus kappa-accessibility, accessible functors and relation to presentable and ordinary accessible categories 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 objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Kappa-compact objects form an essentially small generating subcategory and every object is reconstructed as a kappa-filtered colimit of those generators through an Ind-completion., and type the carrier, state every parameter and convention in the definition, test that the infinity-category or quasi-category C, regular cardinal kappa, kappa-filtered colimits, kappa-compact objects, essentially small generating subcategory, equivalence with Ind-kappa of a small infinity-category, accessibility versus kappa-accessibility, accessible functors and relation to presentable and ordinary accessible categories are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Accessible quasi-category Domain-specific

Parents (1) — outgoing typed relations

Accessible quasi-category is a kind of Aggregation Prime

It remains its own entry because its identity is fixed by the infinity-category or quasi-category C, regular cardinal kappa, kappa-filtered colimits, kappa-compact objects, essentially small generating subcategory, equivalence with Ind-kappa of a small infinity-category, accessibility versus kappa-accessibility, accessible functors and relation to presentable and ordinary accessible categories.

Hierarchy path (1) — routes to 1 parentless root

Accessible quasi-category → is a kind of Aggregation → is a kind of Transformation → is a kind of, typical Function (Mapping)

Neighborhood in Abstraction Space¶

Accessible quasi-category sits in a crowded region of the domain-specific corpus (11^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Category Theory Foundations (48 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25