Globular set¶

A sequence of sets of n-cells with source and target maps satisfying globularity equations, forming the presheaf carrier for many higher-category structures.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4742
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Higher Category Theory → Mathematics

Core Idea¶

A globular set generalizes a directed graph by adding cells in every dimension whose source and target are parallel cells one dimension lower. Paired source and target maps descend dimensions, and the equations ss=st and ts=tt ensure every higher cell's source and target share their own boundaries. 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…

Arrows Between Arrows

Draw some dots, and draw arrows going from one dot to another. Now draw a fat arrow from one arrow to another arrow, but only if both arrows start at the same dot and end at the same dot. You can keep going: even fatter arrows between fat arrows, with the same rule each time.

Towers of Matching Arrows

A globular set starts like a map of dots connected by one-way arrows. Then it adds 'arrows between arrows': a level-2 arrow can go from one arrow to another, but only when the two arrows begin at the same dot and end at the same dot. Level-3 arrows go between level-2 arrows that line up the same way, and so on forever. Every arrow at each level has a start and an end one level down, and they always match up neatly.

Higher-Dimensional Directed Graphs

A directed graph has points and arrows, where each arrow has a source and a target point. A globular set generalizes this by adding cells in every dimension: 2-cells between arrows, 3-cells between 2-cells, and so on. Each n-cell has a source and target that are (n−1)-cells. The key rule is that the source and target must be 'parallel': they themselves share the same source and the same target. Written as equations, s∘s = s∘t and t∘s = t∘t. These are called the globularity conditions, and they make each cell look like a lens or globe shape bounded by two matching halves.

A globular set consists of sets X0, X1, X2, … with source and target maps s, t: Xn → Xn−1 for each n ≥ 1. These satisfy the globular identities s∘s = s∘t and t∘s = t∘t, which ensure that the source and target of any n-cell are parallel (n−1)-cells sharing the same boundary. Truncated at dimension 1, this is exactly a directed graph, so globular sets generalize graphs to all dimensions. They are the underlying shape data for globular approaches to higher categories, where composition is added on top. Note that a globular set itself has no composition; it is purely cells and boundary maps. Checking the structure means verifying typed s and t at each level and both equations at every level.

Scope of Application¶

Globular set belongs to higher category theory and is useful where the analyst can specify the typed higher category theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate each dimension has typed source and target maps to the preceding dimension and both globularity equations hold for every composable level. The scope is broad within that domain but bounded by the need for each dimension has typed source and target maps to the preceding dimension and both globularity equations hold for every composable level. 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 each dimension has typed source and target maps to the preceding dimension and both globularity equations hold for every composable level 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 Globular set 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 Globular set. Globular set 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, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express each dimension has typed source and target maps to the preceding dimension and both globularity equations hold for every composable level 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, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Paired source and target maps descend dimensions, and the equations ss=st and ts=tt ensure every higher cell's source and target share their own boundaries., and type the carrier, state every parameter and convention in the definition, test that each dimension has typed source and target maps to the preceding dimension and both globularity equations hold for every composable level, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Globular set Domain-specific

Parents (1) — outgoing typed relations

Globular set is a kind of Hierarchy Prime

What makes it its own entry: the domain-specific identity determined by each dimension has typed source and target maps to the preceding dimension and both globularity equations hold for every composable level.

Hierarchy paths (4) — routes to 4 parentless roots

Globular set → is a kind of Hierarchy → is a kind of Network

Show alternative paths (3)

Neighborhood in Abstraction Space¶

Globular set sits in a crowded region of the domain-specific corpus (15^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