Coherent category

Equip a regular category with finite unions of subobjects that remain stable under pullback, providing categorical semantics for finite-limit, existential, and finite-disjunctive reasoning.

Version

v2 · 2026-08-30 · History

Domain-specific #

1493

Origin domain

category theory

Subdomain

categorical logic and topos theory

Core Idea

A coherent category is a regular category \(\mathcal C\) in which each subobject poset (Sub(X)) has finite joins and every inverse-image map \(f^*:Sub(Y)\to Sub(X)\) preserves those joins. Finite limits interpret conjunction and equality, regular images interpret existential quantification, and pullback-stable finite joins interpret disjunction; preservation makes these logical operations compatible with substitution along morphisms.

Its autonomous residual is the conjunction of regular categorical structure with pullback-stable finite subobject unions, and its role as semantics for coherent logic, not generic consistency, categorical coherence diagrams, or coherence of sheaves.

Scope of Application

Coherent category applies when the analyst can specify a category with finite limits, regular image factorizations, and a subobject poset for every object and establish that regular-category structure is present and finite unions of subobjects exist and are stable under pullback, equivalently the subobject inverse-image maps preserve finite joins under the adopted convention. The entry uses the standard categorical-logic definition. It does not assert that every functor category built from a coherent category is coherent without additional hypotheses, and all size and exactness assumptions must remain visible.

Clarity

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because coherent describes proof systems, sheaves, rings, configurations and ordinary consistency, while category-theory coherence theorems concern commuting diagrams rather than coherent categories. The disciplined statement is that the object counts as Coherent category exactly when regular-category structure is present and finite unions of subobjects exist and are stable under pullback, equivalently the subobject inverse-image maps preserve finite joins under the adopted convention

Manages Complexity

The abstraction compresses syntactic categories of coherent theories, pretoposes, coherent toposes, categories of algebraic or relational structures, small and locally small settings, coherent functors, Heyting enrichments, geometric infinitary extensions, and alternative regularity axiom packages into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.

Abstract Reasoning

1. Type the carrier. Establish a category with finite limits, regular image factorizations, and a subobject poset for every object and reject examples from a different problem. 2. Lock the rule. Express that regular-category structure is present and finite unions of subobjects exist and are stable under pullback, equivalently the subobject inverse-image maps preserve finite joins under the adopted convention independently of one notation or implementation.

Knowledge Transfer

Transfer within category theory is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from For a coherent first-order theory, its syntactic category has formulas-in-context as objects and provably functional relations as arrows, with conjunction, existential quantification and finite disjunction represented by the coherent categorical operations to A coherent functor from a small coherent category into Set acts as a model whose interpretation preserves finite limits, regular images and finite unions of definable subobjects demonstrates that continuity.

Relationships to Other Abstractions

Current abstraction Coherent category Domain-specific

Parents (1) — more general patterns this builds on

• Coherent category is a kind of Category Prime

  The proposed strict upward parent is prime:category.

Hierarchy paths (3) — routes to 3 parentless roots

• Coherent category → Category → Associativity → Invariance

Show alternative paths (2)

Neighborhood in Abstraction Space

Coherent category sits in a moderately populated region (41^st percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Categorical Algebra & Model Systems (8 abstractions)

Nearest neighbors

• Extensive category — 0.90
• Image (category theory) — 0.90
• Injective object — 0.90
• Filtered category — 0.90
• Tower of objects — 0.89

Computed from structural-signature embeddings · 2026-09-08