Atlas of Fundamental Concepts

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Canonical static concept record

Exact category

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Summary

An additive category with a chosen class of kernel–cokernel pairs behaving like short exact sequences.

Record metadata

Canonical name
Exact category
Concept ID
exact_category
Content version
2.3.0
Primary field
Mathematics
Fields
Mathematics
Primary domain
Homological algebra
Domains
Homological algebra, Category theory
Alternate terminology
No alternate terminology is currently authored for this record.

Carrier(s)

Data

Axioms / constraints

Concept sources

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Abelian categoryExact category

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID
abelian_to_exact_category
Relation type
Induced / forgotten induced-forgotten
Direction
source → target
Endpoint roles
source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation
take all short exact sequences

Authored explanation

Every abelian category has a canonical Quillen exact structure consisting of all short exact sequences.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

Additive categoryExact category

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID
e_additive_exact
Relation type
Add data add-data
Direction
source → target
Endpoint roles
source: Builds toward; target: Built from
Authored annotation
+ conflations

Authored explanation

Choose a class of kernel–cokernel pairs satisfying exact-category axioms.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Exact categoryExact sequence

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID
story_exact_category_to_exact_sequence_induced_forgotten
Relation type
Induced / forgotten induced-forgotten
Direction
source → target
Endpoint roles
source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation
has distinguished exact sequences

Authored explanation

An exact category comes with a specified class of kernel–cokernel pairs, called conflations; retaining one such pair gives an exact sequence in the chosen exact structure.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources