Atlas of Fundamental Concepts

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Canonical static concept record

Finitely complete category

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Summary

A category admitting all finite limits.

Record metadata

Canonical name
Finitely complete category
Concept ID
finitely_complete_category
Content version
2.3.0
Primary field
Mathematics
Fields
Mathematics
Primary domain
Category theory
Domains
Category theory, Logic
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.

CategoryFinitely complete category

Permalink to relation

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

Relation ID
category_to_finitely_complete
Relation type
Impose axiom impose-axiom
Direction
source → target
Endpoint roles
source: Builds toward; target: Built from
Authored annotation
require finite limits

Authored explanation

Require every finite diagram to have a limit.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Finitely complete categoryCartesian category

Permalink to relation

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

Relation ID
finitely_complete_to_cartesian
Relation type
Induced / forgotten induced-forgotten
Direction
source → target
Endpoint roles
source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation
retain finite products

Authored explanation

Finite limits include a terminal object and binary products, yielding cartesian 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

Finitely complete categoryCategorical limit

Permalink to relation

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

Relation ID
story_finitely_complete_category_to_categorical_limit_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
retains its finite limits

Authored explanation

A finitely complete category has limits for every finite diagram; forgetting the ambient axiom leaves individual categorical limits.

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

Finitely complete categoryfinite limits + exponentials(construction junction)

Permalink to relation

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

Relation ID
finite_complete_to_j_topos
Relation type
Combine compatible structures combine-compatible
Direction
source → target
Endpoint roles
source: Builds toward; target: Built from
Authored annotation
supply finite limits

Authored explanation

Use the finite-limit structure.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources