Atlas of Fundamental Concepts

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Canonical static concept record

Adjunction

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Summary

A pair of functors F:C⇄D:GF:\mathcal C\rightleftarrows\mathcal D:G equipped with natural bijections Hom⁑(FX,Y)β‰…Hom⁑(X,GY)\operatorname{Hom}(F X,Y)\cong\operatorname{Hom}(X,GY).

Record metadata

Canonical name
Adjunction
Concept ID
adjunction
Content version
2.3.0
Primary field
Mathematics
Fields
Mathematics
Primary domain
Category theory
Domains
Category theory
Alternate terminology
No alternate terminology is currently authored for this record.

Carrier(s)

Data

Axioms / constraints

Canonically induces

Concept sources

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Cartesian closed categoryAdjunction

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

Relation ID
story_cartesian_closed_category_to_adjunction_canonical_construction
Relation type
Canonical construction canonical-construction
Direction
source β†’ target
Endpoint roles
source: Canonically constructs; target: Canonically constructed from
Authored annotation
product is left adjoint to exponentiation

Authored explanation

In a cartesian closed category, for each object AA the functor (βˆ’)Γ—A(-)\times A is left adjoint to the exponential functor (βˆ’)A(-)^A.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

FunctorAdjunction

Permalink to relation

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

Relation ID
story_functor_to_adjunction_add_data
Relation type
Add data add-data
Direction
source β†’ target
Endpoint roles
source: Builds toward; target: Built from
Authored annotation
pair with an adjoint functor

Authored explanation

An adjunction consists of a left and right adjoint functor together with equivalent unit/counit or hom-set data; a single functor need not possess an adjoint.

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)

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