Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The internal hom and its evaluation morphism
Definition
Assume is right closed. For objects , the right internal hom is the value at of the chosen right adjoint to (Left-closed, right-closed, and biclosed monoidal categories).
The evaluation morphism
is the counit component at of the adjunction . For every object , the adjunction yields the transposition bijection
which sends a morphism to its adjunct and sends to its inverse transpose in the sense of Adjuncts and transposition under an adjunction.
If is left closed, the left internal hom and its evaluation map
are defined dually from the adjunction .
Depends on
Used by
- The underlying ordinary category is change of base along the underlying-hom functor Corollary
- Tensor and cotensor in a V-category Definition
- The internal hom of abelian groups Example
- A supplied symmetry identifies the left and right internal homs Theorem
- Currying and uncurrying are mutually inverse Theorem
- Strong enriched Yoneda lemma as a particular end Theorem
- The free enriched category is left 2-adjoint to the underlying-category construction Theorem
- The internal hom preserves limits in the covariant variable and sends colimits to limits in the contravariant variable Theorem
- The internal-hom composition morphism Theorem
- The tensor unit is an internal-hom unit Theorem
- Weak enriched Yoneda lemma Theorem
Dependency tree · two levels
4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- G. M. Kelly, Basic Concepts of Enriched Category Theory, equations (1.23) and (1.24) (standard reference, not scraped)
- Emily Riehl, Category Theory in Context, 2nd ed., Section 4.4 (standard reference, not scraped)