Natural isomorphism

Depends on

• Natural transformation and its components
• Identity natural transformation and vertical composition
• Functor category $[\mathcal C,\mathcal D]$
• Isomorphism, groupoid, and connected category

Used by

• Additive cocontinuous module functors admit right adjoints Corollary
• Eilenberg-Watts is a schematic equivalence of Hom categories Corollary
• Finite Eilenberg–Watts is a biequivalence Corollary
• Graded bimodule maps classify shift-compatible transformations Corollary
• Objects a and b are isomorphic exactly when C(-,a) and C(-,b) are naturally isomorphic Corollary
• The center is Morita invariant, via natural endomorphisms of the identity Corollary
• The left-to-right exact equivalence need not preserve the identity Counterexample
• Weak actions do not supply pentagon coherence data Counterexample
• Bicategories, pseudofunctors, and biequivalences Definition
• Braiding Definition
• Coherent action of a group on a category Definition
• Coherently shift-compatible functors and natural transformations Definition
• Equivalence, quasi-inverse, and adjoint equivalence of categories Definition
• Exact functor between triangulated categories Definition
• Faithful weak action Definition
• Lax, strong, and strict monoidal functors Definition
• Localization of a category at a class of morphisms Definition
• Monoidal category Definition
• Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors Definition
• Presheaves, covariantly and contravariantly representable functors, and representations Definition
• The category of right-module endofunctors Definition
• Weak action of a group on a category Definition
• Open-set and closed-set functors on Topᵒᵖ are naturally isomorphic by complements Example
• The distributive and exponential laws of sets are natural isomorphisms Example
• The kernel end and coend distinguish the regular and co-regular bimodules Example
• The opposite-group functor is naturally isomorphic to the identity functor by inversion Example
• Bilinear right exact functors are determined by their value on the regular modules Lemma
• Coherently shift-compatible functors and transformations form k-linear hom categories Lemma
• Finite Eilenberg–Watts kernels: explicit end and coend universal maps Lemma
• Finite module duality is exact with commuting bimodule actions Lemma
• Finite vector-space copowers in a k-linear abelian category Lemma
• Homogeneous free presentations prove the graded comparison is an isomorphism Lemma
• Nakayama kernels give well-defined adjoint functors Lemma
• Shift signs and exact-functor maps on triangulated K0 Lemma
• A natural transformation is a natural isomorphism exactly when every component is an isomorphism Proposition
• A translation-compatible natural isomorphism of exact functors respects triangles Proposition
• Equivalences preserve, reflect, and create limits and colimits in the isomorphism-invariant sense Proposition
• The left-to-right exact equivalence sends the identity to the Nakayama functor Proposition
• The projective Nakayama pairing and the symmetric-algebra specialization Proposition
• A category with finite products is monoidal Theorem

…and 8 more results.

Dependency tree · two levels

8 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

• Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)