Sameness, Adjunction, Duality
Classify examples of equivalence and relate them to adjunctions.
- Compare my diagram of adjunction-equivalence-isomorphism with the template that I made for adjunction.
- Collect examples of equivalences that are not isomorphisms. Organize their taxonomy and relate it to the taxonomy of adjunctions.
读物
Lists of equivalences
Examples of equivalences that are not isomorphisms
- Consider a discrete category without arrows. Map twice, that is a relabeling of a relabeling.
- Consider the preorder Z. Shift twice.
- Consider the preorder Z. Take it to the opposite direction. Reverse the arrows.
- Map a group to the opposite group, an element to its inverse.
- Consider various automorphisms, self-equivalences.
Consider the category {$C$} having a single object {$c$} and a single morphism {$1_c$}, and the category {$D$} with two objects {$d_1$}, {$d_2$} and four morphisms: two identity morphisms {$1_{d_1}$}, {$1_{d_2}$} and two isomorphisms {$α : d_1 \rightarrow d_2$} and {$β : d_2 \rightarrow d_1$}. The categories {$C$} and {$D$} are equivalent; we can (for example) have {$F$} map {$c$} to {$d_1$} and {$G$} map both objects of {$D$} to {$c$} and all morphisms to {$1_c$}.
- Isomorphism is based on the internal view of structures from within them. Equality considers the external view upon structures as components.
- Structures may be isomorphic but as changes are made within a system - perhaps one structure is assigned a particular role - then distinctions arise. For example, a structure may have an owner, it may be their personal property.