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Equivalence of categories
Collection
| context | ${\bf C},{\bf D}$ … categories |
| definiendum | $F$ in ${\bf D}\simeq{\bf C}$ |
| inclusion | $F$ in ${\bf D}\longrightarrow{\bf C}$ |
| exists | $G$ in ${\bf C}\longrightarrow{\bf D}$ |
| exists | $\alpha$ in $FG\cong Id_{\bf C}$ |
| exists | $\beta$ in $Id_{\bf D}\cong GF$ |
Discussion
Elaboration
Here $Id_{\bf C}$ denotes the identity functor on ${\bf C}$.
In words
Two categories are equivalent, if there is are functors $F,G$ mediating between them, which don't deform them too much: They are tame enough so that their composite deformations can be repaired back to unity.
Motivation
We want to formalize when two categories are “exchangeable in terms of results which can be workout in them”. We would call a functors $F$ “invertible” if there is a another functor $G$ so that $G\circ F=Id_{\bf D}$ and $F\circ G=Id_{\bf C}$. However, since one is generally interested in statements involving objects up to isomorphism only, we define a functor to give an “equivalence” by the above definition.
Comparison with homotopy theory: The postulate then looks essentially the same as the one for homotopy equivalent spaces. The natural transformations $\alpha,\beta$ play the role of homotopy equivalences. Working up to isomorphism amounts to working with just any one representative of a homotopy type.
See also my equivalence of categories.
Reference
Wikipedia: Equivalence of categories
Parents
Context
Requirements
