===== My equivalence of categories =====
==== Collection ====
| @#55CCEE: context     | @#55CCEE: $F$ in ${\bf D}\longrightarrow{\bf C}$ |
| @#55CCEE: context     | @#55CCEE: $G$ in ${\bf C}\longrightarrow{\bf D}$ |
| @#FFBB00: definiendum | @#FFBB00: $\langle\alpha,\beta\rangle$ in $F\simeq G$ |
| @#AAFFAA: inclusion   | @#AAFFAA: $\alpha, \beta$ ... my nice nats $\left(F,G\right)$ |
| @#AAFFAA: inclusion   | @#AAFFAA: $\alpha,\beta$ ... natural isomorphism |

==== Discussion ====
=== Elaboration===
$\alpha$ in $FG\cong Id_{\bf C}$

$\beta$ in $Id_{\bf D}\cong GF$.

Note the two different symbols $\cong$ and $\simeq$. The first is about equivalences, the second about invertible gadgets.

=== Idea ===
This is like [[equivalence of categories]], except the natural transformations are not just required to exist but must be concretely specified. As such, this is a subset of [[my nice nats]].

=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Equivalence_of_categories|Equivalence of categories]]

nLab: [[http://ncatlab.org/nlab/show/equivalence+of+categories|Equivalence of categories]], [[http://ncatlab.org/nlab/show/principle+of+equivalence|Principle of equivalence]]

==== Parents ====
=== Context ===
[[Categories]]
=== Subset of ===
[[My nice nats]]
=== Refinement of ===
[[Equivalence of categories]]
=== Requirements ===
[[Natural isomorphism]]