## You are here

Homenatural equivalence of categories

## Primary tabs

# natural equivalence of categories

###### Definition 0.1.

Let us consider two arbitrary categories $\mathcal{C}$ and $\mathcal{D}$.
A *natural equivalence of categories** is said to exist between two categories $\mathcal{C}$ and
$\mathcal{D}$ if and only if there is a covariant functor $E:\mathcal{C}\to\mathcal{D}$ which
is full and faithful, and that also has a full and faithful adjoint (that is either
a left- or right- adjoint).

* See ref. $[288]$ in the bibliography for category theory and algebraic topology.

# 0.0.1 Examples:

1. 2. 3. The category of crossed modules of $R$–algebroids is equivalent to the category of double $R$–algebroids with thin structure (Brown and Mosa, 1986, 2008.)

4. The categories of crossed modules of algebroids and of double algebroids with a connection pair are equivalent.

# References

- 1
Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences,
*Transactions of the American Mathematical Society*58: 231-294.

## Mathematics Subject Classification

18-00*no label found*18A25

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections