# Representation theorem

In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure.[1]

## Examples

### Category theory

• The Yoneda lemma provides a full and faithful limit-preserving embedding of any category into a category of presheaves.
• Mitchell's embedding theorem for abelian categories realises every small abelian category as a full (and exactly embedded) subcategory of a category of modules over some ring.[5]
• Mostowski's collapsing theorem states that every well-founded extensional structure is isomorphic to a transitive set with the ∈-relation.
• One of the fundamental theorems in sheaf theory states that every sheaf over a topological space can be thought of as a sheaf of sections of some (étalé) bundle over that space: the categories of sheaves on a topological space and that of étalé spaces over it are equivalent, where the equivalence is given by the functor that sends a bundle to its sheaf of (local) sections.

## References

1. "The Definitive Glossary of Higher Mathematical Jargon". Math Vault. 2019-08-01. Retrieved 2019-12-08.
2. "Cayley's Theorem and its Proof". www.sjsu.edu. Retrieved 2019-12-08.
3. Dirks, Matthew. "The Stone Representation Theorem for Boolean Algebras" (PDF). math.uchicago.edu. Retrieved 2019-12-08.
4. Schneider, Friedrich Martin (November 2017). "A uniform Birkhoff theorem". Algebra universalis. 78 (3): 337–354. doi:10.1007/s00012-017-0460-1. ISSN 0002-5240.
5. "Freyd-Mitchell embedding theorem in nLab". ncatlab.org. Retrieved 2019-12-08.
6. "Notes on the Nash embedding theorem". What's new. 2016-05-11. Retrieved 2019-12-08.