Tannakian Reconstruction
3 days ago
- #Functor
- #Tannakian Reconstruction
- #Category Theory
- Tannakian reconstruction is analogous to Bob's photo project: combining images from multiple functors to recover the original category's structure.
- A functor preserves the structure of a source category when mapping it into a target category, with morphisms between objects also mapped.
- Fiber functors map objects to sets (in the category of sets) and morphisms to functions, allowing probing of objects and their neighborhoods via natural transformations.
- To reconstruct hom-sets, consider natural transformations between fiber functors, expressed as an end, which combines all possible functors like superimposing photos.
- Functoriality ensures the end is non-trivial, as morphisms induce functions between sets, and the Yoneda embedding shows these correspond to original morphisms.
- Using Yoneda lemma and reduction, the original hom-set is recovered, demonstrating Tannakian reconstruction.
- An example applies reconstruction to a one-object category (a monoid), recovering the monoid from all its set representations via natural transformations.