Some Theorems
Table of Contents
Modular categories are not determined by their modular data
A modular category \( \mathcal{C} \) is not determined by its \( S \) and \( T \) matrices.
Comments: If something seems too good to be true, it probably is.
Classification of commutative Frobenius algebras by TQFTs
For a field \( k \), there is an equivalence of categories \( \mathsf{2TQFT}_k \simeq \mathsf{cFrob}_k \) of 2-dimensional topological quantum field theories and commutative Frobenius algebras.
Comments: This was the first result I learned that expressed how some classical tensor algebras arise as categorical constructions. Essential to this equivalence is the classification of closed 1-dimensional manifolds and how well behaved the category \( \mathsf{2Cob} \) is. A significant amount of work is needed to even hypothesize a higher dimensional analogue. This is the cobordism hypothesis, proposed by Baez and Dolan.
The Yoneda Lemma
Let \( \mathsf{C} \) be a locally small category and \( F : \mathsf{C} \to \mathsf{Set} \) be a functor. Then \[ \operatorname*{Hom}(\operatorname*{Hom}(X,-),F) \cong FX \] and this isomorphism is natural in both \(X\) and \(F\).
Comments: This theorem is remarkable. The object on the left, as a collection of natural transformations, is seemingly incalculably large. Not only does this theorem tell us that this collection is a set, but it also gives an explicit description of these transformations, parameterized by \( FX \). When applied to \( F = \operatorname*{Hom}(Y,-) \) (or more generally representable functors), this theorem gives meaning to the intuitively-known idea that an object is uniquely determined by the maps into (our out of) it.