Timeline for Idempotent completion of linear categories and Yoneda

6 events

when toggle format	what		by	license	comment
Mar 28, 2019 at 21:42	history	edited	Qiaochu Yuan	CC BY-SA 4.0	added 144 characters in body
Mar 28, 2019 at 11:29	comment	added	Jeremy Rickard		@Arthur Doesn’t your argument require that $\mathcal{C}$ has only finitely many isomorphism classes of simple objects?
Mar 28, 2019 at 9:43	history	edited	Qiaochu Yuan	CC BY-SA 4.0	deleted 41 characters in body
Mar 27, 2019 at 23:05	comment	added	Arthur		To see that in this case semisimplicity implies the desired property consider a functor $ F $ in $ \text{Fun}(\mathcal{C}^{\text{op}}, \text{Vect}) $. The map $$ F(X) = \bigoplus\limits_S \text{Hom}(S,X)^* \otimes F(S) = \bigoplus\limits_S \text{Hom}(X,S)\otimes \text{Hom}(\mathbb{Y}(S),F)\\ = \bigoplus\limits_S \text{Hom}(\mathbb{Y}(S),F) \otimes \mathbb{Y}(S)(X) $$ identifies $ F $ with an object in the additive subcategory generated by direct summands of $ \mathbb{Y}(X) $.
Mar 27, 2019 at 22:52	comment	added	Arthur		Sorry, I should have specified that $ \text{Fun}(\mathcal{C}^{\text{op}}, \text{Vect}) $ is the linear functors and that $ \text{Vect} $ is the category of finite dimensional vector spaces and that $ \mathcal{C} $ is only enriched over $ \text{Vect} $. I have edited the question appropriately.
Mar 27, 2019 at 20:53	history	answered	Qiaochu Yuan	CC BY-SA 4.0