# The Unapologetic Mathematician

## Categories with Duals

July 7, 2007 - Posted by | Category theory

1. It looks like you’ve got some misplaced dollar signs in the penultimate paragraph, after “so that we can just use $\eta_M$“.

Comment by Blake Stacey | July 8, 2007 | Reply

2. Thanks, fixed.

Comment by John Armstrong | July 8, 2007 | Reply

3. […] this sort of naturality before, though. If we read the diagram in , consider a monoidal category with duals, and use the functor , then this is exactly the sort of naturality we find in the duality arrows […]

Pingback by Extraordinary Naturality « The Unapologetic Mathematician | August 30, 2007 | Reply

4. […] Another thing vector spaces come with is duals. That is, given a vector space we have the dual vector space of “linear functionals” […]

Pingback by Dual Spaces « The Unapologetic Mathematician | May 27, 2008 | Reply

5. […] category of matrices also has duals. In fact, each object is self-dual! That is, we set . We then need our arrows and […]

Pingback by The Category of Matrices II « The Unapologetic Mathematician | June 3, 2008 | Reply

6. typoes in par 4: for every f:N->M an arrow f^*:M*->N*

Comment by Avery Andrews | July 19, 2008 | Reply

7. Thanks.

Comment by John Armstrong | July 19, 2008 | Reply

8. […] of a bialgebra is monoidal. What do we get for Hopf algebras? What does an antipode buy us? Duals! At least when we restrict to finite-dimensional […]

Pingback by Representations of Hopf Algebras I « The Unapologetic Mathematician | November 12, 2008 | Reply

9. […] Okay, I noticed that I never really gave the definition of the coevaluation when I introduced categories with duals, because you need some linear algebra. Well, now we have some linear algebra, so let’s do […]

Pingback by The Coevaluation on Vector Spaces « The Unapologetic Mathematician | November 13, 2008 | Reply

10. […] if that weren’t enough, has duals! Indeed, we have a cone in the dual vector space defined by if and only if for all . Or in other […]

Pingback by Ordered Linear Spaces II « The Unapologetic Mathematician | May 15, 2009 | Reply