Internal categories in monoidal categories

Internal categories in monoidal categories

[Mike Stay]

I'm having trouble understanding cotensors and the role they play in
categories internal to a monoidal category.

Say we take the data from a category internal to Set and apply the
"free complex-linear sums" functor to it.  The result is not quite a
category in Hilb, since the functor doesn't preserve limits: the
product in Set gets mapped to the tensor product in Hilb.  In
particular, the composition function gets mapped to the composition
linear transformation
o: C^{Mor sxt Mor} -> C^Mor
However, since we have a chosen basis, we can take a superposition of
composable morphisms and produce a superposition of compositions.

Is this an example of a category internal to a monoidal category, and
if so, is the Hilbert space C^{Mor sxt Mor} an example of a cotensor?

-- 
Mike Stay - metaweta@gmail.com
http://www.cs.auckland.ac.nz/~mike
http://reperiendi.wordpress.com


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Internal categories in monoidal categories

[Mike Stay]

By the way, the definition I'm using is the one from the nLab page:
https://ncatlab.org/nlab/show/internal+category+in+a+monoidal+category

On Sat, Oct 29, 2016 at 2:02 PM, Mike Stay <metaweta@gmail.com> wrote:
> I'm having trouble understanding cotensors and the role they play in
> categories internal to a monoidal category.
>
> Say we take the data from a category internal to Set and apply the
> "free complex-linear sums" functor to it.  The result is not quite a
> category in Hilb, since the functor doesn't preserve limits: the
> product in Set gets mapped to the tensor product in Hilb.  In
> particular, the composition function gets mapped to the composition
> linear transformation
> o: C^{Mor sxt Mor} -> C^Mor
> However, since we have a chosen basis, we can take a superposition of
> composable morphisms and produce a superposition of compositions.
>
> Is this an example of a category internal to a monoidal category, and
> if so, is the Hilbert space C^{Mor sxt Mor} an example of a cotensor?
>
> --
> Mike Stay - metaweta@gmail.com
> http://www.cs.auckland.ac.nz/~mike
> http://reperiendi.wordpress.com
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]



-- 
Mike Stay - metaweta@gmail.com
http://www.cs.auckland.ac.nz/~mike
http://reperiendi.wordpress.com


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]