Comments on: Spans and Vector Spaces – pt 3 http://theoreticalatlas.wordpress.com/2007/10/26/spans-and-vector-spaces-pt-3/ He had bought a large map representing the sea, / Without the least vestige of land: / And the crew were much pleased when they found it to be / A map they could all understand. Wed, 04 Nov 2009 16:14:49 +0000 http://wordpress.com/ hourly 1 By: David Corfield http://theoreticalatlas.wordpress.com/2007/10/26/spans-and-vector-spaces-pt-3/#comment-44 David Corfield Fri, 02 Nov 2007 09:40:58 +0000 http://theoreticalatlas.wordpress.com/2007/10/26/spans-and-vector-spaces-pt-3/#comment-44 Urs has frequently encouraged us to think about richer 2-vector spaces, e.g., <a href="http://golem.ph.utexas.edu/category/2006/10/topology_in_trondheim_and_kro.html" rel="nofollow">here</a>. Ultimately, should we be looking for non-integer dimensional 'vector spaces' for these matrix entries? Urs has frequently encouraged us to think about richer 2-vector spaces, e.g., here.

Ultimately, should we be looking for non-integer dimensional ‘vector spaces’ for these matrix entries?

]]> By: Jeffrey Morton http://theoreticalatlas.wordpress.com/2007/10/26/spans-and-vector-spaces-pt-3/#comment-43 Jeffrey Morton Thu, 01 Nov 2007 23:15:03 +0000 http://theoreticalatlas.wordpress.com/2007/10/26/spans-and-vector-spaces-pt-3/#comment-43 David: interesting question. I don't think extra labels are needed here. Any 2-linear map can be realized as a matrix of vector spaces, and I believe that any components you want can be realized as coming from some span. The components in the position labelled by some choice of objects in source and target, and by irreps, say, $latex U$ and $latex V$ of their automorphism groups, will get components of the form $latex \oplus_x U \otimes_{\mathbb{C}[G_x]} V$, where the sum is over all the objects $latex x$ in the middle which restrict to the specified ones, and the tensor product is reducing modulo the action of the automorphism group of $latex x$. I think you can choose the groupoid in the middle to get any vector space you like (up to isomorphism!) in this way. The reason you need something extra when you have spans of sets is the fact that components of a linear map need to be arbitrary complex numbers, and cardinalities of sets are too coarse. With KV 2-vector spaces, there is a kind of discreteness already built into the morphisms. The set of isomorphism classes of vector spaces is $latex \mathbb{N}$, rather than $latex \mathbb{C}$ - which is one way that (KV) $latex Vect$ is less than satisfying as a categorification of $latex \mathbb{C}$. Baez-Crans 2-vector spaces avoid this problem, but that's not what we get here. There are also Elgueta's "generalized 2-vector spaces": I don't know if these would have any room for the kind of generalization we're talking about here. There are similar considerations for normalization: you can reduce vector spaces modulo some group action, but "normalizing" them is not so straightforward. At the object level, 2-vector spaces resemble rig modules over $latex \mathbb{N}$ more than complex vector spaces. David: interesting question.

I don’t think extra labels are needed here. Any 2-linear map can be realized as a matrix of vector spaces, and I believe that any components you want can be realized as coming from some span. The components in the position labelled by some choice of objects in source and target, and by irreps, say, U and V of their automorphism groups, will get components of the form \oplus_x U \otimes_{\mathbb{C}[G_x]} V, where the sum is over all the objects x in the middle which restrict to the specified ones, and the tensor product is reducing modulo the action of the automorphism group of x. I think you can choose the groupoid in the middle to get any vector space you like (up to isomorphism!) in this way.

The reason you need something extra when you have spans of sets is the fact that components of a linear map need to be arbitrary complex numbers, and cardinalities of sets are too coarse. With KV 2-vector spaces, there is a kind of discreteness already built into the morphisms. The set of isomorphism classes of vector spaces is \mathbb{N}, rather than \mathbb{C} – which is one way that (KV) Vect is less than satisfying as a categorification of \mathbb{C}. Baez-Crans 2-vector spaces avoid this problem, but that’s not what we get here. There are also Elgueta’s “generalized 2-vector spaces”: I don’t know if these would have any room for the kind of generalization we’re talking about here.

There are similar considerations for normalization: you can reduce vector spaces modulo some group action, but “normalizing” them is not so straightforward.

At the object level, 2-vector spaces resemble rig modules over \mathbb{N} more than complex vector spaces.

]]> By: David Corfield http://theoreticalatlas.wordpress.com/2007/10/26/spans-and-vector-spaces-pt-3/#comment-42 David Corfield Thu, 01 Nov 2007 10:23:27 +0000 http://theoreticalatlas.wordpress.com/2007/10/26/spans-and-vector-spaces-pt-3/#comment-42 You said in part 1 "To get a general matrix, you’d have to give labels to X" when converting a span of sets into a map between vector spaces. Is something similar necessary in this new sitatuation when converting a span of groupoids into a 2-linear map? And, is there any sign that there's scope for some form of row/column normalization in these 2-linear map matrices? You said in part 1 “To get a general matrix, you’d have to give labels to X” when converting a span of sets into a map between vector spaces. Is something similar necessary in this new sitatuation when converting a span of groupoids into a 2-linear map?

And, is there any sign that there’s scope for some form of row/column normalization in these 2-linear map matrices?

]]> By: John Baez http://theoreticalatlas.wordpress.com/2007/10/26/spans-and-vector-spaces-pt-3/#comment-41 John Baez Tue, 30 Oct 2007 03:59:06 +0000 http://theoreticalatlas.wordpress.com/2007/10/26/spans-and-vector-spaces-pt-3/#comment-41 Good stuff! Hope you get healthy! typo: "how to transport a \mathbf{Vect}-valued functor" Good stuff! Hope you get healthy!

typo: “how to transport a \mathbf{Vect}-valued functor”

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