Timeline for Categorification of a vector space such that a functor between these is a linear map?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 11 at 6:11 | comment | added | Mike Shulman | Possibly related to mathoverflow.net/q/476/49 | |
| May 10 at 0:18 | comment | added | Emily | Taking $A=k$ then gives a correspondence between vector spaces and $\mathsf{Vect}_{k}$-enriched presheaves on $\mathbf{B}k$, which extends to an equivalence of categories $\mathsf{Vect}_{k}\cong\mathbf{PSh}_{\mathsf{Vect}_{k}}(\mathbf{B}k)$ (i.e. linear maps will correspond to $\mathsf{Vect}_k$-enriched natural transformations between those). | |
| May 10 at 0:18 | comment | added | Emily | Moving over to the setting of $\mathsf{Vect}_{k}$-enriched categories and replacing $\mathsf{Sets}$ by $\mathsf{Vect}_{k}$, the correspondence between monoids and one-object categories now becomes a correspondence between $k$-algebras $A$ and one-object $\mathsf{Vect}_{k}$-enriched categories $\mathbf{B}A$. Then, $\mathsf{Vect}_{k}$-enriched presheaves on $\mathbf{B}A$ will correspond exactly to left $A$-modules. | |
| May 10 at 0:18 | comment | added | Emily | Hi, @rick! I think a possible way to answer this question is via enriched categories and presheaves. If you take $A$ to be a monoid and write $\mathbf{B}A$ for the one-object category associated to it, then a presheaf on $\mathbf{B}A$, i.e. a functor $\mathbf{B}A^{\mathsf{op}}\to\mathsf{Sets}$, is precisely the data of a left $A$-set. | |
| May 9 at 21:35 | comment | added | rick | @AndrejBauer maybe? As long as everytime I have a linear map $f:\mathbb{R}^n\to\mathbb{R}^m$ I can translate that into a functor between two appropriate categories I'm happy | |
| May 9 at 20:48 | comment | added | Andrej Bauer | Are you asking how to embed the category of vector spaces and linear maps fully and faithfully into the category of categories? | |
| May 9 at 20:47 | comment | added | rick | @AlecRhea thank you! I think is going in the right direction. So in this case the multiplication with scalars would be given by an enriched structure of the groupoid? And we would require the functor to respect this enriched structure | |
| May 9 at 19:58 | comment | added | Alec Rhea | Well, a vector space is just a group with a scalar multiplication by the underlying field, and a group is just a category with one object where every morphism is invertible, so a natural (trivial) answer would be a one object groupoid together with a scalar multiplication by the underlying field. But there may be more interesting ‘categorical’ answers for the real/complex numbers; we can define a ‘field of scalars’ in any monoidal category if i remember correctly from categorical QM, but I’ll have to check my references when I get home tonight. | |
| May 9 at 19:46 | comment | added | Manuel Araújo | Sorry I think I misunderstood your question. I deleted my comment. | |
| May 9 at 19:24 | comment | added | rick | Hi @ManuelAraújo, thank you for your comment! I've looked at the links, but I'm still unclear on how the construction would work... could you please help me clarify what would the category and functor be in the case, say, if I want to categorify alinear map $f:\mathbb{R}^n\to\mathbb{R}^n$ | |
| May 9 at 17:34 | history | asked | rick | CC BY-SA 4.0 |