Can we see FdHlb as a 2Category of groupoids?

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Can we see a finite dimensional Hilbert space, $H$ as a groupoid if we include the unitary endomorphisms of $H$? It would be like a category with a single object and just isos.



If so, can we take a category of all such objects and put them together to form a 2category of groupoids?



Ben




category-theory quantum-mechanics groupoids 2-categories




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    Can we see a finite dimensional Hilbert space, $H$ as a groupoid if we include the unitary endomorphisms of $H$? It would be like a category with a single object and just isos.



    If so, can we take a category of all such objects and put them together to form a 2category of groupoids?



    Ben




    category-theory quantum-mechanics groupoids 2-categories




    share|cite|improve this question





















      up vote
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      up vote
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      down vote

      favorite











      Can we see a finite dimensional Hilbert space, $H$ as a groupoid if we include the unitary endomorphisms of $H$? It would be like a category with a single object and just isos.



      If so, can we take a category of all such objects and put them together to form a 2category of groupoids?



      Ben




      category-theory quantum-mechanics groupoids 2-categories




      share|cite|improve this question











      Can we see a finite dimensional Hilbert space, $H$ as a groupoid if we include the unitary endomorphisms of $H$? It would be like a category with a single object and just isos.



      If so, can we take a category of all such objects and put them together to form a 2category of groupoids?



      Ben





      category-theory quantum-mechanics groupoids 2-categories





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      asked Jul 14 at 19:47









      Ben Sprott

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          This is not viewing the Hilbert space as a groupoid, it's looking at a category whose only object is that space as a groupoid. You can do this, as you can with literally any object in any category, by taking the group of automorphisms. You could put all such categories together into a sub-2-category of the 2-category of groupoids, if you wanted to. But again, this possibility exists in absolutely every category. It's generally of limited interest.






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            This is not viewing the Hilbert space as a groupoid, it's looking at a category whose only object is that space as a groupoid. You can do this, as you can with literally any object in any category, by taking the group of automorphisms. You could put all such categories together into a sub-2-category of the 2-category of groupoids, if you wanted to. But again, this possibility exists in absolutely every category. It's generally of limited interest.






            share|cite|improve this answer

























              up vote
              2
              down vote



              accepted










              This is not viewing the Hilbert space as a groupoid, it's looking at a category whose only object is that space as a groupoid. You can do this, as you can with literally any object in any category, by taking the group of automorphisms. You could put all such categories together into a sub-2-category of the 2-category of groupoids, if you wanted to. But again, this possibility exists in absolutely every category. It's generally of limited interest.






              share|cite|improve this answer























                up vote
                2
                down vote



                accepted







                up vote
                2
                down vote



                accepted






                This is not viewing the Hilbert space as a groupoid, it's looking at a category whose only object is that space as a groupoid. You can do this, as you can with literally any object in any category, by taking the group of automorphisms. You could put all such categories together into a sub-2-category of the 2-category of groupoids, if you wanted to. But again, this possibility exists in absolutely every category. It's generally of limited interest.






                share|cite|improve this answer













                This is not viewing the Hilbert space as a groupoid, it's looking at a category whose only object is that space as a groupoid. You can do this, as you can with literally any object in any category, by taking the group of automorphisms. You could put all such categories together into a sub-2-category of the 2-category of groupoids, if you wanted to. But again, this possibility exists in absolutely every category. It's generally of limited interest.







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                answered Jul 15 at 16:15









                Kevin Carlson

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