Coproduct in the FdHilb category

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Consider the FdHilb category i.e. the category of finite dimensional Hilbert spaces that is important especially in the quantum mechanics.



I have some trouble with identification of coproduct in the category. What is it? Any help, hints are welcome.



Edit: The original question incorrectly identified the tensor product as the categorial product. It is very important in the quantum mechanics because it is used to describe compound systems. It will be also fine to get a categorial construction for the tensor product if it is possible




category-theory hilbert-spaces tensor-products




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  • Tensor product is not the product, at least in the category of finite dimensional vector spaces. Can you clarify what are the maps in your category?
    – Arnaud D.
    2 days ago










  • As I understood the morphisms in the category are the linear maps (between different Hilber spaces) that preserve the structure i.e. such that $f(c_1 a_1 + c_2 a_2) = c_1 f(a_1) + c_2 f(a_2)$ where $c_1, c_2 in mathbbC$
    – Ivan
    2 days ago















up vote
1
down vote

favorite
1












Consider the FdHilb category i.e. the category of finite dimensional Hilbert spaces that is important especially in the quantum mechanics.



I have some trouble with identification of coproduct in the category. What is it? Any help, hints are welcome.



Edit: The original question incorrectly identified the tensor product as the categorial product. It is very important in the quantum mechanics because it is used to describe compound systems. It will be also fine to get a categorial construction for the tensor product if it is possible




category-theory hilbert-spaces tensor-products




share|cite|improve this question





















  • Tensor product is not the product, at least in the category of finite dimensional vector spaces. Can you clarify what are the maps in your category?
    – Arnaud D.
    2 days ago










  • As I understood the morphisms in the category are the linear maps (between different Hilber spaces) that preserve the structure i.e. such that $f(c_1 a_1 + c_2 a_2) = c_1 f(a_1) + c_2 f(a_2)$ where $c_1, c_2 in mathbbC$
    – Ivan
    2 days ago













up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





Consider the FdHilb category i.e. the category of finite dimensional Hilbert spaces that is important especially in the quantum mechanics.



I have some trouble with identification of coproduct in the category. What is it? Any help, hints are welcome.



Edit: The original question incorrectly identified the tensor product as the categorial product. It is very important in the quantum mechanics because it is used to describe compound systems. It will be also fine to get a categorial construction for the tensor product if it is possible




category-theory hilbert-spaces tensor-products




share|cite|improve this question













Consider the FdHilb category i.e. the category of finite dimensional Hilbert spaces that is important especially in the quantum mechanics.



I have some trouble with identification of coproduct in the category. What is it? Any help, hints are welcome.



Edit: The original question incorrectly identified the tensor product as the categorial product. It is very important in the quantum mechanics because it is used to describe compound systems. It will be also fine to get a categorial construction for the tensor product if it is possible





category-theory hilbert-spaces tensor-products





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited 2 days ago
























asked 2 days ago









Ivan

696




696











  • Tensor product is not the product, at least in the category of finite dimensional vector spaces. Can you clarify what are the maps in your category?
    – Arnaud D.
    2 days ago










  • As I understood the morphisms in the category are the linear maps (between different Hilber spaces) that preserve the structure i.e. such that $f(c_1 a_1 + c_2 a_2) = c_1 f(a_1) + c_2 f(a_2)$ where $c_1, c_2 in mathbbC$
    – Ivan
    2 days ago

















  • Tensor product is not the product, at least in the category of finite dimensional vector spaces. Can you clarify what are the maps in your category?
    – Arnaud D.
    2 days ago










  • As I understood the morphisms in the category are the linear maps (between different Hilber spaces) that preserve the structure i.e. such that $f(c_1 a_1 + c_2 a_2) = c_1 f(a_1) + c_2 f(a_2)$ where $c_1, c_2 in mathbbC$
    – Ivan
    2 days ago
















Tensor product is not the product, at least in the category of finite dimensional vector spaces. Can you clarify what are the maps in your category?
– Arnaud D.
2 days ago




Tensor product is not the product, at least in the category of finite dimensional vector spaces. Can you clarify what are the maps in your category?
– Arnaud D.
2 days ago












As I understood the morphisms in the category are the linear maps (between different Hilber spaces) that preserve the structure i.e. such that $f(c_1 a_1 + c_2 a_2) = c_1 f(a_1) + c_2 f(a_2)$ where $c_1, c_2 in mathbbC$
– Ivan
2 days ago





As I understood the morphisms in the category are the linear maps (between different Hilber spaces) that preserve the structure i.e. such that $f(c_1 a_1 + c_2 a_2) = c_1 f(a_1) + c_2 f(a_2)$ where $c_1, c_2 in mathbbC$
– Ivan
2 days ago











2 Answers
2






active

oldest

votes

















up vote
1
down vote



accepted










One of the features of FdHilb (the cateogry of finite dimensional Hilbert spaces) is that it possesses finite biproducts. A biproduct of a finite collection of objects is both a product and a coproduct.



In FdHilb the construction that is both a product and a coproduct is direct sum of Hilbert spaces, not tensor product. According to the definition on Wikipedia, tensor product is not a categorical product: there are not natural maps from a tensor product of two spaces to the two spaces separately, etc.






share|cite|improve this answer























  • Thanks for the answer. As I understood it is not possible to find any categorial construction for the tensor product. It is very strange for me especially because the tensor product is so important for the theory application and without it the categorial view on the quantum mechanic will be incomplete
    – Ivan
    2 days ago






  • 1




    @Ivan - Nature is not a good categorist ;-)
    – Taroccoesbrocco
    2 days ago






  • 1




    @Ivan According to wikipedia, the tensor product $H_1otimes H_2$ represents the covariant functor $K mapsto mathsfHS(H_1,H_2;K)$ where the latter is the set of Hilbert-Schmidt maps from $H_1times H_2$ to $K$.
    – Pece
    2 days ago


















up vote
1
down vote













Be careful! It is not at all clear what the morphisms in $textFdHilb$ should be. If you want them to be all linear maps, then the category you end up with is not really a category of Hilbert spaces, and in fact is equivalent to the category of finite-dimensional complex vector spaces; in other words this choice of morphism effectively forgets the inner product.



You could ask for the morphisms to be unitary maps or isometries, which does take the inner product into account, but this choice of morphism is very restrictive and hardly any categorical constructions exist in this category.



The "correct" thing to do appears to be to regard $textFdHilb$ as something more complicated than a category: it is really a dagger category where the dagger is given by taking adjoints and this is somehow crucial to understanding its categorical nature. I do not really understand how "dagger category theory" is supposed to work, and in particular how to adjust the notion of limits and colimits to take the dagger into account.






share|cite|improve this answer





















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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    1
    down vote



    accepted










    One of the features of FdHilb (the cateogry of finite dimensional Hilbert spaces) is that it possesses finite biproducts. A biproduct of a finite collection of objects is both a product and a coproduct.



    In FdHilb the construction that is both a product and a coproduct is direct sum of Hilbert spaces, not tensor product. According to the definition on Wikipedia, tensor product is not a categorical product: there are not natural maps from a tensor product of two spaces to the two spaces separately, etc.






    share|cite|improve this answer























    • Thanks for the answer. As I understood it is not possible to find any categorial construction for the tensor product. It is very strange for me especially because the tensor product is so important for the theory application and without it the categorial view on the quantum mechanic will be incomplete
      – Ivan
      2 days ago






    • 1




      @Ivan - Nature is not a good categorist ;-)
      – Taroccoesbrocco
      2 days ago






    • 1




      @Ivan According to wikipedia, the tensor product $H_1otimes H_2$ represents the covariant functor $K mapsto mathsfHS(H_1,H_2;K)$ where the latter is the set of Hilbert-Schmidt maps from $H_1times H_2$ to $K$.
      – Pece
      2 days ago















    up vote
    1
    down vote



    accepted










    One of the features of FdHilb (the cateogry of finite dimensional Hilbert spaces) is that it possesses finite biproducts. A biproduct of a finite collection of objects is both a product and a coproduct.



    In FdHilb the construction that is both a product and a coproduct is direct sum of Hilbert spaces, not tensor product. According to the definition on Wikipedia, tensor product is not a categorical product: there are not natural maps from a tensor product of two spaces to the two spaces separately, etc.






    share|cite|improve this answer























    • Thanks for the answer. As I understood it is not possible to find any categorial construction for the tensor product. It is very strange for me especially because the tensor product is so important for the theory application and without it the categorial view on the quantum mechanic will be incomplete
      – Ivan
      2 days ago






    • 1




      @Ivan - Nature is not a good categorist ;-)
      – Taroccoesbrocco
      2 days ago






    • 1




      @Ivan According to wikipedia, the tensor product $H_1otimes H_2$ represents the covariant functor $K mapsto mathsfHS(H_1,H_2;K)$ where the latter is the set of Hilbert-Schmidt maps from $H_1times H_2$ to $K$.
      – Pece
      2 days ago













    up vote
    1
    down vote



    accepted







    up vote
    1
    down vote



    accepted






    One of the features of FdHilb (the cateogry of finite dimensional Hilbert spaces) is that it possesses finite biproducts. A biproduct of a finite collection of objects is both a product and a coproduct.



    In FdHilb the construction that is both a product and a coproduct is direct sum of Hilbert spaces, not tensor product. According to the definition on Wikipedia, tensor product is not a categorical product: there are not natural maps from a tensor product of two spaces to the two spaces separately, etc.






    share|cite|improve this answer















    One of the features of FdHilb (the cateogry of finite dimensional Hilbert spaces) is that it possesses finite biproducts. A biproduct of a finite collection of objects is both a product and a coproduct.



    In FdHilb the construction that is both a product and a coproduct is direct sum of Hilbert spaces, not tensor product. According to the definition on Wikipedia, tensor product is not a categorical product: there are not natural maps from a tensor product of two spaces to the two spaces separately, etc.







    share|cite|improve this answer















    share|cite|improve this answer



    share|cite|improve this answer








    edited 2 days ago


























    answered 2 days ago









    Taroccoesbrocco

    3,31941230




    3,31941230











    • Thanks for the answer. As I understood it is not possible to find any categorial construction for the tensor product. It is very strange for me especially because the tensor product is so important for the theory application and without it the categorial view on the quantum mechanic will be incomplete
      – Ivan
      2 days ago






    • 1




      @Ivan - Nature is not a good categorist ;-)
      – Taroccoesbrocco
      2 days ago






    • 1




      @Ivan According to wikipedia, the tensor product $H_1otimes H_2$ represents the covariant functor $K mapsto mathsfHS(H_1,H_2;K)$ where the latter is the set of Hilbert-Schmidt maps from $H_1times H_2$ to $K$.
      – Pece
      2 days ago

















    • Thanks for the answer. As I understood it is not possible to find any categorial construction for the tensor product. It is very strange for me especially because the tensor product is so important for the theory application and without it the categorial view on the quantum mechanic will be incomplete
      – Ivan
      2 days ago






    • 1




      @Ivan - Nature is not a good categorist ;-)
      – Taroccoesbrocco
      2 days ago






    • 1




      @Ivan According to wikipedia, the tensor product $H_1otimes H_2$ represents the covariant functor $K mapsto mathsfHS(H_1,H_2;K)$ where the latter is the set of Hilbert-Schmidt maps from $H_1times H_2$ to $K$.
      – Pece
      2 days ago
















    Thanks for the answer. As I understood it is not possible to find any categorial construction for the tensor product. It is very strange for me especially because the tensor product is so important for the theory application and without it the categorial view on the quantum mechanic will be incomplete
    – Ivan
    2 days ago




    Thanks for the answer. As I understood it is not possible to find any categorial construction for the tensor product. It is very strange for me especially because the tensor product is so important for the theory application and without it the categorial view on the quantum mechanic will be incomplete
    – Ivan
    2 days ago




    1




    1




    @Ivan - Nature is not a good categorist ;-)
    – Taroccoesbrocco
    2 days ago




    @Ivan - Nature is not a good categorist ;-)
    – Taroccoesbrocco
    2 days ago




    1




    1




    @Ivan According to wikipedia, the tensor product $H_1otimes H_2$ represents the covariant functor $K mapsto mathsfHS(H_1,H_2;K)$ where the latter is the set of Hilbert-Schmidt maps from $H_1times H_2$ to $K$.
    – Pece
    2 days ago





    @Ivan According to wikipedia, the tensor product $H_1otimes H_2$ represents the covariant functor $K mapsto mathsfHS(H_1,H_2;K)$ where the latter is the set of Hilbert-Schmidt maps from $H_1times H_2$ to $K$.
    – Pece
    2 days ago











    up vote
    1
    down vote













    Be careful! It is not at all clear what the morphisms in $textFdHilb$ should be. If you want them to be all linear maps, then the category you end up with is not really a category of Hilbert spaces, and in fact is equivalent to the category of finite-dimensional complex vector spaces; in other words this choice of morphism effectively forgets the inner product.



    You could ask for the morphisms to be unitary maps or isometries, which does take the inner product into account, but this choice of morphism is very restrictive and hardly any categorical constructions exist in this category.



    The "correct" thing to do appears to be to regard $textFdHilb$ as something more complicated than a category: it is really a dagger category where the dagger is given by taking adjoints and this is somehow crucial to understanding its categorical nature. I do not really understand how "dagger category theory" is supposed to work, and in particular how to adjust the notion of limits and colimits to take the dagger into account.






    share|cite|improve this answer

























      up vote
      1
      down vote













      Be careful! It is not at all clear what the morphisms in $textFdHilb$ should be. If you want them to be all linear maps, then the category you end up with is not really a category of Hilbert spaces, and in fact is equivalent to the category of finite-dimensional complex vector spaces; in other words this choice of morphism effectively forgets the inner product.



      You could ask for the morphisms to be unitary maps or isometries, which does take the inner product into account, but this choice of morphism is very restrictive and hardly any categorical constructions exist in this category.



      The "correct" thing to do appears to be to regard $textFdHilb$ as something more complicated than a category: it is really a dagger category where the dagger is given by taking adjoints and this is somehow crucial to understanding its categorical nature. I do not really understand how "dagger category theory" is supposed to work, and in particular how to adjust the notion of limits and colimits to take the dagger into account.






      share|cite|improve this answer























        up vote
        1
        down vote










        up vote
        1
        down vote









        Be careful! It is not at all clear what the morphisms in $textFdHilb$ should be. If you want them to be all linear maps, then the category you end up with is not really a category of Hilbert spaces, and in fact is equivalent to the category of finite-dimensional complex vector spaces; in other words this choice of morphism effectively forgets the inner product.



        You could ask for the morphisms to be unitary maps or isometries, which does take the inner product into account, but this choice of morphism is very restrictive and hardly any categorical constructions exist in this category.



        The "correct" thing to do appears to be to regard $textFdHilb$ as something more complicated than a category: it is really a dagger category where the dagger is given by taking adjoints and this is somehow crucial to understanding its categorical nature. I do not really understand how "dagger category theory" is supposed to work, and in particular how to adjust the notion of limits and colimits to take the dagger into account.






        share|cite|improve this answer













        Be careful! It is not at all clear what the morphisms in $textFdHilb$ should be. If you want them to be all linear maps, then the category you end up with is not really a category of Hilbert spaces, and in fact is equivalent to the category of finite-dimensional complex vector spaces; in other words this choice of morphism effectively forgets the inner product.



        You could ask for the morphisms to be unitary maps or isometries, which does take the inner product into account, but this choice of morphism is very restrictive and hardly any categorical constructions exist in this category.



        The "correct" thing to do appears to be to regard $textFdHilb$ as something more complicated than a category: it is really a dagger category where the dagger is given by taking adjoints and this is somehow crucial to understanding its categorical nature. I do not really understand how "dagger category theory" is supposed to work, and in particular how to adjust the notion of limits and colimits to take the dagger into account.







        share|cite|improve this answer













        share|cite|improve this answer



        share|cite|improve this answer











        answered 2 days ago









        Qiaochu Yuan

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