What is a Hyper-Category?

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I am reading the 2008 book "Institution Independent Model Theory" by Razvan Diaconescu and looking up what the category ℂat was it states in the symbol Index




ℂat: the hyper-category of categories as objects and functors as arrows




But it nowhere defines what a hyper-category is. Is this just a way of saying that it is a Category containing Categories?










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    up vote
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    down vote

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    I am reading the 2008 book "Institution Independent Model Theory" by Razvan Diaconescu and looking up what the category ℂat was it states in the symbol Index




    ℂat: the hyper-category of categories as objects and functors as arrows




    But it nowhere defines what a hyper-category is. Is this just a way of saying that it is a Category containing Categories?










    share|cite|improve this question

























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I am reading the 2008 book "Institution Independent Model Theory" by Razvan Diaconescu and looking up what the category ℂat was it states in the symbol Index




      ℂat: the hyper-category of categories as objects and functors as arrows




      But it nowhere defines what a hyper-category is. Is this just a way of saying that it is a Category containing Categories?










      share|cite|improve this question















      I am reading the 2008 book "Institution Independent Model Theory" by Razvan Diaconescu and looking up what the category ℂat was it states in the symbol Index




      ℂat: the hyper-category of categories as objects and functors as arrows




      But it nowhere defines what a hyper-category is. Is this just a way of saying that it is a Category containing Categories?







      category-theory






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      edited Sep 10 at 20:12

























      asked Sep 10 at 20:08









      Henry Story

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          It's analogous to the idea of a "hyper-class," which is to classes what classes are to sets. Basically, in an appropriate background theory, a hypercategory consists of a hyperclass of objects and a hyperclass of morphisms, as opposed to a class of objects and a class of morphisms (= category) or a set of objects and a set of morphisms (= small category).



          Note that to even talk about such things, we need to enrich our background set theory in a manner similar to how we move from ZFC to NBG. However, we can do this in a "conservative way," so this isn't as big a deal as it sounds.






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          • A so essentially it's very big. :-)
            – Henry Story
            Sep 10 at 20:13






          • 1




            @HenryStory Yup. And the "category" of hypercategories will be a hyperhypercategory, and etc.
            – Noah Schweber
            Sep 10 at 20:13










          • Your answer fits with what the author of the book writes in a more recent 2015 article in Internet Encyclopedia of Philosophy on Institution Theory, which being philosophical, explains some of the basics a lot better than the book. He writes "The Σ-models may be so many that they may not constitute a set anymore. In examples this is closely related to the fact that there does not exists ‘the set of all sets’, which would be a violation of one of the axioms of formal set theory." iep.utm.edu/insti-th
            – Henry Story
            Sep 11 at 11:14










          Your Answer




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          It's analogous to the idea of a "hyper-class," which is to classes what classes are to sets. Basically, in an appropriate background theory, a hypercategory consists of a hyperclass of objects and a hyperclass of morphisms, as opposed to a class of objects and a class of morphisms (= category) or a set of objects and a set of morphisms (= small category).



          Note that to even talk about such things, we need to enrich our background set theory in a manner similar to how we move from ZFC to NBG. However, we can do this in a "conservative way," so this isn't as big a deal as it sounds.






          share|cite|improve this answer






















          • A so essentially it's very big. :-)
            – Henry Story
            Sep 10 at 20:13






          • 1




            @HenryStory Yup. And the "category" of hypercategories will be a hyperhypercategory, and etc.
            – Noah Schweber
            Sep 10 at 20:13










          • Your answer fits with what the author of the book writes in a more recent 2015 article in Internet Encyclopedia of Philosophy on Institution Theory, which being philosophical, explains some of the basics a lot better than the book. He writes "The Σ-models may be so many that they may not constitute a set anymore. In examples this is closely related to the fact that there does not exists ‘the set of all sets’, which would be a violation of one of the axioms of formal set theory." iep.utm.edu/insti-th
            – Henry Story
            Sep 11 at 11:14














          up vote
          2
          down vote



          accepted










          It's analogous to the idea of a "hyper-class," which is to classes what classes are to sets. Basically, in an appropriate background theory, a hypercategory consists of a hyperclass of objects and a hyperclass of morphisms, as opposed to a class of objects and a class of morphisms (= category) or a set of objects and a set of morphisms (= small category).



          Note that to even talk about such things, we need to enrich our background set theory in a manner similar to how we move from ZFC to NBG. However, we can do this in a "conservative way," so this isn't as big a deal as it sounds.






          share|cite|improve this answer






















          • A so essentially it's very big. :-)
            – Henry Story
            Sep 10 at 20:13






          • 1




            @HenryStory Yup. And the "category" of hypercategories will be a hyperhypercategory, and etc.
            – Noah Schweber
            Sep 10 at 20:13










          • Your answer fits with what the author of the book writes in a more recent 2015 article in Internet Encyclopedia of Philosophy on Institution Theory, which being philosophical, explains some of the basics a lot better than the book. He writes "The Σ-models may be so many that they may not constitute a set anymore. In examples this is closely related to the fact that there does not exists ‘the set of all sets’, which would be a violation of one of the axioms of formal set theory." iep.utm.edu/insti-th
            – Henry Story
            Sep 11 at 11:14












          up vote
          2
          down vote



          accepted







          up vote
          2
          down vote



          accepted






          It's analogous to the idea of a "hyper-class," which is to classes what classes are to sets. Basically, in an appropriate background theory, a hypercategory consists of a hyperclass of objects and a hyperclass of morphisms, as opposed to a class of objects and a class of morphisms (= category) or a set of objects and a set of morphisms (= small category).



          Note that to even talk about such things, we need to enrich our background set theory in a manner similar to how we move from ZFC to NBG. However, we can do this in a "conservative way," so this isn't as big a deal as it sounds.






          share|cite|improve this answer














          It's analogous to the idea of a "hyper-class," which is to classes what classes are to sets. Basically, in an appropriate background theory, a hypercategory consists of a hyperclass of objects and a hyperclass of morphisms, as opposed to a class of objects and a class of morphisms (= category) or a set of objects and a set of morphisms (= small category).



          Note that to even talk about such things, we need to enrich our background set theory in a manner similar to how we move from ZFC to NBG. However, we can do this in a "conservative way," so this isn't as big a deal as it sounds.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Sep 10 at 20:15

























          answered Sep 10 at 20:11









          Noah Schweber

          113k9143267




          113k9143267











          • A so essentially it's very big. :-)
            – Henry Story
            Sep 10 at 20:13






          • 1




            @HenryStory Yup. And the "category" of hypercategories will be a hyperhypercategory, and etc.
            – Noah Schweber
            Sep 10 at 20:13










          • Your answer fits with what the author of the book writes in a more recent 2015 article in Internet Encyclopedia of Philosophy on Institution Theory, which being philosophical, explains some of the basics a lot better than the book. He writes "The Σ-models may be so many that they may not constitute a set anymore. In examples this is closely related to the fact that there does not exists ‘the set of all sets’, which would be a violation of one of the axioms of formal set theory." iep.utm.edu/insti-th
            – Henry Story
            Sep 11 at 11:14
















          • A so essentially it's very big. :-)
            – Henry Story
            Sep 10 at 20:13






          • 1




            @HenryStory Yup. And the "category" of hypercategories will be a hyperhypercategory, and etc.
            – Noah Schweber
            Sep 10 at 20:13










          • Your answer fits with what the author of the book writes in a more recent 2015 article in Internet Encyclopedia of Philosophy on Institution Theory, which being philosophical, explains some of the basics a lot better than the book. He writes "The Σ-models may be so many that they may not constitute a set anymore. In examples this is closely related to the fact that there does not exists ‘the set of all sets’, which would be a violation of one of the axioms of formal set theory." iep.utm.edu/insti-th
            – Henry Story
            Sep 11 at 11:14















          A so essentially it's very big. :-)
          – Henry Story
          Sep 10 at 20:13




          A so essentially it's very big. :-)
          – Henry Story
          Sep 10 at 20:13




          1




          1




          @HenryStory Yup. And the "category" of hypercategories will be a hyperhypercategory, and etc.
          – Noah Schweber
          Sep 10 at 20:13




          @HenryStory Yup. And the "category" of hypercategories will be a hyperhypercategory, and etc.
          – Noah Schweber
          Sep 10 at 20:13












          Your answer fits with what the author of the book writes in a more recent 2015 article in Internet Encyclopedia of Philosophy on Institution Theory, which being philosophical, explains some of the basics a lot better than the book. He writes "The Σ-models may be so many that they may not constitute a set anymore. In examples this is closely related to the fact that there does not exists ‘the set of all sets’, which would be a violation of one of the axioms of formal set theory." iep.utm.edu/insti-th
          – Henry Story
          Sep 11 at 11:14




          Your answer fits with what the author of the book writes in a more recent 2015 article in Internet Encyclopedia of Philosophy on Institution Theory, which being philosophical, explains some of the basics a lot better than the book. He writes "The Σ-models may be so many that they may not constitute a set anymore. In examples this is closely related to the fact that there does not exists ‘the set of all sets’, which would be a violation of one of the axioms of formal set theory." iep.utm.edu/insti-th
          – Henry Story
          Sep 11 at 11:14

















           

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