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?
category-theory
asked Sep 10 at 20:08
Henry Story
23517
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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?
category-theory
asked Sep 10 at 20:08
Henry Story
23517
add a comment |Â
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?
category-theory
asked Sep 10 at 20:08
Henry Story
23517
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
category-theory
asked Sep 10 at 20:08
Henry Story
23517
asked Sep 10 at 20:08
Henry Story
23517
edited Sep 10 at 20:12
asked Sep 10 at 20:08
Henry Story
23517
asked Sep 10 at 20:08
Henry Story
23517
asked Sep 10 at 20:08
Henry Story
23517
23517
add a comment |Â
add a comment |Â
1 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.
answered Sep 10 at 20:11
Noah Schweber
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
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
answered Sep 10 at 20:11
Noah Schweber
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
add a comment |Â
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.
answered Sep 10 at 20:11
Noah Schweber
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
add a comment |Â
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.
answered Sep 10 at 20:11
Noah Schweber
113k9143267
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.
answered Sep 10 at 20:11
Noah Schweber
113k9143267
edited Sep 10 at 20:15
answered Sep 10 at 20:11
Noah Schweber
113k9143267
answered Sep 10 at 20:11
Noah Schweber
113k9143267
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
add a comment |Â
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
add a comment |Â
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