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Ordinals in category theory?
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I am just learning category theory and I am wondering if there is some way to define ordinals, and ordinal arithmetic purely through categorical means. I have a notion of taking a category with 0, 1, 2, etc elements but I have no idea how you could define addition on these categories with the usual nice properties (associativity, commutativity in the finite case, etc). I also don't know how this would generalize to categories representing ordinals $geq omega$.
Through the basic notions of category theory, what is an ordinal? And what does ordinal arithmetic look like categorically?
category-theory ordinals
asked Jul 24 at 4:01
TreFox
1,82711031
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up vote
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I am just learning category theory and I am wondering if there is some way to define ordinals, and ordinal arithmetic purely through categorical means. I have a notion of taking a category with 0, 1, 2, etc elements but I have no idea how you could define addition on these categories with the usual nice properties (associativity, commutativity in the finite case, etc). I also don't know how this would generalize to categories representing ordinals $geq omega$.
Through the basic notions of category theory, what is an ordinal? And what does ordinal arithmetic look like categorically?
category-theory ordinals
asked Jul 24 at 4:01
TreFox
1,82711031
Ordinals are partial orders and so in particular are themselves categories. It's surprisingly unclear what ordinal arithmetic looks like in this setup.
– Qiaochu Yuan
Jul 24 at 4:26
1
Ah, no, it's fine, ordinal sum is join: ncatlab.org/nlab/show/join+of+categories
– Qiaochu Yuan
Jul 24 at 6:07
1
Do you want to see ordinals as categories (as Qiaochu Yuan proposes) and reflect ordinal arithmetic at that level, or do you want to define cardinals internally to a fixed category with sufficient structure?
– Pece
Jul 24 at 6:43
1
@Pece I don't understand your point. The ordinals will have to be defined as internal posets, and so $omega$ will have to be defined up to isomorphism of posets. I see no issue with that.
– Kevin Carlson
Jul 24 at 17:00
1
@KevinCarlson Yes you are right, I didn't thought about making them internal posets directly, I was just wondering what would happen if we were to mimic the usual construction of ordinals in, say, a topos and I stumble across the "increasing union" not being so good then. I guess taking the transfinite composition in the category of internal posets would yield a well-behaved $omega$ then.
– Pece
Jul 25 at 8:31
 |Â
show 3 more comments
up vote
5
down vote
favorite
up vote
5
down vote
favorite
I am just learning category theory and I am wondering if there is some way to define ordinals, and ordinal arithmetic purely through categorical means. I have a notion of taking a category with 0, 1, 2, etc elements but I have no idea how you could define addition on these categories with the usual nice properties (associativity, commutativity in the finite case, etc). I also don't know how this would generalize to categories representing ordinals $geq omega$.
Through the basic notions of category theory, what is an ordinal? And what does ordinal arithmetic look like categorically?
category-theory ordinals
asked Jul 24 at 4:01
TreFox
1,82711031
I am just learning category theory and I am wondering if there is some way to define ordinals, and ordinal arithmetic purely through categorical means. I have a notion of taking a category with 0, 1, 2, etc elements but I have no idea how you could define addition on these categories with the usual nice properties (associativity, commutativity in the finite case, etc). I also don't know how this would generalize to categories representing ordinals $geq omega$.
Through the basic notions of category theory, what is an ordinal? And what does ordinal arithmetic look like categorically?
category-theory ordinals
asked Jul 24 at 4:01
TreFox
1,82711031
asked Jul 24 at 4:01
TreFox
1,82711031
asked Jul 24 at 4:01
TreFox
1,82711031
asked Jul 24 at 4:01
TreFox
1,82711031
1,82711031
Ordinals are partial orders and so in particular are themselves categories. It's surprisingly unclear what ordinal arithmetic looks like in this setup.
– Qiaochu Yuan
Jul 24 at 4:26
1
Ah, no, it's fine, ordinal sum is join: ncatlab.org/nlab/show/join+of+categories
– Qiaochu Yuan
Jul 24 at 6:07
1
Do you want to see ordinals as categories (as Qiaochu Yuan proposes) and reflect ordinal arithmetic at that level, or do you want to define cardinals internally to a fixed category with sufficient structure?
– Pece
Jul 24 at 6:43
1
@Pece I don't understand your point. The ordinals will have to be defined as internal posets, and so $omega$ will have to be defined up to isomorphism of posets. I see no issue with that.
– Kevin Carlson
Jul 24 at 17:00
1
@KevinCarlson Yes you are right, I didn't thought about making them internal posets directly, I was just wondering what would happen if we were to mimic the usual construction of ordinals in, say, a topos and I stumble across the "increasing union" not being so good then. I guess taking the transfinite composition in the category of internal posets would yield a well-behaved $omega$ then.
– Pece
Jul 25 at 8:31
 |Â
show 3 more comments
Ordinals are partial orders and so in particular are themselves categories. It's surprisingly unclear what ordinal arithmetic looks like in this setup.
– Qiaochu Yuan
Jul 24 at 4:26
1
Ah, no, it's fine, ordinal sum is join: ncatlab.org/nlab/show/join+of+categories
– Qiaochu Yuan
Jul 24 at 6:07
1
Do you want to see ordinals as categories (as Qiaochu Yuan proposes) and reflect ordinal arithmetic at that level, or do you want to define cardinals internally to a fixed category with sufficient structure?
– Pece
Jul 24 at 6:43
1
@Pece I don't understand your point. The ordinals will have to be defined as internal posets, and so $omega$ will have to be defined up to isomorphism of posets. I see no issue with that.
– Kevin Carlson
Jul 24 at 17:00
1
@KevinCarlson Yes you are right, I didn't thought about making them internal posets directly, I was just wondering what would happen if we were to mimic the usual construction of ordinals in, say, a topos and I stumble across the "increasing union" not being so good then. I guess taking the transfinite composition in the category of internal posets would yield a well-behaved $omega$ then.
– Pece
Jul 25 at 8:31
Ordinals are partial orders and so in particular are themselves categories. It's surprisingly unclear what ordinal arithmetic looks like in this setup.
– Qiaochu Yuan
Jul 24 at 4:26
Ordinals are partial orders and so in particular are themselves categories. It's surprisingly unclear what ordinal arithmetic looks like in this setup.
– Qiaochu Yuan
Jul 24 at 4:26
1
1
Ah, no, it's fine, ordinal sum is join: ncatlab.org/nlab/show/join+of+categories
– Qiaochu Yuan
Jul 24 at 6:07
Ah, no, it's fine, ordinal sum is join: ncatlab.org/nlab/show/join+of+categories
– Qiaochu Yuan
Jul 24 at 6:07
1
1
Do you want to see ordinals as categories (as Qiaochu Yuan proposes) and reflect ordinal arithmetic at that level, or do you want to define cardinals internally to a fixed category with sufficient structure?
– Pece
Jul 24 at 6:43
Do you want to see ordinals as categories (as Qiaochu Yuan proposes) and reflect ordinal arithmetic at that level, or do you want to define cardinals internally to a fixed category with sufficient structure?
– Pece
Jul 24 at 6:43
1
1
@Pece I don't understand your point. The ordinals will have to be defined as internal posets, and so $omega$ will have to be defined up to isomorphism of posets. I see no issue with that.
– Kevin Carlson
Jul 24 at 17:00
@Pece I don't understand your point. The ordinals will have to be defined as internal posets, and so $omega$ will have to be defined up to isomorphism of posets. I see no issue with that.
– Kevin Carlson
Jul 24 at 17:00
1
1
@KevinCarlson Yes you are right, I didn't thought about making them internal posets directly, I was just wondering what would happen if we were to mimic the usual construction of ordinals in, say, a topos and I stumble across the "increasing union" not being so good then. I guess taking the transfinite composition in the category of internal posets would yield a well-behaved $omega$ then.
– Pece
Jul 25 at 8:31
@KevinCarlson Yes you are right, I didn't thought about making them internal posets directly, I was just wondering what would happen if we were to mimic the usual construction of ordinals in, say, a topos and I stumble across the "increasing union" not being so good then. I guess taking the transfinite composition in the category of internal posets would yield a well-behaved $omega$ then.
– Pece
Jul 25 at 8:31
 |Â
show 3 more comments
1 Answer
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The class $bf Ord$ of ordinal numbers can be described as the free algebra on a singleton, with respect to a certain monad $P$ on the category of classes (although not precisely a $P$-algebra).
The reference is this old paper by Rosicky.
answered Jul 25 at 22:58
Fosco Loregian
4,55611945
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
The class $bf Ord$ of ordinal numbers can be described as the free algebra on a singleton, with respect to a certain monad $P$ on the category of classes (although not precisely a $P$-algebra).
The reference is this old paper by Rosicky.
answered Jul 25 at 22:58
Fosco Loregian
4,55611945
add a comment |Â
up vote
5
down vote
The class $bf Ord$ of ordinal numbers can be described as the free algebra on a singleton, with respect to a certain monad $P$ on the category of classes (although not precisely a $P$-algebra).
The reference is this old paper by Rosicky.
answered Jul 25 at 22:58
Fosco Loregian
4,55611945
add a comment |Â
up vote
5
down vote
up vote
5
down vote
The class $bf Ord$ of ordinal numbers can be described as the free algebra on a singleton, with respect to a certain monad $P$ on the category of classes (although not precisely a $P$-algebra).
The reference is this old paper by Rosicky.
answered Jul 25 at 22:58
Fosco Loregian
4,55611945
The class $bf Ord$ of ordinal numbers can be described as the free algebra on a singleton, with respect to a certain monad $P$ on the category of classes (although not precisely a $P$-algebra).
The reference is this old paper by Rosicky.
answered Jul 25 at 22:58
Fosco Loregian
4,55611945
edited Jul 27 at 8:31
answered Jul 25 at 22:58
Fosco Loregian
4,55611945
answered Jul 25 at 22:58
Fosco Loregian
4,55611945
answered Jul 25 at 22:58
Fosco Loregian
4,55611945
4,55611945
add a comment |Â
add a comment |Â
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Ordinals are partial orders and so in particular are themselves categories. It's surprisingly unclear what ordinal arithmetic looks like in this setup.
– Qiaochu Yuan
Jul 24 at 4:26
1
Ah, no, it's fine, ordinal sum is join: ncatlab.org/nlab/show/join+of+categories
– Qiaochu Yuan
Jul 24 at 6:07
1
Do you want to see ordinals as categories (as Qiaochu Yuan proposes) and reflect ordinal arithmetic at that level, or do you want to define cardinals internally to a fixed category with sufficient structure?
– Pece
Jul 24 at 6:43
1
@Pece I don't understand your point. The ordinals will have to be defined as internal posets, and so $omega$ will have to be defined up to isomorphism of posets. I see no issue with that.
– Kevin Carlson
Jul 24 at 17:00
1
@KevinCarlson Yes you are right, I didn't thought about making them internal posets directly, I was just wondering what would happen if we were to mimic the usual construction of ordinals in, say, a topos and I stumble across the "increasing union" not being so good then. I guess taking the transfinite composition in the category of internal posets would yield a well-behaved $omega$ then.
– Pece
Jul 25 at 8:31