Why is an object not defined as identity morphism?References on functorially-defined subgroups“Distributivity” of unary operationsWhat do people mean by “subcategory”?Why do categorical foundationalists want to escape set theory?Tensor product and category theoryHomset vs one collectionIs there a nice application of category theory to functional/complex/harmonic analysis?Is a background in Category Theory enough for starting a PhD in Category Theory?What's there to do in category theory?The “derived drift” is pretty unsatisfying and dangerous to category theory (or at least, to me)
Why is an object not defined as identity morphism?
References on functorially-defined subgroups“Distributivity” of unary operationsWhat do people mean by “subcategory”?Why do categorical foundationalists want to escape set theory?Tensor product and category theoryHomset vs one collectionIs there a nice application of category theory to functional/complex/harmonic analysis?Is a background in Category Theory enough for starting a PhD in Category Theory?What's there to do in category theory?The “derived drift” is pretty unsatisfying and dangerous to category theory (or at least, to me)
$begingroup$
I've seen that there was a single-sorted definition of a category. In some ways, it seems more understandable than the original definition.
I don't know much about category theory. But I would like to know how each definition is useful.
ct.category-theory
asked 8 hours ago
Usin JungUsin Jung
182 bronze badges
New contributor
Usin Jung is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I've seen that there was a single-sorted definition of a category. In some ways, it seems more understandable than the original definition.
I don't know much about category theory. But I would like to know how each definition is useful.
ct.category-theory
asked 8 hours ago
Usin JungUsin Jung
182 bronze badges
New contributor
Usin Jung is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
11
$begingroup$
The two-sorted definition corresponds to how most people think of categories, and how they talk about categories. For example, we speak of the categories of sets (not of functions), of groups (not of group homomorphisms), of topological spaces (not of continuous maps), etc. (If I remember correctly, Ehresmann did write about the categories of functions, of homomorphisms, of continuous maps, etc., but that never caught on.)
$endgroup$
– Andreas Blass
8 hours ago
1
$begingroup$
One can identify an object with its identity morphism. Then we have a category consisting of morphisms that can be composed. Usually the composition is thought of one function for the whole category. However, the composition function can be broken up into a performing compositions, one such place for each object. In higher order category theory there may be 2-morphisms between the usual morphisms. They may also be places or performing compositions. With the appropriate additional axioms higher order category theory fits into this scheme.
$endgroup$
– Jay Kangel
7 hours ago
$begingroup$
My impression is that the single-sorted POV is slightly more widespread in the context of groupoids (considered as small categories where every morphism is iso) but I am not a specialist and would be happy to be corrected here by others. However, one often ends up introducing the "unit space" of the groupoid which is the "set of objects", so even then two sorts seem to emerge
$endgroup$
– Yemon Choi
7 hours ago
3
$begingroup$
Another area where the single sorted definition is more widespread is when working with strict $n$-categories or strict $infty$-categories. Working with the single sorted definition allows to compose arrows of different dimension without writing iterated identities everywhere. It makes the manipulation of expressions a little more bearable, so lots of paper use it (where the "lots" need to be taken relatively as there is not that many paper on that topics anyway)
$endgroup$
– Simon Henry
5 hours ago
add a comment |
$begingroup$
I've seen that there was a single-sorted definition of a category. In some ways, it seems more understandable than the original definition.
I don't know much about category theory. But I would like to know how each definition is useful.
ct.category-theory
asked 8 hours ago
Usin JungUsin Jung
182 bronze badges
New contributor
Usin Jung is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I've seen that there was a single-sorted definition of a category. In some ways, it seems more understandable than the original definition.
I don't know much about category theory. But I would like to know how each definition is useful.
ct.category-theory
ct.category-theory
asked 8 hours ago
Usin JungUsin Jung
182 bronze badges
New contributor
Usin Jung is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
Usin JungUsin Jung
182 bronze badges
New contributor
Usin Jung is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
Usin JungUsin Jung
182 bronze badges
New contributor
Usin Jung is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
Usin JungUsin Jung
182 bronze badges
asked 8 hours ago
Usin JungUsin Jung
182 bronze badges
182 bronze badges
New contributor
Usin Jung is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Usin Jung is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
11
$begingroup$
The two-sorted definition corresponds to how most people think of categories, and how they talk about categories. For example, we speak of the categories of sets (not of functions), of groups (not of group homomorphisms), of topological spaces (not of continuous maps), etc. (If I remember correctly, Ehresmann did write about the categories of functions, of homomorphisms, of continuous maps, etc., but that never caught on.)
$endgroup$
– Andreas Blass
8 hours ago
1
$begingroup$
One can identify an object with its identity morphism. Then we have a category consisting of morphisms that can be composed. Usually the composition is thought of one function for the whole category. However, the composition function can be broken up into a performing compositions, one such place for each object. In higher order category theory there may be 2-morphisms between the usual morphisms. They may also be places or performing compositions. With the appropriate additional axioms higher order category theory fits into this scheme.
$endgroup$
– Jay Kangel
7 hours ago
$begingroup$
My impression is that the single-sorted POV is slightly more widespread in the context of groupoids (considered as small categories where every morphism is iso) but I am not a specialist and would be happy to be corrected here by others. However, one often ends up introducing the "unit space" of the groupoid which is the "set of objects", so even then two sorts seem to emerge
$endgroup$
– Yemon Choi
7 hours ago
3
$begingroup$
Another area where the single sorted definition is more widespread is when working with strict $n$-categories or strict $infty$-categories. Working with the single sorted definition allows to compose arrows of different dimension without writing iterated identities everywhere. It makes the manipulation of expressions a little more bearable, so lots of paper use it (where the "lots" need to be taken relatively as there is not that many paper on that topics anyway)
$endgroup$
– Simon Henry
5 hours ago
add a comment |
11
$begingroup$
The two-sorted definition corresponds to how most people think of categories, and how they talk about categories. For example, we speak of the categories of sets (not of functions), of groups (not of group homomorphisms), of topological spaces (not of continuous maps), etc. (If I remember correctly, Ehresmann did write about the categories of functions, of homomorphisms, of continuous maps, etc., but that never caught on.)
$endgroup$
– Andreas Blass
8 hours ago
1
$begingroup$
One can identify an object with its identity morphism. Then we have a category consisting of morphisms that can be composed. Usually the composition is thought of one function for the whole category. However, the composition function can be broken up into a performing compositions, one such place for each object. In higher order category theory there may be 2-morphisms between the usual morphisms. They may also be places or performing compositions. With the appropriate additional axioms higher order category theory fits into this scheme.
$endgroup$
– Jay Kangel
7 hours ago
$begingroup$
My impression is that the single-sorted POV is slightly more widespread in the context of groupoids (considered as small categories where every morphism is iso) but I am not a specialist and would be happy to be corrected here by others. However, one often ends up introducing the "unit space" of the groupoid which is the "set of objects", so even then two sorts seem to emerge
$endgroup$
– Yemon Choi
7 hours ago
3
$begingroup$
Another area where the single sorted definition is more widespread is when working with strict $n$-categories or strict $infty$-categories. Working with the single sorted definition allows to compose arrows of different dimension without writing iterated identities everywhere. It makes the manipulation of expressions a little more bearable, so lots of paper use it (where the "lots" need to be taken relatively as there is not that many paper on that topics anyway)
$endgroup$
– Simon Henry
5 hours ago
11
11
$begingroup$
The two-sorted definition corresponds to how most people think of categories, and how they talk about categories. For example, we speak of the categories of sets (not of functions), of groups (not of group homomorphisms), of topological spaces (not of continuous maps), etc. (If I remember correctly, Ehresmann did write about the categories of functions, of homomorphisms, of continuous maps, etc., but that never caught on.)
$endgroup$
– Andreas Blass
8 hours ago
$begingroup$
The two-sorted definition corresponds to how most people think of categories, and how they talk about categories. For example, we speak of the categories of sets (not of functions), of groups (not of group homomorphisms), of topological spaces (not of continuous maps), etc. (If I remember correctly, Ehresmann did write about the categories of functions, of homomorphisms, of continuous maps, etc., but that never caught on.)
$endgroup$
– Andreas Blass
8 hours ago
1
1
$begingroup$
One can identify an object with its identity morphism. Then we have a category consisting of morphisms that can be composed. Usually the composition is thought of one function for the whole category. However, the composition function can be broken up into a performing compositions, one such place for each object. In higher order category theory there may be 2-morphisms between the usual morphisms. They may also be places or performing compositions. With the appropriate additional axioms higher order category theory fits into this scheme.
$endgroup$
– Jay Kangel
7 hours ago
$begingroup$
One can identify an object with its identity morphism. Then we have a category consisting of morphisms that can be composed. Usually the composition is thought of one function for the whole category. However, the composition function can be broken up into a performing compositions, one such place for each object. In higher order category theory there may be 2-morphisms between the usual morphisms. They may also be places or performing compositions. With the appropriate additional axioms higher order category theory fits into this scheme.
$endgroup$
– Jay Kangel
7 hours ago
$begingroup$
My impression is that the single-sorted POV is slightly more widespread in the context of groupoids (considered as small categories where every morphism is iso) but I am not a specialist and would be happy to be corrected here by others. However, one often ends up introducing the "unit space" of the groupoid which is the "set of objects", so even then two sorts seem to emerge
$endgroup$
– Yemon Choi
7 hours ago
$begingroup$
My impression is that the single-sorted POV is slightly more widespread in the context of groupoids (considered as small categories where every morphism is iso) but I am not a specialist and would be happy to be corrected here by others. However, one often ends up introducing the "unit space" of the groupoid which is the "set of objects", so even then two sorts seem to emerge
$endgroup$
– Yemon Choi
7 hours ago
3
3
$begingroup$
Another area where the single sorted definition is more widespread is when working with strict $n$-categories or strict $infty$-categories. Working with the single sorted definition allows to compose arrows of different dimension without writing iterated identities everywhere. It makes the manipulation of expressions a little more bearable, so lots of paper use it (where the "lots" need to be taken relatively as there is not that many paper on that topics anyway)
$endgroup$
– Simon Henry
5 hours ago
$begingroup$
Another area where the single sorted definition is more widespread is when working with strict $n$-categories or strict $infty$-categories. Working with the single sorted definition allows to compose arrows of different dimension without writing iterated identities everywhere. It makes the manipulation of expressions a little more bearable, so lots of paper use it (where the "lots" need to be taken relatively as there is not that many paper on that topics anyway)
$endgroup$
– Simon Henry
5 hours ago
add a comment |
1 Answer
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$begingroup$
They're so automatically interchangeable that it doesn't really make sense to say that one is more useful than the other except in minor ways: if we really care about "symbolic parsimoy" then the object-free approach has a minor advantage, while if we care about matching informal discourse then objects are generally essential (per Andreas' comment).
Ultimately I'd say that the second point wins out - it's hard to beat intuitive clarity if the cost is so minor. But since the translations in both directions are really so trivial, you can use whichever one you want. And to be fair, it's not the case that the two-sorted approach is always more intuitive - when thinking of a group as a category, having to consider an utterly pointless object is a bit weird at first. (EDIT: Simon Henry's comment points out a more convincing example of this, where the two-sorted approach results in meaningfully annoying technical overhead.)
answered 6 hours ago
Noah SchweberNoah Schweber
20.4k3 gold badges51 silver badges153 bronze badges
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$begingroup$
They're so automatically interchangeable that it doesn't really make sense to say that one is more useful than the other except in minor ways: if we really care about "symbolic parsimoy" then the object-free approach has a minor advantage, while if we care about matching informal discourse then objects are generally essential (per Andreas' comment).
Ultimately I'd say that the second point wins out - it's hard to beat intuitive clarity if the cost is so minor. But since the translations in both directions are really so trivial, you can use whichever one you want. And to be fair, it's not the case that the two-sorted approach is always more intuitive - when thinking of a group as a category, having to consider an utterly pointless object is a bit weird at first. (EDIT: Simon Henry's comment points out a more convincing example of this, where the two-sorted approach results in meaningfully annoying technical overhead.)
answered 6 hours ago
Noah SchweberNoah Schweber
20.4k3 gold badges51 silver badges153 bronze badges
$endgroup$
add a comment |
$begingroup$
They're so automatically interchangeable that it doesn't really make sense to say that one is more useful than the other except in minor ways: if we really care about "symbolic parsimoy" then the object-free approach has a minor advantage, while if we care about matching informal discourse then objects are generally essential (per Andreas' comment).
Ultimately I'd say that the second point wins out - it's hard to beat intuitive clarity if the cost is so minor. But since the translations in both directions are really so trivial, you can use whichever one you want. And to be fair, it's not the case that the two-sorted approach is always more intuitive - when thinking of a group as a category, having to consider an utterly pointless object is a bit weird at first. (EDIT: Simon Henry's comment points out a more convincing example of this, where the two-sorted approach results in meaningfully annoying technical overhead.)
answered 6 hours ago
Noah SchweberNoah Schweber
20.4k3 gold badges51 silver badges153 bronze badges
$endgroup$
add a comment |
$begingroup$
They're so automatically interchangeable that it doesn't really make sense to say that one is more useful than the other except in minor ways: if we really care about "symbolic parsimoy" then the object-free approach has a minor advantage, while if we care about matching informal discourse then objects are generally essential (per Andreas' comment).
Ultimately I'd say that the second point wins out - it's hard to beat intuitive clarity if the cost is so minor. But since the translations in both directions are really so trivial, you can use whichever one you want. And to be fair, it's not the case that the two-sorted approach is always more intuitive - when thinking of a group as a category, having to consider an utterly pointless object is a bit weird at first. (EDIT: Simon Henry's comment points out a more convincing example of this, where the two-sorted approach results in meaningfully annoying technical overhead.)
answered 6 hours ago
Noah SchweberNoah Schweber
20.4k3 gold badges51 silver badges153 bronze badges
$endgroup$
They're so automatically interchangeable that it doesn't really make sense to say that one is more useful than the other except in minor ways: if we really care about "symbolic parsimoy" then the object-free approach has a minor advantage, while if we care about matching informal discourse then objects are generally essential (per Andreas' comment).
Ultimately I'd say that the second point wins out - it's hard to beat intuitive clarity if the cost is so minor. But since the translations in both directions are really so trivial, you can use whichever one you want. And to be fair, it's not the case that the two-sorted approach is always more intuitive - when thinking of a group as a category, having to consider an utterly pointless object is a bit weird at first. (EDIT: Simon Henry's comment points out a more convincing example of this, where the two-sorted approach results in meaningfully annoying technical overhead.)
answered 6 hours ago
Noah SchweberNoah Schweber
20.4k3 gold badges51 silver badges153 bronze badges
edited 5 hours ago
answered 6 hours ago
Noah SchweberNoah Schweber
20.4k3 gold badges51 silver badges153 bronze badges
answered 6 hours ago
Noah SchweberNoah Schweber
20.4k3 gold badges51 silver badges153 bronze badges
answered 6 hours ago
Noah SchweberNoah Schweber
20.4k3 gold badges51 silver badges153 bronze badges
20.4k3 gold badges51 silver badges153 bronze badges
add a comment |
add a comment |
Usin Jung is a new contributor. Be nice, and check out our Code of Conduct.
Usin Jung is a new contributor. Be nice, and check out our Code of Conduct.
Usin Jung is a new contributor. Be nice, and check out our Code of Conduct.
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11
$begingroup$
The two-sorted definition corresponds to how most people think of categories, and how they talk about categories. For example, we speak of the categories of sets (not of functions), of groups (not of group homomorphisms), of topological spaces (not of continuous maps), etc. (If I remember correctly, Ehresmann did write about the categories of functions, of homomorphisms, of continuous maps, etc., but that never caught on.)
$endgroup$
– Andreas Blass
8 hours ago
1
$begingroup$
One can identify an object with its identity morphism. Then we have a category consisting of morphisms that can be composed. Usually the composition is thought of one function for the whole category. However, the composition function can be broken up into a performing compositions, one such place for each object. In higher order category theory there may be 2-morphisms between the usual morphisms. They may also be places or performing compositions. With the appropriate additional axioms higher order category theory fits into this scheme.
$endgroup$
– Jay Kangel
7 hours ago
$begingroup$
My impression is that the single-sorted POV is slightly more widespread in the context of groupoids (considered as small categories where every morphism is iso) but I am not a specialist and would be happy to be corrected here by others. However, one often ends up introducing the "unit space" of the groupoid which is the "set of objects", so even then two sorts seem to emerge
$endgroup$
– Yemon Choi
7 hours ago
3
$begingroup$
Another area where the single sorted definition is more widespread is when working with strict $n$-categories or strict $infty$-categories. Working with the single sorted definition allows to compose arrows of different dimension without writing iterated identities everywhere. It makes the manipulation of expressions a little more bearable, so lots of paper use it (where the "lots" need to be taken relatively as there is not that many paper on that topics anyway)
$endgroup$
– Simon Henry
5 hours ago