Mixing
Fundamental Theorem of Category Theory appropriate for undergraduates?
Clash Royale CLAN TAG#URR8PPP
up vote
3
down vote
favorite
I have been self-studying category theory (mostly from Lawvere and Schanuel’s Conceptual Mathematics). In my experience in undergraduate mathematics, many one or two quarter classes in mathematics build up to a “Fundamental Theorem†that is a somewhat deep result in the subject, is accessible to undergraduate level students in the time available, and serves as a kind of “capstone†which brings together the techniques and theorems studied in the course. For example, an introduction to abstract algebra and group theory may have the fundamental theorem of finite abelian groups, and an introduction to analysis may have the fundamental theorem of calculus. Is there an analogous result in Category theory that meets these criteria? Does category theory have too many prerequisites for reaching such a result? What would be a good result to conclude a short introduction to category theory with?
ct.category-theory
edited 1 hour ago
David White
10.7k45997
asked 2 hours ago
James Newman
161
New contributor
James Newman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
3
down vote
favorite
I have been self-studying category theory (mostly from Lawvere and Schanuel’s Conceptual Mathematics). In my experience in undergraduate mathematics, many one or two quarter classes in mathematics build up to a “Fundamental Theorem†that is a somewhat deep result in the subject, is accessible to undergraduate level students in the time available, and serves as a kind of “capstone†which brings together the techniques and theorems studied in the course. For example, an introduction to abstract algebra and group theory may have the fundamental theorem of finite abelian groups, and an introduction to analysis may have the fundamental theorem of calculus. Is there an analogous result in Category theory that meets these criteria? Does category theory have too many prerequisites for reaching such a result? What would be a good result to conclude a short introduction to category theory with?
ct.category-theory
edited 1 hour ago
David White
10.7k45997
asked 2 hours ago
James Newman
161
New contributor
James Newman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
Probably the Yoneda lemma is a good candidate.
– Alec Rhea
2 hours ago
3
Special Adjoint Functor Theorem.
– Oskar
2 hours ago
5
The Yoneda lemma is indeed a "deep" result, but not because it's some dramatic denouement or capstone result in the way some of these other theorems are (like quadratic reciprocity in an elementary number theory course, or the Tychonoff theorem in an introductory topology course). The proof of the Yoneda lemma is very short and easy, but its profundity is subtle and takes a while to draw out (for a teacher) or sink in (for a student). Nevertheless, if there is any one result in category theory that deserves to be singled out as fundamental, I'd say that's really the one.
– Todd Trimble♦
1 hour ago
I've read a non-category-theorist say that Beck monadicity is a 'real' theorem. See eg math.stackexchange.com/questions/2046470/… for one application
– David Roberts
16 mins ago
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I have been self-studying category theory (mostly from Lawvere and Schanuel’s Conceptual Mathematics). In my experience in undergraduate mathematics, many one or two quarter classes in mathematics build up to a “Fundamental Theorem†that is a somewhat deep result in the subject, is accessible to undergraduate level students in the time available, and serves as a kind of “capstone†which brings together the techniques and theorems studied in the course. For example, an introduction to abstract algebra and group theory may have the fundamental theorem of finite abelian groups, and an introduction to analysis may have the fundamental theorem of calculus. Is there an analogous result in Category theory that meets these criteria? Does category theory have too many prerequisites for reaching such a result? What would be a good result to conclude a short introduction to category theory with?
ct.category-theory
edited 1 hour ago
David White
10.7k45997
asked 2 hours ago
James Newman
161
New contributor
James Newman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I have been self-studying category theory (mostly from Lawvere and Schanuel’s Conceptual Mathematics). In my experience in undergraduate mathematics, many one or two quarter classes in mathematics build up to a “Fundamental Theorem†that is a somewhat deep result in the subject, is accessible to undergraduate level students in the time available, and serves as a kind of “capstone†which brings together the techniques and theorems studied in the course. For example, an introduction to abstract algebra and group theory may have the fundamental theorem of finite abelian groups, and an introduction to analysis may have the fundamental theorem of calculus. Is there an analogous result in Category theory that meets these criteria? Does category theory have too many prerequisites for reaching such a result? What would be a good result to conclude a short introduction to category theory with?
ct.category-theory
ct.category-theory
edited 1 hour ago
David White
10.7k45997
asked 2 hours ago
James Newman
161
New contributor
James Newman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 1 hour ago
David White
10.7k45997
asked 2 hours ago
James Newman
161
New contributor
James Newman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 1 hour ago
David White
10.7k45997
edited 1 hour ago
David White
10.7k45997
edited 1 hour ago
David White
10.7k45997
10.7k45997
asked 2 hours ago
James Newman
161
New contributor
James Newman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
James Newman
161
asked 2 hours ago
James Newman
161
161
New contributor
James Newman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
James Newman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
James Newman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
Probably the Yoneda lemma is a good candidate.
– Alec Rhea
2 hours ago
3
Special Adjoint Functor Theorem.
– Oskar
2 hours ago
5
The Yoneda lemma is indeed a "deep" result, but not because it's some dramatic denouement or capstone result in the way some of these other theorems are (like quadratic reciprocity in an elementary number theory course, or the Tychonoff theorem in an introductory topology course). The proof of the Yoneda lemma is very short and easy, but its profundity is subtle and takes a while to draw out (for a teacher) or sink in (for a student). Nevertheless, if there is any one result in category theory that deserves to be singled out as fundamental, I'd say that's really the one.
– Todd Trimble♦
1 hour ago
I've read a non-category-theorist say that Beck monadicity is a 'real' theorem. See eg math.stackexchange.com/questions/2046470/… for one application
– David Roberts
16 mins ago
add a comment |Â
2
Probably the Yoneda lemma is a good candidate.
– Alec Rhea
2 hours ago
3
Special Adjoint Functor Theorem.
– Oskar
2 hours ago
5
The Yoneda lemma is indeed a "deep" result, but not because it's some dramatic denouement or capstone result in the way some of these other theorems are (like quadratic reciprocity in an elementary number theory course, or the Tychonoff theorem in an introductory topology course). The proof of the Yoneda lemma is very short and easy, but its profundity is subtle and takes a while to draw out (for a teacher) or sink in (for a student). Nevertheless, if there is any one result in category theory that deserves to be singled out as fundamental, I'd say that's really the one.
– Todd Trimble♦
1 hour ago
I've read a non-category-theorist say that Beck monadicity is a 'real' theorem. See eg math.stackexchange.com/questions/2046470/… for one application
– David Roberts
16 mins ago
2
2
Probably the Yoneda lemma is a good candidate.
– Alec Rhea
2 hours ago
Probably the Yoneda lemma is a good candidate.
– Alec Rhea
2 hours ago
3
3
Special Adjoint Functor Theorem.
– Oskar
2 hours ago
Special Adjoint Functor Theorem.
– Oskar
2 hours ago
5
5
The Yoneda lemma is indeed a "deep" result, but not because it's some dramatic denouement or capstone result in the way some of these other theorems are (like quadratic reciprocity in an elementary number theory course, or the Tychonoff theorem in an introductory topology course). The proof of the Yoneda lemma is very short and easy, but its profundity is subtle and takes a while to draw out (for a teacher) or sink in (for a student). Nevertheless, if there is any one result in category theory that deserves to be singled out as fundamental, I'd say that's really the one.
– Todd Trimble♦
1 hour ago
The Yoneda lemma is indeed a "deep" result, but not because it's some dramatic denouement or capstone result in the way some of these other theorems are (like quadratic reciprocity in an elementary number theory course, or the Tychonoff theorem in an introductory topology course). The proof of the Yoneda lemma is very short and easy, but its profundity is subtle and takes a while to draw out (for a teacher) or sink in (for a student). Nevertheless, if there is any one result in category theory that deserves to be singled out as fundamental, I'd say that's really the one.
– Todd Trimble♦
1 hour ago
I've read a non-category-theorist say that Beck monadicity is a 'real' theorem. See eg math.stackexchange.com/questions/2046470/… for one application
– David Roberts
16 mins ago
I've read a non-category-theorist say that Beck monadicity is a 'real' theorem. See eg math.stackexchange.com/questions/2046470/… for one application
– David Roberts
16 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
The Special Adjoint Functor Theorem has already been recommended, and I agree with that suggestion. I also nominate the idea that "All concepts are Kan Extensions" as a capstone. Classically, you might finish the course with MacLane's coherence theorem, but I prefer to end with something that has lots of applications students would appreciate.
Another example (that I might select, as a homotopy theorist) would be Giraud's theorem.
A good resource is Emily Riehl's book Category Theory in Context, which is aimed at undergraduates and finishes with an Epilogue titled "Theorems in Category Theory".
answered 1 hour ago
David White
10.7k45997
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
The Special Adjoint Functor Theorem has already been recommended, and I agree with that suggestion. I also nominate the idea that "All concepts are Kan Extensions" as a capstone. Classically, you might finish the course with MacLane's coherence theorem, but I prefer to end with something that has lots of applications students would appreciate.
Another example (that I might select, as a homotopy theorist) would be Giraud's theorem.
A good resource is Emily Riehl's book Category Theory in Context, which is aimed at undergraduates and finishes with an Epilogue titled "Theorems in Category Theory".
answered 1 hour ago
David White
10.7k45997
add a comment |Â
up vote
4
down vote
The Special Adjoint Functor Theorem has already been recommended, and I agree with that suggestion. I also nominate the idea that "All concepts are Kan Extensions" as a capstone. Classically, you might finish the course with MacLane's coherence theorem, but I prefer to end with something that has lots of applications students would appreciate.
Another example (that I might select, as a homotopy theorist) would be Giraud's theorem.
A good resource is Emily Riehl's book Category Theory in Context, which is aimed at undergraduates and finishes with an Epilogue titled "Theorems in Category Theory".
answered 1 hour ago
David White
10.7k45997
add a comment |Â
up vote
4
down vote
up vote
4
down vote
The Special Adjoint Functor Theorem has already been recommended, and I agree with that suggestion. I also nominate the idea that "All concepts are Kan Extensions" as a capstone. Classically, you might finish the course with MacLane's coherence theorem, but I prefer to end with something that has lots of applications students would appreciate.
Another example (that I might select, as a homotopy theorist) would be Giraud's theorem.
A good resource is Emily Riehl's book Category Theory in Context, which is aimed at undergraduates and finishes with an Epilogue titled "Theorems in Category Theory".
answered 1 hour ago
David White
10.7k45997
The Special Adjoint Functor Theorem has already been recommended, and I agree with that suggestion. I also nominate the idea that "All concepts are Kan Extensions" as a capstone. Classically, you might finish the course with MacLane's coherence theorem, but I prefer to end with something that has lots of applications students would appreciate.
Another example (that I might select, as a homotopy theorist) would be Giraud's theorem.
A good resource is Emily Riehl's book Category Theory in Context, which is aimed at undergraduates and finishes with an Epilogue titled "Theorems in Category Theory".
answered 1 hour ago
David White
10.7k45997
answered 1 hour ago
David White
10.7k45997
answered 1 hour ago
David White
10.7k45997
answered 1 hour ago
David White
10.7k45997
10.7k45997
add a comment |Â
add a comment |Â
James Newman is a new contributor. Be nice, and check out our Code of Conduct.
James Newman is a new contributor. Be nice, and check out our Code of Conduct.
James Newman is a new contributor. Be nice, and check out our Code of Conduct.
James Newman is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f311996%2ffundamental-theorem-of-category-theory-appropriate-for-undergraduates%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
2
Probably the Yoneda lemma is a good candidate.
– Alec Rhea
2 hours ago
3
Special Adjoint Functor Theorem.
– Oskar
2 hours ago
5
The Yoneda lemma is indeed a "deep" result, but not because it's some dramatic denouement or capstone result in the way some of these other theorems are (like quadratic reciprocity in an elementary number theory course, or the Tychonoff theorem in an introductory topology course). The proof of the Yoneda lemma is very short and easy, but its profundity is subtle and takes a while to draw out (for a teacher) or sink in (for a student). Nevertheless, if there is any one result in category theory that deserves to be singled out as fundamental, I'd say that's really the one.
– Todd Trimble♦
1 hour ago
I've read a non-category-theorist say that Beck monadicity is a 'real' theorem. See eg math.stackexchange.com/questions/2046470/… for one application
– David Roberts
16 mins ago