I'm interested in the connections between Category theory and Model theory. From this video, I understood why the category of vector spaces is well-behaved and that of fields is not. The theory of vector spaces is(sorry I'm not completely sure) uncountably categorical, while that of algebraically closed fields of a fixed characteristic is also uncountably categorical. So, the problem here again lies with mixing fields of different characteristic. Is the category of fields of a fixed characteristic well-behaved? Do you know of any books dealing with Category theory and Model theory? Great video btw!
Ah, that is a great question! (Or, actually, these are great questions!) Not sure what you mean by uncountably categorical. What is true that a both, KVECT and FIELD have uncountable many objects (so vector spaces respectively fields), but I do not think that is what you have in mind. So let me try to answer your question differently by telling you a bit about pFIELD, i.e. fields of a fixed characteristic. Indeed, your hunch is correct and pFIELD is a bit better behaved than FIELD. The categories pFIELD are the connected components of FIELD (in the sense of en.wikipedia.org/wiki/Connected_category). Thus, most of the “problems” with FIELD still hold in pFIELD, see here for some non-properties of FIELD: math.stackexchange.com/questions/3756136/why-is-the-category-of-fields-seemingly-so-poorly-behaved nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/field math.stackexchange.com/questions/1638499/examples-of-a-categories-without-products However, somehow its a bit better. For example, pFIELD has initial objects, namely the prime field of characteristic p (so Z/pZ or Q). So the first “problem” in the linked list is partially gone ;-) Summarized, I think pFIELD is still not my favorite category ;-) Regarding books: Maybe try bookstore.ams.org/conm-104 This mathoverflow.net/questions/11974/is-there-a-relationship-between-model-theory-and-category-theory looks also very relevant.
@@VisualMath Thank you for the reply! In model theory, a complete first order theory is said to be uncountably categorical if for an uncountable cardinal kappa, the theory has exactly one model up to isomorphism(Morley's Categoricity theorem states that if this theory in a countable language is categorical in some uncountable cardinal, then it is categorical in all uncountable cardinals). His work started modern model theory. In contrast, the theory of the natural numbers with the successor function is not countably categorical, as it has infinitely many non-isomorphic models that are countably infinite(which shattered one of Hilbert's dreams of describing completely/pinning down mathematical structures only by their axioms). The first result is maybe the Lowenheim-Skolem theorems in this regard. If you are interested in this subject, I recommend you start with Chang and Keisler's classic book on Model theory.
@@NoNTr1v1aL Ah, I understand. Thanks for the clarification. I do not know too much about model theory, sorry not helpful!, but I think the book in my first answer looks extremely relevant. I would like to know more about model theory and category theory - thanks for the food for thoughts!
@@VisualMath I'll be checking out the book you suggested. Glad that you're interested in Model theory. Keep up the good work! Your channel will grow big soon 👍
I'm interested in the connections between Category theory and Model theory. From this video, I understood why the category of vector spaces is well-behaved and that of fields is not. The theory of vector spaces is(sorry I'm not completely sure) uncountably categorical, while that of algebraically closed fields of a fixed characteristic is also uncountably categorical. So, the problem here again lies with mixing fields of different characteristic. Is the category of fields of a fixed characteristic well-behaved? Do you know of any books dealing with Category theory and Model theory? Great video btw!
Ah, that is a great question! (Or, actually, these are great questions!)
Not sure what you mean by uncountably categorical. What is true that a both, KVECT and FIELD have uncountable many objects (so vector spaces respectively fields), but I do not think that is what you have in mind.
So let me try to answer your question differently by telling you a bit about pFIELD, i.e. fields of a fixed characteristic. Indeed, your hunch is correct and pFIELD is a bit better behaved than FIELD. The categories pFIELD are the connected components of FIELD (in the sense of en.wikipedia.org/wiki/Connected_category). Thus, most of the “problems” with FIELD still hold in pFIELD, see here for some non-properties of FIELD:
math.stackexchange.com/questions/3756136/why-is-the-category-of-fields-seemingly-so-poorly-behaved
nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/field
math.stackexchange.com/questions/1638499/examples-of-a-categories-without-products
However, somehow its a bit better. For example, pFIELD has initial objects, namely the prime field of characteristic p (so Z/pZ or Q). So the first “problem” in the linked list is partially gone ;-)
Summarized, I think pFIELD is still not my favorite category ;-)
Regarding books: Maybe try
bookstore.ams.org/conm-104
This
mathoverflow.net/questions/11974/is-there-a-relationship-between-model-theory-and-category-theory
looks also very relevant.
@@VisualMath Thank you for the reply! In model theory, a complete first order theory is said to be uncountably categorical if for an uncountable cardinal kappa, the theory has exactly one model up to isomorphism(Morley's Categoricity theorem states that if this theory in a countable language is categorical in some uncountable cardinal, then it is categorical in all uncountable cardinals). His work started modern model theory. In contrast, the theory of the natural numbers with the successor function is not countably categorical, as it has infinitely many non-isomorphic models that are countably infinite(which shattered one of Hilbert's dreams of describing completely/pinning down mathematical structures only by their axioms). The first result is maybe the Lowenheim-Skolem theorems in this regard.
If you are interested in this subject, I recommend you start with Chang and Keisler's classic book on Model theory.
@@NoNTr1v1aL Ah, I understand. Thanks for the clarification.
I do not know too much about model theory, sorry not helpful!, but I think the book in my first answer looks extremely relevant.
I would like to know more about model theory and category theory - thanks for the food for thoughts!
@@VisualMath I'll be checking out the book you suggested. Glad that you're interested in Model theory. Keep up the good work! Your channel will grow big soon 👍
@@NoNTr1v1aL Thanks - let us see what the future holds ;-)