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Feb
6
answered Co-domain & Image
Jan
30
comment How are non-bijective morphisms “reversed”?
To show that epis in the category (Sets) are exactly the surjections using definition 4b) is most easily done by pointing out that a monic in Set^op is an epi in Set (which is a purely formal argument that works in any category whatever), and then give the usual proof using the categorical definition of epi that an epi in Set is surjective. It really doesn't make sense to try to use the fact that Set^op has a concrete representation to prove it -- it is bound to be much more complicated doing it that way.
Jan
9
comment Definition of “set” in HoTT
To me, this appears to say that if x and y are elements of a set and x=y then there is only one way to prove that x=y. But I don't understand the intuition behind that, if indeed it is correct.
Jan
2
comment What Is Associativity in Composition
In a category, composition of arrows (morphisms) is defined without reference to elements or graphs, and is associative by definition. Composition can be defined for relations as well as functions, and then it is associative, but if you are following elements there may be more than one path if I understand your meaning correctly.
Sep
18
awarded  Commentator
Sep
18
comment Initial and Final Objects in a Category
One example of a useful and interesting initial object is that the initial object in a category of models is a term algebra. A special case of this is described in Category Theories for Computing Science,Section 4.7, but the fact is true more generally. (CTCS is available at tac.mta.ca/tac/reprints/articles/22/tr22.pdf). Roughly speaking if you have a theory of some sort, build a category with the primitive terms as objects, operations as arrows, impose commutative diagrams to express the axioms, and you get the initial object in the category of models of the theory..
Sep
7
comment What is wrong with ZFC?
I am a category theorist. I sort of agree with you; there is nothing wrong with ZFC as a foundations. It shows that a great many things mathematicians do are consistent. My big complaint is that it is used in teaching to define many mathematical structures (a real number is a Dedekind cut, a pair (a,b) is {a,{a,b}}) in a way that makes them seem bizarre and unintuitive. That works in showing consistency. Such things should never be exposed to students beginning a subject since it makes the learn a difficult idea that will do them no good whatever.
May
12
answered Categorical Foundations text
Apr
21
comment Comparing Category Theory and Model Theory for Master's Thesis.
Here are two references to applications of category theory: n category cafe link talks about many areas of applications with emphasis on physics. David Spivak, Category theory for the sciences link emphasizes applications to database theory, which has suddenly become a big deal among data base people.
Apr
21
comment Comparing Category Theory and Model Theory for Master's Thesis.
Here is a start on references to applications of category theory:
Mar
20
answered Set notation for generalized elements
Mar
2
answered Is it possible to construct ZFC set theory inside category theory?
Apr
29
comment What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)
I discovered a special case of this in third (?) grade: 3 times 37 is 111. It caught my attention -- how could THAT happen?
Apr
10
comment Closing up the elementary functions under integration
I don't know what you mean by "elementary integral". If you mean "if the integral f of a given elementary function g is also elementary, is f elementary? The answer is no: e^(x^2) is the integral of the function 2x e^(x^2), but e^(x^2) is not elementary.
Jul
24
comment How to check whether it is a direct product?
If you are given the operation (A,B)↦A×B, then a product preserving functor between categories with finite products must preserve the specified product. If you require only that there be an object A times B with the required properties, then a product preserving functor must take a product to SOME product of the images of A and B (of course, they are all naturally isomorphic). Note: This is a fine point that hardly ever makes any difference in practice! See Toposes, Triples and Theories at www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf, page 141.
Jun
29
awarded  Yearling
May
20
awarded  Good Answer
Nov
14
awarded  Nice Question
Oct
20
awarded  Nice Answer
Aug
22
awarded  Teacher