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 Mar 16 comment Matching the definition of hom-functor with how these are used when defining adjuncts "F⊢U" means F is left adjoint to U. To say something, say F, has "type C$\to$ D" means that F is a morphism from C to D in some category, in other words F:C$\to$ D. Some mathematicians make use of type notation, for example "S:Set" means S is a set (they say "S is of type Set") and "n:Integer" means n is of type integer --or n is an integer. Mar 10 comment In definition of a category , what is the meaning of 'consists of' The wording given in the original question is a formal definition of category in my opinion. It is clear (once you understand what "consists of" means!) and gives a precise specification of what is required to be a category. But maybe you are asking to put it into a formal language. The Homotopy Type Theory book defines "category" in type theory on pages 298ff. Note that "set", "function" and other words used in the definition are previously defined in the HoTT book. I am quite sure you could define "category" in first order logic as well, but I don't know of a reference. Mar 8 answered In definition of a category , what is the meaning of 'consists of' Feb 26 comment Monos are exactly the “injective homomorphisms” "just proving this for $x,y:1\to|M|$" means exactly that he is proving it for any two elements of $M$, since a map from $1$ to $M$ is the same thing as an element of $M$. Feb 25 comment Free and bound variables in “if” statements In the entry "Open Sentence", The Handbook of Mathematical Discourse says "In many circumstances such an assertion is taken as being true for all instantiates of its variables." If I read the sentence, "If $x\gt 2$ then $x \gt 3$, I would assume that it is a (false) claim that it is universally true, so that $x$ is bound. But you have to be careful; there are places where that is not intended. That is why the book says "in many circumstances". But Ned is right: when the statement is an implication an open sentence like that is usually read as a claim that it is true for all $x$. Feb 25 comment Free and bound variables in “if” statements In The Handbook of Mathematical Discourse, Feb 25 answered Questions regarding bound variables Feb 25 answered Is this a bound variable? Feb 24 comment Functor from group viewed as a category to another category. Just to clarify: The set of arrows of a small category with a single object is automatically a monoid whose binary operation is composition of arrows in the category. Feb 16 awarded Critic 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: