5,516 reputation
1522
bio website poisson.phc.unipi.it/~mossa
location Earth
age 25
visits member for 2 years, 10 months
seen 15 hours ago

I'm math student, in particular I'm interested in algebra, geometry, topology and category theory (especially higher dimensional category theory) and its application in mathematics.


Feb
19
comment How do I prove that the function symbol $\circ$ is not a term by induction in the calculus?
What do you mean by $\circ$, is a defined term or is part of the alphabeth (which I suppose is $S$)? In the second case from you rules it should follow that $\circ$ is a term since for every $f \in S$ it follows that $f$ is term (T2), since in this rule there's no restriction on the applicability. Hope this helps.
Feb
15
comment What are morphisms in the category of sets $\mathbf{Set}$?
@Alexey you're welcome.
Feb
15
comment What are morphisms in the category of sets $\mathbf{Set}$?
@Alexey I do: "Category theory starts... ... Each arrow $f \colon X \to Y$ represent a function; that is, a set $X$, a set $Y$, and a rule $x \mapsto fx$ which ..." at page 1.
Feb
15
comment What are morphisms in the category of sets $\mathbf{Set}$?
For the function is said in the introduction of "Category theory for Working mathematician".
Feb
15
answered What are morphisms in the category of sets $\mathbf{Set}$?
Feb
12
comment If $N$ is an $R/I$-submodule of $M$ can we view $N$ as an $R$-submodule of $M$?
@Edgar $N$ is a $R/I$-submodule of the $R/I$-module...?
Feb
8
reviewed Approve suggested edit on Suggestions on how to verify this partial differential equation solution
Feb
7
reviewed Approve suggested edit on $f(f(\sqrt{2}))=\sqrt{2}$ then f has a fixed point
Feb
7
comment On which structures does the free group 'naturally' act?
Of course a free group naturally act on the set of strings on the generators: which is the Cayley actions, nonetheless I suppose you weren't interested in such trivial action.
Feb
3
comment There are two types of functors; covariant and contravariant. Is it right?
@user112018 I guess so :)
Feb
3
comment There are two types of functors; covariant and contravariant. Is it right?
@user112018 I've added an Edit, let me know if that solves your doubts.
Feb
3
revised There are two types of functors; covariant and contravariant. Is it right?
added 977 characters in body
Feb
3
answered There are two types of functors; covariant and contravariant. Is it right?
Feb
1
comment What does “Arrows are more important than objects” really mean?
@mare_nnoem I don't think so. Here why: yoneda lemma holds for every category while the reduction of points to special kind of arrows is something that holds for groups and topological (and other special kind of concrete categories). To be more detailed the point-arrow fact holds in those concrete categories (i.e. categories with a faithful $\mathbf{Set}$-valued functor) the underlying-set functor is representable (and that's not always the case).
Feb
1
revised What does “Arrows are more important than objects” really mean?
Made some corrections
Feb
1
answered What does “Arrows are more important than objects” really mean?
Jan
31
comment Are there any examples of vector spaces over non-numerical fields? If not, why not?
@twirlobite I see that you didn't like the answer below. Then I guess you should try to make clear what do you mean by non-numerical vector space. I have interpretated as vector spaces whose underlying field isn't either $\mathbb R$, or $\mathbb C$, or something similar. Am I got it right?
Jan
30
comment A question about groups as categories.
@Fernando As I said above I've assumed that I was working in a different foundational theory than ZFC: one in which we have proper classes as set-like object, and classes-of-classes and so on. :)
Jan
30
comment A question about groups as categories.
Of course, the problem is that when doing category theory I always suppose to work with an infinite hierarchy of universes (Tarski-Grothendieck theory, or a rich enough type theory). Thanks for pointing out the needed correction :)
Jan
30
comment A question about groups as categories.
Injectivity on objects is garanted by the fact that if $x \ne y \in \mathbf C$ then $1_x \in F(x) \setminus F(y)$ and $1_y \in F(y) \setminus F(x)$. From this follows that morphisms having different sources or targets are sent in different functions by $F$. So at most parallel morphisms can be identified by $F$, that's not the case since if $f,g \colon x \to y$ and $F(f)=F(g)$ then $f=f\circ 1_X=F(f)(1_x)=F(g)(1_x)=g \circ 1_x=g$. This proves also injectivity on the arrow part, hence $F$ is an embedding.