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Mar
21
comment What does $1a \in Hom(a, a)$ mean?
I think you could have answered this question yourself had you but looked a few pages before (or paragraphs above?) to see what is the definition of $1_a$ and of $\hom(a,a)$.
Mar
21
revised What does $1a \in Hom(a, a)$ mean?
more descriptive title
Mar
21
answered Is complex exact if its Euler characteristic is zero?
Mar
19
comment Every module over a field is free. Is every commutative ring whose modules are all free a field?
Here the question was asked for the more general non-commutative case.
Mar
19
answered Size Issues in Category Theory
Mar
18
revised Application of Composition of Functions: Real world examples?
added 13 characters in body
Mar
18
answered Application of Composition of Functions: Real world examples?
Mar
18
comment Categorical introduction to Algebra and Topology
To see Yoneda's lemma in action you should go into algebraic geometry, for example :) but for that you need to learn your basic abstract algebra first! Also, it is important to remember that not everything can be done/explained categorically. For example, a lot of the material in a standard introduction to group theory.
Mar
18
comment Categorical introduction to Algebra and Topology
It does go beyond that. The mindset of the book is categorical, and as far as I remember every concept that can be introduced and explained categorically is explained in that way. Some concepts are introduced as needed (it is an algebra book, not a category theory book), such as adjoint functors which are introduced to explain the free-forgetful adjunction. It is already much more than can be said of a lot of algebra books that explain free objects (groups, modules...) Yoneda appears in the exercises, maybe because it doesn't appear all that obviously in a first course in algebra.
Mar
18
answered Categorical introduction to Algebra and Topology
Mar
18
answered Why is $\pi_1(\Bbb{R}^n,x_0)$ the trivial group in $\Bbb{R}^n$?
Mar
17
comment If a morphism of pushouts of complexes (with one arrow monic) is composed of quasi-isos, then the induced arrow is one also
@Sunny: you're right, I wasn't clear enough. I mean one arrow monic in both pushout diagrams, yes.
Mar
16
comment How to prove a set with at least one element is a separator?
Also, be careful, you say "a separator object for $f_1,f_2$", but that's not right. A separator is a separator for the category, and it satisfies the condition you wrote for every $f_1,f_2$.
Mar
16
comment How to prove a set with at least one element is a separator?
@Problemania: $f(a)\neq g(a) \Rightarrow f\circ h(x)\neq g\circ h (x)$ since $a=h(x)$. Hence the two functions $f\circ h$ and $g\circ h$ are different at the element $x\in X$. If two functions differ in one point, then they are different, by definition of equality of functions! As for your second comment, yes, exactly, that's actually the definition of generator (or a slight rewording of it).
Mar
15
comment How to prove a set with at least one element is a separator?
I'm not familiar with Lawvere's book, but I take it that he, as Johnstone, takes "separator" to mean what other categorical sources call "generator" . If it is not the case, then tell me and I'll delete the answer.
Mar
15
answered How to prove a set with at least one element is a separator?
Mar
15
reviewed Reject Why is the partial derivative $f_x' = 0 $ is not continous?
Mar
15
revised If a morphism of pushouts of complexes (with one arrow monic) is composed of quasi-isos, then the induced arrow is one also
added 526 characters in body; edited title
Mar
14
comment If a morphism of pushouts of complexes (with one arrow monic) is composed of quasi-isos, then the induced arrow is one also
Thank you. I will read your answer with detail later. I was just aware of "homotopy colimits" in the category of topological spaces, not in any more general sense.
Mar
14
comment If a morphism of pushouts of complexes (with one arrow monic) is composed of quasi-isos, then the induced arrow is one also
@ZhenLin: Thanks. I'll roll up my sleeves and diagram-chase in that particular case then, I guess. I'm intrigued about your comment about homotopy colimits though, if you would like to expand on it (maybe as an answer?) it would be great :)