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Mar
7
comment Why not just define equivalence relations on objects and morphisms for equivalent categories?
First check up on the definition e.g. by glancing at the n-lab (ncatlab.org/nlab/show/skeleton), but do not worry that much about the detailed logical / philosophical aspects unless that interests you or in Oskar's detailed answer. My intuition about skeleta is that picking a skeleton or a coskeleton for a small category crushes the information that before was spread out naturally, down into a 'confined space', so it is less easy to see the beautiful structure there is in the category. It also misses the point that structure (e.g. products) is really only defined up to isomorphism.
Mar
6
comment Why not just define equivalence relations on objects and morphisms for equivalent categories?
The important thing is not to worry about mine. Like everyone here the important thing is explaining that helps everyone (including yourself and me!) and hopefully the OP.
Mar
6
comment Why not just define equivalence relations on objects and morphisms for equivalent categories?
Perhaps you could elaborate on that slightly.
Mar
6
comment Why not just define equivalence relations on objects and morphisms for equivalent categories?
Oskar: perhaps it would be good to point out why it is not a skeleton.
Mar
6
awarded  Editor
Mar
6
revised Why not just define equivalence relations on objects and morphisms for equivalent categories?
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Mar
6
answered Why not just define equivalence relations on objects and morphisms for equivalent categories?
Mar
3
answered Free crossed modules
Mar
3
answered Geometric interpretation of the fundamental theorem for coalgebras?
Jan
24
comment Categorical introduction to Algebra and Topology
This would also be my recommendation. Ronnie's book is wide ranging and very categorical in its approach. Not everything can be proved jst using category theoretic methods, but motivation for constructions can often be given that way. (Sometimes you have to get your hands `dirty' and not just manipulate things from a distance.) As a plus the groupoid stuff is great for understanding more algebra.
Dec
9
awarded  Caucus
Mar
8
comment How can a functor which preserves products not be natural?
To me it is not clear what exactly you mean by saying that the isomorphism between $XZ$ and $X\times Y$ is not natural. You should check up on the precise meaning of natural in this context.
Mar
6
comment Fundamental group of row of spheres
... but are your 5 generators independent of each other? Try to find a homotopy equivalence from your space to the bouquet of n-circles. Look up the van Kampen theorem as well. Finally, for fun what about the infinite (or the doubly infinite rows of circles????
Mar
2
comment Telescopic groups
@Ryan: I think the point is that in the subgroup lattice of a group there may be many different `copies' of a group, so you have to think of them as being different subgroups since they are! There are also logical problems in thinking of a group as an isomorphism calss of groups, as then what does an automorphism of that group give. This, of course, can be avoided but that can get very confusing. You may or may not like a categorical viewpoint but taking a more categorical view does get around this and makes the overall structure more transparent.
Feb
27
comment Topics in combinatorial Group Theory
Loday's paper is a good one to get those aspects of pictures and has the references to Igusa.
Feb
27
comment Topics in combinatorial Group Theory
Igusa is credited with them as well and may have the edge on Steve Pride. My main point is that that material is a very nice intro to some of the outward looking links from Comb. Group Theory. It is fun, and easy to present as it does not require that much background.
Feb
27
answered Topics in combinatorial Group Theory
Feb
10
comment Referencing something that isn't a lemma, proposition etc…
I feel that the second and third are more or less equivalent, and which you use is then a question of the actual fact that is being referred to and how it is presented in ref. [1]. The point is to see in any particular case which form works best for you (and test it on others). Don't worry about unwritten rules, seek clarity of express instead. (Unwritten rules may be right or wrong, and as they are unwritten how can one tell. ;-))
Feb
10
comment Why do people not use “partially directed” graphs?
... but then it is convenient to draw that graph as one with some edges having arrows on them and others not so! The point is thus more that these graphs are very common, but can be reduced to examples of directed graphs.
Dec
27
comment Mathematics and slavery
The question is at present a bit imprecise. The system of slavery used by different civilisations was very varied. The form of what we call slavery in ancient Greece is therefore important in its detail. Athens also had a large network of tribute providers which led to the rise of a class with leisure time. There are other factors that may be just as important however.