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May
24
revised Analogy for Cosets, Ideals, and Quotient Rings
Another technical typo
May
24
revised Analogy for Cosets, Ideals, and Quotient Rings
Clarified the reference to 'coset'
May
24
revised Analogy for Cosets, Ideals, and Quotient Rings
Corrected a technical typo
May
8
comment How to get a group from a semigroup
It seems to be that, in the Added note, all occurrences of "right adjoint" should be changed to "left adjoint".
Apr
7
awarded  Yearling
Apr
6
comment Why is the tensor product important when we already have direct and semidirect products?
Ok, that is nice! Do you have a reference that discusses this?
Apr
5
comment Why is the tensor product important when we already have direct and semidirect products?
I don't think so. The collection $Hom_R(X,Y)$ of $R$-modules for a non-commutative ring $R$ does not have the structure of an $R$-module. See the paragraph 7.5 of my Notes on Semigroups for example.
Apr
4
comment Why is the tensor product important when we already have direct and semidirect products?
What happens if $R$ is not commutative?
Apr
1
comment Homsets of group actions related to fixed points
Fantastic! Thanks very much. I was probably too exhausted to understand it last night. But, I had chat with an 'expert' this morning, who made it all clear to me :-)
Apr
1
accepted Homsets of group actions related to fixed points
Mar
31
asked Homsets of group actions related to fixed points
Mar
29
revised Analogy for Cosets, Ideals, and Quotient Rings
Added a footnote on normal subgroups
Mar
28
answered Analogy for Cosets, Ideals, and Quotient Rings
Feb
7
comment Categorial definition of subsets
I think what you might need is a 10-minute exercise: Prove that, in category Set, the subobjects of $A$ (as clarified above) are one-to-one with the subsets of $A$.
Feb
7
awarded  Commentator
Feb
7
comment Categorial definition of subsets
You cannot make a statement like $\{1,2\} \subseteq \{0,1,2\}$ categorically, let alone distinguish it from another. What you might say is that $\langle \{1,2\}, u\rangle$ is a subject of $\{0,1,2\}$ where $u$ is the injection of $\{1,2\}$ in $\{0,1,2\}$. But then neither $\langle \{0,1\},u \rangle$ nor $\langle\{1,5\},u\rangle$ is a subobject because $u$ doesn't type-check in those cases.
Feb
6
answered Categorial definition of subsets
Feb
4
answered The importance of parallel arrows in a commutative square
Feb
2
comment Unit of adjunction: Epimorphism?
I can hardly think of any cases where the unit of an adjunction is an epimorphism. Perhaps you meant to ask whether the counit of an adjunction is an epimorphism?
Jan
23
answered In the morphisms-are-functions view of category theory, how are poset categories explained?