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"We encourage children to read for enjoyment, yet we never encourage them to 'math' for enjoyment. We teach kids that math is done fast, done only one way and if you don't get the answer right, there's something wrong with you. You would never teach reading this way." - Rachel McAnallen

"Mathematicians create by acts of insight and intuition. Logic then sanctions the conquests of intuition." - Morris Kline


12h
comment Monoidal categories in which $\mathrm{Aut}(X \otimes Y) \cong \mathrm{Aut}(X) \sqcup \mathrm{Aut}(Y).$
In your opinion Martin, what is the most natural and/or important way in which coproducts of groups (in the category $\mathbf{Grp}$) arise?
18h
accepted Is there a commutative ring $R$ with an idempotent endomorphism $f$ that cannot be expressed as $f(x)=sx$ for some idempotent $s \in R$?
19h
comment Is there a commutative ring $R$ with an idempotent endomorphism $f$ that cannot be expressed as $f(x)=sx$ for some idempotent $s \in R$?
This does answer the question, because (for example) we can assume that $R$ is an integral domain. But, is it really true that every idempotent in a polynomial ring is constant? This seems too good to be true. For example, what if $R$ has nilpotent elements? We should be able to find idempotents in $R[X]$ that are not elements of $R$, methinks.
19h
answered Is there a commutative ring $R$ with an idempotent endomorphism $f$ that cannot be expressed as $f(x)=sx$ for some idempotent $s \in R$?
19h
asked Is there a commutative ring $R$ with an idempotent endomorphism $f$ that cannot be expressed as $f(x)=sx$ for some idempotent $s \in R$?
19h
comment Is this a general structure for constructs?
@Lehs, you're welcome. I don't know much about coalgebras, but here's a link to get you started. Don't worry too much about the resistance; most great ideas were resisted initially. Anyway, the important thing is not to get 100% of all kinds of mathematical structures ever considered. The important thing is whether it always gets the morphisms right. So I think you should work out, in detail, what the morphisms are in a large variety of cases. If your method consistently gets the right answer, then consider publishing an article.
20h
comment Is this a general structure for constructs?
Lehs, I think you will find most constructs of coalgebras do not fit this pattern. Its a cool idea though. Does it really give the right notion of morphism for e.g. topological spaces, uniform spaces etc.? If so, I think that's kind of a big deal.
20h
comment Is this a general structure for constructs?
@NajibIdrissim, this is a standard term; it just means a $\mathbf{Set}$-concrete category.
1d
comment Commutative monoids arising from categories with finite coproducts
+1 nice question. The world needs more questions like this.
2d
revised Is there a name for those relations that behave a bit like $<$?
added 104 characters in body; edited title
2d
accepted Monoidal categories in which $\mathrm{Aut}(X \otimes Y) \cong \mathrm{Aut}(X) \sqcup \mathrm{Aut}(Y).$
2d
comment Monoidal categories in which $\mathrm{Aut}(X \otimes Y) \cong \mathrm{Aut}(X) \sqcup \mathrm{Aut}(Y).$
By the free monoidal category, I assume you mean the following: there is a functor $U$ that forgets the monoidal structure of a small monoidal category. I guess it has a left-adjoint $F$ that constructs the "free" monoidal category on a small category. Okay, can we describe $F(\mathbf{C})$ somewhat explicitly? Or at least, what is your intuition about this object?
2d
comment Symbols for “odd” and “even”
@WChargin, true. So maybe $A_e$ and $A_o$ are better.
2d
comment Symbols for “odd” and “even”
I like $A_0$ and $A_1$. Its nice.
2d
comment Groups, neutral elements and uniqueness
Linux, you'll have to be more specific before we can help. Elements of $G$ do not map anything to anything else; that is what functions do, and the elements of $G$ are not (necessarily) functions. Also, it will be clearer for people if you write $x O e = x$ as $xe=x$. And it will be clearer if you consistently use lowercase letters to refer to elements of mathematical structures. And, you want to make your question really STAND OUT. e.g. Start a new line. Write QUESTION. Then write your question is clear English. See what I mean? You have to help us help you.
2d
comment SEVEN - NINE= EIGHT
I don't see why this is getting downvotes, I think it is a cute little riddle, and sufficiently mathematical that posting it here was okay. There's nothing wrong with the occasional not-entirely rigorous puzzle.
2d
asked Monoidal categories in which $\mathrm{Aut}(X \otimes Y) \cong \mathrm{Aut}(X) \sqcup \mathrm{Aut}(Y).$
2d
awarded  Nice Question
2d
revised Is there a name for those relations that behave a bit like $<$?
added 10 characters in body
2d
revised Is there a name for those relations that behave a bit like $<$?
deleted 18 characters in body; edited title