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| visits | member for | 8 months |
| seen | 10 hours ago | |
| stats | profile views | 414 |
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17h |
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Do any authors takes structures and objects as mutually exclusive concepts? @QiaochuYuan also, you ask: is $\mathbb{R}$ a structure? What's its default category? To the first question, I would say "yes!" After all, if $x,y : \mathbb{R}$, then I may write $x+y$ without qualification. So $\mathbb{R}$ is more than just a set. However, it does not belong to any particular category by default. But that's PRECISELY my point! |
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18h |
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Do any authors takes structures and objects as mutually exclusive concepts? @QiaochuYuan, take the following example. We have a group structure. We conservatively extend it with a new predicate symbol $C$ such that $C(a)$ asserts that $a$ commutes with every other element of the group. So in particular, we adjoin the axiom $\forall a[C(a) \leftrightarrow \forall b : ab=ba]$. No 'new information' is required to know that such a predicate $C$ exists. However, $C$ is not preserved by group homomorphism. |
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20h |
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Do any authors takes structures and objects as mutually exclusive concepts? No no, it IS essentially the same structure. We do this sort of thing all the time: like, we start with the ZFC axioms and the $\in$ symbol, and the very next thing we do is conservatively extend our language with a $\subseteq$ symbol. In doing so, I've essentially changed nothing. |
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21h |
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Do any authors takes structures and objects as mutually exclusive concepts? @QiaochuYuan, I'm comfortable with it! All I'm saying is, if you extend a structure with new relations that can be defined in terms of the old ones, well its basically the same structure. However, if structures come packaged with a default category (whose arrows are the structural homomorphisms) then extending those structures with new definitions isn't "allowed" unless we also restrict the homomorphisms and thereby change the category. |
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21h |
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Do any authors takes structures and objects as mutually exclusive concepts? @QiaochuYuan, you misunderstand me. I'm not saying that forgetful functors don't exist! I'm saying that if $F$ is a forgetul functor, then it's most useful to distinguish between $X$ and $F(X)$. |
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21h |
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Do any authors takes structures and objects as mutually exclusive concepts? @QiaochuYuan, I think that subtyping isn't useful or desirable in category theory, because for example the coproduct of objects in the category Ab is different from the coproduct of their forgetful image in the category Grp. With regards to your last comment, check out that link and if its still not clear, I will try to clarify. |
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21h |
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Do any authors takes structures and objects as mutually exclusive concepts? @QiaochuYuan, in type theory, if we write $x : X$ (read: $x$ has type $X$), and if $X$ and $Y$ are distinct types, then its not true that $x : Y$. |
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23h |
asked | Do any authors takes structures and objects as mutually exclusive concepts? |
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1d |
asked | Is the category of *strictly* partially ordered sets interesting? |
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1d |
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Is there even a point in defining the notion of a 'metric' (as opposed to a metric space), etc.? Pretty sure a metric space can only have one metric. I think you're trying to say: a set will typically admit more than one metric. |
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1d |
asked | Is there even a point in defining the notion of a 'metric' (as opposed to a metric space), etc.? |
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1d |
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How to know when a system of axioms is 'complete'? @joriki, the above was in reply to your comment. |
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1d |
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How to know when a system of axioms is 'complete'? Sentence 0: "First, a group is a monoid where every element has a two-sided inverse, not just a one-sided inverse." |
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1d |
revised |
How to know when a system of axioms is 'complete'? added 63 characters in body |
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1d |
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How to know when a system of axioms is 'complete'? @joriki, Ittay criticizes it as well. |
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1d |
asked | How to know when a system of axioms is 'complete'? |
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1d |
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I dont understand equivalence classes with relations Cool, +1. $\;\;\!\!$ |
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1d |
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I dont understand equivalence classes with relations I'd feel more comfortable with your opening statement if you replaced the word 'is' with the phrase 'can be identified with.' As in: "The quotient set of integers modulo your relation R can be identified with the set of natural numbers ≥0." |
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1d |
revised |
I dont understand equivalence classes with relations added 183 characters in body |
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1d |
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I dont understand equivalence classes with relations As @user3533 points out, the first sentence isn't strictly true. |
