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To what extent do functors map "elements" of an object to "elements" of another object? They are usually described as just mapping one object to another.

Thanks

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    What do you mean by 'element'? Are these *concrete* categories?2012-11-11
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    Hi Berci. Yes, they are.2012-11-11
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    There are categories in practice where objects aren't just "sets with enriched structure." For those categories, there aren't really notions of "elements of an object"... Well, maybe one approach you can take is the functor of points approach: Choose an object $A$ in the category, and for each other object $X$, associate to $X$ the set of maps from $A$ to $X$.2012-11-11
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    Yes, the question is good, but it is not so easy because of the interpretation of 'elements'. Also, there is a concept of '*generalized element*' of an object $X$ in a category, and that is none other than an arrow $A\to X$. Of course, in this sense, generalized elements are nicely preserved.2012-11-11
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    I think many (or most) functors between concrete categories that you'll come across naturally will probably descend to underlying sets. But of course, the real power of category theory stems from the fact that we've turned our attention from the individual objects and their elements to morphisms and functors of points, anyways.2012-11-11
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    @AaronMazel-Gee If by "descend to underlying sets" you mean "depends only on the underlying set" then this is absolutely false: just take the functor $\textbf{Grp} \to \textbf{Ab}$ that sends a group to its abelianisation. Or, alternatively, we could consider the (contravariant) functor that takes a complete atomic boolean algebra to its set of atoms: there is no way of getting a natural map from the elements of the complete atomic boolean algebra to its set of atoms (because, for example, the set of atoms of the trivial boolean algebra is empty).2012-11-11
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    @ZhenLin: I certainly don't mean to say that I can define a (set-valued) functor on the essential image of a concretization functor. Of course this would depend on remembering *how* your sets came from objects in each category! All I mean is that if $U_1:\mathcal{C}_1 \rightarrow \mbox{Set}$ and $U_2:\mathcal{C}_2 \rightarrow \mbox{Set}$ are two "naturally-appearing" concrete categories and we have a "naturally-appearing" functor $F:\mathcal{C}_1 \rightarrow \mathcal{C}_2$, then chances are good that we will have functorial-in-$\mathcal{C}_1$ functions $f_X:U_1(X) \rightarrow U_2(FX)$.2012-11-11
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    I don't agree. The reason why that claim sounds reasonable is because many functors of interest are part of an adjunction, in which case there is indeed such a natural transformation $U_1 \Rightarrow U_2 F$. But there are also important functors where there is no such natural transformation: consider, for example, the homology functors on the category of chain complexes, or more generally, derived functors...2012-11-11
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    I edited the title for clarity2012-11-11
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    @ZhenLin: Right, this touches on helopticor's answer below; by the time we pass to homotopy categories, of course we're going to have already forgotten about elements of underlying sets. Still, the first examples one sees of categories are usually concrete, and the first examples one sees of functors of concrete categories usually fit into what I said above.2012-11-11

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