Timeline for A weak Yoneda-type lemma for certain nonrepresentable functors?
Current License: CC BY-SA 2.5
11 events
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| Dec 31, 2010 at 1:09 | history | edited | Todd Trimble | CC BY-SA 2.5 |
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| Dec 31, 2010 at 1:08 | comment | added | Todd Trimble | I would like whoever downvoted this answer to explain why it was downvoted. If there is a good reason for the downvote, it should be made explicit. | |
| Dec 29, 2010 at 14:19 | history | edited | Todd Trimble | CC BY-SA 2.5 |
added a few words of explanation regarding sheaves
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| Dec 29, 2010 at 14:11 | comment | added | Todd Trimble | You're right: much of the time $Nat(F, G)$ is hard-to-understand, and you're right also that it's often not even a set if the domain $C$ is large. Basically it's a set if $F: C \to Set$ ($Set$ can be replaced by other bases of hom-enrichment) can be presented as a small colimit of representables. I don't know off the bat if $H_3(\ast, Z)$ is so presentable (say if $C$ is the homotopy category of CW complexes), but that would be something to determine in these calculations. | |
| Dec 29, 2010 at 5:19 | comment | added | David Feldman | @Todd I've been saddled since graduate school with the misconception that ${\rm Nat}(F,G)$ was somehow deeply intractable even just at the conceptual level. Do there exist criteria (e.g. interesting necessary condition or sufficient conditions weaker than Yoneda) to determine when ${\rm Nat}(F,G)$ forms a set (rather than a proper class)? Can one describe, say $Nat(H_3(\ast,Z),\pi_5(\ast))$, or at least reduce it to a purely topological calculation? | |
| Dec 29, 2010 at 3:25 | history | edited | Todd Trimble | CC BY-SA 2.5 |
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| Dec 29, 2010 at 2:40 | comment | added | Todd Trimble | It is slightly scandalous that I cannot find an immediate online link for this (classic) material, but there is brief reference to this in the nLab at ncatlab.org/nlab/show/end#enriched_functor_categories_39. I might recommend reading about ends in Categories for the Working Mathematician, and then working out the meaning of the end formula for $Nat(F, G)$ (in the Set-valued case). You'll probably be able to work out the sheaf-y patching formulas for the end from these hints, but I'd be happy to supply them as an edit to this answer as well. | |
| Dec 29, 2010 at 2:35 | comment | added | Todd Trimble | It makes a lot of sense! But also this stuff is well worked out. In outline, each presheaf $F$ is a colimit of representables, in fact a union (a special type of filtered colimit) of quotients $F_x$ of representables as you are saying. (There is another way of expressing this via the calculus of coends; see for instance ncatlab.org/nlab/show/co-Yoneda+lemma) This means $Nat(F, G)$ is a limit of sets $Nat(\hom(A, -), G)$, in other words a limit of sets $G(A)$ or: an end $\int_A G(A)^{F(A)}$ where the exponential is the $F(A)$-indexed product of copies of $F(A)$. (cont.) | |
| Dec 29, 2010 at 1:31 | comment | added | David Feldman | Thanks, Todd. My original aim was understanding the class of all natural transformations between two arbitrary set-valued functors. So natural transformations from $F$ to $G$ restrict to natural transformation from the various sub-functors $F_x$ to G. Then one wants to cover $F$ with the various $F_x$, keep track of how things patch together, and ultimately identify the set of natural transformations with some kind of possibly large limit of subsets of various sets $G(A)$. Does this make any sense, and can I find it worked out somewhere? I would ask a whole new MO question if so advised. | |
| Dec 29, 2010 at 1:03 | vote | accept | David Feldman | ||
| Dec 28, 2010 at 22:35 | history | answered | Todd Trimble | CC BY-SA 2.5 |