Timeline for How is the right adjoint $f_*$ to the inverse image functor $f^*$ described for functor categories $Set^C$, $Set^D$ and $f : C \to D$
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Jul 21, 2010 at 22:41 | comment | added | Peter LeFanu Lumsdaine | Answering your questions on the topological case: their $f$ is different from yours — it's a map $f: X \to Y$ between topological spaces, and so $f^{-1}$ is a functor $f^{-1}: \mathcal{O}(Y) \to \mathcal{O}(X)$. Then $(f_*F)(V) := (f^{-1}(V))$ makes sense as a presheaf on $Y$, since $f^{-1}(V)$ is an open set in $X$ (for $V$ an open set in $Y$), and an inclusion $V' \subseteq V$ induces an inclusion $f^{-1}(V') \subseteq f^{-1}(V)$. So this really is just the composition $F \circ f^{-1} : \mathcal{O}(Y) \to \mathcal{O}(X) \to \mathbf{Set}$. | |
| Jul 21, 2010 at 22:33 | comment | added | Peter LeFanu Lumsdaine | @vincenzo: Mac Lane and Moerdijk do the topological case in Ch.I as you say, but later they do the fully general presheaves version in Ch.VII, in Theorem 2.2 and Exercise 4 of that chapter. | |
| Jul 21, 2010 at 18:37 | comment | added | Pietro Majer | vincenzo: sorry you're right. Not concentrated! | |
| Jul 21, 2010 at 17:27 | vote | accept | vincenzoml | ||
| Jul 21, 2010 at 15:28 | answer | added | Michael A Warren | timeline score: 7 | |
| Jul 21, 2010 at 14:23 | comment | added | vincenzoml | Peter: Mac Lane and Moerdijk define $f_*$ in the "topological" way, that is (p. 68 of the copy you linked) $(f_*F)V=F(f^{−1})V$. But it is not clear to me how this definition gives us (from a presheaf $F : Set^C$) a presheaf $f_*F: Set^D$. That is: 1) What is now $F(f^{−1})(V)$? f may be non-injective on objects. 2) What is the action of $f_*F$ on arrows? $C$ may even be a discrete category, so $f^{−1}$ of a arrow in $D$ may be undefined. | |
| Jul 21, 2010 at 14:09 | comment | added | Andreas Blass | The adjoints (left and right) to such a pre-composition functor are called Kan extensions. They're the subject of the last chapter of Mac Lane's "Categories for the Working Mathematician." | |
| Jul 21, 2010 at 14:09 | comment | added | vincenzoml | Pietro: I might be wrong here, but $f^*$ seems to be covariant, with $f^*(F : Set^D) = F\circ f$ and $f^*(\gamma : F \to G)_{c : C} = \gamma_{fc} : Ffc \to Gfc$. | |
| Jul 21, 2010 at 13:15 | comment | added | Peter LeFanu Lumsdaine | If I remember right, there's a good section or two on this in the Mac Lane and Moerdijk book: gen.lib.rus.ec/… Unfortunately I'm in a rush at the moment — hopefully someone else can provide chapter and verse and perhaps a digest... | |
| Jul 21, 2010 at 11:40 | history | asked | vincenzoml | CC BY-SA 2.5 |