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2005-07-24

limit and evaluation


Let D be a category with functorial limits of functors from some category C. That is, there is a functor

lim : Cat(C, D) -> D.

For each object x from C, there is a functor

ev_x : Cat(C, D) -> D,
sending functor
F : C -> D
to F(x), and sending natural transformation
r : F -> G : C ->D
to its component at x (r_x : F(x) -> G(x)). By mapping x to ev_x and morphism f : x -> y to an obvious natural transformation from ev_x to ev_y we obtain a functor
EV : C -> Cat(Cat(C, D), D).

Amusing and trivially checkable fact is that

lim = Lim EV.
That is, limits are always point-wise.

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