Difference between the simplex category (Δ)and the opposite simplex category
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From: Omega John on 6 Feb 2010 08:19 Hi, Sorry, but I am not completely following. Using your example, consider the morphism 0 -> 0, 1 -> 2 from (1) -> (2) in Δ. Exactly which would the associated opposite morphism (2) -> (1) be in Δ^op? Thank you. On Sat, 06 Feb 2010 04:23:05 -0800, victor_meldrew_666(a)yahoo.co.uk wrote: > On 5 Feb, 20:07, Omega John <omega_john_userfrien...(a)gmail.com> wrote: >> Hi, >> >> Exactly what is the difference? I know that morphisms in the former are >> non-decreasing maps. Are the morphisms of the latter non-increasing >> maps? > > No. > > One category is the opposite of the other. If your supposition were > correct, then there would be as many morphisms from _m_ to _n_ in both > categories (as there are the same number of weakly increasing maps from > _m_ = {0,1,...,m} to _n_ as there are weakly increasing maps). But in > the opposite to the simplex category the morphisms from _m_ to _n_ > correspond to weakly increasing maps from _n_ to _m_ and there aren't > the same number of these as there are from _m_ to _n_ in general. For > instance there are six weakly increasing maps from _1_ = {0,1} to _2_ = > {0,1,2} but only for from _2_ to _1_. |