Re: direct sums and products from category theory

From: Arturo Magidin (magidin_at_math.berkeley.edu)
Date: 09/20/04

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    Date: 19 Sep 2004 20:15:00 -0400

    In article <cikukt$4ah$1@dizzy.math.ohio-state.edu>,
    Arturo Magidin <magidin@math.berkeley.edu> wrote:

    Sigh... I always get these two confused. My most sincere apologies!

    >More generally, left adjoints respect limits, and right adjoints
    >respect colimits;

    Exactly the other way around: left adjoints respect colimits; right
    adjoints respect limits.

    Since "underlying set" is the right adjoint of the "underlying
    set-free object" pair, that means that the underlying set of the limit
    is the limit of the underlying sets. E.g., "the underlying set of the
    product is the product of the underlying sets." Free object is the
    left adjoint, which is why the free object on a disjoint union of sets
    is the coproduct of the free objects on the sets; e.g., the free group
    on X (disjoint union) Y is the free product of F(X) and F(Y). 

    -- 
======================================================================
"It's not denial. I'm just very selective about
 what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu
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