Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Category Theory

  1. Sep 13, 2009 #1
    What does it mean to quotient a category by an object of the category?

    In particular, the problem in front of me specifies the category I ( a full subcategory of Top which is compact, contains coproducts, and the one point space), an object X of I, and asks me to do stuff with I/X.
     
  2. jcsd
  3. Sep 13, 2009 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    This isn't a quotient, but a "slice category", a particular kind of "comma category".

    Given any category E and object A, the category E/A is defined by:
    . The objects of E/A are arrows of C of the form B --> A
    . The arrows of E/A are commutative triangles of C with a distinguished vertex A (This arrow "points" in the same direction as the edge opposite A)

    Or a more algebraic description
    . Objects are arrows f of E such that codom(f) = A
    . HomE/A(f,g) is the class of arrows h of E such that gh=f
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook