2
$\begingroup$

A question in Tennison's Sheaf Theory is about the category of pointed sets and its characteristics. I have that

  • its zero object is given by $(\{x\},x)$
  • the kernel of $f\colon (A,a)\to (B,b)$ is given by $(f^{-1}(b),a)$
  • the cokernel is given by $(f(A),b)$
  • epimorphisms are surjective maps

but I fail to see why this breaks down cokernels.

  • 2
    What do you mean by "this breaks down cokernels"? What does "this" refer to, and what's breaking down?2011-06-17
  • 1
    Am I wrong? It seems that the cokernel is $(B-f(A)\cup\{b\},b)$.2011-06-17
  • 1
    It sounds like surjectivity breaks for cokernels. No, a cokernel is a colimit, then an epimorphism, then it is surjective.2011-06-17
  • 0
    @Alon: It means not every epimorphism (giving a quotient object) is a cokernel. So the "this" refers to "what the (co)kernels and epis of the category of pointed sets are". But it might've been a colourful and informal description :).2011-06-19
  • 0
    @wxu: You're right, and that was why I couldn't see the obvious. As @beroal mentioned, the surjectivity fails in the case of split epimorphisms. Thanks for the answers!2011-06-19

0 Answers 0