On 25/09/09 6:23 AM, "Tony Meman" <tonymeman1@googlemail.com> wrote:
Dear category theorists, I have two questions concerning comma categories.
If C is a category with a terminal object *, is the comma category (C,*) consisting of arrows from C to * isomorphic to the category C itself? If this is true, the same should apply to the dual case with an initial object.
Dear Tony, Yes, this is true. You could even take it as a definition of terminal object.
The category sSet of simplicial sets is the category of functors from the opposite delta category Delta^op to Set. The category of pointed simplicial sets sSet* is defined as the comma category (delta0, sSet), where delta0=hom(-,[0]). Is sSet* isomorphic to the category of functors from ([0],Delta)^op to Set?
No, this is not true. The category sSet* is pointed (it has a terminal object which is also initial), while the category of functors from ([0],Delta)^op to Set is not. I'm not sure if there was supposed to be a connection between the two questions, but just in case, I might point out that [0] is not initial in Delta (in fact it is terminal). Steve Lack.
Thank you in advance for any help. Tony
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