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. 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? Thank you in advance for any help. Tony [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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Steve Lack -
Tony Meman