Re: Comma categories

[Steve Lack <s.lack@uws.edu.au>]

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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