Thread: Coslice category

i am wanting to know what a terminal (final) object in the coslice category

might be (technically, where "A" is the functor:

and where the other functor of the comma category is



alternatively, if it is easier to explain this as an intial object in the slice category:

, that's fine as well.

Originally Posted by Deveno

i am wanting to know what a terminal (final) object in the coslice category

might be (technically, where "A" is the functor:

and where the other functor of the comma category is



alternatively, if it is easier to explain this as an intial object in the slice category:

, that's fine as well.

if we ignore the single object of , which i'll also denote it by , and look at the objects of your slice category as pairs , where is a set and , then the initial object should be , where is the empty function.

so the initial object in the slice category is the unique map from the initial object of Set (ok, technically the initial object and the map, but i'm trying to "forget about objects")?

if that is so, then the final object in the coslice category should be...(*,{*}) where {*} is a "generic" singleton set, and * is the only possible arrow (constant map), yes?

what is motivating these questions is that i know that for a given function f:A-->Z, the projection p:A-->A/~f is initial in the coslice category (A ↓ Set),

and i am wondering about the possible "dual statements".