# Coslice category

• Jun 8th 2011, 01:15 AM
Deveno
Coslice category
i am wanting to know what a terminal (final) object in the coslice category

$\displaystyle (A\downarrow Set)$ might be (technically, where "A" is the functor:

$\displaystyle A:1\rightarrow Set$ and where the other functor of the comma category is

$\displaystyle id_{Set}$

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

$\displaystyle (Set \downarrow A)$, that's fine as well.
• Jun 8th 2011, 05:09 AM
NonCommAlg
Quote:

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

$\displaystyle (A\downarrow Set)$ might be (technically, where "A" is the functor:

$\displaystyle A:1\rightarrow Set$ and where the other functor of the comma category is

$\displaystyle id_{Set}$

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

$\displaystyle (Set \downarrow A)$, that's fine as well.

if we ignore the single object of $\displaystyle 1$, which i'll also denote it by $\displaystyle 1$, and look at the objects of your slice category as pairs $\displaystyle (\alpha,g)$, where $\displaystyle \alpha$ is a set and $\displaystyle g: \alpha \longrightarrow A(1)$, then the initial object should be $\displaystyle (\emptyset, f)$, where $\displaystyle f$ is the empty function.
• Jun 8th 2011, 07:49 AM
Deveno
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".