If C and D are full subcategories of each other, are they equal or equivalent?

[phoenixthoth]

One question for now:
1. If C is a full subcategory of D and D is a full subcategory of C, what can be said, if anything, about C and D being either equal or equivalent in some way?

 

[Hurkyl]

They'd have to be equal.

Every object of C is an object of D, and vice cersa. Full subcategories are determined by their objects.

 

[phoenixthoth]

2. Could you explain slice categories?

I have a quote that I don't get:

It is useful to think of an object of Set/A as an A-indexed family of disjoint sets (the inverse images of the elements of A). The commutivity of the above diagram means that the function h is consistent with the decomposition of B and C into disjoint sets.

In the "above diagram," f:B-->A, g:C-->A, and h:B-->C s.t. gh=f.

Thanks in advance...

 

[Hurkyl]

Have you worked much with fiber bundles? I think they're a good model to understand what's going on.

You can imagine the objects of the slice category are the projections from objects of the original category to the base object.

For a given projection, each "point" X of the base object corresponds to a "fiber", those "points" of the source object that project onto X. (Thus, we have a B-indexed family of fibers)

The morphisms of the slice category, then, are the morphisms that act "fiber-wise". That is, if P is in a fiber of X, then f(P) is also in a fiber of X.

Of course, in general you won't have points and fibers to manipulate, but I think this is the spirit behind it.