Slice category is a specific case of comma category.
Given a category C we can 'slice' it over some object XC which we fix in C.
|This construction allows us to start with one category 'C' and generate a different category 'C/X' by fixing a given element 'X' in C. The elements in C/X are pairs (A,P) where AC and P is a morphism from A to X.
Slice Category Examples
Colouring of labeled set.
Here we choose as our fixed object 'X' the 3 element set containing R, G and B (for red green and blue). Every object (in this case set) has an arrow (function) to this set so all the elements are assigned a colour.
When we add functions between these set objects these functions must now commute. This restricts us to functions which only map R to R, G to G and B to B.
This effectively partitions the structure into 3 separate sets, one for each colour.
In category theory we are not really supposed to look inside objects or arrows, but we do so here to attempt to give some intuition.
Adding a distinguished point (the origin) to Euclidean space to give a vector space.
'co-slice' is the dual concept, reverse all the arrows.