The General Construction and Further Examples

The construction from the previous subsection can put into a more general context, which, gratifyingly, will then specialize to further useful and interesting notions. 

There are several special cases of the comma category construction. First, taking A = 1, B = C, G = id , and taking F 1 — G to be the functor mapping the unique object of 1 to some given object x G 0 C) gives the category [x [ C) described in Definition 4.35. 

Another example is provided when we take A = B = C and F = G = idc-This gives the category of all morphisms of G, often called the category of arrows of G, with the new morphisms being all commuting diagrams of old morphisms. 

Finally, let us remark that it is possible to take one of the functors F and G in Definition 4.36 contravariant. This will also yield a category, which we denote by 0(F°p G) if F is taken to be contravariant, and 0 F J. G°p) if G is taken to be contravariant. This construction will come in handy when we define the analogue of intervals for acyclic categories. 

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