Comma ∞-Categories

Abstract

Definition. In the case where either F1 or F2 is actually the identity functor on B, then one typically uses the notations B ↓F2 or F1 ↓B rather than IdB ↓F2 or F1 ↓ IdB. In general, as an abuse of notation, one often denotes the identity functor on a category with the category itself. Similarly, one often denotes the identity morphism on an object with the object itself. Definition. 1 is the category with a single object (?) and a single morphism (?) on that object. Example. Given functors 1 1 −→ Set IdSet ←−−− Set (where the former is the constant functor picking out the singleton set 1), the comma category 1 ↓Set is also known as pSet, the category of pointed sets. Unfolding definitions, an object in pSet is a set A and an element a of A. A morphism in pSet from 〈A, a〉 to 〈B, b〉 is a function f : A→ B such that f(a) = b. In other words, the following diagrams commute: 1(?) IdSet(A) a