Topoi Ch 4, Part 1

Topoi Ch 4, now we’re getting somewhere. First section is a discussion on subobjects. A subobject of an object d is an equivalence class of monic arrows with codomain d. The section initially defines subobjects as a monic arrow with domain d, but the inclusion relation extracted from this definition is not antisymmetric; only reflexive and transitive. Golblatt states an antisymmetric inclusion relation beneficial in later sections. When we define a subobject of an object d as a monic arrow with codomain d, we can define inclusion by saying f is included in g if f “factors through” g. Specifically, if f: a >-> d and g: b >-> d are monic arrows with codomain d, and their exists an arrow h: a -> b such that f = g o h (“g of h” or “g composed with h” or “g after h”). The section also goes on to discuss elements and naming. An element of a set can be defined categorially from the outside as an arrow from the terminal object 1 to the containing set. Effectively, if x is an element of A, we can define x as x: 1 -> A. (1 = {0} is a terminal object in set). The name of a function is described similarly as an arrow from the terminal object 1 to the exponential B^A.