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Additional resources for Algebra, Chapter 0, Version 2009.01.18

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13This (reasonable) statement is the axiom of choice, cf. 3. 18 I. 11. 10, it is not unreasonable to use 2A to denote the set of functions from an arbitrary set A to a set with 2 elements (say {0, 1}). Prove that there is a bijection between 2A and the power set of A (cf. 2). 3] 3. Categories The language of categories is aﬀectionately known as abstract nonsense, so named by Norman Steenrod. This term is essentially accurate, and not necessarily derogatory: categories refer to nonsense in the sense that they are all about the ‘structure’, and not about the ‘meaning’, of what they represent.

Now assume that f is a monomorphism. This says something about arbitrary sets Z and arbitrary functions Z → A; we are going to use a microscopic portion of this information, choosing Z to be any singleton {p}. Then assigning functions α , α : Z → A amounts to choosing to which elements a = α (p), a = α (p) we should send the single element p of Z. For this particular choice of Z, the property deﬁning monomorphisms, f ◦α =f ◦α =⇒ α = α becomes f ◦ α (p) = f ◦ α (p) =⇒ α = α , that is . f (a ) = f (a ) =⇒ α = α Now two functions from Z = {p} to A are equal if and only if they send p to the same element, so this says: f (a ) = f (a ) =⇒ a = a .

These categories are called discrete. As another example, consider the category corresponding to endowing Z with the relation ≤: for example, G3 G5 UG 7 2b bb ppÐpÐd p p bb 13 pp Ð bb ppp ÐÐÐ 0  pppp Ð G4 3 22 I. Preliminaries: Set theory and categories is a (randomly chosen) commutative diagram in this category. It would still be a (commutative) diagram in this category if we reversed the vertical arrow 3 → 3, or if we added an arrow from 3 to 4; while we are not allowed to draw an arrow from 4 to 3, since 4 ≤ 3.