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Notes by RBJon

Categories for the Working Mathematician

by Saunders Mac Lane

CONTENTS
ChapSectionTitle
Introduction
ICategories, Functors and Natural Transformations
1Axioms for Categories
2Categories
3Functors
4Natural Transformations
5Monics, Epics, and Zero
6Foundations
7Large Categories
8Hom-sets
IIConstructions on Categories
IIIUniversals and Limits
IVAdjoints
VLimits
VIMonads and Algebras
VIIMonoids
VIIIAbelian Categories
IXSpecial Limits
XKan Extensions

Categories, Functors and Natural Transformations

Axioms for Categories

Mac Lane begins by giving a first order axiomatisation of the notion of category (which he here calls metacategory). Essentially, a metacategory is a graph with an associative operation of composition and an identity operator (which yields an identity under composition). He then asserts that:
A metacategory is to be any interpretation which satsifies all these axioms.
..as if that made sense independently of any specific set theoretic context.

Categories

Next a category is defined as:
any interpretation of the category axioms within set theory.
so the notion of category (unlike that of metacategory) is aknowleged to be relative to some set theory.

Functors

Natural Transformations

Monics, Epics, and Zero

Foundations

Now we return to puzzle over the foundational issues.

First he points out that Category Theory is "to discuss properties of totalities" such as the "set" of all groups. These means that category theory would like to be able to use an unrestricted principle of comprehension, were it not that this is known to give rise to problems of consistency (e.g. Russell's paradox).

Consequently we have to settle for separation (subset formation) instead of comprehension and we end up with many interesting metacategories which are not categories. Mac Lane ends up plumping for either Zermelo-Fraenkel set theory with a single universe U thrown in for good measure, or else Gödel-Bernays set theory (where classes serve the same purpose as U does in ZF). He then draws the distinction between small and large categories, the former being the ones whose sets of morphisms and of objects are members of U (or in NBG are sets rather than classes).

So here we have exposed classical set theory as cumbersome for category theory because it lacks unrestricted comprehension. We then put a bit of sticky plaster on it and carry on, making little of the missing categories.

Large Categories

Having drawn the distinction (between small and large categories), most of the interesting ones turn out to be large.

Hom-sets


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