An Invitation to General Algebra and Universal Constructions by George M. Bergman

Author:George M. Bergman
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

The Axiom of Choice clearly allows us to construct a skeleton for every category.

Lemma 7.9.8.

Let C and D be categories, with skeleta C 0 and D 0 . Then C and D are equivalent if and only if C 0 and D 0 are isomorphic. □

Exercise 7.9:18.

Write out the proof of Lemma 7.9.8.

Lemma 7.9.8 shows that equivalent categories agree in all properties that respect isomorphism of categories and “don’t depend on how many isomorphic copies each object has”; that is, intuitively speaking, in all “genuinely category-theoretic” properties.

Exercise 7.9:19.

Show that Set is not equivalent to Set op by finding a category-theoretic property possessed by one of these categories but not the other, and proving that equivalent categories must agree with respect to whether this property holds. For additional credit, demonstrate the non-equivalence of a few other pairs of familiar categories, e.g., show that Set is not equivalent to Group.

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