Round 9: Tossup 5

Functors into these structures are described as colimits by a “density theorem” stated by Mac Lane. Examples of these structures that are “hereditarily finite” form a prototypical Grothendieck universe. A category is “concrete” if it has a faithful forgetful functor into the category of these structures. A functor on a locally small category whose target is these structures admits a natural isomorphism according to the Yoneda lemma. Categories are “small” if their proper class of objects can be represented as one of these structures. Independence from the standard theory of these structures can be proved using Paul Cohen’s technique of “forcing.” By the axiom of foundation, (10[1])the collection of all of these structures is not itself one of these structures. For 10 points, name these structures described by the Zermelo–Fraenkel axioms. ■END■

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