Category Theory in Context (Aurora: Dover Modern Math Originals) by Emily Riehl

Author:Emily Riehl [Riehl, Emily]
Language: eng
Format: azw3
Publisher: Dover Publications
Published: 2017-03-09T05:00:00+00:00

Because Ord and Sym are objectwise isomorphic, the sets of labeled Sym-structures and labeled Ord-structures are isomorphic. However, the set of unlabeled Sym-structures on n is the set of conjugacy classes of permutations of n-elements, while the set of unlabeled Ord-structures on n is trivial: all linear orders on n are isomorphic. See [Joy81] for more.

A discussion of the functoriality of chosen limits and colimits should come with a few warnings. Limits, when they exist, are unique up to a unique isomorphism commuting with the maps in the limit cone. But this is not the same thing as saying that limits are unique on the nose: even if there is a unique object satisfying the universal property of the limit, each of its automorphisms gives rise to a distinct limit cone. Moreover, choices of limits of diagrams of fixed shape, as required to define the limit functor of Proposition 3.6.1, can seldom be made compatibly. For instance, Freyd has shown that in any category with pullbacks, it is possible to choose canonical pullbacks so that the “horizontal” composite of canonical pullback squares, in the sense of Exercise 3.1.viii, is the canonical pullback of the composite rectangle; however, it is not possible to also arrange so that the “vertical” composites of canonical pullback squares are canonical pullbacks [FS90, §1.4].

Even when the chosen limit objects are equal, the various natural isomorphisms associated to the limit constructions might not be identities. To illustrate this, first observe:

LEMMA 3.6.6. For any triple of objects X, Y, Z in a category with binary products, there is a unique natural isomorphism X × (Y × Z) ≅ (X × Y) × Z commuting with the projections to X, Y, and Z.

PROOF. Exercise 3.6.ii.

Lemma 3.6.6 asserts that the product is naturally associative. It follows that any iteration of binary products can be used to define n-ary products. However, even in a skeletal category, in which the objects X × (Y × Z) and (X × Y) × Z are necessarily equal, the natural isomorphism may not be the identity. The following example, from [ML98a, p. 164], is due to Isbell.

EXAMPLE 3.6.7. Consider sk(Set), a skeletal category of sets. Since sk(Set) is equivalent to a complete category it has all limits and, in particular, has products. Let C denote the countably infinite set. Its product C × C = C and the product projections π1, π2 : C → C are both epimorphisms. Suppose the component of the natural isomorphism C × (C × C) ≅ (C × C) × C were the identity. Naturality would then imply that for any triple of maps f, g, h : C → C that f × (g × h) = (f × g) × h, on account of commutativity of the square

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