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By Peter J. Olver

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If a functor F : C − → Set takes its values in a category deﬁned by some algebraic structure (we do not intend to give a precise meaning to such a sentence) and if this functor is representable by some object X , then X will be endowed with morphisms which will mimic this algebraic structure. For example if F takes its values in the category Group of groups, then X will be endowed with a structure of a “group-object”. This notion will be discussed in Sect. 1. We shall see in Chap. 2 that the notion of representable functor allows us to deﬁne projective and inductive limits in categories.

Then there exists a unique iso→ G 1 such that θ1 = (hC ◦θ) ◦ θ0 . morphism of functors θ : G 0 − 28 1 The Language of Categories Proof. 11 to the functor F∗ : C − → C ∧ and the full sub∼ → C such that F∗ − category C of C ∧ , we get a functor G : C − → hC ◦G, and this functor G is unique up to unique isomorphism, again by this lemma. d. 2) Hom C (X, G(Y )) F∗ (Y )(X ) Hom C (F(X ), Y ) . Consider the functor ∨ G∗ : C − → C , X → Hom C (X, G( • )) . Then for each X ∈ C, G ∗ (X ) is representable by F(X ).

The simplicial category ∆ is deﬁned as follows. The objects of ∆ are the ﬁnite totally ordered sets and the morphisms are the orderpreserving maps. Let ∆ be the subcategory of ∆ consisting of non-empty sets and Hom ∆ (σ, τ ) = ⎧ ⎫ u sends the smallest (resp. the largest)⎬ ⎨ u ∈ Hom ∆ (σ, τ ) ; element of σ to the smallest (resp. the . ⎩ ⎭ largest) element of τ For integers n, m denote by [n, m] the totally ordered set {k ∈ Z; n ≤ k ≤ m}. (i) Prove that the natural functor ∆ − → Set f is half-full and faithful.