By Frederick S. Woods

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**Extra info for A course in mathematics,: For students of engineering and applied science, **

**Example text**

The proof of proposition 11 relies on an alternative description of S, as follows. Let J be the category with objects n = {1, . . , n} for n ≥ 1, and morphisms the surjections. Then any permutative category A determines a lax functor A∗ : J → Cat, where A∗ (n) = An , and if p : m → n is a surjection, then A∗ (p) : Am → An is given on objects by the formula A∗ (p)(a1 , . . , am ) = ×nj=1 ai , p(i)=j 28 Permutative categories using the natural order on the set p−1 (j), and similarly for morphisms.

The free EndA (P )-module of rank one is the image of P . Since A(P, −) commutes with sums, every free module is in the image, up to isomorphism. Finally, every EndA (P )-module X has a presentation, so it occurs in an exact sequence of EndA (P )-modules g EndA (P ) − → I EndA (P ) −→ X −→ 0 . J Since A(P, −) is full, the homomorphism g is isomorphic to A(P, f ) for some morphism f : J P in A. Since the functor A(P, −) is exact, I P −→ X is the image of the cokernel of f . Thus A(P, −) is an equivalence of categories.

The derived category. In this section, R is any ring. , the differential decreases the degree by 1. 1. A chain complex C of R-modules is cofibrant if there exists an exhaustive increasing filtration by subcomplexes 0 = C0 ⊆ C1 ⊆ · · · ⊆ Cn ⊆ · · · such that each subquotient C n /C n−1 consists of projective modules and has trivial differential. The (unbounded) derived category D(R) of the ring R has as objects the cofibrant complexes of R-modules and as morphisms the chain homotopy classes of chain maps.

### A course in mathematics,: For students of engineering and applied science, by Frederick S. Woods

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