By Werner Greub, Stephen Halperin, Ray Vanstone

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**Extra info for Connections, Curvature, and Cohomology: De Rham Cohomology of Manifolds and Vector Bundles**

**Sample text**

Thus we have a short exact sequence p →Y − → 0. 0− →K− →X− Since p is a weak equivalence, we have H∗ K = 0. First choose w ∈ Xn such that pw = y. Then dw − x ∈ Zn−1 K, since p(dw) = d(pw) = dy = px, and d(dw − x) = dx = 0. Since H∗ K = 0, there is a v ∈ Kn such that dv = dw − x. Let z = w − v. Then pz = y and dz = x, as required. Our next goal is to characterize the coﬁbrations. We begin with the coﬁbrant objects. 6. Suppose R is a ring. If A is a coﬁbrant chain complex, then An is a projective R-module for all n.

That is, if σ : f − →f and τ : g − → g are 2-morphisms of K such that g ◦ f makes sense, then the following diagram commutes. mgf F g ◦ F f −−−−→ F (g ◦ f ) ⏐ ⏐ ⏐ ⏐ F (τ ∗σ) F τ ∗F σ mg F g ◦ F f −−−−→ F (g ◦ f ) Note that pseudo-2-functors need not preserve isomorphisms, but they do preserve equivalences. Similarly, if we apply a pseudo-2-functor to a commutative diagram of morphisms, the resulting diagram need no longer be commutative; but it is commutative up to natural isomorphism. 9 in the following way.

Either W ∩ I-cof ⊆ J-cof or W ∩ J-inj ⊆ I-inj. Proof. These conditions certainly hold in a coﬁbrantly generated model category. Conversely, suppose we have a category C with a subcategory W and sets of maps I and J satisfying the hypotheses of the theorem. Deﬁne a map to be a ﬁbration if and only if it is in J-inj, and deﬁne a map to be a coﬁbration if and only if it is in I-cof. Then certainly the coﬁbrations and ﬁbrations are closed under retracts. It follows from the hypotheses that every map in J-cell is a trivial coﬁbration, and hence that every map in J-cof is a trivial coﬁbration.