New PDF release: A radical approach to algebra

By Mary W Gray

ISBN-10: 020102568X

ISBN-13: 9780201025682

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If sheaf morphisms f , g : F → G on C induce maps f∗ , g∗ : π ∗ (F ) → π ∗ (G) such that f∗ = g∗ , then f∗ = g∗ : π ∗ F ({x}) → π ∗ G({x}) for all x ∈ C. This means, then, that f = g. 42 The functor π ∗ preserves local epimorphisms and local monomorphisms of presheaves. Proof Suppose m : P → Q is a local monomorphism of presheaves on C. This means that if m(α) = m(β) for α, β ∈ P (x) there is a covering φi : yi → x such that φi∗ (α) = φi∗ (β) for all φi . Suppose that α, β ∈ π ∗ P (σ ) such that m∗ (α) = m∗ (β) in π ∗ Q(σ ).

This class of cofibrations α therefore contains all projective cofibrations. Observe further that the map (α, j ) is a sectionwise weak equivalence if either α is a sectionwise equivalence or j is a weak equivalence of simplicial sets. The I -diagram category therefore has a simplicial model structure in the sense that, if α : A → B is a projective cofibration and j : K → L is a cofibration of simplicial sets, then the map (α, j ) is a projective cofibration, which is a sectionwise weak equivalence if either α is a sectionwise weak equivalence or j is a weak equivalence of simplicial sets.

Say that a Grothendieck topos B is localic if it is equivalent to Shv(A) for some frame A. 32 A Grothendieck topos E is localic if and only if it is equivalent to Shv(P ) for some topology on a poset P . Proof [Outline] Starting with the poset P , the corresponding frame is the poset of subobjects of the terminal object 1 = ∗. For the reverse implication, these subobjects generate E, and then Giraud’s theorem is used to finish the proof. 5]. e. preserves all finite meets and all infinite joins, hence preserves both 0 and 1.

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A radical approach to algebra by Mary W Gray

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