Model Categories and Their Localizations by Philip S. Hirschhorn

By Philip S. Hirschhorn

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Model Categories and Their Localizations

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Associated with any space is a cosheaf D(U ) = |U | that carries an open set U to its underlying set. If X is also locally connected, but not discrete (such as the reals), then the unique morphism G π0 to the terminal cosheaf (connected components) is not an isomorD G π0 is both G π0 to an isomorphism, so that D phism. But Φ carries D a monomorphism and an epimorphism in Cosh(X). Thus, Cosh(X) is not a balanced category in this case. In particular, it cannot even be an elementary topos. 15 The following comments may be of interest.

G1 equal to X ✷ This shows that the two conditions are equivalent. 2. 4 If a dense geometric morphism has subopen domain, then its codomain topos is also subopen. Proof . Let F diagram in E . ψ G E be dense with F subopen. Consider the following e∗ (ΩS ) G G ψ∗ f ∗ (ΩS )  τE  ΩE ψ∗ (τF )  G ψ∗ (ΩF ) The τ ’s are the transposes of the canonical frame morphisms. We conclude ✷ that τE is a monomorphism, so E is subopen. 5 If a geometric morphism F G G any definable subobject U Y in E , the adjunction square 38 2 Complete Spread Maps of Toposes U  G ρ∗ ρ∗ U   Y  G ρ∗ ρ∗ Y is a pullback.

Proof . Suppose that ρ is dense. Let U G G Y be definable, ‘classified’ by a G e∗ ΩS (possibly not unique). Both squares in the following morphism Y diagram are pullbacks. U  G1  G1   Y  G e∗ ΩS G  G ρ∗ ρ∗ (e∗ ΩS ) The outer square above is equal to the outer square of U  G ρ∗ ρ∗ U  G1   Y  G ρ∗ ρ∗ Y  G ρ∗ ρ∗ (e∗ ΩS ) which is therefore a pullback. The right hand square is also a pullback so we are done. For the converse, we assume E is subopen. If the stated pullback condition holds, then sending a definable subobject U G G Y to ρ∗ U G G ρ∗ Y is injective.

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