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strongly connected topos

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Definition

A sheaf topos \mathcal{E} is called strongly connected if it is a locally connected topos

(Π 0ΔΓ):ΓΔΠ 0Set (\Pi_0 \dashv \Delta \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}} Set

such that the extra left adjoint Π 0\Pi_0 in addition preserves finite products (the terminal object and binary products).

This means it is in particular also a connected topos.

If Π 0\Pi_0 preserves even all finite limits then \mathcal{E} is called a totally connected topos.

If a strongly connected topos is also a local topos, then it is a cohesive topos.

Terminology

The “strong” in “strongly connected” may be read as referring to f !f *f_! \dashv f^* being a strong adjunction in that we have a natural isomorphism for the internal homs in the sense that

[f !X,A]f *[X,f *A]. [f_! X, A] \simeq f_* [X, f^* A] \,.

This follows already for ff connected and essential if f !f_! preserves products, because this already implies the equivalent Frobenius reciprocity isomorphism. See here for more.

and

Revised on December 7, 2011 19:47:10 by Urs Schreiber (131.174.40.86)