totally connected geometric morphism


Topos Theory

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Totally connected geometric morphism


A geometric morphism f:ESf\colon E\to S is totally connected if

  1. It is locally connected, i.e. its inverse image functor f *f^* has a left adjoint f !f_! which is SS-indexed, and

  2. The functor f !f_! is left exact, i.e. preserves finite limits.

When thinking of EE as a topos over SS via ff, we say that it is a totally connected SS-topos. In particular, when S=SetS=Set and f=(LConst,Γ)f = (L Const, \Gamma) is the unique global sections geometric morphism, we call EE a totally connected topos.


Of course, any totally connected geometric morphism is connected, since the terminal object is a particular finite limit. It is also strongly connected, since finite products are also finite limits.


  • A topos Sh(X)Sh(X) of sheaves on a topological space is totally connected iff XX has a dense point (a single point whose closure is all of XX).

  • A presheaf topos Psh(C)Psh(C) is totally connected iff CC is cofiltered.

Totally connected sites

A small site CC is called totally connected if

  1. CC is cofiltered, and

  2. Every covering sieve in CC is connected, when regarded as a subcategory of a slice category.

The second condition implies that all constant presheaves are sheaves, and hence that the left adjoint Colim:Psh(C)SetColim\colon Psh(C) \to Set of Const:SetPsh(C)Const\colon Set\to Psh(C) restricts to Sh(C)Sh(C) to give a left adjoint of LConstL Const. Cofilteredness of CC is exactly what is needed for left exactness of Colim:Psh(C)SetColim\colon Psh(C) \to Set, essentially by definition. Hence the topos of sheaves on any totally connected site is totally connected.

Conversely, one can show that any totally connected topos can be (but need not be) presented by some totally connected site.



Chapter C3.6 in

Revised on January 6, 2011 01:10:44 by Urs Schreiber (