totally connected geometric morphism (changes) in nLab

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Totally connected geometric morphism

Idea

Totally connectedness is a stronger form of local connectedness that arose in the work of M. Bunge and J. Funk on topos distributions.

The properties of totally connected geometric morphisms are largely dual to those of local geometric morphisms.

Definition

A geometric morphism f:E→Sf\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.

Properties

• 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.

• Totally connected geometric morphisms are closed under composition and stable under pullback.

• ff is totally connected iff ff is connected and has a right adjoint in 𝔗𝔬𝔭\mathfrak{Top}, the 2-category of toposes and geometric morphisms.

• Totally connected geometric morphisms are orthogonal to grouplike morphismsgrouplike geometric morphisms (cf. Johnstone (2002)).

• A connected, locally connected p:ℰ→𝒮p:\mathcal{E}\to\mathcal{S} that satisfies the Lawvere Nullstellensatz (i.e. the canonical θ:p *→p !\theta:p_*\to p_! is epic) is totally connected precisely iff it is a quality type over 𝒮\mathcal{S} (i.e. θ\theta is iso; cf. Johnstone (2011)).

Examples

• 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:Set→Psh(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.

• locally connected topos / locally ∞-connected (∞,1)-topos

  • connected topos / ∞-connected (∞,1)-topos

  • strongly connected topos / strongly ∞-connected (∞,1)-topos

  • totally connected topos / totally ∞-connected (∞,1)-topos

• local topos / local (∞,1)-topos.

• cohesive topos / cohesive (∞,1)-topos

and

• locally connected site / locally ∞-connected site

  • connected site / ∞-connected site

  • strongly connected site / strongly ∞-connected site

  • totally connected site / totally ∞-connected site

• local site / ∞-local site

• cohesive site / ∞-cohesive site

References

• Marta Bunge, Jonathon Funk, Spreads and the Symmetric Topos , JPAA 113 (1996) pp.1-38.

• Marta Bunge, Jonathon Funk, Spreads and the Symmetric Topos II , JPAA 130 (1998) pp.49-84.

• Peter Johnstone, Sketches of an Elephant , Oxford UP 2002. (vol. 2, Section C.3.6, pp.707-710)

• P. Johnstone, Remarks on Punctual Local Connectedness , TAC 25 no.3 (2011) pp.51-63. (pdf)