nLab
strongly connected topos

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Geometry

higher geometry / derived geometry

Ingredients

Concepts

Constructions

Examples

Theorems

Contents

Definition

A sheaf topos ℰ is called strongly connected if it is a locally connected topos

(Π 0⊣Δ⊣Γ):ℰ→Γ←Δ→Π 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 in addition preserves finite products (the terminal object and binary products).

This means it is in particular also a connected topos.

If Π 0 preserves even all finite limits then 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 * 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 f connected and essential if 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)
Edit | Back in time (3 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: classifying topos, sheaf and topos theory, topos, sheaf, presheaf, site, smooth infinity-groupoid, ETCS, Grothendieck topos, subobject classifier, Heyting algebra, global element, cover, sieve, Grothendieck topology, Lawvere-Tierney topology, geometric morphism, locale, Heyting category, pretopos, Elephant, internal logic, coverage, Grothendieck pretopology, subcanonical coverage, large site, Isbell envelope, fundamental group of a topos, frame, Grothendieck's Galois theory, universe in a topos, abelian sheaf, local isomorphism, base change, category of sheaves, sheafification, category of open subsets, dense monomorphism, Sheaves in Geometry and Logic, ringed site, inverse image, natural numbers object, diffeological space, co-Yoneda lemma, D-module, quasicoherent sheaf, ringed space, Verdier site, geometric embedding, synthetic differential geometry, separated presheaf, quasitopos, stalk, point of a topos, smooth space, concrete sheaf, effective topos, big and little toposes, Models for Smooth Infinitesimal Analysis, flabby sheaf, soft sheaf, fine sheaf, higher topos theory, localic topos, (0,1)-topos, locally ringed space, structure sheaf, topos theory - contents, Q-category, 1-topos, ringed topos, Kock-Lawvere axiom, ionad, geometric theory, canonical topology, Johnstone's topological topos, fully formal ETCS, open cover, inhabited object, logical functor, Euclidean-topological infinity-groupoid, locally n-connected (n+1,1)-topos, bounded geometric morphism, locally connected geometric morphism, etale geometric morphism, fpqc site, DistLat, open map, Loc, global section, essential geometric morphism, category of presheaves, subtopos, Whitehead tower in an (infinity,1)-topos, connected topos, forcing, locally connected topos, cohesive infinity-toposes - contents, big site, Cahiers topos, topological locale, double negation, posite, Diaconescu's theorem, base topos, homotopy equivalence of toposes, infinity-Lie group, localic geometric morphism, hyperconnected geometric morphism, indecomposable object, circle n-bundle with connection, coherent coverage, poset of subobjects, line object, super infinity-groupoid, cohesive (infinity,1)-topos, geometric realization of cohesive infinity-groupoids, Topos, local geometric morphism, strongly connected site, cohesive topos, totally connected geometric morphism, cohesive site, locally connected site, infinity-cohesive site, discrete group, Barr's theorem, Galois topos, local site, constructive Gelfand duality theorem, van Kampen theorem for toposes, surjective geometric morphism, Cohesive Toposes -- Combinatorial and Infinitesimal Cases, cosheaf, etale site, strongly connected topos, over-topos, topos of algebras over a monad, cohesive (infinity,1)-topos -- structure sheaves, strongly infinity-connected (infinity,1)-topos, cohesive, infinity-connected (infinity,1)-site, locally infinity-connected (infinity,1)-site, Tannaka duality for geometric stacks, smooth infinity-groupoid -- structures, locally ringed topos, Some Thoughts on the Future of Category Theory, infinity-local site, cohesive (infinity,1)-topos -- structures, standard site, super-groupoid, fppf site, strongly infinity-connected site, totally infinity-connected (infinity,1)-topos, connected site, totally infinity-connected site, (geometric surjection, embedding) factorization system, crystalline site, regular coverage, Handbook of Categorical Algebra, synthetic differential infinity-groupoid, discrete infinity-groupoid, dense sub-site, small-generated site, syntactic site, cartesian site, locally contractible topological infinity-groupoid, smooth super infinity-groupoid, cohesive (infinity,1)-topos -- infinitesimal cohesion, Bohr topos, atomic geometric morphism, syntactic category, coherent topos, regular topos, cohesive (infinity,1)-topos -- structures II, internal site, indexed topos, geometric transformation, concrete smooth infinity-groupoid, n-topos, formally smooth object, internal sheaf, real numbers object, contents of contents, cohesive homotopy type theory, shape modality, proper geometric morphism, infinitesimal shape modality, open geometric morphism, (hyperconnected, localic) factorization system, Artin gluing, topos of types, Giraud's theorem, closed cover, separated geometric morphism, unbounded topos, plus construction on presheaves, coreduced object, smooth groupoid, sheaf of meromorphic functions, morphism of sites, synthetic differential super infinity-groupoid, sheaf of abelian groups, realizability topos, geometry of physics, infinitesimal flat modality, predicative topos, reduced object, Galois cohomology, reduction modality, differential homotopy type theory, concretification, higher Atiyah groupoid