Redirected from "terminal geometric morphisms".
Contents
Context
Topos Theory
topos theory
Background
Toposes
Internal Logic
Topos morphisms
Cohomology and homotopy
In higher category theory
Theorems
-Topos Theory
(∞,1)-topos theory
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elementary (∞,1)-topos
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(∞,1)-site
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reflective sub-(∞,1)-category
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(∞,1)-category of (∞,1)-sheaves
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(∞,1)-topos
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(n,1)-topos, n-topos
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(∞,1)-quasitopos
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(∞,2)-topos
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(∞,n)-topos
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hypercomplete (∞,1)-topos
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over-(∞,1)-topos
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n-localic (∞,1)-topos
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locally n-connected (n,1)-topos
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structured (∞,1)-topos
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locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
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local (∞,1)-topos
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cohesive (∞,1)-topos
structures in a cohesive (∞,1)-topos
Contents
For the time being, the following speaks in -topos theory, for definiteness; but all statements and proofs apply verbatim also to Grothendieck toposes, since they just depend on general abstract category-theoretic properties.
Idea
For an -topos over the base , it has an essentially unique geometric morphism to
(1)
where:
In other words, is the terminal object in the -category of -stack -toposes and -geometric morphisms between these.
Properties
Proposition
The geometric morphism (1) exists essentially uniquely.
(e.g.
Lurie 2009, HTT 6.3.4.1)
Proof
Since every -groupoid is an -colimit (over itself) of the point (see there):
(2)
and since the inverse image of a geometric morphism preserves finite limits (by definition), such as the terminal object, and all -colimits (since left adjoints preserve colimits), we have that must be given by forming the corresponding -colimit of copies of the terminal object in , which does exist:
Proposition
The direct image of the terminal geometric morphism (1) is given by the hom-space out of the terminal object, in that for there is a natural equivalence
where denotes the terminal object.
Proof
For all we have the following sequence of natural equivalences:
(Here we used (2) and that hom-functor preserves limits and that left adjoints preserve colimits and that preserves finite limits such as the terminal object, by definition.)
But this is a hom-equivalence which exhibits (see here) as a right adjoint -functor to . This implies the claim by essential uniqueness of adjoints.
Literature
For discussion in plain topos theory see any of the references listed there.
Discussion in -topos theory includes: