Contents
Context
-Topos Theory
(∞,1)-topos theory
Background
Definitions
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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
Characterization
Morphisms
Extra stuff, structure and property
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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
Models
Constructions
structures in a cohesive (∞,1)-topos
Contents
Idea
By is denoted the collection of all (∞,1)-toposes. This is the (∞,1)-category-theory analog of Topos.
Definition
is the (non-full) sub-(∞,1)-category of (∞,1)Cat on (∞,1)-toposes and (∞,1)-geometric morphisms between them.
Morally, this should actually be an (∞,2)-category, just as Topos is a 2-category, but since the technology of -categories is not well developed, this point of view is not often taken yet.
Properties
Existence of sites of definition
(…)
Existence of limits and colimits
We discuss existence of (∞,1)-limits and (∞,1)-colimits in .
Proposition
The -category has all small -colimits and functor
preserves these.
This is HTT, prop. 6.3.2.3.
Proposition
The -category has all small -colimits and the inclusion
sends (∞,1)-limits to (∞,1)-limits.
Propoisition
The -category has filtered (∞,1)-limits and the inclusion
preserves these.
This is HTT, prop. 6.3.3.1.
Propoisition
The -category has all small (∞,1)-limits.
This is HTT, prop. 6.3.4.7.
This is HTT, remark 6.3.4.10.
Computation of limits and colimits
We discuss more or less explicit descriptions of (∞,1)-limits and (∞,1)-colimits in .
Proposition
Let
be a diagram of (∞,1)-toposes. Then its (∞,1)-pullback is computed, roughly, by the (∞,1)-pushout of their (∞,1)-sites of definition (see above).
More in detail: there exist (∞,1)-sites , , and with finite (∞,1)-limit and morphisms of sites
such that
Let then
be the (∞,1)-pushout of the underlying (∞,1)-categories in the full sub-(∞,1)-category (∞,1)Cat of -categories with finite -limits.
Let moreover
be the reflective sub-(∞,1)-category obtained by localization at the class of morphisms generated by the inverse image of the coverings of and the inverse image of the coverings of .
Then
is an (∞,1)-pullback square.
This is HTT, prop. 6.3.4.6.
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Topos
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Topos
References
section 6.3 in