Redirected from "(∞,1)-bisite".
Contents
Context
-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
Idea
The notion of -quasitopos is the (∞,1)-topos-analog of the notion of quasitopos.
Definition
Definition
Let be an (∞,1)-bisite. Say an (∞,1)-presheaf is -biseparated if it is an (∞,1)-sheaf for and for every -covering sieve in we have that the induced morphism
in ∞Grpd is a full and faithful (∞,1)-functor.
We say it is -biseparated if
the induced morphism
is an (n-1)-truncated object in the (∞,1)-overcategory .
Definition
A (Grothendieck) -quasitopos is an (∞,1)-category that is equivalent to the full sub-(∞,1)-category of some on the -biseparated -presheaves, on some (∞,1)-bisite .
Examples
For a local (∞,1)-topos
and be a site of definition for , the -quasitopos on that factors the geometric embedding
is that of concrete objects in , the analog of concrete sheaves.
References
The definition as it stands, originated out of a discussion between Urs Schreiber and David Carchedi. The suggestion to rephrase the definition in terms of bisites came from Mike Shulman.