sheaf and topos theory
(∞,1)-category of (∞,1)-presheaves
localization of an (∞,1)-category
(∞,1)-category of (∞,1)-sheaves
groupoid object in an (∞,1)-topos
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
models for ∞-stack (∞,1)-toposes
model structure on functors
model structure on simplicial presheaves
descent for simplicial presheaves
descent for presheaves with values in strict ∞-groupoids
structures in a cohesive (∞,1)-topos
shape / coshape
fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos/of a locally ∞-connected (∞,1)-topos
categorical/geometric homotopy groups
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A totally ∞-connected site is a site satisfying sufficient conditions to make the (∞,1)-sheaf (∞,1)-topos over it a totally ∞-connected (∞,1)-topos.
Let C be a locally and globally ∞-connected site; we say it is a strongly ∞-connected site if it is also a cofiltered (∞,1)-category.
If C is a totally ∞-connected site, then the (∞,1)-sheaf (∞,1)-topos Sh (∞,1)(C) over it is a totally ∞-connected (∞,1)-topos.
We need to check that the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos-functor Π:Sh (∞,1)(C)→∞Grpd preserves finite (∞,1)-limits.
By the discussion at ∞-connected site we have that Π is given by the (∞,1)-colimit (∞,1)-functor lim →:Func(C op,∞Grpd)→∞Grpd. On the opposite and therefore filtered (∞,1)-category C op these preserve finite (∞,1)-limits.
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
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