Discrete and concrete objects
(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
Inside a local (∞,1)-topos there are objects that may be thought of as ∞-groupoids equipped with extra structure (“cohesive structure” if is even a cohesive (∞,1)-topos). These are the concrete objects in .
Let be a local (∞,1)-topos.
Since is by definition a full and faithful (∞,1)-functor this means that
is a geometric embedding. By the discussion at reflective sub-(∞,1)-category this means that is the localization of an (∞,1)-category at a class of morphisms. It factors therefore canonically through the (∞,1)-quasitopos of -separated -sheaves
We call the (∞,1)-quasitopos of concrete objects of the local -topos .
We say an object is -concrete if the canonical morphism is (n-1)-truncated.
If a 0-truncated object is -concrete, we call it just concrete.
For and , the -concretification of is the morphism
that is the left factor in the decomposition with respect to the n-connected/n-truncated factorization system of the -unit
This entry goes back to some observations by David Carchedi.
Revised on November 23, 2011 17:50:00
by Urs Schreiber