nLab
very large (infinity,1)-sheaf (infinity,1)-topos

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Definition

Fix a Grothendieck universe 𝒰\mathcal{U} and a smaller universe 𝒱𝒰\mathcal{V} \in \mathcal{U}. Let 𝒱\mathcal{V} be the reference-universe, so that sets in 𝒱\mathcal{V} are called small sets, sets in 𝒰\mathcal{U} are called large, and sets not necessarily in 𝒰\mathcal{U} are called very large.

Write ∞Grpd for the (large) (∞,1)-category of small ∞-groupoids and GRPD\infty GRPD for the very-large (,1)(\infty,1)-category of large \infty-groupoids.

Then the general procedures of universe enlargement can be applied to any large (,1)(\infty,1)-category to produce a very-large one. Specifically, we have the locally presentable enlargement: for CC a large (∞,1)-category with small (∞,1)-colimits, write

CFunc(C op,GRPD) \Uparrow C \subset Func(C^{op}, \infty GRPD)

for the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors that preserves small (∞,1)-limits. As described at universe enlargement, if CC is locally presentable, then C\Uparrow C can be identified with the naive enlargement, which consists of the large models of the “theory” of which CC consists of the small models.

In particular, when CC is an (∞,1)-sheaf (∞,1)-topos Sh(S)Sh(S) of small (∞,1)-sheaves, then C\Uparrow C can be identified with a (∞,1)-category of large (∞,1)-sheaves on the same site. (That is, with a suitable accessible left-exact-reflective subcategory of PSH(S)PSH(S) rather than Psh(S)Psh(S)—it is not yet known how to specify such a reflective subcategory purely in terms of data on SS.) Thus, we refer to C\Uparrow C as the very large (,1)(\infty,1)-sheaf (,1)(\infty,1)-topos on CC.

Note that since every topos can be identified with the category of sheaves on itself for the canonical topology, it is also reasonable to denote H\Uparrow \mathbf{H} by SH(H)SH(\mathbf{H}). C\Uparrow C can also also be identified with the category of ind-objects of H\mathbf{H}, for a suitable regular cardinal κ\kappa (namely, the cardinal of 𝒱\mathcal{V}).

Properties

Lemma

For every (,1)(\infty,1)-topos H\mathbf{H} there is an (∞,1)-functor

((,1)Topos)H \Uparrow ((\infty,1)Topos) \to \Uparrow \mathbf{H}

that preserves large (∞,1)-colimits and finite (∞,1)-limits. It is defined by sending F:(,1)Topos opGRPDF : (\infty,1)Topos^{op} \to \infty GRPD to the composite

H op((,1)Topos/H) et op(,1)Topos opFGRPD \mathbf{H}^{op} \simeq ((\infty,1)Topos/\mathbf{H})_{et}^{op} \to (\infty,1)Topos^{op} \xrightarrow{F} \infty GRPD

This is HTT, lemma 6.3.5.21.

References

This is discussed in section 6.3 of

The definition of H\Uparrow\mathbf{H} is in Notation 6.3.5.16 and Remark 6.3.5.17. The relation to ind-objects appears as remark 6.3.6.18.

Revised on December 11, 2010 17:58:50 by Mike Shulman (71.137.3.108)