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

Definition

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

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

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

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

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

In particular, when $C$ is an (∞,1)-sheaf (∞,1)-topos $Sh(S)$ of small (∞,1)-sheaves, then $\Uparrow C$ 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 $PSH(S)$ rather than $Psh(S)$—it is not yet known how to specify such a reflective subcategory purely in terms of data on $S$.) Thus, we refer to $\Uparrow C$ as the very large $(\infty,1)$-sheaf $(\infty,1)$-topos on $C$.

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 $\Uparrow \mathbf{H}$ by $SH(\mathbf{H})$. $\Uparrow C$ can also also be identified with the category of ind-objects of $\mathbf{H}$, for a suitable regular cardinal $\kappa$ (namely, the cardinal of $\mathcal{V}$).

Properties

Lemma

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

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

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

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

This is HTT, lemma 6.3.5.21.

References

This is discussed in section 6.3 of

• Jacob Lurie, Higher Topos Theory .

The definition of $\Uparrow\mathbf{H}$ 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.