(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
By $(\infty,1)Topos$ is denoted the collection of all (∞,1)-toposes. This is the (∞,1)-category-theory analog of Topos.
$(\infty,1)\,Topos$ is the (non-full) sub-(∞,1)-category of (∞,1)Cat on (∞,1)-toposes and (∞,1)-geometric morphisms between them.
Morally, this should actually be an (∞,2)-category, just as Topos is a 2-category, but since the technology of $(\infty,2)$-categories is not well developed, this point of view is not often taken yet.
(…)
We discuss existence of (∞,1)-limits and (∞,1)-colimits in $(\infty,1)Topos$.
The $(\infty,1)$-category $(\infty,1)Topos$ has all small $(\infty,1)$-colimits and functor
preserves these.
This is HTT, prop. 6.3.2.3.
The $(\infty,1)$-category $(\infty,1)Topos$ has all small $(\infty,1)$-colimits and the inclusion
sends (∞,1)-limits to (∞,1)-limits.
The $(\infty,1)$-category $(\infty,1)Topos$ has filtered (∞,1)-limits and the inclusion
preserves these.
This is HTT, prop. 6.3.3.1.
The $(\infty,1)$-category $(\infty,1)Topos$ has all small (∞,1)-limits.
This is HTT, prop. 6.3.4.7.
The $(\infty,1)$-limits in $(\infty,1)Topos$ coincide actually with the proper $(\infty,2)$-limits.
This is HTT, remark 6.3.4.10.
We discuss more or less explicit descriptions of (∞,1)-limits and (∞,1)-colimits in $(\infty,1)Topos$.
Let
be a diagram of (∞,1)-toposes. Then its (∞,1)-pullback is computed, roughly, by the (∞,1)-pushout of their (∞,1)-sites of definition (see above).
More in detail: there exist (∞,1)-sites $\tilde \mathcal{D}$, $\mathcal{D}$, and $\mathcal{C}$ with finite (∞,1)-limit and morphisms of sites
such that
Let then
be the (∞,1)-pushout of the underlying (∞,1)-categories in the full sub-(∞,1)-category (∞,1)Cat${}^{lex} \subset (\infty,1)Cat$ of $(\infty,1)$-categories with finite $(\infty,1)$-limits.
Let moreover
be the reflective sub-(∞,1)-category obtained by localization at the class of morphisms generated by the inverse image $Lan_{f'}(-)$ of the coverings of $\mathcal{D}$ and the inverse image $Lan_{g'}(-)$ of the coverings of $\tilde \mathcal{D}$.
Then
is an (∞,1)-pullback square.
This is HTT, prop. 6.3.4.6.
$(\infty,1)$Topos
section 6.3 in