(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The notion of $(\infty,1)$-sheaf (or ∞-stack or geometric homotopy type) is the analog in (∞,1)-category theory of the notion of sheaf (geometric type?) in ordinary category theory.
See (∞,1)-category of (∞,1)-sheaves for more.
Given an (∞,1)-site $C$, let $S$ be the class of monomorphisms in the (∞,1)-category of (∞,1)-presheaves $PSh_{(\infty,1)}(C)$ that correspond to covering (∞,1)-sieve?s
on objects $c \in C$, where $j$ is the (∞,1)-Yoneda embedding.
Then an (∞,1)-presheaf $A \in PSh_{(\infty,1)}(C)$ is an $(\infty,1)$-sheaf if it is an $S$-local object. That is, if for all such $\eta$ the morphism
is an equivalence. For a presheaf $A : C^{\op} \to E$ with values in an arbitrary ∞-category, we say it is a sheaf iff $E(e, A(-))$ is a sheaf for every object $e$ of $E$.
This is the analog of the ordinary sheaf condition for covering sieves. The ∞-groupoid $PSh_C(U,A)$ is also called the descent-∞-groupoid of $A$ relative to the covering encoded by $U$.
As in the 1-categorial case, the sheaf condition for a covering sieve can be translated into a condition on a covering family that generates it:
Let $\{ u_i \to c \}$ be a family of morphisms of $C$ that generate the sieve corresponding to $\eta : U \hookrightarrow j(c)$, and let $r_\bullet : \mathbf{\Delta}^{\op} \to PSh_C$ be the Čech nerve of $\amalg_i j(u_i) \to j(c)$. Then a presheaf $A$ is local with respect to $\eta$ iff the induced map $A(c) \to \lim A(r_\bullet)$ is an equivalence.
Thus, a presheaf $A$ is a sheaf iff every covering sieve contains a generating family satisfying this condition. Spelling out the description of the Čech nerve, the condition is that we have
If $C$ has pullbacks, this simplifies to
and furthermore this formulation applies to presheaves with values in an arbitrary ∞-category.
Taking colimits of Čech nerve computes $(-1)$-truncations in $(PSh_C)/j(X)$, so $\colim(r_\bullet)$ is the subobject of $j(c)$ corresponding to the sieve $\eta$. We have
and so the theorem follows.
An ($\infty$,1)-sheaf is also called an ∞-stack with values in ∞-groupoids.
The practice of writing “$\infty$-sheaf” instead of ∞-stack is a rather reasonable one, since a stack is nothing but a 2-sheaf.
Notice however that there is ambiguity in what precisely one may mean by an $\infty$-stack: it can be an $(\infty,1)$-sheaf or more specifically a hypercomplete $(\infty,1)$-sheaf. This is a distinction that only appears in (∞,1)-topos theory, not in (n,1)-topos theory for finite $n$.
$(\infty,1)$-sheaf / ∞-stack
homotopy level | n-truncation | homotopy theory | higher category theory | higher topos theory | homotopy type theory |
---|---|---|---|---|---|
h-level 0 | (-2)-truncated | contractible space | (-2)-groupoid | true/unit type/contractible type | |
h-level 1 | (-1)-truncated | contractible-if-inhabited | (-1)-groupoid/truth value | (0,1)-sheaf/ideal | mere proposition/h-proposition |
h-level 2 | 0-truncated | homotopy 0-type | 0-groupoid/set | sheaf | h-set |
h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid |
h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | (3,1)-sheaf/2-stack | h-2-groupoid |
h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf/3-stack | h-3-groupoid |
h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h-$n$-groupoid |
h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |
Section 6.2.2 in