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

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos

Locality and descent



The notion of (,1)(\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 CC, let SS be the class of monomorphisms in the (∞,1)-category of (∞,1)-presheaves PSh (,1)(C)PSh_{(\infty,1)}(C) that correspond to covering (∞,1)-sieve?s

η:Uj(c) \eta : U \hookrightarrow j(c)

on objects cCc \in C, where jj is the (∞,1)-Yoneda embedding.

Then an (∞,1)-presheaf APSh (,1)(C)A \in PSh_{(\infty,1)}(C) is an (,1)(\infty,1)-sheaf if it is an SS-local object. That is, if for all such η\eta the morphism

A(c)PSh C(j(c),A)PSh C(η,A)PSh(U,A) A(c) \simeq PSh_C(j(c),A) \stackrel{PSh_C(\eta,A)}{\to} PSh(U,A)

is an equivalence. For a presheaf A:C opEA : C^{\op} \to E with values in an arbitrary ∞-category, we say it is a sheaf iff E(e,A())E(e, A(-)) is a sheaf for every object ee of EE.

This is the analog of the ordinary sheaf condition for covering sieves. The ∞-groupoid PSh C(U,A)PSh_C(U,A) is also called the descent-∞-groupoid of AA relative to the covering encoded by UU.

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 ic}\{ u_i \to c \} be a family of morphisms of CC that generate the sieve corresponding to η:Uj(c)\eta : U \hookrightarrow j(c), and let r :Δ opPSh Cr_\bullet : \mathbf{\Delta}^{\op} \to PSh_C be the Čech nerve of ⨿ ij(u i)j(c)\amalg_i j(u_i) \to j(c). Then a presheaf AA is local with respect to η\eta iff the induced map A(c)limA(r )A(c) \to \lim A(r_\bullet) is an equivalence.

Thus, a presheaf AA 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

A(c)lim( iA(u i) i,jPSh C(j(u i)× j(c)j(u j),A)) A(c) \simeq \lim\left( \prod_i A(u_i) \stackrel{\to}{\to} \prod_{i,j} PSh_C(j(u_i) \times_{j(c)} j(u_j), A) \stackrel{\to}{\stackrel{\to}{\to}} \cdots \right)

If CC has pullbacks, this simplifies to

A(c)lim( iA(u i) i,jA(u i× cu j)) A(c) \simeq \lim\left( \prod_i A(u_i) \stackrel{\to}{\to} \prod_{i,j} A(u_i \times_c u_j) \stackrel{\to}{\stackrel{\to}{\to}} \cdots \right)

and furthermore this formulation applies to presheaves with values in an arbitrary ∞-category.


Taking colimits of Čech nerve computes (1)(-1)-truncations in (PSh C)/j(X)(PSh_C)/j(X), so colim(r )\colim(r_\bullet) is the subobject of j(c)j(c) corresponding to the sieve η\eta. We have

PSh C(colim(r ),A)limPSh C(r ,A) PSh_C(\colim(r_\bullet), A) \simeq \lim PSh_C(r_\bullet, A)

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 (,1)(\infty,1)-sheaf or more specifically a hypercomplete (,1)(\infty,1)-sheaf. This is a distinction that only appears in (∞,1)-topos theory, not in (n,1)-topos theory for finite nn.

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/​unit type/​contractible type
h-level 1(-1)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-\infty-groupoid


Section 6.2.2 in

Last revised on October 21, 2020 at 11:33:45. See the history of this page for a list of all contributions to it.