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

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos




For H\mathbf{H} an (∞,1)-topos and XHX \in \mathbf{H} any object, the over-(∞,1)-category H/X\mathbf{H}/X is itself an (,1)(\infty,1)-topos: the over-(,1)(\infty,1)-topos .

If we think of H\mathbf{H} as a big topos, then for XHX \in \mathbf{H} we may think of H/X\mathbf{H}/X \in (∞,1)-topos as the little topos-incarnation of XX. The objects of H/X\mathbf{H}/X we may think of as (∞,1)-sheaves on XX.

This correspondence between objects of XX and their little-topos incarnation is entirly natural: H\mathbf{H} is equivalently recovered as the (∞,1)-category whose objects are over-(,1)(\infty,1)-toposes H/X\mathbf{H}/X and whose morphisms are (∞,1)-geometric morphisms over H\mathbf{H}.



For H\mathbf{H} an (∞,1)-topos and XHX \in \mathbf{H} an object also the over-(∞,1)-category H/X\mathbf{H}/X is an (,1)(\infty,1)-topos. This is the over-(,1)(\infty,1)-topos of H\mathbf{H} over XX.

This is HTT, prop 1).


Base change to the point


There is a canonical (∞,1)-geometric morphism

H/XX *X *X !H \mathbf{H}/X \stackrel{\overset{X_!}{\longrightarrow}}{\stackrel{\overset{X^*}{\longleftarrow}}{\underset{X_*}{\longrightarrow}}} \mathbf{H}

where the extra left adjoint X !X_! is the obvious projection X !:(YX)XX_! : (Y \to X) \mapsto X, and X *X_* is given by forming the (∞,1)-product with XX.

This is called an etale geometric morphism. See there for more details.


The fact that (X !X *)(X_! \dashv X^*) follows from the universal property of the products. The fact that X *X^* preserves (∞,1)-colimits and hence has a further right adjoint X *X_* by the adjoint (∞,1)-functor theorem follows from that fact that H\mathbf{H} has universal colimits.


If H\mathbf{H} is a locally ∞-connected (∞,1)-topos then for all XHX \in \mathbf{H} also the over-(,1)(\infty,1)-topos H/X\mathbf{H}/X is locally \infty-connected.


The composite of (∞,1)-geometric morphisms

H/XX *X *X !HΓ HLConst HΠ HGrpd \mathbf{H}/X \stackrel{\overset{X_!}{\longrightarrow}}{\stackrel{\overset{X^*}{\longleftarrow}}{\underset{X_*}{\longrightarrow}}} \mathbf{H} \stackrel{\overset{\Pi_{\mathbf{H}}}{\longrightarrow}}{\stackrel{\overset{LConst_{\mathbf{H}}}{\longleftarrow}}{\underset{\Gamma_{\mathbf{H}}}{\longrightarrow}}} \infty Grpd

is itself a geometric morphism. Since ∞Grpd is the terminal object in (∞,1)Topos this must be the global section geometric morphism for H/X\mathbf{H}/X. Since it has the extra left adjoint ΠX !\Pi \circ X_! it is locally \infty-connected.


Let ((,1)Topos/H) et(,1)Topos/H((\infty,1)Topos/\mathbf{H})_{et} \subset (\infty,1)Topos/\mathbf{H} be the full sub-(∞,1)-category on the etale geometric morphisms H/XH\mathbf{H}/X \to \mathbf{H}. Then there is an equivalence

((,1)Topos/H) etH ((\infty,1)Topos/\mathbf{H})_{et} \simeq \mathbf{H}

See etale geometric morphism for more details.

General base change

See base change geometric morphism.

As (,1)(\infty,1)-sheaves on the big (,1)(\infty,1)-site of an object

We spell out how H/X\mathbf{H}/X is the (∞,1)-category of (∞,1)-sheaves over the big site of XX.


(forming overcategories commutes with passing to presheaves)

Let CC be a small (∞,1)-category and X:KCX : K \to C a diagram. Write C /XC_{/X} and PSh(C)/ XPSh(C)/_{X} for the corresponding over (∞,1)-categories, where – notationally implicitly – we use the (∞,1)-Yoneda embedding CPSh(C)C \to PSh(C).

Then we have an equivalence of (∞,1)-categories

PSh(C/X)PSh(C)/X. PSh(C/X) \stackrel{\simeq}{\longrightarrow} PSh(C)/X \,.

This appears as HTT, For more on this see (∞,1)-category of (∞,1)-presheaves.


Here we may think of C/XC/X as the big site of the object cPSh(C)c \in PSh(C), hence of PSh(C/X)PSh(C/X) as presheaves on XX.


Let CC be equipped with a subcanonical coverage, let XCX \in C and regard C/XC/X as an (∞,1)-site with the big site-coverage. Then we have

Sh(C/X)Sh(C)/X. Sh(C/X) \simeq Sh(C)/X \,.

By the discussion of adjoint (∞,1)-functors on over-(∞,1)-categories adjoint (∞,1)-functors we have that the adjunction

(Fi):Sh(C)FPSh(C) (F \dashv i) : Sh(C) \stackrel{\overset{F}{\leftarrow}}{\hookrightarrow} PSh(C)

passes to an adjunction on the over-(∞,1)-categories

(F/Xi/X):Sh(C)/XFPSh(C)/X, (F/X \dashv i/X) : Sh(C)/X \stackrel{\overset{F}{\leftarrow}}{\hookrightarrow} PSh(C)/X \,,

(where we are using that FiXXF i X \simeq X by the assumption that the coverage is subcanonical, so that the representable XX is a (∞,1)-sheaf), such that i/Xi/X is still a full and faithful (∞,1)-functor (where we are using that the unit XFiXX \to F i X is an equivalence, since XX is a sheaf).

Since moreover the (∞,1)-limits in Sh(C)/XSh(C)/X are computed as limits in Sh(C)Sh(C) over diagrams with a bottom element adjoined (as discussed at limits in over-(∞,1)-categories) it follows that with FF preserving all finite limits, so does F/XF/X.

In summary we have that (F/Xi/X)(F/X \dashv i/X) is a reflective sub-(∞,1)-category of PSh(C/X)PSh(C/X) hence is the (∞,1)-category of (∞,1)-sheaves on the category C/XC/X for some (∞,1)-site-structure. But since F/XF/X inverts precisely those morphisms that are inverted by FF under the projection PSh(C)/XPSh(C)PSh(C)/X \to PSh(C), it follows that this is the big site structure on C/XC/X (this is the defining property of the big site).

Specifically for the (,1)(\infty,1)-topos H=\mathbf{H} = ∞Grpd we also have the following characterization.


For H=\mathbf{H} = ∞Grpd we have that for XGrpX \in \infty Grp any ∞-groupoid the corresponding over-(,1)(\infty,1)-topos is equivalent to the (∞,1)-category of (∞,1)-presheaves on XX:

Grpd/XPSh(X)[X,Grpd]. \infty Grpd/X \simeq PSh(X) \simeq [X, \infty Grpd] \,.

This is a special case of the (∞,1)-Grothendieck construction. See the section (∞,0)-fibrations over ∞-groupoids.

The following proposotion asserts that the over-(,1)(\infty,1)-topos over an nn-truncated object indeed behaves like a generalized n-groupoid


For nn \in \mathbb{N} and 𝒳\mathcal{X} an n-localic (∞,1)-topos, then the over-(,1)(\infty,1)-topos 𝒳/U\mathcal{X}/U is nn-localic precisely if the object UU is nn-truncated.

This is (StrSp, lemma 2.3.16).

Object classifier

If Obj κHObj_\kappa \in \mathbf{H} is an object classifier for κ\kappa-small objects, then the projection Obj Κ×XoXObj_\Kappa \times X \o X regarded as an object in the slice is a κ\kappa-small object classifier in H /X\mathbf{H}_{/X}.

In homotopy type theory

If a homotopy type theory is the internal language of H\mathbf{H}, then then theory in context x:Xx : X \vdash \cdots is the internal language of H /X\mathbf{H}_{/X}.


The general notion is discussed in section 6.3.5 of

Some related remarks are in

Revised on September 8, 2015 17:06:29 by Urs Schreiber (