Sheaves of E 8 -Rings

We now study sheaves with values in the 8-category CAlg of E8-rings.

Remark 1.3.5.1. The forgetful functor CAlg “ CAlgpSpq Ñ Sp is conservative and preserves small limits (see Lemma HA.3.2.2.6 and Corollary HA.3.2.2.5 ). It follows that for any8-toposX, we have a canonical equivalence of 8-categories (even an isomorphism of simplicial sets)Shv_CAlgpXq »CAlgpShv_SppXqq.

Remark 1.3.5.2. Let X be an 8-topos and O : X^op Ñ CAlg a sheaf of E8-rings onX. Composing with the forgetful functor CAlgÑSp, we obtain a sheaf of spectra onX; we will generally abuse notation by denoting this sheaf of spectra also by O. In particular, we can define homotopy groups π_nO as in Notation 1.3.2.3. These homotopy groups have a bit more structure in this case: π0O is a commutative ring object in the underlying topos of X, while each π_nO has the structure of a π₀O-module.

Definition 1.3.5.3. Let X be an 8-topos. We will say that a sheafO of E8-rings on X is connective if it is connective when regarded as a sheaf of spectra on X: that is, if the homotopy groups πnO vanish for nă0. We letShv_CAlgpXq^cn denote the full subcategory of Shv_CAlgpXq spanned by the connective sheaves ofE8-rings onX.

1.3. SHEAVES OF SPECTRA 129 Remark 1.3.5.4. Let X be an 8-topos. Combining Proposition 1.3.4.7 with Remark HA.2.2.1.5 , we deduce that the inclusion Shv_CAlgpXq^cn ãÑ Shv_CAlgpXq admits a right adjoint. In other words, if O is an arbitrary sheaf of E8-rings onX, then we can find a connective sheaf of E8-rings O¹ equipped with a map α : O¹ Ñ O having the following universal property: for every object A P Shv_CAlgpXq^cn, composition with α induces a homotopy equivalence

Map_Shv

CAlgpXqpA,O¹q ÑMap_Shv

CAlgpXqpA,Oq.

In this case, we will say that O¹ is a connective cover of O, or that α exhibits O¹ as a connective cover ofO. Moreover, the mapα exhibitsO¹ as a connective cover of O in the 8-category Shv_SppXq; in particular, it induces isomorphisms

π_mO¹ »

#π_mO ifmě0 0 ifmă0.

We will generally denote the connective cover of O byτ_ě0O.

Definition 1.3.5.5. Let X be an 8-topos, let O be a connective sheaf ofE8-rings on X, and let ně0 be an integer. We will say that O is n-truncatedif the underlying spectrum-valued sheaf of O is n-truncated. We will say that O is discrete if it is 0-truncated. We letShv_CAlgpXq^cn_ďn denote the full subcategory ofShv_CAlgpXq^cn spanned by then-truncated objects ofShv_CAlgpXq^cn.

Remark 1.3.5.6. Let X be an 8-topos and let n ě 0 be an integer. Then we can identifyShv_CAlgpXq^cn_ďnwith the8-category of commutative algebra objects of the symmetric monoidal8-categoryShv_SppXq^cn_ďn. In particular, whenn“0, we can identifyShv_CAlgpXq^cn_ďn with the ordinary category of commutative ring objects of the underlying topos of X (see Proposition 1.3.2.7).

Combining Proposition 1.3.4.7 with Proposition HA.2.2.1.9 , we deduce that the inclusion functor

Shv_CAlgpXq^cn_ďnãÑShv_CAlgpXq^cn_ďn

admits a left adjoint. In other words, if O is an arbitrary connective sheaf of E8-rings on X, then we can find an n-truncated connective sheaf ofE8-rings O¹ equipped with a map α:O Ñ O¹ having the following universal property: for every objectA PShv_CAlgpXq^cn_ďn, composition with α induces a homotopy equivalence

Map_Shv

CAlgpXqpO¹,Aq ÑMap_Shv

CAlgpXqpO,Aq.

In this case, we will say that O¹ is an n-truncation of O, or that α exhibits O¹ as an n-truncation ofO. Moreover, the mapα exhibitsO¹ as ann-truncation ofO in the8-category

Shv_SppXq; in particular, it induces isomorphisms π_mO¹ »

#π_mO ifmďn

0 ifmąn.

We will generally denote the n-truncation ofO byτďnO.

Let X be an 8-topos and let F be a spectrum-valued sheaf on X. The condition thatF be connectivedoes not guarantee that FpUq is connective for each objectU PX. Nevertheless, there is a close relationship between connective Sp-valued sheaves on X and Sp^cn-valued sheaves onX:

Proposition 1.3.5.7. Let X be an 8-topos. Then composition with the truncation functor τě0 : SpÑSp^cn induces an equivalence of 8-categories Shv_SppXq^cnÑShv_Sp^cnpXq.

We will deduce Proposition 1.3.5.7 from the following more general principle:

Proposition 1.3.5.8. LetC be a compactly generated presentable8-category. LetC₀ĎC be a full subcategory which is closed under the formation of colimits and which is generated under small colimits by compact objects of C. LetX be an 8-topos. Then:

p1q The8-categoryC₀ is presentable and compactly generated.

p2q The inclusion C₀ĎC admits a right adjoint g which commutes with filtered colimits.

p3q Composition withg determines a functorG:Shv_CpXq ÑShv_C0pXq. p4q The functorG admits a fully faithful left adjoint F.

Remark 1.3.5.9. In the situation of Proposition 1.3.5.8, an objectF PShv_CpXqbelongs to the essential image of the full faithful embedding Shv_C0pXq ÑShv_CpXq if and only if the canonical mapGpFq ÑF is an L-equivalence in FunpX^op,Cq, whereLdenotes a left adjoint to the inclusion Shv_CpXqãÑFunpX^op,Cq.

Proof of Proposition 1.3.5.7. Let Sp^cn denote the full subcategory of Sp spanned by the connective spectra. Then Sp^cn is stable under small colimits in Sp, and is generated under small colimits by the sphere spectrum S PSp^cn (which is a compact object of the 8-category Sp). Consequently, Proposition 1.3.5.8 supplies a fully faithful embedding F :Shv_Sp^cnpXq ÑShv_SppXq for every8-topos X, which is right homotopy inverse to the functorShv_SppXq ÑShv_Sp^cnpXq given by composition with τě0 : SpÑSp^cn. To complete the proof, it will suffice to show that Shv_SppXq^cn is the essential image of the functor F.

Let F PShv_SppXq, so that we have a fiber sequence τě0F Ñ^φ F Ñτď´1F in the8 -category FunpX^op,Spq. Let L: FunpX^op,Spq ÑShv_SppXq be a left adjoint to the inclusion.

1.3. SHEAVES OF SPECTRA 131