Sheaves of Spectra

Proof. Let f :V ÑW be a morphism between affine objects of X; we wish to show that the induced map OXpVq b_OXpWqFpWq Ñ FpVq is an isomorphism. Replacing X by pX_{W,OX|Wq, we may assume that pX,OXq “Sp´etAfor some commutative ring A. Let us identify X with Shv_SetpCAlg^´et_Aq, so that we can view F as a functor from CAlg^´et_A to the category of sets. To prove thatF is quasi-coherent, it will suffice to show that for every

´

etaleA-algebra B, the natural map BbAFpAq ÑFpBq is an isomorphism (see Example 1.2.6.2).

We may assume without loss of generality that each Uα is affine and therefore associated to some ´etale A-algebra A_α. Since the objectsU_α coverX, we may choose a finite set of indicesα1, . . . , αnfor which the productś

1ďiďnAαi is faithfully flat overA. Our assumption onF guarantees that for any morphism of ´etaleA-algebrasC ÑC¹, ifCadmits the structure of an A_α-algebra for some index α, then the induced map C¹ bC FpCq Ñ FpC¹q is an isomorphism.

For any ´etaleA-algebra B, the hypothesis that F is an ´etale sheaf (and the flatness of B as an A-module) supplies a commutative diagram of short exact sequences

0 ^//BbAFpAq ^//

Since θ¹ andθ² are isomorphisms, we conclude that θis also an isomorphism.

1.3 Sheaves of Spectra

In §1.1, we introduced the notion of aspectrally ringed space: that is, a pair pX,OXq where X is a topological space and OX is a sheaf of E8-rings on X. The language of spectrally ringed spaces is adequate for describing many of the algebro-geometric objects which we are interested in studying (such as the spectral schemes of Definition 1.1.2.8).

However, to accommodate more exotic objects (such as thespectral Deligne-Mumford stacks of §1.4), it will be useful to work with a more general notion of CAlg-valued sheaf. In this section, we will study pairs pX,OXqwhere X is an8-topos andOX is a sheaf ofE8-rings on X.

1.3.1 Sheaves on 8-Topoi

Let C be an 8-category. In §1.1, we introduced the notion of a C-valued sheaf on a topological spaceX (Definition 1.1.2.1). This definition can be generalized to an arbitrary Grothendieck site:

Definition 1.3.1.1. LetT be an essentially small8-category. Recall that aGrothendieck topology onT is a Grothendieck topology on the homotopy category hT, in the sense of classical category theory (see§HTT.6.2.2 for a detailed discussion). Let C be an arbitrary 8-category. We will say that a functor O :T^opÑC is aC-valued sheaf on T if the following condition is satisfied: for every object U P T and every covering sieve T⁰_{U Ď T_{U, the composite map

pT⁰_{Uq^ŸĎ pT_{Uq^ŸÑT ^OÑ^opC^op

is a colimit diagram in C^op. We let Shv_CpTq denote the full subcategory of FunpT^op,Cq spanned by the C-valued sheaves on T.

More informally, a functor O :T^op ÑC is a C-valued sheaf if, for every object U PT and every covering sieveT⁰_{U ofU, the canonical map

OpUq Ñ lim

VÐÝPT⁰_{U

OpVq

is an equivalence in C.

Example 1.3.1.2. LetXbe a topological space, letCbe an8-category, and letShv_CpXqbe the8-category ofC-valued sheaves onX(Definition 1.1.2.1). ThenShv_CpXq “Shv_CpUpXqq, where UpXq is the partially ordered set of all open subsets ofX, which is endowed with the usual Grothendieck topology (so that a collection of inclusions tUα ĎUu generates a covering sieve onU if and only if U “Ť

U_α).

Example 1.3.1.3. Let T be a small8-category equipped with a Grothendieck topology and letS denote the8-category of spaces. Then we will denote the 8-categoryShv_SpTq simply by ShvpTq and refer to it as the 8-category of sheaves on T. The 8-category ShvpTq is an accessible left-exact localization of the presheaf8-category FunpT^op,Sq, and is therefore an 8-topos (see §HTT.6.2 ).

In the situation of Definition 1.3.1.1, the 8-categoryShv_CpTqdoes not depend on the exact details of the Grothendieck site T: it depends only on the associated8-toposShvpTq.

To see this, it will be convenient to introduce a site-independent version of Definition 1.3.1.1 (which also makes sense for8-topoi which do not arise as sheaves on a Grothendieck site).

Definition 1.3.1.4. LetX be an8-topos and letC be an arbitrary8-category. AC-valued sheaf on X is a functorX^op ÑC which preserves small limits. We letShv_CpXqdenote the full subcategory of FunpX^op,Cq spanned by the C-valued sheaves on X.

Warning 1.3.1.5. LetX be an8-topos, and let C be an arbitrary8-category. Then the 8-category Shv_CpXq introduced in Definition 1.3.1.4 generally doesnot coincide with the 8-category C-valued sheaves with respect to a Grothendieck topology on X (for example,

1.3. SHEAVES OF SPECTRA 117 the canonical topology onX). Consequently, the conventions of Definition 1.3.1.4 and 1.3.1.1 conflict with one another. However, there should be little danger of confusion: for example, an8-toposX is never essentially small as an8-category, unless X is a contractible Kan complex.

Remark 1.3.1.6. Let C be a presentable 8-category and X an 8-topos. Then the 8 -category Shv_CpXq can be identified with the tensor product CbX introduced in§HA.4.8.1 . In particular, Shv_CpXqis a presentable 8-category.

We now show that Definitions 1.3.1.1 and 1.3.1.4 are compatible with one another, at least when the8-categoryC admits small limits. For any8-categoryT, we letPpTq denote the 8-category FunpT^op,Sq of S-valued presheaves on T, and j :T Ñ PpTq the Yoneda embedding.

Proposition 1.3.1.7. Let T be a small 8-category equipped with a Grothendieck topology.

Let j:T ÑPpTq denote the Yoneda embedding and L:PpTq ÑShvpTq a left adjoint to the inclusion. LetC be an arbitrary8-category which admits small limits. Then composition withL˝j induces an equivalence of8-categories Shv_CpShvpTqq ÑShv_CpTq.

Corollary 1.3.1.8. Let X be a topological space and let C be an 8-category which ad-mits small liad-mits. Then there is a canonical equivalence of 8-categories Shv_CpXq » Shv_CpShvpXqq, where the left hand side is given by Definition 1.1.2.1 and the right hand side by Definition 1.3.1.4.

Proof of Proposition 1.3.1.7. According to Theorem HTT.5.1.5.6 , composition with j in-duces an equivalence of8-categories Fun₀pPpTq^op,Cq ÑFunpT^op,Cq, where Fun₀pPpTq^op,Cq denotes the full subcategory of FunpPpTq^op,Cq spanned by those functors which preserve small limits. According to Proposition HTT.5.5.4.20 , composition withL induces a fully faithful embedding Shv_CpShvpTqq ÑFun₀pPpTq^op,Cq. The essential image of this embed-ding consists of those limit-preserving functorsF :PpTq^op ÑC such that, for everyX PT and every covering sieveT⁰_{X ĎT_{X, the induced mapFpjXq ÑFpYqis an equivalence in C, whereY is the subobject ofjX corresponding to the sieveT⁰_{X. Unwinding the definitions, this translates into the condition that the composition

pT⁰_{Xq^ŹĎ pT_{Xq^ŹÑT Ñ^j PpTqÑ^F C^op is a colimit diagram. It follows that the composition

ShvCpShvpTqq ÑFun₀pPpTq^op,Cq ÑFunpT^op,Cq is fully faithful, and its essential image is the full subcategoryShv_CpTq.

1.3.2 Sheaves of Spectra

In this book, are primarily interested inC-valued sheaves when C“Sp is the 8-category of spectra.

Definition 1.3.2.1. Let X be an 8-topos. A sheaf of spectraon X is a sheaf on X with values in the8-category Sp of spectra. We letShv_SppXq denote the full subcategory of FunpX^op,Spqspanned by the sheaves of spectra on X.

Remark 1.3.2.2. Let X be an8-topos and let Shv_SpXq denote the full subcategory of FunpX^op,Sq spanned by those functors which preserve small limits. Recall that the 8 -category Sp of spectra can be defined as the full sub-category of FunpS_˚^fin,Sq spanned by those functorsE:S_˚^fin ÑS which are reduced and excisive; hereS_˚^findenotes the8-category of pointed finite spaces (Definition HA.1.4.3.1 ). We therefore obtain an isomorphism of Shv_SppXq with the full subcategory of FunpS_˚^fin,Shv_SpXqq spanned by those functors which are reduced and excisive. Since the Yoneda embedding induces an equivalence of 8-categories X Ñ Shv_SpXq (Proposition HTT.5.5.2.2 ), we obtain an equivalence of Shv_SppXq with the8-category SppXq of spectrum objects of X (see Definition HA.1.4.2.8 ).

In particular, Shv_SppXq is a presentable stable8-category. Moreover, we have a forgetful functor Ω⁸:Shv_SppXq ÑX, which is obtained by pointwise composition with the forgetful functor Ω⁸: SpÑS (together with the identificationX »Shv_SpXq).

Notation 1.3.2.3. Let X be an 8-topos and let X^♥ “τď0X denote its underlying topos.

Composing the forgetful functor functor Shv_SppXq ÑShv_SpXq » X with the truncation functor τ_ď0 :X ÑX^♥, we obtain a functor π₀ :Shv_SppXq ÑX^♥. More generally, for any integer n, we let πn :Shv_SppXq Ñ τď0X denote the composition of the functor π0 with the shift functor Ωⁿ :Shv_SppXq ÑShv_SppXq. Note that π_n can also be described as the composition

Mod_SppXq^Ω

n´2

ÝÑ Mod_SppXq ÑShv_S˚pXq »X_˚ Ñ^π2 τ_ď0X.

It follows that π_n can be regarded as a functor fromShv_SppXq to the category of abelian groups objects ofX^♥.

Example 1.3.2.4. In the situation of Notation 1.3.2.3, suppose thatX “ShvpXq for some topological space X, and let F be an object of Shv_SppShvpXqq » Shv_SppXq. For each integern, we can identifyπnF with the sheaf of abelian groups onX given by sheafifying the presheafU ÞÑπnpFpUqq.

Definition 1.3.2.5. For every integer n, the functor Ω^8´n : Sp Ñ S induces a functor Shv_SppXq ÑShvSpXq »X, which we will also denote by Ω^8´n. We will say that an object F PShv_SppXq is n-truncated if Ω^8`nF is a discrete object of X. We will say that a sheaf of spectraF PShv_SppXq is n-connective if the homotopy groups π_mF vanish formăn.

1.3. SHEAVES OF SPECTRA 119