Sheafification and Tensor Products

Our next objective is to describe a symmetric monoidal structure on the 8-category Shv_SppXq. Roughly speaking, this symmetric monoidal structure is given by levelwise smash product, followed by sheafification. We begin by discussing the latter procedure.

1.3. SHEAVES OF SPECTRA 125 Remark 1.3.4.1. Let D and C be small 8-categories, and assume that D admits finite colimits. Composition with the Yoneda embeddings D^opÑPpD^opqand C ÑIndpCq yields functors

Fun^˚pPpD^opq,PpC^opqq ÑFun^lexpD^op,PpC^opqq »FunpC,IndpDqq ÐFun¹pIndpCq,IndpDqq.

Here Fun^˚pPpD^opq,PpC^opqq denotes the full subcategory of FunpPpD^opq,PpC^opqq spanned by those functors which preserve small colimits and finite limits, Fun^lexpD^op,PpC^opqqthe full subcategory of FunpD^op,PpC^opqqspanned by those functors which preserve finite limits, and Fun¹pIndpCq,IndpDqqthe full subcategory of FunpIndpCq,IndpDqqspanned by those functors which preserve filtered colimits. Each of these functors is an equivalence of 8-categories (see Propositions HTT.6.1.5.2 and HTT.5.3.5.10 ; the middle equivalence is an isomorphism of simplicial sets obtained by identifying both sides with a full subcategory of FunpD^opˆC,Sq).

Assume that bothCandDadmit finite colimits, so that IndpCqand IndpDqare compactly generated presentable 8-categories. The presheaf 8-categories PpC^opq and PpD^opq are classifying 8-topoi for IndpCq-valued and IndpDq-valued sheaves, respectively. The above argument shows that every geometric morphism between classifying 8-topoi arises from a functor IndpCq Ñ IndpDq which preserves filtered colimits. Put more informally, every natural operation which takes IndpCq-valued sheaves and produces IndpDq-valued sheaves is determined by a functor IndpCq ÑIndpDq which preserves filtered colimits.

Suppose now that we are given 8-categoriesC andD which admit finite colimits, and let f : IndpCq Ñ IndpDq be a functor which preserves filtered colimits. Remark 1.3.4.1 guarantees the existence of an induced functor θ : Shv_IndpCqpXq Ñ Shv_IndpDqpXq for an arbitrary8-topos X, which depends functorially onX. In the special case whereX “PpUq is an 8-category of presheaves on some small8-categoryU, we can write down the functor θ very explicitly: it fits into a homotopy commutative diagram

Shv_IndpCqpXq ^//

Shv_IndpCqpXq

FunpU^op,IndpCqq ^{˝f //}FunpU^op,IndpDqq,

where the vertical maps are equivalences of 8-categories given by composition with the Yoneda embedding U Ñ PpUq. More generally, if we assume only that we are given a

geometric morphismPpUq ÑX, then we obtain a larger (homotopy commutative) diagram Shv_IndpCqpXq ^//Shv_IndpDqpXq

Shv_IndpCqpPpUqq

OO //

Shv_IndpDqpPpUqq

OO

FunpU^op,IndpCqq ^{˝f //}FunpU^op,IndpDqq.

The existence of this diagram immediately implies the following result:

Lemma 1.3.4.2. Let U be a small 8-category and suppose we are given a geometric morphism of 8-topoi g^˚ : PpUq Ñ X. Let C be a small 8-category which admits finite colimits, and let TC denote the functorFunpU^op,IndpCqq »Shv_IndpCqpPpUqq ÑShv_IndpCqpXq induced by g^˚. Let D be another small 8-category which admits finite colimits, and define TD similarly. Suppose that f : IndpCq ÑIndpDq is a functor which preserves small filtered colimits. Then if α : F Ñ G is a morphism in FunpU^op,IndpCqq such that TCpαq is an equivalence, then the induced mapα¹ :pf˝Fq Ñ pf ˝Gq also has the property thatTDpα¹q is an equivalence.

Lemma 1.3.4.3. Let X be an 8-topos and C a presentable 8-category. Then the inclusion i:ShvCpXq ĎFunpX^op,Cq admits a left adjoint L.

Proof. The proof does not really require thatX is an8-topos, only thatX is a presentable 8-category. Under this assumption, we may suppose without loss of generality that X “ Ind_κpX₀q, whereκ is a regular cardinal and X₀ is a small 8-category which admitsκ-small colimits. Theniis equivalent to the composition

Shv_CpXq^GÑ^CFun¹pX^op₀ ,Cq ⁱ

1

ĎFunpX^op₀ ,Cq^G

1

ÑCFunpX^op,Cq,

where Fun¹pX^op₀ ,Cq is the full subcategory of FunpX^op₀ ,Cq spanned by those functors which preserve κ-small limits, GC is the functor given by restriction along the Yoneda embedding j : X₀ Ñ X, and G¹_C is given by right Kan extension along j. The functor GC is an equivalence of8-categories (Proposition HTT.5.5.1.9 ), and the functorG¹_C admits a left adjoint (given by composition withj). Consequently, it suffices to show that the inclusioni¹ admits a left adjoint. This follows immediately from Lemmas HTT.5.5.4.17 , HTT.5.5.4.18 , and HTT.5.5.4.19 .

Lemma 1.3.4.4. Let X be an 8-topos, and letf :CÑD be a functor between compactly generated presentable8-categories. Assume thatf preserves small filtered colimits. Let LC : FunpX^op,Cq ÑShv_CpXq and LD: FunpX^op,Dq ÑShv_DpXqbe left adjoints to the inclusion

1.3. SHEAVES OF SPECTRA 127 functors. Then composition with f determines a functor F : FunpX^op,Cq ÑFunpX^op,Dq which carries LC-equivalences to LD-equivalences.

Remark 1.3.4.5. In the situation of Lemma 1.3.4.4, the functorF descends to a functor Shv_CpXq ÑShv_DpXq, given by the composition LD˝F. This is simply another description of the construction arising from Remark 1.3.4.1.

Proof. We use notation as in the proof of Lemma 1.3.4.3. For κ sufficiently large, the full subcategoryX₀ ĎX is stable under limits, so that (by Proposition HTT.6.1.5.2 ) we have a geometric morphismg^˚:PpX₀q ÑX. Then the functorLC can be realized as the composition of the restriction functor rC : FunpX^op,Cq Ñ FunpX^op₀ ,Cq with the functor TC : FunpX^op₀ ,Cq » ShvCpPpX₀qq Ñ ShvCpXq induced by g^˚, and we can similarly write LD “TD˝rD. Ifαis a morphism in the8-category FunpX^op,Cqsuch thatLCpαq “TCprCpαqq is an equivalence, then Lemma 1.3.4.2 shows thatLDpFpαqq “TDprDpF αqqis an equivalence, as required.

We will regard the 8-category Sp of spectra as endowed with the smash product monoidal structure defined in §HA.4.8.2 . This symmetric monoidal structure induces a symmetric monoidal structure on the 8-category FunpK,Spq, for any simplicial set K (Remark HA.2.1.3.4 ); we will refer to this symmetric monoidal structure as thepointwise smash product monoidal structure.

Proposition 1.3.4.6. LetX be an 8-topos, and let L: FunpX^op,Spq ÑShv_SppXq be a left adjoint to the inclusion. ThenL is compatible with the pointwise smash product monoidal structure, in the sense of Definition HA.2.2.1.6 : that is, if f :F ÑF¹ is anL-equivalence in FunpX^op,Spq and G P FunpX^op,Spq, then the induced map FbG Ñ F¹bG is also an L-equivalence in FunpX^op,Spq. Consequently, the 8-category Shv_SppXq inherits the structure of a symmetric monoidal 8-category, with respect to which L is a symmetric monoidal functor (Proposition HA.2.2.1.9 ).

Proof. Apply Lemma 1.3.4.4 to the tensor product functorb: SpˆSpÑSp.

We will henceforth regard the 8-category Shv_SppXq as endowed with the symmetric monoidal structure of Proposition 1.3.4.6, for any8-toposX. We will abuse terminology by referring to this symmetric monoidal structure as thesmash product symmetric monoidal structure.

Proposition 1.3.4.7. Let X be an 8-topos, and let L : FunpX^op,Spq ÑShv_SppXq be a left adjoint to the inclusion. Regard FunpX^op,Spq as endowed with the t-structure induced by the natural t-structure onSp. Then:

p1q The functor L is t-exact: that is, L carries FunpX^op,Sp_ě0q into Shv_SppXqě0 and FunpX^op,Sp_ď0q into Shv_SppXqď0.

p2q The smash product symmetric monoidal structure on Shv_SppXq is compatible with the t-structure onShv_SppXq. In other words, the full subcategoryShv_SppXq^cnĎShv_SppXq contains the unit object and is stable under tensor products.

Proof. The construction of Lemma 1.3.4.3 shows that (for sufficiently large κ) we can factor L as the composition of a restriction functor FunpX^op,Spq Ñ FunpX^op₀ ,Spq with the functor FunpX^op₀ ,Spq »Shv_SppPpX₀qq ÑShv_SppXq induced by a geometric morphism g^˚ :PpX₀q ÑX. Assertion p1q now follows from Remark 1.3.2.8. To provep2q, we show that if we are given a finite collection of connective objectstFiu1ďiďn of Shv_SppXq, then the tensor productF1b ¨ ¨ ¨ bFn is connective. Choose fiber sequencesF¹i ÑFi ÑF²i

in FunpX^op,Spq, where F¹i PFunpX^op,Sp_ě0q and F²i PFunpX^op,Sp_ď´1q. It follows from p1qthat LF¹iPShv_SppXqě0 and LF²i PShv_SppXqď´1. We have fiber sequences

LF¹iÑLFiÑLF²i

inShv_SppXq. Since LFi »Fi is connective, we deduce that the mapLF¹_iÑLFi »Fi

is an equivalence for every index i. Using Proposition 1.3.4.6, we deduce that the tensor productF1b ¨ ¨ ¨ bFnin the8-categoryShv_SppXqcan be written asLpF¹₁b ¨ ¨ ¨ bF¹nq. By virtue ofp1q, it will suffice to show thatF¹1b ¨ ¨ ¨ bF¹nis a connective object of FunpX^op,Spq, which follows because the smash product monoidal structure on Sp is compatible with its t-structure (Lemma HA.7.1.1.7 ).