Tensor triangular geometry of filtered objects and sheaves

We compute the Balmer spectra of compact objects of tensor triangulated categories whose objects are filtered or graded objects of (or sheaves valued in) another tensor triangulated category. Notable examples include the filtered derived category of a scheme as well as the homotopy category of filtered spectra. We use an ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-categorical method to properly formulate and deal with the problem. Our computations are based on a point-free approach, so that distributive lattices and semilattices are used as key tools. In the Appendix, we prove that the ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-topos of hypercomplete sheaves on an ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-site is recovered from a basis, which may be of independent interest.


Introduction
In the subject called tensor triangular geometry, a basic object to study is a ttcategory, which is a triangulated category equipped with a compatible symmetric monoidal structure.Balmer introduced a way to associate to each tt-category T ⊗ a topological space Spc(T ), which we call the Balmer spectrum.We refer the reader to [2] for a survey on tensor triangular geometry.
In this paper, we compute the Balmer spectra of mainly two families of ttcategories, whose objects are diagrams in another tt-category.Although those two computations are logically independent, many techniques used in them are similar.
To state the main results, we introduce some terminology.See Remark 3.2 for the comparison with the common setting of tensor triangular geometry.Definition 1.1.A big tt-∞-category is a compactly generated stable ∞-category equipped with an E 2 -monoidal structure whose tensor products preserve (small) colimits separately in each variable and restrict to compact objects.
We here state only a main consequence of the first computation because stating it in full generality requires some notions.
Theorem I. Suppose that C ⊗ is a big tt-∞-category.
(1) For a nonzero Archimedean group A (for example, Z, Q, or R, equipped with their usual orderings), there is a canonical homeomorphism where S denotes the Sierpiński space (that is, the Zariski spectrum of a discrete valuation ring); see Figure 1.(2) For an abelian group A, considered as a discrete symmetric monoidal poset, there is a canonical homeomorphism Spc(Fun(A, C) ω ) Spc(C ω ).
In each statement, we consider the Day convolution E 2 -monoidal structure on the ∞-category Fun(A, C).
Figure 1.As a set, Spc(Fun(Z, C) ω ) consists of two copies of Spc(C ω ).Its closed subsets correspond to inclusions between two closed subsets of Spc(C ω ).
Specializing to the case A = Z, this theorem has several consequences: Example 1.2.For any quasicompact quasiseparated scheme X, the ∞-category QCoh(X), whose homotopy category is the derived category of discrete quasicoherent sheaves, becomes a big tt-∞-category when considering the (derived) tensor products.The reconstruction theorem (see [2,Theorem 54]) implies that Spc(QCoh(X) ω ) is the underlying topological space X.Applying our result to the case C ⊗ = QCoh(X) ⊗ , we have that the Balmer spectrum of perfect filtered complexes on X is the product of the Sierpiński space and the underlying topological space of X.We note that in the special case where X is an affine scheme, this result is obtained by Gallauer in [4] using a different method.
Example 1.4.One advantage of the generality is that it can be applied iteratively.For example, it can be used when objects are Z-filtered in several directions.Actually, what we prove in Section 5 is so general that we can determine the Balmer spectrum of Z κ -filtered objects for any cardinal κ.
We note that this theorem has a geometric interpretation: Example 1.5.For an E ∞ -ring R, Moulinos proved in [13] that there exist the following equivalences of symmetric monoidal ∞-categories: , where Z disc denotes an abelian group of integers without a poset structure.Applying our result to the case when A = Z, Z disc and C ⊗ = Mod ⊗ R , we get the Balmer spectra of perfect complexes on these geometric stacks (but note that the determination of Spc(Mod ω R ) for general R is a difficult problem).At least when R is a field, our computations reflect a naive intuition on how these stacks look like.
The second result is the following: Theorem II.Suppose that C ⊗ is a big tt-∞-category and X is a coherent topological space (that is, a space which arises as the underlying topological space of a quasicompact quasiseparated scheme).Let Shv C (X) denote the ∞-category of Cvalued sheaves on X. Considering the pointwise E 2 -monoidal structure on it, we have a canonical homeomorphism Spc(Shv C (X) ω ) X cons × Spc(C ω ), where X cons denotes the set X endowed with the constructible topology of X.
One big problem in tensor triangular geometry is to compute the Balmer spectrum of compact objects of the stable motivic homotopy theory SH(X) ⊗ associated to a quasicompact quasiseparated scheme X.The first step would be the determination of that of spectrum-valued sheaves on the smooth-Nisnevich site of X. Theorem II can be considered as a toy case of that calculation.Nevertheless, it is an interesting fact in its own right.
In this paper, we use distributive lattices to deal with the Balmer spectra.More precisely, we introduce the notion of the Zariski lattice of a tt-category, which turns out to be just the opposite of the distributive lattice of quasicompact open sets of the Balmer spectrum.Thus it contains the same information as the Balmer spectrum, but it has a more algebraic nature, which makes our computations possible.We note that (upper) semilattices are also used in the proof.
Outline.In Section 2, we study ∞-categorical machinery (mainly) related to functor categories.In Section 3, we review basic notions in tensor triangular geometry using the language of distributive lattices.Its "tensorless" variant is also introduced there.Sections 4 and 5 are devoted to the proofs of Theorem II and a general version of Theorem I, respectively.These two sections are logically independent, but we arrange them in this way because the former is quite simpler.In Appendix A, we develop some technical material on ∞-toposes which we need in Section 4.
Acknowledgments.I would like to thank Jacob Lurie for kindly answering several questions on ∞-categories.I thank Martin Gallauer and Shane Kelly for commenting on an early draft of this paper.

Functor categories
Concerning ∞-categories, we will follow terminology and notation used in [9,10,11] with minor exceptions, which we will explicitly mention.
In this section, we study general properties of the ∞-category Fun(K, C) for a small ∞-category K and a presentable ∞-category C.

2.1.
Tensor products of presentable ∞-categories.Let Pr denote the very large ∞-category of presentable ∞-categories whose morphisms are those functors that preserve colimits.This is what is denoted by Pr L in [9].Similarly, for an infinite regular cardinal κ, we let Pr κ denote the very large ∞-category of κ-compactly generated ∞-categories whose morphisms are those functors that preserve colimits and κ-compact objects.
Here we list basic properties of the symmetric monoidal structure on Pr as one theorem; see [10,Section 4.8.1] for the proofs.where the large ∞-category Cat ∞ of ∞-categories is considered to be equipped with the cartesian symmetric monoidal structure.(4) The tensor product operations preserve small colimits in each variable.
(5) For an infinite regular cardinal κ and C 1 , . . ., C n , D ∈ Pr κ , we have spanned by those functors preserving colimits and κ-compact objects is canonically equivalent to that of Fun(C spanned by those functors that preserve colimits in each variable and restrict to determine functors from The useful consequence for us is the following: Corollary 2.2.For a presentable ∞-category C and a small ∞-category K, we have a canonical equivalence Fun(K, S) ⊗ C Fun(K, C).
Proof.According to (1) of Theorem 2.1, the left hand side can be regarded as the full subcategory of Fun(Fun(K, S) op , C) spanned by those functors preserving limits.Hence the equivalence follows from [9, Theorem 5.1.5.6].
We have the following result from this description, although it can be proven more concretely: Corollary 2.3.Let κ be an infinite regular cardinal.If C is a κ-compactly generated ∞-category, so is Fun(K, C) for any small ∞-category K.

2.2.
Compact objects in a functor category.In this subsection, we fix an infinite regular cardinal κ.
For a κ-compactly generated ∞-category C, according to Corollary 2.3, the ∞category Fun(K, C) is also κ-compactly generated for any small ∞-category K.The aim of this subsection is to determine κ-compact objects of Fun(K, C) under some assumptions on K. Definition 2.4.Let K be a small ∞-category.We say that K is κ-small if there exist a simplicial set K with < κ nondegenerate simplices and a Joyal equivalence K → K.We say that K is locally κ-compact if the mapping space functor Map : K op × K → S factors through the full subcategory S κ spanned by κ-compact spaces.
Lemma 2.6.Let C be a κ-compactly generated ∞-category and K a κ-small ∞category.Then every functor K → C that factors through C κ is a κ-compact object of Fun(K, C).
Lemma 2.7.Let C be a κ-compactly generated ∞-category and K a locally κcompact ∞-category.Then every κ-compact object K → C of Fun(K, C) factors through C κ .
Proof.Let k be an object of K.The inclusion i : * → K corresponding to k induces a functor i * : Fun(K, C) → C by composition.Since F (k) i * F holds for every functor F : K → C, it suffices to show that the functor i * preserves κ-compact objects.By [9, Proposition 5.5.7.2], this is equivalent to the assertion that its right adjoint i * preserves κ-filtered colimits.The functor i * can be concretely described as the assignment C → l → C Map K (l,k) using the cotensor structure on C. Since C is κ-compactly generated and Map K (l, k) is κ-compact for every object l ∈ K, we can see that i * preserves κ-filtered colimits.
Combining these two lemmas, we get the following result: Proposition 2.8.Let C be a κ-compactly generated ∞-category and K a κ-small locally κ-compact ∞-category.Then an object F ∈ Fun(K, C) is κ-compact if and only if it takes values in κ-compact objects.Remark 2.9.Proposition 2.8 does not hold without the local condition: Take K = S 1 and consider the object X of Fun(S 1 , S) corresponding to the universal covering * → S 1 by the Grothendieck construction.By [9, Lemma 5.1.6.7] the object X is compact, but X( * ) Z is not compact.
Remark 2.10.In the case κ = ω, the assumption on K in the statement of Proposition 2.8 is restrictive.For example, if K is also assumed to be equivalent to a space, K must be equivalent to a finite set: When K is simply connected, this can be deduced by considering the homological Serre spectral sequence associated to the fiber sequence ΩK → * → K.The general case follows by taking the universal cover of each connected component of K and using the fact that the classifying space of a nontrivial finite group is not finite.
Later in this paper, we consider a slightly more general situation than that of Proposition 2.8.At that moment, the following result is useful: Corollary 2.11.Let K be a locally κ-compact ∞-category and C a κ-compactly generated ∞-category.Suppose that there is a κ-directed family (K j ) j∈J of full subcategories of K such that K j is κ-small for j ∈ J.We let (i j ) ! denote the left Kan extension functor along the inclusion i j : K j → K. Then we have an equality of full subcategories of Fun(K, C).
Conversely, let F ∈ Fun(K, C) be a κ-compact object.By assumption, F is the colimit of the κ-filtered diagram j → (i j ) ! (F | Kj ).Hence we can take j ∈ J such that F is a retract of (i j ) ! (F | Kj ).Since the essential image of (i j ) ! is closed under retracts, F is in fact equivalent to (i j ) ! (F | Kj ).From Lemma 2.7 and Proposition 2.8, we have that F | Kj is κ-compact, which completes the proof of the other inclusion.2.3.Recollement.We refer the reader to [10,Section A.8] for the theory of recollement for ∞-categories.Definition 2.12.Suppose that C is an ∞-category and i : C 0 → C and j : C 1 → C are fully faithful functors.We say that C is a recollement of i and j if C is a recollement of the essential images of i and j in the sense of [10,Definition A.8.1].
There are many ways to write a functor category as a recollement due to the following observation: Proposition 2.13.Let C be a presentable ∞-category and K 1 ⊂ K a full inclusion of ∞-categories.Suppose that K 1 is a cosieve on K; that is, k ∈ K 1 implies l ∈ K 1 if there exists a morphism k → l in K. Let K 0 denote the full subcategory of K spanned by objects not in K 1 .Let i * : Fun(K 0 , C) → Fun(K, C) and j * : Fun(K 1 , C) → Fun(K, C) denote the functors defined by right Kan extensions.Then Fun(K, C) is a recollement of i * and j * .
Proof.The only nontrivial point is to verify that j * i * sends every object to the final object, where we write j * for the functor given by restriction along the inclusion K 1 ⊂ K.For k ∈ K 1 , the cosieve condition implies (K 0 ) k/ = ∅.Hence for any F ∈ Fun(K 0 , C) and any k ∈ K 1 , we have (j which completes the proof.
We state a lemma on recollements in the stable setting.
Lemma 2.14.Let C be a stable ∞-category, which is a recollement of i * and j * .We write i * , j * for the left adjoints of i * , j * , respectively, and j ! for that of j * .Then for an object C ∈ C, the cofiber sequence j !j * C → C → i * i * C splits if and only if the map i * C → i * j * j * C is zero.
Proof.We write i ! for the right adjoint of i * .The "only if" direction follows from the fact that i * (j !j * C) and i * j * j * (i * i * C) are both zero.We wish to prove the converse.By applying i * to the cofiber sequence i * i !C → C → j * j * C and shifting, we obtain a cofiber sequence Σ −1 i * j * j * C → i !C → i * C, which splits by assumption.Then the map i * i * C → C corresponding to the section i * C → i !C by adjunction induces the desired splitting.
2.4.The two monoidal structures on a functor category.In this subsection, we fix an ∞-operad E ⊗ k , where k is a positive integer or the symbol ∞.Recall that an E k -monoidal ∞-category can be regarded as an E k -algebra object of the symmetric monoidal category Cat × ∞ .In the following discussion, we often use this identification implicitly.
The following is a special case of [10, Proposition 3.2.4.4]: Lemma 2.15.There exist a symmetric monoidal structure on Alg E k (Pr) such that the forgetful functor Alg E k (Pr) → Pr has a symmetric monoidal refinement.Also, the same holds for Pr ⊗ κ , where κ is an infinite regular cardinal.First we use this to construct the pointwise E k -monoidal structure.Definition 2.16.Let K be a small ∞-category and C ⊗ a presentable E k -monoidal ∞-category whose tensor products preserve colimits in each variable.Then combining with the cartesian E k -monoidal structure on Fun(K, S), we obtain an E kmonoidal structure on Fun(K, C) Fun(K, S) ⊗ C by using Lemma 2.15.We call this the pointwise E k -monoidal structure.
The pointwise tensor products can be computed pointwise, as the name suggests.We consider a condition under which this construction is compatible with the compact generation property.See also Corollary 2.22 for another result in this direction.
Proposition 2.17.Let κ be an infinite regular cardinal.In the situation of Definition 2.16, suppose furthermore that K is κ-small and locally κ-compact, C is κ-compactly generated, and the tensor products on C restrict to C κ .Then the pointwise tensor products on Fun(K, C) also restrict to Fun(K, C) κ .
Proof.Without loss of generality we may assume that C = S. Since finite products of κ-compact spaces are again κ-compact, the desired result follows from Proposition 2.8.
We then consider the Day convolution E k -monoidal structure.We first note that for an E k -monoidal ∞-category K ⊗ , we can equip a canonical E k -monoidal structure on the opposite K op ; see [10,Remark 2.4.2.7].Therefore, for such K ⊗ , we have an E k -monoidal structure on Fun(K, S) PShv(K op ) by ( 3) of Theorem 2.1.
Definition 2.18.Let K ⊗ be a small E k -monoidal ∞-category and C ⊗ a presentable E k -monoidal ∞-category whose tensor products preserve colimits in each variable.Then considering the E k -monoidal structure on Fun(K, S) explained above, we obtain an E k -monoidal structure on Fun(K, C) Fun(K, S) ⊗ C by using Lemma 2.15.We call this the Day convolution E k -monoidal structure.
Concretely, the Day convolution tensor products can be computed as follows: Lemma 2.19.In the situation of Definition 2.18, the Day convolution tensor product of F 1 , . . ., F n ∈ Fun(K, C) is equivalent to the left Kan extension of the composite We note that in the case C = S, this is claimed in [10,Remark 4.8.1.13].
Proof.By universality, we have a canonical map from the functor constructed in the statement to the tensor product.Since both constructions are compatible with colimits in each variable, we can assume that C = S and that F 1 , . . ., F n are in the image of the Yoneda embedding K op → PShv(K op ).In this case, the desired claim is trivial.
The author learned the following fact from Jacob Lurie, which says that the Day convolution counterpart of Proposition 2.17 does not need any assumption on K ⊗ : Lemma 2.20.Let κ be an infinite regular cardinal.In the situation of Definition 2.18, suppose furthermore that C is κ-compactly generated and the tensor products on C restrict to C κ .Then the Day convolution tensor products on Fun(K, C) also restrict to Fun(K, C) κ .
Proof.Without loss of generality we may assume that C = S.According to [9,Proposition 5.3.4.17], we have that PShv(K op ) κ is the smallest full subcategory of PShv(K op ) that contains the image of the Yoneda embedding and is closed under κ-small colimits and retracts.Since the Yoneda embedding has an E k -monoidal refinement, the desired result follows.Now we give a comparison result of these two E k -monoidal structures.
Proposition 2.21.In the situation of Definition 2.18, suppose furthermore that the E k -monoidal structure on K is cocartesian.Then on Fun(K, C) the pointwise and Day convolution E k -monoidal structures are equivalent.
Proof.Without loss of generality we may assume that C = S and k = ∞.Since both tensor products preserve colimits in each variable and restrict the image of the Yoneda embedding, it suffices to show that they are equivalent on the image.Hence the result follows from the uniqueness of cartesian symmetric monoidal structures on K op .
Combining this with Lemma 2.20, we have the following: Corollary 2.22.Let κ be an infinite regular cardinal.In the situation of Definition 2.16, suppose furthermore that K has finite coproducts and C is κ-compactly generated.Then the conclusion of Proposition 2.17 holds.
Remark 2.23.We cannot completely remove assumptions on K: When K = Z, the final object of Fun(Z, S) is not compact.See also Example A.13.

Latticial approach to tensor triangular geometry
In this section, first we review the notion of the Balmer spectrum using distributive lattices.In Subsection 3.4 we introduce a variant of tensor triangular geometry.
3.1.Our setting.Let Pr st ω denote the full subcategory of Pr ω spanned by compactly generated stable ∞-categories, to which the symmetric monoidal structure on Pr ω explained in Theorem 2.1 restricts.A big tt-∞-category, which is defined in Definition 1.1, can be seen as an E 2 -algebra object of (Pr st ω ) ⊗ .We let Cat perf ∞ denote the large ∞-category of idempotent complete stable ∞categories whose morphisms are exact functors.The equivalence Ind : Cat perf ∞ → Pr st ω induces a symmetric monoidal structure on it.Definition 3.1.A tt-∞-category is an E 2 -algebra object of (Cat perf ∞ ) ⊗ ; in concrete terms, a tt-∞-category is an idempotent complete stable ∞-category equipped with an E 2 -monoidal structure whose tensor products are exact in each variable.
By definition, the large ∞-category Alg E2 (Cat perf ∞ ) of tt-∞-categories and the very large ∞-category Alg E2 (Pr st ω ) of big tt-∞-categories are equivalent.Remark 3.2.There are several differences between our setting and that of the usual theory of tensor triangular geometry, as found in [2, Hypothesis 21]: (1) We use an ∞-categorical enhancement.We note that by [10, Lemma 1.2.4.6] the idempotent completeness assumptions in both settings are equivalent.(2) We consider an E 2 -monoidal structure, so that the induced monoidal structure on the underlying triangulated category is not necessarily symmetric, but braided.Actually, the arguments of this paper works with slight modifications even if calling an (E 1 -)algebra object of (Cat perf ∞ ) ⊗ a tt-∞-category, mainly since many notions including the Balmer spectrum only depend on the underlying (E 1 -)monoidal structure.However, the author does not know if such a generalization is useful.
(3) We do not impose any rigidity condition.This is because we do not need it for our computations.
3.2.The Stone duality.We review the Stone duality for distributive lattices.
For the basic theory, we refer the reader to [6, Section II.3].We write DLat, Loc, Loc coh , Top and Top coh for the category of distributive lattices, locales, coherent locales, topological spaces, and coherent topological spaces (also called spectral spaces), respectively.Note that both inclusions Loc coh ⊂ Loc and Top coh ⊂ Top are not full since only quasicompact maps are considered as morphisms in them.The Stone duality for distributive lattices states that the ideal frame functor Idl : DLat → (Loc coh ) op and the spectrum functor Spec : DLat op → Top coh are equivalences, and these two are compatible with the functor pt : Loc → Top that sends a locale to its space of points.
Proof.Since the functor pt : Loc → Top has a left adjoint, which sends a topological space to its frame of open sets, it preserves limits.From Lemma 3.3 and the fact that coherent locales are spatial, we obtain the result.We denote the smallest radical ideal containing an object C ∈ T by √ C.
We note that the notion of radical ideal of a tt-∞-category T ⊗ only depends on the underlying tensor triangulated category (hT ) ⊗ .Definition 3.6.Let T ⊗ be a tt-∞-category.A support for T ⊗ is a pair (L, s) of a distributive lattice L and a function s : T → L satisfying the following: (0) The function s takes the same values on equivalent objects.Hence we can evaluate s(C) In particular, we have s(0) = 0. (2) For any cofiber sequence In particular, we have s(1) = 1, where 1 denotes the unit.They form a category, with morphisms (L, s) → (L , s ) defined to be morphisms of distributive lattices f : L → L satisfying f • s = s .Proposition 3.9.For a tt-∞-category T ⊗ , the following assertions hold: (1) The Zariski lattice Zar(T ) is a distributive lattice.
(2) The pair Zar(T ), C → √ C is a support for T ⊗ .(3) It is an initial support; that is, an initial object of the category described in Definition 3.6.
Although this can be proven directly, here we give a proof using several results obtained in [7, Section 3].
Proof.According to [7, Theorem 3.1.9],all radical ideals of T ⊗ form a coherent frame by inclusion and its compact objects are precisely the elements of Zar(T ), so (1) holds.Also, (2) follows from this observation, together with [7, Lemma 3. We conclude this subsection by proving a property of the functor Zar.
Lemma 3.12.The functor Zar : Alg E2 (Cat perf ∞ ) → DLat preserves filtered colimits.The classical version of this result is [4, Proposition 8.2], which Gallauer proved as a corollary of a more general result there.This lemma might be seen as a consequence of its variants, but we here give a different proof using supports.
Proof.Suppose that I is a directed poset and that T ⊗ is the colimit of a diagram I → Alg E2 (Cat perf ∞ ), which maps i to T ⊗ i .We wish to show that the morphism lim − →i Zar(T i ) → Zar(T ) is an equivalence.By [10, Corollary 3.2.2.5] and the fact that Hence it suffices to prove that a function from T to a distributive lattice L is a support if the composite T i → T → L is a support for each i.This follows from the definition of a support.

3.4.
Tensorless tensor triangular geometry.In this subsection, we develop the "tensorless" counterpart of the theory described in the previous section.This is used in Section 5.
First recall that an upper semilattice is a poset that has finite joins.A morphism between upper semilattices is defined to be a function that preserves finite joins.We let SLat denote the category of upper semilattices.Definition 3.13.Suppose that T is an idempotent complete stable ∞-category.
(1) A semisupport for T is a pair (U, s) of an upper semilattice U and a function s : T → U satisfying conditions (0) to (2) of Definition 3.6, which make sense in this situation.(2) A thick subcategory of T is an idempotent complete stable full replete subcategory of T .It is called principal if it is generated, as a thick subcategory, by one object.
The following is the counterpart of Proposition 3.9 for semisupports: Proposition 3.14.For an idempotent complete stable ∞-category T , the set of principal thick subcategories of T ordered by inclusion is a semilattice, which is (the target of ) the initial semisupport.Proof.In this proof, we refer to the conditions given in Definition 3.6.From (0) we have that I is a full replete subcategory.Condition (1) implies 0 ∈ I and (2) implies that I is closed under shifts and (co)fibers.Hence I is a stable subcategory.Also, from (1) we have that I is idempotent complete, which completes the proof.
Proof of Proposition 3.14.For C 1 , . . ., C n ∈ T , it is easy to see that the join of C 1 , . . ., C n can be computed as Remark 3.16.We can also prove the tensorless counterpart of Lemma 3.12 by the same argument.
We state a well-known concrete description of the free functor Free : SLat → DLat, which is defined as the left adjoint of the forgetful functor.Lemma 3.17.For an upper semilattice U , we let P(U ) denote the power set of U ordered by inclusion.Then the morphism U → P(U ) that maps u to {v ∈ U | u v} induces a monomorphism of distributive lattices Free(U ) → P(U ).

Tensor triangular geometry of sheaves
In this section, we prove Theorem II, which is stated in Section 1.

4.1.
Tensor triangular geometry of the pointwise monoidal structure.First we define a class of categories.Beware that there are other usages of the word "acyclic category" in the literature.Note that every finite acyclic category is ω-finite and locally ω-compact as an ∞-category, so that we can apply Propositions 2.8 and 2.17.Theorem 4.3.Let C ⊗ be a big tt-∞-category and K a finite acyclic category.Then we have a canonical isomorphism where the right hand side denotes the power of Zar(C ω ) indexed by the set of objects of K, computed in the category DLat.
Remark 4.4.In the language of usual tensor triangular geometry, the conclusion of Theorem 4.3 just says that the Balmer spectrum of Fun(K, C) ω is homeomorphic to that of C ω to the power of the cardinality of objects of K.
Example 4.5.In the case C ⊗ = Mod ⊗ k for some field k, this result is the special case of [8, Theorem 2.1.5.1] when the quiver is not equipped with relations.
To give the proof of Theorem 4.3, we introduce a notation.Definition 4.6 (used only in this subsection).In the situation of Theorem 4.3, suppose that k is an object of K. We let K denote the cosieve generated by k.Let X(k) ∈ Fun(K, C) denote the left Kan extension of the object of Fun(K , C) which is obtained as the right Kan extension of the unit 1 ∈ C Fun({k}, C).This object satisfies X(k)(k) 1 and X(k)(l) 0 for l = k.Lemma 4.7.In the situation of Theorem 4.3, suppose that (L, s) is a support for (Fun(K, C) ω ) ⊗ .Then we have k∈K s(X(k)) = 1.In other words, the object k∈K X(k) generates Fun(K, C) ω as a radical ideal.Proof.First, we name objects of K as k 1 , . . ., k n so that we have Hom K (k j , k i ) = ∅ for any i < j.This is possible since K is acyclic.For 0 ≤ i ≤ n, let K i denote the full subcategory of K whose set of objects is {k j | j ≤ i}.We write F i ∈ Fun(K, C) ω for the right Kan extension of 1| Ki , where 1 denotes the unit of Fun(K, C) ⊗ .
We wish to prove s(F i ) = s(F i−1 ) ∨ s(X(k i )) for 1 ≤ i ≤ n, which completes the proof since s(F n ) = s(1) = 1 and s(F 0 ) = s(0) = 0 holds.Now since K \ K i−1 is a cosieve, we get a cofiber sequence X(k i ) → F i → F i−1 by applying Proposition 2.13.Combining this with an equivalence X(k i ) F i ⊗ X(k i ), we obtain the desired equality.
Proof of Theorem 4.3.Let P(K 0 ) denote the power set of the set of objects of K ordered by inclusion.Then there exists a canonical isomorphism Zar(C ω )⊗P(K 0 ) Zar(C ω ) K0 of distributive lattices.
First, we claim that there exists a (unique) morphism of distributive lattice P(K 0 ) → Zar(Fun(K, C) ω ) that maps {k} to X(k) for k ∈ K.This follows from the following two observations: (1) For k = l, we have X(k) ⊗ X(l) 0; this can be checked pointwise.
To complete the proof, we wish to prove that g • f and f • g are identities.By construction, it is easy to see that g • f is the identity, so it remains to prove the other claim.We take an object F ∈ Fun(K, C) ω and for k ∈ K let F k denote the object of Fun(K, C) ω obtained by precomposing the functor K → {k} → K with F .Unwinding the definitions, the assertion (g • f ) √ F = √ F is equivalent to the assertion that F and k∈K F k ⊗ X(k) generate the same radical ideal.Since we have an equivalence F ⊗ X(k) F k ⊗ X(k) for each k ∈ K, this follows from Lemma 4.7.

Main result.
In order for Theorem II to make sense, we need to clarify what the pointwise E 2 -monoidal structure on Shv C (X) is.Proposition 4.8.For a coherent topological space X, the ∞-topos Shv(X) is compactly generated.Moreover, finite products of compact objects are again compact.
Proof.The first assertion is the content of [9, Proposition 6.5.4.4].Let L denote the distributive lattice of quasicompact open subsets of X.We have Shv(L) Shv(X), where L is equipped with the induced Grothendieck topology (see Definition A.2). We let L : PShv(L) → Shv(L) denote the sheafification functor.It follows from the proof of [9, Proposition 6.5.4.4] that Shv(L) ω is the smallest full subcategory that contains the image of PShv(L) ω under L and is closed under finite colimits and retracts.Since finite products preserve (finite) colimits in each variable in Shv(L) and the functor L preserves finite products, it suffices to show that finite products of compact objects are again compact in PShv(L).This follows from Corollary 2.22 since L has finite products.
Using the equivalence Shv C (X) Shv(X) ⊗ C and Lemma 2.15, we can equip an E 2 -monoidal structure on Shv C (X). Corollary 4.9.For a big tt-∞-category C and a coherent topological space X, the E 2 -monoidal ∞-category Shv C (X) ⊗ , defined as above, is a big tt-∞-category.
To state the main theorem, we recall basic facts on Boolean algebras.See [6, Section II.4] for details.First recall that Boolean algebras form a reflective subcategory of DLat, which we denote by BAlg.The left adjoint of the inclusion BAlg → DLat is called the Booleanization functor, which we denote by B. For a coherent topological space X, the spectrum of its Booleanization of the distributive lattice of quasicompact open subsets of X is the Stone space whose topology is the constructible topology (also referred to as the patch topology) of X. Hence using Proposition 3.4, Theorem II can be rephrased as follows: Theorem 4.10.Let C ⊗ be a big tt-∞-category and L a distributive lattice.Then we have a canonical isomorphism The proof uses the following notion: Definition 4.11.For a poset P , the Alexandroff topology is a topology on the underlying set of P whose open sets are cosieves (or equivalently, upward closed subsets).Let Alex(P ) denote the set P equipped with this topology.Lemma 4.12.For a big tt-∞-category C ⊗ and a finite poset P , there exists a canonical equivalence between big tt-∞-categories The proof relies on a result obtained in Appendix A.
Proof.We have the desired equivalence by taking the tensor product of (the cartesian E 2 -monoidal refinement of) the equivalence PShv(P op ) → Shv(Alex(P )) obtained in Example A.12 with C ⊗ .Lemma 4.13.For a big tt-∞-category C ⊗ , the functor from DLat to Alg E2 (Pr st ω ) that maps L to (Shv C (Spec(L))) ⊗ preserves filtered colimits.
Proof.We first note that the composite of forgetful functors Alg E2 (Pr st ω ) → Pr st ω → Pr ω preserves sifted colimits and conservative.Since limits in the large ∞-categories Pr op and Pr op ω are both computed in the very large ∞-category of large ∞-categories, the forgetful functor Pr ω → Pr preserves colimits and obviously conservative.Hence we are reduced to showing that the following composite preserves filtered colimits: where Top ∞ denote the very large ∞-category of ∞-toposes whose morphisms are geometric morphisms.Now we can check that each functor preserves filtered colimits as follows: (1) The first functor preserves colimits by Lemma 3.3.
(2) The second functor preserves colimits since 0-localic ∞-toposes form a reflective subcategory of Top ∞ .(3) The third functor preserves filtered colimits since cofiltered limits in Top op ∞ can be computed in the very large ∞-category of large ∞-categories.(4) The fourth functor preserves colimits by ( 5) of Theorem 2.1.Hence the composite also preserves filtered colimits.
Proof of Theorem 4.10.Since finitely generated distributive lattice has a finite number of objects, we can write L as a filtered colimit of finite distributive lattices.Hence by Lemmas 3.12 and 4.13, it suffices to consider the case when L is finite.Let P be a poset of points of Spec(L) with the specialization order; that is, the partial order in which p ≤ q if and only if the point p is contained in the closure of the singleton {q}.Since Spec(L) is finite, there is a canonical homeomorphism Spec(L) Alex(P ).Booleanizing their associated distributive lattices, we have that B(L) is canonically isomorphic to the power set of P ordered by inclusion.Then applying Lemma 4.12, we obtain the desired equivalence as a corollary of Theorem 4.3.

Tensor triangular geometry of the Day convolution
In this section, we prove Theorem 5.17, which is a generalization of Theorem I.

5.1.
Partially ordered abelian groups.We begin with reviewing some notions used in the theory of partially ordered abelian groups.Definition 5.1.A partially ordered abelian group is an abelian group object of the category of posets; that is, an abelian group A equipped with a partial order in which the map a + -is order preserving for any a ∈ A.
We can regard a partially ordered abelian group as a symmetric monoidal poset.Definition 5.2.Let A be a partially ordered abelian group.
(1) The submonoid (2) The subgroup As the name suggests, this is the connected component containing 0 when A is regarded as a category by its partial order.(3) If A • = A holds and A ≥0 has finite joins, A is called lattice ordered.This is equivalent to the condition that A has binary joins (but beware that A does not have the nullary join unless A is trivial).Note that in this case A also has binary meets, which are computed as Example 5.3.For a cardinal κ, the (categorical) product ordering on Z κ defines a lattice ordered abelian group.The assignment f : ( gives a morphism Z 2 → Z 3 of partially ordered abelian groups, but does not define a morphism of unbounded lattice: Indeed, we have f Example 5.4.There are many partially ordered abelian groups that are connected and not lattice ordered.We here give two relatively simple examples.The first is Z with the ordering that makes its positive cone to be the submonoid generated by 2 and 3.The second is Z × Z/2 with the ordering that makes its positive cone to be the submonoid generated by (1, 0) and (1, 1).

The Archimedes semilattice.
In this subsection, we introduce the notion of the Archimedes semilattice of a partially ordered abelian group.Definition 5.5.For a partially ordered abelian group A, a submonoid and b ∈ J, we have a ∈ J.An ideal J is called principal if it is generated as an ideal by an element of A ≥0 .Proposition 5.6.For a partially ordered abelian group A, the set of principal ideals of A ≥0 ordered by inclusion is an upper semilattice.
Proof.It is easy to see that a 1 +• • •+a n is a join of a 1 , . . ., a n for a 1 , . . ., a n ∈ A ≥0 , where a denotes the ideal of A ≥0 generated by an element a ∈ A ≥0 .Definition 5.7.For a partially ordered abelian group A, the upper semilattice of principal ideals of A ≥0 is denoted by Arch(A).We call this the Archimedes semilattice of A. Note that this only depends on its positive cone A ≥0 , regarded as a partially ordered abelian monoid.
Remark 5.8.There is a characterization of the Archimedes semilattice similar to that of the Zariski lattice given in Proposition 3.9: Namely, the Archimedes semilattice Arch(A) is initial among pairs (U, s) where U is an upper semilattice and s : A ≥0 → U is a function satisfying the following conditions: (1) For a 1 , . . ., a n ∈ A ≥0 , we have s(a Example 5.9.If A is a totally ordered abelian group, its Archimedes semilattice Arch(A) consists of all Archimedean classes of A and the singleton {0}.This observation justifies the name.In particular, if A is nonzero Archimedean, we have Arch(A) {0 < 1}.
Example 5.10.Any Riesz space R can be regarded as a lattice ordered abelian group in a trivial way.There is a bijective (and order preserving) correspondence between (principal) ideals of R ≥0 and those of R in the usual sense.

5.3.
Tensorless tensor triangular geometry with actions.In this subsection, we construct the comparison map, which we prove to be an isomorphism under some mild assumptions.
Proposition 5.11.Let C ⊗ be a big tt-∞-category and A a partially ordered abelian group.Then we have a canonical morphism of distributive lattices where Free : SLat → DLat denotes the left adjoint to the forgetful functor.
It is convenient to regard the ∞-category Fun(A, C) as equipped with the action of A, which is described in the following definition: Definition 5.12.Suppose that C is a compactly generated stable ∞-category and A is a partially ordered abelian group.
Then precomposition with the map (a, b) → b − a induces a functor Fun(A, C) → Fun(A op × A, C), which can be seen as a functor from Fun(A, C) × A op to Fun(A, C).We write F {a} for the value of this functor at (F, a) and F {a/b} for the cofiber of the map F {b} → F {a}, which is only defined when a ≤ b; concretely, F {a} is an object satisfying We call a semisupport s for Fun(A, C) ω an A-semisupport if for a ∈ A and F ∈ Fun(A, C) ω , we have s(F {a}) = s(F ).Similarly, we call a thick subcategory of Fun(A, C) ω a thick A-subcategory if it is stable under the operation F → F {a} for any a ∈ A.
From now on, we abuse notation by identifying the object C ∈ C ω with its left Kan extension along the inclusion 0 → A of partially ordered abelian groups.Especially, we do not distinguish the units of C ⊗ and Fun(A, C) ⊗ , which we denote by 1.
Example 5.13.Suppose that C ⊗ is a big tt-∞-category.For any object F ∈ Fun(A, C) ω and any element a ∈ A we have F {a} F ⊗ 1{a} and that 1{a} is invertible with inverse 1{−a}.Thus any support for (Fun(A, C) ω ) ⊗ can be regarded as an A-semisupport.More generally, for any object G ∈ Fun(A, C) ω and any support s, the assignment F → s(G ⊗ F ) defines an A-semisupport for Fun(A, C) ω .Lemma 5.14.In the situation of Definition 5.12, suppose that s is an A-semisupport for Fun(A, C) ω and F is an object of Fun(A, C) ω .Then the following assertions hold: (1) For a 1 , . . ., a n ∈ A ≥0 , we have s(F {0/(a Proof.We first prove (2).Consider the following diagram, in which all the rows and columns are cofiber sequences: Since there exists an equivalence F {a/(a + b)} F {0/b}{a}, we have s(F {a/(a + b)}) = s(F {0/b}).Similarly we have s(F {b/(a + b)}) = s(F {0/a}).Using the right cofiber sequence, we have s(C) ≤ s(F {0/b}).We complete the proof by showing s(C) = s(F {0/a}).To prove this, it suffices to show that the morphism f in the diagram is zero.This follows from the fact that the left top square can be decomposed as follows: We then prove (1).The case n = 0 is trivial.Hence it suffices to consider the case n = 2. Consider the following diagram: Since all the other rows and columns are cofiber sequences, so is the bottom row.Hence we have s(F {0/(a 1 +a 2 )}) ≤ s(F {a 2 /(a 1 +a 2 )})∨s(F {0/a 2 }) = s(F {0/a 1 })∨ s(F {0/a 2 }).On the other hand, applying (2), we have s(F {0/a 1 }) ∨ s(F {0/a 2 }) ≤ s(F {0/(a 1 + a 2 )}).Therefore, the desired equality follows.
Proof of Proposition 5.11.The left Kan extension along the inclusion 0 → A defines a morphism Zar(C ω ) → Zar(Fun(A, C) ω ).By Lemma 5.14, the assignment a → 1{0/a} satisfies the conditions given in Remark 5.8, so we have a morphism Free(Arch(A)) → Zar(Fun(A, C) ω ).Combining these two, we obtain the desired morphism.
Now we study A-semisupports in more detail.First, by mimicking the proof of Proposition 3.14, we can deduce the following: Proposition 5.15.In the situation of Definition 5.12, principal thick A-subcategories form an upper semilattice by inclusion and the assignment that takes an object of Fun(A, C) ω to the thick A-subcategory generated by it defines an A-semisupport.Furthermore, it is an initial A-semisupport.
If A is lattice ordered, the initial A-semisupport has simple generators.Proposition 5.16.In the situation of Definition 5.12, we furthermore assume that A is lattice ordered.Then the target of the initial A-semisupport described in Proposition 5.15 is generated as an upper semilattice by thick A-subcategories generated by an object of the form C{0/a 1 } • • • {0/a n } with C ∈ C ω and a 1 , . . ., a n ∈ A ≥0 satisfying a i ∧ a j = 0 if i = j.
We prove this in Subsection 5.5.Note that the conclusion also holds if we only require the positive cone to have binary joins; see the proof of Proposition 5.20.5.4.Main theorem.We now state the main result of this section.
Theorem 5.17.In the situation of Proposition 5.11, suppose furthermore that A ≥0 has finite joins.Then the morphism f is an isomorphism.Question 5.18.In Theorem 5.17, what happens if A does not satisfy the hypothesis?By Proposition 5.20, we may assume that A is connected.To consider the case A is one of the examples given in Example 5.4 would be a starting point.
Example 5.19.Claim (2) of Theorem I is a direct consequence of Theorem 5.17.
Since Free({0 < 1}) is the linearly ordered set consisting of three elements, Spec(Free({0 < 1})) is homeomorphic to the Sierpiński space.Hence by using Proposition 3.4 and Example 5.9, we can deduce (1) of Theorem I.
Since the inclusion A • → A induces an isomorphism Arch(A • ) Arch(A), Theorem 5.17 is a consequence of the following two results: Proposition 5.20.Let C ⊗ be a big tt-∞-category and A a partially ordered abelian group.Then the morphism i : Zar(Fun(A • , C) ω ) → Zar(Fun(A, C) ω ) induced by the inclusion A • → A is an equivalence.Proposition 5.21.In the situation of Proposition 5.11, suppose furthermore that A is lattice ordered.Then the morphism f is an isomorphism.
We conclude this subsection by showing Proposition 5.20; we prove Proposition 5.21 in the next subsection.
Proof of Proposition 5.20.In this proof, we regard Fun(A • , C) ω as a full subcategory of Fun(A, C) ω by left Kan extension.
Let J denote the quotient A/A • .For each j ∈ J we choose an element a j ∈ j.Then as a category, we can identify A with j∈J (A • + a j ).Hence combining the translations A + a j → A for all j ∈ J, we have a functor A → A • .This defines a functor Fun(A, C) ω → Fun(A • , C) ω by left Kan extension; concretely, it is given by the formula j∈J F j {a j } → j∈J F j for any family (F j ) j∈J of objects of Fun(A • , C) ω such that F j is zero except a finite number of indices j ∈ J.
Let s denote the composite Fun(A, C) ω → Fun(A • , C) ω → Zar(Fun(A • , C) ω ), where the second map is the initial support.We wish to show that s is a support for (Fun(A, C) ω ) ⊗ by checking that the conditions given in Definition 3.6 are satisfied.The only nontrivial point is to prove that s satisfies condition (3).Since the case n = 0 follows from the fact that 1{−a [0] } is invertible, it is sufficient to consider the case n = 2.We take two objects F, G ∈ Fun(A, C) ω and decompose them as F j∈J F j {a j } and G j∈J G j {a j } using two families of objects (F j ) j∈J , (G j ) j∈J of Fun(A • , C) ω such that both F j and G j are zero except a finite number of indices j ∈ J.To prove s(F ⊗ G) = s(F ) ∧ s(G), unwinding the definitions, we need to show that the equality holds in Zar(Fun(A • , C) ω ).This follows from the observation made in Example 5. 13.
Hence we obtain a morphism r : Zar(Fun(A, C) ω ) → Zar(Fun(A • , C) ω ) from s.The composite r • i is the identity since we have (r To prove that i•r is the identity, unwinding the definitions, it suffices to show that j∈J F j {a j } equals to j∈J F j for any family (F j ) j∈J of objects of Fun(A • , C) ω such that F j is zero except a finite number of indices j ∈ J.This also follows from the observation made in Example 5.13.5.5.Postponed proofs.In this subsection, we give the proofs of Propositions 5.16 and 5.21.First we introduce some terminology.Definition 5.22 (used only in this subsection).Suppose that A is a lattice ordered abelian group.
(1) We call a subset B ⊂ A saturated if it is finite and closed under binary joins.Note that for every finite set B ⊂ A we can find the smallest saturated subset of A containing B.
(2) Let C be a compactly generated stable ∞-category.By applying Proposition 2.13 to the cosieve of A generated by a single element a ∈ A, we obtain a presentation of Fun(A, C) as a recollement.Hence for F ∈ Fun(A, C) we have a cofiber sequence j !j * F → F → i * i * F , where we use the notation of Lemma 2.14.We write F ≥a and F a for j !j * F and i * i * F , respectively.
The proof of Proposition 5.16 relies on the following lemma: Lemma 5.23.Suppose that A is a lattice ordered abelian group, C is a compactly generated stable ∞-category, and s is an A-semisupport for Fun(A, C) ω .Then for a ∈ A and F ∈ Fun(A, C) ω , we have We first prove the following special case: Proof.We may assume that a = 0 and B contains 0, which automatically becomes the least element of B.
We again consider the recollement description of Fun(A, C) given in (2) of Definition 5.22 and continue to use the notation of Lemma 2.14.
We first show that j * ((i !i * F ){0/a}) is zero, where i !denote the left adjoint of i * .This is equivalent to the assertion that for any c ∈ A ≥0 the morphism (i Then we consider the following diagram: By what we have shown above, the bottom left object is zero, so that the right vertical morphism is zero.By applying Lemma 2.14, we have that F a (F {0/a}) a is a direct sum of F {0/a}.Hence we have s(F ) ≥ s(F {0/a}) ≥ s(F a ), which completes the proof.
Proof of Lemma 5.23.We take a saturated subset B ⊂ A such that F is equivalent to the left Kan extension of F | B .We may assume that B contains 0 as the least element.By replacing a with a ∨ 0, we may assume that a ≥ 0 and also a ∈ B. Now we take a maximal chain 0 Then we can apply Lemma 5.24 iteratively to obtain an inequality which completes the proof.We may assume that b = 0 and F = F ≥0 .We define a subset of A by B = i∈I a i I ⊂ {1, . . ., n} ; note that here i∈I a i equals to the join i∈I a i taken in A ≥0 .By induction we can see that F (0){0/a 1 } • • • {0/a n } is equivalent to the left Kan extension of the object of Fun(B , C) which is the right Kan extension of F (0) ∈ Fun({0}, C).Using the equivalence for some i.This follows from the above description.We claim that this set is in the image of the monomorphism Free(Arch(A)) → P(Arch(A)) described in Lemma 3.17.We note that Θ({a, b, a ∨ b}, a) is in the image for any b ∈ A because an ideal J belongs to this set if and only if J does not contain the ideal generated by (a ∨ b) − a. Hence Θ(B, a) is also in the image since it can be written as b∈B Θ({a, b, a ∨ b}, a).
For an object F ∈ Fun(A, C) ω , we can take a saturated subset B such that F is equivalent to the left Kan extension of F | B .Then we define s(F ), which is a priori dependent on B, as follows: Here we abuse the notation by identifying Free(Arch(A)) with its image under the monomorphism Free(Arch(A)) → P(Arch(A)).
First we wish to prove that s(F ) is independent of the choice of B. Let B be another saturated subset such that F is equivalent to the left Kan extension of F | B .We need to prove the following equality: Hence we have a function s : Fun(A, C) ω → Zar(C ω )⊗Free(Arch(A)).We wish to prove that s is a support.Since it is an A-semisupport and s( 1 To prove this, by using Proposition 5.16 again, we may assume that F can be written as C{0/a 1 } • • • {0/a n } with C ∈ C ω and a 1 , . . ., a n ∈ A ≥0 satisfying a i ∧ a j = 0 if i = j.We may furthermore suppose that a i > 0 for each i and b > 0; otherwise the claim is trivial.For I ⊂ {1, . . ., n} let a I denote the sum i∈I a i , which is equal to the join i∈I a i taken in A ≥0 by assumption.We now take the following two subsets of A, both of which are saturated by assumption: We omit the proof of this fact because this is proven in the same way as in the 1-categorical setting; see the second and third paragraphs of the proof of [12,Proposition B.6.6], but beware that some arguments in the first paragraph cannot be translated to our setting.
The main result is the following: Theorem A.6.Let B be a basis for an ∞-site C and G a presheaf on C. Then G is a hypercomplete object of the ∞-topos Shv(C) if and only if the following conditions are satisfied: (1) The restriction G| B op is a hypercomplete object of the ∞-topos Shv(B).
(2) The functor G is a right Kan extension of G| B op .
We note that Jacob Lurie let the author know that this result could be proven using hypercoverings.Here we will give a different proof, which does not use (semi)simplicial machinery.
Before giving the proof of this theorem, let us collect its formal consequences.
Corollary A.7.Let B be a basis for an ∞-site C. Then the geometric embedding Shv(B) → Shv(C) obtained in Proposition A.5 is cotopological.In particular, it induces equivalences between their hypercompletions, their Postnikov completions, their bounded reflections, and their n-localic reflections for any n.
Corollary A.8. Let B be a basis for an ∞-site C. Suppose that both B and C are n-category for some n and have finite limits (but the inclusion need not preserve them).Then the geometric embedding obtained in Proposition A.5 is an equivalence.
Remark A.9.In [5, Lemma C.3], Hoyois gave another sufficient condition under which the geometric embedding Shv(B) → Shv(C) itself is an equivalence.As a special case of his result, if B and C both admit pullbacks and the inclusion preserves them, it becomes an equivalence.
Our proof uses the following relative variant of [11,Lemma 20.4.5.4]: Lemma A.10.Let f * : Y → X be the left adjoint of a geometric morphism between ∞-toposes and D ⊂ Y an essentially small full subcategory.Suppose that for every Y ∈ Y there exists a family of objects (V i ) i∈I of D and a morphism i∈I V i → Y whose image under f * is an effective epimorphism.Then the object Proof.This follows from a slight modification of the proof of [11,Lemma 20.4.5.4] by using the fact that f * preserves finite limits and colimits and it determines a functor Y /Y → X /f * Y for any object Y ∈ Y, which is again the left adjoint of a geometric morphism.
Proof of Theorem A.6.Let i * : PShv(C) → PShv(B) denote the restriction functor.We write i * for its right adjoint.We let L be the sheafification functor associated to the ∞-site C and j : C → PShv(C) the Yoneda embedding.

Theorem 2 . 1 .
There exists a symmetric monoidal structure on Pr which satisfies the following properties: (1) For C, D ∈ Pr, we have that C ⊗ D is canonically equivalent to the full subcategory of Fun(C op , D) spanned by those functors preserving limits.(2) For C 1 , . . ., C n , D ∈ Pr, the full subcategory of Fun(C 1 ⊗ • • • ⊗ C n , D) spanned by those functors preserving colimits is equivalent to that of Fun(C 1 × • • • × C n , D) spanned by those functors preserving colimits in each variable.(3) The functor PShv : Cat ∞ → Pr has a symmetric monoidal refinement,

3. 3 .
The Zariski lattice and the Balmer spectrum.In this subsection, we introduce the notion of the Zariski lattice of a tt-∞-category.Definition 3.5.Let T ⊗ be a tt-∞-category.A radical ideal of T ⊗ is a stable full replete subcategory I ⊂ T that satisfies the following conditions:(1) If C ⊕ D ∈ I for some C, D ∈ T , we have C, D ∈ I. (2) For any C ∈ T and D ∈ I, we have C ⊗ D ∈ I. (3) If C ∈ T satisfies C ⊗k ∈ I for some k ≥ 0, we have C ∈ I.

Remark 3 . 7 .
The notion of support introduced here is different from what is called "support on T " in [7, Definition 3.2.1],which values in a frame.Definition 3.8.The Zariski lattice Zar(T ) of a tt-∞-category T ⊗ is the partially ordered set √ C C ∈ T ordered by inclusion.
2.2].Assertion (3) follows from essentially the same argument as that given in the proof of [7, Theorem 3.2.3].Remark 3.10.This construction determines a functor Zar : Alg E2 (Cat perf ∞ ) → DLat.Now we can give a definition of the Balmer spectrum in this paper; this is equivalent to the original definition by [7, Theorem 3.1.9and Corollary 3.4.2].Definition 3.11.For a tt-∞-category T ⊗ , we let Spc(T ) denote the coherent topological space Spec(Zar(T ) op ) and call it the Balmer spectrum of T ⊗ .

Lemma 3 .
15.For any semisupport (U, s) and any object C ∈ T , the full subcategory I ⊂ C spanned by objects D satisfying s(D) ≤ s(C) is a thick subcategory of T .
where C denotes the thick subcategory of T generated by an object C ∈ T .Hence it suffices to show that if objects C, D ∈ T satisfy s(C) = s(D) for some semisupport s, they generate the same thick subcategory.This follows from Lemma 3.15.

Definition 4 . 1 .
An (ordinary) category is called acyclic if only identity morphisms are isomorphisms or endomorphisms in it.

Example 4 . 2 .
Any poset, considered as a category, is an acyclic category.

Lemma 5 .
24.In the situation of Lemma 5.23, suppose that a ≤ a are elements of A and B ⊂ A is a saturated subset satisfying b ∧ a ∈ {a , a} for b ∈ B. If an object F ∈ Fun(A, C) ω is obtained as the left Kan extension of F | B , we have s(F ) = s(F ≥a ) ∨ s(F a ).
is an equivalence.We may assume that b∧a = 0 holds for any b ∈ B to prove this.Unwinding the definitions, it is enough to show that {b ∈ B | b ≤ c} and {b ∈ B | b ≤ c + a} have the same greatest element.Let b be the greatest element of the latter set.Then we have b = b ∧ (c + a) = c + ((b − c) ∧ a) ≤ c + (b ∧ a) = c, which means that b belongs to the former set.

Proof of Proposition 5 . 16 .
First we take a finite subset B ⊂ A such that B is closed under binary joins and meets and F is equivalent to the left Kan extension of F | B .For b ∈ B, we can take distinct elements a 1 , . . ., a n ∈ A ≥0 (possibly n = 0) such that {b + a 1 , . . ., b + a n } is the set of minimal elements of {c ∈ B | b < c}.By the assumption on B, we have that a i ∧ a j = 0 for i = j.Let F b denote the object (• • • (F ≥b ) b+a1 • • • ) b+an .Then applying Lemma 5.23 iteratively, we have s(F ) = b∈B s(F b ).Thus we wish to show that F b {−b} is equivalent to F (b){0/a 1 } • • • {0/a n } for b ∈ B to complete the proof.

a∈BF
(a) ∧ Θ(B, a) = a∈B F (a) ∧ Θ(B , a).By considering a saturated set containing B ∪ B , we may assume that B ⊂ B .For any minimal element b of B \ B, the subset B \ {b } is also saturated.Hence by induction we may also assume that B \B = {b } for some b ∈ B .If b is a minimal element of B , then we get the equality since in this case F (b ) is a zero object and Θ(B, a) = Θ(B , a) for any a ∈ B. Let us consider the case when b is not minimal.Since B is saturated, we can take the greatest element b of the set {a ∈ B | a ≤ b }.Consider an element a ∈ B. It is clear that Θ(B, a) ⊃ Θ(B , a) holds and this inclusion becomes an equality if a b .In fact, it also becomes an equality if a ≤ b and a = b: Suppose that J ∈ Θ(B, a) fails to belong to Θ(B , a).Then we have (a + J) ∩ B = {a, b }.But in this case, from the inequality 0 ≤ b − a ≤ b − a we get b ∈ (a + J) ∩ B, which contradicts our assumption.Therefore, it is enough to show that Θ(B, b) = Θ(B , b) ∪ Θ(B , b ) holds since we have F (b) F (b ).First we prove Θ(B, b) ⊂ Θ(B , b) ∪ Θ(B , b ).For J ∈ Θ(B, b) \ Θ(B , b), we have (b + J) ∩ B ⊂ (b + J) ∩ B = {b, b } and so J ∈ Θ(B , b ).To prove the other inclusion, it remains to show that Θ(B, b) ⊃ Θ(B , b ) holds.For J ∈ Θ(B , b ) and a ∈ (b + J) ∩ B, by 0 ≤ (a ∨ b ) − b = a − (a ∧ b ) ≤ a − b ∈ J, we have a ∨ b ∈ (b + J) ∩ B = {b }.This means a = b, which completes the proof of the well-definedness of s(F ).

B
= {a I | I ⊂ {1, . . ., n}}, B = B ∪ {a I + (a I ∨ b) | I, I ⊂ {1, . . ., n} satisfying I ∩ I = ∅}.Then F and F {0/b} are left Kan extensions of F | B and F {0/b}| B , respectively.We first prove that s(F {0/b}) ≤ s(F ) ∧ s(1{0/b}).Since we have s(F {0/b}) ≤ s(F ) by the fact that s is an A-semisupport, we need to show s(F {0/b}) ≤ s(1{0/b}).Unwinding the definitions, we are reduced to proving that F {0/b}(c)0 or Θ(B , c) ⊂ Θ({0, b}, 0) holds for each c ∈ B .If F {0/b}(c) is not zero, F (c) or F {b}(c) F (c− b) is not zero.These two cases are treated separately as follows: (1) If F (c) is not zero, then we have either c = 0 or c = b.In the former case, we have indeed Θ(B , 0) ⊂ Θ({0, b}, 0) since b ∈ B .In the latter case, we have that the morphism F (b) → F (0) is equivalent to the identity of C. Hence F {0/b}(b) is zero, which is a contradiction.(2) If F {b}(c) is not zero, then we have that c ∈ B or c = a I ∨ b for some I ⊂ {1, . . ., n}.In the former case, we have Θ(B , c) ⊂ Θ({0, b}, 0) since c + b ∈ B .In the latter case, we have c = a I ∨ b < a I + b ≤ c + b; if the Proposition A.5.Let C be an ∞-site.Suppose that B is a basis for C and F is a presheaf on B. Then F is a sheaf on B if and only if its right Kan extension is a sheaf on C. Especially, by restricting the right Kan extension functor PShv(B) → PShv(C), we obtain a geometric embedding Shv(B) → Shv(C).