Posets for which Verdier duality holds

We discuss two known sheaf-cosheaf duality theorems: Curry’s for the face posets of finite regular CW complexes and Lurie’s for compact Hausdorff spaces, i.e., covariant Verdier duality. We provide a uniform formulation for them and prove their generalizations. Our version of the former works over the sphere spectrum and for more general finite posets, which we characterize in terms of the Gorenstein* condition. Our version of the latter says that the stabilization of a proper separated ∞\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 is rigid in the sense of Gaitsgory. As an application, for stratified topological spaces, we clarify the relation between these two duality equivalences.


Introduction
For the face poset P of a locally finite regular CW complex, Curry proved in [6,Theorem 7.7] that a canonical equivalence (1.1) D : hD b (Fun(P, Vect)) −→ hD b (Fun(P op , Vect)) between triangulated categories 1 exists, where Vect denotes the category of vector spaces over a field.Note that the polysimplicial case was proven before by Schneider in [13,Proposition 2].This leads to the following: Under what conditions on P do we have such a duality equivalence?
To answer this question, we must give a definition of "such a duality equivalence" so that the desired equivalence is not an extra datum but a property.
Example 1.3.Let P be the poset generated by the relations 0 ≤ 1 ≤ 2 and 0 ≤ 1 ≤ 2 on the set {0, 1, 1 , 2}.An isomorphism P P op gives an equivalence Fun(P, Sp) Fun(P op , Sp), but the two isomorphisms give different equivalences.We can also use the suspension functor to get many more equivalences.
Curry's proof rules out this example, but his argument depends on a somewhat arbitrary choice of a dualizing complex.
To get some insights, let us look at a similar equivalence in general topology.Let X be a locally compact Hausdorff space.In [12, Section 5.5.5],Lurie states Verdier duality as a canonical equivalence D : Shv Sp (X) −→ cShv Sp (X) Date: February 14, 2022. 1 Here h denotes the underlying triangulated category of a stable ∞-category.
between the ∞-categories of spectrum-valued sheaves and cosheaves.The original construction is complicated, but as we see in Section 6, a simpler explanation exists: We assume that X is compact for simplicity and write p : X → * and d : X → X × X for the projection and the diagonal, respectively.Then the composites constitute a duality datum for the self-duality in Pr st , the symmetric monoidal ∞-category of presentable stable ∞-categories. 2Now let us get back to our problem and take a finite poset P .It is known (see Section 5.2) that when X is its Alexandroff space Alex(P ) (see Definition 5.15), we have Shv Sp (X) Fun(P, Sp) and cShv Sp (X) Fun(P op , Sp).Moreover, in this case, both (1.4) and (1.5) are in Pr st .However, these do not form a duality datum unless P is discrete: Example 1.6.Assume X = Alex(P ) for a finite nondiscrete poset P .Pick an element p ∈ P that is minimal among nonminimal elements.Let F : P → Sp be the extension of S by zero along {p} → P .Then the value at p of the image of F under is given by the limit lim ← − (F ), which is a coproduct of (#(P <p ) − 1) copies of Σ −1 S and never equivalent to F (p) = S.
Therefore we give up to use (1.4) and consider if the composite (1.7) Shv Sp (X) where [-, -] denotes the internal mapping object in Pr st , is an equivalence.Thus we reach the following: Definition 1.8.We call a finite poset P Verdier if the pair (Fun(P, Sp), Γ) is a commutative Frobenius algebra (cf.Section 2.1) in Pr st . 3n other words, P is Verdier if and only if (1.5) is a "perfect pairing".Our main result rephrases the Verdier property in terms of the Gorenstein* property, a concept used in combinatorial commutative algebra: Theorem A. For a finite poset P , the following are equivalent: (i) The poset P is Verdier.
(ii) For each p ∈ P , the full subposet P <p is Gorenstein* (over Z), i.e., its geometric realization is a generalized homology sphere (see Section 3.2).(iii) For each p < q in P , the limit of Z [p,q] , the extension of the constant functor Z by zero along [p, q] → P vanishes.(Or equivalently, the limit of S [p,q] vanishes; see Lemma 3.16.) The existence of an equivalence Fun(P, Sp) Fun(P op , Sp) for P satisfying (ii) or its variant where Sp is replaced by D(Z) may not surprise experts.What is novel is the formulation itself, with which an if-and-only-if statement becomes possible.
Corollary 1.9.A finite poset P is Gorenstein* if and only if P is Verdier.
Example 1.10.As proven in [4, Proposition 3.1], a finite poset P is the face poset of some regular CW complex if and only if the geometric realization of P <p is homeomorphic to a sphere for each p. Hence any finite face poset is Verdier.In particular, we have an equivalence Fun(P, Sp) Fun(P op , Sp).In light of this example, the equivalence (i) ⇔ (ii) in Theorem A can be informally summarized by the following: Slogan 1.11.A finite poset enjoys Verdier duality if and only if it is homologically CW.
Of course, there is an example that is not a face poset: Example 1.12.Let P be the face poset of the triangulation of a homology sphere that is not a sphere.Then P is Verdier, but it is not the face poset of any regular CW complex.Remark 1.13.In this paper, we work over S (or Z) for simplicity, but our argument is valid over other coefficients.For example, for a field k and a finite poset P , the functor lim ← − : Fun(P, D(k)) → D(k) makes Fun(P, D(k)) a commutative Frobenius algebra in the ∞category of k-linear (presentable) stable ∞-categories if and only if P <p is Gorenstein* over k for any p ∈ P .
As a byproduct of our proof, we find the following generalization of [ We can handle the locally finite case by a limit argument; precisely, we show the following: Theorem C. Let P a poset such that P ≥p is finite and P <p is finite and Gorenstein* for each p ∈ P (e.g., the face poset of a locally finite regular CW complex).Then there is a canonical equivalence D : Fun(P, Sp) −→ Fun(P op , Sp), which is pointwise given by D(F ) : p → lim ← −q∈P Map(p, q) ⊗ F (q), where ⊗ denotes the copower.
Remark 1.14.In Theorem C, the requirement that P ≥p be finite cannot be dropped: Let P be the poset given in [1,Example A.13].Then P <p is Gorenstein* for any p; in fact, P is the face poset of a regular CW structure of S ∞ .However, the assignment described in the statement does not preserve compact objects, thus does not lift to an equivalence.
Our formulation gives us more than aesthetic satisfaction.For instance, this unified view to the two duality theorems enables us to study the interaction between startification and Verdier duality.A sample application is the following: Theorem D. Let X → Alex(P ) be a stratification of a compact Hausdorff space, where P is a Verdier finite poset.Suppose that the inverse image Shv Sp (Alex(P )) → Shv Sp (X) is fully faithful.Then our duality functor Fun(P, Sp) → Fun(P op , Sp) can be canonically identified with the composite Shv Sp (Alex(P )) where D is the Verdier duality equivalence for X and f + is the cosheaf pushforward.
Example 1.15.In a nice situation, the (space-valued) inverse image Shv(Alex(P )) → Shv(X) is fully faithful and its image consists of constructible sheaves; see [5, Section 3]4 for a precise statement.For example, we can show that the assumption of Theorem D is satisfied when X is a finite regular CW complex and P is its face poset.This paper is organized as follows: We develop necessary tools on duality in Section 2 and on poset (co)homology in Section 3. Then we prove Theorem A by showing (ii) ⇒ (iii), (iii) ⇔ (i), and (i) ⇒ (ii) in Sections 3.3, 4.2, and 4.3, respectively.We also show Theorem B in Section 4.3.After that, we study its variants in Section 5 and in particular prove Theorem C. In Section 6, we study Verdier duality for locally compact Hausdorff spaces from a formal standpoint.It motivates our formulation and is used to obtain Theorem D.
Conventions.For a poset P , we write P ⊥ and P for the posets obtained by adding the least element ⊥ and the greatest element , respectively.When we regard P as an ∞-category, these correspond to its left and right cones (P and P ) in [11,Notation 1.2.8.4].We also write P ⊥, for the one obtained by adding both.The empty face (or (−1)-face) is not included in our face poset, but we regard S −1 = ∅ as a sphere.
We use the closed symmetric monoidal structure on Pr given in [12,Section 4.8.1].For an ∞-topos X and a presentable ∞-category C, the ∞-categories of C-valued sheaves Shv C (X ) and cosheaves cShv C (X ) are identified with C ⊗ X and [X , C], respectively.Concretely, their objects can be regarded as limit-preserving functors X op → C and colimit-preserving functors X → C, respectively.We write f + f + for the pushforward-pullback adjunction for cosheaves.The global section functor, i.e., the cohomology functor, is denoted by Γ, not by RΓ.
Acknowledgments.While working on this project, I was at the University of Tokyo and the Max Planck Institute for Mathematics and was partially supported by the Hausdorff Center for Mathematics.I thank them for the hospitality.
2. General facts on duality 2.1.A useful criterion.Recall that for objects A, A ∨ and a morphism e : A ∨ ⊗ A → 1 in a symmetric monoidal ∞-category, we say that e is a counit of a duality between A and A ∨ if for any objects C and D the composite is an equivalence.Lemma 2.1.Let A and A ∨ be objects and e : A ∨ ⊗ A → 1 a morphism in a closed symmetric monoidal ∞-category.If A is dualizable and the composite is an equivalence, then e is a counit.Here [-, -] denotes the mapping object functor.
Proof.By the definition of [-, -], it suffices to show that the morphism is an equivalence for every C. Since A is dualizable, this morphism is equivalent to the one obtained by applying C ⊗ -to (2.2).
2.2.Functorialities.For a commutative algebra A and a morphism l : A → 1 (of objects) in a closed symmetric monoidal ∞-category, we can form a morphism A → [A, 1] as in (2.2) by letting e be the composite l • m where m : A ⊗ A → A is the multiplication.We discuss the (1-categorical) naturality of this assignment (A, l) ]).Note that the pair (A, l) is called a commutative Frobenius algebra if e is a counit.
Proposition 2.3.Suppose that f : A → B is a morphism of commutative algebras in a symmetric monoidal ∞-category and that g : B → A is an A-linear morphism.Then for every morphism l : A → 1 and any objects C and D, there are commutative squares where the horizontal morphisms are the ones associated to (A, l) and (B, l • g), respectively.Moreover, if the symmetric monoidal structure is closed, the same thing holds when the mapping spaces are replaced by the internal mapping objects.
Example 2.4.Let K be an ∞-category and i : K 0 → K be an inclusion of a sieve.A direct computation shows that f = i * : Fun(K, Sp) → Fun(K 0 , Sp) and its right adjoint g satisfy the assumptions of Proposition 2.3 in Pr st .
Example 2.5.Let K be an ∞-category and j : K 1 → K be an inclusion of a cosieve.A direct computation shows that f = j * : Fun(K, S) → Fun(K 1 , S) and its left adjoint g satisfy the assumptions of Proposition 2.3 in Pr.This is a special case of Example 2.7 below.
Example 2.6.Let p : Y → X be a proper geometric morphism between ∞-toposes.According to Lemma 6.7, f = p * : Shv Sp (X ) → Shv Sp (Y) and its right adjoint g satisfy the assumptions of Proposition 2.3 in Pr st .
Proof of Proposition 2.3.In this proof, in order to simplify the notation, we write (-, -) for For the first square, we construct the 2-cells in the diagram We obtain the triangle and the four rectangles by naturality.We also obtain the pentagon by the linearity of g and the functoriality of (B, -).
For the second square, we construct the 2-cells in the diagram We obtain the upper triangle and the four rectangles by naturality.We also obtain the lower triangle by the linearity of g and the functoriality of (A, -).
We record the following obvious consequence: Corollary 2.8.In the situation of Proposition 2.3, assume furthermore that (A, l) is Frobenius and that f • g is homotopic to the identity.Then (B, l • g) is also Frobenius.
We also have the following variant: Lemma 2.9.Suppose that f : A → B is a morphism of commutative algebras in a symmetric monoidal ∞-category.Then for every morphism m : B → 1 and any objects C and D, there is a commutative square where the horizontal morphisms are the ones associated to (A, m • f ) and (B, m), respectively.Moreover, if the symmetric monoidal structure is closed, the same thing holds when the mapping spaces are replaced by the internal mapping objects.
Proof.We use the same notation as in the proof of Proposition 2.3.We construct the 2-cells in the diagram We obtain the upper square since f is a morphism of commutative algebras.We also obtain the other cells by naturality.
Definition 3.1.For a poset P , we write |P | for the geometric realization (as a topological space) of its nerve and ∆(P ) for its order complex, i.e., the abstract simplicial complex consisting of finite (nonempty) chains in P .Note that |P | is canonically homeomorphic to the geometric realization of ∆(P ).
In this subsection, we study how the cohomology of |P | and that of P , i.e., the sheaf cohomology of Fun(P, S), are related.We first recall the following from [12, Section A.1]: Definition 3.2.We say that an ∞-topos X has constant shape if the shape Sh X is corepresentable.If X /X has constant shape for every X ∈ X , we say that X is locally of constant shape.According to [12,Proposition A.1.18],this is equivalent to the condition that the constant sheaf functor S → X admits a left adjoint.
Example 3.3.The presheaf ∞-topos of an ∞-category is locally of constant shape.Its shape is the image under the left adjoint of S → Cat ∞ .
Example 3.4.The sheaf ∞-topos of a CW complex is locally of constant shape.Its shape is the homotopy type.In fact, any CW complex is locally of singular shape in the sense of [12,Section A.4] as any open subspace is homotopy equivalent to a CW complex.Proposition 3.5.If an ∞-topos X is locally of constant shape, for any spectrum E, the canonical morphism [Σ ∞ + Sh X , E] → Γ(X ; E) is an equivalence.Here [-, -] denotes the mapping spectrum.Proof.Let p : X → S denote the projection.By assumption, p * admits a left adjoint p ! .If we regard objects in Shv Sp (-) as limit-preserving functors (-) op → Sp, the spectrum-valued pullback Sp → Shv Sp (X ) is given as the precomposition with (p ! ) op .Therefore, Γ(X ; Remark 3.7.In fact, at least if P ≥p is finite for p ∈ P , we can construct a canonical geometric morphism Shv(|P |) → Fun(P, S) whose inverse image functor is fully faithful.This shows that we can take any functor E : P → Sp as a coefficient in the statement of Corollary 3.6, but we do not need this generality in this paper.
3.2.Gorenstein* posets.We first recall the following notion from combinatorial commutative algebra.See [14, Chapter II] for a textbook account, which in particular explains where the name comes from.Definition 3.8.We call an n-dimensional 5 finite abstract simplicial complex Gorenstein* if its geometric realization is a generalized homology n-sphere, i.e., an (integral) homology n-manifold having the (integral) homology of an n-sphere.
The following definition is a variant of the definition of a Cohen-Macaulay poset given in [3, Section 3].Definition 3.9.We call a finite poset P Gorenstein* if for every p < q in P ⊥, the interval (p, q) has the (integral) homology of a sphere 6 .
By definition if P is a Gorenstein* finite poset then (p, q) is Gorenstein* for every p < q ∈ P ⊥, .Corollary 3.12.For a finite abstract simplicial complex, it is Gorenstein* if its underlying poset is Gorenstein*.
We later need the following lemma, as we prefer cohomology: Lemma 3.13.For a finite poset, the Gorenstein* condition can be checked via cohomology instead of homology; i.e., P is Gorenstein* if and only if (p, q) has the cohomology of a sphere for p < q in P ⊥, .
Proof.By definition P is Gorenstein* if and only if so is P op .Hence the desired result follows from the self-duality of the ∞-category of perfect complexes over Z.

3.3.
A vanishing result.Definition 3.14.Let P be a poset and E a spectrum.For p ≤ q in P , we let E [p,q] ∈ Fun(P, Sp) denote the functor obtained from the constant functor E ∈ Fun([p, q], Sp) by left Kan extending along [p, q] → P ≤q and then right Kan extending along P ≤q → P .If E ∈ D(Z), we use the same symbol for the element in Fun(P, D(Z)) determined similarly.Proposition 3.15.For a Gorenstein* finite poset P , for every p < q in P the cohomology Γ(P ; Z [p,q] ) vanishes.
We later prove the converse.
Proof.Since Γ(P ; Z [p,q] ) Γ((P ) ≤q ; Z [p,q] ) holds and (P ) <q is also Gorenstein*, we can assume q = .By Proposition 3.11, there is a unique rank function r : P ⊥, → Z satisfying r(⊥) = −1.Then |P | is a generalized homology (r( ) − 1)-sphere.If r( ) = 1 holds, P is the discrete poset with two elements and the result can be directly checked.So we henceforth assume r( In what follows, we repeatedly use Corollary 3.6.Let Z ≥p denote the left Kan extension of the constant functor with value Z along P ≥p → P .Then the pullback diagram Γ(P ; Z) Here n can be −1, so that the empty complex is Gorenstein*. 6We regard S −1 = ∅ as a sphere. 7A rank function on a finite poset P is a function r : P → Z such that r(q) = r(p) + 1 if q is an immediate successor of p.
can be formed in D(Z).Here f is induced by P → P .As this morphism of posets induces an isomorphism on H 0 (|-|; Z), we see that f induces an isomorphism on π 0 .On the other hand, Γ(P ; Z ≥p ) is computed as the relative cohomology fib(Γ(P ; Z) → Γ(P \ P ≥p ; Z)).By the Lefschetz duality theorem, it is the dual of Σ r( )−1 Γ(P ≥p ; Z) in D(Z), which is Σ 1−r( ) Z, and g induces an isomorphism on π 1−r( ) as |P ≥p | is connected.Therefore, f and g can be identified with the two direct summand inclusions of Γ(P ; Z) Z ⊕ Σ 1−r( ) Z, from which Γ(P ; Z [p, ] ) 0 follows.
Proof of (ii) ⇒ (iii) of Theorem A. This follows from Proposition 3.15.
We note that this vanishing also holds when Z is replaced by S: Lemma 3.16.For a finite poset P and p ≤ q in P , the cohomology Γ(P ; Z [p,q] ) vanishes if and only if so does Γ(P ; S [p,q] ).
Proof.Note that if a spectrum E is nonzero and bounded below, E ⊗ Z is also nonzero; this can be seen by considering the smallest i such that π i E is nonzero.Hence the desired result follows from Γ(P ; Z [p,q] ) Γ(P ; S [p,q] ) ⊗ Z and the fact that Γ(P ; S [p,q] ) is bounded below, both of which follow from the finiteness of P .
4. Verdier duality for finite posets 4.1.Recollements.We refer the reader to [12, Section A.8] for a discussion on recollements using ∞-categories.When we say C 0 and C 1 form a recollement, C 0 is supposed to be the "closed" part; i.e., the C 1 -localization annihilates C 0 .We abuse terminology to say the two functors C 0 → C and C 1 → C determine a recollement when C is a recollement of their images.We recall the following standard fact: Lemma 4.1.Consider a presentable stable ∞-category C and suppose that j : P 1 → P be an upward-closed full subposet with complement i : P 0 → P .Let i * and j * denote the right Kan extension along i and j and j ! the left Kan extension along j.Then the following hold for the functor ∞-category Fun(P, C): (1) The functors i * and j * form a recollement.
(2) The functors j ! and i * also form a recollement.
Proof.Both can be easily checked by using [12,Proposition A.8.20] and observing that i * and j ! are given as the extension-by-zero functors.induced by the composite Fun(K, Sp)⊗Fun(K, Sp) where ⊗ denotes the copower.
Proof.By definition, D is the composite Fun(K, Sp) → Fun(K op × K × K, Sp) → Fun(K op , Sp), where the first and second maps are objectwise given by F Proof of (iii) ⇒ (i) of Theorem A. We show by induction on #P that D P is an equivalence, which is equivalent to the Verdier property by Lemma 2. We then consider applying Lemma 4.2 to conclude the proof.Since D P \{m} is an equivalence by our inductive hypothesis, it remains to check that D P restricts to define the dashed arrow and that it is an equivalence.Equivalently, we need to show that the composite Sp j * − → Fun(P, Sp) is zero for p = m and an equivalence for p = m.Note that since this functor is colimit-preserving, it is determined by its value at S, for which we write E p .Lemma 4.3 says that E p is computed as lim ← −q∈P Map(p, q) ⊗ (j * S)(q).If p m, the spectrum E p is zero as (j * S)(q) = 0 holds for q m.If p ≤ m, the spectrum E p is equivalent to the cohomology Γ(P ; S [p,m] ).Hence E p is zero for p < m by the assumption (iii) and Lemma 3.16.Therefore, it remains to compute E m Γ(P ; S [m,m] ).We pick a maximal chain p 0 < • • • < p r = m in P .For i = 1, . . ., n, by using Γ(P ; S [p i−1 ,p i ] ) = 0, we have ).
Thus we have Γ(P ; S [m,m] ) = Σ −r S. Hence the dashed arrow in (4.5) is identified with the functor Σ −r , which is an equivalence.
Proof of (i) ⇒ (iii) of Theorem A. We proceed by induction on #P .If P = ∅, the claim is obvious.We assume P = ∅ and pick a maximal element m.Then P \ {m} is also Verdier by Corollary 2.8 applied to Example 2.4.Hence by our inductive hypothesis, it suffices to show that Γ(P ; S [p,m] ) vanishes for any p < m.We now form the diagram (4.5), but the dashed arrow already exists in this case since D P is an equivalence.As we have observed in the above proof, the existence of the dashed arrow in particular means that Γ(P ; S [p,m] ) vanishes for p < m, which is what we wanted to show by Lemma 3.16.

4.3.
The Gorenstein* condition from duality.We finally complete the proof of Theorem A. The main ingredient is the following nontrivial observation: Proposition 4.6.If a finite poset P is Verdier, P >p is also Verdier for any p ∈ P .
The proof requires the following trivial observations: Lemma 4.7.Let P be a finite poset.For p ∈ P , let S ≤p denote the right Kan extension of the constant functor with value S along P ≤p → P .Then Fun(P, Sp) is generated by S ≤p under colimits and shifts.
Proof.We proceed by induction on #P .There is nothing to prove if P = ∅.Assume otherwise and pick a maximal element m ∈ P .Let C ⊂ Fun(P, Sp) be the full subcategory generated by S ≤p under colimits and shifts.The inductive hypothesis implies that F ∈ Fun(P, Sp) is in C if F (m) is zero.Hence any F ∈ Fun(P, Sp) the fiber of F → F (m) ⊗ S ≤m is in C. Since F (m) ⊗ S ≤m is also in C by assumption we have F ∈ C. Lemma 4.8.Let P be a (not necessarily finite) poset Then E [⊥,p] : P ⊥ → Sp is a limit diagram for any p ∈ P and any E ∈ Sp.
Proof.We can assume that p is the greatest element by replacing P with P ≤p .Then the result follows from Corollary 3.6 or, more directly, the observation that now P is weakly contractible.
Proof of Proposition 4.6.By induction, it suffices to consider the case where p is minimal.
Let j denote the inclusion P >p → P and j !: Fun(P >p , Sp) → Fun(P, Sp) the left Kan extension functor.Corollary 2.8 applied to Example 2.5 says that the pair (Fun(P >p , Sp), Γ P • j ! ) is a commutative Frobenius algebra.Hence it suffices to construct an equivalence We write j as the composite where (-) ! and (-) * denote the left and right Kan extension functors.By applying Γ P • -, we obtain Γ P • j !→ Γ P • l !• k * .Since its cofiber can be computed as where we use the minimality of p, we are reduced to showing that Let C denote the full subcategory of Fun(P ≥p , Sp) spanned by the limit diagrams.We need to show that Γ P • l ! is zero on C. We now observe that C is generated under colimits and shifts by S [p,q] for q ∈ P >p : First, they are indeed limit diagrams by Lemma 4.8.Then it follows from Lemma 4.7 that Fun(P >p , Sp) is generated by their restrictions.Therefore, we need to show that ) is zero, but this follows from (iii) of Theorem A; note that we have already proven (i) ⇒ (iii) in Section 4.2.
Proof of (i) ⇒ (ii) of Theorem A. Let P be a Verdier finite poset.According to Lemma 3.13, it suffices to show that (p, q) has the (integral) cohomology of a sphere for p < q in P ⊥ .Since we know that P ≤q is Verdier from (i) ⇔ (iii), we can assume that q is the greatest element of P .We can also assume that p = ⊥ by Proposition 4.6.Hence it remains to compute the cohomology of P <q , which is the fiber of Γ(P ; Z) → Γ(P ; Z [q,q] ).If P is a singleton, it is obviously zero.Otherwise, Lemma 4.8 says Γ(P ; Z) Z and the last part of the proof of (iii) ⇒ (i) says that Z [q,q] is some positive desuspension of Z. Therefore, P <q has the cohomology of a sphere.
We then obtain Theorem B as a bonus: Proof of Theorem B. By the standard (stable) Yoneda argument, we can assume C = Sp.Since P op is also Gorenstein*, it suffices to show that any limiting diagram P ⊥, → Sp is colimiting.Then as in the proof of Proposition 4.6, it suffices to show that S [⊥,p] ∈ Fun(P ⊥, , Sp) is colimiting for p ∈ P .Since this is trivial for p = as P ⊥ is weakly contractible, we assume otherwise.By the self-duality of the ∞-category of finite spectra, we are reduced to showing that S [p,⊥] ∈ Fun((P ⊥, ) op , Sp) is limiting.As p = , this is equivalent to the vanishing of the cohomology of S [p,⊥] ∈ Fun((P ⊥ ) op , Sp), which follows from Theorem A.

Variants
In this section, we prove Theorem C and show that our duality can be regarded as a topological sheaf-cosheaf duality.

5.1.
For locally finite posets.The equivalence (1.1) exists for the face poset of a locally finite regular CW complex.We extend our duality to cover that case.Definition 5.1.We say that a poset P is locally finite if P ≥p is finite for every p ∈ P .This terminology is justified by considering the Alexandroff topology of P (see Definition 5.15).Lemma 5.10.Let P be a locally finite poset and C a compactly generated pointed ∞-category.Then every compact object in Fun(P, C) is a right Kan extension of its restriction to s∈S P ≤s for some S ∈ P fin (P ).
Proof.Let F be a compact object.By Lemma 5.8, there is a finite full subposet Q such that F can be identified with the left Kan extension of F | Q along Q → P .We take S = q∈Q P ≥q , which is finite since P is locally finite, and consider the inclusion i : P S = s∈S P ≤s → P .Since P S contains Q, the morphism i !i * F → F is an equivalence.Hence it suffices to show that the composite i !i * F → F → i * i * F is an equivalence.As its restriction to P S is an equivalence, we consider p / ∈ P S .Then (i !i * F )(p) is initial since no q ∈ Q satisfies q ≤ p and (i * i * F )(p) is final since P S is downward closed.Since C is pointed, the desired claim follows.
Proof of Proposition 5.5.According to Lemma 5.9, the diagram actually lands in Pr ω , the ∞category of compactly generated ∞-categories and functors preserving colimits and compact objects.Since the inclusion Pr ω → Pr preserves colimits by [11,Theorem 5.5.3.18 and Proposition 5.5.7.6], it suffices to show that its restriction P fin (P ) → Pr ω is colimiting.Furthermore, since P fin (P ) is filtered, it suffices to show that its composite with (-) ω : Pr ω → Cat ∞ is colimiting.Then the desired claim follows from Lemmas 5.6 and 5.10.
One might think that the desired equivalence could be immediately obtained from Propositions 5.4 and 5.5 by taking the colimit of the assignment given by P fin (P ) S −→ D P S : Fun(P S , Sp) → Fun(P op S , Sp) ∈ Fun(∆ 1 , Pr), where P S denotes s∈S P ≤s .However, what we have proven in Section 2.2 is not sufficient in order to construct such a functor directly.We avoid this issue by first constructing the desired functor D for P : Definition 5.11.For a locally finite poset P , we define Γ cpt : Fun(P, Sp) → Sp as the colimit of the functor P fin (P ) → Fun(∆ 1 , Pr) given by Q → (Γ : Fun(Q, Sp) → Sp).Note that the source is identified with Fun(P, Sp) by Proposition 5.5 and the target is identified with Sp by Lemma 5.6 and the fact that Down fin (P ) is weakly contractible.From this, we obtain D : Fun(P, Sp) → Fun(P op , Sp) as in Section 2.2.Now the following two results imply Theorem C: Proposition 5.12.Let P be a locally finite poset and F : P → Sp a functor.If P ≤p is finite for each p ∈ P , the functor D(F ) : P op → Sp is pointwise given by p → lim ← −q∈P Map(p, q) ⊗ F (q).
Proof.We fix p and vary F .Then F → lim ← −q∈P Map(p, q) ⊗ F (q) preserves colimits by the finiteness assumption on P .Hence we can assume that F is compact.By Lemma 5.10, we can find S ∈ P fin (P ) such that F is a right Kan extension of its restriction to s∈S P ≤s .By replacing S with S ∪ {p}, we can assume p ∈ S. Then the desired result follows from Lemma 4.3 since s∈S P ≤s is finite by assumption.Theorem 5.13.Let P be a locally finite poset.If P <p is finite and Verdier for each p ∈ P , the pair (Fun(P, Sp), Γ cpt ) is a commutative Frobenius algebra in Pr st .In particular, D is an equivalence.
Proof.By Lemma 2.1, we only need to show that D is an equivalence.For S ∈ P fin (P ), let P S denote s∈S P ≤s ∈ Down(P ).We regard D as an object of Fun(∆ 1 , Pr st ) and consider the (essential) poset of subobjects Sub(D).By Proposition 2.3 applied to Example 2.4, each S ∈ P fin (P ) determines D P S ∈ Sub(D).Hence we obtain the morphism of posets P fin (P ) → Sub(D).Then we consider the composite where we set ∞ → D in the first arrow.This is colimiting by Propositions 5.5 and 5.4.By assumption and Theorem A, the functor D P S is an equivalence for S ∈ P fin (P ).Therefore, D is also an equivalence.
Remark 5.14.We can define the lower shriek functor for a morphism between posets satisfying the condition of Theorem C as the functor corresponding to the cosheaf pushforward under the duality equivalences.See Remark 6.12 for the locally compact Hausdorff case.5.2.In terms of sheaves.We explain that our duality for a poset can be interpreted as a sheaf-cosheaf duality over its Alexandroff space, which we recall as follows: Definition 5.15.The Alexandroff space Alex(P ) of a poset P is the topological space whose underlying set is that of P and whose open sets are the upward-closed subsets.
We recall the following fact, which was first proven in [1,Example A.11].
This identifies Shv(Alex(P )) as the bounded reflection of PShv(P op ) and PShv(P op ) as the hypercompletion of Shv(Alex(P )).
Note that this geometric morphism is not an equivalence in general; see [1,Example A.13].However, it is an equivalence in the situation we are interested in: Proposition 5.18.If P is a locally finite poset, (5.17) is an equivalence.
Remark 5.19.Note that by using [2, Corollary 2.6] instead of [1,Example A.12] in the proof, we can obtain this result for a wider class of posets.
Remark 5.20.It is a consequence of [5,Theorem 3.4] that the morphism (5.17) is an equivalence for a poset satisfying the ascending chain condition.However, they use the "geometric morphism" Shv(X) → PShv(P op ) constructed in [5, page 27] for a stratification X → Alex(P ), which is not geometric in general; the trivial stratification on Alex(P ) for the poset P in [1, Example A.13] gives a counterexample.Nevertheless, when P is locally finite, Proposition 5.18 shows that the morphism is indeed geometric.
Hence Theorem C, which we have seen in Section 5.1, says the following: Theorem 5.21.Let P be a locally finite poset such that P <p is finite and Gorenstein* for each p ∈ P .Then there is a canonical equivalence D : Shv Sp (Alex(P )) −→ cShv Sp (Alex(P )).

Verdier duality for proper separated ∞-toposes
The sheaf-cosheaf duality for locally compact Hausdorff spaces, which is often called covariant Verdier duality, was studied in [12,Section 5.5.5].In this section, we first prove its generalization using more abstract methods.Then we prove Theorem D using our formulation.In future work, we will study a relative variant.6.1.Proper separated ∞-toposes.Following [10, C2.4.16], we say that a geometric morphism is Beck-Chevalley if any pullback satisfies the Beck-Chevalley condition; i.e., the (unstable) proper base change theorem holds.Recall that in [11,Section 7.3.1]a geometric morphism is called proper if its arbitrary base change is Beck-Chevalley.Definition 6.1.An ∞-topos X is called separated if its diagonal X → X × X is proper.Remark 6.2.Consider a geometric morphism Y → X between n-toposes.If the geometric morphism Shv(Y) → Shv(X ) between ∞-toposes is proper, its arbitrary base change is Beck-Chevalley in the (n + 1)-category of n-toposes, but not vice versa.This is why in 1-topos theory we usually call a geometric morphism tidy when its arbitrary base change is Beck-Chevalley in the 2-category of 1-toposes.The same remark applies to the notion of separatedness.
We recall the following notion, which was introduced in [7, Appendix D]: Definition 6.4 (Gaitsgory).A presentably symmetric monoidal stable ∞-category C is called rigid if the unit u : Sp → C admits a colimit-preserving right adjoint and the multiplication m : C ⊗ C → C admits a C ⊗ C-linear9 right adjoint.
If C is rigid, it is easy to see that u R • m and m R • u constitute a duality datum in Pr, where -R indicates the right adjoint.In particular, (C, u R • m) is a commutative Frobenius algebra.Theorem 6.5.If X is a proper separated ∞-topos, then Shv Sp (X ) is rigid.Corollary 6.6.The pair (Shv Sp (X ), Γ) is a commutative Frobenius algebra in Pr st for any proper separated ∞-topos X .
Proof of Theorem 6.5.According to [12,Example 4.8.1.19],the binary product of ∞-toposes can be computed as their tensor product in Pr.Hence the result follows from Lemma 6.7 below.Here the composite Γ X • j !depends on j, not only on U, but there is a canonical choice for locally compact spaces: Definition 6.9.Let X be a locally compact Hausdorff space.We define the global section with compact support Γ cpt as the composite p * • j !where j : X → X ∞ is the inclusion to its one-point compactification and p : X ∞ → * is the projection.Then Theorem 6.8 says that (Shv Sp (X), Γ cpt ) is a commutative Frobenius algebra.We let D : Shv(X) → cShv(X) denote the associated equivalence (cf.Section 2.2).Remark 6.10.One can prove Verdier duality for locally compact Hausdorff spaces by a similar method to the one we have used in Section 5.1 for locally finite posets: Namely, the sheaf and cosheaf ∞-categories of a locally compact Hausdorff space can be written as colimits in Pr as those of compact subspaces.We leave the details to the interested reader.
For a poset P and a spectrum E, we have a functorial (both in P and in E) equivalence Γ(P ; E) Γ(|P |; E), where E denotes the constant sheaves on the ∞-toposes Fun(P, S) and Shv(|P |), respectively.Proof.This follows from Examples 3.3 and 3.4 and Proposition 3.5.

Lemma 3 . 10 .
Any maximal chain of a Gorenstein* finite poset P has the same length.In other words, P ⊥, admits a rank function 7 .Proof.This holds more generally for Cohen-Macaulay finite posets; see [3, Proposition 3.1].These two definitions are compatible: Proposition 3.11.For a finite poset P , it is Gorenstein* if and only if ∆(P ) is Gorenstein*.We omit the proof since it is a straightforward variant of[3, Proposition 3.3].

Lemma 4 . 2 . 4 . 2 .
Consider a left exact functor C → C and suppose that C and C are recollements of C 0 and C 1 and C 0 and C 1 , respectively.Furthermore, assume the following:• The functor f restricts to define equivalencesC 0 → C 1 and C 0 → C 1 .• The morphism L 0 • f → f • L 0 obtainedby the above condition is an equivalence.Then f itself is an equivalence.Proof.This follows from[12, Proposition A.8.14].Duality and the vanishing condition.We prove (i) and (iii) in Theorem A are equivalent.We start with a pointwise description of D: Lemma 4.3.Let K be a finite ∞-category so that lim ← − : Fun(K, Sp) → Sp is a morphism in Pr st .Then the potential duality functor D : Fun(K, Sp) −→ [Fun(K, Sp), Sp] Fun(K op , Sp)

Lemma 6 . 7 .pr 1 6 . 2 . 7 : 6 . 8 .
Let f : Y → X be a proper morphism of ∞-toposes, then f * : Shv Sp (X ) → Shv Sp (Y) admits a Shv Sp (X )-linear right adjoint.Proof.According to[11, Remark 7.3.1.5],the direct image functor Y → X preserves filtered colimits.Hence f * : Shv Sp (Y) → Shv Sp (Y) preserves colimits.Now we consider the diagram in the ∞-category of ∞-toposes.Since the right and outer squares are cartesian, so is the left one.According to[12, Example 4.8.1.19],the binary product of ∞-toposes can be computed as their tensor product in Pr.Therefore, since f × id is Beck-Chevalley, for any F ∈ Shv Sp (X ) andG ∈ Shv Sp (Y) the canonical morphism f * G ⊗ F → f * (G ⊗ f * F) is an equivalence.The locally compact case.The following result is derived from Theorem 6.5 by using Corollary 2.8 applied to Example 2.Theorem Let j : U → X be an open subtopos of a proper separated ∞-topos.Then the pair (Shv Sp (U), Γ X • j ! ) is a commutative Frobenius algebra in Pr st .
12, Proposition 1.2.4.3], which may be of independent interest.Theorem B. Let P be a Gorenstein* finite poset.Then for any stable ∞-category C, a diagram (P ) (P ) → C is a limit if and only if it is a colimit.
1.If P = ∅, the claim is obvious.We assume #P > 0 and pick a maximal element m ∈ P .Since {m} is upward closed, by applying Lemma 4.1 to j : {m} → P and i : P \ {m} → P , we can form two recollements, which fit into a diagram Sp) Sp Fun({m} op , Sp) is made and j ?denotes the right adjoint of j + .Proposition 2.3 applied to Example 2.4 says that the left square commutes and that the canonical morphism i ?• D P → D P \{m} • i * is an equivalence, where i ?denotes the left adjoint of i + .