Locally Constant Sheaves (Lecture 21) March 21, 2011 Let X be a finite polyhedron with a triangulation T and let C be an 1-category. We will say that T -constructible sheaf F : T ! C is locally constant if F(τ) ! F(τ 0) is invertible whenever τ ≤ τ 0. We let Shvlc(X; C) denote the full subcategory of ShvT (X; C) spanned by the locally constant sheaves. This 1-category does not depend on the choice of triangulation: if S is a refinement of T , then the pullback functor ShvT (X; C) ! ShvS(X; C) induces an equivalence on the full subcategories spanned by the locally constant sheaves. Let R be an A1-ring, fixed for the remainder of this lecture. Our goal is to study local systems of R-modules on C: that is, locally constant sheaves on C with values in the 1-category of R-modules. Fix a finite polyhedron X and a triangulation T of X. Let ShvT (X : R) denote the 1-category of fp op fp T -constructible sheaves on X with values in LModR : that is, contravariant functors T ! LModR . The fp fp formation of R-linear duals gives a contravariant equivalence of LModR with RModR ; we may therefore fp identify ShvT (X; R) with the opposite of the 1-category coShvT (X; RModR ) of T -constructible cosheaves fp op fp with values in RModR (that is, functors T ! RModR ). This is contained in the larger 1-category coShvT (X; RModR) of cosheaves with values in RModR. In fact, we can identify coShvT (X; RModR) with fp the 1-category of Ind-objects Ind(coShvT (X; RModR ). Let coShvlc(X : RModR) denote the full subcategory of coShvT (X; RModR) spanned by the locally constant cosheaves (that is, those cosheaves for which F(τ) ! F(τ 0) is an equivalence for every τ 0 ⊆ τ 2 T ). Note that coShvlc(X; RModR) is closed under all limits and colimits in coShvT (X; RModR). It follows that the inclusion coShvlc(X; RModR) ,! coShvT (X; RModR) 0 admits both left and right adjoints. We will denote a left adjoint to this inclusion by L. Let coShvT (X; RModR) denote the full subcategory of coShvT (X; RModR) spanned by those objects F such that T (F) ' 0: that is, 0 fp those objects F such that MorcoShvT (X;RModR)(F; G) ' 0 whenever G is locally constant. We let coShvT (X; RModR ) 0 fp denote the intersection coShvT (X; RModR) \ coShvT (X; RModR ). 0 0 fp Lemma 1. The full subcategory coShvT (X; RModR) is generated (under filtered colimits) by coShvT (X; RModR ). Consequently, we have a canonical equivalence 0 0 fp coShvT (X; RModR) ' Ind coShvT (X; RModR ): 0 fp Proof. Since every object of coShvT (X; RModR ) is a compact object of coShvT (X; RModR), we get a fully faithful embedding 0 fp Ind(coShvT (X; RModR ) ! coShvT (X; RModR): Let C denote the essential image of this embedding; it is a full subcategory of ShvT (X; RModR). We clearly 0 0 have C ⊆ coShvT (X; RModR). Let us prove the reverse inclusion. For any object F 2 coShvT (X; RModR), we can choose a fiber sequence F0 ! F !α F00 1 where F0 2 C and Mor(G; F00) ' 0 for every G 2 C (here F0 is given by the colimit of the filtered diagram 0 fp of all objects of coShvT (X; RModR ) equipped with a map to F). For every simplex τ 2 T , let Fτ 2 coShvT (X; RModR) be given by the formula ( R if σ ⊆ τ Fτ (σ) = 0 otherwise. 0 For any cosheaf G, we have Mor(Fτ ; G) ' G(τ). Let τ ⊆ τ, and form a cofiber sequence Fτ 0 ! Fτ ! Fτ = Fτ 0 : 0 0 fp If G is locally constant, we have Mor(Fτ = Fτ 0 ; G) ' fib(G(τ) ! G(τ )) ' 0, so that Fτ = Fτ 0 2 coShvT (X; RModR ). 00 00 00 0 0 It follows that Mor(Fτ = Fτ 0 ; F ) ' 0, so that F (τ) ' F (τ ). Since τ and τ are arbitrary, we deduce that 00 0 F is locally constant. Since F 2 coShvT (X; RModR), the map α is nullhomotopic. Then F is a direct summand of F0, and therefore belongs to C as desired. fp fp Under the contravariant equivalence of 1-categories coShvT (X; RModR ) ' ShvT (X; LModR ), the sub- 0 fp 0 fp fp category coShvT (X; RModR ) corresponds to a full subcategory ShvT (X : LModR ) ⊆ ShvT (X; LModR ), which is evident closed under the formation of direct summands. Several lectures ago, we constructed a quotient 1-category fp 0 fp ShvT (X; LModR )= ShvT (X; LModR ) fp fp op op as a full subcategory of Pro(ShvT (X; LModR )) ' Ind(coShvT (X; RModR )) ' coShvT (X; RModR) . Unwinding the definitions, we see that this subcategory consists precisely of those objects of the form L F, fp fp where F 2 coShvT (X; RModR) ). Let us denote this subcategory by coShvlc (X; RModR). We now study the 1-category coShvlc(X; RModR) ' Shvlc(X; RModR) in more detail. For sim- plicity, let us restrict our attention to the case where X is connected. For every point x 2 X, let ∗ ix : fxg ! X denote the inclusion map. Pullback along ix determines a functor x : Shvlc(X; RModR) ! RModR (given by evaluation at the unique simplex τ 2 T containing x in its interior). This functor commutes with all limits and colimits. In particular, it admits a left adjoint, which we will denote by ∗ x+ : RModR ! Shvlc(X; RModR). Since x commutes with filtered colimits, x+(R) is a compact object of Shvlc(X; RModR). Moreover, it is a compact generator of Shvlc(X; RModR): if F 2 Shvlc(X; RModR), ∗ ∗ then Mor(x+(R); F) ' 0 if and only if MorRModR (R; x F) = x F vanishes. Since X is connected, this is equivalent to the vanishing of all stalks of F: that is, to the condition that F ' 0. It follows that 0 Shvlc(X; RModR) is equivalent to the 1-category RModR0 , where R is the A1-ring whose underlying ∗ ∗ spectrum is MorShvlc(X;RModR)(x+(R); x+(R)) ' MorRModR (R; x x+R) ' x x+R. We can describe R0 more explicitly. More generally, suppose we are given any pair of points x; y 2 X. We can form a homotopy pullback diagram of topological spaces (commutative up to canonical homotopy) φ Px;y / fxg x y fyg / X; ∗ ∗ where Px;y is the path space fp : [0; 1] ! X : p(0) = x; p(1) = yg. Let φ and denote the pullback functors on locally constant sheaves of right R-modules, and let φ+ and + be their left adjoints. There is a ∗ ∗ ∗ natural \base-change" isomorphism y x+ ' +φ of functors from RModR to itself. Consequently, y x+(R) is given by + of the constant sheaf on Px;y with values in R. This is given by the smash product spectrum Px;y ^ R (here we regard Px;y as an unpointed space) whose homotopy groups are given by the R-homology 0 groups R∗(Px;y) of the space Px;y. In particular, we have R = Px;x ^ R. If R is connective, we obtain 0 M π0R = R0(Px;x) = π0R ' (π0R)[π1X]: η2π0Px;x 2 fp θ Let us now describe the full subcategory coShvlc (X; RModR) ⊆ coShvlc(X; RModR) ' RModR0 . This fp is a stable subcategory, consisting of those objects of the form L(F), where F 2 coShvT (X; RModR ). Note fp that coShvT (X; RModR ) is generated, as a stable 1-category, by objects of the form Fτ;M , where ( M if σ ⊆ τ Fτ;M (σ) = 0 otherwise and M is a finitely presented right R-module. It is therefore generated as a stable 1-category by objects of the form Fτ;R = Fτ . We observe that L Fτ ' y+(R), where y is any point in the interior of τ: indeed, for any object G 2 coShvlc(X; RModR) we have ∗ Mor(L Fτ ; G) ' Mor(Fτ ; G) ' G(τ) ' y G ' Mor(y+(R); G): Let x be our fixed base point of X. Since X is connected, for any point y 2 X there is an isomorphism x+(R) ' y+(R) in coShvlc(X; RModR) (obtained by choosing a path joining x and y). Consequently, the fp full subcategory coShvlc (X; RModR) ⊆ coShvlc(X; RModR) is generated, as a stable subcategory, by the object x+(R). In particular, it corresponds to the full subcategory fp RModR0 ⊆ RModR under the equivalence θ. Passing to opposite 1-categories, we obtain the following result: Theorem 2. Let X be a connected finite polyhedron with base point x, T a triangulation of X, R an A1- 0 0 fp fp ring, and R = Px;x ^ R the A1-ring constructed above. Let ShvT (X; LModR ) ⊆ ShvT (X; LModR ) be defined as above. Then there is a canonical equivalence of 1-categories fp 0 fp fp ShvT (X; LModR )= ShvT (X; LModR ) ' LModR0 : Let Shv0 (X; LModfp) ' lim Shv0 (X; LModfp. Passing to the direct limit over T , we obtain an const R −!T T R equivalence fp 0 fp fp Shvconst(X; LModR )= Shvconst(X; LModR ) ' LModR0 : Warning 3. Working with perfect module spectra in place of finitely presented module spectra, one can construct a fully faithful embedding perf 0 fp perf ShvT (X; LModR )= ShvT (X; LModR ) ' LModR0 : This embedding is generally not an equivalence, which is why we have generally confined our attention to the study of finitely presented modules rather than perfect modules.