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Draft lesson.

Relative pullback of locally module stacks

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

Pulling a local algebra to a new space should give the category of all modules over the new algebra. The ordinary inverse image of a stack need not contain all of those modules. Morita presentations give a larger pullback with the expected local algebra, and relative bimodule reconstruction characterizes it by functors with actual stack right adjoints.

Let \(f:Y\to X\) be any continuous map, \(R\) a commutative ring sheaf on \(X\), and \(S=f^{-1}R\). All algebra scalar maps are central and unital. A locally module stack has local module models on an open cover; all restrictions below are to opens, with their canonical comparisons. The direct image of an \(S\)-linear stack is \[ (f_*\mathsf N)(W)=\mathsf N(f^{-1}W). \tag{0.1} \] It is an \(R\)-linear stack: an open cover pulls back to an open cover, so both descent axioms follow from those of \(\mathsf N\), and the scalar action uses \(R\to f_*S\).

We use Bimodules and adjoints of module-stack functors for same-space reconstruction, Noncommutative gluing by invertible bimodules for coherent presentations and inverse transports, and Dual and tensor products of locally module stacks for the stack \(\mathsf H\) of linear functors with a stack right adjoint. Its arrows are all natural transformations.

1. The inverse image of module sheaves

For a sheaf \(F\) on \(X\), the inverse image is the sheafification on \(Y\) of \[ V\longmapsto\operatorname*{colim}_{V\subset f^{-1}W}F(W), \tag{1.1} \] where \(W\) runs through opens of \(X\). Intersections refine two such choices. A presheaf map from (1.1) to a sheaf \(G\) is the same as compatible maps \(F(W)\to G(f^{-1}W)\): one direction uses the term for \(V=f^{-1}W\), and the other restricts that term to each \(V\subset f^{-1}W\). These rules are inverse, so sheafification proves \(f^{-1}\dashv f_*\). The same construction transports additions, algebra multiplications and module actions, and the bijection respects them.

Every germ at \(y\) in (1.1) comes from a section near \(f(y)\). Conversely any such section pulls back on the inverse image of its neighborhood. Equal germs become equal after intersecting neighborhoods. Sheafification leaves germs unchanged, since its sections are locally represented by presheaf sections. Thus \[ (f^{-1}F)_y\simeq F_{f(y)}. \tag{1.2} \] A sheaf map is an isomorphism if it is one on every stalk: a section in the target has local lifts by stalk surjectivity, those lifts agree locally by stalk injectivity, and the sheaf axiom glues them uniquely. This also proves the criterion for injectivity and local surjectivity. Filtered neighborhood colimits preserve exact module sequences: a finite equation holds in the colimit exactly when it holds at a common later neighborhood. Consequently \(f^{-1}\) is exact for module sheaves and preserves small colimits, the latter also following from its adjunction.

For a possibly noncommutative \(R\)-algebra \(A\), put \(A'=f^{-1}A\). Tensor sheaves are sheafifications of balanced tensor presheaves. Taking a stalk gives the balanced tensor product of the stalks: every tensor uses finitely many representatives and relations, which can be placed on a common neighborhood. Formula (1.2) therefore identifies the canonical map \[ f^{-1}(P\otimes_A M)\xrightarrow{\sim} f^{-1}P\otimes_{A'}f^{-1}M. \tag{1.3} \] It is an isomorphism on every stalk and hence globally. Its map sends the germ of \(p\otimes m\) to the tensor of the two germs. This proves its compatibility with associativity, units, permutations of scalar tensor factors and bimodule actions. No flatness of these modules is assumed.

2. Open extensions, generators and Morita transport

For an open inclusion \(j:W\hookrightarrow X\), direct image is \((j_*N)(T)=N(T\cap W)\); its right-adjoint relation to restriction follows by restricting or extending the same sectionwise maps. Extension by zero \(j_!N\) is the sheaf whose local sections in \(j_*N\) vanish on a neighborhood of every point outside \(W\). Maps from it to \(M\) are exactly maps \(N\to M|_W\): extend a local value by zero wherever the section vanishes. Hence \(j_!\dashv j^*\dashv j_*\). Its stalk is \(N_x\) for \(x\in W\) and zero otherwise. The stalk criterion proves the natural comparisons \[ f^{-1}j_!N\simeq j'_!f_W^{-1}N, \qquad P\otimes_Aj_!N\simeq j_!(P|_W\otimes_{A|_W}N), \tag{2.1} \] where \(j':f^{-1}W\hookrightarrow Y\).

The module \(A[W]=j_!(A|_W)\) represents sections on \(W\). Thus the coproduct indexed by all pairs \((W,m\in M(W))\) maps epimorphically to \(M\): every local section has its named lift. Applying the same construction to its kernel gives a presentation by two coproducts of these generators. Use fixed universes so the indexing sets are small.

We will also need restriction on a locally module stack to have a stack right adjoint. Here is the boundary comparison that permits this. Suppose \(L\) is an invertible \((A,B)\)-bimodule, with coherent inverse \(K\). Locally write the inverse image of \(1_A\) under \(L\otimes_BK\simeq A\) as a finite sum \(\sum_r\ell_r\otimes k_r\). The inverse-pair triangle gives \[ \ell=\sum_r\ell_r\,d(k_r\otimes\ell), \tag{2.2} \] where \(d:K\otimes_AL\to B\) is the other contraction. Each coefficient is right \(B\)-linear in \(\ell\). The maps \(B^n\to L\) defined by the \(\ell_r\), and \(L\to B^n\) defined by these coefficients, compose to identity. Thus \(L\) is locally a retract of a finite free right \(B\)-module.

The canonical map \[ L\otimes_Bj_*N\longrightarrow j_*(L|_W\otimes_{B|_W}N) \tag{2.3} \] is therefore an isomorphism. It is so for \(L=B^n\), because direct image commutes with finite sums of module sheaves, which are also finite products. Naturality makes it so for a retract: the two sides and the comparison retain the same splitting maps. This verifies it locally, hence globally, with its left \(A\)-action. The same argument with (2.1) handles \(j_!\). These canonical comparisons respect composition of Morita transports by balanced tensor associativity. In local models of a stack, the module functors \(j_*\) consequently agree on overlaps, including their unit and counit maps. They glue to the stack right adjoint of restriction. This proof uses only the supplied invertible transports and their triangles.

3. Relative bimodule reconstruction

Let \(A\) be an \(R\)-algebra on \(X\) and \(C\) an \(S\)-algebra on \(Y\). An \((C,A')\)-bimodule \(P\) defines \[ T_P(M)=P\otimes_{A'}f^{-1}M, \qquad H_P(N)=f_*[P,N]_C. \tag{3.1} \] The right action on \(P\) makes its internal Hom a left \(A'\)-module, and \(A\to f_*A'\) makes the second expression a left \(A\)-module. Tensor–Hom followed by inverse-image adjunction gives \[ \operatorname{Hom}_C(T_PM,N) \simeq\operatorname{Hom}_{A'}(f^{-1}M,[P,N]_C) \simeq\operatorname{Hom}_A(M,H_PN). \tag{3.2} \] These are inverse evaluation/currying bijections. On every open \(W\subset X\), direct image means that of \(f_W:f^{-1}W\to W\); its section formula commutes with further restriction in \(W\). Thus (3.2) is an actual stack adjunction \(T_P:\mathsf{Mod}(A)\to f_*\mathsf{Mod}(C)\), not only a global-category adjunction.

Theorem 3.1. Every \(R\)-linear stack functor \(\Phi:\mathsf{Mod}(A)\to f_*\mathsf{Mod}(C)\) with a stack right adjoint is uniquely reconstructed, up to the canonical isomorphism, from the bimodule \(P=\Phi(A)\). All transformations are precisely \((C,A')\)-bimodule maps.

Proof. A section \(a\in A(W)\) gives the source-module map \(r_a:x\mapsto xa\). Applying \(\Phi_W\) gives a compatible right \(A\)-action on \(P|_{f^{-1}W}\), commuting with \(C\). In sheaf terms this is an algebra map \(A^{\mathrm{op}}\to f_*\underline{\operatorname{End}}_C(P)\). Adjunction (1.1) gives a right \(A'\)-action on \(P\). Its central \(S\)-action agrees with the given one by \(R\)-linearity and the same adjunction. The equation \(r_b r_a=r_{ab}\) gives the correct right-action order.

A section \(m\in M(W)\) gives \(\ell_m:A|_W\to M|_W\). The rule \(p\otimes m\mapsto\Phi_W(\ell_m)(p)\) defines \[ \theta_M:P\otimes_{A'}f^{-1}M\longrightarrow\Phi(M). \tag{3.3} \] Sections of the inverse image are locally represented by such sections. The rule commutes with their refinements, is additive, \(C\)-linear and balanced because \(\ell_{am}=\ell_mr_a\). It consequently sheafifies to (3.3), natural in \(M\) and compatible with source restriction.

Both functors preserve coproducts and cokernels in every source fibre by their stack adjunctions. They also commute with extension from a source open. If \(G\) is the stack right adjoint of \(\Phi\), then, naturally in \(Z\), \[ \begin{aligned} \operatorname{Hom}_C(\Phi j_!N,Z) &\simeq\operatorname{Hom}_A(j_!N,GZ)\\ &\simeq\operatorname{Hom}_{A|_W}(N,(GZ)|_W)\\ &\simeq\operatorname{Hom}_{C|_{f^{-1}W}}(\Phi_WN,Z|_{f^{-1}W})\\ &\simeq\operatorname{Hom}_C(j'_!\Phi_WN,Z). \end{aligned} \tag{3.4} \] The third step uses the stack restriction comparison of \(G\). This gives \(\Phi j_!\simeq j'_!\Phi_W\). For \(N=A|_W\), formula (2.1) identifies the tensor-functor value with the same \(j'_!P|_{f^{-1}W}\). The actual map \(\theta_{A[W]}\) is this isomorphism: transpose its inverse-direction extension map under \(j'_!\dashv j'^*\). It is \(\Phi_W(e)\), where \(e:A|_W\to j^*A[W]\) is the extension unit. Formula (3.3) takes the universal section of this generator to precisely \(\Phi_W(e)(p)\). The two maps are therefore equal by that adjunction.

Apply both functors and \(\theta\) to the two-generator presentation of Section 2. The comparison is an isomorphism on both coproduct terms, hence on their cokernels. This proves \(\theta_M\) invertible for every \(M\), with its restriction compatibility.

For a transformation, its component at \(A\) is a \(C\)-linear map \(h:P\to Q\). Naturality at every \(r_a\) makes it right \(A'\)-linear, using the action adjunction just established. Naturality at every \(\ell_m\) forces its component to be \(h\otimes1\) on local pure tensors in (3.3). Conversely a bimodule map gives those balanced natural components. These assignments are inverse and preserve identity and composition. They prove full faithfulness on all transformations and complete the reconstruction. \(\square\)

4. Transporting a Morita presentation

Choose an open-cover presentation \(\mathsf M=\mathsf T(A,L,\mu)\) on \(X\). Pull the algebras, invertible bimodules and multiplication maps to the inverse-image cover on \(Y\). Define \[ f^{\boxtimes}\mathsf M =\mathsf T(f^{-1}A,f^{-1}L,f^{-1}\mu), \tag{4.1} \] using (1.3) to identify the tensor domains. The old coherent inverse contractions pull back to inverse contractions; the triangles and four-coordinate equation remain equalities after pullback. Thus these are valid gluing data. In particular \[ f^{\boxtimes}\mathsf{Mod}(A)\simeq\mathsf{Mod}(A'). \tag{4.2} \]

This construction is independent of the chosen presentation, with coherent comparison equivalences. To prove this, take a common refinement of two presentations of \(\mathsf M\). Their comparison equivalence is represented by an invertible bimodule family with its comparison equations, by the gluing theorem. Pulling that family and its inverse to \(Y\), and using (1.3), preserves its equations and both inverse contractions. It therefore gives an equivalence of the two pullback presentations. Comparisons of three presentations come from composition of their families. Since pulling a balanced product preserves its associator and units, these comparisons satisfy the same composition and identity laws. The refinement equivalences also pull back to their explicit restricted data. This proves independence without choosing unrelated equivalences on each overlap.

Pullback of object modules and transitions gives a natural functor \[ \eta_f:\mathsf M\longrightarrow f_*f^{\boxtimes}\mathsf M. \tag{4.3} \] Locally it is ordinary sheaf inverse image \(\mathsf{Mod}(A)\to f_*\mathsf{Mod}(A')\), whose stack right adjoint is sheaf direct image by (3.2) with regular \(P=A'\). On source overlaps the adjoints agree by the adjunction comparisons: transpose the comparison of the inverse-image functors through their Hom bijections. Both model changes are equivalences, so inserting their inverse comparisons makes these Hom bijections invertible and gives invertible represented comparisons of the adjoints. Their cocycles follow by composing those bijections and cancelling the inverse pairs by their triangles. The source stack therefore glues them and their units and counits, as in the functor-stack descent lemma. Thus \(\eta_f\) has a stack right adjoint as well. It gives the direct-image functor from pulled-back twisted objects to the original stack.

5. The relative universal property

Theorem 5.1. If \(\mathsf M\) is locally a module stack on \(X\) and \(\mathsf N\) is locally a module stack over \(S\) on \(Y\), composition with (4.3) gives an equivalence of \(R\)-linear stacks \[ f_*\mathsf H_S(f^{\boxtimes}\mathsf M,\mathsf N) \xrightarrow{\sim}\mathsf H_R(\mathsf M,f_*\mathsf N), \qquad\Psi\longmapsto(f_*\Psi)\eta_f. \tag{5.1} \]

Proof. The displayed composite has a stack right adjoint: direct image of a stack adjunction retains its units, counits and restriction comparisons, and composition of two stack adjunctions is a stack adjunction.

First let \(\mathsf M=\mathsf{Mod}(A)\). A right-adjoint functor on the right of (5.1) restricts on any target-model open \(V\subset Y\) to a functor into \((f|_V)_*\mathsf{Mod}(C_V)\) with a stack right adjoint. Indeed target restriction has a stack right adjoint by Section 2; composing with it gives the required right adjoint. Theorem 3.1 reconstructs this restricted functor from its local value \(P_V\) at \(A\), with right \(A'|_V\)-action. These are the restrictions of the single object \(P=\Phi(A)\) of \(\mathsf N\); the right actions glue because they were induced by the same source maps \(r_a\). Conversely, giving this object and right action gives local tensor functors on \(\mathsf{Mod}(A')\). Their comparison maps are those of \(P\), and their equations agree on overlaps. They glue to \(\Psi:\mathsf{Mod}(A')\to\mathsf N\), with its stack right adjoint by the coherent functor-gluing theorem. Full faithfulness in Theorem 3.1 makes all the identifications with \(\Phi\) and all transformations compatible on overlaps. This proves object and arrow reconstruction in (5.1).

For clarity, the relative right adjoint is also explicit. The local sheaves \([P_V,N_V]_{C_V}\) agree across a Morita model change: that equivalence is fully faithful on the restricted module maps, including the right \(A'\)-action on \(P\). They glue to an \(A'\)-module sheaf \(\underline{\operatorname{Hom}}_{\mathsf N}(P,N)\). The right adjoint on \(X\) is \(f_*\underline{\operatorname{Hom}}_{\mathsf N}(P,N)\). Its Hom bijection is local tensor–Hom followed by (1.1); maps and their equations glue, so this is (3.2) for an arbitrary locally module target. It commutes with source-open restriction.

Finally choose source-model opens for \(\mathsf M\). The preceding equivalence applies over each of them. Changing the source model tensors with its invertible transport; changing the target model tensors with its own transport. Theorem 3.1 identifies each functor comparison with its actual bimodule map, and (1.3) identifies composition of those maps after pullback. Thus the reconstructed functors and their inverse comparisons satisfy the source-overlap equations. They descend on both sides, which are stacks, and the local inverse bijections on transformations descend uniquely too. This proves (5.1) globally, with its naturality and all source restriction comparisons. \(\square\)

6. Composition, functors and internal operations

A stack-adjoint functor \(F:\mathsf M\to\mathsf M'\) is represented locally by a coherent bimodule family \(P\). Pulling that family gives \(f^{\boxtimes}F\). It is a stack-adjoint functor by the gluing theorem, and a transformation pulls back to the same component bimodule maps. For two such functors, (1.3) identifies the pulled-back composite bimodule with the composite of the pulled-back bimodules. Their associators and identity maps agree on pure tensors, so this construction retains composition and all its coherence.

For \(g:Z\to Y\), the germ construction gives the canonical sheaf comparison \(g^{-1}f^{-1}\simeq(f\circ g)^{-1}\): both send a germ at \(z\) to the germ at \(f(g(z))\). The sectionwise map is induced by the two successive neighborhood restrictions, so these comparisons respect further restriction and composition. Applying them to the algebras, transports, multiplication and all bimodule maps gives \[ (f\circ g)^{\boxtimes}\mathsf M \simeq g^{\boxtimes}f^{\boxtimes}\mathsf M. \tag{6.1} \] For three maps both comparisons use the same three neighborhood restrictions. Their coherence follows already before gluing. The identity comparison and the comparison of the unit functors (4.3) follow in the same way.

There are natural equivalences \[ \begin{aligned} f^{\boxtimes}(\mathsf D_R\mathsf M)&\simeq\mathsf D_S(f^{\boxtimes}\mathsf M),\\ f^{\boxtimes}(\mathsf M\boxtimes_R\mathsf M') &\simeq f^{\boxtimes}\mathsf M\boxtimes_Sf^{\boxtimes}\mathsf M'. \end{aligned} \tag{6.2} \] The dual presentation has algebra \(A_i^{\mathrm{op}}\) and transport \((L_{ji})^{\mathrm{op}}\); their inverse images are exactly the dual of the inverse-image presentation. The product presentation has algebra \(A_i\otimes_RB_i\) and transport \(L_{ij}\otimes_RK_{ij}\); (1.3) gives exactly the product of the inverse-image presentations. In each case the maps preserve the displayed multiplication, since they preserve reversal and balanced interchange. They therefore descend to (6.2). Compatibility with unit, symmetry, associativity and (6.1) follows by taking the same germs and the same ordered tensor factors on both sides.

7. What happens to the internal Hom of modules

For source module sheaves there is a canonical comparison \[ f^{-1}[P,Q]_A\longrightarrow [f^{-1}P,f^{-1}Q]_{A'}. \tag{7.1} \] Pull back evaluation and use (1.3), then curry, to construct it. A local map on the source sends the two pulled-back modules by its inverse-image map. This specifies its values and restriction compatibility.

If \(P\) is locally a retract of a finite free left \(A\)-module, (7.1) is an isomorphism. For \(P=A^n\), both sides identify with \((f^{-1}Q)^n\); the comparison is that identification, because evaluation at each basis vector agrees. The Hom sheaves for a retract inherit the corresponding splitting of these finite powers. Inverse image preserves the splitting maps, so the comparison is an isomorphism on that retract as well. Local verification proves the assertion globally.

Without this finiteness assumption the comparison can fail, as Exercise 8.4 proves. The relative functor constructed from \(P\) always has right adjoint \(f_*[P,-]_C\), and the pulled-back tensor functor has the new right adjoint computed from its pulled-back bimodule. Neither statement needs an assertion that arbitrary internal Homs commute with inverse image. The dual-stack and product-stack equivalences in (6.2) concern their Morita presentations, not the invalid replacement of (7.1) by an unconditional isomorphism.

8. Graded exercises with complete solutions

Exercise 8.1 (foundation). Map the two-point space \(Y=\{0,1\}\), with opens \(\varnothing,\{1\},Y\), to a point \(X\). Over a field \(k\), compare ordinary inverse image of the stack of vector spaces with the transported-algebra pullback. Exhibit a module missing from the ordinary inverse image.

Solution. A sheaf of modules over the constant ring \(k\) on \(Y\) is an arrow \(V\to W\), giving its sections on \(Y\) and on \(\{1\}\). There are no nontrivial covering conditions on either nonempty open beyond this restriction: a cover of \(Y\) contains \(Y\), and a cover of \(\{1\}\) contains \(\{1\}\). Thus all arrows occur.

The complete category-valued stackification and ordinary inverse-image construction, including the adjunction and stalk comparison, are proved in Stackification, local morphisms and ordinary inverse images, Sections 2–6. The ordinary inverse-image prestack pulls a vector space \(V\) to its constant sheaf \((V\xrightarrow{1}V)\); its stackification adds objects locally of this form. Since a cover of \(Y\) itself contains \(Y\), an object locally of this form on \(Y\) is already of that form there. The object \((k\to0)\) cannot be isomorphic to one: its two vector spaces would force \(V\simeq k\) and \(V\simeq0\). In contrast, (4.2) is the full stack of modules over the constant ring, so includes \((k\to0)\). The transported-algebra construction is therefore necessary even for this continuous map.

Exercise 8.2 (calculation). In the same example let the relative bimodule be \(P=(k\to0)\). Compute \(T_P\) and its right adjoint on \(N=(N_0\xrightarrow aN_1)\). Verify the Hom bijection and both triangles explicitly.

Solution. The source vector space pulls back to \((V\xrightarrow1V)\), so \(T_P(V)=(V\to0)\). A morphism from this arrow to \(N\) consists of \(b:V\to N_0\) with \(ab=0\), and the unique map from zero. It is exactly a map \(V\to\ker a\). Hence \(H_P(N)=\ker a\), naturally in both arguments, agreeing with (3.1).

The unit at \(V\) is identity \(V\to\ker(V\to0)=V\). The counit at \(N\) is the arrow map \((\ker a\to0)\to(N_0\to N_1)\) given by inclusion and zero; it commutes because \(a\) kills its kernel. On \(T_P(V)\), counit after the image of the unit is identity on \(V\) and on zero. On \(H_P(N)\), the kernel of the counit map is the identity of \(\ker a\), after the unit identity. These prove the two triangles. Thus relative reconstruction includes this noninvertible, restriction-sensitive bimodule.

Exercise 8.3 (synthesis). Keep the same map but use the regular bimodule \(P=(k\xrightarrow1k)\). Compute its right adjoint, unit and counit, and explain how its counit differs from that of Exercise 8.2.

Solution. Now \(T_P(V)=(V\xrightarrow1V)\). A map from it to \(N=(N_0\xrightarrow aN_1)\) is a pair \((b_0,b_1)\) with \(ab_0=b_1\). Thus it is freely determined by \(b_0:V\to N_0\), and \(H_P(N)=N_0\). This is sheaf direct image to the point. The unit is identity \(V\to V\). The counit \((N_0\xrightarrow1N_0)\to(N_0\xrightarrow aN_1)\) has components \((1,a)\). Its image under \(H_P\) is identity on \(N_0\), proving that triangle; at the constant arrow its second component is identity too, proving the other triangle.

Exercise 8.2 used a bimodule whose restriction at \(\{1\}\) vanishes. Its relative right adjoint consequently tests whether the global value is killed by the restriction map. The regular bimodule instead gives the ordinary inverse-image/direct-image adjunction and tests the whole global value \(N_0\). Both are instances of the same reconstruction, with no interchange of their two right adjoints.

Exercise 8.4 (obstruction). Let \(f:\{0\}\hookrightarrow\mathbb R\), let \(R\) be the constant sheaf \(\mathbb R\), and let \(Q\) be the sheaf of continuous real functions. Set \(P=\bigoplus_{n\geq1}R\). Prove that comparison (7.1) is not injective by using \(q_n(t)=\max(0,|t|-1/n)\).

Solution. On every open, a map from the direct sum \(P\) to \(Q\) is specified by its values on the indicated basis vectors. These are arbitrary sections of \(Q\), so \([P,Q]_R=\prod_{n\geq1}Q\), with sectionwise products. The family \((q_n)\) is a section of this product on all of \(\mathbb R\). Each \(q_n\) vanishes near zero, so its individual germ at zero is zero.

The germ of the whole family is nevertheless nonzero. If it vanished, there would be a single neighborhood of zero on which every \(q_n\) vanished. Choose a nonzero \(t\) in that neighborhood and then \(n>1/|t|\); the value \(q_n(t)=|t|-1/n\) is positive, a contradiction. Thus it is a nonzero element of \((\prod_nQ)_0\).

Inverse image preserves the direct sum, so \(f^{-1}P=\mathbb R^{(\mathbb N)}\), and \(f^{-1}Q=Q_0\). A map between these point modules is determined by the sequence of its basis values in \(Q_0\). The comparison (7.1) sends the family germ to its individual germs, all zero. This gives a nonzero kernel element. The counterexample uses a common neighborhood for a product germ and separate neighborhoods for its component germs; that distinction persists even though sheaf inverse image is exact.

9. Further reading