Draft lesson.

Coefficient morphism sheaves and local isomorphisms

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

A local morphism specifies maps on every further restriction. Its component on the largest region can forget much of that information. We construct the sheaf of these morphisms for a general coefficient category, prove its gluing and naturality, and use it to glue local inverses. Self-overlaps matter when a covering arrow is not monic.

Prerequisites: Natural transformations and composition of functors, Sections 1–2, and the coefficient matching-object definition. For the module specialization, compare Tensor and Hom for twisted module stacks, Lemma 1.1.

1. Matching limits and maps on slices

Fix a universe. Let \(\mathcal C\) be a small site specified by small covering families. Singleton isomorphisms cover, covering composites cover, and pullbacks of covering families along arbitrary arrows exist and cover. Only fibre products involving covering arrows are required. Empty covers are included wherever specified. Let \(\mathcal A\) be locally small in this universe.

For a coefficient presheaf \(E\) and a cover \(R=\{u_i\colon U_i\to U\}\), a matching object \(M_R(E)\) represents compatible families of arrows into the \(E(U_i)\): for every \(B\in\mathcal A\), maps \(B\to M_R(E)\) correspond bijectively to families \(b_i\colon B\to E(U_i)\) with equal composites into every \(E(U_i\times_UU_j)\). Its maps \(p_i\colon M_R(E)\to E(U_i)\) are jointly monic, by uniqueness of that factor. If products and equalizers exist, this is their usual matching equalizer. For an empty cover it is terminal.

We call \(E\) separated or a sheaf only when these matching objects exist for its covers. The restriction comparison \(\mu_{R,E}\colon E(U)\to M_R(E)\) must respectively be monic or invertible. The composite maps \(E(U)\to E(U_i)\) are therefore jointly monic for a separated \(E\): equality after them gives equality after every \(p_i\), then equality after \(\mu_{R,E}\), and finally equality of the original arrows. Finite matching limits suffice if all generating covers are finite.

For \(U\in\mathcal C\), let \(\mathcal C/U\) be the slice. Its objects are arrows \(v\colon V\to U\), and an arrow from \(w\colon W\to U\) to \(v\) is \(a\colon W\to V\) with \(va=w\). Write \(F|_U\) for the presheaf on this slice obtained from \(F\) by forgetting the label into \(U\). Set

Thus an element \(\phi\) has a component \(\phi_v\colon F(V)\to E(V)\) for every label \(v\colon V\to U\), with

Different labels into \(U\) are different slice objects, even when their domains coincide. The set (1.1) is small: the slice has a small object set, and its transformation set is the subset of the small product of the component Hom sets satisfying (1.2). The coefficient category need not have a small object set.

For \(t\colon T\to U\), postcomposition gives \(\mathcal C/T\to\mathcal C/U\), \(v\mapsto tv\). Precomposing a transformation with this functor defines its restriction to \(T\). Identity and composition hold on every component, so \(\mathcal H(F,E)\) is a set presheaf. No full-faithfulness assertion about this postcomposition functor is needed. The canonical identification \((\mathcal C/U)/(T\xrightarrow t U)\simeq\mathcal C/T\) shows that this is also the intrinsic morphism presheaf on each slice.

2. Separatedness and the actual gluing factor

Proposition 2.1. If \(E\) is separated, then \(\mathcal H(F,E)\) is separated for every coefficient presheaf \(F\).

Proof. Suppose two transformations on \(U\) restrict equally to a cover \((U_i\to U)\). At any \(v\colon V\to U\), pull it back to \(p_i\colon V_i=V\times_UU_i\to V\), with second projection \(q_i\colon V_i\to U_i\). Naturality gives

The right sides agree because \(vp_i=u_iq_i\) and the restricted transformations agree at \(q_i\). The restriction maps into the \(E(V_i)\) are jointly monic, so \(\phi_v=\psi_v\). This holds at every label, proving equality of transformations. If the cover is empty, its pullback is empty and the separated comparison \(E(V)\to1\) is monic; hence any two arrows into \(E(V)\) agree. The same argument covers this case. \(\square\)

Theorem 2.2. If \(E\) is a coefficient sheaf, then \(\mathcal H(F,E)\) is a sheaf of sets for every coefficient presheaf \(F\).

Proof. Let \(R=(u_i\colon U_i\to U)\) be a cover, and let \(\phi_i\in\mathcal H(F,E)(U_i)\) have equal restrictions along both projections of each overlap \(U_i\times_UU_j\). This includes \(i=j\). For a label \(v\colon V\to U\), use the pulled-back cover \(p_i\colon V_i\to V\) and form actual coefficient arrows

They are matching. On \(W=V_i\times_VV_j\), with maps \(r_i,r_j\) to the two factors, there is a map \(W\to U_i\times_UU_j\). The overlap compatibility equates \((\phi_i)_{q_ir_i}\) and \((\phi_j)_{q_jr_j}\). Naturality of the local transformations therefore equates \(E(r_i)a_{v,i}\) and \(E(r_j)a_{v,j}\), since \(p_ir_i=p_jr_j\). The universal property gives

Here the \(p_i\) in the second equation denote matching-object projections. The proposed global component is

It is constructed inside \(\mathcal A\). A matching object for \(F\) is unnecessary.

We check naturality at every arrow \(a\colon(W,w)\to(V,v)\), so \(va=w\). It induces \(\widetilde a_i\colon W_i\to V_i\) between the pulled-back covering members, over \(U_i\). If \(p_i^W,p_i^V\) are their maps to \(W,V\), then \(ap_i^W=p_i^V\widetilde a_i\). Composing the desired equation \(E(a)\phi_v=\phi_wF(a)\) with \(E(p_i^W)\) gives

The sheaf restrictions of \(E\) are jointly monic, proving (1.2). For an empty pulled-back cover \(E(W)\) is terminal, giving the same equality. Thus (2.4) is a natural transformation on the whole slice.

We must recover each \(\phi_i\) on its whole slice. Fix a label \(k\colon W\to U_i\), and regard \(u_ik\) as a label into \(U\). Pull \(R\) back to \(W\). Each member \(W_j\to W\) has a map to \(U_i\times_UU_j\), with its first label \(k\) restricted from \(W\). Overlap compatibility and local naturality show that the restrictions of \((\phi_i)_k\) are exactly the maps prescribed by (2.2) for \(u_ik\). Joint monicity of the restrictions into \(E(W_j)\) equates it with \(\phi_{u_ik}\). Hence the global transformation restricts to the entire chosen \(\phi_i\), not just its component at \(U_i\). This uses self-overlap compatibility when \(j=i\).

Proposition 2.1 gives uniqueness. If \(R\) is empty, it pulls back to an empty cover at every label \(V\to U\); each \(E(V)\) is terminal. The unique component maps from \(F(V)\) form the unique transformation. Thus the empty matching family has exactly one gluing too. We have proved both inverse directions of the set sheaf comparison. \(\square\)

3. Composition and local inverses

For maps \(s\colon F'\to F\) and \(t\colon E\to E'\), componentwise precomposition and postcomposition give natural maps of the corresponding morphism presheaves. They commute with every restriction. Vertical composition gives

For coefficient sheaves this is a map of set sheaves. Associativity and units hold at each slice component, so hold as sheaf identities. In particular the identity transformations give compatible sections over all regions.

Theorem 3.1. For coefficient sheaves \(F,E\), the presheaf

is a sheaf of sets. A fixed morphism of coefficient sheaves is an isomorphism on a slice if its restrictions are isomorphisms after a covering of that slice.

Proof. Restrictions of a natural isomorphism preserve its inverse componentwise. Compatible local isomorphisms glue by Theorem 2.2. Their inverses agree on overlaps, since they are inverses of the same restricted transformation. Apply the same theorem to \(\mathcal H(E,F)\) to glue the inverses. Both composites restrict to identities. Separation of \(\mathcal H(F,F)\) and \(\mathcal H(E,E)\) makes them identity transformations globally. The glued map is therefore an isomorphism; uniqueness follows from the separated morphism sheaf. This also proves the empty-cover case, where both coefficient sheaves have terminal values at every slice label.

For a fixed morphism whose restrictions are isomorphisms, glue their inverse transformations by this argument. The two composites equal the identities after the cover, hence globally. Apply the same reasoning on \(\mathcal C/U\) to obtain the slice statement. A restriction being an isomorphism means that every component on its slice is invertible, by the natural-inverse lemma in the prerequisites. A bijection only at the covering member need not establish this condition. \(\square\)

Local existence of unrelated isomorphisms is not a compatible descent datum. Theorem 3.1 glues chosen isomorphisms with their actual overlap equations, or inverses of one fixed morphism. Exercise 4 exhibits incompatible local choices.

4. Additive coefficient categories

Suppose \(\mathcal A\) is preadditive: each Hom set is an abelian group and composition is additive in both variables. The transformations in (1.1) form an abelian group under componentwise operations. Naturality is preserved under sums and negatives because composition is additive. Restrictions are group homomorphisms. For a sheaf target, the bijective set sheaf comparison of Theorem 2.2 is therefore a bijective group homomorphism, hence a group isomorphism. Thus \(\mathcal H(F,E)\) is a sheaf of abelian groups.

If \(\mathcal A\) is linear over a commutative ring \(k\), the same component argument includes scalar multiplication and makes it a sheaf of \(k\)-modules. Composition (3.1) is bilinear. This applies to module coefficients with their valid ground-ring linearity, and agrees with the module Hom construction in the linked prerequisite. It does not assign an action of a noncommutative coefficient ring to its module Hom groups merely by multiplying outputs.

5. Four graded exercises with complete solutions

Exercise 1 — basic. Let \(X=\{a,b\}\) have open sets \(\varnothing,V=\{b\},X\). Define set sheaves \(A,B\) by \(A(X)=\varnothing\), \(A(V)=\{*\}\), \(B(X)=\{*\}\), \(B(V)=\{0,1\}\), and singleton values at the empty set. The restriction \(B(X)\to B(V)\) has value zero. Compute \(\mathcal H(A,B)(X)\) and its evaluation at the identity label. Then replace \(A\) by the terminal sheaf and explain why a bijection at \(X\) need not make its morphism to \(B\) an isomorphism.

Solution. Every cover of \(X\) contains \(X\) and every cover of \(V\) contains \(V\). Compatibility with that member determines the other local sections; singleton empty values give the empty-cover condition. Consequently the prescribed presheaves are sheaves. A transformation \(A\to B\) has the unique component \(\varnothing\to\{*\}\) at \(X\), either map \(\{*\}\to\{0,1\}\) at \(V\), and the unique map at the empty set. The naturality square from \(X\) to \(V\) imposes no further condition because its source is empty. Thus \(\mathcal H(A,B)(X)\) has two elements, while \(\operatorname{Hom}(A(X),B(X))\) has one; evaluation forgets the smaller-region component.

For the terminal source, its unique map to \(B\) has a bijective component \(\{*\}\to\{*\}\) at \(X\). Naturality forces its component at \(V\) to be the map with value zero, which is not surjective onto \(\{0,1\}\). Hence it is not a natural isomorphism, despite the bijective component at the sole member of the identity cover of \(X\). The local hypothesis in Theorem 3.1 concerns the transformation on every further restriction.

Exercise 2 — intermediate. On the discrete two-point space \(X=\{0,1\}\), take the coefficient category \(\{0<1\}\) as a poset. Let \(F(U)=1\), and let \(E(U)=\bigwedge_{x\in U}e_x\), with \(e_0=0\), \(e_1=1\), and empty meet equal to one. Prove they are coefficient sheaves and compute \(\mathcal H(F,E)\).

Solution. A limit in this poset is the meet of its objects, and all parallel maps agree. For a cover, the matching object for \(E\) is the meet of the values on its members: the overlap objects impose no additional inequality, since each member value maps to its overlap value. This meet is \(\bigwedge_{x\in U}e_x=E(U)\), as the covering members have union \(U\). The same calculation gives value one for \(F\). On the empty cover both values are terminal, so both are sheaves.

A transformation over \(U\) exists precisely when there is a map \(1\to E(W)\) for every \(W\subset U\). Such a map exists exactly when \(E(W)=1\); when it exists it is unique and all naturality equations agree. Including \(W=U\) shows that this happens exactly when \(U\subset\{1\}\). Therefore \(\mathcal H(F,E)(U)\) is a singleton for \(U\subset\{1\}\) and empty otherwise. It is the subterminal set sheaf represented by the open set \(\{1\}\). The calculation uses the coefficient order and matching limits, without choosing elements of the coefficient objects.

Exercise 3 — advanced. On the same discrete two-point space, let \(C_S\) be the presheaf with value \(S\) on nonempty opens, singleton value on the empty open, and identity restrictions between nonempty opens. First use a terminal source and target \(C_{\{0,1\}}\) to show that the target-sheaf hypothesis is needed. Then use source \(C_{\{0,1\}}\) and the sheaf \(G(U)=\{0,1\}^{U}\) to show that the source need not be a sheaf.

Solution. For a terminal source and any set presheaf \(P\), a transformation on \(\mathcal C/U\) is determined by its value at the identity label: all other selected sections are its restrictions. Conversely any element of \(P(U)\) gives precisely this natural family. Thus \(\mathcal H(1,P)\simeq P\), including at the empty set. The two singleton opens give a cover of \(X\). Its comparison for \(C_{\{0,1\}}\) is the diagonal \(\{0,1\}\to\{0,1\}^2\), which misses \((0,1)\) and \((1,0)\). Hence both this target and its morphism presheaf fail the sheaf condition.

For the second pair, \(G\) is a sheaf because compatible functions glue uniquely on a union of opens. The source has the same non-surjective diagonal and is not a sheaf. Nevertheless a local transformation at \(\{x\}\) is any function \(h_x\colon\{0,1\}\to\{0,1\}\). Given independent choices at the two points, define the component at \(X\) by \(s\mapsto(h_0(s),h_1(s))\), and use \(h_x\) at each singleton and the unique empty component. The coordinate restrictions verify every naturality square. This is the unique gluing. More generally \(\mathcal H(C_{\{0,1\}},G)(U)\) is the set of families \((h_x)_{x\in U}\), with ordinary restriction. Its value at \(X\) has \(4^2=16\) elements. Target gluing suffices even though source sections cannot all be assembled.

Exercise 4 — synthesis. Use the small category of labelled finite sets and all functions, with finite jointly surjective covering families. For finite sets \(S,T\), put \(F_S(W)=\operatorname{Map}(W,S)\) and \(F_T(W)=\operatorname{Map}(W,T)\). Compute \(\mathcal H(F_S,F_T)(U)\). For \(S=T=\{0,1\}\), study the singleton cover \(v\colon\{0,1\}\to\{*\}\). Can independently chosen identity and swap maps on its two fibres descend?

Solution. The represented function presheaves are sheaves: a matching family of functions on a jointly surjective cover gives one value at each point of the target, independent of its chosen preimage by the equations on pairwise fibre products, including self-products. This produces the unique glued function. The empty-cover case is the unique function on the empty set.

For a transformation over \(U\), evaluate at the one-point object labelled by each \(x\in U\). It gives a function \(h_x\colon S\to T\). For \(w\colon W\to U\), \(a\colon W\to S\), and \(y\in W\), naturality at the labelled one-point inclusion forces

Conversely this formula defines a function on every \(W\) and commutes with precomposition by every arrow over \(U\). The two constructions are inverse, so \(\mathcal H(F_S,F_T)(U)=\operatorname{Map}(U,\operatorname{Map}(S,T))\), with restrictions by precomposition on \(U\). Empty source or target sets satisfy the same formula, since each required component function is accounted for by the one-point test; at an empty region the transformation is unique.

In the stated singleton cover, a local transformation is a pair \((h_0,h_1)\), with 16 choices. Its self-overlap is \(\{0,1\}\times\{0,1\}\). The two restrictions have respectively values \(h_i\) and \(h_j\) at \((i,j)\), so compatibility is exactly \(h_0=h_1\). Only four local transformations descend, corresponding to the four global functions \(S\to T\). Taking identity on one fibre and swap on the other gives an invertible local transformation on the whole slice over \(\{0,1\}\), because (5.1) has pointwise inverses. It fails the self-overlap equation and cannot descend as that chosen transformation. For local isomorphisms there are four pairs but only the two constant pairs descend. Omitting \(i=j\) when there is a single nonmonic covering member would lose the entire compatibility test.

References

• Pierre Schapira, An Introduction to Sheaves on Grothendieck Topologies, version 1 August 2026, Theorem 1.5.5 and Corollary 1.5.6. The construction above gives all slice naturality and restriction equations explicitly under the required matching-limit hypotheses.
• The linked course prerequisites give natural transformation size, restriction and inverse formulas, the coefficient matching convention, and the existing module specialization.

Original exposition and worked solutions in this lesson: CC0-1.0.