Homotopy colimits and global observables in Abelian gauge theory

We study chain complexes of field configurations and observables for Abelian gauge theory on contractible manifolds, and show that they can be extended to non-contractible manifolds by using techniques from homotopy theory. The extension prescription yields functors from a category of manifolds to suitable categories of chain complexes. The extended functors properly describe the global field and observable content of Abelian gauge theory, while the original gauge field configurations and observables on contractible manifolds are recovered up to a natural weak equivalence.


Introduction and summary
in the category Sets. Here denotes the categorical product in Sets and as usual we denote the intersections by U ij := U i ∩ U j .
Another essential ingredient of a classical field theory is the characterization of the observables of the theory, which is usually done by specifying for each manifold M a suitable algebra A(M ) of functions on F(M ). 1 Following [BFV03], an important guiding principle for the choice of the observable algebras A(M ) is the requirement of functoriality of the assignment A : Man → Alg, where Alg is a suitable category of algebras whose details depend on the context. Another reasonable requirement is the cosheaf property of A, which would allow us to capture all information about the global observables A(M ) on a manifold M in terms of the local observables in an open cover {U i : i ∈ I} of M . More precisely, the cosheaf property demands that the algebra A(M ) can be recovered (up to isomorphism) from any open cover {U i : i ∈ I} of M by taking the colimit in the category Alg. Here denotes the categorical coproduct in Alg.
When studying explicit examples of classical (and also quantum) field theories it might very well happen that one can construct rather easily the field configurations F(M ) and the observables A(M ) of the theory on a special class of manifolds, e.g. on the category of contractible manifolds Man c , but that the construction becomes much more involved for non-contractible manifolds. Reasons for this might be global features, such as non-trivial bundles and topological charges, which are related to topologically non-trivial manifolds. In such a situation one would obtain two functors F : Man op c → Sets and A : Man c → Alg describing the field configurations and observables of the theory only on contractible manifolds, and the goal is to then 'extend' these functors to the category of all manifolds Man. In view of the desired sheaf and cosheaf properties, a reasonable procedure for obtaining such extensions is to define the field configurations F(M ) and the observables A(M ) on a generic manifold M in terms of the limit or, respectively, the colimit of a diagram induced by a suitable open contractible cover of M . In the context of algebraic quantum field theory, such a procedure has been proposed by Fredenhagen and it is called the 'universal algebra', see e.g. [Fre90,Fre93,FRS92].
In gauge theories the structures discussed above become considerably more complicated. First of all, the gauge field configurations on a manifold M are not described by a set, but by a groupoid. For example, the field configurations of gauge theory with structure group G on a manifold M are described by the groupoid with objects given by all principal G-bundles over M endowed with a connection and morphisms given by all principal G-bundle isomorphisms compatible with the connections (i.e. gauge transformations). Instead of forming a sheaf, the collection of these groupoids for all manifolds M forms a stack, see e.g. [Fan01,Vis05] for an introduction. For our purposes, a more explicit and also more suitable characterization of stacks in terms of homotopy sheaves of groupoids has been developed by Hollander [Hol08a]. A stack is the same thing as a functor G : Man op → Groupoids to the category of groupoids which satisfies the homotopy sheaf property, i.e. for any manifold M the groupoid G(M ) can be recovered (up to weak equivalence) from any open cover {U i : i ∈ I} of M by taking the homotopy limit in the category Groupoids. 2 Note that forming the coarse moduli spaces (i.e. the gauge orbit spaces) of a homotopy sheaf in general does not yield a sheaf. Hence the groupoid point of view is essential for 'gluing' local gauge field configurations to global ones.
Classical observables for gauge theories may be described by taking suitable 'function algebras' on groupoids, which can be modeled by cosimplicial algebras or differential graded algebras, see Section 2 below for details. A natural requirement for the choice of such 'function algebras' is again functoriality, i.e. we seek a functor B : Man → cAlg to the category of cosimplicial algebras (or a functor B : Man → dgAlg to the category of differential graded algebras). Instead of the cosheaf property which appears in ordinary field theories, this functor should satisfy the homotopy cosheaf property, i.e. the cosimplicial (or differential graded) algebra B(M ) can be recovered (up to weak equivalence) from any open cover {U i : i ∈ I} of M by taking the homotopy colimit in the category cAlg (or in the category dgAlg). 3 In gauge theories we are exactly in the situation where the groupoids of field configurations are rather simple and explicit for contractible manifolds M , while they are much harder to describe for non-contractible manifolds. This is because on a contractible manifold M all principal G-bundles are isomorphic to the trivial principal G-bundle M × G → M . Consequently, gauge field observables on contractible manifolds are also much easier to describe than those on noncontractible manifolds. Hence we can rather easily get two functors G : Man op c → Groupoids and B : Man c → cAlg (or B : Man c → dgAlg) which describe gauge field configurations and observables only on contractible manifolds. The goal is then to extend these two functors to the category Man by taking homotopy limits or, respectively, homotopy colimits of diagrams induced by suitable open contractible covers of manifolds M . From the perspective of algebraic quantum field theory, these constructions may be interpreted as a gauge theoretic (or homotopy theoretic) version of Fredenhagen's 'universal algebra' construction. Let us emphasize again the importance of describing gauge field configurations in terms of groupoids and observables in terms of cosimplicial (or differential graded) algebras, instead of working with gauge orbit spaces and gauge invariant observable algebras. As an explicit example of what may go wrong when not doing so, see [DL12,FL14] where the 'universal algebra' has been constructed for gauge invariant observable algebras of Abelian gauge theory. These 'universal algebras' fail to produce the correct global gauge invariant observable algebras [BSS14] because they neglect flat connections and violate the quantization condition for magnetic charges in the integral cohomology H 2 (M, Z). See also [BDHS14,BDS14] for a presentation of the global gauge invariant observable algebras for a fixed but arbitrary principal bundle and [FP03,DS13,SDH14,CRV13,Ben14] for the trivial principal bundle. Certain aspects of non-Abelian gauge theories and also gravity in this context have been studied in [Hol08b,FR13,CRV12,BFR13,Kha14,Kha15].
In this paper we shall make explicit and test the above ideas for constructing global gauge field configurations and observables by homotopy theoretic techniques. We shall consider the simplest example of a classical gauge theory, namely that whose structure group is the circle group G = T = U (1). From the perspective of differential cohomology, there already exist several models for the groupoid of gauge potentials in this case which have been discussed from the perspective of 'locality' of (generalized) Abelian gauge theories: The category of differential cocycles constructed by [HS05] is based on singular cochains (see also [Sza12,Section 2.4]), while theČech theoretic model of [FW99] is somewhat closer in spirit to our approach (see also [Fre00,Example 1.11]); see also [BM06] for a more heuristic proposal. To simplify our constructions, we shall study this gauge theory on a purely kinematical level, leaving out both dynamical aspects (i.e. Maxwell's equations) and quantization for the moment.
The outline of the remainder of this paper is as follows: In Section 2 we give an explicit and very useful description of the groupoids of gauge field configurations on contractible manifolds in terms of chain complexes of Abelian groups. This formulation allows us to identify a simple class of gauge field observables, given by smooth group characters, which also forms a chain complex of Abelian groups. Our constructions are functorial in the sense that we obtain a functor C : Man op c → Ch ≥0 (Ab) describing gauge field configurations and a functor O : Man c → Ch ≤0 (Ab) describing observables on the category of contractible manifolds Man c . In Section 3 we shall construct an extension of the functor C : Man op c → Ch ≥0 (Ab) to the category Man of all manifolds by using homotopy limits. For this we first show that any manifold M has a canonical open cover by contractible subsets, which induces a canonical diagram of gauge field configurations on M . We compute explicitly the homotopy limit of this diagram and show that it is isomorphic to the Deligne complex in the canonical cover. This will imply that our homotopy limit describes all possible gauge field configurations on M , including also non-trivial principal T-bundles whenever H 2 (M, Z) = 0. As the canonical cover is functorial, it is easy to prove that the global field configurations given by the homotopy limits are described by a functor C ext : Man op → Ch ≥0 (Ab). We shall show that this functor is an extension (up to a natural quasi-isomorphism) of our original functor C : Man op c → Ch ≥0 (Ab). In Section 4 we shall focus on the gauge field observables and construct an extension of the functor O : Man c → Ch ≤0 (Ab) to the category Man by using homotopy colimits. Similarly to the gauge field configurations, we obtain a canonical diagram of gauge field observables on any manifold M and we compute explicitly the homotopy colimits. Functoriality of the global observables O ext : Man → Ch ≤0 (Ab) is again a simple consequence of functoriality of the canonical cover. We then show that this functor is an extension (up to a natural quasiisomorphism) of our original functor O : Man c → Ch ≤0 (Ab). Finally, we construct a natural pairing between global gauge field configurations and observables, which allows us to show that our class of observables separates all possible gauge field configurations. Two appendices at the end of the paper summarize some of the more technical details which are used in the main text. In Appendix A we review the (dual) Dold-Kan correspondence, which is an important technical tool for our constructions. In Appendix B we summarize the explicit techniques to compute homotopy (co)limits for chain complexes of Abelian groups given in [Dug, Section 16.8] and [Rod14].

Local gauge field configurations and observables
In this section we consider gauge fields on contractible manifolds; in this paper all manifolds considered are oriented.

Groupoids and cosimplicial algebras
Let G be a (matrix) Lie group, g its Lie algebra and M a contractible manifold. Then all principal G-bundles over M are isomorphic to the trivial G-bundle, and the field configurations of gauge theory with structure group G on M are described by the g-valued one-forms Ω 1 (M, g); elements A ∈ Ω 1 (M, g) are typically called 'gauge potentials'. Recall that gauge theory comes with a distinguished notion of gauge group, the group of vertical automorphisms of the principal G-bundle. In the present case the gauge group may be identified with the group of G-valued smooth functions C ∞ (M, G) and it acts on gauge potentials from the left via where d denotes the exterior derivative.
Having available both gauge potentials and gauge transformations, one can ask which mathematical structure is suitable for describing the relevant field content of gauge theory on M . The most obvious option is to take the orbit space Ω 1 (M, g)/C ∞ (M, G) under the ρ-action, which identifies all gauge potentials that differ by a gauge transformation; this is often called the 'gauge orbit space'. However, there are several problems with the orbit space construction: First, even though both Ω 1 (M, g) and C ∞ (M, G) can be equipped with a suitable (locally convex infinite-dimensional) smooth manifold structure, the orbit space is in general singular [ACMM86,ACM89]. Second, forming the orbit space inevitably leads to a substantial loss of information; even though we can still decide whether or not two gauge potentials A and A ′ are gauge equivalent, we cannot keep track of the gauge transformation g that identifies A with A ′ when they are equivalent. The latter information is essential whenever one wants to obtain global field configurations of gauge theory on a topologically non-trivial manifold M by 'gluing' local configurations in contractible patches. A classic example is Dirac's famous magnetic monopole which represents the Chern class in Abelian gauge theory with structure group the circle group G = T = U (1): Its construction is based on gauge potentials A 1 and A 2 on an open cover {U 1 , U 2 } of a topologically non-trivial manifold M subject to the requirement A 2 | U 12 − A 1 | U 12 = g 12 dg −1 12 for some fixed g 12 ∈ C ∞ (U 12 , T) on the overlap U 12 = U 1 ∩ U 2 . This operation of 'gluing up to gauge transformations' cannot be described in terms of gauge orbit spaces.
In order to solve these and other problems, a more modern perspective suggests that, instead of looking at gauge orbits, one should organize the gauge potentials and gauge transformations into a groupoid. Recall that a groupoid is a small category in which every morphism is invertible. The groupoid corresponding to gauge theory with structure group G on a contractible manifold M is simply the action groupoid C ∞ (M, G) ⋉ Ω 1 (M, g) ⇒ Ω 1 (M, g): The set of objects is Ω 1 (M, g) and the set of morphisms is C ∞ (M, G) × Ω 1 (M, g), an element (g, A) of which should be interpreted as an arrow starting at A and ending at the gauge transform (2.1) of A by g. Two morphisms (g ′ , A ′ ) and (g, A) are composable whenever A ′ = ρ(g, A) and the composition reads as (g ′ , , where e is the identity element in C ∞ (M, G), i.e. the constant function to the identity element of the structure group G. As an aside, note that one can use the techniques of [ACMM86] to realize that this action groupoid is moreover a (locally convex infinite-dimensional) Lie groupoid, i.e. a groupoid carrying a smooth structure. This smooth structure will not play a role in the present paper, since we will shortly restrict ourselves to Abelian gauge theory with structure group G = T, which can be studied in purely algebraic terms. However, in studies of non-Abelian gauge theory the smooth structures will play an important role.
Instead of using groupoids, we may equivalently organize the gauge potentials and gauge transformations into a simplicial set (or even a simplicial manifold if we use the smooth structure discussed above). Recall that a simplicial set is a collection {S n } n∈N 0 of sets together with face maps ∂ n i : S n → S n−1 , for n ≥ 1 and i = 0, 1, . . . , n, and degeneracy maps ǫ n i : S n → S n+1 , for n ≥ 0 and i = 0, 1, . . . , n, satisfying simplicial identities, see e.g. [GJ99, Section I.1]. The simplicial set corresponding to our groupoid (which is called its nerve) may be depicted as where the arrows are the face maps and we have suppressed the degeneracy maps for notational convenience. Explicitly, the face maps read as , for i = n , (2.4a) and the degeneracy maps read as The simplicial set perspective has the advantage of making clear how to describe gauge theory observables. Interpreting (2.3) as the simplicial set of gauge field configurations, it is natural to model (classical) observables as functions on it. This can be done by taking the algebra of complex-valued functions C( − , C) on each degree of (2.3). Doing so, a cosimplicial algebra is obtained by dualizing the face and degeneracy maps to co-face and co-degeneracy maps under the contravariant functor C( − , C) : Sets → Alg between sets and algebras. Restricting to infinitesimal gauge transformations, this picture reduces nicely to the well-known BRST formalism, see [FR12] for a presentation of this topic in the context of the algebraic approach to field theory. By the dual Dold-Kan correspondence (see Appendix A) we can regard our cosimplicial algebra as a differential graded algebra (dg-algebra), see e.g. [CC04]. This dg-algebra can be 'linearized' via a procedure called the van-Est map (here we require the smooth structure mentioned above) to yield the BRST algebra (Chevalley-Eilenberg dgalgebra) of gauge theory, see e.g. [Cra03]. It is important to stress that this linearization procedure neglects finite gauge transformations and hence leads to an incomplete description of gauge theory. Our cosimplicial algebra (or its associated dg-algebra) should be interpreted as an improvement of the usual BRST algebra, which keeps track of all gauge transformations and not only of the infinitesimal ones; in fact, finite gauge transformations are essential for gluing local field configurations to global ones. Using the analogy with the BRST formalism, we may call the factors C ∞ (M, G) in (2.3) the 'ghost fields'. Notice that these ghost fields are non-linear in the sense that they are functions with values in the structure group, while the ordinary ghost fields in the BRST formalism are described by the tangent space C ∞ (M, g) at the identity e ∈ C ∞ (M, G) and hence they are linear.

Abelian gauge theory
In the remainder of this paper we shall fix the structure group G = T with Lie algebra t = i R and hence consider only Abelian gauge theory. Then the structures described above simplify considerably. In particular, all sets appearing in (2.3) naturally become Abelian groups with respect to the direct product group structure given by Moreover, the action of the gauge group on gauge potentials (2.1) simplifies to ρ(g, A) = A + g dg −1 , and it is easy to show that the face and degeneracy maps (2.4) are group homomorphisms. It follows that the simplicial set (2.3) is a simplicial Abelian group, which under the Dold-Kan correspondence can be identified with a non-negatively graded chain complex of Abelian groups, see Appendix A. This chain complex is called the normalized Moore complex and in the present case it reads explicitly as where Ω 1 (M, t) sits in degree 0 and C ∞ (M, T) sits in degree 1. As a convenient sign convention, we shall take as the differential (of degree −1) in C(M ) the negative of the differential in the normalized Moore complex (A.2), i.e.
M being contractible, the homology H * of the chain complex C(M ) is given by which gives the gauge orbit space in degree 0 and the global constant gauge transformations in degree 1. Notice that the first homology group is not a vector space, but only an Abelian group. This feature naturally distinguishes between the Abelian gauge theories with structure groups G = T and G = R: Both theories have the same zeroth homology (i.e. the same gauge orbit space) on contractible manifolds, but they differ in the first homology which is always isomorphic to G.
For Abelian gauge theory we also obtain a distinguished class of observables: Since in this case (2.3) is a simplicial Abelian group, instead of all complex-valued functions C( − , C) on each degree, we can take only those functions which are group characters, i.e. homomorphisms of Abelian groups Hom Ab ( − , T) to the circle group. The group characters do not form an algebra, but rather an Abelian group called the character group; of course one can generate an algebra by the group characters, but this will not be done in the present paper. The character group should be interpreted as a generalization of the vector space of linear observables for a real scalar field theory, which also does not form an algebra, but which generates a polynomial algebra. Taking the character groups Hom Ab ( − , T) in each degree of (2.3) gives rise to a cosimplicial Abelian group because all face and degeneracy maps dualize to co-face and codegeneracy maps. Under the dual Dold-Kan correspondence this can be identified with a non-positively graded chain complex of Abelian groups, see Appendix A, which for our model explicitly reads as where − * := Hom Ab ( − , T). Here C ∞ (M, T) * sits in degree −1 and Ω 1 (M, t) * sits in degree 0, while the differential δ * (of degree −1) is the dual of the differential (2.7). Using the smooth character groups as in [BSS14], the chain complex (2.9) can be restricted to (2.11) We observe that (2.11) is a bi-character, i.e.
and that it is compatible with the gradings of O(M ) and C(M ) if we take the target T to sit in degree 0. The differential δ * in O(M ) is defined via the duality induced by (2.11); we compute where f * is the pull-back of functions/differential forms along f . Similarly, the chain complexes in (2. where f * is the push-forward of compactly supported differential forms along f . It follows that the pairing (2.11) is natural in the sense that the diagram with the same source and target category. The pull-back (2.15) then defines a natural transformation (denoted by the same symbol)

Homotopy limit of canonical diagrams
In this subsection we shall fix an arbitrary object M in Man and study the homotopy limit of the canonical diagram for all U ⊂ V , and for all U ⊂ V ⊂ W . The differential δ : C ext (M ) 1 → C ext (M ) 0 is given by

Deligne complex
We shall show that the chain complex C ext (M ), given by the homotopy limit (3.4), is isomorphic to the Deligne complex for the canonical cover D(M ) 0 of M ; see [Bry07,Bou10,Sza12] for details on the Deligne complex and Deligne cohomology. In the canonical cover D(M ) 0 , the Deligne complex reads as is the subgroup of all elements U A U × U,V g U V satisfying the conditions for all U, V , and for all U, V, W . The degree 1 component is given by and the differential δ Del : We define a Ch ≥0 (Ab)-morphism by setting the identity ψ 1 = id on U C ∞ (U, T) in degree 1 and in degree 0. Using (3.10), it is easy to show that the image of ψ 0 lies in C ext (M ) 0 , i.e. that the conditions (3.6) are fulfilled. Using also (3.7) and (3.12) one easily shows that ψ preserves the differentials, i.e. δ • ψ 1 = ψ 0 • δ Del .
Let us now define a Ch ≥0 (Ab)-morphism by setting the identity ϕ 1 = id on U C ∞ (U, T) in degree 1 and for all i ∈ I. Given now i, j ∈ I such that U i ∩ U j = ∅, there exists a subset K ⊆ I such that {U k : k ∈ K} is an open cover of U i ∩ U j . Given any element U k of that cover, the conditions (3.6b) imply that Hence ( g U V ) i and ( g U V ) j coincide on the overlap U i ∩U j . Using now the fact that C ∞ ( − , T) is a sheaf of Abelian groups, there exists an element for all i ∈ I. This is the element appearing on the right-hand side of (3.16). Using

Functoriality
We can assign to any object M in Man the chain complex C ext (M ) given by the homotopy limit (3.4). Using (3.3) and functoriality of the homotopy limit it immediately follows that this assignment is a functor (3.20) Explicitly, for any morphism f op : given in degree 0 by and in degree 1 by (3.22b)

Functor extension
We shall show that the functor C ext : Man op → Ch ≥0 (Ab) given in (3.20) is an extension of our original functor C : Man op c → Ch ≥0 (Ab), i.e. that there is a diagram of functors which commutes up to a natural transformation η. The functor Man op c → Man op is simply the full subcategory embedding. We further show that the natural transformation η is a natural quasi-isomorphism, so that the functors C and C ext give weakly equivalent descriptions of the gauge field configurations on contractible manifolds. Our extension C ext of C is distinguished by the fact that it gives a correct description of the global gauge field configurations on noncontractible manifolds, see (3.19).
For any object M in Man c , there is a Ch ≥0 (Ab)-morphism given by It remains to show that η is a natural quasi-isomorphism, i.e. that any component η M is a quasi-isomorphism in Ch ≥0 (Ab). For this, we define a (non-natural) Ch ≥0 (Ab)-morphism by setting The push-forward (2.16) then defines a natural transformation (denoted by the same symbol) where the coproducts are respectively over all objects U in D(M ) and over all proper subset inclusions U ⊂ V . The quotient in O ext (M ) 0 is by the Abelian subgroup I(M ) that is generated by the elements for all U ⊂ V and ϕ ∈ Ω m−1 c (U ), and for all U ⊂ V ⊂ W and χ ∈ Ω m c,Z (U ). Here ι − denote the inclusion morphisms in the coproducts and as before ext − denote the extension maps. The differential δ * : O ext (M ) 0 → O ext (M ) −1 is given by where we suppress the equivalence classes in O ext (M ) 0 .

Functoriality
We can assign to any object M in Man the chain complex O ext (M ) given by the homotopy colimit (4.3). Using (4.2) and functoriality of the homotopy colimit, it immediately follows that this assignment is a functor Explicitly, for any morphism f : given in degree 0 by and in degree −1 by It remains to show that ζ is a natural quasi-isomorphism, i.e. that any component ζ M is a quasi-isomorphism in Ch ≤0 (Ab). For this, we define a (non-natural) Ch ≤0 (Ab)-morphism (4.14) by setting Hence any Ch ≤0 (Ab)-morphism (4.11) is a quasi-isomorphism.  Notice that the pairing (4.17) is non-degenerate in the right entry, i.e. the observables O ext (M ) separate all possible global gauge field configurations C ext (M ) on M . In particular, when H 2 (M, Z) = 0, our homotopy colimit construction has produced enough observables to measure and distinguish all possible principal T-bundles on M .

A Dold-Kan correspondence and Moore complex
We shall briefly review the Dold-Kan correspondence between simplicial Abelian groups and non-negatively graded chain complexes of Abelian groups. In particular, we shall give explicit formulas for the normalized Moore complex, which is used at various instances in this paper. The dual Dold-Kan correspondence is an equivalence between the categories of cosimplicial Abelian groups cAb and non-positively graded chain complexes of Abelian groups Ch ≤0 (Ab).
For our purposes we only have to explain the functor N * : cAb → Ch ≤0 (Ab), which is called the co-normalized Moore complex. Let A = {A n } n∈N 0 be any cosimplicial Abelian group with co-face and co-degeneracy maps denoted by d i n : A n → A n+1 , for i = 0, 1, . . . , n + 1 and n ≥ 0, and e i n : A n → A n−1 , for i = 0, 1, . . . , n − 1 and n ≥ 1. Then the functor N * assigns to A the non-positively graded chain complex of Abelian groups Ker e i n : A n → A n−1 , for all n ≥ 0, and the differential δ * (of degree −1) is defined as the alternating sum of the co-face maps, i.e. we set Note that the normalized Moore complex (A.2) is still defined when we replace the category of Abelian groups Ab by other Abelian categories, such as the category of (possibly unbounded) chain complexes of Abelian groups Ch(Ab). In this case the normalized Moore complex N assigns to simplicial chain complexes of Abelian groups sCh(Ab) double chain complexes of Abelian groups Ch ≥0 (Ch(Ab)), where the first grading is non-negative. Similarly, the co-normalized Moore complex (A.3) is still defined when we replace the category of Abelian groups Ab by Ch(Ab). Then the co-normalized Moore complex N * assigns to cosimplicial chain complexes of Abelian groups cCh(Ab) double chain complexes of Abelian groups Ch ≤0 (Ch(Ab)), where the first grading is non-positive.

B Homotopy limits and colimits for chain complexes
We shall briefly explain how to compute homotopy limits and colimits of diagrams of chain complexes of Abelian groups. Our presentation follows mainly [Dug, Section 16.8], but we also refer the reader to [Rod14] for more technical details.

B.1 Homotopy limits for non-negatively graded chain complexes
Let D be a small category. Given any functor X : D → Ch ≥0 (Ab), which we interpret as a diagram in Ch ≥0 (Ab) of shape D, we would like to compute the homotopy limit holim(X). This construction is a three step procedure: First, one takes the cosimplicial replacement of the diagram X : D → Ch ≥0 (Ab), which gives a cosimplicial object in Ch ≥0 (Ab). Then one assigns a double chain complex in Ch ≤0 (Ch ≥0 (Ab)) via the co-normalized Moore complex, where the first grading is non-positive and the second grading is non-negative. Finally one forms the -total complex, which gives the homotopy limit holim(X) after truncation to non-negative degrees. We shall now explain these steps in more detail and give explicit formulas.
The nerve of the small category D is the simplicial set {D n } n∈N 0 , where D 0 is the set of objects in D and D n , for n ≥ 1, is the set of all composable n-arrows in D. For n ≥ 1, we shall denote an element of D n by an n-tuple (f 1 , . . . , f n ) of morphisms in D such that the source of f i is the target of f i+1 (i.e. the compositions f i •f i+1 exist). The face maps are given by composing two subsequent arrows (or throwing away the first/last arrow) and the degeneracy maps are given by inserting the identity morphisms. The cosimplicial replacement of X : D → Ch ≥0 (Ab) is the cosimplicial object in Ch ≥0 (Ab) given by d∈D 0 where the arrows are the co-face maps and we have suppressed the co-degeneracy maps for notational convenience. Here denotes the product in the category Ch ≥0 (Ab) and we have denoted by t(f ) the target of a morphism f in D. The co-face maps d i n : (f 1 ,...,fn)∈Dn X(t(f 1 )) → (f 1 ,. ..,f n+1 )∈D n+1 X(t(f 1 )), for n ≥ 0 and i = 0, 1, . . . , n + 1, are defined by using the universal property of the product; explicitly, for i = 0, and for i = n + 1, (f 1 ,...,fn)∈Dn X(t(f 1 )) π (f 1 ,. ..,fn) ' ' P P P P P P P P P d n+1 where π − are the projection morphisms from the products. The co-degeneracy maps e i n : (f 1 ,...,fn)∈Dn X(t(f 1 )) → (f 1 ,...,f n−1 )∈D n−1 X(t(f 1 )), for n ≥ 1 and i = 0, 1, . . . , n − 1, are also defined by using the universal property of the product; explicitly, for i = 0, 1, . . . , n − 1, for all n ≥ 1, where the second product is taken over all composable n-arrows (f 1 , . . . , f n ) such that none of the f i is an identity morphism. The vertical differential δ v : X * , * → X * −1, * is given by the alternating sum of the co-face maps, i.e. δ v = n+1 i=0 (−1) i d i n on X −n, * , and the horizontal differential δ h : X * , * → X * , * −1 is given by the product of the differentials in the chain complexes X(d), for d an object in D. The double complex X * , * may be visualized as

(B.5)
We now form the -total complex X Tot := n∈Z X Tot n , δ Tot := n∈Z p+q=n X p,q , δ Tot := δ v + (−1) p δ h (B.6) and we notice that X Tot is a Z-graded chain complex of Abelian groups, in particular it is nontrivial in negative degrees. The homotopy limit holim(X) of the diagram X : D → Ch ≥0 (Ab) is then the truncation of X Tot to non-negative degrees. Explicitly, holim(X) = for all n ≥ 1, and the differential is given by δ = δ Tot .

B.2 Homotopy colimits for non-positively graded chain complexes
Let D be a small category. Given any functor Y : D → Ch ≤0 (Ab), the homotopy colimit hocolim(Y) is constructed in a three step procedure: First, one takes the simplicial replacement of the diagram Y : D → Ch ≤0 (Ab), which gives a simplicial object in Ch ≤0 (Ab). Then one assigns a double chain complex in Ch ≥0 (Ch ≤0 (Ab)) via the normalized Moore complex, where the first grading is non-negative and the second grading is non-positive. Finally one forms the -total complex, which gives the homotopy colimit hocolim(Y) after truncation to nonpositive degrees. Notice that this is precisely the dual of the construction for homotopy limits presented in Subsection B.1. However, we find it useful to go through these steps in more detail and give explicit formulas.
Denoting as before the nerve of the small category D by {D n } n∈N 0 , the simplicial replacement of Y : D → Ch ≤0 (Ab) is the simplicial object in Ch ≤0 (Ab) given by We now form the -total complex for all n ≤ −1, and the differential is given by δ = δ Tot .