An Introduction to Simplicial Sets

An introduction to simplicial sets 1 Introduction This is an elementary introduction to simplicial sets, which are generalizations of ∆-complexes from algebraic topology. The theory of simplicial sets provides a way to express homotopy and homology without requiring topology. This paper is meant to be accessible to anyone who has had experience with algebraic topology and has at least basic knowledge of category theory. An important part of simplicial homology is the idea of using ∆-complexes instead of simplicial complexes (see [6, Ch. 2]). They allow one to deal with the combinatorial data associated with a simplicial complex (which is important for homology) instead of the actual topological structure (which is not). Another simplex-based homology theory is singular homology, whose singular maps (see [6, Ch. 2]) represent simplices in a given topological space. While singular maps have properties analogous to simplices, such as a sensible definition for the faces of a singular map, singular maps are not in general injective, which means the data for gluing the faces of a singular map together might not be able to be described as a ∆-complex. The theory of simplicial sets generalizes the idea of ∆-complexes to encompass other objects with simplex structure, such as singular maps. The theory provides a realization functor |−| from simplicial sets to topological spaces which preserves homotopy. This functor is left adjoint to the functor S which takes topological spaces and gives a simplicial set consisting of the singular maps. Perhaps the most difficult part for a newcomer to the subject of simplicial sets is getting used to the category theory involved. Because of this, this paper limits discussion to simplicial sets and algebraic topology. The author found [1] very useful when trying to understand the idea of simplicial sets and [4] illuminating for the derivation of the relations (8), (9), and (10). Much of the material comes from [3], but it was corroborated with [2] to determine what is modern notation. 1 2 Simplicial sets In this section, we define simplicial sets without providing motivation, and we describe the combinatorial data necessary for specifying a simplicial set. We then try to build intuition by bringing in the geometric notion of simplices from algebraic topology. We first define the category ∆, a visualization of which is Example 3. Definition 1. Let ∆ be the category whose objects are finite sets {0, 1, 2, . . . , n} and whose morphisms are order-preserving functions (i.e., functions f : {0, . . . , n} → {0, . . . , m} such that i ≤ j implies f (i) ≤ f (j), for 0 ≤ i, j ≤ n). Recall that a contravariant functor from a category C to a category D is a covariant functor from the opposite category C op (whose morphisms are reversed) to D. That is, a contravariant functor is a functor which reverses the directions of the morphisms. See [6, Ch. 2.3] for an overview of categories and functors. The central definition of this paper is the following. Definition 2. A simplicial set is a contravariant functor ∆ → Set. It is sometimes profitable to generalize to arbitrary categories C and define a simplicial object in C as a contravariant functor ∆ → C. The phrase “object in C” is replaced by a reasonable name when appropriate, leading to simplicial objects beyond simplicial sets, such as simplicial groups, which will be discussed when we get to homology of simplicial sets. A fact which greatly aids in describing a simplicial object is Proposition 5, which says that any morphism in the category ∆ is the composite of coface maps di : {0, . . . , n} → {0, . . . , n + 1} and codegeneracy maps si : {0, . . . , n + 1} → {0, . . . , n}, for 0 ≤ i ≤ n. The names of these two kinds of maps come from the names of the corresponding face maps and degeneracy maps on simplices which will be recalled later in this section. The coface map di is defined to be the unique order-preserving bijection {0, . . . , n} → {0, . . . , ı̂, . . . , n + 1}, where the notation ı̂ means i is omitted from {0, . . . , n + 1}. That is, di (j) = j if 0 ≤ j < i, and otherwise di (j) = j + 1. The codegeneracy map si is defined to be the map which duplicates i, which is to say si (j) = j if 0 ≤ j ≤ i, and si (j) = j − 1 if i < j ≤ n. Example 3. Objects in ∆ have a geometric realization (which should not be confused with the realization of a simplicial set) given by the covariant functor {0, . . . , n} 7→ |∆n |, where |∆n | is the standard n-dimensional simplex in Top given by P |∆n | = {(t0 , . . . , tn ) ∈ Rn+1 | ni=0 ti = 1 and ti ≥ 0}, and morphisms f∗ : |∆n | → |∆m | induced by f : {0, . . . , n} → {0, . . . , m} are defined by X f∗ (t0 , . . . , tn ) = (s0 , . . . , sm ), sj = ti . f (i)=j 2 That is, the ith vertex of |∆n | is sent to the f (i)th vertex of |∆m |, and the barycentric coordinates are mapped linearly. It is left to the reader to show that this is, in fact, a functor ∆ → Top. We see that the coface map di∗ sends |∆n | to the ith face of |∆n+1 | and that the codegeneracy map sj∗ sends |∆n | to |∆n−1 | by collapsing together vertices j and j + 1. As in [4, Ch. VII.5], we have the following decomposition lemma, which we use in Proposition 5 to show the morphisms in ∆ are the composites of all di and sj . Lemma 4. In ∆, any morphism f : {0, . . . , n} → {0, . . . , n0 } has a unique representation f = dik · · · di1 sj1 · · · sjh , (1) where the non-negative integers h, k satisfy n + k − h = n0 and the subscripts i and j satisfy n0 ≥ ik > · · · > i1 ≥ 0, 0 ≤ j1 < · · · < jh < n. Proof. A monotonically non-decreasing function f is determined by its image in {0, . . . , n0 } and by those j ∈ {0, . . . , n − 1} for which f does not increase (that is, j such that f (j) = f (j + 1)). Let i1 , . . . , ik , in increasing order, be the elements of {0, . . . , n0 } which are not in the image of f , and let j1 , . . . , jh be the elements of {0, . . . , n} where f does not increase. It follows that both sides of equation (1) are equal. Thus, any composite of two di ’s or sj ’s may be put into the form (1). This fact along with the definitions of di and sj give the following identities, whose verification is left to the reader. di dj = dj+1 di i ≤ j, (2) sj si = si sj+1  i j−1  d s sj di = 1   i−1 j d s i ≤ j, (3) if i < j if i = j, j + 1 otherwise. (4) Proposition 5. The morphisms of ∆ consist of all composites of all arrows di and sj subject to the relations (2), (3), and (4). Proof. By Lemma 4, any morphism in ∆ is a composite of di ’s and sj ’s, and the relations (2), (3), and (4) suffice to put any composite of di ’s and sj ’s into the form (1). In the opposite category ∆op , we label the morphisms corresponding to di and sj by di and sj , respectively. In category theoretic spirit, di is called a face map and sj is called a 3 degeneracy map. Since (−)op is a contravariant functor, we have the following identities from relations (2), (3), and (4): dj di = di dj+1 si sj = sj+1 si   sj−1 di di sj = 1   sj di−1 i ≤ j, i ≤ j, (5) (6) if i < j if i = j, j + 1 otherwise, (7) and, corresponding to Proposition 5, we have that all morphisms in ∆op are generated by composites of di and sj , subject to relations (5), (6), and (7). Thus, a contravariant functor from ∆ is completely specified by where it sends the objects {0, . . . , n} and by where it sends the di and sj . This gives us a more concrete definition for simplicial sets (and, likewise, simplicial objects by replacing “sets” with “objects from C”). To simplify notation, we write Xn = X{0, . . . , n} and confuse di and sj with Xdi and Xsj , respectively, when given a simplicial object X. Proposition 6. A simplicial set X is a collection of sets Xn for each n ≥ 0 together with functions di : Xn → Xn−1 and si : Xn → Xn+1 , for all 0 ≤ i ≤ n and for each n, satisfying the following relations: di dj = dj−1 di si sj = sj+1 si   sj−1 di di sj = 1   sj di−1 i < j, i ≤ j, if i < j if i = j, j + 1 otherwise. (8) (9) (10) This is the standard way to write down the data of a simplicial set according to [2]. The elements of X0 are called the vertices of the simplicial set, and elements of all Xn are called simplices. A simplex x is degenerate if x is in the image of some sj . Example 7. The standard ∆-simplex [0, 1, . . . , ] of the simplex |∆n | has a representation as a simplicial set, which we write as ∆n . The data for the simplicial set are as follows: ∆nm = {[k0 , k1 , . . . , km ] | 0 ≤ k0 ≤ . . . ≤ km ≤ n} di [k0 , . . . , km ] = [k0 , . . . , kˆi , . . . , km ] sj [k0 , . . . , km ] = [k0 , . . . , kj , kj , . . . , km ]. The relations (8), (9), and (10) are straightforward to check. For instance, di dj [k0 , . . . , km ] for i < j is [k0 , . . . , kˆi , . . . , kˆj , . . . , km ], and this is dj−1 di since removing ki first shifts the index of kj by one. Checking the other relations is done in a similar manner and is left to the reader. 4 This example illustrates where the names of the maps come from. The map di gives the ith face of a simplex, while sj gives the jth degeneracy (as in, sj [0, 1, . . . , n] = [0, . . . , j, j, . . . , n] is the (n + 1)-simplex which is degenerate by doubling the j-th vertex). For instance, ∆nn contains [0, . . . , n], and the di [0, . . . , n] are all the standard faces of the simplex. We denote this [0, . . . , n] by En . Note that this simplicial set is essentially the result of taking the smallest simplicial set which contains En as an n-simplex and which is closed under the expected face and degeneracy maps. When we look at the geometric realization functor, we will see that this definition of ∆n is, indeed, a reasonable choice. An alternative definition from [2, Ch. 1] is that ∆n is the contravariant functor given by hom∆ (−, {0, . . . , n}) : ∆ → Set, which takes maps f : {0, . . . , m} → {0, . . . , m0 } from ∆ to g defined by g(σ 0 (i)) = (σ ◦ f )(i), where σ ∈ hom∆ ({0, . . . , m}, {0, . . . , n}) and σ 0 ∈ hom∆ ({0, . . . , m0 }, {0, . . . , n}). Example 8. As an illustration of the degenerate simplices which are present, the simplicial set ∆0 contains one element in each ∆0m , namely [0, . . . , 0] with m zeros. As these examples indicate, there are many degenerate simplices in the data for a simplicial complex, and writing all these data can be unwieldy. However, the following proposition, modified from [1], shows that we only need to specify the nondegenerate simplicies since the degenerate simplicies are the images of composites of sj ’s under the relation (9). Proposition 9. For a simplex z in a simplicial set X there is a unique representation z = sjh · · · sj1 x where x is a nondegenerate simplex in X and the subscripts j satisfy j1 < · · · < jh . Proof. Suppose z is a degenerate simplex. Then z = si1 x1 for some simplex x in X. By induction, if xj is degenerate, we replace it with sij1 xj+1 , and therefore there is a representation z = si1 · · · sih x for some nondegenerate x. Suppose z has two representations z = Sx and z = S 0 x0 for composites of degeneracy maps S and S 0 and nondegenerate simplices x and x0 . If S = si1 · · · sik , let D = dik · · · di1 . ˜ and S˜0 be the result of applying (10) repeatedly so By (10), DS = 1, thus x = DS 0 x0 . Let D ˜ where D ˜ is a composite of face maps and S˜0 is a composite of degeneracy that DS 0 = S˜0 D ˜ 0 , it must be the case that S˜0 = 1. Thus, x is maps. Since x is nondegenerate and x = S˜0 Dx 0 0 a face of x . By symmetry, x is also a face of x, so it follows that x = x0 . The uniqueness of S and the condition on the subscripts follow from Lemma 4 when dualized. Thus, in our examples above, ∆n can be described by [0, . . . , n] along with its faces (and faces’ faces, etc.), as in the traditional description of a ∆-complex. A map of simplicial sets f : X → Y is a natural transformation of the functors X and Y . These are the morphisms of the category S of simplicial sets. Unraveling the definition of natural transformation, and tying into the second definition of a simplicial set, another way to write f is as a sequence of functions fn : Xn → Yn for each n ≥ 0 such that fn−1 di = di fn and fn+1 sj = sj fn . 5 3 Basic constructions This section gives several basic contructions on simplicial sets which are analogous to constructions on topological spaces, as in [3]. The degree to which some of these constructions hold as analogues is examined in Section 4. The cartesian product of simplicial sets is the categorical product in S. Explicitly, given simplicial sets X and Y , the product X × Y is given by (X × Y )n = Xn × Yn di (x, y) = (di x, di y) si (x, y) = (si x, si y). The relations (8), (9), and (10) for X × Y follow from them holding for X and Y . A subsimplicial set of a simplicial set X is a simplicial set Y which satisfies Yn ⊂ Xn for each n ≥ 0, and which inherits the same di ’s and sj ’s. In [3], this is also known as a subcomplex because simplicial sets there are called complexes (which is a use we avoid because “complex” is used already in use for ∆-complexes and chain complexes). A basepoint ∗ of X is a subsimplicial set consisting of a vertex and all its degeneracies; that is, ∗ is the inclusion of ∆0 into X via a map of simplicial sets. The union of subsimplicial sets X and Y of Z is a subsimplicial set, where (X ∪ Y )n = Xn ∪ Yn . The reader may verify that the resulting simplicial set is closed under the di and sj since the di and sj agree on each (X ∩Y )n . This suggests a construction for the simplicial set of an arbitrary ∆-complex by taking a union of constituent simplicial sets ∆n with relabeled vertices. The wedge product of simplicial sets X and Y is the union of ∗ × Y and X × ∗ as subsimplicial sets of X × Y , where ∗ is taken to be the basepoints of X and Y . The quotient X/Y for Y a subsimplicial set of X is the result of identifying the simplices in Yn together. The resulting maps di and sj after the quotient continue to satisfy the relations (8), (9), and (10). The boundary ∂∆n of ∆n is the smallest subsimplicial set of ∆n which contains all of the faces di En . While ∂∆n could provide a model for the (n − 1)-sphere, we may be more economical with simplices and define the n-sphere S n to be the quotient ∆n /∂∆n . This S n has two nondegenerate simplices: ∗ ∈ S0n and σn ∈ Snn , which is the image of En via the quotient. We note that di σn is a degeneracy of ∗, and we will see that the geometric realization of S n identifies these faces with the realization of ∗. 4 Realization In this section, we describe the realization functor |−| : S → Top, and we also describe the functor S : Top → S which is right adjoint to |−| and which connects singular homology to the theory of simplicial sets. We follow [3], but with more modern notation. 6 It is somewhat unfortunate that we are overloading our notation |−|, since we already used it for the covariant functor from ∆ to Top, but this is the practice in [1] and [2]. Positively, it ends up being the case that |∆n | as the image of the ∆ → Top functor is homeomorphic to |∆n | as the image of ∆n via the realization functor, so the notation is deliberately confused. Definition 10. Let X be a simplicial set, and give each Xn the discrete topology. Let |∆n | be the standard n-dimensional simplex in Rn+1 . The realization |X| of X is given by ! ∞ a |X| = Xn × |∆n | /(∼), n=0 where |X| is given the quotient topology, and where (∼) is an equivalence relation defined by (x, p) ∼ (y, q) if either • di x = y and di q = p; or • sj x = y and sj q = p. Here, di and sj are the coface and codegeneracy maps induced by the functor ∆ → Top. We now describe the action of the realization functor so we can see it makes sense for ∆-complexes and so we can see it is reasonable to confuse the notation for |−|. In an intuitive sense, the equivalence relation collapses degeneracies and glues together faces. Say x is an n-simplex in a simplicial set X, and let us treat X as the simplicial set of a ∆-complex. That is, di and sj have their geometric notions as face and degeneracy maps. First we will inspect the second relation: that (x, sj p) ∼ (sj x, p), where p ∈ |∆n+1 |. Recall that sj maps |∆n+1 | to |∆n | by collapsing the j and (j + 1)th vertices together, and sj maps x to the (n + 1)-simplex by doubling the (j + 1)th vertex. This relation captures the notion that a point p in a realization |∆n+1 | of sj x as ought to be collapsed via sj since sj x is degenerate. Now for the first relation: that (x, dj p) ∼ (dj x, p), for p ∈ |∆n−1 |. Recall that dj includes |∆n−1 | onto the jth face of |∆n |, and dj x gives the jth face of x. This relation glues points p on the boundary of x to their corresponding location in the realization |∆n | of x, and thus does the work of gluing the faces of the realization of x to the realizations of the faces of x. Thus, the simplicial complex of a ∆-complex and the realization of a simplicial set of a ∆-complex are homeomorphic since they perform the same gluings, and we can essentially ignore the degenerate simplices in the simplicial set. In particular, |∆n | as the image of the realization functor S → Top is homeomorphic to the image of {0, . . . , n} under the functor ∆ → Top. Example 11. The realization |S n | is homeomorphic to the topological n-sphere. Since the faces of σn are degeneracies of ∗, and ∗ is realized as the point in (∗, |∆0 |), the equivalence relations say that the faces (di σn , |∆n−1 |) are all mapped to that point. This is a standard construction of the n-sphere using CW-complexes from topology. 7 One would hope that, for simplicial sets X and Y , that |X × Y | and |X| × |Y | under the product topology are homeomorphic. While there is a natural map |X × Y | → |X| × |Y | which is continuous, and bijective, the inverse is not necessarily continuous. The map is a homeomorphism if either • Both X and Y are countable; or • One of |X| and |Y | are locally finite (it is a fact proved in [2, Ch. 1] that the realization of a simplicial set is a CW-complex). The reader is invited to look at [2, Ch. 1] or the references in [1] for a proof. The following functor is important because it puts singular homology into the language of simplicial sets. Definition 12. Let S : Top → S be the functor which sends a topological space Y to a simplicial set defined by (SY )n = homTop (|∆n | , Y ), where di and sj are defined by di σ = σdi : |∆n−1 | → Y and sj σ = σsj : |∆n+1 | → Y , for σ ∈ S(Y )n . Note that the relations (8), (9), and (10) for SY follow from di and sj satisfying the opposite relations in ∆. It turns out that |−| and S are adjoint functors, as described in the following theorem. Theorem 13. If X is a simplicial set and Y is a topological space, then homTop (|X| , Y ) ∼ = homS (X, SY ). The proof of this is in [2, Ch. 1], and it is basically category theory. 5 Homotopy In this section, we define homotopy in the abstract setting of simplicial sets. It is not the general case that homotopy is an equivalence relation. However, simplicial sets satisfying the Kan extension condition, which we describe, admit such an equivalence relation. We first let I be the simplicial set ∆1 (whose realization is a line segment), and let 0 and 1 denote the two vertices of I. Then, analogously to homotopy from algebraic topology, we give the following definition from [3]. Definition 14. Let f, g : X → Y be maps of simplicial sets for simplicial sets X and Y . We say f is homotopic to g if there is a simplicial map F : X × I → Y with F (x, 0) = f (x) and F (x, 1) = g(x), in which case we write f ' g. If A ⊂ X is a subsimplicial set, then we write f ' g (rel A) if F is constant on A. As we mentioned, we need an extra condition on simplicial sets for homotopy to be an equivalence relation. Let Λnk be the smallest subsimplicial set of ∆n which contains the faces di En for all i 6= k (see Example 7 for the definition of En ). 8 Definition 15. A simplicial set X satisfies the Kan extension condition if any map of simplicial sets f : Λnk → X extends to a map of simplicial sets g : ∆n → X. Example 16. Note that ∆1 does not satisfy this condition. Let f : Λ20 → ∆1 be defined by f ([0, 1]) = [0, 1] and f ([0, 2]) = [0, 0]. This function f is well-defined because we have specified what happens to all nondegenerate simplices of Λ20 (that is, we can compute f ([0]) = f (d1 [0, 1]) = d1 f ([0, 1]) = [0], and also f (sj x) = sj f (x), and so on). Since f ([1]) = [1] and f ([2]) = [0], we cannot extend f , since g([1, 2]) is forced to be [1, 0], which is not an element of ∆1 . Example 17. Importantly, SY , where S is the functor Top → S, satisfies the Kan extension condition. Let f : Λnk → SY be a map of simplicial sets. By Theorem 13, there is a corresponding continuous f˜ : |Λnk | → Y . There is a continuous map r : |∆n | → |Λnk | which retracts the kth face and interior onto the remaining faces. Thus, f˜r : |∆n | → Y is a continuous map which, again by Theorem 13, gives the required extension g : ∆n → SY . The following property that the Kan extension condition also works when Λnk appears in a product will be used to show that (') is an equivalence relation. Lemma 18. If X and Y are simplicial sets, and Y satisfies the Kan extension condition, then a map of simplicial sets f : X × Λnk → Y can be extended to G : X × ∆n → Y . Sketch of proof. This amounts to repeated application of the Kan extension condition to fill N n in occurances of Λm l within X × Λk ⊂ X × ∆ by induction on m. The following lemma can also be generalized to show homotopy relative to a subsimplicial set A ⊂ X is an equivalence relation. Lemma 19. If X and Y are simplicial sets, and Y satisfies the Kan extension condition, then (') is an equivalence relation on maps of simplicial sets X → Y . Proof. For the following, let f, g, h : X → Y be maps of simplicial sets. We will proceed by checking each axiom. Reflexivity. Let F : X × I → Y be defined by F = f ◦ π1 , where π1 : X × I → X is a projection. Thus, F is a homotopy, and f ' f . Symmetry. Assume there is a homotopy F from f to g. Then, we may define F 0 : X × Λ21 → Y to be F 0 (x, [0, 1]) = F (x, I) and F 0 (x, [0, 2]) = f (x) for all x ∈ X, which is well-defined since F 0 (x, [0]) = f (x) where the two legs meet. By the Kan extension condition, there is an extension G0 : X × ∆2 → Y , and restricting G0 to X × [0, 2] gives the required homotopy. Transitivity. Assume that F is a homotopy from f to g and that G is a homotopy from g to h. Define F 0 : X × Λ21 → Y to be the homotopy F on the edge X × {[0, 1]} and the homotopy G on the edge X × {[1, 2]}. By the Kan extension condition, F 0 extends to G0 : X × ∆2 → Y , and restricting this to X × {[0, 2]} gives a homotopy from f to h. 9 We will not prove the following two theorems from [3], but they provides the link between the homotopy theory of simplicial sets and that of topological spaces: Theorem 20. For simplicial sets X and Y where Y satisfies the Kan extension condition, there is a bijective correspondence between homotopy classes of maps X → Y of simplicial sets and homotopy classes of continuous maps |X| → |Y |. Theorem 21. For topological spaces X and Y , where X is a CW complex, there is a bijection correspondence between homotopy classes of continuous maps X → Y and homotopy classes of maps SX → SY of simplicial sets. 6 Homotopy groups In this section, we will not prove anything, but instead briefly mention homotopy groups. We assume X is a simplicial set which satisfies the Kan extension condition so that homotopy is an equivalence relation. There are a couple of ways to define the homotopy groups πn (X, ∗). One is as the set of homotopy classes of basepoint-preserving maps ∂∆n+1 → X of simplicial sets. This notion should be intuitive from algebraic topology since ∂∆n+1 is homotopy equivalent to an n-sphere. Some writers define homotopy groups without appealing to homotopy classes of maps and instead define homotopy classes of simplices. The following definition is from [1]. Definition 22. We say that two n-simplices x, x0 ∈ Xn are homotopic if 1. di x = di x0 for 0 ≤ i ≤ n; and 2. there exists a simplex y ∈ Xn+1 such that 3. (a) dn y = x, (b) dn+1 y = x0 , and (c) di y = sn−1 di x = sn−1 di x0 for 0 ≤ i ≤ n − 1. The homotopy group πn (X, ∗) is defined to be all n-simplices which contain the basepoint ∗ as a vertex, where the group operation involves using the Kan extension condition to get a third simplex from one. Alternatively, there are more combinatoric definition of these sets. Interested readers are invited to read [1]. 7 Homology We first talk about homology of simplicial abelian groups, and then apply it to arbitrary simplicial sets. The material in this section comes from [3]. 10 Let G be a abelian simplicial group. That is, each Gn is an abelian group, and there are homomorphisms di : Gn → Gn−1 and sj : Gn → Gn+1 which satisfy the identities for simplicial objects. Interestingly, G actually is a chain complex with the boundary operator ∂n : Gn → Gn−1 defined by ∂n = n X (−1)i di . i=0 It is straightforward to verify that ∂ 2 = 0: ∂2x = n−1 X (−1)i di i=0 = n n−1 X X n X ! (−1)j dj x j=0 (−1)i+j di dj x = 0, i=0 j=0 by using the identity di dj = dj −1 di for i < j. We define the homology groups of this chain complex to be Hn (G) = ker ∂n / im ∂n+1 . Let Z : Set → Ab be the functor which takes sets and gives the free abelian groups on those sets. Then, if X is a simplicial set, Z(X) is a simplicial free abelian group where (Z(X))n is the free abelian group with the elements of Xn as generators. Definition 23. The homology groups Hn (X) P of a simplicial set X are the homology groups Hn (Z(X)) using the boundary operator ∂n = ni=0 (−1)i di . Example 24. For a topological space Y , it is clear that the homology groups Hn (SY ) are isomorphic to the singular homology groups of Y from algebraic topology since (Z(SY ))n is just the singular n-chain group. References [1] G. Friedman, An elementary illustrated introduction to simplicial sets, eprint arXiv:0809.4221v2, 2011. [2] P.G. Goerss and J.F. Jardine, Simplicial homotopy theory, Progress in Mathematics, vol. 174, Birkhauser Verlag, Basel, 1999. [3] E.B. Curtis, Simplicial homotopy theory, Advances in Math. 6, (1971), 107-209. [4] S. Mac Lane, Categories for the Working Mathematician, Second Edition, Graduate Texts in Mathematics 5, Springer-Verlag, 1971. [5] E. Riehl, A leisurely introduction to http://math.uchicago.edu/~eriehl/cat/, 2008. simplicial sets, [6] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. 11 eprint