Semi-simplicial combinatorics of cyclinders and subdivisions

arXiv:2307.13749v1 [math.CO] 25 Jul 2023 Semi-simplicial combinatorics of cyclinders and subdivisions José Manuel Garcı́a Calcines, Luis Javier Hernández Paricio and Marı́a Teresa Rivas Rodrı́guez Abstract In this work, we analyze the combinatorial properties of cylinders and subdivisions of augmented semi-simplicial sets. These constructions are obtained as particular cases of a certain action from a co-semi-simplicial set on an augmented semi-simplicial set. We also consider cylinders and subdivision operators in the algebraic setting of augmented sequences of integers. These operators are defined either by taking an action of matrices on sequences of integers (using binomial matrices) or by taking the simple product of sequences and matrices. We compare both the geometric and algebraic contexts using the sequential cardinal functor | · |, which associates the augmented sequence |X| = (|Xn |)n≥−1 to each augmented semi-simplicial finite set X. Here, |Xn | stands for the finite cardinality of the set of n-simplices Xn . The sequential cardinal functor transforms the action of any co-semi-simplicial set into the action of a matrix on a sequence. Therefore, we can easily calculate the number of simplices of cylinders or subdivisions of an augmented semi-simplicial set. Alternatively, instead of using the action of a matrix on a sequence, we can also compute suitable matrices and consider the product of an augmented sequence of integers and an infinite augmented matrix of integers. The calculation of these matrices is related mainly to binomial, chain-power-set, and Stirling numbers. From another point of view, these matrices can be considered as continuous automorphisms of the Baer-Specker topological group. Keywords: Augmented semi-simplicial set, augmented integer sequence, simplicial combinatorics, cylinder, barycentric subdivision, Baer-Specker group. Mathematics Subject Classification (2020): 05E45, 18M05, 55U10. Contents 1 Introduction 2 1 2 Augmented semi-simplicial sets and integer sequences 2.1 Presheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Augmented semi-simplicial sets . . . . . . . . . . . . . . 2.3 Augmented integer sequences and matrices . . . . . . . 2.4 The sequential cardinal functor . . . . . . . . . . . . . . . . . . . . . . 3 Cylinders and barycentric subdivisions 3.1 Cylinders of an augmented semi-simplicial set . . . . . . . . 3.1.1 The standard cylinder . . . . . . . . . . . . . . . . . 3.1.2 The 0-cylinder . . . . . . . . . . . . . . . . . . . . . 3.1.3 The 2-cylinder . . . . . . . . . . . . . . . . . . . . . 3.2 Barycentric subdivision of an augmented semi-simplicial set 3.3 Cylinders for integer sequences . . . . . . . . . . . . . . . . 3.3.1 The standard cylinder for integer sequences . . . . . 3.3.2 The 0-cylinder for integer sequences . . . . . . . . . 3.3.3 The 2-cylinder for integer sequences . . . . . . . . . 3.4 Subdivision for integer sequences . . . . . . . . . . . . . . . 3.5 Comparing geometric and arithmetic constructions through sequential cardinal functor . . . . . . . . . . . . . . . . . . . 4 Conclusions and future work 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the . . . 7 7 9 12 17 20 20 20 25 29 33 34 34 35 37 38 41 44 Introduction In our previous work [10], we analyzed some combinatorial properties of the category of augmented semi-simplicial sets. We studied the sequential cardinal functor, which associates an augmented sequence |X|n = |Xn | of nonnegative integers with each augmented semi-simplicial finite set X. The categories of augmented semi-simplicial sets and augmented sequences of integers admit monoidal structures induced by adequate join products and unit objects. This fact enabled us to easily calculate the number of simplices of cones and suspensions of an augmented semi-simplicial set, as well as other augmented semi-simplicial sets that are built by joins. The more standard category of simplicial sets is also used as a model to study homotopy invariants. The tools used to model sets of homotopy classes in the category of simplicial sets are based on cylinders and subdivisions (the subdivision approximation theorem). For this reason, the main objective of this paper is to analyze the combinatorial properties of subdivisions and cylinders in the category of augmented semi-simplicial sets. Among others, we can highlight the following goals: • The study of some functorial procedures for constructing cylinders and subdivisions for augmented semi-simplicial sets. • The analysis of the numerical sequences (sequential cardinals) arising from 2 the combinatorial structure of cylinders and subdivisions of augmented semi-simplicial sets. • The search for methods of counting the number of simplices of cylinders and subdivisions of an augmented semi-simplicial set. In order to reach these targets we consider two different kinds of mathematical objects: (i) Augmented semi-simplicial and co-semi-simplicial objects. (ii) Augmented integer sequences and matrices. For further information on semi-simplicial sets, we recommend consulting the following references: [7], [17], [9], [21], [3], [10]. To learn more about the realization of semi-simplicial sets, [6] is a good resource. For category models related to homotopy theory of simplicial sets and topological spaces, [15] is a relevant reference. One noteworthy aspect of our study is that we are focusing on augmented semisimplicial objects instead of standard semi-simplicial objects. This minor modification yields more symmetric and simplified structures and formulas, making computations much easier. As basic combinatorial elements we consider the following ones:
(bc) Binomial coefficients (if q > p, then take pq = 0):   p p(p − 1) · · · (p − q + 1) p! = = q!(p − q)! (p − q)! q which give the number of strictly increasing maps from the ordered set with q elements {1 < · · · < q} to the ordered set of p elements {1 < · · · < p}. These numbers occur as coefficients in Newton’s binomial formula p (a + b) = p   X p q=0 q ap−q bq and the coefficients in Pascal’s triangle 1 1 1 1 1 ··· 1 .. . .. . 3 .. . 1 3 4 5 .. . 1 2 6 10 .. . 3 .. . 1 4 10 .. . .. . 1 5 .. . .. . 1 .. . ··· which can also be represented by the matrix  0
1 2 3 4 0  0| 0
0 0 0 0
 1 1 | 1 0 0 0 1
0

2 2 2  2| 0 0 2
1
0

3 3 3 3  3| 0
 3
2
1
0
4 4 4 4 4  4|  4
3
2
1
0
5 5 5 5 5  5|  4 3 2 1 0 .. .. .. .. .. . . . . . 6| 5 0 0 0 0 0
5 5 .. . 6 ··· 0 ··· 0 ··· 0 ··· 0 ··· 0 ··· 0 ··· .. . . . .              In this work, we have considered some results about binomial numbers from [11], [12], [4], [5] and [16]. (cps) The chain-power-set numbers are studied in [14]. They are defined as the cardinality of the set of all chains of length k in the poset P(Xn ), where Xn = {1, . . . , n}, and a chain of length k has the form ∅ ⊆ N0 ⊂ N1 ⊂ · · · ⊂ Nk ⊆ Xn , where Ni ̸= Ni+1 for 0 ≤ i ≤ k − 1. These numbers are denoted by an,k and analyzed in [14], which provides interesting results on recurrence formulas and relations between chain-power-set and Stirling numbers ([2]). In subsection 3.4 of our work, we use the notation cad[n]k = an+1,k for these numbers. We also introduce other chain numbers denoted by ˘ + [n]p , and establish some relations between these differcad+ [n]p and cad ent chain numbers and Stirling numbers, which are given in Lemma 1 and Theorem 3. (s2c) Consider a finite set Fp = {1, · · · , p} and q ∈ N. A q-partition P of Fp is a family of non-empty subsets P = {C1 , · · · , Cq } such that C1 ∪· · ·∪Cq = Fp and Ci ∩ Cj = ∅ for i ̸= j. Notice that there are no q-partitions of Fp when q > p. For p, q ∈ N, the Stirling numbers of second class are given by    |{P | P is a q-partition of Fp }| if p ≥ q ≥ 1,      p 1 if p = 0, q = 0 = q      0 otherwise This paper introduces new construction techniques for cylinders and subdivisions, as follows: Given an augmented semi-simplicial set X and a co-semisimplicial object Y (in the category of augmented semi-simplicial sets), we take e , the right action of Y on X to obtain the augmented semi-simplicial set X ▷Y as defined in Definition 1. In particular, we have introduced co-semi-simplicial objects, Cil0 , Cil, Cil2 to establish three kinds of cylinders of an augmented semi-simplicial set X: e 0, X ▷Cil e X ▷Cil, 4 e 2. X ▷Cil The same procedure have been used to create the barycentric subdivision of an augmented semi-simplicial set X. We deal with a certain co-semi-simplicial object Sd, and its action on X is denoted by e X ▷Sd which represents the barycentric subdivision of X. As far as the algebraic context is concerned, we consider categories of augmented sequences of integers with a categorical ring structure admitting actions of certain augmented matrices. This action is determined by using the inverse matrix of the matrix “bin” associated with binomial transformations. Given a finite augmented sequence a and an augmented matrix B, the action of B on a is described in Definition 6 by the formula ae ▷B := (a · bin−1 ) · B. One of the main techniques used in this paper is based on the fact that the sequential cardinal functor is compatible with action operators (Theorem 2), see Theorem 5 in [10]. In other words, if X is an augmented semi-simplicial set and Z is an augmented co-simplicial-object of augmented semi-simplicial sets, then we have ˜ = |X|˜▷|Z|. |X ▷Z| ˜ stands for the action in the geometOn the left side of the formula above, ▷ ric setting, whereas on the right side ˜▷ represents the action in the algebraic one. In subsection 3.1 of this paper, we compute the corresponding augmented matrices of the co-semi-simplicial objects Cil, Cil0 , Cil2 : cil = |Cil|, cil0 = |Cil0 |, cil2 = |Cil2 |. Using properties of chain-power-set numbers and Stirling numbers applied to the co-semi-simplicial object Sd, we prove in Theorem 5 that cad+ = |Sd| where cad+ , defined in subsection 3.4, is given as a slightly different version of the chain numbers. These results allow us to count effortlessly the number of simplices of an augmented semi-simplicial finite set which is built through action operations. Regarding cylinder objects one can take the matrices ˘ = bin−1 · cil, cil ˘ 0 = bin−1 · cil0 , cil ˘ 2 = bin−1 · cil2 cil and, with respect to barycentric subdivisions, ˘ = cad ˘ + = bin−1 · cad+ sd 5 bin being the augmented matrix of binomial coefficients. Now, for a given augmented semi-simplicial set X whose cardinal sequence is a = |X|, we can compute the cardinal sequences of Cil(X), Cil0 (X), Cil2 (X) and Sd(X) by considering the matrix products ˘ a · cil, ˘ 0, a · cil ˘ 2, a · cil ˘ a · sd. The first arrows and columns of these matrices are given by ˘ cil −1 0 1 2 3 .. . −1 0 1 0 0 2 0 0 0 0 0 0 .. .. . . 1 0 1 3 0 0 .. . 2 0 0 2 4 0 .. . 3 0 0 0 3 5 .. . 4 0 0 0 0 4 .. . 5 0 0 0 0 0 .. . ... ... ... ... ... ... .. . .. . −1 0 1 0 0 2 0 0 0 0 0 0 .. .. . . 1 0 0 3 0 0 .. . 2 0 0 0 4 0 .. . 3 0 0 0 0 5 .. . 4 0 0 0 0 0 .. . 5 0 0 0 0 0 .. . ... ... ... ... ... ... .. . 2 3 0 0 0 0 4 1 8 12 0 16 .. .. . . 4 0 0 0 6 32 .. . 5 0 0 0 1 24 .. . 3 0 0 0 0 24 .. . 4 0 0 0 0 0 .. . ˘0 cil −1 0 1 2 3 ˘2 cil −1 0 1 2 3 .. . −1 0 1 0 0 2 0 0 0 0 0 0 .. .. . . ˘ + = sd ˘ cad −1 0 1 2 3 .. . 1 0 1 4 0 0 .. . −1 0 1 1 0 0 0 1 0 0 1 2 0 1 6 0 1 14 .. .. .. . . . 2 0 0 0 6 36 .. . ... ... ... ... ... ... .. . 5 0 0 0 0 0 .. . ··· ··· ··· ··· ··· .. . We note that Π∞ i=−1 Z, equipped with the product topology, is a topological ˘ cil ˘ 0 and cil ˘ 2 induce continuous group automorphisms group and the matrices cil, 6 ˘ only induces a continuous group autoof this group. However, the matrix sd Z. For more information on the properties morphism on the subgroup ⊕∞ i=−1 of continuous homomorphisms of these Abelian topological groups, we refer the reader to [1], [20] and [8]. Finally, we also want to highlight the connection between the structure of cylinders and subdivisions and certain crystal molecular structures, where the dominating features are trans-edge connected tetrahedra. For instance, if we apply the funtor Cil2 to the barycentric subdividision Sd2 (Γ+ [1]) (see Figure 1), we obtain simplicial models for some crystal structures that are analyzed in [19] and [22]. Figure 1: Simplicial structure of Cil2 Sd2 (Γ+ [1]) 2 Augmented semi-simplicial sets and integer sequences In this section, we establish some notation and results regarding the categories used in this work. 2.1 Presheaves We denote the category of sets as Sets. Given a small category C we consider op the usual functor category SetsC which has functors X : Cop → Sets as objects, and natural transformations f : X → X ′ as arrows. The category op SetsC is commonly referred to as the category of presheaves on C. For a given object c in C, we can consider the presheaf Y(c) on C, defined as the contravariant Hom-functor Y(c)(−) := HomC (−, c). This construction gives rise to the well-known Yoneda embedding: op Y : C → SetsC , c 7→ HomC (−, c). In this setting, an embedding is a full and faithfull functor. Any presheaf that is isomorphic to a presheaf of the form Y(c) is called representable. The Yoneda lemma asserts that for any presheaf X, there exists a bijection between the natural transformations Y(c) → X and the elements of X(c): ∼ = Nat(Y(c), X) → X(c), α 7→ αc (1c ). 7 op Associated with X ∈ SetsC , we have the so-called category of elements of X, R denoted by C X. Its objects are pairs of the formR(c, x) where c is an object in C and x ∈ X(c); a morphism (c, x) → (c′ , x′ ) in C X consists of a morphism φ : c → c′ in C such that X(φ)(x′ ) = x. Observe that we have a projection functor: Z X → C, (c, x) 7→ c. πX : C The proof of the following theorem can be found in [13]. Theorem 1. Let Z : C → E be a functor from a small op category C to a cocomplete category E. Then, the functor SingZ : E → SetsC given by SingZ (Y )(c) := HomE (Z(c), Y ) op admits a left adjoint functor LZ : SetsC → E, defined for each presheaf X as Z πX Z LZ (X) := colim( X −→ C −→ E) C The functor LZ preserves colimits and makes commutative the following diagram Z C _ ;/ E Y  op SetsC LZ In other words, LZ is an extension of Z that preserves colimits. Moreover LZ is, up to natural isomorphism, the only extension of Z preserving all colimits. Remark 1. Observe that, actually, we have a functor: SetsC op × E C → E, (X, Z) 7→ LZ (X). op We note that, as a consequence of Theorem 1 above with E = SetsC , certain induced functors arise, which are called action functors: Definition 1. The action functors are the following ones: e 1. (−)▷(−) : SetsC op op × (SetsC )C → SetsC op op e := LY (X). X ▷Y op op e 2. (−)▷(−) : (SetsC )C × (SetsC )C → (SetsC )C e e = LZ (Y (c)). (Y ▷Z)(c) := Y (c)▷Z 8 op op op e ∈ SetsC is the Given X ∈ SetsC and Y, Z ∈ (SetsC )C we say that X ▷Y op C e ∈ (Sets )C is the (right) action (right) action of Y on X, and similarly, Y ▷Z of Z on Y . Remark 2. If X ∈ SetsC op op and Y, Z ∈ (SetsC )C , then it follows that e )▷Z e ∼ e ▷Z). e (X ▷Y = X ▷(Y This formula inspires the name given: “action functors”. 2.2 Augmented semi-simplicial sets Now consider the small category Γ, whose objects are all non-empty totally ordered sets [p] = {0 < · · · < p} for p ≥ 0, and whose morphisms are strictly increasing maps [p] → [q]. We can add the empty set ∅ = [−1] to this category together with all the strictly increasing maps. The resulting augmented category will be denoted by Γ+ . op We can consider SetsΓ+ and SetsΓ+ , whichopare the presheaf categories on Γ+ Γ+ and Γop , C = Γ+ , and using the no+ , respectively. Taking E = Sets Y e tation X ▷Y for L (X) given in Definition 1 we obtain the following action functors: op op op e SetsΓ+ × (SetsΓ+ )Γ+ → SetsΓ+ , (X, Y ) 7→ X ▷Y, op op op e (SetsΓ+ )Γ+ × (SetsΓ+ )Γ+ → (SetsΓ+ )Γ+ , (Y, Z) 7→ Y ▷Z. We obviously have the corresponding Yoneda embeddings: op Y : Γ+ → SetsΓ+ , Γ+ . Yop : Γop + → Sets For any object [n] in Γ+ we denote by Γ+ [n] the representable presheaf op Γ+ [n] := Y([n]) = HomΓ+ (−, [n]) ∈ SetsΓ+ . Analogously, we consider the notation Γ+ op . Γop + [n] := Y ([n]) = HomΓ+ ([n], −) ∈ Sets Definition 2. Any presheaf on Γ+ X : Γop + → Sets will be called augmented semi-simplicial set or Γop + -set. Analogously, any presheaf on Γop + , Z : Γ+ → Sets, will be called augmented co-semi-simplicial set or Γ+ set. 9 op Giving an augmented semi-simplicial set X ∈ SetsΓ+ is equivalent to giving a collection of sets {Xn }n≥−1 together with a collection of set maps dni : Xn → Xn−1 (n ≥ 0 and 0 ≤ i ≤ n) satisfying dni ◦ dn+1 = dnj−1 ◦ dn+1 , if i < j. j i op Moreover, giving an arrow f : X → X̄ in SetsΓ+ (i.e., a natural transformation) is equivalent to giving a collection of set maps {fn : Xn → X̄n }n≥−1 such that fn−1 ◦ dni = d¯in−1 ◦ fn , for n ≥ 0 and 0 ≤ i ≤ n. There is a similar description for the category of augmented co-semi-simplicial sets by just reversing the arrows in the representation above. We can also consider the subcategory of finite sets Setsfin and the corresponding op category of presheaves (Setsfin )Γ+ . op An augmented semi-simplicial set X ∈ SetsΓ+ is said to have finite dimension if there is k ∈ N+ such that Xi = ∅ for every i > k. Given a non-empty finite dimensional semi-simplicial set X, we denote dim(X) := min{k | Xi = ∅ for all i > k, i ∈ N+ }. For the empty semi-simplicial set, we set dim(∅) := −∞ op The subcategory of finite dimensional semi-simplicial sets is denoted by (SetsΓ+ )fd . op An augmented semi-simplicial set X ∈ SetsΓ+ is said to be finite if it has finite dimension and for every i ∈ N+ , Xi is a finite set. The subcategory of finite op semi-simplicial sets is denoted by (SetsΓ+ )fin . The join of any pair of Γop + -sets, X, Y , is given by the formula: G (X ⊞ Y )m := Xp × Yq p+q=m−1 where p and q are integers greater than or equal to −1. In particular, we have (X ⊞ Y )−1 = X−1 × Y−1 , (X ⊞ Y )0 = (X−1 × Y0 ) ⊔ (X0 × Y−1 ), (X ⊞ Y )1 = (X−1 × Y1 ) ⊔ (X0 × Y0 ) ⊔ (X1 × Y−1 ) and so on. Additionally, the operators Y diX⊞Y of X ⊞ Y are defined naturally from the operators dX k and dl of X and Y respectively. The definition of ⊞ on morphisms is straightforward. This gives the join functor op op op ⊞ : SetsΓ+ × SetsΓ+ → SetsΓ+ op For any pair X, Y ∈ SetsΓ+ we have that the functors op op X ⊞ (−), (−) ⊞ Y : SetsΓ+ → SetsΓ+ 10 op preserve colimits (see [10]). Furthermore, if X, Y, Z ∈ SetsΓ+ , then there exist canonical natural isomorphisms X ⊞ (Y ⊞ Z) ∼ = (X ⊞ Y ) ⊞ Z, Γ+ [−1] ⊞ X ∼ =X∼ = X ⊞ Γ+ [−1], X ⊞Y ∼ = Y ⊞ X. Moreover, it is shown in [10] that for any n, m ∈ N+ , the following isomorphism holds true Γ+ [n] ⊞ Γ+ [m] ∼ = Γ+ [n + m + 1]. op The category SetsΓ+ together with the join functor ⊞ and the unit object Γ+ [−1] forms a symmetric monoidal category [10]. Furthermore, considering the coproduct (the ordinal sum) in the category Γ+ [p] ⊔ [q] := [p + q + 1] we obtain a symmetric monoidal category structure on Γ+ having [−1] as a unit object. It is also shown in [10] that the Yoneda embedding, Y : (Γ+ , ⊔, [−1]) → op op (SetsΓ+ , ⊞, Γ+ [−1]), is monoidal. Moreover, ((SetsΓ+ )fin , ⊞, Γ+ [−1]) is a monoidal op subcategory of ((Setsfin )Γ+ , ⊞, Γ+ [−1]) and the latter is a monoidal subcateop gory of (SetsΓ+ , ⊞, Γ+ [−1]). We have the left-cone functor Conl := Γ+ [0] ⊞ (−) and the right-cone functor Conr := (−) ⊞ Γ+ [0]: op op Conl , Conr : SetsΓ+ → SetsΓ+ Conl (X) = Γ+ [0] ⊞ X, Conr (X) = X ⊞ Γ+ [0]. These functors satisfy Conl (Γ+ [k]) = Γ+ [k + 1] = Conr (Γ+ [k]), for all k ≥ −1. e defined in Definition 1, we have the action Recall that, using the notation X ▷Y functors: op op op e SetsΓ+ × (SetsΓ+ )Γ+ → SetsΓ+ , (X, Y ) 7→ X ▷Y op op op e (SetsΓ+ )Γ+ × (SetsΓ+ )Γ+ → (SetsΓ+ )Γ+ , (Y, Z) 7→ Y ▷Z op Here, we take E = SetsΓ+ and C = Γ+ . The following result is also proved in [10]. Recall that we are using the particular e for the construction LZ (−): notation (−)▷Z 11 op Proposition 1. Let Z : (Γ+ , ⊔, [−1]) → (SetsΓ+ , ⊞, Γ+ [−1]) be a monoidal op op e : SetsΓ+ → SetsΓ+ , functor. Then the colimit-preserving functor (−)▷Z which makes the following diagram commute Γ+ _ Y  op SetsΓ+ Z op / SetsΓ+ 9 e (−)▷Z op is monoidal. In particular, for all X, Y ∈ SetsΓ+ , we have e ∼ e ⊞ (Y ▷Z). e (X ⊞ Y )▷Z = (X ▷Z) Moreover, up to isomorphism, it is the unique colimit-preserving functor making this diagram commute. Remark 3. We also obtain: op (i) If Z : (Γ+ , ⊔, [−1]) → ((Setsfin )Γ+ , ⊞, Γ+ [−1]) is a monoidal functor, op op e : (Setsfin )Γ+ → (Setsfin )Γ+ is a monoidal functor. then (−)▷Z op (ii) If Z : (Γ+ , ⊔, [−1]) → ((SetsΓ+ )fin , ⊞, Γ+ [−1]) is a monoidal functor, op op e : (SetsΓ+ )fin → (SetsΓ+ )fin is a monoidal functor. then (−)▷Z 2.3 Augmented integer sequences and matrices We study the second kind of mathematical objects in this work: augmented integer sequences (and matrices). The set of integer numbers Z admits the structure of a discrete category. However, we can also consider it as a groupoid Z where the cardinal |HomZ (n, m)| = 1 and the unique morphism from n to m is denoted by m − n : n → m, for every pair of integers n, m. The sum of integers can be easily extended to a functor + : Z × Z → Z, (n, m) 7→ n + m Taking + as a tensor product and 0 as a unit object, it is immediate to check that (Z, +, 0) has the structure of a strict symmetric monoidal category and also of a strict categorical group. Given any small category J, the functor category ZJ has an induced strict symmetric monoidal category structure (in addition, it is a strict symmetric categorical group). We consider (ZJ )fin the full subcategory of ZJ consisting of functors c : J → Z such that there exists a finite set of objects Fc satisfying that c(j) = 0, for all j ∈ J \ Fc . Now, if N denotes the set of natural numbers (0 is also included as a natural number), take the discrete category N+ = N ∪ {−1}. This category can be considered as a non-full subcategory of both Γ+ and Γop + through the (inclusion) 12 functor n → [n]. Observe that the category N+ is self-dual, that is, N+ = Nop +. op Given a functor c ∈ (ZN+ )fin with c ̸= 0, the dimension of c is the integer dim(c) := min{k | ci = 0 for all i > k, i ∈ N+ }. If c = 0, then we set dim(c) = −∞. op The Yoneda embedding Y : N+ → SetsN+ associated to N+ (note that this op functor is a restriction of the Yoneda functor Y : Γ+ → SetsΓ+ ) induces, by applying the cardinal operator to the corresponding hom-sets, a functor op y : N+ → ZN+ satisfying y(n)(j) = δn,j , for all n, j ∈ N+ . Here δn,j denotes the Kronecker delta ( 1, n = j δn,j = 0, n ̸= j op Therefore, y(n) ∈ (ZN+ )fin and dim(y(n)) = n, for all n ∈ N+ . This way, we op actually have a functor y : N+ → (ZN+ )fin . Definition 3. An augmented integer sequence is a functor a : Nop + → Z. We will denote an augmented sequence a by means of a row matrix a = (a−1 a0 a1 a2 · · · ). However, in some cases, this row matrix will be denoted by (using commas): a = (a−1 , a0 , a1 , a2 , · · · ). Analogously, an augmented integer co-sequence is a functor b : N+ → Z. We will denote an augmented co-sequence b by means of a column matrix   b−1  b0    b =  b1    .. . or b = (b−1 b0 b1 · · · )T , where T denotes the transposition operator. Definition 4. An augmented integer matrix is a functor U : N+ × Nop + → Z. An augmented matrix U will be denoted by its usual form   U−1,−1 U−1,0 U−1,1 · · ·  U0,−1 U0,0 U0,1 · · ·   U =  U1,−1 U U · · · 1,0 1,1   .. .. .. .. . . . . 13 op ∼ ZN+ ×Nop ∼ + Note that, as there are isomorphisms of categories (ZN+ )N+ = = op op Nop N+ + ZN+ ×N+ ∼ (Z ) , any augmented matrix U : N × N → Z may be consid= + + ered as an object in any of the categories above. op If a ∈ (ZN+ )fin and b : N+ → C is any functor, then we have that they are of the form a = (a−1 , a0 , a1 , · · P · an , 0, 0, 0, · · · ) and b = (b−1 , b0 , b1 , b2 , b3 , · · · )T , +∞ respectively. Therefore, a·b = i=−1 ai bi is well defined and we have an induced bifunctor op (−)·(−) : (ZN+ )fin × ZN+ → Z, (a, b) 7→ a · b. Similarly, we have a bifunctor op (−)·(−) : ZN+ × (ZN+ )fin → Z, (c, d) 7→ c · d. op ∼ = Taking into account the transposition isomorphism (−)T : ZN+ → ZN+ , the composition induced by the identity on the first variable and the transposition on the second variable induces the scalar (or inner) product: op op ⟨−, −⟩ : (ZN+ )fin × (ZN+ )fin → Z. Namely, if a = (a−1 , a0 , a1 , · · · an , 0 · · · ) and b = (b−1 , b0 , b1 , · · · bm , 0 · · · ), then min{n,m} X ⟨a, b⟩ = a · bT = a i bi . i=−1 One has the following canonical extended bifunctors of the dot-product: op op op (−) · (−) : (ZN+ )fin × (ZN+ )N+ → ZN+ , (a, B) 7→ a · B, op op op (−) · (−) : ZN+ × ((ZN+ )fin )N+ → ZN+ , (a, B) 7→ a · B, op (−) · (−) : (ZN+ )N+ × (ZN+ )fin → ZN+ , (A, b) 7→ A · b, op (−) · (−) : ((ZN+ )fin )N+ × ZN+ → ZN+ , (A, b) 7→ A · b, (a · B)j = X (A · b)i = ak Bk,j , k∈N+ X Ai,k bk . k∈N+
Nop op op (−) · (−) : ((ZN+ )fin )N+ × (ZN+ )N+ → ZN+ + , (A, B) 7→ A · B,
Nop op op (−) · (−) : (ZN+ )N+ × (ZN+ )fin + → (ZN+ )N+ , (A, B) 7→ A · B, (A · B)i,j = X k∈N+ 14 Ai,k Bk,j . op op The category ZN+ has the structure of a ring (ZN+ , +, ×), which is induced by the ring structure of (Z, +, ×) by pointwise operation. For a, b ∈ Z we will also use the notation a × op b = ab. However, we may consider a new symmetric monoidal structure on ZN+ . This is given by the join product: op The join product of a, b ∈ ZN+ , denoted as a ⊞ b, is given by the following formula: X ap bq , p, q ∈ N+ . (a ⊞ b)m := p+q=m−1 In this case the unit object is given as 1−1 where (1−1 )i = δ−1,i is the Kronecker delta. In this work, for k ∈ N+ , 1k will denote the augmented sequence given by (1k )i = δk,i . op The category ZN+ equipped with the join product ⊞ and the unit object 1−1 has the structure of a strict symmetric monoidal category ([10]). Moreover, the op induced functor y : (N+ , ⊔, [−1]) → (ZN+ , ⊞, 1−1 ) is monoidal. op op Obviously, ((ZN+ )fin , ⊞, 1−1 ) is a monoidal subcategory of (ZN+ , ⊞, 1−1 ). op If a, b ∈ ZN+ are fixed, then we easily obtain functors op op a ⊞ (−), (−) ⊞ b : ZN+ → ZN+ given by c 7→ a ⊞ c and c 7→ c ⊞ b, respectively. op Now, for any k ∈ Z, we define an operator in the category ZN+ (actually, a functor) op op Dk : ZN+ → ZN+ , b 7→ Dk (b) op as follows: given b ∈ ZN+ and i ∈ N+ , If k ≥ 0, then (Dk (b))i = bi+k , If k ≤ 0, then: (Dk (b))i = ( bi+k , 0, if i + k ≥ −1, if i + k < −1.
Nop op Definition 5. For any given b ∈ ZN+ , we define R(b) ∈ (ZN+ )fin + , the shifting of b, by the formula (R(b))i = D−(i+1) (b), for all i ∈ N+ . We may also see it as the matrix     D0 (b) b−1 b0 b1 b2 · · · D−1 (b)  0 b−1 b0 b1 · · ·      R(b) = D−2 (b) =  0  0 b b · · · −1 0     .. .. .. .. .. . . . . . 15 op This construction naturally gives rise to a functor R : ZN+ → (ZN+ )fin which satisfies the equalities
Nop + a ⊞ b = a · R(b) = b · R(a) op for all a, b ∈ ZN+ . Now we presentopan interesting construction: the cone of a sequence. Indeed, the op cone of c ∈ ZN+ is the sequence con(c) ∈ ZN+ defined as con(c) := c + D−1 (c). That is to say, ( ci + ci−1 , if i ≥ 0 con(c)i = c−1 , if i = −1. op op We obtain the cone functor con : ZN+ → ZN+ . Considering c = con(1−1 ) = op 1−1 + 10 ∈ ZN+ the cone functor is related to the join and the dot product through the following formula: con(b) = c ⊞ b = b ⊞ c = b · R(c). To finish this section we consider actions of sequences and matrices. We first op establish the augmented binomial matrix bin ∈ ((ZN+ )fin )N+ defined as   i+1 bini,j = , i, j ∈ N+ j+1 op and its inverse matriz bin−1 ∈ ((ZN+ )fin )N+ given by   i−j i + 1 , i, j ∈ N+ . bin−1 = (−1) i,j j+1 op op Definition 6. Given a sequence a ∈ (ZN+ )fin and a matrix B ∈ (ZN+ )N+ , it is defined the action of B on a by the formula ae ▷B := (a · bin−1 ) · B. The resulting sequence ae ▷B is also said to be the tilde-triangle product of a and B. This construction gives rise to an action functor op op op (−)e ▷(−) : (ZN+ )fin × (ZN+ )N+ → ZN+ . We point out that we also have the identity ae ▷bin = (a · bin−1 ) · bin = a. op Remark 4. There is an obvious extension of the dot product when a ∈ ZN+ is Nop N+ + a general sequence and C ∈ ((Z )fin ) : op op op (−) · (−) : ZN+ × ((ZN+ )fin )N+ → ZN+ , (a, C) 7→ a · C 16 op Note that C ∈ ((ZN+ )fin )N+ if, and only if, the columns of the matrix C are eventually constant atop 0. Then, the tilde-triangle product ae ▷B can also be op defined when a ∈ ZN+ is a general sequence and the matrix B ∈ (ZN+ )N+ op satisfies that bin−1 · B ∈ ((ZN+ )fin )N+ . Indeed, in this case we can define ae ▷B = a · (bin−1 · B) We oppoint out that this definition is compatible with the first one when a ∈ (ZN+ )fin since the matricial product is associative whenever it has sense. 2.4 The sequential cardinal functor Now we recall from [10] the relationship between the category of augmented semi-simplicial finite sets and the category of augmented integer sequences. The key point is the sequential cardinal functor, which applies every finite augmented semi-simplicial set to the sequence constituted by the cardinal of the set of n-simplices. This sequential cardinal functor also preserves certain structures. Given any functor X : Γop + → Setsfin we may consider the diagram Nop + ZO |−| in  Γop + X / Setsfin where in denotes the inclusion functor and |−| the functor that gives the cardinal of any finite set. op Γ+ Definition 7. The sequential cardinal of a Γop , is + - finite set, X ∈ (Setsfin ) defined as the augmented sequence: |X| : Nop + →Z given by the composite |X| := | − | ◦ X ◦ in, that is, |X|n := |Xn |, n ∈ N+ . op op We observe that there is an induced functor | − | : (Setsfin )Γ+ → ZN+ where, for morphisms f : X → Y , it is defined as |f | := |Y | − |X|; that is, |f |n = |Y |n − |X|n , n ∈ N+ . On the one hand, we showed in [10] that op ((Setsfin )Γ+ , ⊔, Γ∅+ ) and 17 op ((Setsfin )Γ+ , ×, Γ1+ ) are monoidal categories. On the other, the ring structure (Z, +, ×) induces op op monoidal structures (ZN+ , +, 0), (ZN+ , ×, 1). Taking into account the identities: |X ⊔ Y | = |X| + |Y |, |X × Y | = |X|×|Y | |Γ∅+ | = 0, |Γ1+ | = 1 op op we have that the functor | − | : (Setsfin )Γ+ → ZN+ preserves the monoidal structures induced by coproducts and products (see [10]): op op | − | : ((Setsfin )Γ+ , ⊔, Γ∅+ ) → (ZN+ , +, 0) op op | − | : ((Setsfin )Γ+ , ×, Γ1+ ) → (ZN+ , ×, 1). Definition 8. Associated to the Γop + -sets, Γ+ [n] and S+ [n − 1] = Γ+ [n] \ {ιn }, where ιn is the identity of [n] ∈ Γ+ , we consider the sequential cardinals: γ+ [n] := |Γ+ [n]|, s+ [n − 1] := |S+ [n − 1]|. For every n ∈ N+ , the sequential cardinal γ+ [n] = |Γ+ [n]| is given by the binomial coefficients: |Γ+ [n]| = = (|Γ −1 |, |Γ+ [n]0 |, |Γ[n]1 |, · · · , |Γ+ [[n]n|, |∅|, |∅|, · · · )  + [n]
n+1
n+1

n+1 n+1 0 , 1 , 2 , · · · , n+1 , 0, 0, · · · . op Now we consider ((Setsfin )Γ+ , ⊞, Γ+ [−1]), which is a monoidal subcategory of op ((Sets)Γ+ , ⊞, Γ+ [−1]). The sequential cardinal functor preserves the monoidal structure, that is |X ⊞ Y | = |X| ⊞ |Y |, |Γ+ [−1]| = 1−1 Γop + for all X, Y ∈ (Setsfin ) . In other words (see [10]) the sequential cardinal functor op op | · | : ((Setsfin )Γ+ , ⊞, Γ+ [−1]) → (ZN+ , ⊞, 1−1 ) is monoidal. The cone functors for semi-simplicial sets and augmented sequences are also related through the sequential cardinal functor: Proposition 2. The following diagrams are commutative: op (Setsfin )Γ+ Conl op |·| (Setsfin )Γ+ |·|  Z op / (Setsfin )Γ+ Nop + con /Z Conr op / (Setsfin )Γ+ |·|  |·|  Nop + Z 18 Nop + con /Z  Nop + Moreover, as γ+ [0] = |Γ+ [0]| = c, we have γ+ [n] = |Γ+ [n]| = ⊞n+1 c (the (n + 1)-fold join of c with itself). op The cardinal functor |·| : Setsfin → Z induces a canonical funtor |·| : ((Setsfin )Γ+ )Γ+ → op (ZN+ )N+ , Z 7→ |Z|, where |Z| = | · | ◦ Z ◦ in is the composite N+ in Z / Γ+ |·| op / (Setsfin )Γ+ op / Z N+ . In particular, the augmented Pascal matrix is defined as the augmented cosequence given by the composite: N+ in Y / Γ+ |·| op / (Setsfin )Γ+ op / Z N+ . Note that | · | ◦ Y ◦ in = bin. As a consequence, see [10], the functor op op | · | : ((Setsfin )Γ+ )Γ+ → (ZN+ )N+ op carries the Yoneda augmented semi-cosimplicial set Y : Γ+ → SetsΓ+ , matricially represented as   Γ+ [−1]   Y =  Γ+ [0]  .. . to the augmentend Pascal matrix, where each row is the cone of the previous one.     γ+ [−1] |Γ+ [−1]|     |Y| =  |Γ+ [0]|  =  γ+ [0]  = bin .. .. . . To finish this section, we establish an important result which asserts that, under mild restrictions, the sequential cardinal functor carries the action toopthe triangle product. We recall from [10] that a functor Z : Γ+ → SetsΓ+ is said to be regular if Z(φ) is injective (on each dimension) for every morphism φ in Γ+ . op op Γ + Theorem 2. If (SetsΓ+ )reg stands for the full subcategory of (SetsΓ+ )Γ+ consisting of regular functors, then the following diagram is commutative: op op Γ + (SetsΓ+ )fin × ((Setsfin )Γ+ )reg e (−)▷(−) op / (Setsfin )Γ+ |−| |−|×|−|   op op (ZN+ )fin × (ZN+ )N+ op op / Z N+ (−)e ▷(−) op In other words, if X ∈ (SetsΓ+ )fin and Z ∈ ((Setsfin )Γ+ )reg , then ˜ = |X|˜▷|Z|. |X ▷Z| e + [−]| = |X|˜▷|Γ+ [−]| = |X|. Moreover, if we specialize Z := Γ+ [−], then |X ▷Γ 19 op Remark 5. In the diagram above, instead of the categories (SetsΓ+ )fin and op op op (ZN+ )fin , we can take the larger categories (Setsfin )Γ+ and ZN+ if we reduce op Γ + ((Setsfin )Γ+ )reg to co-simplicial objects Z verifying that for each q ∈ N+ , there is nq ∈ N+ such that |Z̆([n])q | = 0 for every n ≥ nq ; here the set Z̆([n])q is given by Z̆([−1])q := Z([−1])q Z̆([n])q := Z([n])q \ (∪ni=0 (φi )∗ (Z([n − 1])q ), n ≥ 0 where φi : [n − 1] → [n] is the canonical i-th increasing inclusion i ∈ {0, · · · , n}. 3 Cylinders and barycentric subdivisions 3.1 3.1.1 Cylinders of an augmented semi-simplicial set The standard cylinder Given σ ∈ Γ+ [n]p and τ ∈ Γ+ [n]q , we will write σ ≼ τ whenever the following two conditions hold: (i) |σ([p]) ∩ τ ([q])| ≤ 1 (ii) σ(i) ≤ τ (j), for all i ∈ [p] and j ∈ [q] Observe that if σ ∈ Γ+ [n]−1 or τ ∈ Γ+ [n]−1 , then σ ≼ τ . For each object [n] ∈ Γ+ we define the (standard) cylinder of Γ+ [n], denoted by CilΓ+ [n], as the following augmented semi-simplicial set. If m ∈ Γ+ we consider two possibilities: m ≥ 0 and m = −1. Now, if m ≥ 0 we take CilΓ+ [n]m := {(σ, τ ) ∈ Γ+ [n]p × Γ+ [n]q : p, q ≥ 0, p + q = m − 1 and σ ≼ τ } On the other hand, if m = −1 we take CilΓ+ [n]−1 = {(∅, ∅)}. Here, ∅ denotes the unique element ∅ = [−1] → [n] belonging to Γ+ [n]−1 . Remark 6. With this definition observe that CilΓ+ [−1]−1 = {(∅, ∅)} = {∗} and CilΓ+ [−1]m = ∅, for all m ≥ 0. Now, for any m ≥ 0, (σ, τ ) ∈ CilΓ+ [n]m and 0 ≤ i ≤ m, the face operator di : CilΓ+ [n]m → CilΓ+ [n]m−1 is given by  if 0 ≤ i ≤ p,  (di (σ), τ ) di ((σ, τ )) =  (σ, di−p−1 (τ )) if p + 1 ≤ i ≤ m. It easy to check that we have an induced augmented semi-simplicial set CilΓ+ [n] : Γop + → Sets Notation: For the sake of conciseness and clarity we will consider the following notation. For m ≥ 0, since any strictly increasing map σ : [m] → [n] is given 20 by 0 ≤ σ(0) < σ(1) < · · · < σ(m) ≤ n, then we may write σ as the tuple σ = (σ(0), σ(1), · · · , σ(m)). As far as a pair (σ, τ ) is concerned, it will be denoted according to the following possibilities: • If σ = (σ(0), σ(1), · · · , σ(p)) and τ = (τ (0), τ (1), · · · , τ (q)), then we will write the pair (σ, τ ) as (σ(0), σ(1), · · · , σ(p), τ (0)′ , τ (1)′ , · · · , τ (q)′ ). • If σ ̸= ∅ and τ = ∅, then (σ, ∅) is reduced to (σ(0), σ(1), · · · , σ(p)). Similarly, if σ = ∅ and τ ̸= ∅, then (∅, τ ) is reduced to (τ (0)′ , τ (1)′ , · · · , τ (q)′ ). • Finally, if σ = ∅ and τ = ∅, we use the notation (∅, ∅) = ∗ Example 1. Using the notation described above, we always have CilΓ+ [n]−1 = {∗}(= {(∅, ∅)}) for all n. Next we describe CilΓ+ [0], CilΓ+ [1] and CilΓ+ [2]: 1. For n = 0 we have CilΓ+ [0]−1 = {∗} and CilΓ+ [0]0 = {((0), ∅), (∅, (0))} = {0, 0′ }. Now, if m = 1, then p and q must satisfy p + q = 0 and we have (p, q) ∈ {(−1, 1), (1, −1), (0, 0)}. Since Γ+ [0]1 is the empty set we obtain CilΓ+ [0]1 = {((0), (0))} = {(0, 0′ )}. Moreover, CilΓ+ [0]m = ∅, for all m ≥ 2. 2. For n = 1 we have CilΓ+ [1]−1 = {∗} CilΓ+ [1]0 = {((0), ∅), ((1), ∅), (∅, (0)), (∅, (1))} = {0, 1, 0′ , 1′ } CilΓ+ [1]1 = {((0, 1), ∅), (∅, (0, 1)), ((0), (1)), ((0), (0)), ((1), (1))} = {(0, 1), (0′ , 1′ ), (0, 1′ ), (0, 0′ ), (1, 1′ )} If m = 2, then p and q must satisfy p+q = 1 so (p, q) ∈ {(−1, 2), (0, 1), (1, 0)}. Since Γ+ [1]2 is the empty set we obtain CilΓ+ [1]2 = {(0, (0, 1)), ((0, 1), 1)} = {(0, 0′ , 1′ ), (0, 1, 1′ )} If m = 3, then p and q must satisfy p+q = 2 so (p, q) ∈ {(−1, 3), (0, 2), (1, 1)}. Note that Γ+ [1]3 , Γ+ [1]2 are empty sets. For (p, q) = (1, 1), we have σ = (0, 1) = τ and σ ̸≼ τ in this case. This implies that CilΓ+ [1]3 = ∅; moreover, CilΓ+ [1]m = ∅, for all m ≥ 3. 3. For n = 2 (see Figure 3) we have the following sets: CilΓ+ [2]−1 = {∗} CilΓ+ [2]0 = {0, 1, 2, 0′ , 1′ , 2′ } CilΓ+ [2]1 = {(0, 1), (0, 2), (0, 0′ ), (0, 1′ ), (0, 2′ ), (1, 2), (1, 1′ ), (1, 2′ ), (2, 2′ ), (0′ , 1′ ), (0′ , 2′ ), (1′ , 2′ )} CilΓ+ [2]2 = {(0, 1, 2), (0, 1, 1′ ), (0, 1, 2′ ), (0, 2, 2′ ), (0, 0′ , 1′ ), (0, 0′ , 2), (0, 1′ , 2′ ), (1, 2, 2′ ), (1, 1′ , 2′ ), (0′ , 1′ , 2′ )} CilΓ+ [2]3 = {(0, 1, 2, 2′ ), (0, 1, 1′ , 2′ ), (0, 0′ , 1′ , 2′ )} CilΓ+ [2]m = ∅, m ≥ 4. 21 0’ 1’ 2’ 0 1 2 Figure 2: The cylinder of Γ+ [2] One can straightforwardly check that we have an induced functor Cil : Γ+ → op SetsΓ+ , Cil([n]) = CilΓ+ [n]. By Theorem 1 it is obtained the colimit-preserving op op e functor (−)▷Cil : SetsΓ+ → SetsΓ+ making commutative the diagram Γ+ _ Y  op SetsΓ+ Cil op / SetsΓ+ , 9 e (−)▷Cil op Definition 9. For any augmented semi-simplicial set X ∈ SetsΓ+ we define its (standard) cylinder as the augmented semi-simplicial set e Cil(X) := X ▷Cil e Similarly for augmented semi-simplicial maps, Cil(f ) := f ▷Cil. The functor Cil has a right adjoint op op rCil : SetsΓ+ → SetsΓ+ . If we have an augmented simplicial subset X ⊂ Γ+ [n], then we can take CilX as CilXm = {(σ, τ ) ∈ CilΓ+ [n]m | there is γ ∈ Xm with (Im(σ) ∪ Im(τ )) ⊂ Im(γ)}. Therefore, we can define inductively the cylinder CilΓ+ [n] as follows. Suppose that σ : [m] → [n] is given by σ(0) < σ(1) < · · · < σ(m). Then: 22 • For each 0 ≤ p ≤ m, we have the (m + 1)-simplex (σ(0), 0) < (σ(1), 0) < · · · < (σ(p), 0) < (σ(p), 1) < · · · < (σ(m), 1) = (σ(0), · · · , σ(p), σ(p)′ , · · · , σ(m)′ ) • and for each −1 ≤ q ≤ m + 1 we have the m-simplex (σ(0), 0) < (σ(1), 0) < · · · < (σ(q), 0) < (σ(q + 1), 1) < · · · < (σ(m), 1) = (σ(0), · · · , σ(q), σ(q + 1)′ , · · · , σ(m)′ ), where for q = −1, (σ(0), 1) < (σ(1), 1) < · · · < (σ(m), 1) = (σ ′ (0), · · · , σ(m)′ ), and for q = m+1, (σ(0), 0) < (σ(1), 0) < · · · < (σ(m), 0) = (σ(0), · · · , σ(m)). Remark 7. When studying simplicial theory, the standard way to triangulate the product prism ∆[p]×∆[1] is by taking the (p+1)-simplices (0, · · · , k, k ′ , · · · , p′ ), where the numbers without the prime symbol represent vertices in ∆[p] × {0} and the numbers with the prime symbol represent vertices in ∆[p] × {1}. The simplex (0, · · · , k, k ′ , · · · , p′ ) corresponds to k + 1 zeros and p − k + 1 ones. ˘ + [n] := CilΓ+ [n]\Cil∂Γ+ [n]. One Take ∂Γ+ [n], the boundary of Γ+ [n], and CilΓ ˘ + [n]|: can find a pretty straightforward pattern for the cardinal sequences |CilΓ ˘ + [−1]| |CilΓ ˘ + [0]| |CilΓ ˘ + [1]| |CilΓ ˘ + [2]| |CilΓ ˘ + [3]| |CilΓ ˘ + [4]| |CilΓ ˘ + [5]| |CilΓ ˘ + [6]| |CilΓ .. . −1 0 1 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 .. .. . . 1 0 1 3 0 0 0 0 0 .. . 2 0 0 2 4 0 0 0 0 .. . 3 0 0 0 3 5 0 0 0 .. . 4 0 0 0 0 4 6 0 0 .. . 5 0 0 0 0 0 5 7 0 .. . 6 0 0 0 0 0 0 6 8 .. . ... ... ... ... ... ... ... ... ... .. . This permits us the computation of the cardinal sequences |Cil∂Γ+ [n]| and |CilΓ+ [n]|. Example 2. For instance, for n = 2 we have (see proposition below):     ˘ + [1]|. ˘ + [0]| + 2 + 1 |CilΓ ˘ + [−1]| + 2 + 1 |CilΓ |Cil∂Γ+ [2]| = |CilΓ 1+1 0+1 ˘ + [−1]| = (1, 0, 0, 0, 0, 0, 0, 0, · · · ), |CilΓ ˘ + [0]| = (0, 2, 1, 0, 0, 0, 0, 0, · · · ) As |CilΓ ˘ + [1]| = (0, 0, 3, 2, 0, 0, 0, 0, · · · ), then and |CilΓ |Cil∂Γ+ [2]| = (1, 6, 12, 6, 0, 0, 0, 0, · · · ). 23 Moreover, |CilΓ+ [2]| = =
˘ + [−1]| + 2+1 |CilΓ ˘ + [0]| + |CilΓ 0+1 ˘ + [2]|. |Cil∂Γ+ [2]| + |CilΓ 2+1 1+1
˘ + [1]| + |CilΓ 2+1 2+1
˘ + [2]| |CilΓ ˘ + [2]| = (0, 0, 0, 4, 3, 0, · · · ), As |Cil∂Γ+ [2]| = (1, 6, 12, 6, 0, 0, 0, 0, · · · ) and |CilΓ then |CilΓ+ [2]| = (1, 6, 12, 10, 3, 0, 0, 0, · · · ). The simple algorithm above has a proper generalization: Proposition 3. For every n ∈ N+ , we have: (i) (ii) |Cil∂Γ+ [n]| =  n + 2,      ˘ + [n]|k = n + 1, |CilΓ      0, Pn−1 i=−1 n+1 i+1
if k = n, if k = n + 1 if k ̸∈ {n, n + 1} ˘ + [i]|, |CilΓ ˘ + [n]|. (iii) |CilΓ+ [n]| = |Cil∂Γ+ [n]| + |CilΓ Now, using the result above, we are able to compute the cardinal sequences |Cil∂Γ+ [n]| and |CilΓ+ [n]|. Here, we display the following tables: |Cil∂Γ+ [−1]| |Cil∂Γ+ [0]| |Cil∂Γ+ [1]| |Cil∂Γ+ [2]| |Cil∂Γ+ [3]| |Cil∂Γ+ [4]| |Cil∂Γ+ [5]| |Cil∂Γ+ [6]| .. . −1 0 1 1 1 1 1 1 1 .. . 0 1 0 0 0 0 4 2 6 12 8 22 10 35 12 51 14 70 .. .. . . 2 0 0 0 6 28 60 110 182 .. . 24 3 0 0 0 0 12 55 135 280 .. . 4 0 0 0 0 0 20 96 266 .. . 5 0 0 0 0 0 0 30 154 .. . 6 0 0 0 0 0 0 0 42 .. . ... ... ... ... ... ... ... ... ... .. . |CilΓ+ [−1]| |CilΓ+ [0]| |CilΓ+ [1]| |CilΓ+ [2]| |CilΓ+ [3]| |CilΓ+ [4]| |CilΓ+ [5]| |CilΓ+ [6]| .. . −1 1 1 1 1 1 1 1 1 .. . 0 1 0 0 2 1 4 5 6 12 8 22 10 35 12 51 14 70 .. .. . . 2 0 0 2 10 28 60 110 182 .. . 3 0 0 0 3 17 55 135 200 .. . 4 0 0 0 0 4 26 96 266 .. . 5 0 0 0 0 0 5 37 154 .. . 6 0 0 0 0 0 0 6 50 .. . 0 0 0 0 0 0 0 0 7 .. . ... ... ... ... ... ... ... ... ... .. . Remark 8. For n ≥ 1, a(n) = |CilΓ+ [n − 1]|1 is the sequence of pentagonal (https://oeis.org/A000326). numbers A000326, where a(n) = n(3n−1) 2 Now, taking a(n) = |CilΓ+ [n]|2 one has the sequence A006331, where a(n) = n(n + 1) (2n+1) (https://oeis.org/A006331). 3 We also have that a(n) = |CilΓ+ [n]|3 is the sequence A212415, where a(n) = (n − 1)n(n + 1) (5n+2) (https://oeis.org/A212415). 24 3.1.2 The 0-cylinder Similarly to the case of the standard cylinder, given σ ∈ Γ+ [n]p and τ ∈ Γ+ [n]q we will write σ ≺ τ whenever the following two conditions hold: (i) |σ([p]) ∩ τ ([q])| = 0 (ii) σ(i) < τ (j), for all i ∈ [p] and j ∈ [q] Note that, if σ ∈ Γ+ [n]−1 or τ ∈ Γ+ [n]−1 , then σ ≺ τ . For each object [n] ∈ Γ+ , we introduce Cil0 Γ+ [n], the augmented semi-simplicial subset of CilΓ+ [n] defined as: Cil0 Γ+ [n]m = {(σ, τ ) ∈ CilΓ+ [n]m : σ ≺ τ } ⊂ CilΓ+ [n]m for m ≥ 0 and Cil0 Γ+ [n]−1 = {(∅, ∅)} = {∗}. We consider the same notation as the one introduced for the standard cylinder. Example 3. Note that Cil0 Γ+ [−1]−1 = {∗} and Cil0 Γ+ [−1]m = ∅ for m ≥ 0. Next, we describe Cil0 Γ+ [0], Cil0 Γ+ [1] and Cil0 Γ+ [2]: 1. For n = 0 we have Cil0 Γ+ [0]−1 = {(∅, ∅)} = {∗}, Cil0 Γ+ [0]0 = {(∅, (0)), ((0), ∅)} = {0′ , 0}. 25 If m = 1, then p and q must satisfy p + q = 0, and therefore (p, q) ∈ {(−1, 1), (1, −1), (0, 0)}. Since Γ+ [0]1 is the empty set we have Cil0 Γ+ [0]1 = ∅. Moreover, Cil0 Γ+ [0]m = ∅, for all m ≥ 1. 2. For n = 1 we have Cil0 Γ+ [1]−1 = {∗}, Cil0 Γ+ [1]0 = {(∅, (0)), (∅, (1)), ((0), ∅), ((1), ∅)} = {0′ , 1′ , 0, 1}, Cil0 Γ+ [1]1 = {(∅, (0, 1)), ((0), (1)), ((0, 1), ∅)} = {(0′ , 1′ ), (0, 1′ ), (0, 1)}. If m = 2, then p and q must satisfy p+q = 1, so (p, q) ∈ {(−1, 2), (0, 1), (1, 0)}. In this case, Γ+ [1]2 is the empty set so Cil0 Γ+ [1]2 = {(0, (0, 1)), ((0, 1), 1)} = ∅. Moreover, Cil0 Γ+ [1]m = ∅, for all m ≥ 2. 3. For n = 2 (see Figure 3) we have the following sets of simplices which do not contain “vertical” 1-simplices: Cil0 Γ+ [2]−1 = {∗}, Cil0 Γ+ [2]0 = {0′ , 1′ , 2′ , 0, 1, 2}, Cil0 Γ+ [2]1 = {(0′ , 1′ ), (0′ , 2′ ), (1′ , 2′ ), (0, 1′ ), (0, 2′ ), (1, 2′ ), (0, 1), (0, 2), (1, 2)}, Cil0 Γ+ [2]2 = {(0′ , 1′ , 2′ ), (0, 1′ , 2′ ), (0, 1, 2′ ), (0, 1, 2)}, Cil0 Γ+ [2]m = ∅, m ≥ 3. op Again, we have an induced functor Cil0 : Γ+ → SetsΓ+ , Cil0 ([n]) = Cil0 Γ+ [n] e 0 Γ+ : and, by Theorem 1, one obtains the colimit-preserving functor (−)▷Cil Γop Γop + + Sets → Sets , making commutative the diagram Γ+ _ Y  op SetsΓ+ Cil0 op / SetsΓ+ , 9 e (−)▷Cil 0 op Definition 10. For any augmented semi-simplicial set X ∈ SetsΓ+ we define its 0-cylinder as e 0 Cil0 (X) := X ▷Cil e 0. and similarly, for augmented semi-simplicial maps Cil0 f := f ▷Cil op op This functor Cil0 has a right adjoint rCil0 : SetsΓ+ → SetsΓ+ . 26 0’ 0’ 1’ 1’ 2’ 2’ 0 0 1 1 2 2 Figure 3: The 0-cylinder of Γ+ [2] is on the right (it is obtained from the left by removing simplices which contain some vertical 1-simplex) Note that, if we have an augmented semi-simplicial subset X ⊂ Γ+ [n], we can take Cil0 X = {(σ, τ ) ∈ Cil0 Γ+ [n] | there is γ ∈ Xm with (Im(σ) ∪ Im(τ )) ⊂ Im(γ)}. ˘ 0 Γ+ [n] = Cil0 Γ+ [n] \ In the same manner as the standard cylinder, denote Cil Cil0 ∂Γ+ [n]. In this case, we even find a simpler pattern for the cardinal se˘ 0 Γ+ [n]|: quences |Cil ˘ 0 Γ+ [−1]| |Cil ˘ 0 Γ+ [0]| |Cil ˘ 0 Γ+ [1]| |Cil ˘ 0 Γ+ [2]| |Cil ˘ 0 Γ+ [3]| |Cil ˘ 0 Γ+ [4]| |Cil ˘ 0 Γ+ [5]| |Cil ˘ 0 Γ+ [6]| |Cil .. . −1 0 1 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 .. .. . . 1 0 0 3 0 0 0 0 0 .. . 2 0 0 0 4 0 0 0 0 .. . 3 0 0 0 0 5 0 0 0 .. . 4 0 0 0 0 0 6 0 0 .. . 5 0 0 0 0 0 0 7 0 .. . 6 0 0 0 0 0 0 0 8 .. . ... ... ... ... ... ... ... ... ... .. . which allows us to compute the sequences |Cil0 ∂Γ+ [n]| and |Cil0 ∂Γ+ [n]|. Proposition 4. For every n ∈ N+ , we have: 27 (i) ˘ 0 Γ+ [n]|k = |Cil (ii) |Cil0 ∂Γ+ [n]| = Pn−1 i=−1 n+1 i+1
  n + 2,  0, if k = n, if k ̸= n ˘ 0 Γ+ [i]|. |Cil ˘ 0 Γ+ [n]|. (iii) |Cil0 Γ+ [n]| = |Cil0 ∂Γ+ [n]| + |Cil Example 4. For instance, using Proposition 4 parts (i) and (ii), we have for n = 2 that ˘ 0 Γ+ [−1]| + 3|Cil ˘ 0 Γ+ [0]| + 3|Cil ˘ 0 Γ+ [1]| |Cil0 ∂Γ+ [2]| = |Cil and ˘ 0 Γ+ [−1]| = (1, 0, 0, 0, 0, 0, 0, 0, · · · ) |Cil ˘ 0 Γ+ [0]| = (0, 2, 0, 0, 0, 0, 0, 0, · · · ) |Cil ˘ 0 Γ+ [1]| = (0, 0, 3, 0, 0, 0, 0, 0, · · · ) |Cil Therefore, |Cil0 ∂Γ+ [2]| = (1, 6, 9, 0, 0, 0, 0, 0, · · · ). Finally, using again Proposition 4, and taking into account that the cardinal ˘ 0 Γ+ [2]| = (0, 0, 0, 4, 0, 0, · · · ), we have sequence |Cil ˘ 0 Γ+ [2]| = (1, 6, 9, 4, 0, 0, 0, · · · ). |Cil0 Γ+ [2]| = |Cil0 ∂Γ+ [2]| + |Cil Using this proposition, we are able to compute the cardinal sequences |Cil0 ∂Γ+ [n]| and |Cil0 Γ+ [n]|. Here we list the first examples: |Cil0 ∂Γ+ [−1]| |Cil0 ∂Γ+ [0]| |Cil0 ∂Γ+ [1]| |Cil0 ∂Γ+ [2]| |Cil0 ∂Γ+ [3]| |Cil0 ∂Γ+ [4]| |Cil0 ∂Γ+ [5]| |Cil0 ∂Γ+ [6]| .. . −1 0 1 1 1 1 1 1 1 .. . 0 1 0 0 0 0 4 0 6 9 8 18 10 30 12 45 14 63 .. .. . . 2 0 0 0 0 16 40 80 140 .. . 28 3 0 0 0 0 0 25 75 175 .. . 4 0 0 0 0 0 0 36 126 .. . 5 0 0 0 0 0 0 0 49 .. . 6 0 0 0 0 0 0 0 0 .. . ... ... ... ... ... ... ... ... ... .. . |Cil0 Γ+ [−1]| |Cil0 Γ+ [0]| |Cil0 Γ+ [1]| |Cil0 Γ+ [2]| |Cil0 Γ+ [3]| |Cil0 Γ+ [4]| |Cil0 Γ+ [5]| |Cil0 Γ+ [6]| .. . −1 1 1 1 1 1 1 1 1 .. . 0 1 0 0 2 0 4 3 6 9 8 18 10 30 12 45 14 63 .. .. . . 2 0 0 0 4 16 40 80 140 .. . 3 0 0 0 0 5 25 75 175 .. . 4 0 0 0 0 0 6 36 126 .. . 5 0 0 0 0 0 0 7 49 .. . 6 0 0 0 0 0 0 0 8 .. . ... ... ... ... ... ... ... ... ... .. . Remark 9. Note that a(n) = |Cil0 Γ+ [n]|1 is the sequence A045943 of triangular (https://oeis.org/A045943). matchstick numbers: a(n) = 3n (n+1) 2 Taking a(n) = |Cil0 Γ+ [n + 1]|2 , we have the sequence A210440, where a(n) = 2n(n + 1) (n+2) (https://oeis.org/A210440). 3 3.1.3 The 2-cylinder Now, we give the third kind of cylinder we are considering in this work. Definition 11. For each [n] ∈ Γ+ , we introduce Cil2 Γ+ [n], the augmented semi-simplicial set defined as the join: Cil2 Γ+ [n] := Γ+ [n] ⊞ Γ+ [n]. Recall that, for each m ∈ Γ+ , Cil2 Γ+ [n]m is constituted by pairs of the form (σ, τ ) ∈ Γ+ [n]p × Γ+ [n]q with p + q = m − 1. We can follow the notation established in previous subsections. Moreover, if σ ∈ Γ+ [n]p , τ ∈ Γ+ [n]q , p+q = m − 1, and 0 ≤ i ≤ m, the face operator is given by  if 0 ≤ i ≤ p,  (di (σ), τ ) di ((σ, τ )) =  (σ, di−p−1 (τ )) if p + 1 ≤ i ≤ m. Example 5. We describe Cil2 Γ+ [1]: Cil2 Γ+ [1]−1 = {∗}, Cil2 Γ+ [1]0 = {(∅, (0)), (∅, (1)), ((0), ∅), ((1), ∅)} = {0′ , 1′ , 0, 1}, Cil2 Γ+ [1]1 = {(∅, (0, 1)), ((0), (0)), ((0), (1)), ((1), (0)), ((1), (1)), ((0, 1), ∅)} = {(0′ , 1′ ), (0, 0′ ), (0, 1′ ), (1, 0′ ), (1, 1′ ), (0, 1)}, Cil2 Γ+ [1]2 = {((0), (0, 1)), ((1), (0, 1)), ((0, 1), (0)), ((0, 1), (1)))} = {(0, 0′ , 1′ ), (1, 0′ , 1′ ), (0, 1, 0′ ), (0, 1, 1′ )}, Cil2 Γ+ [1]3 = {((0, 1), (0, 1))} = {(0, 1, 0′ , 1′ ))}. And Cil2 Γ+ [1]m = ∅ for m ≥ 4. 29 We have an obvious induced functor Cil2 : Γ+ → Sets, Cil2 ([n]) = Cil2 Γ+ [n] op e 2 : SetsΓ+ → and, by Theorem 1, the colimit-preserving functor (−)▷Cil op SetsΓ+ , making commutative the diagram Cil2 Γ+ _ Y  op SetsΓ+ op / SetsΓ+ , 9 e (−)▷Cil 2 op Definition 12. For any augmented semi-simplicial set X ∈ SetsΓ+ , we define its 2-cylinder as e 2 Cil2 (X) := X ▷Cil e 2. Similarly, for augmented semi-simplicial maps Cil2 (f ) := f ▷Cil op op This functor Cil2 has a right adjoint rCil2 : SetsΓ+ → SetsΓ+ . If we have an augmented semi-simplicial subset X ⊂ Γ+ [n], we can take: Cil2 X = {(σ, τ ) ∈ Cil2 Γ+ [n] | there is γ ∈ Xm with (Im(σ)∪Im(τ )) ⊂ Im(γ)}. ˘ 2 Γ+ [n] = Cil2 Γ+ [n] \ Cil2 ∂Γ+ [n], then it is not hard to prove If we denote Cil
the following result. Recall our convention that the combinatorial number pq is zero whenever p < q or q < 0. Proposition 5. For any n, m ≥ −1 we have ˘ 2 Γ+ [n] is (i) The number total of simplices in Cil   i  n  X n+1 X i+1 . i + 1 j=−1 j + 1 i=−1 ˘ 2 Γ+ [n]|m = (ii) |Cil Pn i=−1 n+1 i+1
i+1 m−n
. Proof. It suffices to count the pair of subsets {v0 , · · · , vp }, {w0 , · · · , wq } of {0, 1, · · · , n} = [n] such that their union is equal to [n] and (1 + p) + (1 + q) = (1 + m) − (1 + n). ˘ 2 Γ+ [n]|, Using the proposition above, we can describe the cardinal sequences |Cil 30 for all n. Here we provide a list of the six first examples: ˘ 2 Γ+ [−1]| |Cil ˘ 2 Γ+ [0]| |Cil ˘ 2 Γ+ [1]| |Cil ˘ 2 Γ+ [2]| |Cil ˘ 2 Γ+ [3]| |Cil ˘ 2 Γ+ [4]| |Cil ˘ 2 Γ+ [5]| |Cil ˘ 2 Γ+ [6]| |Cil .. . −1 0 1 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 .. .. . . 1 0 1 4 0 0 0 0 0 .. . 2 3 4 5 6 ... 0 0 0 0 0 ... 0 0 0 0 0 ... 4 1 0 0 0 ... 8 12 6 1 0 ... 0 16 32 24 8 ... 0 0 32 80 80 . . . 0 0 0 64 192 . . . 0 0 0 0 128 . . . .. .. .. .. .. .. . . . . . . This will enable the calculation of the sequences |Cil2 ∂Γ+ [n]| and |Cil2 ∂Γ+ [n]|. We need the following result, which provides an easy algorithm. Proposition 6. For any n, m ≥ −1 we have:
Pn−1 ˘ (i) |Cil2 ∂Γ+ [n]| = i=−1 n+1 i+1 |Cil2 Γ+ [i]|. ˘ 2 Γ+ [n]|. (ii) |Cil2 Γ+ [n]| = |Cil2 ∂Γ+ [n]| + |Cil Example 6. For instance, for n = 2, we have

˘ 2 Γ+ [1]|. ˘ 2 Γ+ [0]| + 3 |Cil ˘ 2 Γ+ [−1]| + 3 |Cil |Cil2 ∂Γ+ [2]| = |Cil 2 1 ˘ 2 Γ+ [−1]| = (1, 0, 0, 0, · · · ), |Cil ˘ 2 Γ+ [0]| = (0, 2, 1, 0, 0, 0, · · · ) and |Cil ˘ 2 Γ+ [1]| = As |Cil (0, 0, 4, 4, 1, 0, 0, 0, · · · ) we conclude that |Cil2 ∂Γ+ [2]| = (1, 6, 15, 12, 3, 0, 0, 0, · · · ). ˘ 2 Γ+ [2]| = (0, 0, 0, 8, 12, 6, 1, 0, 0, · · · ) we have that Finally, as |Cil ˘ 2 Γ+ [2]| = (1, 6, 15, 20, 15, 6, 1, 0, · · · ). |Cil2 Γ+ [2]| = |Cil2 ∂Γ+ [2]| + |Cil Using this simple algorithm, we can compute the sequence cardinals |Cil2 ∂Γ+ [n]| and |Cil2 Γ+ [n]|, for all n. Here, we provide a table for the first examples: 31 |Cil2 ∂Γ+ [−1]| |Cil2 ∂Γ+ [0]| |Cil2 ∂Γ+ [1]| |Cil2 ∂Γ+ [2]| |Cil2 ∂Γ+ [3]| |Cil2 ∂Γ+ [4]| |Cil2 ∂Γ+ [5]| |Cil2 ∂Γ+ [6]| |Cil2 ∂Γ+ [7]| .. . −1 0 1 1 1 1 1 1 1 1 .. . 0 0 0 4 6 8 10 12 14 16 .. . 1 0 0 2 15 28 45 66 91 120 .. . 2 0 0 0 12 56 120 220 364 560 .. . 3 0 0 0 3 54 210 495 1001 1820 .. . 4 0 0 0 0 24 220 792 2002 4368 .. . 5 0 0 0 0 4 130 860 3003 8008 .. . 6 0 0 0 0 0 40 600 3304 11440 .. . |Cil2 Γ+ [−1]| |Cil2 Γ+ [0]| |Cil2 Γ+ [1]| |Cil2 Γ+ [2]| |Cil2 Γ+ [3]| |Cil2 Γ+ [4]| |Cil2 Γ+ [5]| |Cil2 Γ+ [6]| .. . −1 1 1 1 1 1 1 1 1 .. . 0 1 0 0 2 1 4 6 6 15 8 28 10 45 12 66 14 91 .. .. . . 2 0 0 4 20 56 120 220 364 .. . 3 0 0 1 15 70 210 495 1001 .. . 4 0 0 0 6 56 252 792 2002 .. . 5 0 0 0 1 28 120 495 3003 .. . 6 0 0 0 0 8 45 220 3432 .. . op ... ... ... ... ... ... ... ... ... ... .. . ... ... ... ... ... ... ... ... ... .. . op Remark 10. Observe thatopthe functor Cil2 : SetsΓ+ → SetsΓ+ and the funcΓop tor Dup : Sets + → SetsΓ+ given as Dup(X) := X ⊞ X verify that DupY = Cil2 Y (Y is the Yoneda functor). However, these functors are different. For instance, if X = ∂Γ+ [2], then one has that |Cil2 (∂Γ+ [2])| = (1, 6, 15, 12, 3, 0, · · · ) and |Dup(∂Γ+ [2])| = (1, 6, 15, 18, 9, 0, · · · ). This implies that Dup does not preserve colimits. op op Remark 11. For the functors Cil0 , Cil, Cil2 , Dup : SetsΓ+ → SetsΓ+ , there are natural transformacions Cil0 ⊂ Cil ⊂ Cil2 ⊂ Dup. Remark 12. Note that, for n ≥ 0, a(n) = |CilΓ+ [n − 1]|1 is the sequence AA000384 of hexagonal numbers: a(n) = n(2n−1) (https://oeis.org/AA000384). Taking a(n) = |CilΓ+ [n]|2 , we have the sequence A002492 which is the sum (https://oeis.org/ of the first even squares, where a(n) = 2n(n + 1) (2n+1) 3 A002492). 32 3.2 Barycentric subdivision of an augmented semi-simplicial set In this subsection we describe the barycentric subdivision of an augmented semisimplicial set. For any [n] ∈ Γ+ , we first introduce the barycentric subdivision of Γ+ [n], denoted by SdΓ+ [n], as the following augmented semi-simplicial set: If −1 ≤ m ≤ n, an element a = (φ−1 , φ0 , · · · , φm ) ∈ SdΓ+ [n]m is just a chain of composable morphisms in Γ+ , of the form φ−1 φ0 φm−1 φ1 φm ∅ = [−1] −→ [k0 ] −→ [k1 ] −→ · · · −→ [km ] −→ [km+1 ] = [n] where 0 ≤ k0 < k1 < · · · < km ≤ n. We set SdΓ+ [n]m = ∅, if m > n. Now, if m ≥ 0 and 0 ≤ i ≤ m, then the face operator di : SdΓ+ [n]m → SdΓ+ [n]m−1 is given as di (a) = di ((φ−1 , φ0 , · · · , φm )) := (φ−1 , ..., φi−2 , φi ◦ φi−1 , φi+1 , · · · , φm ) Observe that SdΓ+ [n]−1 = {∅ = (∅ → [n])}. In particular, SdΓ+ [−1]−1 = {∅ = (∅ → [−1])}; moreover SdΓ+ [−1]m = ∅, for all m. We also have SdΓ+ [0]−1 = {∅ = (∅ → [−1])}, SdΓ+ [0]0 = {∅ → [0]} and SdΓ+ [0]m = ∅, for all m ≥ 1. This implies that SdΓ+ [−1] = Γ+ [−1] and SdΓ+ [0] = Γ+ [0]. (1) op Then, we can consider the canonical functor Sd : Γ+ → SetsΓ+ , Sd([n]) = op e SdΓ+ [n] and, by Theorem 1, the colimit-preserving functor (−)▷Sd : SetsΓ+ → op SetsΓ+ , making commutative the diagram Γ+ Y  op SetsΓ+ Sd op / SetsΓ+ 9 e (−)▷Sd and it has a right adjoint op op rSd : SetsΓ+ → SetsΓ+ op Definition 13. For any augmented semi-simplicial set X ∈ SetsΓ+ we define its barycentric subdivision as ˜ Sd(X) = X ▷Sd ˜ Similarly, for augmented semi-simplicial maps, Sd(f ) = f ▷Sd. For the next result, we recall that S+ [n − 1] = Γ+ [n] \ {ιn }, where ιn : [n] → [n] is the identity map. 33 op op Proposition 7. Consider the cone funtors Conl , Conr : SetsΓ+ → SetsΓ+ . Then there are canonical isomorphisms Conl (Sd(S+ [n − 1])) ∼ = SdΓ+ [n] ∼ = Conr (Sd(S+ [n − 1]) for k ≥ 0. Remark 13. The isomorphisms above can also be written as: Γ+ [0] ⊞ (Sd(S+ [n − 1])) 3.3 3.3.1 ˜ Γ+ [0] ⊞ (S+ [n − 1]▷Sd) ˜ (Γ+ [0] ⊞ S+ [n − 1])▷Sd ˜ Γ+ [n]▷Sd ˜ (S+ [n − 1] ⊞ Γ+ [0])▷SdΓ + (Sd(S+ [n − 1])) ⊞ Γ+ [0]. = ∼ = ∼ = ∼ = ∼ = Cylinders for integer sequences The standard cylinder for integer sequences Associated with the augmented semi-simplicial sets CilΓ+ [n], Cil∂Γ+ [n], we consider the following augmented integer sequences: ˘ + [n] ∈ ZNop + is given by Definition 14. The augmented sequence cilγ  n + 2 if m = n,      ˘ + [n]m := n + 1 if m = n + 1 cilγ      0 if m ̸∈ {n, n + 1} (2) ˘ = (cil ˘ nm ) by cil ˘ nm := cilγ ˘ + [n]m . and the augmented matrix cil op The augmented sequences cil∂γ+ [n], cilγ+ [n] ∈ ZN+ are defined as  1+n ˘ cilγ+ [j] 1+j (3)  n  X 1+n ˘ cilγ+ [j] 1+j j=−1 (4) cil∂γ+ [n] := n−1 X j=−1 cilγ+ [n] :=  and the augmented matrices cil∂ = (cil∂nm ) , cil = (cilnm ) by cil∂nm := cil∂γ+ [n]m , cilnm := cilγ+ [n]m . Recall that, for k ∈ N+ , 1k denotes the augmented sequence given by (1k )i = δk,i . For k, l ∈ N+ , we also consider the augmented matrices 1k,l where (1k,l )i,j := δk,i δl,j ˘ cil∂, cil can be described as folProposition 8. The augmented matrices cil, lows: 34 ˘ nm = (n + 2)δn,m + (n + 1)δn+1,m (i) cil
Pn−1 (ii) cil∂nm = j=−1 1+n 1+j ((j + 2)δj,m + (j + 1)δj+1,m )
Pn (iii) cilnm = j=−1 1+n 1+j ((j + 2)δj,m + (j + 1)δj+1,m ) ˘ + [j]m = ((j + 2)δj,m + (j + 1)δj+1,m ) we have cilγ+ [n]m = Proof. (iii) Since cilγ

Pn Pn 1+n 1+n ˘ j=−1 1+j ((j + 2)δj,m + (j + 1)δj+1,m ). j=−1 1+j cilγ+ [j]m = Proposition 9. The following equalities hold true: cilγ+ [n] = |Cil(Γ+ [n])|, cil = |Cil| (5) where Cil is the co-semi-simplicial object given in subsection 3.1 Proof. It is a direct consequence of Proposition 3 and Definition 14. op Definition 15. Given an augmented integer sequence a ∈ ZN+ , the standard cylinder of a is the sequence obtained as cil(a) := a˜▷cil. This construction gives rise to a functor op op cil : ZN+ → ZN+ ˘ and therefore a˜▷cil has sense, for all Remark 14. Observe that bin−1 · cil = cil Nop a∈Z +. op Corollary 1. Given an augmented integer sequence a ∈ ZN+ , its standard ˘ cylinder can be described as cil(a) = a · cil. op op N+ Remark op15. Note that the functor cil : ZN+ → Z has an inverse functor op −1 −1 N+ Nop N −1 ˘ cil : Z → Z + , cil (a) = a · (cil) , a ∈ Z + . For k ∈ Z, one can also ˘ k , a ∈ ZNop + . consider the iteration functor cilk given by cilk (a) = a · (cil) 3.3.2 The 0-cylinder for integer sequences This subsection will follow a similar structure to the previous one for standard cylinders of sequences. Indeed, associated with Cil0 Γ+ [n], Cil0 ∂Γ+ [n], we consider the following augmented integer sequences; ˘ 0 γ+ [n] ∈ ZNop + is given by Definition 16. The sequence cil ˘ 0 γ+ [n]m = (n + 2)δn,m cil (6) ˘ 0 = ((cil ˘ 0 )nm ) is given by (cil ˘ 0 )nm = cil ˘ 0 γ+ [n]m . and the augmented matrix cil op The sequences cil0 ∂γ+ [n], cil0 γ+ [n] ∈ ZN+ are defined by n−1 X 1 + n  ˘ 0 γ+ [j] cil cil0 ∂γ+ [n] = 1+j j=−1 35 (7)  n  X 1+n ˘ cil0 γ+ [j] cil0 γ+ [n] = 1+j j=−1 (8) and the augmented matrices cil0 ∂ = ((cil0 ∂)nm ) , cil0 = ((cil0 )nm ) are given by (cil0 ∂)nm = cil0 ∂γ+ [n]m , (cil0 )nm = cil0 γ+ [n]m . ˘ 0 , cil0 ∂, cil0 can be described as Proposition 10. The augmented matrices cil follows: ˘ 0 )nm = (n + 2)δn,m , (i) (cil
Pn−1 (ii) (cil0 ∂)nm = j=−1 1+n 1+j (j + 2)δj,m ,
Pn (iii) (cil0 )nm = j=−1 1+n 1+j (j + 2)δj,m . Proof. It is a routine check. Proposition 11. For the augmented simplicial set Cil0 Γ+ [n] we have cil0 γ+ [n] = |Cil0 Γ+ [n]|, cil0 = |Cil0 | (9) where Cil0 is the co-semi-simplicial object given in subsection 3.1 Proof. It is a consequence of Proposition 4 and Definition 16. We define the notion of a 0-cylinder of an augmented sequence analogously to that of its standard cylinder. op Definition 17. The 0-cylinder of an augmented integer sequence a ∈ ZN+ is defined as the tilde-triangle product cil0 (a) := a˜▷cil0 . This construction gives rise to a functor op op cil0 : ZN+ → ZN+ ˘ 0 , this definition is well-given. In this case, as bin−1 · cil0 = cil op Corollary 2. Given an augmented integer sequence a ∈ ZN+ its 0-cylinder can be described as ˘ 0. cil0 (a) = a · cil op op has an inverse Remark 16. Note that op the functor cil0 : ZN+ → ZN+ also −1 Nop N ˘ 0 )−1 , a ∈ ZNop + . For k ∈ Z, one functor cil0 : Z + → Z + , cil−1 (a) = a · ( cil 0 ˘ 0 )k , a ∈ ZNop + . can also consider the iteration functor cilk0 given by cilk0 (a) = a·(cil In this case, (cilk0 (a))m = am (2 + m)k . 36 3.3.3 The 2-cylinder for integer sequences Again, following the same structure of previous subsection, and associated with the augmented semi-simplicial sets CilΓ+ [n], Cil∂Γ+ [n], we can consider the following integer sequences: ˘ 2 γ+ [n] ∈ ZNop + is given by Definition 18. The augmented sequence cil ˘ 2 γ+ [n]m := cil   n  X i+1 n+1 m−n i+1 i=−1 (10) op and the augmented sequences cil2 ∂γ+ [n], cil2 γ+ [n] ∈ ZN+ by  1+n ˘ cilγ+ [j] 1+j (11)  n  X 1+n ˘ cilγ+ [j] 1+j j=−1 (12) cil2 ∂γ+ [n] := n−1 X j=−1 cil2 γ+ [n] :=  ˘ 2 , cil2 ∂, cil2 can be described as Proposition 12. The augmented matrices cil follows:
i+1
n+1 ˘ 2 )nm = Pn (i) (cil i=−1 i+1 m−n ,
j+1
i+1
Pn−1 Pj (ii) (cil2 ∂)nm = j=−1 i=−1 1+n i+1 m−j , 1+j
i+1

Pn Pj j+1 (iii) (cil2 )nm = j=−1 i=−1 1+n i+1 m−j . 1+j
i+1
j+1 ˘ 2 γ+ [j]m = Pj Proof. (iii) Since cil i=−1 i+1 m−j , we have

Pn Pn 1+n Pj ˘ cil2 γ+ [n]m = j=−1 1+n i=−1 j=−1 1+j 1+j cil2 γ+ [j]m = j+1 i+1
i+1 m−j
Proposition 13. For the augmented simplicial set Cil2 Γ+ [n] we have cil2 γ+ [n] = |Cil2 Γ+ [n]|, cil2 = |Cil2 | (13) where Cil2 is the co-semi-simplicial object given in subsection 3.1 Proof. It is a consequence of Proposition 6 and Definition 18. op Definition 19. The 2-cylinder of an augmented integer sequence a ∈ ZN+ , is defined as the sequence cil2 (a) = a˜▷cil2 . This construction gives us a functor op op cil2 : ZN+ → ZN+ 37 ˘ 2 this is well defined. Moreover As bin−1 · cil2 = cil op Corollary 3. The 2-cylinder of a ∈ ZN+ can be described as ˘ 2. cil2 (a) = a · cil op op op N+ Remark 17. The functor cil2 : ZN+ → ZN+ has an inverse functor cil−1 → 2 : Z op op −1 N+ N+ −1 ˘ given as cil2 (a) = a · (cil2 ) , for all a ∈ Z . For k ∈ Z, one can also Z ˘ 2 )k , for all a ∈ ZNop + . consider the iteration functor cilk2 given by cilk2 (a) = a · (cil 3.4 Subdivision for integer sequences Now, we analyse (barycentric) subdivisions for sequences. Recall that, for n ≥ 0 we have [n] = {0, · · · , n} whereas [−1] = ∅. As the empty set is always included, the cardinal of the power set P([n]) is 2n+1 . For all n, p ∈ N+ , we consider: Cad+ [n]p := {∅ = N−1 ⊂ N0 ⊂ N1 ⊂ · · · ⊂ Np ⊆ [n] : Ni ̸= Ni+1 , −1 ≤ i ≤ p−1}} ˘ + [n]p := {∅ = N−1 ⊂ N0 ⊂ N1 ⊂ · · · ⊂ Np = [n] : Ni ̸= Ni+1 , −1 ≤ i ≤ p−1}} Cad Cad[n]p := {∅ = N−1 ⊆ N0 ⊂ N1 ⊂ · · · ⊂ Np ⊆ [n] : Ni ̸= Ni+1 , 0 ≤ i ≤ p−1}} where the notation Cad comes from the Spanish word “Cadena”, which means chain. Taking cardinals, we obtain the augmented sequences cad+ [n]p := |Cad+ [n]p |, ˘ + [n]p := |Cad ˘ + [n]p |, cad cad[n]p := |Cad[n]p | op and the corresponding matrices in (ZN+ )N+ :     ˘ + [−1] cad+ [−1] cad     .. ..     + . .   + ˘   cad :=  , cad :=  , + +  cad [n] ˘   cad [n]     .. .. . . op   cad[−1)   ..   .  cad :=   cad[n]    .. . op Lemma 1. The cone functor con : ZN+ → ZN+ satisfies the following relation: con(cad+ [n]) = cad[n] = cad+ [n] ⊞ c = cad+ [n] · R(c) and, therefore, cad = cad+ · R(c). Theorem 3. The following equality holds true:   + n+1 ˘ cad [n]p = (p + 1)! . p+1 38 ˘ + [n]p , the elements in Proof. Observe that, by definition of Cad {N0 , N1 \ N0 , · · · , Np \ Np−1 } are non empty and disjoint subsets. Consider the set Partitions[n]p consisting of all sets of the form {S0 , · · · , Sp } where every Si is not empty, Si ∩ Sj = ∅ for all i ̸= j, and S0 ∪ · · · ∪ Sp = [n]. Then, from the definition of Stirling number of second kind (see the introduction section 1), it follows that   n+1 |Partitions[n]p | = . p+1 There is a canonical surjective map ˘ + [n]p → Partitions[n]p Dif : Cad sending each chain ∅ = N−1 ⊂ N0 ⊂ N1 ⊂ · · · ⊂ Np to {N0 , N1 \ N0 , · · · , Np \ Np−1 }. Since, in order to construct all the strictly increasing chains associated to a partition, we have to take into account all possible permutations of the members of this partition, we obtain:   + n+1 ˘ cad [n]p = (p + 1)! . p+1 Proposition 14. The following properties hold true: + ˘ [k]p = 0. (i) If k < p, then cad
Pn Pn ˘ + (ii) cad+ [n]p = k=−1 n+1 k=p k+1 cad [k]p = + + n+1 k+1 ˘ and bin−1 · cad+ = cad ˘ . (iii) cad+ = bin · cad
˘ + [k]p . cad ˘ + [k]p = ∅. Proof. (i) Just observe that, if k < p, then Cad ˘ + [S]p (ii) For every finite set S = {s0 , · · · , sk }, we can obviously define Cad ˘ + [S]p and Cad ˘ + [k]p so that there is a bijective correspondence between Cad (|S| = k + 1). We can consider the bijection distribute : Cad+ [n]p −→ G ˘ + [S]p Cad S⊂[n],|S|≥(p+1) where each chain ∅ = N−1 ⊂ N0 ⊂ N1 ⊂ · · · ⊂ Np ⊆ [n]) is carried to + ˘ [S]p . (∅ = N−1 ⊂ N0 ⊂ N1 ⊂ · · · ⊂ Np = S) ∈ Cad 39 Therefore, we obtain + + cad [n]p = |Cad [n]p | =  n  X n+1 k=p k+1 ˘ + [k]p | = |Cad  n  X n+1 k=p k+1 ˘ + [k]p . cad (iii) This item follows from (i), (ii) and the definition of matrix multiplication. Using the proposition above, one can easily compute the coefficients in the ˘ + and cad. Here, we are displaying the first rows augmented matrices cad+ , cad of such matrices: cad+ cad+ [−1] cad+ [0] cad+ [1] cad+ [2] cad+ [3] cad+ [4] cad+ [5] cad+ [6] .. . −1 1 1 1 1 1 1 1 1 .. . ˘ + cad ˘ + [−1] cad 0 0 1 3 7 15 31 63 127 .. . −1 0 1 0 0 2 12 50 180 602 1932 .. . 2 0 0 0 6 60 390 2100 10206 .. . 3 0 0 0 0 24 360 3360 25200 .. . 4 0 0 0 0 0 120 2520 31920 .. . 5 0 0 0 0 0 0 720 20160 .. . 6 0 0 0 0 0 0 0 5040 .. . 7 ··· ··· ··· ··· ··· ··· ··· ··· .. . 1 2 3 4 5 6 7 1 0 0 0 0 0 0 0 ··· ˘ + [0] cad ˘ + [1] cad 0 1 0 0 0 0 0 0 ··· 0 1 2 0 0 0 0 0 ··· ˘ + [2] cad ˘ + [3] cad 0 1 6 6 0 0 0 0 ··· 0 1 14 36 24 0 0 0 ··· ˘ + [4] cad ˘ + [5] cad 0 1 30 150 240 120 0 0 ··· 0 1 62 540 1560 1800 720 0 ··· ˘ + [6] cad .. . 0 .. . 1 126 .. .. . . 1806 .. . 8400 .. . 16800 .. . 15120 .. . cad cad[−1] cad[0] cad[1] cad[2] cad[3] cad[4] .. . −1 1 1 1 1 1 1 .. . 0 1 2 4 8 16 32 .. . 1 0 1 5 19 65 211 .. . 2 0 0 2 18 110 570 .. . 40 3 0 0 0 6 84 750 .. . 4 0 0 0 0 24 480 .. . 5 6 0 0 0 0 0 0 0 0 0 0 120 0 .. .. . . 5040 · · · .. .. . . 7 ··· ··· ··· ··· ··· ··· .. . op Definition 20. The barycentric subdivision of a ∈ (ZN+ )fin is the augmented sequence defined as sd(a) := a˜▷cad+ . This construction gives rise to a functor op op sd : (ZN+ )fin → (ZN+ )fin ˘ + , we also denote sd ˘ = cad ˘ + and we have an alternative As bin−1 · cad+ = cad description of sd(a): Corollary 4. The barycentric subdivision of an augmented integer sequence op ˘ + = a · sd. ˘ a ∈ (ZN+ )fin can be described by sd(a) = a · cad Remark 18. Given any augmented sequence c = (c−1 , c0 , c1 , · · · , cn , 0, · · · ) ∈ op (ZN+ )fin we have the formula (sdc)j = n X (j + 1)!ci i=−1   i+1 . j+1 In particular, the first three terms of this sequence are:   Pn i+1 ) = c−1 + 0 + · · · 0 = c−1 (sdc)−1 = i=−1 0!ci  0 Pn i+1 (sdc)0 = i=−1 1!ci = c0 + · · · + cn      1  Pn n+1 3 i+1 . + · · · + 2cn = 2c1 + 2c2 (sdc)1 = i=−1 2!ci 2 2 2 3.5 Comparing geometric and arithmetic constructions through the sequential cardinal functor From propositions 9, 11 13, we already know that cil = |Cil|, cil0 = |Cil0 | and cil2 = |Cil2 |. The following theorem gives an interesting relationship between geometric and arithmetic cylinders: op Theorem 4. If X ∈ (Setsfin )Γ+ , then ˜ (i) |Cil(X)| = |X ▷Cil| = |X|˜▷|Cil|, ˜ 0 | = |X|˜▷|Cil0 |, (ii) |Cil0 (X)| = |X ▷Cil ˜ 2 | = |X|˜▷|Cil2 |. (iii) |Cil2 (X)| = |X ▷Cil Therefore, the following diagrams are commutative: op (Setsfin )Γ+ Cil Z Nop + (Setsfin )Γ+ cil /Z Cil0 op / (Setsfin )Γ+ |·| |·| |·|  op op / (Setsfin )Γ+   Nop + Z 41 Nop + |·| cil0 /Z  Nop + op (Setsfin )Γ+ Cil2 op / (Setsfin )Γ+ |·|  |·|  cil2 op Z N+ op / Z N+ . Proof. (i), (ii) and (iii) follow from Theorem 2, the commutativity of the diagrams being a direct consequence. Example 7. Consider H the augmented semi-simplicial hexaedron. Then (see Figure 4): ˘ and this • |Cil(H)| = Cil(|H|) = |H| · Cil,  1 0 0  0 2 1 ˘ = (1, 6, 6, 0, 0, 0, · · · )  |H|·Cil  0 0 3  .. .. .. . . . ˘ 0 and • |Cil0 (H)| = Cil0 (|H|) = |H| · Cil  1 0  0 2 ˘ 0 = (1, 6, 6, 0, 0, 0, · · · )  |H|·Cil  0 0  .. .. . . ˘ 2 and • |Cil2 (H)| = Cil2 (|H|) = |H| · Cil  1 0  0 2 ˘ 2 = (1, 6, 6, 0, 0, 0, · · · )  |H|·Cil  0 0  .. .. . . matrix multiplication is  0 ··· 0 ···   = (1, 12, 24, 12, 0, 0, 0, · · · ) 2 ···   .. . . . . 0 0 0 0 3 0 .. .. . . ··· ··· ··· .. .     = (1, 12, 18, 0, 0, 0, · · · )  ··· ··· ··· .. . 0 0 0 1 0 0 4 4 1 .. .. .. . . .     = (1, 12, 30, 24, 6, 0, 0, 0, · · · )  Now we compare geometric and arithmetic subdivisions through the sequential cardinal functor. We start with this simple lemma: Lemma 2. |SdΓ+ [n]| = cad+ [n], for all n ∈ N+ . Proof. Associated to any a ∈ SdΓ+ [n]m , represented as φ−1 φ0 φm−1 φ1 φm ∅ = [−1] −→ [k0 ] −→ [k1 ] −→ · · · −→ [km ] −→ [km+1 ] = [n], we consider the subset chain ∅ = [−1] ⊂ im(φ0 ) ⊂ im(φ1 ) ⊂ · · · ⊂ im(φm ) ⊂ [n] 42 3 3 4 4 2 2 3’ 4’ 4 2 3’ 2’ 5’ 0’ 5 3’ 4’ 1’ 5’ 3 2’ 4’ 1’ 5’ 0’ 1 0 1’ 0’ 5 1 2’ 5 0 1 0 Figure 4: From left to right Cil0 (H) ⊂ Cil(H) ⊂ Cil2 (H) This correspondence gives a bijection between SdΓ+ [n]m and Cad+ [n]m , for all m. Finally, we give a result giving a nice relationship between geometric and arithmetic subdivisions. We use the notation sdγ+ [n] := |Sd(Γ+ [n])| and take sd := |Sd|, the correspondent matrix. op op Theorem 5. If X ∈ (SetsΓ+ )fin and a ∈ (ZN+ )fin , then ˜ (i) |Sd(X)| = |X ▷Sd| = |X|˜▷|Sd|, (ii) sd = |Sd| = cad+ , ˘ (iii) sd(a) = a˜ ▷sd = a · sd. Moreover, the following diagram is commutative: op (SetsΓ+ )fin Sd op / (SetsΓ+ )fin |·|  op (ZN+ )fin |·| sd  op / (ZN+ )fin Proof. (i) follows by Definition 13 and Theorem 2. By Lemma 2, we have |SdΓ+ [n]m | = cad+ [n]m , and therefore |SdΓ+ [n]| = cad+ [n] and |Sd| = cad+ ; so (ii) holds. Finally, (iii) and the commutativity of the diagram are consequences of (i) and (ii). 43 4 Conclusions and future work In this work, we have demonstrated how the subdivision and cylinder constructions for semi-simplicial sets can be obtained by taking certain actions of appropriate co-semi-simplicial objects. Additionally, we have discussed the computation of the sequential cardinality of cylinders and the sequential cardinality of barycentric subdivisions. The sequential cardinality of cylinders can be easily computed using locally finite matrices, which are matrices with rows and columns that are eventually zero. Furthermore, the sequential cardinality of the barycentric subdivision can be computed using chain-power numbers and Stirling numbers (see [2] and [18]). Another interesting objective would be to construct a categorical semi-ring (or symmetric bimonoidal category structure) K op ((Setsfin )Γ+ , ⊞, , Γ+ [−1], Γ+ [0]) that satisfies the following properties: op If X, Y ∈ (Setsfin )Γ+ and dim(X), dim(Y ) are finite, then K dim(X Y ) = (dim(X) + 1)(dim(Y ) + 1) − 1. op If X, Y, Z ∈ (Setsfin )Γ+ , then K K K (X ⊞ Y ) Z∼ Z) ⊞ (Y Z). = (X If we are able to develop this construction, we will analyze the following: J (i) the relationship between |X Y | and |X| and |Y |. Another intriguing aim is to consider the Betti sequences of a finite augmented semi-simplicial set β(X) = (β−1 (X), β0 (X), β1 (X), · · · ) and to study (iii) the relationship between β(X ⊞ Y ) and β(X) and β(Y ), and J (iii) the relationship between β(X Y ) and β(X) and β(Y ). 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