Exponentials of non-singular simplicial sets
aa r X i v : . [ m a t h . A T ] J a n E XPONENTIALS OF N ON - SINGULAR S IMPLICIAL S ETS
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Vegard Fjellbo
Department of MathematicsUniversity of OsloOslo, Norway [email protected]
John Rognes
Department of MathematicsUniversity of OsloOslo, Norway [email protected]
January 28, 2020 A BSTRACT
A simplicial set is non-singular if the representing maps of its non-degenerate simplices are de-greewise injective. The category of simplicial sets has a simplicial mapping set X K whose set of n -simplices are the simplicial maps ∆[ n ] × K → X . We prove that X K is non-singular whenever X is non-singular.MSC-class: 18D15 (Primary), 55U10 (Secondary) K eywords Cartesian Closed · Non-singular · Simplicial sets
There are times when one would like to know whether a category behaves similarly, in some sense, to the categoryof sets and functions. As an example, for homotopy-theoretical purpose the author would like to know whether theendofunctor − × ∆[1] of non-singular simplicial sets preserves colimits. Here, ∆[1] denotes the standard -simplex.Let sSet denote the category of simplicial sets. The full subcategory nsSet whose objects are the non-singularsimplicial sets sits strictly between sSet and the category of ordered simplicial complexes. Despite the fact that non-singular simplicial sets have a natural PL structure [1, p. 126–127] they almost never appear in the literature, thoughthey do play a role in the book Spaces of PL Manifolds and Categories of Simple Maps by Waldhausen, Jahren andRognes [1].The endofunctor ( − ) K : sSet → sSet is designed so that the Yoneda lemma makes it right adjoint to − × K . Ourmain result is the following. Theorem 1.1.
Let K be some simplicial set. Then X K is non-singular whenever X is.Part of the author’s interest in this result comes from the case when K non-singular. Then the restriction of ( − ) K to nsSet corestricts to an endofunctor of non-singular simplicial sets. Moreover, ( − ) K viewed as a functor nsSet → nsSet is right adjoint to the endofunctor − × K of nsSet . This means that we can derive the following consequenceof Theorem 1.1. Corollary 1.2.
Taking the product − × K : nsSet → nsSet with a non-singular simplicial set K preserves colimits.In particular, taking the product − × ∆[1] with an interval is a cocontinous endofunctor of non-singular simplicial sets.The case of the interval is not only of practicle concern, but it is also the theoretical focus of this article as it is nothard to argue that Theorem 1.1 follows from the following result. Proposition 1.3.
The simplicial set X ∆[1] is non-singular whenever X is.The proof of the latter result is the subject of Section 4, whereas Theorem 1.1 is derived from Proposition 1.3 inSection 3. PREPRINT - J
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28, 2020In Section 2, we will discuss applications of Theorem 1.1 beyond Corollary 1.2. We explain how Theorem 1.1follows from Proposition 1.3 in Section 3. Finally, the case of the interval is discussed Section 4.
The inclusion U : nsSet → sSet admits a left adjoint functor called desingularization [1, Rem. 2.2.12., p. 39], whichis denoted D . Note that the unit η X : X → U DX is degreewise surjective and that desingularization has the universal property that any simplicial map f : X → Y whose target Y is non-singular factors through the unit by a unique map U DX → Y .In general, we say that a full subcategory of some category is a reflective subcategory if the inclusion admits aleft adjoint, which is then referred to as a reflector . Thus nsSet is a reflective subcategory of sSet . Note that theword reflective is not quite standard terminology. For example, Mac Lane [2, §IV.3] Adámek and Rosický [3, p. 1306]do not include fullness as an assumption in their definition, although some other authors do. Proposition 1.3 and itsgeneralization Theorem 1.1 has a noteworthy application and a couple of consequences.The main theorem of [4] establishes a model structure on nsSet that is right-induced a la Thomason [5] from sSet equipped with the standard model structure due to Quillen [6]. Moreover, the theorem says that ( D, U ) is a Quillenequivalence. Proposition 1.3 is used as a technical ingredient in the proof of that theor.Another way to state Theorem 1.1 is to say that the non-singular simplicial sets form an exponential ideal in sSet .The category of simplicial sets is cartesian closed and even a topos. Part of this is the fact that ( − ) K is right adjointto − × K . Here, the author has in mind the notions, terminology and notation from [2, §IV.6–§IV.10]. Note that theconstruction X K is bifunctorial. A generalized result known as the parameter theorem ensures this [2, p. 102]. Corollary 2.1.
Desingularization preserves finite products.It seems that Corollary 2.1 follows from Day’s reflection theorem [7, Thm. 1.2] and its corollary [7, Cor. 2.1]. Day’sreflection theorem concerns a more general setting, although he does refer to the condition that the reflective subcat-egory is closed under exponentiation [7, §0]. Another phrase that is used in the literature is that the non-singularsimplicial sets form an exponential ideal in sSet , which is exactly the content of Theorem 1.1.In case one does not want to unravel the general form of Day’s reflection theorem, we provide the following elemen-tary proof. Proof of Corollary 2.1.
It is enough to consider two factors. Suppose X and Y simplicial sets.Consider the map Y × X η Y × X −−−−→ D ( Y × X ) . Here, we omit the redundant symbol U for the inclusion functor. By Theorem 1.1, the simplicial set D ( Y × X ) X isnon-singular, so we obtain a factorization Y % % ❑❑❑❑❑❑❑❑❑❑ η Y / / DY y y r r r r r D ( Y × X ) X (1)of the adjoint. Next, we switch the two factors of the adjoint DY × X → D ( Y × X ) of the dashed map in (1) and factor the adjoint of the resulting map by means of the diagram X % % ▲▲▲▲▲▲▲▲▲▲ η X / / DX x x q q q q q D ( X × Y ) DY (2)in which the dashed map arises by Theorem 1.1 as D ( X × Y ) DY is non-singular.2 PREPRINT - J
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28, 2020By adjunction, we can combine (1) and (2) into the solid commutative diagram X × Y η Y × X & & ▼▼▼▼▼▼▼▼▼▼ id × η X / / X × DY x x ♣♣♣♣♣♣♣♣♣♣♣ η X × id & & ◆◆◆◆◆◆◆◆◆◆◆ D ( X × Y ) ( D ( pr ) ,D ( pr )) ❳ ❩ ❬ ❭ ❫ ❴ ❵ ❜ ❝ ❞ ❢ DX × DY o o (3)in which a dashed map arises because DX × DY is non-singular, being a product of non-singular simplicial sets.Indeed, the dashed map must be equal to the canonical map ( D ( pr ) , D ( pr )) due to the universal property of desin-gularization.Because the map η X × Y is degreewise surjective and because (3) commutes, it follows immediately that DX × DY → D ( X × Y ) is degreewise surjective.Furthermore, by the universal property of desingularization, it follows that the composite DX × DY → D ( X × Y ) ( D ( pr ) ,D ( pr )) −−−−−−−−−−→ DX × DY is the identity. This implies that the first of the two maps of the composite is even degreewise injective, which impliesthat it is degreewise bijective and hence an isomorphism. In this way, we see that ( D ( pr ) , D ( pr )) is degreewisebijective and hence an isomorphism. (cid:4) Another consequence of Theorem 1.1 is the following result.
Corollary 2.2.
The category of non-singular simplicial sets is cartesian closed.
In this section we will prove Theorem 1.1, assuming that Proposition 1.3 holds. First we will point out that the latterresult can be generalized fairly easily from the interval to the standard n -simplex, for all n ≥ . Lemma 3.1.
Suppose n ≥ . The simplicial set X ∆[ n ] is non-singular if X is.To verify Lemma 3.1 we note that Proposition 1.3 implies that X ∆[1] n is non-singular if X is. This is by induction on n , which is made possible by the exponential law ( X K ) L ∼ = X L × K , which holds because sSet is cartesian closed.Let [ n ] denote the totally ordered set { < < · · · < n } . Following [8, p. 132], we shall refer to an operator as afunction α : [ m ] → [ n ] such that α ( i ) ≤ α ( j ) if i ≤ j . Observe that ∆[ n ] embeds in ∆[1] n in such a way that ∆[1] n retracts onto ∆[ n ] . The embedding i that we have in mind is induced by the operator [ n ] → [1] n given by j . . . . . . where the string . . . . . . starts with j ’s and the rest are ’s. One can make a retraction r : ∆[1] n → ∆[ n ] bytaking the string k . . . k n from [1] n and then finding the lowest index j such that k j = 0 . Then one defines an operatorby the rule k . . . k n j − , which induces the announced r . We get that the composite ri is the identity as this is true on the level of operators.There are induced maps X ∆[ n ] X i ←−− X ∆[1] n X r ←−− X ∆[ n ] such that the composite is equal to the identity. In other words, the simplicial set X ∆[ n ] is identified with a simplicialsubset of X ∆[1] n , which is non-singular if X is. Hence, the simplicial set X ∆[ n ] is non-singular if X is. This concludesour proof of Lemma 3.1, given that Proposition 1.3 holds.By means of Lemma 3.1, we can derive our main result.3 PREPRINT - J
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Proof of Theorem 1.1.
Suppose K is some simplicial set and let X be non-singular. Let ∆ ↓ K denote the simplexcategory , meaning the category whose objects are the pairs ( x, n ) , where x is a simplex of K whose degree is n , andwhose morphisms ( y, m ) → ( x, n ) are the pairs ( x, α ) with α an operator such that y = xα .The simplicial set K can be viewed as the colimit of the diagram Υ K : ∆ ↓ K → sSet that sends a simplex of degree n to the standard n -simplex ∆[ n ] [8, Lem. 4.2.1]. We explain that X K is the limit of the composite ∆ ↓ K Υ K == ⇒ sSet X ( − ) === ⇒ sSet, denoted X Υ K , or in other words that the cone X K ⇒ X Υ K is universal.Assume that Z ⇒ X Υ K is a cone. Recall that sSet is cartesian closed. Via the natural bijection sSet ( Z × ∆[ n ] , X ) ∼ = −→ sSet ( Z, X ∆[ n ] ) , we can consider the cocone Z × X Υ K ⇒ X illustrated in the diagram Z × ∆[ m ] id × α (cid:15) (cid:15) id × ¯ y & & ▲▲▲▲▲▲▲▲▲▲ (cid:27) (cid:27) Z × K ∃ ! / / ❴❴❴ XZ × ∆[ n ] id × ¯ x rrrrrrrrrr D D instead. Because Z × − is a cocontinous endofunctor of simplicial sets, the simplicial set Z × K is the colimit of Z × Υ K . Hence, there exists a (unique) map Z × K → X that gives rise to a factorization of the cocone Z × X Υ K ⇒ X .By adjointness, we obtain a map Z → X K such that the given, arbitrary cone on X Υ K factors through X K ⇒ X Υ K .On the other hand, any map Z → X K that gives rise to such a factorization corresponds to a map Z × K → X thatfactors the cocone Z × Υ K ⇒ X through the universal cocone. However, there is only one map Z × K → X of thelatter type. By adjointness, the map Z → X K is therefore unique.The diagram X Υ K is by Lemma 3.1 a diagram whose objects are non-singular. Because nsSet is a reflectivesubcategory of sSet , it follows that X K is non-singular [3, p. 1306]. (cid:4) In the proof of Theorem 1.1, we used the non-trivial fact that a reflective subcategory inherits limits from its surround-ing category, although we could have argued in more elementary terms.According to Adámek and Rosický [3, p. 1306], the earliest proof that appears in the literature, of the inheritance oflimits by reflective subcategories, is to be found in the works of H. Herrlich [9].
We give a proof that X ∆[1] is non-singular whenever X is non-singular. This is the claim presented in Proposition 1.3.An informal way of stating this result is to say that prisms on non-singular simplicial sets are very rigid. Recall thatSection 3 explains how to derive Theorem 1.1 from Proposition 1.3. Thus the work of this section finishes the proofof our main result.For convenience, we introduce some terminology and notation before we present the proof. An injective operatoris said to be a face operator and a surjective operator is said to be a degeneracy operator . Special face operators arethe elementary face operators δ ni : [ n − → [ n ] that omit the index i and vertex operators ε ni : [0] → [ n ] that hitthe indices i . Special degeneracy operators are the elementary degeneracy operators σ ni : [ n + 1] → [ n ] that send i and its successor i + 1 to i . Frequently, we omit the upper index in the notation. Similar to the terminology in [1], wewill refer to δ nn . . . δ qq : [ q − → [ n ] , < q ≤ n , as the q -th front face of [ n ] and to δ np . . . δ n − p : [ n − ( p + 1)] → [ n ] , ≤ p < n , as the p -th back face of [ n ] .A face operator or degeneracy operator is proper if it is not the identity. Consider a simplicial set. A simplex y is a (proper) face of another simplex x if y = xµ for a (proper) face operator µ . Analogously, a simplex y is a (proper) PREPRINT - J
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28, 2020 degeneracy of another simplex x if y = xρ for a (proper) degeneracy operator ρ . A simplex is degenerate if it is aproper degeneracy of some simplex. Otherwise, it is said to be non-degenerate .In the proof, we will use the Eilenberg-Zilber lemma [8, Thm. 4.2.3], which says that any simplex x of any simplicialset X is uniquely a degeneration x = x ♯ x ♭ of some non-degenerate simplex x ♯ . We say that x ♯ is the non-degeneratepart of x , following [1], and that x ♭ is the degenerate part of x . Note that x and x ♯ are objects in the category ∆ ↓ X while x ♭ can be regarded as a morphism x → x ♯ . Thus the terminology is not perfect, however it is useful. Accordingto the Yoneda lemma, the n -simplices x of a simplicial set X are in natural bijective correspondence x ¯ x with thesimplicial maps ∆[ n ] → X . The map ¯ x is the representing map of x . We say that a simplex is embedded if itsrepresenting map is degreewise injective.Because of the new terminology, we get a shorter definition of non-singular in the second condition of Lemma 4.1,below. Furthermore, there is another formulation that is useful in the proof of Proposition 1.3, though a bit awkward.It is given as the third condition Lemma 4.1 Lemma 4.1.
The following statements are equivalent.1. The simplicial set X is non-singular.2. Each non-degenerate simplex of X is embedded.3. Eeach simplex of X is degenerate provided its vertices are not pairwise distinct.The equivalence of the second and third statement is somewhat refined by the next lemma. Lemma 4.2.
Let X be a non-singular simplicial set and x some simplex with zε k = zε l . Then the degenerate part x ♭ of x factors uniquely through the degeneracy operator σ k . . . σ l − . Proof.
Write ρ = σ k . . . σ l − . The uniqueness of a factorization of x ♭ through ρ is automatic as ρ is epic in Cat . It isthe existence part that requires an argument.Because X is non-singular it follows that the non-degenerate part x ♯ is embedded, which is the same as sayingthat its vertices are pairwise distinct. This means that x ♭ ( k ) = x ♭ ( l ) . As x ♭ is order-preserving, it follows that x ♭ ( j ) = x ♭ ( k ) if k ≤ j ≤ l . Thus ρ ( i ) = ρ ( j ) implies x ♭ ( i ) = x ♭ ( j ) . Take a section µ of ρ . We get that x ♭ = ( x ♭ µ ) ρ . (cid:4) Lemma 4.2 will be used to break down the proof of Proposition 1.3 into two parts.If x is some simplex, say of degree n , whose degenerate part factors through the elementary degeneracy operator σ k for some k with ≤ k < n , then we will say that x splits off σ k . In particular, if X is non-singular and if x is asimplex of X such that xε k = xε k +1 , then x splits off σ k according to Lemma 4.2.The canonical identification N ([ n ] × [1]) ∼ = −→ ∆[ n ] × ∆[1] gives us a preferred set of generators of the prism ∆[ n ] × ∆[1] , namely the n + 1 non-degenerate ( n + 1) -simplices γ n +1 j : [ n + 1] → [ n ] × [1] , ≤ j ≤ n , given by γ n +1 j ( i ) = (cid:26) ( i, , ≤ i ≤ j ( i − , , j < i ≤ n. Coming from the diagram . . . / / ( j, / / ( j + 1 , / / . . . / / ( n, , / / . . . / / ( j, O O / / : : ttttttttt ( j + 1 , / / O O . . . are the conditions γ n +1 j δ j +1 = γ n +1 j +1 δ j +1 (4)for ≤ j ≤ n . These conditions, which can be thought of glueing conditions for constructing the prism from n + 1 copies of the standard ( n + 1) -simplex, generate all relations that the generators satisfy.5 PREPRINT - J
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28, 2020We are done with the setup and are ready to prove Proposition 1.3. Suppose X non-singular. Keep in mind the thirdand equivalent way to state this, as formulated in Lemma 4.1. The proof is divided into two parts, the first of which isthe following result. Lemma 4.3.
Assume that Φ is an n -simplex of X ∆[1] such that the k -th vertex and the l -th vertex are equal, for some k and some l with ≤ k < l ≤ n . Then Φ ε k = Φ ε k +1 = · · · = Φ ε l . The second part is Lemma 4.4, where we prove that any given n -simplex Φ of X ∆[1] is degenerate if it is such that the k -th vertex is equal to the ( k + 1) -th vertex, for some k with ≤ k < n .Thus, by Lemma 4.3 and Lemma 4.4, any simplex of X ∆[1] is degenerate provided its vertices are not pairwisedistinct. Lemma Lemma 4.1 then says that X ∆[1] is non-singular. We can therefore conclude that Proposition 1.3holds when we have proven the two lemmas. Proof of Lemma 4.3.
Suppose Φ an n -simplex of X ∆[1] such that Φ ε k = Φ ε l for some k and some l with ≤ k
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28, 2020that establishes Φ ε j +1 as a face of the -simplex z = Φ ◦ ( N µ × ◦ γ and Φ ε j as a face of the -simplex z = Φ ◦ ( N µ × ◦ γ , in such a way that z δ = z δ .Recall that the j -th and the ( j + 1) -th vertex of the simplex Φ( γ nj +1 ) of X are equal. This implies that z = w σ . Similarly, the ( j + 1) -st and the ( j + 2) -nd vertex of Φ( γ nj ) are equal, implying that z = w σ . It follows that Φ ε j = Φ ε j +1 as δ and δ are sections of σ and δ and δ are sections of σ . (cid:4) Lemma 4.4.
Let Φ be an n -simplex of X ∆[1] such that the k -th vertex is equal to the ( k + 1) -th vertex, for some k with ≤ k < n . Then there is an ( n − -simplex Ψ such that Φ = Ψ σ k . Proof.
For the purpose of constructing Ψ we apply N σ k × id to the elements of the preferred set { γ n +10 , . . . , γ n +1 n } of generators of the prism. The result of the calculation is the set of equations ( N σ k × id )( γ n +1 j ) = (cid:26) γ nj σ k +1 , ≤ j ≤ kγ nj − σ k , k < j ≤ n. Should Ψ exist, then it must therefore satisfy Φ( γ n +1 j ) = (cid:26) Ψ( γ nj ) σ k +1 , ≤ j ≤ k Ψ( γ nj − ) σ k , k < j ≤ n. As δ k +1 is a section of both σ k and σ k +1 we are lead to define a function ψ : { γ n , . . . , γ nn − } → X n by ψ ( γ nj ) = ( Φ( γ n +1 j ) δ k +1 , ≤ j ≤ k Φ( γ n +1 j +1 ) δ k +1 , k < j < n that specifies where Ψ sends the generators, if it exists.Note the following regarding the definition of ψ . First, we have made the choices of the section δ k +1 of σ k +1 andthe section δ k +1 of σ k . These choices seem to make the argument below as simple as possible. Second, we have that ψ ( γ nk ) = Φ( γ n +1 k ) δ k +1 = Φ( γ n +1 k δ k +1 ) = Φ( γ n +1 k +1 δ k +1 ) = Φ( γ n +1 k +1 ) δ k +1 due to (4). This ensures that there is some compatibility between the two clauses of the definition of ψ by cases. Wetake advantage of the equation below.Crucially, the function ψ obeys the compatibility criterion ψ ( γ nj ) δ j +1 = ψ ( γ nj +1 ) δ j +1 (5)for ≤ j < n − , as we now explain. There are three cases. Either j < k , j = k or j > k .First, we verify (5) in the case when j = k . For this we use (4) and the general rule δ i δ j = δ j δ i − for j < i . Weget that ψ ( γ nk ) δ k +1 = (Φ( γ n +1 k ) δ k +1 ) δ k +1 = (Φ( γ n +1 k +1 ) δ k +1 ) δ k +1 = Φ( γ n +1 k +1 )( δ k +1 δ k +1 )= Φ( γ n +1 k +1 )( δ k +2 δ k +1 )= (Φ( γ n +1 k +1 ) δ k +2 ) δ k +1 = (Φ( γ n +1 k +1 δ k +2 )) δ k +1 = (Φ( γ n +1 k +2 δ k +2 )) δ k +1 and that ψ ( γ nk +1 ) δ k +1 = (Φ( γ n +1 k +2 ) δ k +1 ) δ k +1 = Φ( γ n +1 k +2 )( δ k +1 δ k +1 )= Φ( γ n +1 k +2 )( δ k +2 δ k +1 ) , PREPRINT - J
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28, 2020which confirms that (5) holds in the case when j = k .Second, consider the case when j < k . We get that ψ ( γ nj ) δ j +1 = (Φ( γ n +1 j ) δ k +1 ) δ j +1 = Φ( γ n +1 j )( δ k +1 δ j +1 )= Φ( γ n +1 j )( δ j +1 δ k )= (Φ( γ n +1 j ) δ j +1 ) δ k = (Φ( γ n +1 j δ j +1 )) δ k = (Φ( γ n +1 j +1 δ j +1 )) δ k and that ψ ( γ nj +1 ) δ j +1 = (Φ( γ n +1 j +1 ) δ k +1 ) δ j +1 = Φ( γ n +1 j +1 )( δ k +1 δ j +1 )= Φ( γ n +1 j +1 )( δ j +1 δ k ) , which confirms that (5) holds in the case when j < k .Third, consider the case when j > k . We get that ψ ( γ nj ) δ j +1 = (Φ( γ n +1 j +1 ) δ k +1 ) δ j +1 = Φ( γ n +1 j +1 )( δ k +1 δ j +1 )= Φ( γ n +1 j +1 )( δ j +2 δ k +1 )= (Φ( γ n +1 j +1 ) δ j +2 ) δ k +1 = (Φ( γ n +1 j +1 δ j +2 )) δ k +1 = (Φ( γ n +1 j +2 δ j +2 )) δ k +1 and that ψ ( γ nj +1 ) δ j +1 = (Φ( γ n +1 j +2 ) δ k +1 ) δ j +1 = Φ( γ n +1 j +2 )( δ k +1 δ j +1 )= Φ( γ n +1 j +2 )( δ j +2 δ k +1 ) . This confirms that (5) holds in the case when j > k and concludes our verification of (5) for any j with ≤ j < n − .We define Ψ : ∆[ n − × ∆[1] → X by letting Ψ( γ nj α ) = ψ ( γ nj ) α for all j with ≤ j < n . The map Ψ is well defined and a simplicial map as ψ satisfies the glueing condition (5).Thus it remains to argue that Φ = Ψ ◦ ( N σ k × id ) . (6)It suffices to check that the equation holds on the generators γ n +10 , . . . , γ n +1 n for the prism ∆[ n ] × ∆[1] .We use the calculation of ( N σ k × id )( γ n +1 j ) , ≤ j ≤ n , above. There are three cases. Either ≤ j ≤ k , j = k + 1 or j > k + 1 .If ≤ j ≤ k , then Ψ ◦ ( N σ k × id )( γ n +1 j ) = Ψ( γ nj σ k +1 )= ψ ( γ nj ) σ k +1 = (Φ( γ n +1 j ) δ k +1 ) σ k +1 = Φ( γ n +1 j ) , which confirms (6) for the generators γ n +10 , . . . γ n +1 k . This is because the vertices of Φ( γ n +1 j ) that are numbered k + 1 and k + 2 are equal. Thus the simplex splits off σ k +1 by Lemma 4.2 as X is non-singular. Furthermore, δ k +1 is asection of σ k +1 .Note that Φ( γ n +1 j ) splits off σ k when j > k . This is because the vertices of Φ( γ n +1 j ) that are numbered k and k + 1 are equal. Thus the simplex splits off σ k by Lemma 4.2 as X is non-singular. Furthermore, δ k +1 is a section of σ k .Consider the case when j = k + 1 . We get that Ψ ◦ ( N σ k × id )( γ n +1 k +1 ) = Ψ( γ nk σ k )= ψ ( γ nk ) σ k = (Φ( γ n +1 k ) δ k +1 ) σ k = (Φ( γ n +1 k +1 ) δ k +1 ) σ k = Φ( γ n +1 k +1 ) , PREPRINT - J
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28, 2020which confirms (6) for the generator γ n +1 k +1 .Finally, we consider the case when j > k + 1 . Then Ψ ◦ ( N σ k × id )( γ n +1 j ) = Ψ( γ nj − σ k )= ψ ( γ nj − ) σ k = (Φ( γ n +1 j ) δ k +1 ) σ k = Φ( γ n +1 j ) , which confirms (6) for the generators γ n +1 k +2 , . . . , γ n +1 n . This concludes our verification of (6). Thus Φ is a degeneratesimplex of X ∆[1] . (cid:4) References [1] Friedhelm Waldhausen, Bjørn Jahren, and John Rognes.
Spaces of PL manifolds and categories of simple maps ,volume 186 of
Annals of Mathematics Studies . Princeton University Press, 2013.[2] Saunders Mac Lane.
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Graduate texts in mathematics .Springer-Verlag New York, Inc., 1998.[3] J. Adámek and J. Rosický. On reflective subcategories of locally presentable categories.
Theory and Applicationsof Categories , 30(41):1306–1318, 2015.[4] R. Vegard S. Fjellbo. Homotopy theory of non-singular simplicial sets. arXiv , arXiv:2001.05032v1, 2020. TBA2020.[5] R. W. Thomason. Cat as a closed model category.
Cahiers Topologie Géom. Différentielle , 21(3):305–324, 1980.[6] Daniel G. Quillen.
Homotopical algebra . Lecture Notes in Mathematics, No. 43. Springer-Verlag, 1967.[7] Brian Day. A reflection theorem for closed categories.
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