HOMOTOPY INVARIANCE OF CONVOLUTION PRODUCTS

Transcription

1 HOMOTOPY INVARIANCE OF CONVOLUTION PRODUCTS STEFAN SCHWEDE Introduction The purpose of this note is to establish the homotopy invariance of two particular kinds of convolution products, the operadic product of tame M-spaces, and the box product of I-spaces, see Corollary 2.6 and Theorem 4.7. A related result is that the respective classes of semistable objects are closed under the two convolution products in full generality, see Theorems 3.6 and 4.11, respectively. Before we explain our results in more detail, we wish to put them into perspective. The convolution product of functors between symmetric monoidal categories was introduced by category theorist Brian Day in [3]. It made a prominent appearance in homotopy theory when Jeff Smith and Manos Lydakis simultaneously and independently introduced the smash products of symmetric spectra [7] and the smash product of Γ-spaces [11], respectively. Since then, convolution products have become an essential ingredient in the homotopy theory toolkit, and many more examples were introduced and studied in the context of stable homotopy theory [10, 12], unstable homotopy theory [9, 15], equivariant homotopy theory [4, 13], and motivic homotopy [5, 8], to name just a few. Convolution products also appear in the higher categorical approach to homotopy theory [6], but in this context, all notions are intrinsically homotopy invariant. In order for convolutions products to be homotopically meaningful, they must interact nicely with some relevant notion of equivalence. Ideally, the products would preserve weak equivalences in both variables in full generality, but in practice, one usually has to settle for less. In typical situations, homotopically useful convolution products come with one or several compatible closed model structures. The compatibility then includes the pushout product property which implies that the convolution product is homotopical when both factors are cofibrant. Often there is a naturally occurring, broader class of flat objects, characterized by a latching property, such that the convolution product is homotopical as soon as one factor is flat. The proof of the homotopy invariance for flat objects typically proceeds by a cellular reduction argument to free or representable objects. Since homotopy invariance tends to require some flatness property, many subsequent statements will have flatness or cofibrancy hypothesis. Given the history of the subject, one would not expect convolution products to be fully homotopical, i.e., preserve the relevant kind of weak equivalences in both variables without any hypothesis. Nevertheless, there are already two non-obvious instances of full homotopy invariance: the box product of orthogonal spaces (also known as I -functors, I -spaces or I-spaces) and the operadic product of L-spaces (i.e., spaces with a continuous action of the topological monoid L of linear self-isometries of R ) are homotopical for weak equivalences and global equivalences by [18, Thm ] and [19, Thm. 1.21], respectively. The proofs make essential use of explicit homotopies that are not available for more discrete or combinatorial index categories. In this paper we prove the homotopy invariance of two other kinds of convolution products, the operadic product of tame M-simplicial sets under homotopy orbit equivalences (Corollary 2.6), and the box product of I-simplicial sets under I-equivalences (Theorem 4.7). Here M is the monoid of injective self-maps of the set of positive natural numbers, and tameness is a certain local finiteness condition, see Definition 1.2. The category I is a skeleton of the category of finite sets and injections. Both situations model the homotopy theory of topological spaces under weak homotopy equivalences; the important feature is that strictly commutative multiplications for these two convolution products model E -spaces. Our present results can be viewed as discrete analogs of the invariance properties in [18, Thm ] and [19, Thm. 1.21]; the arguments, however, are completely different. Date: December 20, 2018; 2010 AMS Math. Subj. Class.: 55P. 1

2 2 STEFAN SCHWEDE To the authors knowledge, homotopical properties of the operadic product of tame M-spaces have not been studied before, but there is a substantial literature about the continuous analog, i.e., for L-spaces [1, 9, 19]. The homotopy invariance of the box product of I-spaces was previously established by Sagave and Schlichtkrull under the additional hypothesis that one of the factors is flat, see [14, Prop. 8.2] or [15, Cor. 2.29]. Similarly, the closure property of the class of semistable I-spaces under box product was known under the hypothesis that both factors are flat, see [15, Prop. 2.23]. Since we now know these properties in full generality, several flatness hypotheses required in the work of Sagave and Schlichtkrull turn out to be unnecessary. For example, Proposition 2.20, Proposition 2.23, Lemma 2.25, Proposition 2.27, Corollary 2.29, Proposition 4.2, Corollary 4.3 and Proposition 4.8 of [15] hold without the flatness hypotheses, and Theorem 1.2 and Lemma 4.12 of [15] hold without the cofibrancy hypothesis on the commutative I-space monoid. Open questions. At the time of this writing, and to the best of the author s knowledge, it is still unknown whether the smash product of symmetric spectra (of simplicial sets, say) and the smash product of orthogonal spectra are fully homotopical for stable equivalences or not. I would like to see this being sorted out. Conventions. In this paper we follow a common abuse of language in that the term I-space means a functor from the injection category I to the category of simplicial sets (as opposed to a functor to some category of topological spaces). Similarly, an M-space is a simplicial set with an action of the injection monoid M. The use of simplicial sets is essential for several of our arguments, and we can offer no new insight about the convolution products of I-topological spaces and M-topological spaces. Acknowledgments. I am indebted to Steffen Sagave and Christian Schlichtkrull for several helpful discussion on the topics of this paper. 1. The structure of tame M-sets In this section we discuss tame M-sets, i.e., sets equipped with an action of the injection monoid that satisfy a certain local finiteness condition. The main result is the structure theorem for tame M-set in Theorem 1.10, which says that every tame M-set W decomposes as a disjoint union of M-subsets arising in a specific way from certain Σ m -sets associated with W. The arguments of this section are combinatorial in nature, but they crucially enter into the homotopical analysis is Section 2. Definition 1.1. The injection monoid M is the monoid, under composition of injective self-maps of the set ω = {1, 2, 3,... } of positive natural numbers, Definition 1.2. Let W be an M-set. An element x of W is supported on a subset A of ω if the following condition holds: for every injection f M that fixes A elementwise, the relation fx = x holds. The M-set W is tame if every element is supported on some finite subset of ω. Clearly, if x is supported on A and A B ω, then x is supported on B. Every element is supported on all of ω. An element is supported on the empty set if and only if it is fixed by M. Proposition 1.3. Let x be an element of an M-set W. If x is supported on two finite subsets A and B of ω, then x is supported on the intersection A B. Proof. We let f M be an injection that fixes A B elementwise. We let m be the maximum of the finite set A B f(a) and define σ M as the involution that interchanges j with j + m for all j B A, i.e., j + m for j B A, σ(j) = j m for j (B A) + m, and j for j (B A) ((B A) + m). In particular, the map σ fixes the set A elementwise. Since A and f(a) are both disjoint from B + m, we can choose a bijection γ M such that { f(j) if j A, and γ(j) = j for j B + m.

3 HOMOTOPY INVARIANCE OF CONVOLUTION PRODUCTS 3 Then f can be written as the composition f = σ (σγσ) (σγ 1 f). In this decomposition the factors σ and σγ 1 f fix A pointwise, and the factor σγσ fixes B pointwise. So because x is supported on A and on B. This gives σ x = (σγσ) (x) = (σγ 1 f) (x) = x f x = (σ (σγσ) (σγ 1 f) x = x. Since f was any injection fixing A B elementwise, the object x is supported on A B. Definition 1.4. Let x be an element of an M-set. The support of x is the intersection of all finite subsets of ω on which x is supported. We write supp(x) for the support of x; Proposition 1.3 then shows that x is supported on its support supp(x). If x is not finitely supported, then we agree that supp(x) = ω. It is important that in Definition 1.4 the intersection is only over finite supporting subsets. Indeed every object is supported on the set ω {j} for every j ω (because the only injection that fixes ω {j} elementwise is the identity). So without the finiteness condition the intersection in Definition 1.4 would always be empty. Proposition 1.5. Let W be an M-set and x W. (i) If the injections u, ū M agree on supp(x), then ux = ūx. (ii) For every injection u M, the relation supp(ux) u(supp(x)) holds. If x is finitely supported, then supp(ux) = u(supp(x)), and ux is also finitely supported. Proof. (i) If x is not finitely supported, then supp(x) = ω, so u = ū and there is nothing to show. If x is finitely supported, we can choose a bijection h M that agrees with u and ū on supp(x). Then h 1 u and h 1 ū fix the support of x elementwise and hence (h 1 u) x = x = (h 1 ū) x. Thus ux = h ((h 1 u) x) = h x = h ((h 1 ū) x) = ūx. (ii) We let v M be an injection that fixes u(supp(x)) elementwise. Then vu agrees with u on supp(x), so v (ux) = (vu)x = ux by part (i). We have thus shown that ux is supported on u(supp(x)). If x is finitely supported, then u(supp(x)) is finite, and this proves that ux is finitely supported and supp(ux) u(supp(x)). For the reverse inclusion we choose h M such that hu fixes supp(x) elementwise; then (hu) x = x. Applying the argument to h and ux (instead of u and x) gives and thus supp(x) = supp((hu) x) = supp(h (ux)) h(supp(ux)), u(supp(x)) u(h(supp(ux))) = (uh)(supp(ux)) = supp(ux). The last equation uses that uh is the identity on the set u(supp(x)), hence also the identity on the subset supp(ux). This proves the desired relation when x is finitely supported. Proposition 1.6. Let W be a tame M-set and f M. Then the left multiplication map f : W W is injective, and its image consists precisely of those elements that are supported on the set f(ω). Proof. For injectivity we consider any x, y W with fx = fy. Since f is injective and x and y are finitely supported, we can choose h M such that hf is the identity on the support of x and the support of y. Then x = hfx = hfy = y. It remains to identify the image of multiplication by f. For all x W we have supp(fx) f(supp(x)) f(ω), so fx is supported on f(ω). Now suppose that z W is supported on f(ω). The f restricts to a bijection from f 1 (supp(z)) to supp(z). We choose a bijection u M such that fu is the identity on supp(z);, then fuz = z, and hence z is in the image of multiplication by f.

4 4 STEFAN SCHWEDE Proposition 1.6 can fail for non-tame M-sets: we can let f M act on the set {0, 1} as the identity if the image of f has finite complement, and we set f(0) = f(1) = 0 if its image has infinite complement. Example 1.7. We let A be a finite subset of ω, and we denote by I A the set of injective maps from A to ω. The monoid M acts from the left on I A by composition. The M-set I has only one element, so the M-action is necessarily trivial. If A is non-empty, then I A is countably infinite and the M-action is non-trivial, but tame: the support of an injection α : A ω is its image α(a). The M-set I A represents the functor of taking elements with support in A: the inclusion ι A : A ω is supported on A, and for every M-set W, the evaluation map = Hom M (I A, W ) {x W : supp(x) A}, ϕ ϕ(ι A ) is bijective. For A = m = {1,..., m}, we write I m = I m. The M-set I m comes with a commuting right action of the symmetric group Σ m by precomposition. So we can and will view I m as a Σ m -object in the category of tame M-sets. If K is any left Σ m -set, we can form the tame M-set I m Σm K by coequalizing the two Σ m -actions on the product I m K. If W is any M-set, we write W (m) for the set of elements supported on {1,..., m}, which is Σ m -invariant by Proposition 1.5 (ii). We define a morphism (1.8) ψ W m : I m Σm W (m) W by [f, x] fx, where f M is any injection that extends f : m ω. This assignment is well-defined by Proposition 1.5 (i). The natural transformation ψ m is the counit of an adjunction between the functor I m Σm : Σ m -set M-set (the left adjoint) and the functor sending W to the Σ m -set W (m). The following lemma would be obvious if the monoid M were a group. However, M is not a group, and without the tameness hypothesis, complements of M-invariant subsets need not be M-invariant. The lemma can interpreted as saying that the tameness condition makes M-actions behave like actions of groups. Lemma 1.9. Let Y be an M-invariant subset of a tame M-set Z. Then the complement Z Y is M- invariant. Proof. We consider an element z Z and an injection f M such that fz Y. We choose an injection u M such that uf is the identity on the support of z. Since fz belongs to Y, so does z = ufz. This proves the Z Y is M-invariant. Now we prove that every tame M-set is a disjoint union of M-sets of the form I m Σm A m for varying m 0 and Σ m -sets A m. For m 0, we denote by sk m W the subset of those elements of W that are supported on a subset of ω of cardinality m. Then sk m W is invariant under the M-action by Proposition 1.5 (ii). Moreover, sk 0 W is the set of M-fixed elements, sk m W sk m+1 W, and M is tame if and only if W = m 0 skm W. The morphism ψ W m : I m Σm W (m) W defined in (1.8) takes values in sk m W. For an M-set W, we set, C m (W ) = sk m (W ) sk m 1 (W ) the complement of the (m 1)-skeleton in the m-skeleton. This is an M-invariant subset by Lemma 1.9. We alert the reader that the M-sets C m (W ) are not functors in W : a morphism f : V W of M-sets need not take C m (V ) to C m (W ). Proposition Let W be an M-set. (i) Suppose that W is purely m-dimensional, i.e., sk m 1 W = and sk m W = W. Then the morphism is an isomorphism of M-sets. ψ W m : I m Σm W (m) W

5 HOMOTOPY INVARIANCE OF CONVOLUTION PRODUCTS 5 (ii) If the M-action on W is tame, then the canonical morphism I m Σm (C m (W )) (m) W is an isomorphism of M-sets. m 0 Proof. (i) Since W = sk m W, every element x W is supported on a set of cardinality m. So we can choose a bijection u M such that x is supported on u({1,..., m}). Then u 1 x is supported on {1,..., m} by Proposition 1.5 (ii), so it belongs to W (m). Moreover, ψ W m [u {1,...,m}, u 1 x] = u(u 1 x) = x, so the map ψm W is surjective. For injectivity we consider f, g I m and x, y W such that ψm W [f, x] = ψm W [g, y]. Then supp(ψ W m [f, x]) = f(supp(x)) f({1,..., m}), and similarly supp(ψm W [g, y]) g({1,..., m}). So the class ψm W [f, x] = ψm W [g, y] is supported on the intersection of f({1,..., m}) and g({1,..., m}). The assumption sk m 1 W = means that no element of W is supported on a set of cardinality strictly less than m, so we conclude that f({1,..., m}) = g({1,..., m}). There is thus a unique element σ Σ m such that g = fσ. We extend σ to an element σ M by fixing all numbers greater than m. We choose a bijection u M that extends f; then u σ extends fσ = g. Hence x = u 1 (ux) = u 1 ψ W m [f, x] = u 1 ψ W m [g, y] = u 1 u σy = σy. So we conclude that [f, x] = [f, σy] = [fσ, y] = [g, y]. This proves that the map ψm W is injective, hence bijective. (ii) By induction on n 0, Lemma 1.9 shows that the canonical morphism n C m (W ) sk n W m=0 is an isomorphism of M-sets. Since W is tame, taking the union over all n 0 shows that the canonical morphism C m (W ) W m 0 is an isomorphism. Since C m (W ) is purely m-dimensional, part (i) finishes the proof. We recall the operadic product of M-sets. Definition Let V and W be M-sets. The operadic product V M W is the subset of the product consisting of those pairs (x, y) V W such that supp(x) supp(y) =. Proposition Let V and W be M-sets. (i) The map operadic product V M W is an M-invariant subset of V W. (ii) If V and W are tame, then so is V M W. Proof. (i) We a pair (x, y) V W of elements with disjoint supports. Then for every f M we have supp(f x) supp(f y) f(supp(x)) f(supp(y)) = f(supp(x) supp(y)) =. as the subset of the product consisting of those pairs such that supp(x) supp(y) =. So the pair f (x, y) = (fx, fy) belongs to V M W. Part (ii) follows from the relation supp V W (x, y) = supp V (x) supp W (y). Example For m 0, the tame M-set I m of injective maps from m = {1,..., m} to ω was discussed in Example 1.7. We define a morphism of M-sets I m+n I m I n, f (fi 1, fi 2 ), where i 1 : m m + n is the inclusion and i 2 : n m + n sends i to m + i. This morphism is injective and its image consists precisely of those pairs (α, β) such that supp(α) supp(β) = α(m) β(n) =.

6 6 STEFAN SCHWEDE So the map restricts to an isomorphism of M-sets ρ : I m+n = Im M I n. 2. Homotopy theory of M-spaces In this section we introduce two notions of equivalence for M-spaces, the weak equivalences and the M-equivalences. Then we show that the operadic product of M-spaces is fully homotopical for both kinds of equivalences, see Theorem 2.4 and Corollary 2.6. Definition 2.1. An M-space is a simplicial set equipped with a left action of the injection monoid M. An M-space is tame if for every simplicial dimension k, the M-set of k-simplices is tame. Definition 2.2. A morphism f : X Y of M-spaces is (i) a weak equivalence if it is a weak equivalence of underlying simplicial sets after forgetting the M-action, and (ii) an M-equivalence if the induced morphism on homotopy colimits (bar constructions) f hm : X hm Y hm is a weak equivalence of simplicial sets. The M-equivalences are the more important class of equivalences, and the homotopy theory represented by tame M-spaces relative to M-equivalences is the homotopy theory of spaces. The weak equivalences of M-spaces are relevant for the discussion of semistable M-spaces in Section 3. Every weak equivalence of M-spaces induces a weak equivalence on homotopy M-orbits; hence weak equivalences are in particular M-equivalences. The converse is not true: for every injective map α : m n the induced restriction morphism α : I n I m, f f α is an M-equivalence because the M-homotopy orbits of source and target are weakly contractible, by the following Example 2.3. However, for m n, the morphism is not a weak equivalence. Example 2.3. The M-set I m of injective maps m ω was discussed in Example 1.7. We claim that the simplicial set (I m ) hm is weakly contractible. To this end we observe that (I m ) hm is the nerve of the translation category T m whose object set is I m, and where morphisms from f to g are all u M such that uf = g. Given any two f, g I m, we can choose a bijection u M such that uf = g, and then u is an isomorphism from f to g in T m. Since all objects of the category T m are isomorphic, its nerve is weakly equivalent to the classifying space of the endomorphism monoid of any of its objects. All these endomorphism monoids are isomorphic to the injection monoid M, so we conclude that the simplicial set (I m ) hm = N(Tm ) is weakly equivalent to the classifying space of the injection monoid M. The classifying space of M is weakly contractible by [17, Lemma 5.2], hence so is the simplicial set (I m ) hm. The next results show that the operadic product with any tame M-space preserves weak equivalences and M-equivalences. Theorem 2.4. For every tame M-space X, the functor X M preserves weak equivalences between tame M-spaces. Proof. We start with the special case where X is a tame M-set (as opposed to an M-space). For an M-space Y and a finite subset A of ω we write Y [A] for the simplicial subset of element that are supported on ω A. By definition, the underlying simplicial set of X M Y is the disjoint union, over x X, of the simplicial sets {x} Y [supp(x)], and this decomposition is natural for M-equivariant morphisms in Y. We let f M be any injection with f(ω) = ω supp(x). Then left multiplication by f is an isomorphism from Y onto the simplicial subset of Y [supp(x)], by Proposition 1.6. So Y [supp(x)] is naturally isomorphic to Y, and we conclude that the underlying simplicial set of X M Y is isomorphic to the underlying simplicial set of X Y. The reader should beware that this isomorphism in neither M-equivariant nor natural in the X-variable. In any case, the isomorphism of simplicial sets X M Y = X Y is natural in Y, so it witnesses that X M preserves weak equivalences between tame M-spaces.

7 HOMOTOPY INVARIANCE OF CONVOLUTION PRODUCTS 7 Now we treat the general case. The operadic product is formed dimensionwise in the simplicial direction, and realization (i.e., diagonal) of bisimplicial set preserves weak equivalences. So the general case follows from the special case of M-sets. Since X M Y was defined as an M-invariant subspace of the product X Y, the two projections restrict to morphisms of M-spaces p 1 : X M Y X and p 2 : X M Y Y. Theorem 2.5. For all tame M-spaces X and Y, the morphism is a weak equivalence of simplicial sets. (p 1 hm, p 2 hm ) : (X M Y ) hm X hm Y hm Proof. We start with the special case where X = I m /H and Y = I n /K for some m, n 0, for some subgroup H of Σ m, and some subgroup K of Σ n. By Example 1.13, the operadic product I m M I n is isomorphic to I m+n. Hence by Example 2.3, source and target of the (H K)-equivariant morphism (p 1 hm, p 2 hm ) : (I m M I n ) hm (I m ) hm (I n ) hm are weakly contractible. The actions of H on (I m ) hm and of K on (I n ) hm are free. Similarly, the isomorphism (I m M I n ) hm = (Im+n ) hm provided by Example 1.13 shows that the action of H K on (I m M I n ) hm is free. Any equivariant map between free (H K)-spaces that is a weak equivalence of underlying simplicial sets induces a weak equivalence on orbit spaces. So the morphism (p 1 hm, p 2 hm )/(H K) : (I m M I n ) hm /(H K) (I m ) hm /H (I n ) hm /K is a weak equivalence of simplicial sets. The claim now follows because the operadic product preserves colimits in each variable, and because the canonical morphism ((I m ) hm )/H (I m /H) hm is an isomorphism of simplicial sets. Now we assume that X and Y are M-sets. Then X is isomorphic to a disjoint union of M-sets of the form I m Σm A m, for varying m 0 and Σ m -sets A m, by Proposition 1.10 (iii). The Σ m -set A, in turn, is isomorphic to a disjoint union of Σ m -sets of the form Σ m /H, for varying subgroups H of Σ m. The operadic product of M-spaces and the cartesian product of simplicial sets distribute over disjoint unions, and homotopy orbits preserve disjoint unions. So source and target of the morphism under consideration decompose as disjoint unions of M-set considered in the previous paragraph. Weak equivalences of simplicial sets are stable under disjoint unions, so this reduces the case of M-sets to the case of the previous paragraph. Now we treat the general case. All relevant construction commute with realization of simplicial objects in M-spaces (i.e., diagonal of simplicial sets). So the general case follows from the special case of M-sets because realization of bisimplicial sets is homotopical. The product of simplicial sets preserves weak equivalences in both variables. So Theorem 2.5 directly implies: Corollary 2.6. For every tame M-spaces X, the functor X M preserves M-equivalences between tame M-spaces. Remark 2.7. The cartesian product of M-spaces, with diagonal M-action, is not homotopical for M- equivalences. For example, the unique morphism I 1 to the terminal I-space is an M-equivalence by Example 2.3. However, the product I 1 I 1 is the disjoint union of I 1 M I 1 (which is isomorphic to I 2 ) and the diagonal (which is isomorphic to I 1 ). So (I 1 I 1 ) hm consists of two weakly contractible components, and the projection to the first factor I 1 I 1 I 1 is not an M-equivalence. The situation improves when we restrict to semistable tame M-spaces, as defined in Definition 3.2 below. For semistable tame M-spaces X and Y, the inclusion X M Y X Y is both a weak equivalence and an I-equivalence, by Theorem 3.6 below, and hence the product of semistable tame M-spaces is fully homotopical for M-equivalences.

8 8 STEFAN SCHWEDE 3. Semistability of tame M-spaces Semistability is a property of a tame M-space W that guarantees that the canonical morphism W W hm to the homotopy M-orbits is a weak equivalence of simplicial sets. Hence morphisms between semistable tame M-spaces are M-equivalences if and only if they are weak equivalences. In this section we prove that the class of semistable tame M-spaces is closed under the operadic product. The shift homomorphism sh : M M is defined by (sh f)(i) = { 1 for i = 1, and f(i 1) + 1 for i 2. The shift sh W of an M-space W has the same underlying set as W, but the injection monoid acts through the shift homomorphism. We define d M by d(i) = i + 1. The relation holds for all f M, and it shows that the map (sh f)d = df λ W : W sh W, λ W (x) = dx is a homomorphism of M-spaces. By Proposition 1.6, the homomorphism λ W is injective, and its image consists exactly of those elements x whose support in W does not contain 1. Example 3.1 (Shift of I m ). The (M Σ m )-equivariant morphism λ Im : I m sh(i m ) is injective and its image consists of all injections f : m ω whose image does not contain the element 1. The complement of the image of λ Im is generated as a (M Σ m )-set by the inclusion ι : m ω whose support in sh(i m ) is the set m 1. Moreover, an isomorphism of (M Σ m )-sets I m (I m 1 Σm 1 Σ m ) = sh(i m ) is given by λ Im on the first summand; on the second summand it sends [g, σ] to sh( g)σ, where g M extends g : m 1 ω. Definition 3.2. An M-space W is semistable if the morphism of M-spaces λ W : W sh W is a weak equivalence. The set of k-simplices of the homotopy orbits W hm of an M-space is the set M k W k. The embeddings (1,..., 1, ) : W k M k W k = (W hm ) k are compatible with the simplicial structure maps and define a natural monomorphism of simplicial sets e W : W W hm. Theorem 3.3. A tame M-space W is semistable if and only if the embedding e W : W W hm is a weak equivalence of simplicial sets. Proof. Adapt the argument for I-spaces given in [15, Prop. 2.1] to the context of tame M-spaces; or give an independent proof. Example 3.4. We let M act on the set ω = {1, 2, 3,... } by evaluation. With this action, ω is isomorphic to the M-set I 1. We claim that for every non-empty tame M-space Y, the M-space ω M Y is not semistable. By definition, ω M Y is the M-subspace of ω Y consisting of those pairs (i, y) such that i supp(y). We let d i : ω ω be the injection defined by { j if j < i, and d i (j) = j + 1 if j i. By Proposition 1.6, left multiplication by d i is a bijection from Y onto the simplicial subset of Y consisting of those elements y such that i supp(y). So the underlying simplicial set of ω M Y is isomorphic to ω Y (but this isomorphism is not M-equivariant); under this isomorphism, multiplication by d M takes the summand indexed by i to the summand indexed by i + 1. So unless Y is empty, multiplication by d cannot be a weak equivalence, and so ω M Y is not semistable.

9 HOMOTOPY INVARIANCE OF CONVOLUTION PRODUCTS 9 Proposition 3.5. For all tame M-spaces X and Y, the commutative square X M Y λ X Y (sh X) M Y X λ Y ξ X (sh Y ) ξ sh(x Y ) is a pushout, and the composite equals the morphism λ X M Y : X Y sh(x Y ). Proof. Pushouts of simplicial sets are formed dimensionwise, so we can assume that X and Y are tame M- sets. The operadic product preserves colimits in each variable, and the shift functor preserves colimits. So by the structure theorem for tame M-sets (Proposition 1.10 (ii)), it suffices to prove the claim for X = I m and Y = I n for all m, n 0. In this case we exploit the isomorphism I m M I n = Im+n of Example 1.13 and the isomorphism sh(i m ) = I m (I m 1 Σm 1 Σ m ) of Example 3.1. After this rewriting, the pushout property follows by direct inspection. Theorem 3.6. Let X and Y be semistable tame M-spaces. Then the operadic product of X M Y is again semistable, and the inclusion X M Y X Y is a weak equivalence, and hence also an M-equivalence, of M-spaces. Proof. We let X and Y be two semistable tame M-spaces. Because λ X : X sh X is a weak equivalence, so is λ X M Y : X M Y (sh X) M Y, by Theorem 2.4. Similarly, the morphism X M λ Y : X M Y X M (sh Y ) is a weak equivalence. Since the operadic product with a fixed M-space preserves monomorphisms, both of these morphisms are also injective. Hence the morphisms ξ and ξ in the pushout square of Proposition 3.5 are also weak equivalences, and hence so is the composite morphism λ X M Y. This proves that X M Y is again semistable. For the second claim we consider the commutative square of simplicial sets: X M Y incl X Y e X M Y e X e Y (X M Y ) hm (p 1 hm,p2 hm ) X hm Y hm The vertical maps are weak equivalences by Theorem 3.3 because X M Y, X and Y are semistable. The lower horizontal map is a weak equivalence by Theorem 2.5. So the inclusion is a weak equivalence. Example 3.7 (Homotopy infinite symmetric product). We review an interesting class of tame M-spaces due to Schlichtkrull [16]. Schlichtkrull s paper is written for topological spaces, but we work with simplicial sets instead. We let X be a based simplicial set, with basepoint denoted by. We define X as the simplicial subset of X ω consisting, in each simplicial dimension, of those functions α : ω X such that α(i) = for almost all i ω. The injection monoid M acts on X by { α(i) if f(i) = j, and (f α)(j) = if j f(ω). So informally speaking, one can think of f α as the composite α f 1, with the caveat that f need not be invertible. An element α X is supported on the finite set ω α 1 ( ), so the M-action on X is tame. Schlichtkrull s main result about this constriction is that for connected X, the M-homotopy orbit space XhM is weakly equivalent to Q(X), the underlying infinite loop space of the suspension spectrum of X, see [16, Thm. 1.2]. In other words, Q(X) is a non-semistable model for Q(X) in the world of tame M-spaces. The M-space X has additional structure: it is a commutative monoid for the operadic product of M-spaces. Indeed, a multiplication µ X : X M X X

10 10 STEFAN SCHWEDE is given by α(i) if α(i), and µ X (α, β)(i) = β(i) if β(i), and if α(i) = β(i) =. This assignment makes sense because whenever α and β have disjoint support, then for all i ω, at least one of the elements α(i) and β(i) must be the basepoint of X. The multiplication map µ X is clearly associative and commutative, and the constant map to the basepoint of X is a neutral element. With this additional structure, X has the following universal property. We let M act on ω X by f (i, x) = (f(i), x). Then X is the free commutative M -monoid, in the category of tame M-spaces, generated by the tame M-space ω X. This universal property, or direct inspection, shows that the functor (based simplicial sets) (tame M-spaces), X X takes disjoint unions to operadic products (which are coproducts in the category of commutative M - monoids). More precisely, for all based simplicial sets X and Y, the composite map X M Y i X M i Y (X Y ) M (X Y ) µ X Y (X Y ) is an isomorphism of tame M-spaces, where i X : X X Y and i Y : Y X Y are the inclusions. The analogous statement for I-space is Lemma 3.1 of [16]. 4. The box product of I-spaces In this section we prove that the box product of I-spaces is fully homotopical for the classes of I- equivalences and N -equivalences, and that the box product of semistable I-spaces is again semistable. Our strategy is to reduce each of these claims to the corresponding one for the operadic product of tame M-spaces. The key tool is a certain functor from I-spaces to tame M-spaces that matches the box product with the operadic product, and that matches the relevant notions of equivalences and semistability. We let I denote the category whose objects are the sets n = {1,..., n} for n 0, with 0 being the empty set. Morphisms in I are all injective maps. The category I is thus a skeleton of the category of finite sets and injections. Construction 4.1 (From I-spaces to tame M-spaces). We let F be an I-space. We denote by N the non-full subcategory of I containing all objects, but only the inclusions as morphisms. We write F (ω) = colim N F for the colimit of F formed over the non-full subcategory N. We observe that the simplicial set F (ω) supports a natural tame action by the injection monoid M, defined as follows. We let [x] F (ω) be represented by x F (m), and we let f M be an injection. We set n = max(f(m)) and write f : m n for the restriction of f. Then we set f [x] = [F ( f)(x)]. By construction, the M-action is tame. Definition 4.2. An I-space is a functor from the category I to the category of simplicial sets. A morphism of I-spaces is a natural transformation of functors. Definition 4.3. A morphism f : X Y of I-spaces is an N -equivalence if the induced morphism f(ω) : A(ω) B(ω) on colimits over the non-full subcategory N is a weak equivalence of underlying simplicial sets, and an I-equivalence if the induced morphism f hi : A hi B hi on Bousfield-Kan homotopy colimits [2, Ch. XII] is a weak equivalence of simplicial sets. Remark 4.4. The above definition of N -equivalence is not the same as the one given in [15, Sec. 2.5], where the homotopy colimit over N is used instead of the categorical colimit. However, sequential colimits of simplicial sets are fully homotopical for weak equivalences, and for every functor F : N (simplicial sets), the canonical morphism hocolim N F colim N F is a weak equivalence, compare [2, Ch. XII, 3.5]. So the two definitions of N -equivalences of I-spaces are equivalent.

11 HOMOTOPY INVARIANCE OF CONVOLUTION PRODUCTS 11 We recall that the homotopy colimit of an I-space X is related to the homotopy colimit of the M-space F (ω) by a chain of three natural weak equivalences; this observation is due to Jeff Smith, and recorded in [20, Prop ]. We I ω denote the category whose objects are the sets n = {1,..., n} for n 0, and the object ω = {1, 2, 3,... }. Morphisms in I ω are all injective maps. The restriction functor has a left adjoint I ω -spaces I-spaces I-spaces I ω -spaces, X X, given by left Kan extension. Since I is a full subcategory of I ω, we can take X = X on the subcategory I, and X(ω) = X(ω) = colim N X. The inclusion j : BM I ω is cofinal, so the canonical morphism ((L h X)(ω)) hm = hocolim BM (L h X j) hocolim Iω L h X is another weak equivalence of simplicial sets, see the dual version of [2, Ch. XI, 9.2]. Because sequential colimits of simplicial sets are fully homotopical, the natural morphism from the homotopy Kan extension of X at ω to the (categorical) Kan extension at ω is a weak equivalence. So we obtain a chain of three natural weak equivalences X hi (Lh X) hiω ((L h X)(ω)) hm X(ω)hM. This chain of equivalences proves the second item of the following proposition. The first item is a restatement of definitions. Proposition 4.5. (i) A morphism f : X Y of I-spaces is an N -equivalence if and only if the morphism f(ω) : X(ω) Y (ω) is a weak equivalence of M-spaces. (ii) A morphism f : X Y of I-spaces is an I-equivalence if and only if the morphism f(ω) : X(ω) Y (ω) is an M-equivalence of M-spaces. The category I supports a permutative structure + : I I I by concatenation of finite sets. As the notation suggests, the functor + is defined on objects by addition. The concatenation of two morphisms α : m n and β : k l is given by { α(i) for 1 i m, and (α + β)(i) = β(i m) + n for m + 1 i m + n. The box product X Y of two I-spaces X and Y is the convolution product for the concatenation monoidal structure, i.e., the left Kan extension of the functor I I (simplicial sets), (m, n) X(m) Y (n) along + : I I I. The box product of I-spaces comes with two projections, i.e., morphism of I-spaces q 1 : X Y X and q 2 : X Y Y. The morphism q 1 corresponds, via the universal property of the box product, to the bimorphism given by the composite X(m) Y (n) project X(m) X(i 1 ) X(m + n), for m, n 0, where i 1 : m m + n is the inclusion. The morphism q 2 corresponds, via the universal property of the box product, to the bimorphism given by the composite X(m) Y (n) project Y (n) Y (i 2 ) Y (m + n), where i 2 : n m + n sends i to m + i. We introduced the operadic product of M-spaces in Definition 1.11 as the M-invariant subspace of disjointly supported pairs inside the product. The image of the morphism of M-spaces (q 1 (ω), q 2 (ω)) : (X Y )(ω) X(ω) Y (ω)

12 12 STEFAN SCHWEDE lands in the subspace X(ω) M Y (ω), and we write τ X,Y : (X Y )(ω) X(ω) M Y (ω) for the restriction of (q 1 (ω), q 2 (ω)) to this image. The next proposition says that the transformation τ is a strong monoidal structure on the functor ( )(ω) : (I-spaces) (tame M-spaces). Proposition 4.6. For all I-spaces X and Y, the morphism τ X,Y : (X Y )(ω) X(ω) M Y (ω) is an isomorphism of M-spaces. Proof. All structure is sight is defined dimensionwise in the simplicial direction, so it suffices to prove the claim for I-sets (as opposed to I-spaces). Both sides of the transformation τ preserve colimits in each variable, which reduces the claim to representable I-sets. The convolution product of represented functors is represented, i.e., I(m, ) I(n, ) = I(m + n, ), and I(m, )(ω) is isomorphic to the M-set I m discussed in Example 1.7. So the case of represented functors is taken care of by Example Theorem 4.7. For every I-space X, the functor X preserves N -equivalences and I-equivalences of I-spaces. Proof. We let f : Y Z be a morphism of I-spaces that is an N -equivalences or I-equivalences, respectively. Then by Proposition 4.5, the morphism of tame M-spaces f(ω) : Y (ω) Z(ω) is a weak equivalence or M-equivalence, respectively. So the morphism X(ω) M f(ω) : X(ω) M Y (ω) X(ω) M Z(ω) is a weak equivalence by Theorem 2.4, or an M-equivalence by Corollary 2.6, respectively. By the isomorphism of Proposition 4.6, this means that (X f)(ω) : (X Y )(ω) (X Z)(ω) is a weak equivalence or an M-equivalence, respectively. Another application of Proposition 4.5 proves the claim. Theorem 4.8. For all I-spaces X and Y, the morphism is a weak equivalence of simplicial sets. (q 1 hi, q 2 hi) : (X Y ) hi X hi Y hi Proof. The following diagram of simplicial sets commutes by naturality: (q 1 hi (X Y ),q2 hi ) hi X hi Y hi (L h (X Y )) hiω ((L h q 1 ) hiω,(l h q 2 ) hiω ) (L h X) hiω (L h Y ) hiω (((L h q 1 )(ω)) hm,((l h q 2 )(ω)) hm (L h (X Y )(ω)) ) hm ((L h X)(ω)) hm ((L h Y )(ω)) hm ((X Y )(ω)) hm (q 1 (ω) hm,q 2 (ω) hm ) X(ω) hm Y (ω) hm All vertical morphisms in the diagram are weak equivalences. So to prove the theorem we may show that the lower horizontal morphism is a weak equivalence. This morphism factors as the composite ((X Y )(ω)) hm 1 hm,p2 hm ) (τ X,Y ) hm (p ((X(ω) M Y (ω)) hm = X(ω) hm Y (ω) hm. The first morphism is the isomorphism of Proposition 4.6, and the second morphism is the weak equivalence of Theorem 2.5. This completes the proof.

13 HOMOTOPY INVARIANCE OF CONVOLUTION PRODUCTS 13 We recall the notion of semistability for I-spaces. Given an I-space X, we define a new I-space RX as the composite functor 1+ X I I (simplicial sets). The morphisms j m : m 1 + m given by j m (i) = 1 + i form a natural transformation from the identity functor to the functor 1 + : I I. Postcomposition with X provides a natural morphism of I-spaces j X : X X (1 + ) = RX. Definition 4.9. An I-space X is semistable if the morphism j X : X RX is an N -equivalence. Our definition of semistability is not the same as the one in [15, Prop. 2.7], but the two definitions are equivalent by [15, Prop. 2.10]. Proposition An I-space X is semistable if and only if the M-space X(ω) is semistable. Proof. The identity maps j m : X(1 + m) = (RX)(m) are compatible with stabilization over N, and they induce an isomorphism of simplicial sets κ : X(ω) (RX)(ω). This isomorphism does not, however, arise from a morphism of I-spaces, and it is in general not M-equivariant. The isomorphism is M-equivariant for the M-action on X(ω) through the shift homomorphism sh : M M. So κ is an isomorphism of M-spaces from sh(x(ω)) to (RX)(ω) Moreover, the following diagram of M-spaces commutes: sh(x(ω)) λ X(ω) X(ω) = κ j X (ω) (RX)(ω) So λ X(ω) is a weak equivalence of underlying simplicial sets if and only j X (ω) is a weak equivalence, and this proves the claim. Theorem Let X and Y be semistable I-spaces. Then the I-space X Y is semistable and the morphism (q 1, q 2 ) : X Y X Y is both an N -equivalence and an I-equivalence of I-spaces. Proof. Proposition 4.10 shows that the tame M-spaces X(ω) and Y (ω) are semistable. Hence their operadic product X(ω) M Y (ω) is semistable by Theorem 3.6. Proposition 4.6 provides an isomorphism of this operadic product to the M-space (X Y )(ω), which is thus semistable. Proposition 4.10 then shows that X Y is semistable. Filtered colimits preserves finite products, and hence the canonical morphism can : (X Y )(ω) X(ω) Y (ω) is an isomorphism of M-spaces. The upper horizontal morphism in the commutative square of M-spaces τ X,Y (X Y )(ω) X(ω) M Y (ω) (q 1,q 2 )(ω) (X Y )(ω) = = can incl X(ω) Y (ω) is an isomorphism by Proposition 4.6. The right vertical inclusion is a weak equivalence of M-spaces by Theorem 3.6, because X(ω) and Y (ω) are both semistable. We conclude that the morphism (q 1, q 2 )(ω) : (X Y )(ω) (X Y )(ω) is a weak equivalence of M-spaces. Hence the morphism (q 1, q 2 ) : X Y X Y is an N -equivalence of I-spaces, by Proposition 4.5. References [1] A J Blumberg, R L Cohen, C Schlichtkrull, Topological Hochschild homology of Thom spectra and the free loop space. Geom. Topol. 14 (2010), no. 2, [2] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, V1ol Springer-Verlag, v+348 pp. [3] B Day, On closed categories of functors. Reports of the Midwest Category Seminar, IV pp Lecture Notes in Mathematics, Vol Springer-Verlag, Berlin, [4] B I Dundas, O Röndigs, P A Østvær, Enriched functors and stable homotopy theory. Doc. Math. 8 (2003), [5] B I Dundas, O Röndigs, P A Østvær, Motivic functors. Doc. Math. 8 (2003), [6] S Glasman, Day convolution for -categories. Math. Res. Lett. 23 (2016), no. 5, [7] M Hovey, B Shipley, J Smith, Symmetric spectra. J. Amer. Math. Soc. 13 (2000),

14 14 STEFAN SCHWEDE [8] J F Jardine, Motivic symmetric spectra. Doc. Math. 5 (2000), [9] J A Lind, Diagram spaces, diagram spectra and spectra of units. Algeb. Geom. Topol. 13 (2013), [10] M Lydakis, Simplicial functors and stable homotopy theory. Preprint, 1998, hopf.math.purdue.edu/lydakis/s_functors.pdf [11] M Lydakis, Smash products and Γ-spaces. Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 2, [12] M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra. Proc. London Math. Soc. (3) 82 (2001), no. 2, [13] M. A. Mandell, J. P. May, Equivariant orthogonal spectra and S-modules. Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108 pp. [14] S Sagave, Ch Schlichtkrull, Diagram spaces and symmetric spectra. Adv. Math. 231 (2012), no. 3-4, [15] S Sagave, Ch Schlichtkrull, Group completion and units in I-spaces. Algebr. Geom. Topol. 13 (2013), no. 2, [16] Ch Schlichtkrull, The homotopy infinite symmetric product represents stable homotopy. Algebr. Geom. Topol. 7 (2007), [17] S Schwede, On the homotopy groups of symmetric spectra. Geom. Topol. 12 (2008), [18] S Schwede, Global homotopy theory. New Mathematical Monographs 34. Cambridge University Press, Cambridge, xvi+828 pp. [19] S Schwede, Orbispaces, orthogonal spaces, and the universal compact Lie group. arxiv: [20] B Shipley, Symmetric spectra and topological Hochschild homology, K-Theory 19 (2) (2000), Mathematisches Institut, Universität Bonn, Germany address: schwede@math.uni-bonn.de