Theory and Applications of Categories, Vol. 30, No. 3, 2015, pp. 40–48.

WEAK BRAIDED MONOIDAL CATEGORIES AND THEIR HOMOTOPY COLIMITS

MIRJAM SOLBERG

Abstract. We show that the homotopy colimit construction for diagrams of categories with an operad action, recently introduced by Fiedorowicz, Stelzer and Vogt, has the desired homotopy type for diagrams of weak braided monoidal categories. This provides a more flexible way to realize E_{2} spaces categorically.

1. Introduction

Braided monoidal categories have been much studied and are used extensively in many areas of mathematics, for instance in knot theory, representation theory and topological quantum field theories. It has been known for a long time that the nerve of a braided monoidal category is an E_{2} space, and it was shown recently [FSV13] that all homotopy types of E_{2} spaces arise in this way. In this article we study a weaker categorical structure, namely weak braided monoidal categories. These are monoidal categories with a family of natural morphisms X ⊗Y → Y ⊗X satisfying the axioms for a braiding, except that they are not required to be isomorphisms. We will see that weak braided monoidal categories give a more flexible way to realize E_{2} spaces categorically.

Homotopy colimit constructions have become increasingly important in homotopy theory. In order for the equivalence between weak braided monoidal categories and E_{2} spaces to be really useful, one should be able to construct homotopy colimits on the categorical level. Such a homotopy colimit construction was defined in [FSV13] in general for diagrams of categories with an operad action. The question of the homotopy properties of the homotopy colimit was left open for weak braided monoidal categories. In this paper we provide an answer to that question. Let Br^{+}-Cat denote the category of weak braided monoidal categories and let X be a diagram of weak braided monoidal categories. Applying the nerve N to a weak braided monoidal category yields a space with an action of the E_{2} operad NBr^{+}, see Subsection 3.1. Let hocolim^{Br^{+}}X denote the homotopy colimit of X defined in [FSV13], and let hocolim^{NBr^{+}}NX denote the homotopy colimit of NX, for details see Subsection 3.1. Then our main result, Theorem 3.2, can be stated as follows.

1.1. Theorem. There is a natural weak equivalence

hocolim^{NBr^{+}}NX → N(hocolim^{Br^{+}}X)

Received by the editors 2014-10-24 and, in revised form, 2015-01-23.

Transmitted by Clemens Berger. Published on 2015-01-28.

2010 Mathematics Subject Classification: Primary 18D10, 18D50; Secondary 55P48.

Key words and phrases: Weak braided monoidal categories, homotopy colimits, double loop spaces.

\begin{matrix}c \\ ◯ \end{matrix} Mirjam Solberg, 2015. Permission to copy for private use granted.

WEAK BRAIDED MONOIDAL CATEGORIES AND THEIR HOMOTOPY COLIMITS 41

of NBr^{+}-algebras.

1.2. Organization. We begin by giving the definition of weak braided monoidal categories in Section 2 and provide some examples. In Section 3 we set up and prove our main result, Theorem 3.2. The proof involves an analysis of braid monoids which is interesting in its own right.

2. Weak braided monoidal categories

Let D be a monoidal category with monoidal product ⊗, monoidal unit i, associativity isomorphism a and left and right unit isomorphisms l and r respectively. A weak braiding for D consists of a family of morphisms

b_{d,e}: d ⊗ e → e ⊗ d

in D, natural in d and e, such that l_{d}b_{d,i} = r_{d}, and r_{d}b_{i,d} = l_{d} and the following two diagrams

\xymatrix{ & (e ⊗ d) ⊗ f \ar[r]^{a} & e ⊗ (d ⊗ f) \ar[dr]^{id⊗b} &  &  & d ⊗ (f ⊗ e) \ar[r]^{a^{−1}} & (d ⊗ f) ⊗ e \ar[dr]^{b⊗1} \\ (d ⊗ e) ⊗ f \ar[ur]^{b⊗id} \ar[dr]_{a} &  &  & e ⊗ (f ⊗ d) & d ⊗ (e ⊗ f) \ar[ur]^{id⊗b} \ar[dr]_{a^{−1}} &  &  & (f ⊗ d) ⊗ e \\  & d ⊗ (e ⊗ f) \ar[r]_{b} & (e ⊗ f) ⊗ d \ar@{-->}[ur]_{a} &  &  & (d ⊗ e) ⊗ f \ar[r]_{b} & f ⊗ (d ⊗ e) \ar@{-->}[ur]_{a^{−1}}}

commute for all d, e and f in D. Here the sub indices of the weak braiding b and the associativity isomorphism a have been omitted. A weak braided monoidal category is a monoidal category equipped with a weak braiding. Note that if all the morphisms b_{d,e} are isomorphisms, then b is a braiding for the monoidal category.

2.1. Remark. The notion of a weak braided monoidal category found in [BFSV03] and [FSV13] differs from the definition given here, in that the underlying monoidal structure is required to be strictly associative and strictly unital. This is not a significant difference, since each weak braided monoidal category is equivalent to a weak braided strict monoidal category, along monoidal functors preserving the weak braiding. The proof of this is similar to the proof of the analogous result for braided monoidal structures.

Weak braided monoidal categories have not been much studied in the literature, so before we proceed we will look at some examples to show how such structures naturally arise. The first example, the disjoint union of the braid monoids, is somehow the canonical example.

2.2. Example. Let B_{m}^{+} denote the braid monoid on m strings with the following presentation:

⟨σ_{1}, …, σ_{m−1} | σ_{i}σ_{j} = σ_{j}σ_{i} if |i − j| > 1 and σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1}⟩.

The elements in B_{m}^{+} are called positive braids on m strings, or just positive braids.

Let B^{+} denote the category with an object m for each integer m ≥ 0, with endomorphisms of m the elements of the braid monoid B_{m}^{+}, and no other morphisms. This is a strict monoidal category with monoidal product given by m ⊔ n = m+n and juxtaposition of positive braids, the unit is 0. The weak braiding from m ⊔ n to n ⊔ m is given by the positive braid

(σ_{n} ⋯ σ_{m+n−1}) ⋯ (σ_{2} ⋯ σ_{m+1})(σ_{1} ⋯ σ_{m}),

braiding the first m strings over the last n strings. This is the same as the usual braiding in the classical braid category, which is the disjoint union of the braid groups, see [JS93, Example 2.1].

2.3. Example. We consider the category of non-negatively graded abelian groups. An object G is a collection of abelian groups G_{n} for n ≥ 0. A morphism f: G → H consists of group homomorphisms f_{n}: G_{n} → H_{n} for n ≥ 0. This category has a monoidal product given by

(G ⊗ H)_{n} = ⨁_{n_{1}+n_{2}=n} G_{n_{1}} ⊗ H_{n_{2}}.

Now fix an integer k. For g ∈ G_{n_{1}} and h ∈ H_{n_{2}} the assignment g ⊗ h ↦ k^{n_{1}n_{2}}h ⊗ g induces a map from G_{n_{1}} ⊗ H_{n_{2}} to H_{n_{2}} ⊗ G_{n_{1}}, which in turn induces a homomorphism (G⊗H)_{n} → (H ⊗G)_{n}. The collection of such maps gives a weak braiding for the category of non-negatively graded abelian groups. Note that if k is a unit, i.e. ±1, then the weak braiding is an actual braiding.

This example may be generalized to the category of non-negatively graded R-modules for any commutative ring R. Pick an element in R to play the role of k in the weak braiding.

A much studied construction is the center of a monoidal category, which can be endowed with a braided monoidal structure, see for instance Example 2.3 in [JS93]. Our next example is a weak version of this.

2.4. Example. Let D be a strict monoidal category with monoidal unit i. We consider pairs (d, δ) where d is an object in D and δ is a natural transformation δ: d⊗(−) → (−)⊗d such that δ_{i} = id_{d} and such that for any two objects x, y ∈ D the triangle

\xymatrix{d ⊗ x ⊗ y \ar[rr]^{δ_{x}⊗id_{y}} \ar[dr]_{δ_{x⊗y}} &  & x ⊗ d ⊗ y \ar[dl]^{id_{x}⊗δ_{y}} \\  & x ⊗ y ⊗ d}

commutes. An arrow between two pairs (d, δ) → (e, ϵ) consists of a morphism ϕ: d → e such that for all x ∈ D the identity ϵ_{x} ∘ (ϕ ⊗ id_{x}) = (id_{x} ⊗ ϕ) ∘ δ_{x} holds. We can define a monoidal product of two such pairs by setting

(d, δ) ⊗ (e, ϵ) = (d ⊗ e, (δ ⊗ id_{e}) ∘ (id_{d} ⊗ ϵ)).

The collection of morphisms

δ_{e}: (d, δ) ⊗ (e, ϵ) → (e, ϵ) ⊗ (d, δ)

satisfies the conditions for a weak braiding on this category of pairs and arrows. We call this the weak center of D.

The requirement that D should be strictly associative and strictly unital was only a matter of convenience. A similar construction works for any monoidal category, details are left to the interested reader.

2.5. Operadic interpretation of weak braided monoidal structures. When the underlying monoidal multiplication is strict, weak braided monoidal categories are the algebras over a certain Cat-operad. By a Cat-operad we understand an operad internal to the category Cat of small categories. Following [FSV13, Section 8] we will introduce the Cat-operad Br^{+} such that Br^{+}-algebras are weak braided strict monoidal categories. The objects of Br^{+}(k) are the elements A of the symmetric group Σ_{k}. Let p: B_{k}^{+} → Σ_{k} denote the projection of the braid monoid onto the corresponding symmetric group. Then a morphism α: A → B in Br^{+}(k) is a positive braid α ∈ B_{k}^{+} such that p(α)A = B. Composition in Br^{+}(k) is given by multiplication in B_{k}^{+}. The category Br^{+}(k) has a right action of Σ_{k} defined on objects and morphisms by sending α: A → B to α: Ag → Bg for g ∈ Σ_{k}. The operad structure map

γ: Br^{+}(k) × Br^{+}(j_{1}) × ⋯ × Br^{+}(j_{k}) → Br^{+}(j_{1} + ⋯ + j_{k})

takes the tuple (A, B_{1}, …, B_{k}) to

A(j_{1}, …, j_{k}) ∘ (B_{1} ⊔ ⋯ ⊔ B_{k}).

Here A(j_{1}, …, j_{k}) denotes the canonical block permutation obtained from A by replacing the ith letter with j_{i} letters. The action on morphisms is analogous except for the obvious permutation of the indices. It is easy to check that the category of weak braided monoidal categories with weak braiding preserving strict monoidal functors is isomorphic to Br^{+}- algebras. See for instance the argument given in Section 5.1 in [SS14] for the braided monoidal version. We denote the category of Br^{+}-algebras by Br^{+}-Cat.

3. Homotopy colimits of weak braided monoidal categories

In [FSV13, Definition 4.10] there is a general homotopy colimit construction for a diagram of algebras over a Cat-operad. Let L be a small category and consider the category (Br^{+}-Cat)^{L} of functors L → Br^{+}-Cat and natural transformations. The above mentioned construction gives in particular a functor

hocolim_{L}^{Br^{+}} : (Br^{+}-Cat)^{L} → Br^{+}-Cat.

3.1. The homotopy type of the homotopy colimit. Let S be the category of simplicial sets and let N be the nerve functor from Cat to S. If we apply N levelwise to the Cat-operad Br^{+}, we get an operad NBr^{+} internal to the category S. We denote the category of algebras over NBr^{+} as NBr^{+}-S. A morphism of NBr^{+}-algebras is called a weak equivalence if the underlying simplicial set map is a weak equivalence. These are the weak equivalences in the standard model structure on NBr^{+}-algebras, for a reference to the topological case, see for instance [SV91, Theorem B]. Given a diagram W: L → (NBr^{+}-S), let hocolim_{L}^{NBr^{+}} W denote the coend construction N(−/L) ⊗_{L} QW, where Q is an object wise cofibrant replacement functor in the category of NBr^{+}-algebras. This is the homotopy colimit of QW from Definition 18.1.2 in [Hir03]. If X is in (Br^{+}-Cat)^{L}, then there is a natural map

hocolim_{L}^{NBr^{+}}NX → N(hocolim_{L}^{Br^{+}}X),

see the paragraph before Definition 6.7 [FSV13]. This is an operadic version of Thomason’s map in Lemma 1.2.1 [Tho79]. The question if this map is a weak equivalence or not, was left open in [FSV13]. Our main result provides a positive answer to this problem.

3.2. Theorem. The diagram

\xymatrix{(Br^{+}-Cat)^{L} \ar[r]^{N} \ar[d]_{_{hocolim\begin{matrix}Br & ^{+} \\ L \end{matrix}}} & (NBr^{+}-S)^{L} \ar[d]^{_{hocolim{NBr^{+} \atop L}}} \\ Br^{+}-Cat \ar[r]^{N} & NBr^{+}-S}

commutes up to weak equivalence of NBr^{+}-algebras.

The operad NBr^{+} is an E_{2} operad, see Proposition 8.13 in [FSV13]. The above theorem gives one way to relate weak braided monoidal categories and E_{2} spaces as seen in the corollary below. Fiedorowicz, Stelzer and Vogt obtain the same equivalence without using the homotopy colimit construction of Br^{+}-algebras in [FSV].

3.3. Corollary. We have an equivalence of localized categories

(Br^{+}-Cat)[we^{−1}] ≃ (NBr^{+}-S)[we^{−1}].

Proof. Theorem 3.2 shows that Theorem 7.6 in [FSV13] applies to the operad Br^{+}. The corollary then follows from the latter theorem with the added observation that the localization (Br^{+}-Cat)[we^{−1}] exists, see Proposition A.1 in [SS14].

By general theory (details will be provided later), the proof of the theorem reduces to showing that certain categories have the property that each connected component has an initial object. Fix an A ∈ Σ_{m}, a B ∈ Σ_{n}, and non-negative integers r_{1}, …, r_{n} such that r_{1} + ⋯ + r_{n} = m. Let B̃ denote the canonical block permutation B(r_{1}, …, r_{n}) ∈ Σ_{m} obtained from B by replacing the ith letter with r_{i} letters. We define a poset category C depending on A, B and r_{1}, …, r_{n} . The objects in C are the positive braids α ∈ B_{m}^{+} such that

p(α)AB̃ ∈ (Σ_{r_{1}} × ⋯ × Σ_{r_{n}}) ⊆ Σ_{m}.

There is a morphism α ≤ β from α to β in C if there exist γ_{i} ∈ B_{r_{1}}^{+} for i = 1, …, n such that (γ_{1} ⊔ ⋯ ⊔ γ_{n})α = β in B_{m}^{+}.

3.4. Analysis of minimal positive braids in C. First we note that all references to the presentation of a braid monoid, will be to the standard presentation, see Example 2.2. We call an object ν in C a minimal object if for all objects ν′ in C, ν′ ≤ ν implies ν′ = ν. Define the norm |β| ∈ ℕ_{0} of an element β in B_{m}^{+}, as the length of any word representing β, see Section 6.5.1 in [KT08]. This is well defined because each of the relations in the presentation identifies words of equal length. It is immediate from the definition that ν is a minimal object if and only if ν ≠ (γ_{1} ⊔ ⋯ ⊔ γ_{n})ν′ for all γ_{i} ∈ B_{r_{i}}^{+} and all ν′ ∈ C with |ν′| < |ν|.

3.5. Proposition. Given an object α in C there is a unique minimal object ν_{α} such that ν_{α} ≤ α.

This proposition is the key ingredient in the proof of our main result. But before we prove either, we will derive some auxiliary results from the nature of the standard presentation of a braid monoid.

Let w and w′ be two words representing the same positive braid. According to Section 6.1.5 in [KT08], w′ can be obtained from w by a finite number of consecutive substitutions of the form

w_{1}rw_{2} = w_{1}r′w_{2}

where r = r′ is one of the relations in the presentation. Observe that for each of the relations r = r′ in the presentation, r and r′ contain the same letters, only the order and number of occurrences of each letter differ. This implies that a letter σ_{i} is in w if and only if σ_{i} is in w′.

For integers r_{1}, …, r_{n} adding up to m, consider B_{r_{1}}^{+} × ⋯ × B_{r_{n}}^{+} as a submonoid of B_{m}^{+} consisting of the positive braids in B_{m}^{+} which can be represented by a word not containing the letters σ_{r_{i}} for i = 1, …, n − 1. The above discussion shows that α ∈ B_{m}^{+} lies in B_{r_{1}}^{+} × ⋯ × B_{r_{n}}^{+} if and only if no word representing α contains any of the letters σ_{r_{i}} for i = 1, …, n − 1. Also, a positive braid α ∈ B_{m}^{+} does not lie in B_{r_{1}}^{+} × ⋯ × B_{r_{n}}^{+} if and only if any word representing α contains at least one of the letters σ_{r_{i}} for i = 1, …, n − 1. This immediately implies the next lemma.

3.6. Lemma. Given two positive braids α and β in B_{m}^{+} such that their product βα lies in B_{r_{1}}^{+} × ⋯ × B_{r_{n}}^{+}, then both α and β lie in B_{r_{1}}^{+} × ⋯ × B_{r_{n}}^{+} as well.

A right common multiple of two elements x and x′ in a monoid M, is an element in M that is of the form xy = x′y′ for some y and y′ in M. A right least common multiple of x and x′ is an element lcm(x, x′) ∈ M such that lcm(x, x′) is a right common multiple of x and x′, and such that any right common multiple of x and x′ is of the form lcm(x, x′)z for some z ∈ M. A unique right least common multiple of γ and γ′ exists for any two positive braids γ and γ′ on k strings, see Theorem 6.5.4 in [KT08]. Since we will only be dealing with right least common multiples, and not left least common multiples, the notation lcm will not be ambiguous. We will however use a subindex k to indicate that the least common multiple lcm_{k} is taken in the braid monoid B_{k}^{+}.

3.7. Corollary. Given γ_{i}, γ′_{i} ∈ B_{r_{i}}^{+} for i = 1, …, n, let γ = γ_{1} ⊔ ⋯ ⊔ γ_{n} and similarly γ′ = γ′_{1} ⊔ ⋯ ⊔ γ′_{n}. Then the least common multiple lcm_{m}(γ, γ′) in B_{m}^{+} of γ and γ′ is equal to

lcm_{r_{1}}(γ_{1}, γ′_{1}) ⊔ ⋯ ⊔ lcm_{r_{n}}(γ_{n}, γ′_{n}).

Proof. It is clear that lcm_{r_{1}}(γ_{1}, γ′_{1})⊔⋯⊔lcm_{r_{n}}(γ_{n}, γ′_{n}) is the right least common multiple of γ and γ′ in the monoid B_{r_{1}}^{+} × ⋯ × B_{r_{n}}^{+}. Now lcm_{r_{1}}(γ_{1}, γ′_{1}) ⊔ ⋯ ⊔ lcm_{r_{n}}(γ_{n}, γ′_{n}) is a right multiple of both γ and γ′, so we get

lcm_{m}(γ, γ′)ϕ = lcm_{r_{1}}(γ_{1}, γ′_{1}) ⊔ ⋯ ⊔ lcm_{r_{n}}(γ_{n}, γ′_{n})

for some ϕ ∈ B_{m}^{+}. Since the product lcm_{m}(γ, γ′)ϕ lies in B_{r_{1}}^{+} × ⋯ × B_{r_{n}}^{+}, then so does lcm_{m}(γ, γ′), and the result follows.

Proof of Proposition 3.5. We first prove the existence of ν_{α}. If α is not a minimal object, there exists an object α_{1} ∈ C such that α_{1} ≤ α and |α_{1}| < |α|. We repeat this process as many times as necessary until we obtain a minimal α_{k} with α_{k} ≤ α_{k−1} and |α_{k}| < |α_{k−1}|. The process terminates after a finite number of steps since the norm of the α_{i}’s decrease strictly each time. We set ν_{α} = α_{k}, so by construction ν_{α} ≤ α.

We now turn to the uniqueness of ν_{α}. Suppose there are two minimal objects ν_{α} and ν′_{α} such that both ν_{α} ≤ α and ν′_{α} ≤ α. Then α equals both (γ_{1} ⊔ ⋯ ⊔ γ_{n})ν_{α} and (γ′_{1} ⊔ ⋯ ⊔ γ′_{n})ν′_{α} for some γ_{i}, γ′_{i} ∈ B_{r_{i}}^{+}, i = 1, …, n. Abbreviating γ_{1} ⊔ ⋯ ⊔ γ_{n} to γ and γ′_{1} ⊔ ⋯ ⊔ γ′_{n} to γ′, we recall that

lcm_{m}(γ, γ′) = lcm_{r_{1}}(γ_{1}, γ′_{1}) ⊔ ⋯ ⊔ lcm_{r_{n}}(γ_{n}, γ′_{n}).

Since α is a right common multiple of both γ and γ′,

α = (lcm_{r_{1}}(γ_{1}, γ′_{1})) ⊔ ⋯ ⊔ (lcm_{r_{n}}(γ_{n}, γ′_{n}))ω

for some ω ∈ B_{m}^{+}. The right least common multiple of γ_{i} and γ′_{i} is in particular a right common multiple of γ_{i} and γ′_{i}, so lcm(γ_{i}, γ′_{i}) = γ_{i}ϕ_{i} = γ′_{i}ϕ′_{i} for some ϕ_{i}, ϕ′_{i} ∈ B_{r_{i}}^{+}, i = 1, …, n. Combining this we get that

(γ_{1} ⊔ ⋯ ⊔ γ_{n})ν_{α} = α = (γ_{1} ⊔ ⋯ ⊔ γ_{n})(ϕ_{1} ⊔ ⋯ ⊔ ϕ_{n})ω and
(γ′_{1} ⊔ ⋯ ⊔ γ′_{n})ν′_{α} = α = (γ′_{1} ⊔ ⋯ ⊔ γ′_{n})(ϕ′_{1} ⊔ ⋯ ⊔ ϕ′_{n})ω.

The braid monoid injects into the corresponding braid group [FSV13, Theorem 6.5.4], so we can apply left cancellation to the above equations to obtain

ν_{α} = (ϕ_{1} ⊔ ⋯ ⊔ ϕ_{n})ω and ν′_{α} = (ϕ′_{1} ⊔ ⋯ ⊔ ϕ′_{n})ω.

It is straightforward to check that p(ω)AB̃ is in Σ_{r_{1}} × ⋯ × Σ_{r_{n}}, so that ω is an object in C. Then the above equations say that ω ≤ ν_{α} and ω ≤ ν′_{α} in C. But since ν_{α} and ν′_{α} are minimal objects these maps have to be identities. This proves the uniqueness of ν_{α}.

3.8. Lemma. Each connected component in C has an initial object.

Proof. Given a morphism α ≤ β in C, Proposition 3.5 associates to α and β unique minimal objects ν_{α} and ν_{β} respectively. The two objects must be equal since ν_{α} ≤ α ≤ β, but ν_{β} is the unique minimal object with ν_{β} ≤ β. Hence the minimal objects associated to any two objects in the same connected component has to be equal, and we have a unique minimal object in each connected component of C. The minimal objects are initial in their respective connected components.

Fix an M ∈ Σ_{m}, an N ∈ Σ_{n}, and non-negative integers s_{1}, …, s_{n} such that s_{1} + ⋯ + s_{n} = m. The factorization category C(M, N, s_{1}, …, s_{n}), as defined in [FSV13, Section 6], has as objects tuples (C_{1}, …, C_{n}, α) consisting of C_{i} ∈ Σ_{s_{i}} for i = 1, … n, and α ∈ B_{m}^{+} such that

p(α)M = Ñ(C_{1} ⊔ ⋯ ⊔ C_{n}).  (1)

A morphism from (C_{1}, …, C_{n}, α) to (D_{1}, …, D_{n}, β) consists of elements γ_{i} in B_{s_{i}}^{+} for i = 1, … n such that (γ_{1} ⊔ ⋯ ⊔ γ_{n})α = β.

3.9. Lemma. The factorization category C(A, B^{−1}, r_{B^{−1}(1)}, …, r_{B^{−1}(n)}) is isomorphic to the category C considered in this section.

Proof. Here B̃^{−1}(C_{1} ⊔ ⋯ ⊔ C_{n}) = (C_{B(1)} ⊔ ⋯ ⊔ C_{B(n)})B̃^{−1}, so Equation (1) can be rewritten as

p(α)AB̃ = C_{B(1)} ⊔ ⋯ ⊔ C_{B(n)}.

This equation determines the C_{i}’s uniquely given α with p(α)AB̃ in Σ_{r_{1}} × ⋯ × Σ_{r_{n}}. The two categories therefore have isomorphic objects, and the morphism sets are easily seen to be isomorphic as well.

Proof of Theorem 3.2. Together Lemmas 3.8 and 3.9 show that each factorization category has an initial object in each of its connected components. Thus the operad Br^{+} satisfies the factorization condition [FSV13, Definition 6.8] and the result follows from Theorem 6.10 [FSV13].

References

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[FSV] Z. Fiedorowicz, M. Stelzer, and R. M. Vogt. Rectification of weak product algebras over an operad in Cat and Top and applications. arXiv:1311.2817.

[FSV13] Z. Fiedorowicz, M. Stelzer, and R. M. Vogt. Homotopy colimits of algebras over Cat-operads and iterated loop spaces. Adv. Math., 248:1089–1155, 2013.

[Hir03] Philip S. Hirschhorn. Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2003.

[JS93] André Joyal and Ross Street. Braided tensor categories. Adv. Math., 102(1):20– 78, 1993.

[KT08] Christian Kassel and Vladimir Turaev. Braid groups, volume 247 of Graduate Texts in Mathematics. Springer, New York, 2008.

[SS14] C. Schlichtkrull and M. Solberg. Braided injections and double loop spaces. Accepted for publication in Trans. Amer. Math. Soc. Available at arXiv:1403.1101, 2014.

[SV91] R. Schwänzl and R. M. Vogt. The categories of A_{∞}- and E_{∞}-monoids and ring spaces as closed simplicial and topological model categories. Arch. Math. (Basel), 56(4):405–411, 1991.

[Tho79] R. W. Thomason. Homotopy colimits in the category of small categories. Math. Proc. Cambridge Philos. Soc., 85(1):91–109, 1979.

Department of Mathematics, University of Bergen, P.O. Box 7800, N-5020 Bergen, Norway

Email: mirjam.solberg@math.uib.no
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Clemens Berger, Université de Nice-Sophia Antipolis: cberger@math.unice.fr Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown(at)btinternet.com

Valeria de Paiva: valeria.depaiva@gmail.com

Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Kathryn Hess, Ecole Polytechnique Fédérale de Lausanne: kathryn.hess@epfl.ch Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, Macquarie University: steve.lack@mq.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Tom Leinster, University of Edinburgh: Tom.Leinster@ed.ac.uk Ieke Moerdijk, Radboud University Nijmegen: i.moerdijk@math.ru.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Giuseppe Rosolini, Università di Genova: rosolini@disi.unige.it Alex Simpson, University of Edinburgh: Alex.Simpson@ed.ac.uk James Stasheff, University of North Carolina: jds@math.upenn.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: rfcwalters@gmail.com R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca