January 01, 1970
Using the homotopy theory of polynomial monads developed by Batanin and Berger and extended to the \(2\)-categorical context by Weber, we prove the cofinality of a particular morphism of polynomial \(2\)-monads. We apply our result to give a new proof of the delooping of derived mapping spaces of infinitesimal bimodules due to Ducoulombier and Turchin.
There is a remarkable connection between embedding spaces and mapping spaces of operads. This connection is exhibited by the following weak equivalence of spaces, which holds for \(n \geq m+3\): \[\label{equationdeloopingembeddingspace} \Omega^{m+1} \mathrm{SOp}^\mathsf{h} (\mathcal{C}_m,\mathcal{C}_n) \xrightarrow{\sim} \overline{\mathrm{Emb}}(\mathbb{R}^m,\mathbb{R}^n).\tag{1}\] In the formula above, \(\Omega^{m+1}\) is the \(m+1\)-fold loop space functor, \(\mathrm{SOp}^\mathsf{h}(-,-)\) is the derived mapping space in the category of symmetric topological operads and \(\mathcal{C}_m\) is the little \(m\)-cubes operad. The space on the right hand side is the homotopy fibre of the canonical inclusion \[\mathrm{Emb}(\mathbb{R}^m,\mathbb{R}^n) \to \mathrm{Imm}(\mathbb{R}^m,\mathbb{R}^n)\] where \(\mathrm{Emb}(-,-)\) (resp. \(\mathrm{Imm}(-,-)\)) is the space of embeddings (resp. immersions) which agree with the standard inclusion outside a compact. This \(m+1\)-fold delooping was proved by Boavida de Brito and Weiss [1] using configuration categories. It can also be deduced from the results of [2]–[4]. Indeed, Arone and Turchin proved that there is a weak equivalence \[\mathrm{IBimod}_{\mathcal{C}_m}^\mathsf{h}(\mathcal{C}_m,\mathcal{C}_n) \xrightarrow{\sim} \overline{\mathrm{Emb}}(\mathbb{R}^m,\mathbb{R}^n)\] where \(\mathrm{IBimod}_{\mathcal{C}_m}^\mathsf{h}(-,-)\) is the derived mapping space in the category of infinitesimal \(\mathcal{C}_m\)-bimodules. This weak equivalence is the higher dimensional analogue of Sinha’s result [5] concerning the space of long knots. Combining this weak equivalence with the delooping [3] \[\label{equationducoulombier} \Omega \mathrm{SOp}^\mathsf{h}(\mathcal{C}_m,\mathcal{C}_n) \xrightarrow{\sim} \mathrm{Bimod}_{\mathcal{C}_m}^\mathsf{h} (\mathcal{C}_m,\mathcal{C}_n),\tag{2}\] where \(\mathrm{Bimod}_{\mathcal{C}_m}^\mathsf{h}(-,-)\) is the derived mapping space in the category of \(\mathcal{C}_m\)-bimodules, and the \(m\)-fold delooping [4] \[\label{equationducoulombierturchin} \Omega^m \mathrm{Bimod}_{\mathcal{C}_m}^\mathsf{h} (\mathcal{C}_m,\mathcal{C}_n) \xrightarrow{\sim} \mathrm{IBimod}_{\mathcal{C}_m}^\mathsf{h} (\mathcal{C}_m,\mathcal{C}_n),\tag{3}\] one does indeed recover 1 .
The deloopings 2 and 3 are the higher dimensional version of the deloopings due to Dwyer-Hess [6] and Turchin [7]. In [8], Batanin and the author gave a new proof of the deloopings of Dwyer-Hess and Turchin, using the homotopy theory of polynomial monads. Unfortunately, the framework of polynomial monads does not allow us to tackle the higher dimensional delooping problems. The issue is that structures related to the little \(m\)-cubes operad can not be encoded by polynomial monads. The solution to overcome this issue is to work with polynomial \(2\)-monads, which is what we will do in this paper.
The central notion in the homotopy theory of polynomial monad is Batanin’s classifier of internal \(S\)-algebras inside categorical \(T\)-algebras, associated to a cartesian morphism between cartesian monads \(f: S \to T\) [9]. A prototypical example of an internal algebra classifier is the category \(\Delta_+\) of finite ordinals and order-preserving maps, associated to \(id: \mathbf{M} \to \mathbf{M}\), where \(\mathbf{M}\) is free monoid monad. The addition of ordinals gives a strict monoidal structure on \(\Delta_+\). In fact, it is the free strict monoidal category containing a monoid [10]. This means that it has the following universal property: for any strict monoidal category \(A\), monoids in \(A\) are equivalent to strict monoidal functors from \(\Delta_+\) to \(A\). It was proved by Batanin [9] that classifiers can be computed as a bar-construction. For example, the truncated nerve of the category \(\Delta_+\) is: \[\xymatrix@C=1.5pc{ \mathbf{M}^3 1 \ar@<1.5ex>[rr]^{\mu_{\mathbf{M}1}} \ar[rr]|{\mathbf{M}\mu_1} \ar@<-1.5ex>[rr]_{\mathbf{M}^2 !} && \mathbf{M}^2 1 \ar@<1.5ex>[rr]^-{\mu_1} \ar@<-1.5ex>[rr]_-{\mathbf{M}!} && \mathbf{M}1, \ar[ll]|-{\mathbf{M} \eta_1} }\] where \(\eta\) and \(\mu\) are the unit and multiplication of the monad and \(1\) is the singleton set. This formula has been extended by Weber to compute the classifier associated to any cartesian morphism between cartesian \(2\)-monads. Indeed, Weber [11] has proved that such classifiers can be computed as the category of corners (see Definition 7) of a crossed double category (see Definition 6).
Batanin’s classifiers allowed us to introduce a notion of a homotopically cofinal morphism of polynomial monads [8], extending the notion of a homotopy left cofinal functor between small categories [12]. Just like (homotopy) cofinal functors preserve homotopy (limits), homotopically cofinal morphisms of polynomial monads preserve derived mapping spaces. One of the main results of [8] is the cofinality of two specific morphisms of polynomial monads, from which we recovered the deloopings of Dwyer-Hess and Turchin. Similarly, in this paper, we will construct a morphism of polynomial \(2\)-monads \[\label{equationhomotopycofinal} f:\mathcal{K}_m^\fivedots \to \mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}.\tag{4}\] Applying Weber’s formula for the computation of internal algebra classifiers, we will prove (see Theorem 13) that this morphism is homotopically cofinal.
The morphism 4 actually corresponds to a morphism of (symmetric coloured) topological operads which we will describe now in the case \(m=2\). The topological operad \(\mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\) is the operad with set of colours \(I=\{A,B,C,D,E\}\). The operations of this operad are configurations of non-overlapping little squares inside the unit square, where each little square is coloured with an element of \(I\), as in the following picture: \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/yoagzxds.png}\label{muyodtfs}\end{figure}\tag{5}\] The unit square is also coloured with an element of \(I\), called target colour. There is the extra condition that there should be an arrow from the colour of any little cube to the target colour as in the following diagram: \[\label{equationdiamond} \xymatrix{ & D \ar[d] \\ A \ar[r] \ar[ru] \ar[rd] & E & B \ar[l] \ar[lu] \ar[ld] \\ & C \ar[u] }\tag{6}\] So, if the target colour is \(A\) (resp. \(B\)), then each little square is coloured with \(A\) (resp. \(B\)) and if the target colour is \(C\) (resp. \(D\)), then each little square is coloured with \(A\), \(B\) or \(C\) (resp. \(A\), \(B\) or \(D\)). Now the topological operad \(\mathcal{C}_m^\fivedots\) is the suboperad of \(\mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\) where the configurations of little cubes satisfy extra conditions. For example, if the target colour is \(E\), then the little cubes coloured with \(A\) should lie in the half plane \(x<0\), the ones coloured with \(B\) in the half plane \(x>0\), the ones coloured with \(C\) on the half line \((x=0,y<0)\), the ones coloured with \(D\) on the half line \((x=0,y>0)\) and there should be at most one little cube coloured with \(E\), centred at the origin, as in the following picture: \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/prfnsctq.png}\label{ypbxoawr}\end{figure}\tag{7}\] It is interesting to note that, when \(\mathcal{C}_m^\fivedots\) is restricted for example to the colours \(A\) and \(C\), one recovers Voronov’s Swiss cheese operad [13]. The morphism 4 is the polynomial \(2\)-monad version of the canonical inclusion of \(\mathcal{C}_m^\fivedots\) into \(\mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\).
The paper is organised as follows. In Section 1, we will recall the notions of polynomial \(2\)-monads, Batanin’s classifiers and Weber’s formula for their computation. We will also recall some results and definitions of the homotopy theory for polynomial monads from [8], [14], [15], which will be used in this paper. Section 2 is dedicated to proving the cofinality of the morphism 4 . Finally, we will explain in Section 3 how our cofinality result can be used to recover Ducoulombier-Turchin’s delooping 3 . The arguments are similar to the ones in [16].
I would like to thank George Raptis and Christina Vasilakopoulou for our interesting discussions on this topic. I acknowledge that this work was implemented in the framework of H.F.R.I call “3rd Call for H.F.R.I.’s Research Projects to Support Faculty Members & Researchers” (H.F.R.I. Project Number: 23249).
Let \(\mathrm{Cat}\) be the category of small categories. Recall that a functor \(p: E \to B\) is exponentiable when the base change functor \(p^*: \mathrm{Cat}/B \to \mathrm{Cat}/E\), given by taking pullbacks, has a right adjoint.
Definition 1. [17] A polynomial in \(\mathrm{Cat}\) is a diagram of shape \[\label{equationpolynomial} \xymatrix{ I & E \ar[l]_s \ar[r]^p & B \ar[r]^t & J }\tag{8}\] where \(p\) is exponentiable. \(s\) is the source map, \(p\) is the middle map and \(t\) is the target map.
Theorem 1. [17] There is a bicategory \(\mathrm{Poly}\) whose
objects are small categories,
morphisms \(I \to J\) are polynomials,
composition of morphisms looks as follows: \[\xymatrix@C=1pc{ &&&& E \ar[rr] \ar[lllldd] \ar[lldd] \ar@/^.8pc/[rrrr]^{} && Q \ar@{}[rrrdd]|{dpb} \ar@{}[llldd]|{pb} \ar[rr] \ar[d] && B \ar[rrrrdd] \ar[rrdd] \\ &&&&&& P \ar@{}[d]|{pb} \ar[lld] \ar[rrd] \\ I && E_1 \ar[rr] \ar[ll] && B_1 \ar[rr] && J && E_2 \ar[ll] \ar[rr] && B_2 \ar[rr] && K \\ }\] where the diagram on the right is the distributivity pullback [17],
\(2\)-cells are diagrams \[\xymatrix{ I \ar@{=}[d] & E \ar[l]_s \ar[d] \ar[r]^p \ar@{}[rd]|{pb} & B \ar[d] \ar[r]^t & J \ar@{=}[d] \\ I & E' \ar[l]^{s'} \ar[r]_{p'} & B' \ar[r]_{t'} & J }\]
Definition 2. A polynomial \(2\)-monad is a monad in \(\mathrm{Poly}\).
Remark 2. There is a functor between bicategories [17] \[P: \mathrm{Poly} \to \mathrm{CAT},\] where \(\mathrm{CAT}\) is the \(2\)-category of (large) categories, functors and natural transformations. \(P\) sends a category \(I\) to \(\mathrm{Cat}/I\) and a polynomial as in 8 to the composite \[\xymatrix{ \mathrm{Cat}/I \ar[r]^{s^*} & \mathrm{Cat}/E \ar[r]^{p_*} & \mathrm{Cat}/B \ar[r]^{t_!} & \mathrm{Cat}/J, }\] where \(s^*\) is the base change functor, \(p_*\) is the dependent product and \(t_!\) is the dependent sum, which is given by composition with \(t\). In particular, any polynomial \(2\)-monad induces an actual \(2\)-monad.
Example 1. [11]Let \(\mathbf{M}\) be the free strict monoidal category monad. For a category \(A\), \(\mathbf{M}(A)\) is the category whose objects are finite sequences of objects of \(A\) and morphisms are levelwise maps, that is, a morphism \((a_1,\ldots,a_k) \to (b_1,\ldots,b_k)\) is given by morphisms \(f_i: a_i \to b_i\), for \(i=1,\ldots,k\). This monad is induced by the polynomial \[\xymatrix{ 1 & \mathbb{N}^* \ar[l] \ar[r] & \mathbb{N} \ar[r] & 1, }\] where \(\mathbb{N}\) is the set of non-negative integers, seen as a discrete category. \(\mathbb{N}^*\) is the set of pairs \((i,n)\), where \(n \in \mathbb{N}\) and \(1 \leq i \leq n\). The middle map forget \(i\). The multiplication of the monad is of course given by concatenation. Note that, since \(\mathbb{N}\) and \(\mathbb{N}^*\) are discrete, this polynomial \(2\)-monad is actually a polynomial monad.
Example 2. [11] Let \(\mathbf{S}\) be the free symmetric strict monoidal category monad. For a category \(A\), \(\mathbf{S}(A)\) is the category whose objects are finite sequences of objects of \(A\) and morphisms \((a_1,\ldots,a_k) \to (b_1,\ldots,b_k)\) are given by a permutation \(\rho \in \Sigma_k\) and morphisms \(f_i: a_i \to b_{\rho i}\), for \(i=1,\ldots,k\). This monad is induced by the polynomial \[\xymatrix{ 1 & \mathbb{P}^* \ar[l] \ar[r] & \mathbb{P} \ar[r] & 1, }\] where \(\mathbb{P}\) is the permutation category, whose objects are non-negative integers and the set of morphisms \(\mathbb{P}(m,n)\) is the symmetric group \(\Sigma_n\) if \(m = n\), and is empty if \(m \neq n\). \(\mathbb{P}^*\) is the category of pairs \((i,n)\), where \(n \in \mathbb{P}\) and \(1 \leq i \leq n\). A morphism \((i,n) \to (j,n)\) is given by \(\sigma \in \Sigma_n\) such that \(\sigma(i) = j\). The rest of the description of the monad is as in the previous example.
Note that Weber [11] also considers the example of the free braided strict monoidal category monad \(\mathbf{B}\).
Definition 3. An algebra over a polynomial \(2\)-monad \(T\) is an algebra for the induced monad on \(\mathrm{Cat}/I\).
Definition 4. [9] Let \(f: S \to T\) be a morphism of polynomial \(2\)-monads and \(A\) a (strict) \(T\)-algebra. An internal \(S\)-algebra in \(A\) is a lax morphism of \(S\)-algebras \(1 \to f^*(A)\), where \(1\) is the terminal \(S\)-algebra and \(f^*\) is the restriction functor.
Definition 5. Let \(f: S \to T\) be a morphism of polynomial \(2\)-monads. The internal algebra classifier \(T^S\) is the representing object of the functor \[\mathrm{Int}_S: \mathrm{Alg}_T(\mathrm{Cat}) \to \mathrm{Cat}\] sending a \(T\)-algebra \(A\) to the category of internal \(S\)-algebras in \(A\).
Weber proved the existence and gave the formula to compute internal algebra classifiers induced by a morphism of polynomial \(2\)-monads, extending [9]. Weber’s result will be stated in Theorem 3. Before that, we need to recall his notion of category of corners of a crossed double category.
Recall that a (strict) double category is an internal category in \(\mathrm{Cat}\). Explicitly, it consists of a truncated simplicial set in \(\mathrm{Cat}\) \[\label{equationdoublecategory} \xymatrix@C=1.5pc{ X_1 \times_{X_0} X_1 \ar@<1.5ex>[rr]^-{p_2} \ar[rr]|-m \ar@<-1.5ex>[rr]_-{p_1} && X_1 \ar@<1.5ex>[rr]^s \ar@<-1.5ex>[rr]_t && X_0 \ar[ll]|i }\tag{9}\] where \(m\) satisfy an extra associativity condition. \(X_0\) is the category of objects and vertical morphisms, \(X_1\) is the category of horizontal morphisms and squares.
Recall that for a functor \(f: A \to B\), a morphism \(\alpha: a_1 \to a_2\) in \(A\) is \(f\)-opcartesian when for all \(\gamma: a_1 \to a_3\) and \(\beta: f(a_2) \to f(a_3)\) such that \(\beta \cdot f(\alpha) = f(\gamma)\), there is a unique \(\overline{\beta}: a_2 \to a_3\) such that \(f(\overline{\beta}) = \beta\) and \(\overline{\beta} \cdot \alpha = \gamma\): \[\xymatrix@R=1pc@C=2pc{ a_1 \ar[dd]_\alpha \ar[rdd]^\gamma &&&& f(a_1) \ar[dd]_{f(\alpha)} \ar[rdd]^{f(\gamma)} \\ && \ar@{|->}[r]^f & \\ a_2 \ar@{.>}[r]_-{\overline{\beta}} & a_3 &&& f(a_2) \ar[r]_\beta & f(a_3) }\] A cleavage is given by, for each \(a \in A\) and \(\beta: f(a) \to b\), an \(f\)-opcartesian lift of \(\beta\), that is an \(f\)-opcartesian morphism \(\alpha\) in \(A\) such that \(f(\alpha)=\beta\). The functor \(f\) is a split opfibration if there is a cleavage where the \(f\)-opcartesian lifts preserve the identity maps and compositions. Finally, a morphism between split opfibrations \(f_1: A_1 \to B\) and \(f_2: A_2 \to B\) is given by a functor \(g:A_1 \to A_2\) which preserves the opcartesian lifts and such that \(f_2 g = f_1\).
Definition 6. [11]A double category 9 is crossed if \(t: X_1 \to X_0\) is a split opfibration and \(i\) and \(m\) are morphisms of split opfibrations (note that \(tm=t p_1\) is a split opfibration because split opfibrations are preserved under pullbacks and compositions): \[\xymatrix{ X_0 \ar[r]^i \ar@{=}[rd] & X_1 \ar[d]^t & X_2 \ar[l]_m \ar[ld] \\ & X_0 }\]
Definition 7. [11]Let \(X\) be a crossed double category. The \(2\)-category of corners \(\mathrm{Cnr}(X)\) is the \(2\)-category whose
objects are the objects of \(X\),
an arrow \(x \to y\) is given by a pair \((f,g)\), where \(f: x \to a\) is a vertical arrow and \(g: a \to y\) is a horizontal arrow,
composition of \((f,g): x \to y\) and \((h,k): y \to z\) is given as follows: \[\xymatrix{ x \ar[d]^f \\ a \ar[r]^g \ar@{.>}[d] \ar@{}[rd]|\kappa & y \ar[d]^h \\ c \ar@{.>}[r] & b \ar[r]^k & z }\] where \(\kappa\) is the \(t\)-opcartesian lift,
a \(2\)-cell \((f,g) \Rightarrow (h,k)\) is given by a pair \((\alpha,\beta)\), where \(\alpha\) is a vertical arrow such that \(\alpha f = h\) and \(\beta\) is a square as in the following diagram: \[\xymatrix{ x \ar[d]_f \ar@/_1.3pc/[dd]_h \\ a \ar[r]^g \ar[d]_\alpha \ar@{}[rd]|\beta & y \ar@{=}[d] \\ b \ar[r]_k & y }\]
Let \(f: S \to T\) be a morphism of polynomial \(2\)-monads, given by \[\xymatrix{ J \ar[d]_\phi & D \ar[l] \ar[d]_\psi \ar[r] \ar@{}[rd]|{pb} & C \ar[r] \ar[d]^\pi & J \ar[d]^\phi \\ I \ar[r] & E \ar[r] \ar[l] & B \ar[r] & I }\] Recall that \(f\) induces a square of adjunctions: \[\xymatrix{ \mathrm{Alg}_S \ar@/^/[r]^{f_!} \ar@/^/[d]^{U_S} \ar@{}[r]|\perp \ar@{}[d]|\dashv & \mathrm{Alg}_T \ar@/^/[d]^{U_T} \ar@/^/[l]^{f^*} \ar@{}[d]|\dashv \\ \mathrm{Cat}/J \ar@/^/[r]^{\phi_!} \ar@/^/[u]^{F_S} \ar@{}[r]|\perp & \mathrm{Cat}/I \ar@/^/[l]^{\phi^*} \ar@/^/[u]^{F_T} }\] For a \(2\)-category \(\mathbb{C}\), let \(\pi_{0*} \mathbb{C}\) be the category whose objects are objects of \(\mathbb{C}\) and for \(X,Y \in \mathbb{C}\), the set of morphisms from \(X\) to \(Y\) in \(\pi_{0*} \mathbb{C}\) is the set of connected components of the category of morphisms from \(X\) to \(Y\) in \(\mathbb{C}\).
Theorem 3. [11]Let \(f: S \to T\) be a morphism of polynomial \(2\)-monads. Then the bar construction \[\label{equationcldoublecat} \xymatrix{ F_T \phi_! S^2 1 \ar@<1.5ex>[rr]^{\mu_{\phi_! S 1} \cdot F_T f_{S1}} \ar[rr]|{F_T \phi_! \mu_1} \ar@<-1.5ex>[rr]_{F_T \phi_! S!} && F_T \phi_! S 1 \ar@<1.5ex>[rr]^-{\mu_{\phi_! 1} \cdot F_T f_1} \ar@<-1.5ex>[rr]_-{F_T \phi_! !} && F_T \phi_! 1 \ar[ll]|-{F_T \phi_! \eta_1} }\tag{10}\] forms a crossed double category \(X\) and \(T^S\) can be computed as \(\pi_{0*}\mathrm{Cnr}(X)\).
Example 3. [11] Let \(f: \mathbf{M} \to \mathbf{S}\) be the obvious morphism given by inclusion functors. Then 10 becomes \[\label{equationexample} \xymatrix{ \mathbf{S} \mathbf{M} (\mathbb{N}) \ar@<1.5ex>[rr]^{\mu_{\mathbb{N}} \cdot \mathbf{S}f_{\mathbb{N}}} \ar[rr]|{\mathbf{S} \mu_1} \ar@<-1.5ex>[rr]_{\mathbf{S} \mathbf{M} !} && \mathbf{S} (\mathbb{N}) \ar@<1.5ex>[rr]^-{\mu_1 \cdot \mathbf{S}f_1} \ar@<-1.5ex>[rr]_-{\mathbf{S} !} && \mathbb{P}. \ar[ll]|-{\mathbf{S} \eta_1} }\tag{11}\] The objects of this double category are non-negative integers. The category of objects and vertical morphisms is \(\mathbb{P}\). The category of objects and horizontal morphisms is \(\Delta_+\). The squares are given by \[\label{squaredoublesm} \xymatrix{ m \ar[r]^f \ar[d]_\sigma & n \ar[d]^\rho \\ m \ar[r]_g & n }\tag{12}\] where \(f\) and \(g\) are order-preserving maps and \(\sigma\) and \(\rho\) are permutations, as illustrated below [18]: \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/vgmxyauw.png}\label{qhebgpia}\end{figure}\tag{13}\] The category of corners is \(\Sigma \Delta_+\), the free symmetric strict monoidal category containing a monoid [11], [18].
Remark 4. Note that in the previous example, the functor \(\mathbf{S}!\) is even a discrete opfibration. So the double category 11 is codomain-discrete [19]. It was proved that there is a correspondence between codomain-discrete double categories and categories equipped with a strict factorization system [19]. The factorization system in \(\Sigma \Delta_+\) is given by the order-preserving maps and the permutations.
In this subsection we will recall some results and notions of homotopy theory for polynomial monads from [8], [14], [15]. We will assume that these results still hold for polynomial \(2\)-monads. We plan to give the proofs in [20].
With sufficient assumption, the category of algebras over a polynomial monad in a monoidal category admits a transferred model structure [14]. Also recall that for a morphism of polynomial monads \(f: S \to T\) and a \(S\)-algebra \(X\) in a monoidal category \(\mathcal{E}\), there is a functor \(\widetilde{X}: T^S \to \mathcal{E}\) representing \(X\) [14].
Theorem 5. [14]Let \(\mathcal{E}\) be a monoidal model category with a “good” realisation functor for simplicial objects. Let \(f: S \to T\) be a morphism of polynomial monads and \(X\) a \(S\)-algebra in \(\mathcal{E}\) whose underlying collection is pointwise cofibrant. Then \[\mathbb{L}f_!(X) \simeq \operatornamewithlimits{hocolim}_{T^S} \widetilde{X},\] where \(\mathbb{L}f_!\) is the left derived Quillen functor and \(\widetilde{X}: T^S \to \mathcal{E}\) represents \(X\).
Definition 8. [8] A morphism of polynomial \(2\)-monads \(f: S \to T\) is homotopically cofinal if \(N(T^S)\) is contractible.
For a monad \(T\) on a category \(\mathbb{C}\), let \(T+1\) be the monad on \(\mathbb{C} \times \mathbb{C}\) given by \(T \times id\), with evident multiplication and unit. If \(T\) is a polynomial monad, so is \(T+1\) [14], and there is a morphism of polynomial monads \(T+1 \to T\) such that the restriction functor is given by the diagonal composed with the forgetful functor. Also recall that the fundamental groupoid of a category is the groupoid obtained from this category by freely inverting all the morphisms.
Definition 9. [15] A polynomial monad \(T\) is quasi-tame if the fundamental groupoid of the classifier \(T^{T+1}\) is equivalent to a discrete groupoid.
Definition 10. [14] A morphism \(f: X \to Y\) in a model category is an \(h\)-cofibration if pushouts along this morphism preserve weak equivalence. Explicitly, for all diagrams \[\xymatrix{ X \ar[r] \ar[d]_f \ar@{}[rd]|{po} & A \ar[r]^w \ar[d] \ar@{}[rd]|{po} & B \ar[d] \\ Y \ar[r] & A' \ar[r]_{w'} & B' }\] where both squares are pushouts, if \(w\) is a weak equivalence, then so is \(w'\).
Definition 11. [14] A monoidal model category is strongly \(h\)-monoidal if for each cofibration (resp. weak equivalence) \(f: X \to Y\) and each object \(Z\), \(f \otimes id: X \otimes Z \to Y \otimes Z\) is an \(h\)-cofibration (resp. weak equivalence).
For a given monoidal model category, let \(\mathcal{K}\) be the monoidal saturation of the class of cofibrations, that is the smallest class containing all cofibrations and which is closed under pushouts, transfinite compositions, retracts and tensoring with arbitrary objects. A monoidal model category is compactly generated if any object is \(\mathcal{K}\)-small and weak equivalences are closed under filtered colimits along morphisms in \(\mathcal{K}\) [21]. Also recall that a model category is left proper if weak equivalences are preserved under pushouts along cofibrations.
Theorem 6. [15]Let \(T\) be a quasi-tame polynomial monad and \(\mathcal{E}\) a compactly generated strongly \(h\)-monoidal model category. Then the category of \(T\)-algebras in \(\mathcal{E}\) is left proper.
Definition 12. For \(m \geq 0\), let \(\mathfrak{D}^m\) be the poset \[\label{equationposet} (\{-1,1\} \times \{1,\ldots,m\}) \cup \{(0,m+1)\}\tag{14}\] where the order is defined by \((\epsilon,l) < (\epsilon',l')\) whenever \(l < l'\).
Remark 7. There is a canonical correspondence between the objects of \(\mathfrak{D}^m\) and \(2m+1\) points in \(\mathbb{R}^m\). These \(2m+1\) points are the centres of each of the \(2m\) faces of the unit \(m\)-cube plus the centre of the cube. From this observation, it is easy to see that the classifying space of \(\mathfrak{D}^m\) is given by the convex hull of these \(2m+1\) points in \(\mathbb{R}^m\). For \(m=2\), we get a square as pictured in the diagram 6 of the introduction, where \(A=(-1,1)\), \(B=(1,1)\), \(C=(-1,2)\), \(D=(1,2)\) and \(E=(0,3)\). For \(m=3\), we get an octahedron. The notation \(\mathrm{Oct}(m)\) was in fact used for the category \(\mathfrak{D}^m\) in [22].
Recall that a coloured operad \(\mathcal{P}\) in a symmetric monoidal category \((\mathcal{V},\otimes,v)\) is given by a set of colours \(I\) and an object \(\mathcal{P}(i_1,\ldots,i_k;i) \in \mathcal{V}\) of \(k\)-ary operations together with
multiplication maps \[\mathcal{P}(i_1,\ldots,i_k;i) \otimes \mathcal{P}(i_{11},\ldots,i_{1l_1};i_1) \otimes \ldots \otimes \mathcal{P}(i_{k1},\ldots,i_{kl_k};i_k) \to \mathcal{P}(i_{11},\ldots,i_{kl_k};i),\]
unit maps \[v \to \mathcal{P}(i;i),\]
symmetric group actions \[\sigma \in \Sigma_k \mapsto \left[\mathcal{P}(i_1,\ldots,i_k;i) \to \mathcal{P}(i_{\sigma^{-1}(1)},\ldots,i_{\sigma^{-1}(k)};i)\right],\]
satisfying associativity, unitality and equivariance axioms. When \(I\) is the singleton set, we write \[\mathcal{P}(k) := \mathcal{P}(\underbrace{*,\ldots,*}_k;*).\] A \(\mathcal{P}\)-algebra in a symmetric monoidal \(\mathcal{V}\)-category \(\mathcal{M}\) is given by an \(I\)-indexed collection \(A := (A_i)_{i \in I}\) of objects in \(\mathcal{M}\), together with maps \[\mathcal{P}(i_1,\ldots,i_k;i) \to \mathcal{M}(A_{i_1} \otimes \ldots \otimes A_{i_k},A_i),\] satisfying axioms. Finally, recall that for two coloured operad \(\mathcal{P}\) and \(\mathcal{Q}\), the Boardman-Vogt tensor product \(\mathcal{P} \otimes_{BV} \mathcal{Q}\) is the operad whose algebras are \(\mathcal{P}\)-algebras in the category of \(\mathcal{Q}\)-algebras, or equivalently \(\mathcal{Q}\)-algebras in the category of \(\mathcal{P}\)-algebras.
Definition 13. Let \[\mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}} := \mathcal{C}_m \otimes_{BV} \mathfrak{D}^m,\] where \(\mathcal{C}_m\) is the little \(m\)-cubes operad and the category \(\mathfrak{D}^m\) is seen as an operad with only unary operations.
The following is immediate:
Proposition 8. \(\mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\) is the operad whose set of colours is the set of objects of \(\mathfrak{D}^m\). The \(k\)-ary operations are ordered configurations of \(k\) non-overlapping coloured little cubes inside the coloured unit \(m\)-cube. The colours satisfy the following condition. If the \(i\)-th little cube is coloured with \(c_i \in \mathfrak{D}^m\) and the unit cube is coloured with \(c \in \mathfrak{D}^m\), then \(c_i \leq c\). The multiplication is given by plugging cubes of the same colour.
To a pair \((\epsilon,l) \in \{-1,1\} \times \{1,\ldots,m\}\), we associate the half-space \[H(\epsilon,l) = \{ (x_1,\ldots,x_m) \in V(l) \mid \epsilon x_l > 0 \}\] and to \(l \in \{1,\ldots,m+1\}\), we associate the subspace \[V(l) = \{(x_1,\ldots,x_m) \in \mathbb{R}^m \mid x_1 = \ldots = x_{l-1} = 0\}.\]
Definition 14. Let \(\mathcal{C}_m^{\fivedots}\) be the suboperad of \(\mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\) of operations such that for all \(i=1,\ldots,k\), if the \(i\)-th little disk is coloured with \(c_i = (\epsilon_i,l_i)\), then its centre lies in \(V(l_i)\). Moreover, if \(c_i\) is not equal to the target colour, then the \(i\)-th little disk lies entirely in \(H(\epsilon_i,l_i)\).
The complete graph operad \(\mathcal{K}_m\) [23] is the following one-coloured operad in \(\mathrm{Cat}\). For \(k \geq 0\),\[\mathcal{K}_m(k) := \{1,\ldots,m\}^{k \choose 2} \times \Sigma_k,\] where \({k \choose 2}\) is the set of pairs \(ij\) with \(1 \leq i < j \leq k\). It has a poset structure given by \((\mu,\sigma) \leq (\nu,\tau)\) if for all \(ij \in {k \choose 2}\), \(\mu_{ij} < \nu_{ij}\) or \((\mu_{ij},\sigma_{ij}) = (\nu_{ij},\tau_{ij})\), where for \(\sigma \in \Sigma_k\), \(\sigma_{ij} \in \Sigma_2\) is the permutation of \(i\) and \(j\) under \(\sigma\). The multiplication is given by \[\label{equationmultiplicationkm} \begin{align} \mathcal{K}_m(k) \times \mathcal{K}_m(n_1) \ldots \mathcal{K}_m(n_k) &\to \mathcal{K}_m(n_1+\ldots+n_k) \\ ((\mu,\sigma),(\mu_1,\sigma_1),\ldots,(\mu_k,\sigma_k)) &\mapsto (\mu(\mu_1,\ldots,\mu_k),\sigma(\sigma_1,\ldots,\sigma_k)) \end{align}\tag{15}\] where \(\sigma(\sigma_1,\ldots,\sigma_k)\) is given by the permutation operad and \[\mu(\mu_1,\ldots,\mu_k)_{ij} = \begin{cases} \mu_{ij} &\text{if i,j belong to different blocks of {n_1+\ldots+n_k \choose 2},} \\ (\mu_s)_{ij} &\text{if i,j belong to the same block {n_s \choose 2} of {n_1+\ldots+n_k \choose 2}.} \end{cases}\]
Definition 15. Let \[\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}} := \mathcal{K}_m \otimes_{BV} \mathfrak{D}^m.\]
Proposition 9. \(\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\) is the operad whose set of colours is the set of objects of \(\mathfrak{D}^m\). The \(k\)-ary operations are elements of \(\mathcal{K}_m(k)\) together with a source colour* \(c_i \in \mathfrak{D}^m\) for \(i=1,\ldots,k\) and a target colour \(c \in \mathfrak{D}^m\). The colours satisfy the same condition as in Proposition 8. The multiplication is as in 15 .*
For \((\mu,\sigma) \in \mathcal{K}_m(k)\) and \(ij \in {k \choose 2}\), let \(s_{ij} := \mathrm{sgn}(\sigma_{ij})\).
Definition 16. Let \(\mathcal{K}_m^{\fivedots}\) be the suboperad of \(\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\) of operations \((\mu,\sigma)\) such that for all \(ij \in {k \choose 2}\), \(c_i \leq (-s_{ij},\mu_{ij})\) or \(c_j \leq (s_{ij},\mu_{ij})\).
Let \(X\) be a topological space and \(\mathcal{A}\) a poset. Recall that a cellular \(\mathcal{A}\)-decomposition [23] of \(X\) is a collection \((X_\alpha)_{\alpha \in \mathcal{A}}\) of subspaces of \(X\) satisfying the following properties:
\(X_\alpha \subset X_\beta\) if and only if \(\alpha \leq \beta\),
\(\operatornamewithlimits{colim}_{\alpha \in \mathcal{A}} X_\alpha \simeq X\),
for each \(\alpha \in \mathcal{A}\), the canonical map \(\operatornamewithlimits{colim}_{\beta < \alpha} X_\beta \to X_\alpha\) is a closed fibration,
for each \(\alpha \in \mathcal{A}\), \(X_\alpha\) is contractible.
If \(X\) admits a cellular \(\mathcal{A}\)-decomposition, then there is a homotopy equivalence \(X \simeq B\mathcal{A}\) [23], where \(B\) is the classifying space functor, obtained by taking the geometric realization of the nerve.
Lemma 1. There is a homotopy equivalence of topological operads \(\mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}} \simeq B \mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\).
Proof. The proof goes as for [23]. The argument is that for all \(k \geq 0\), there is a cellular \(\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}(k)\)-decomposition of \(\mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}(k)\). For \(x_1\) and \(x_2\) two little cubes, we write \(x_1 \square_\mu x_2\) if \(x_1\) and \(x_2\) are separated by a hyperplane \(P_i\) perpendicular to the \(i\)-th coordinate axis for some \(i \leq \mu\) such that, whenever there is no separating hyperplane \(P_i\) for \(i < \mu\), \(x_1\) lies on the negative side of \(P_\mu\) and \(x_2\) on the positive side of \(P_\mu\). For \(\alpha = (\mu,\sigma) \in \mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}(k)\), the desired cellular decomposition is given by \[\mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}(k)_\alpha := \{x \in \mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}(k) \mid x_i \square_{\mu_{ij}} x_j \text{ if } \sigma_{ij}=id \text{ and } x_j \square_{\mu_{ij}} x_i \text{ if } \sigma_{ij} = (12) \},\] where \(x_i\) is the \(i\)-th little cube of \(x\) for \(i=1,\ldots,k\). ◻
Lemma 2. There is a homotopy equivalence of topological operads \(\mathcal{C}_m^\fivedots \simeq B \mathcal{K}_m^\fivedots\).
Proof. For \(\alpha \in \mathcal{K}_m^\fivedots(k)\), let \[\label{equationcellulation} \mathcal{C}_m^\fivedots(k)_\alpha := \mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}(k)_\alpha \cap \mathcal{C}_m^\fivedots(k).\tag{16}\] Let us prove that this space is contractible, the rest of the proof is straightforward. We will show that it is homotopy equivalent to \(\mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}(k)_\alpha\), which is contractible by the previous lemma. Let \(c = (\epsilon,l)\) be the target colour of \(\alpha=(\mu,\sigma)\) and \(c_i = (\epsilon_i,l_i)\) be the \(i\)-th source colour for \(i=1,\ldots,k\). For \(t \in \{1,\ldots,l\}\), let \(\mathcal{C}_t\) be the space of operations \(x \in \mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}(k)_\alpha\) such that for all \(i=1,\ldots,k\), the centre of \(x_i\) lies in \(V(\lambda)\), where \(\lambda = \min(t,l_i)\), and if \(l_i < t\), then \(x_i\) lies entirely in \(H(\epsilon_i,l_i)\). Then \(\mathcal{C}_1 = \mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}(k)_\alpha\) and \(\mathcal{C}_l = \mathcal{C}_m^\fivedots(k)_\alpha\). Note that for \(t \in \{1,\ldots,l-1\}\), there is a canonical inclusion map \(i_t: \mathcal{C}_{t+1} \to \mathcal{C}_t\). One can also construct a map \(r_t: \mathcal{C}_t \to \mathcal{C}_{t+1}\) which re-centres the little cubes with respect to the \(t\)-th coordinate, as in the following picture: \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/gyztmwhu.png}\label{jgzecvtu}\end{figure}\tag{17}\] Let us prove that the map \(r_t\) can be constructed. Let \(x \in \mathcal{C}_t\) and \(ij \in {k \choose 2}\) such that there is a line in the direction of the \(t\)-th coordinate which intersects both \(x_i\) and \(x_j\). Then \(x_i\) and \(x_j\) are separated by a hyperplane \(P_t\) perpendicular to the \(t\)-th coordinate axis and there is no separating hyperplane perpendicular to another coordinate. To simplify, let us assume without loss of generality that \(\sigma_{ij}=id\), the case \(\sigma_{ij}=(12)\) being completely symmetric. Since \(x \in \mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}(k)_\alpha\), \(x_i \square_{\mu_{ij}} x_j\), which means that \(\mu_{ij} = t\) and \(x_i\) lies on the negative side of \(P_t\) and \(x_j\) on the positive side of \(P_t\). If \(x_i\) needs to move to the positive side in order to be re-centred with respect to the \(t\)-th coordinate, then \(c_i > (-1,\mu_{ij})\). Similarly, if \(x_j\) needs to move to the negative side, then \(c_j > (1,\mu_{ij})\). Since \(\alpha \in \mathcal{K}_m^\fivedots(k)\), \(c_i \leq (-1,\mu_{ij})\) or \(c_j \leq (1,\mu_{ij})\). So both cubes won’t need to move to the opposite side at the same time and there is no obstruction to constructing the map \(r_t\) (the cubes can be made smaller if necessary). The maps \(i_t\) and \(r_t\) are homotopy inverses of each other, which proves that the space 16 is contractible.
In order to have \(\operatornamewithlimits{colim}_{\alpha} \mathcal{C}_m^\fivedots(k)_\alpha \simeq \mathcal{C}_m^\fivedots(k)\), we need to check that for all \(x \in \mathcal{C}_m^\fivedots(k)\), there is \(\alpha \in \mathcal{K}_m^\fivedots(k)\) such that \(x \in \mathcal{C}_m^\fivedots(k)_\alpha\). If \(x \in \mathcal{C}_m^\fivedots(k)\), then \(x \in \mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}(k)\) and we know from the previous lemma that there is \(\alpha \in \mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}(k)\) such that \(x \in \mathcal{C}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}(k)_\alpha\). It remains to prove that \(\alpha \in \mathcal{K}_m^\fivedots(k)\) automatically. Let \(ij \in {k \choose 2}\) and let us prove that \(c_i \leq (-s_{ij},\mu_{ij})\) or \(c_j \leq (s_{ij},\mu_{ij})\). We can assume to simplify that \(\sigma_{ij}=id\), without loss of generality. Since \(x \in \mathcal{C}_m^\fivedots(k)\), the centres of both \(x_i\) and \(x_j\) lie in \(V(\lambda)\), where \(\lambda=\min(l_i,l_j)\). In particular there is no separating hyperplane for \(t < \lambda\). Since \(x_i \square_{\mu_{ij}} x_j\), we must have \(\mu_{ij} \geq \lambda\). If \(\mu_{ij} > \lambda\), the conclusion is immediate. If \(\mu_{ij}=\lambda\), this means that \(x_i\) lies on the negative side of a hyperplane perpendicular to the \(\lambda\)-th coordinate and \(x_j\) on the positive side. Then \(x_i\) lies entirely in \(H(-1,\lambda)\) or \(x_j\) lies entirely in \(H(1,\lambda)\). Using the fact that \(x \in \mathcal{C}_m^\fivedots(k)\) again, this implies that \(c_i \leq (-1,\lambda)\) or \(c_j \leq (1,\lambda)\).
In conclusion, the conditions in the definition of \(\mathcal{K}_m^{\fivedots}\) are indeed the ones necessary to have the equivalence of operads of the lemma. The remaining arguments are as in the proof of [24]. ◻
Definition 17. Let \(\mathcal{K}_m\) be the polynomial \(2\)-monad given by the polynomial \[\xymatrix{ 1 & \mathtt{CG}^* \ar[r] \ar[l] & \mathtt{CG} \ar[r] & 1, }\] where \(\mathtt{CG} = \coprod_{k \geq 0} \mathtt{CG}(k)\) and \(\mathtt{CG}(k)\) is the category whose set of objects is \[\{1,\ldots,m\}^{k \choose 2}\] and a morphism \(\mu \to \nu\) is given by \(\sigma \in \Sigma_k\) such that for all \(ij \in {k \choose 2}\), if \(\sigma(j) > \sigma(i)\) then \(\mu_{ij} < \nu_{ij}\). Similarly, \(\mathtt{CG}^* = \coprod_{k \geq 0} \mathtt{CG}^*(k)\), where \(\mathtt{CG}^*(k)\) is the category whose objects are given by an object of \(\mathtt{CG}(k)\) together with \(i \in \{1,\ldots,k\}\) and morphisms \((\mu,i) \to (\nu,j)\) are given by a morphism \(\sigma: \mu \to \nu\) such that \(\sigma(i)=j\). As before, the middle map forgets \(i\). Multiplication is as in 15 .
Proposition 10. The algebras of the complete graph operad \(\mathcal{K}_m\) are the same as the algebras of the polynomial \(2\)-monad \(\mathcal{K}_m\).
Proof. The \(2\)-monad induced by the polynomial \(2\)-monad \(\mathcal{K}_m\) sends a category \(A\) to the category whose objects are given by \(\mu \in \{1,\ldots,m\}^{k \choose 2}\) together with objects \((a_1,\ldots,a_k)\) in \(A\). The morphisms are morphisms \(\sigma: \mu \to \nu\) in \(\mathtt{CG}(k)\) together with a morphism \(a_i \to b_{\sigma i}\) for \(i=1,\ldots,k\). Abusing notations, for \(\sigma \in \Sigma_k\), let \(\sigma: A^k \to A^k\) be the functor sending \((a_1,\ldots,a_k)\) to \((a_{\sigma 1},\ldots,a_{\sigma k})\). A strict categorical algebra of the polynomial \(2\)-monad \(\mathcal{K}_m\) is given by a category \(A\) together with a functor \(\hat{\mu}: A^k \to A\) for each \(\mu \in \{1,\ldots,m\}^{k \choose 2}\) and a natural transformation \(\hat{\mu} \Rightarrow \hat{\nu} \cdot \sigma\) for each morphism \(\sigma: \mu \to \nu\) in \(\mathtt{CG}(k)\), satisfying axioms. On the other hand, an algebra of the complete graph operad is given by a category \(A\), a functor \((\hat{\mu},\hat{\sigma}): A^k \to A\) for each \((\mu,\sigma) \in \mathcal{K}_m(k)\) and a natural transformation \((\hat{\mu},\hat{\sigma}) \Rightarrow (\hat{\nu},\hat{\tau})\) for each morphism \((\mu,\sigma) \to (\nu,\tau)\) in \(\mathcal{K}_m(k)\), again satisfying axioms. From an algebra of \(\mathcal{K}_m\) as a categorical operad, we get an algebra of \(\mathcal{K}_m\) as a polynomial \(2\)-monad by taking \(\hat{\mu} = (\hat{\mu},\hat{id})\) and \(\hat{\mu} \Rightarrow \hat{\nu} \cdot \sigma\) associated to \((\mu,id) \to (\nu,\sigma)\). In the other direction, we take \((\hat{\mu},\hat{\sigma})\) as the composite \(\hat{\mu} \cdot \sigma\) and \((\hat{\mu},\hat{\sigma}) \Rightarrow (\hat{\nu},\hat{\tau})\) associated to \(\tau\sigma^{-1}: \mu \to \nu\). This gives us two functors between the categories of algebras which are clearly inverse of each other. ◻
Remark 11. In [25], Weber constructed a \(2\)-monad \(T/\Sigma\) from an operad \(T\) in \(\mathrm{Set}\). He proved that if \(T\) is \(\Sigma\)-free [25], then \(T/\Sigma\) is a polynomial \(2\)-monad. Moreover, the algebras of \(T\) and \(T/\Sigma\) coincide. We could extend this construction to when \(T\) is an operad in \(\mathrm{Cat}\). Then the polynomial \(2\)-monad \(\mathcal{K}_m\) is actually \(T/\Sigma\) when \(T\) is the complete graph operad.
Definition 18. Let \(\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\) be the polynomial \(2\)-monad given by the polynomial \[\xymatrix{ \mathrm{ob}(\mathfrak{D}^m) & \mathtt{CCG}^* \ar[l] \ar[r] & \mathtt{CCG} \ar[r] & \mathrm{ob}(\mathfrak{D}^m) }\] where \(\mathrm{ob}(\mathfrak{D}^m)\) is the set of objects of \(\mathfrak{D}^m\). \(\mathtt{CCG} = \coprod_{k \geq 0} \mathtt{CCG}(k)\) and \(\mathtt{CCG}(k)\) is the category whose objects are given by \(\mu \in \mathtt{CG}(k)\) equipped with \(c_i \in \mathfrak{D}^m\) for \(i=1,\ldots,k\) and \(c \in \mathfrak{D}^m\). The morphisms \((\mu,c_1,\ldots,c_k,c) \to (\nu,d_1,\ldots,d_k,d)\) are given by morphisms \(\sigma: \mu \to \nu\) in \(\mathtt{CG}(k)\) such that \(c_i=d_{\sigma i}\) for \(i=1,\ldots,k\) and \(c=d\). Again, \(\mathtt{CCG}^* = \coprod_{k \geq 0} \mathtt{CCG}^*(k)\), where \(\mathtt{CCG}^*(k)\) is the category whose objects are given by an object of \(\mathtt{CCG}(k)\) together with \(i \in \{1,\ldots,k\}\). The source map returns the colour \(c_i\). The middle map forgets \(i\). The target map returns the colour \(c\).
Definition 19. Let \(\mathcal{K}_m^{\fivedots}\) be the polynomial \(2\)-monad given by the polynomial \[\xymatrix{ \mathrm{ob}(\mathfrak{D}^m) & \mathtt{PCG}^* \ar[l] \ar[r] & \mathtt{PCG} \ar[r] & \mathrm{ob}(\mathfrak{D}^m) }\] where \(\mathtt{PCG}\) is the full subcategory of \(\mathtt{CCG}\) of objects \((\mu,c_1,\ldots,c_k,c) \in \mathtt{CCG}(k)\) such that for all \(ij \in {k \choose 2}\), \(c_i \leq (-1,\mu_{ij})\) or \(c_j \leq (1,\mu_{ij})\). The rest of the description is as in Definition 18.
Proposition 12. The algebras of the categorical operads \(\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\) and \(\mathcal{K}_m^\fivedots\) are the same as the algebras of \(\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\) and \(\mathcal{K}_m^\fivedots\) as polynomial \(2\)-monads.
Proof. We can proceed as in the proof of Proposition 10. ◻
Let \[\label{equationmappolymon} f: \mathcal{K}_m^\fivedots \to \mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\tag{18}\] be the morphism of polynomial \(2\)-monads given by inclusion functors.
For \(\mu \in \mathtt{CCG}(k)\) and \(S := \{l_1,\ldots,l_r\} \subset \{1,\ldots,k\}\), the restriction of \(\mu\) to \(S\) is defined as \(\mu' \in \mathtt{CCG}(r)\) with \(\mu'_{ij}:=\mu_{l_i l_j}\).
Lemma 3. The classifier associated to the morphism of polynomial \(2\)-monads 18 is the category whose objects are objects of \(\mathtt{CCG}\) and there is a morphism \((\mu,c_1,\ldots,c_k,c) \to (\nu,d_1,\ldots,d_l,d)\) if \(c=d\) and it is given by any function \(g: \{1,\ldots,k\} \to \{1,\ldots,l\}\) together with morphisms \(c_i \to d_{g(i)}\) in \(\mathfrak{D}^m\) for \(i=1,\ldots,k\) such that
for \(ij \in {k \choose 2}\), if \(g(i)<g(j)\), then \(\mu_{ij}=\nu_{g(i)g(j)}\), and if \(g(j)>g(i)\), then \(\mu_{ij}<\nu_{g(j)g(i)}\),
for \(j=1,\ldots,l\), the restriction of \(\mu\) to \(g^{-1}(j)\) is in \(\mathtt{PCG}\).
Proof. The formula 10 becomes \[\label{equationdoublecat} \xymatrix{ \mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}} \mathcal{K}_m^\fivedots (\mathtt{PCG}) \ar@<1.5ex>[r] \ar[r] \ar@<-1.5ex>[r] & \mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}} (\mathtt{PCG}) \ar@<1.5ex>[r] \ar@<-1.5ex>[r] & \mathtt{CCG} \ar[l] }\tag{19}\] The objects of the double category are the objects of \(\mathtt{CCG}\). The vertical morphisms are the morphisms of \(\mathtt{CCG}\). Let \(\mathbb{C}\) be the category described in the statement of the lemma, which we want to prove is the classifier. The morphisms of \(\mathtt{CCG}\) correspond to morphisms of \(\mathbb{C}\) where \(g\) is a bijection and the morphisms \(c_i \to d_{g(i)}\) are the identities. The category \(\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}} (\mathtt{PCG})\) is the category whose objects are given by \(\nu \in \{1,\ldots,m\}^{l \choose 2}\) together with objects \((\mu_1,\ldots,\mu_l)\) in \(\mathtt{PCG}\), objects \((c_1,\ldots,c_l)\) in \(\mathfrak{D}^m\) and \(c \in \mathfrak{D}^m\) such that \(\mu_i\) has target colour \(c_i\) for \(i=1,\ldots,l\). This is equivalent to giving a morphism in \(\mathbb{C}\) where the function \(g\) is order-preserving. Indeed, \((\mu_1,\ldots,\mu_l)\) correspond to the restrictions of \(\mu\) to the fibres of \(g\). As in [11], a square of the double category, that is a morphism in \(\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}} (\mathtt{PCG})\), is completely determined by its boundary, and a square will exist if it forms a commutative diagram in \(\mathbb{C}\). As in [11], the opcartesian lifts are given by bijective-monotone factorisation [11] (see also [26]). So, similarly to [11], the \(2\)-category of corner of 19 is the following. The objects are objects of \(\mathtt{CCG}\), the morphisms are pairs \((\rho,g)\) of morphisms in \(\mathbb{C}\) where \(\rho\) is a vertical morphism (given by a permutation) and \(g\) is a horizontal morphism. A \(2\)-cell between \((\rho_1,g_1)\) and \((\rho_2,g_2)\) exists if and only if \(g_2 \rho_2 \rho_1^{-1} = g_1\), that is if and only if \(g_2 \rho_2 = g_1 \rho_1\). ◻
Theorem 13. The morphism of polynomial \(2\)-monads 18 is homotopically cofinal.
We will proceed as in [8]. We have the commutative square of polynomial \(2\)-monads \[\label{equationsquare} \xymatrix{ \mathcal{K}_m^\fivedots \ar[r]^{pf} \ar[d]_f & \mathcal{K}_m \ar@{=}[d] \\ \mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}} \ar[r]_p & \mathcal{K}_m }\tag{20}\] According to [8], this induces a functor \[\label{equationsmoothfunctor} (\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}})^{\mathcal{K}_m^\fivedots} \to p^*\left((\mathcal{K}_m)^{\mathcal{K}_m}\right).\tag{21}\] Recall [27] that a functor \(F\) is smooth if for all \(y \in \mathcal{Y}\), the canonical inclusion \[\label{equationcanonicalinclusion} F_y \to y/F,\tag{22}\] where \(F_y\) is the fibre over \(y\), induces a weak equivalence between nerves.
Lemma 4. The functor 21 is smooth.
Proof. To simplify the notations, let us denote the functor 21 by \(F: \mathcal{X} \to \mathcal{Y}\). For \(g_1: y \to y_1\) in \(\mathcal{Y}\) and \(x_1 \in \mathcal{X}\) such that \(F(x_1) = y_1\), let \(\mathcal{X}(x_1,g_1)\) be the category whose objects are morphisms \(g: x \to x_1\) such that \(F(g)=g_1\) and a morphism from \(g:x \to x_1\) to \(g':x' \to x_1\) is given by \(h: x \to x'\) in \(F_y\) such that \(g'h=g\). According to [27], \(F\) is smooth if \(\mathcal{X}(x_1,g_1)\) has contractible nerve for all \(x_1\) and \(g_1\).
First let us describe \(F\) explicitly. Note that the category \(\mathcal{Y}\) admits a description similar to the description of \(\mathcal{X}\) given in Lemma 3. The objects are elements of \(\mathtt{CG}\), so they are given by \(k \geq 0\) and \(\mu \in \{1,\ldots,m\}^{k \choose 2}\). The morphisms \(\mu \to \nu\) are given by functions \(g: \{1,\ldots,k\} \to \{1,\ldots,l\}\) such that for \(ij \in {k \choose 2}\), if \(g(i) < g(j)\), then \(\mu_{ij}=\nu_{g(i)g(j)}\), and if \(g(j) > g(i)\), then \(\mu_{ij}<\nu_{g(j)g(i)}\). Note that there is a terminal object given by \(k=1\). The functor \(F\) just forgets the colours \(c_i \in \mathfrak{D}^m\) for \(i=1,\ldots,k\) and \(c \in \mathfrak{D}^m\).
Let us now prove that \(\mathcal{X}(x_1,g_1)\) has contractible nerve for all \(x_1\) and \(g_1\) as above. The objects of this category are given by \(x \in F_y\) together with \(g: x \to x_1\) such that \(F(g)=g_1\). Note that \(g\) is entirely determined by \(g_1\) and is given by a function \(\{1,\ldots,k\} \to \{1,\ldots,l\}\), so \(\mathcal{X}(x_1,g_1)\) can be seen as a subcategory of \(F_y\). Moreover, objects of \(F_y\) are given by the object \(y \in \mathcal{Y}\), that is \(y \in \mathtt{CG}(k)\) for some \(k \geq 0\), together with a colouring of \(y\), that is \(c_i \in \mathfrak{D}^m\) for \(i=1,\ldots,k\) and \(c \in \mathfrak{D}^m\). So \(\mathcal{X}(x_1,g_1)\) is the category whose objects \(x\) are given by colourings of \(y\) such that for \(j=1,\ldots,l\), the restriction of \(x\) to the \(j\)-th fibre of \(g\) gives an element of \(\mathtt{PCG}\). We can therefore assume without loss of generality that \(y_1\) is the terminal object. Indeed, the category \(\mathcal{X}(x_1,g_1)\) in the general case will be isomorphic to the product over the fibres of \(g_1\) of \(\mathcal{X}(x_1^j,g_1^j)\), where \(x_1^j\) and \(g_1^j\) are the restrictions of \(x_1\) and \(g_1\) to the \(j\)-th fibre of \(g_1\), so \(g_1^j\) is just the unique map to the terminal object. When \(y_1\) is the terminal object, \(x_1\) is given by \(d_1 \in \mathfrak{D}^m\) and \(d \in \mathfrak{D}^m\). \(\mathcal{X}(x_1,g_1)\) is the category whose objects are given by \(c_i \in \mathfrak{D}^m\) such that \(c_i \leq d_1\) for \(i=1,\ldots,k\) and the obtained colouring of \(y\) is an element of \(\mathtt{PCG}\). The morphisms are just given by morphisms \(c_i \to c'_i\) in \(\mathfrak{D}^m\) for \(i=1,\ldots,k\).
Since \(\mathcal{X}(x_1,g_1)\) depends on \(y \in \mathtt{CG}(k)\) and \(d_1 \in \mathfrak{D}^m\), let \(\chi(d_1,y) := \mathcal{X}(x_1,g_1)\) (using the notation in the proof of [16]). Let us now prove that the nerve of \(\chi(d_1,y)\) is contractible. We will prove a more general statement. Let \(S \subset {k \choose 2}\), \(y \in \{1,\ldots,m\}^S\) and \(d:=(d_1,\ldots,d_k)\), where \(d_i=(\epsilon_i,l_i) \in \mathfrak{D}^m\) for \(i=1,\ldots,k\). Let us assume that for all \(ij \in S\), \(\epsilon_i \leq \epsilon_j\). Let \(\chi(d,y)\) be the category whose objects are given by \(c_i \in \mathfrak{D}^m\) such that \(c_i \leq d_i\) for \(i=1,\ldots,k\) and for \(ij \in S\), \(c_i \leq (-1,y_{ij})\) or \(c_j \leq (1,y_{ij})\). As above, the morphisms are given by morphisms \(c_i \to c'_i\) in \(\mathfrak{D}^m\) for \(i=1,\ldots,k\). We will prove that \(\chi(d,y)\) has contractible nerve.
Let us proceed by induction on the number of elements in \(S\). The case where \(S\) is empty is trivial. If \(S\) is non-empty, let \(ij \in S\). Let us assume that \(\epsilon_i \neq 1\), otherwise \(\epsilon_j \neq -1\) and we can apply a similar argument. Let \(d'_r := d_r\) for \(r=1,\ldots,k\) with \(r \neq i\) and let \(d'_i := (-1,\lambda)\), where \(\lambda=\min(l_i,y_{ij})\). There is a functor \(\alpha: \chi(d,y) \to \chi(d',y)\) which replaces \(c_i\) by \(d'_i\) if \(c_i \nless d'_i\). Let us prove that \(\alpha\) induces a weak equivalence between nerves. For \(c' \in \chi(d',y)\), the fibre category \(\alpha_{c'}\) is trivial if \(c'_i \neq d'_i\). If \(c'_i = d'_i\), it is equivalent to the category whose objects are \(c_i \in \mathfrak{D}^m\) such that \(c_i \leq d_i\) but \(c_i \nless d'_i\). This last category has a terminal object given by \(d_i\). Moreover the canonical inclusion \(\iota: \alpha_{c'} \to c'/\alpha\) has a right adjoint. This right adjoint sends \(c' \to \alpha(c)\) to \(c''\) given by \(c''_r := c'_r\) for \(r=1,\ldots,k\) with \(r \neq i\) and \(c''_i := c'_i\) if \(c'_i < d'_i\) and \(c''_i := c_i\) otherwise. The unit is the identity, the counit is the unique map. So \(\iota\) induces a weak equivalence between nerves, which means that \(c'/\alpha\) has a contractible nerve for all \(c'\). We deduce, using Quillen’s Theorem A, that \(\alpha\) induces a weak equivalence between nerves. Note that the category \(\chi(d',y)\) is equivalent to \(\chi(d',y')\), where \(y'\) is the restriction of \(y\) to \(S'\), and \(S'\) is the set \(S\) minus \(ij\). By induction the nerve \(\chi(d',y')\) is contractible, so the nerve of \(\chi(d,y)\) is also contractible. This concludes the proof. ◻
Proof of Theorem 13. We want to prove that \(\mathcal{X}\) has a contractible nerve. As it was pointed out in the proof of Lemma 4, \(\mathcal{Y}\) has a contractible nerve since it has a terminal object. It remains to prove that \(F\) induces a weak equivalence between nerves.
Let us prove that for all \(y \in \mathcal{Y}\), \(F_y\) has a contractible nerve. To be precise, \(\mathcal{Y}\) is a collection of categories indexed by the objects of \(\mathfrak{D}^m\). Let \(y \in \mathcal{Y}\) indexed by \(c \in \mathfrak{D}^m\). If \(y \in \mathtt{CG}(k)\), the fibre category \(F_y\) is the category whose objects are given by \(c_i \in \mathfrak{D}^m\) such that \(c_i \leq c\) for \(i=1,\ldots,k\). The morphisms are given by morphims \(c_i \to c_{i'}\) in \(\mathfrak{D}^m\) for \(i=1,\ldots,k\). This category has an obvious terminal object given by \(c_i=c\) for all \(i\). So \(F_y\) indeed has a contractible nerve.
According to Lemma 4, \(F\) is smooth, which means by definition that the functor 22 induces a weak equivalence between nerve. So \(y/F\) has a contractible nerve for all \(y \in \mathcal{Y}\). The conclusion that \(F\) induces a weak equivalence between nerves follows from Quillen’s Theorem A. ◻
Lemma 5. The polynomial \(2\)-monad \(\mathcal{K}_m\) is quasi-tame.
Proof. We need to compute the classifier \(T^{T+1}\) when \(T=\mathcal{K}_m\). Using the description given in [14], we can prove, as in Lemma 3, that we get the following category. The objects are given by \(k \geq 0\) and an object of \(\mathtt{CG}(k)\), together with a colour \(X\) or \(K\) for \(i=1,\ldots,k\). The morphisms are given by functions \(g: \{1,\ldots,k\} \to \{1,\ldots,l\}\) satisfying the condition [conditionclassifier] of Lemma 3. Moreover, the function \(g\) should send an element \(i \in \{1,\ldots,k\}\) coloured with \(X\) or \(K\) to an element of \(\{1,\ldots,l\}\) with the same colour, and \(g\) restricted to the set of elements coloured with \(K\) should be injective and order-preserving. This category is a coproduct of categories with an initial object. The initial object in each component is given the object where each \(i \in \{1,\ldots,k\}\) is coloured with \(K\). In particular, the fundamental groupoid of this category is equivalent to a discrete groupoid. ◻
Remark 14. It was proved in [14] that the polynomial monad \(\mathbf{M}\) of example 1 is tame [14]. More explicitly, it was proved that the classifier \(\mathbf{M}^{\mathbf{M}+1}\) is a coproduct of categories with a terminal object. In fact, each connected component has both an initial and a terminal object. In the proof of the lemma above however, the connected components of the classifier have an initial but not always a terminal object.
Let \(\mathcal{A}\) and \(\mathcal{B}\) be two (one-coloured) operads in a symmetric monoidal category \((\mathcal{V},\otimes,v)\). Recall that a left \(\mathcal{A}\)-module is given by a collection \(\mathcal{C} := (\mathcal{C}(k))_{k \geq 0}\) of objects in \(\mathcal{V}\) together with maps \[\mathcal{A}(k) \otimes \mathcal{C}(n_1) \otimes \ldots \otimes \mathcal{C}(n_k) \to \mathcal{C}(n_1+\ldots+n_k)\] satisfying axioms. Similarly, a right \(\mathcal{B}\)-module is given by a collection \(\mathcal{C} := (\mathcal{C}(k))_{k \geq 0}\) of objects in \(\mathcal{V}\) together with maps \[\mathcal{C}(k) \otimes \mathcal{B}(n_1) \otimes \ldots \otimes \mathcal{B}(n_k) \to \mathcal{C}(n_1+\ldots+n_k)\] satisfying axioms. Finally, \(\mathcal{C}\) is an \(\mathcal{A}-\mathcal{B}\)-bimodule if it has left \(\mathcal{A}\)-module and right \(\mathcal{B}\)-module structure satisfying an extra compatibility axiom. An \(\mathcal{A}\)-bimodule is an \(\mathcal{A}-\mathcal{B}\)-bimodule where \(\mathcal{A}=\mathcal{B}\).
Lemma 6. The category of \(\mathcal{K}_m\)-bimodules is left proper.
Proof. First note that the category of bimodules over an operad \(\mathcal{P}\) is equivalent to the category of algebras in the category of right modules over this operad [28]. The monoidal product \(\otimes\) in the category of right modules is given by convolution. So we need to apply Theorem 6 to the case where the polynomial \(2\)-monad \(T\) is \(\mathcal{K}_m\) and \(\mathcal{E}\) is the category of right \(\mathcal{K}_m\)-modules. We know from Lemma 5 that \(\mathcal{K}_m\) is quasi-tame. It remains to check that the category of right \(\mathcal{K}_m\)-modules is strongly \(h\)-monoidal. Since it is a category of presheaves, any cofibration \(f\) is in particular a pointwise cofibration. Therefore, \(f \otimes id\) is still a pointwise cofibration. This means that pushouts along \(f \otimes id\) preserve weak equivalences, since pushouts and weak equivalences are given pointwise and \(\mathrm{Cat}\) is left proper. So the category of right \(\mathcal{K}_m\)-modules is indeed strongly \(h\)-monoidal.
Note that an alternative proof could have been to show directly that the polynomial \(2\)-monad for \(\mathcal{K}_m\)-bimodules is quasi-tame. ◻
Definition 20. [16] Let \(\mathcal{M}\) be a model category and \(A \in \mathcal{M}\). For \(m \geq 0\), let \(S^{m-1}(A) \in \mathcal{M}\) defined by induction as follows. \(S^{-1}(A)\) is the initial object and, for \(m \geq 0\), \(S^m(A)\) is the pushout \[\xymatrix{ S^{m-1}(A) \ar@{>->}[r] \ar@{>->}[d] \ar@{}[rd]|{po} & D^m(A) \ar[d] \\ D^m(A) \ar[r] & S^m(A) }\] where \(D^m(A)\) is a factorisation \[\xymatrix@C=1pc{ S^{m-1}(A) \ar[rr] \ar@{>->}[rd] && A \\ & D^m(A) \ar[ru]_\sim }\]
To simplify the notations, let \(t=(0,m+1)\) be the terminal object of \(\mathfrak{D}^m\).
Definition 21. Let \(\mathfrak{S}^{m-1}\) be the full subcategory of \(\mathfrak{D}^m\) of all the objects of \(\mathfrak{D}^m\) except \(t\).
Lemma 7. Let \(\mathcal{M}\) be a model category. For \(A \in \mathcal{M}\), \(S^{m-1}(A)\) can be computed as the homotopy colimit of the functor \(\delta_A: \mathfrak{S}^{m-1} \to \mathcal{M}\) which is constant to \(A\).
Proof. Let \(\delta'_A: \mathfrak{S}^{m-1} \to \mathcal{M}\) be the functor sending \((\epsilon,l)\) to \(D^l(A)\). Then \(\delta'_A\) is a cofibrant replacement \(\delta_A\) in the category of covariant presheaves over \(\mathfrak{S}^{m-1}\) in \(\mathcal{M}\). Therefore \(\operatornamewithlimits{hocolim}\delta_A = \operatornamewithlimits{colim}\delta'_A = S^{m-1}(A)\). ◻
For a model category \(\mathcal{M}\), let \(\mathcal{M}^\mathsf{h}(-,-)\) be the derived mapping space in \(\mathcal{M}\). Let \(\mathrm{Bimod}_{\mathcal{K}_m}\) be the category of \(\mathcal{K}_m\)-bimodules.
Lemma 8. For any morphism \(f: X \to Y\) in \(\mathrm{Bimod}_{\mathcal{K}_m}\), there is a weak equivalence \[\Omega^m \mathrm{Bimod}_{\mathcal{K}_m}^\mathsf{h}(X,Y) \xrightarrow{\sim} (S^{m-1}(X)/\mathrm{Bimod}_{\mathcal{K}_m})^\mathsf{h} (X,Y).\]
Proof. This delooping actually holds not only for \(\mathrm{Bimod}_{\mathcal{K}_m}\) but in general for any left proper model category. This was proved in [16] when \(X\) is contractible. However the theorem actually remains true for any \(X\), since the same arguments still apply. Since \(\mathrm{Bimod}_{\mathcal{K}_m}\) is left proper according to Lemma 6, it also holds in this particular case. ◻
Let \(\mathcal{M}\) be the category of right \(\mathcal{K}_m\)-modules.
Definition 22. Let \(\mathcal{K}_m^{\diamonddots}\) be the polynomial \(2\)-monad \(\mathcal{K}_m^{\fivedots}\) restricted to the set of objects of \(\mathfrak{S}^{m-1}\). Let \(\mathbb{B}\) be the category of algebras over \(\mathcal{K}_m^{\diamonddots}\) and \[\Phi: \mathbb{B} \to \mathrm{CAT}\] be the contravariant functor which sends \(b \in \mathbb{B}\) to the category of objects \(X \in \mathcal{M}\) together with the structure of a \(\mathcal{K}_m^{\fivedots}\)-algebra on the pair \((b,X)\).
Definition 23. Let \(\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\square}}\) be the polynomial \(2\)-monad \(\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\) restricted to the set of objects of \(\mathfrak{S}^{m-1}\). Let \(\mathbb{C}\) be the category of algebras over \(\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\square}}\) and \[\Psi: \mathbb{C} \to \mathrm{CAT}\] be the contravariant functor which sends \(c \in \mathbb{C}\) to the category of objects \(Y \in \mathcal{M}\) together with the structure of a \(\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\)-algebra on the pair \((c,Y)\).
Lemma 9. Let \(f: S \to T\) be a morphism of polynomial \(2\)-monads. If the functor \(T^S \to T^T\) preserves homotopy colimits, then for \(T\)-algebra \(X\), there is a weak equivalence \[\mathbb{L} f_! f^*(X) \simeq X.\]
Proof. It is a direct consequence of Theorem 5. ◻
Let \(f: \mathcal{K}_m^\fivedots \to \mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\) and \(p: \mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}} \to \mathcal{K}_m\) be the morphisms of polynomial \(2\)-monads as in the square 20 . Abusing notations, let \(\mathcal{K}_m\) be the algebra over itself in \(\mathcal{M}\) and \(\mathcal{K}_m := p^*(\mathcal{K}_m)\) and \(\mathcal{K}_m := f^* p^* (\mathcal{K}_m)\).
Lemma 10. If \((b,X) = D^0(\mathcal{K}_m) \in \int \Phi\), then for any map \(\mathcal{K}_m \to Y\) in \(\Psi(b)\), there is a weak equivalence \[\Psi(b)^\mathsf{h} (\mathcal{K}_m,Y) \xrightarrow{\sim} \Phi(b)^\mathsf{h} (\mathcal{K}_m,Y).\]
Proof. We follow the same arguments as in the proof of [16]. The map of polynomial \(2\)-monads \(f: \mathcal{K}_m^{\fivedots} \to \mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\) restricts to a map \(g: \mathcal{K}_m^{\diamonddots} \to \mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\square}}\). It is also homotopically cofinal since the classifier is also given by restriction. Let \(H: \Phi(b) \to \Psi g_! (b)\) be the functor such that \(f_!(b,X) = (g_!(b),H(X))\). Since \((b,X)\) is a cofibrant replacement of \(\mathcal{K}_m\), \(f_!(b,X) \simeq \mathbb{L}f_! (\mathcal{K}_m)\). It was proved in Theorem 13 that the functor \(F: \mathcal{X} \to \mathcal{Y}\) defined in 21 is such that for all \(y \in \mathcal{Y}\), the nerve of \(y/F\) is contractible. This means that \(F\) preserves homotopy colimits. So we can apply Lemma 9, to get the weak equivalence \(\mathbb{L}f_! (\mathcal{K}_m) \simeq \mathcal{K}_m\). We deduce that \(f_!(b,X) \simeq \mathcal{K}_m\), and in particular \(H(X) \simeq \mathcal{K}_m\). The conclusion follows from an adjunction argument similar to [16]. ◻
Definition 24. Let \(\mathcal{K}_m^{\mathbin{\tikz{ \draw (0,0) circle (0.65pt); \fill (0,-.067) circle (0.45pt); \fill (0,.067) circle (0.45pt); \fill (-.067,0) circle (0.45pt); \fill (.067,0) circle (0.45pt); }}}\) be the polynomial \(2\)-monad given by the polynomial \[\xymatrix{ \mathrm{ob}(\mathfrak{D}^m) & \mathtt{UCG}^* \ar[l] \ar[r] & \mathtt{UCG} \ar[r] & \mathrm{ob}(\mathfrak{D}^m) }\] where \(\mathtt{UCG}\) is the full subcategory of objects \((\mu,c_1,\ldots,c_k,c) \in \mathtt{PCG}\) such that if \(c = t\), then there is exactly one \(i \in \{1,\ldots,k\}\) such that \(c_i = t\). The rest of the description is as in Definition 18.
Definition 25. Let \(\mathbb{B}\) be as in Definition 22 and \[\Phi^\circ: \mathbb{B} \to \mathrm{CAT}\] be the contravariant functor which sends \(b \in \mathbb{B}\) to the category of objects \(X \in \mathcal{M}\) together with the structure of a \(\mathcal{K}_m^{\mathbin{\tikz{ \draw (0,0) circle (0.65pt); \fill (0,-.067) circle (0.45pt); \fill (0,.067) circle (0.45pt); \fill (-.067,0) circle (0.45pt); \fill (.067,0) circle (0.45pt); }}}\)-algebra on the pair \((b,X)\).
Lemma 11. Any weak equivalence \(b_1 \to b_2\) in \(\mathbb{B}\) induces a Quillen equivalence between \(\Phi^\circ(b_1)\) and \(\Phi^\circ(b_2)\).
Proof. For any \(b \in \mathbb{B}\), \(\Phi^\circ(b)\) is a category of presheaves, so we can apply for example [29]. ◻
Let \(I\) be the unit of \(\mathcal{M}\), which is given by the collection \((1,0,0,\ldots)\), since \(\mathcal{M}\) is the category of right \(\mathcal{K}_m\)-modules.
Lemma 12. There is a Quillen equivalence between the category \(\Phi(b)\) and the category of elements \(X \in \Phi^\circ(\mathcal{K}_m)\) together with a morphism \(I \to X\).
Proof. We follow the same arguments as in the proof of [16]. First note that there is an obvious inclusion map of polynomial \(2\)-monads \(\mathcal{K}_m^{\mathbin{\tikz{ \draw (0,0) circle (0.65pt); \fill (0,-.067) circle (0.45pt); \fill (0,.067) circle (0.45pt); \fill (-.067,0) circle (0.45pt); \fill (.067,0) circle (0.45pt); }}} \to \mathcal{K}_m^{\fivedots}\) which induces a forgetful functor \(U: \Phi(b) \to \Phi^\circ(b)\). Let \(\alpha \in \Phi^\circ(b)\) be the image of the initial object in \(\Phi(b)\) through \(U\). As in [30], the category \(\Phi(b)\) is isomorphic to \(\alpha/\Phi^\circ(b)\). Let \(V: \Phi^\circ(b) \to \mathcal{M}\) be the forgetful functor and \(\gamma \in \Phi^\circ(b)\) the image of \(I\) through the left adjoint of \(V\). We want to prove that there is a weak equivalence \(\gamma \to \alpha\).
Fortunately both \(\alpha\) and \(\gamma\) can be computed using classifiers. Firstly, \(\alpha\) can be computed using the classifier associated to the inclusion morphism of polynomial \(2\)-monads \(\iota: \mathcal{K}_m^{\diamonddots} \to \mathcal{K}_m^{\fivedots}\). Indeed, \(\int \Phi\) is the category of algebras over \(\mathcal{K}_m^{\fivedots}\) and the initial object in \(\Phi(b)\) is given by \(\iota_!(b)\), since \(\iota_!\) is the left adjoint of the restriction functor \(\iota^*: \int \Phi \to \mathbb{B}\) given by projection to the base. Secondly, \(\gamma\) can be computed using the classifier associated to the inclusion morphism of polynomial \(2\)-monads \(\kappa: \mathcal{K}_m^{\mathbin{\tikz{ \draw (-1pt,0) -- (1pt,0); \draw (0,-1pt) -- (0,1pt); \fill (0,-.067) circle (0.45pt); \fill (0,.067) circle (0.45pt); \fill (-.067,0) circle (0.45pt); \fill (.067,0) circle (0.45pt); }}} \to \mathcal{K}_m^{\mathbin{\tikz{ \draw (0,0) circle (0.65pt); \fill (0,-.067) circle (0.45pt); \fill (0,.067) circle (0.45pt); \fill (-.067,0) circle (0.45pt); \fill (.067,0) circle (0.45pt); }}}\). Here, \(\mathcal{K}_m^{\mathbin{\tikz{ \draw (-1pt,0) -- (1pt,0); \draw (0,-1pt) -- (0,1pt); \fill (0,-.067) circle (0.45pt); \fill (0,.067) circle (0.45pt); \fill (-.067,0) circle (0.45pt); \fill (.067,0) circle (0.45pt); }}}\) is the polynomial \(2\)-monad given by the polynomial \[\xymatrix{ \mathrm{ob}(\mathfrak{D}^m) & \mathtt{SCG}^* \ar[l] \ar[r] & \mathtt{SCG} \ar[r] & \mathrm{ob}(\mathfrak{D}^m) }\] where \(\mathtt{SCG}\) is the full subcategory of objects \((\mu,c_1,\ldots,c_k,c) \in \mathtt{CCG}\) such that if \(c=t\), then \(k=1\), \(\mu\) is the unique object of \(\mathtt{CG}(1)\) and \(c_1=t\). This time, \(\int \Phi^\circ\) is the category of algebras over \(\mathcal{K}_m^{\mathbin{\tikz{ \draw (0,0) circle (0.65pt); \fill (0,-.067) circle (0.45pt); \fill (0,.067) circle (0.45pt); \fill (-.067,0) circle (0.45pt); \fill (.067,0) circle (0.45pt); }}}\) and \(\kappa^*(b,X)=(b,V(X))\), so \(\kappa_!(b,I) = (b,\gamma)\).
To simplify the notations, let \(\mathcal{X}\) and \(\mathcal{Y}\) be the classifiers associated to \(\kappa\) and \(\iota\) respectively. Both \(\mathcal{X}\) and \(\mathcal{Y}\) are actually full subcategories of the classifier associated to the morphism of polynomial \(2\)-monads 18 which was described in Lemma 3. \(\mathcal{X}\) is the full subcategory of objects \((\mu,c_1,\ldots,c_k,c) \in \mathtt{PCG}\) such that if \(c = t\), then there is exactly one \(i \in \{1,\ldots,k\}\) such that \(c_i = t\). \(\mathcal{Y}\) is the full subcategory of objects \((\mu,c_1,\ldots,c_k,c) \in \mathtt{PCG}\) such that for all \(i \in \{1,\ldots,k\}\), \(c_i \neq t\). There is a functor \(F: \mathcal{X} \to \mathcal{Y}\) which removes the unique \(i\) such that \(c_i=t\). In other words, it sends \(\mu \in \mathtt{CG}(k)\) to its restriction to \(\{1,\ldots,k\} \backslash \{i\}\). Let us sketch a proof that \(y/F\) has a contractible nerve. For \(y = (\nu,d_1,\ldots,d_l,t) \in \mathcal{Y}\), an object in \(y/F\) is given by \((\mu,c_1,\ldots,c_{l+1},t) \in \mathcal{X}\) together with \(i \in \{1,\ldots,l+1\}\) and a map \(\nu \to \mu'\), where \(\mu'\) is the restriction of \(\mu\) to \(\{1,\ldots,l+1\} \backslash \{i\}\). The morphisms are just morphisms in \(\mathcal{X}\). For example, if \(m=2\), and let \(A,B,C,D,E\) corresponding to the objects of \(\mathfrak{D}^2\) as in Remark 7. Let \(y=(\nu,B,A,E)\), with \(\nu_{12}=2\). Let us draw \((\mu,c_1,c_2,c_3,E) \in \mathcal{X}\) as follows: \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/rlunskhw.png}\label{qxiykrlc}\end{figure}\tag{23}\] Then \(y/F\) is the following category: \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/rcxhkzgi.png}\label{yawiceus}\end{figure}\tag{24}\] For any \(y \in \mathcal{Y}\), \(y/F\) is non-empty and any two elements in this category are connected by a zigzag as above. There are no loops because between any two elements there is always a shortest possible zigzag.
According to Theorem 5, \(\gamma\) and \(\alpha\) can be computed as homotopy colimits over the classifiers \(\mathcal{X}\) and \(\mathcal{Y}\) respectively. The map \(\xi: \gamma \to \alpha\) is induced by \(F: \mathcal{X} \to \mathcal{Y}\). Since \(y/F\) has a contractible nerve for all \(y \in \mathcal{Y}\), \(F\) preserves homotopy limits, which means that \(\xi\) is indeed a weak equivalence.
Let us finish by gathering all the arguments together. The category \(\Phi(b)\) is isomorphic to \(\alpha/\Phi^\circ(b)\). Since \(\Phi^\circ(b)\) is a category of presheaves, is it left proper. So, according to [31], there is a Quillen equivalence between \(\alpha/\Phi^\circ(b)\) and \(\gamma/\Phi^\circ(b)\). Finally, using Lemma 11, there is a Quillen equivalence between \(\gamma/\Phi^\circ(b)\) and \(\gamma/\Phi^\circ(\mathcal{K}_m)\). ◻
Lemma 13. For any map \(X \to Y\) in \(\Phi(b)\), there is a fibration sequence \[\Phi(b)^\mathsf{h}(X,Y) \to \Phi^\circ(\mathcal{K}_m)^\mathsf{h}(X,Y) \to Y_0.\]
Let \(\mathcal{A}\) and \(\mathcal{B}\) be two (one-coloured) symmetric operads in a symmetric monoidal category \((\mathcal{V},\otimes,v)\). Let \(\mathcal{C}\) and \(\mathcal{D}\) be two \(\mathcal{A}-\mathcal{B}\)-bimodules. Recall [16] that an infinitesimal \(\mathcal{C}-\mathcal{D}\)-bimodule \(\mathcal{E}\) is given by a collection \(\mathcal{E} := (\mathcal{E}(k))_{k \geq 0}\) of objects in \(\mathcal{V}\) together with maps \[\label{equationleftaction} \mathcal{A}(k) \times \mathcal{C}(n_1) \times \ldots \times \mathcal{C}(n_{i-1}) \times \mathcal{E}(n_i) \times \mathcal{D}(n_{i+1}) \times \ldots \times \mathcal{D}(n_k) \to \mathcal{E}(n_1+\ldots+n_k)\tag{25}\] and \[\mathcal{E}(k) \times \mathcal{B}(n_1) \times \ldots \times \mathcal{B}(n_k) \to \mathcal{E}(n_1+\ldots+n_k)\] satisfying axioms. Also recall that an infinitesimal \(\mathcal{A}\)-bimodule \(\mathcal{E}\) is given by a collection \((\mathcal{E}(k))_{k \geq 0}\) of objects in \(\mathcal{V}\) together with maps \[\circ_i : \mathcal{A}(k) \otimes \mathcal{E}(n) \to \mathcal{E}(k+n-1)\] and \[\bullet_i : \mathcal{E}(k) \otimes \mathcal{A}(n) \to \mathcal{E}(k+n-1),\] for \(1 \leq i \leq k\), satisfying axioms.
Lemma 14. The category \(\Phi^\circ(\mathcal{K}_m)\) is equivalent to the category of infinitesimal \(\mathcal{K}_m\)-bimodules.
Proof. We will prove that the category \(\Phi^\circ(\mathcal{K}_m)\) is equivalent to the category of infinitesimal \(\mathcal{K}_m-\mathcal{K}_m\)-bimodules in the sense of [16]. The conclusion follows from [16], which says that we recover the classical notion of infinitesimal \(\mathcal{K}_m\)-bimodules in this case.
An object of \(\Phi^\circ(\mathcal{K}_m)\) is given by a right \(\mathcal{K}_m\)-module \(X\) together with the structure of a \(\mathcal{K}_m^{\mathbin{\tikz{ \draw (0,0) circle (0.65pt); \fill (0,-.067) circle (0.45pt); \fill (0,.067) circle (0.45pt); \fill (-.067,0) circle (0.45pt); \fill (.067,0) circle (0.45pt); }}}\)-algebra on the pair \((\mathcal{K}_m,X)\). Let us unpack the definition of a \(\mathcal{K}_m^{\mathbin{\tikz{ \draw (0,0) circle (0.65pt); \fill (0,-.067) circle (0.45pt); \fill (0,.067) circle (0.45pt); \fill (-.067,0) circle (0.45pt); \fill (.067,0) circle (0.45pt); }}}\)-algebra. For \((\mu,c_1,\ldots,c_k,c) \in \mathtt{UCG}\) where \(c=t\), there is exactly one \(i \in \{1,\ldots,k\}\) such that \(c_i=t\). This induces a map \[\underbrace{\mathcal{K}_m \otimes \ldots \otimes \mathcal{K}_m}_{i-1} \otimes X \otimes \underbrace{\mathcal{K}_m \otimes \ldots \otimes \mathcal{K}_m}_{k-i} \to X\] in the category of right \(\mathcal{K}_m\)-modules. Since the tensor product is given by convolution, we get functors \[\label{equationconvolution} \mathcal{K}_m(n_1) \times \ldots \times X(n_i) \times \ldots \times \mathcal{K}_m(n_k) \to X(n_1+\ldots+n_k).\tag{26}\]
On the other hand, an infinitesimal \(\mathcal{K}_m-\mathcal{K}_m\)-bimodule is given by a right \(\mathcal{K}_m\)-module \(X\) together with left actions as in 25 . In our case, \(\mathcal{A}(k)=\mathcal{K}_m(k)\), so we get functors as in 26 but this time for any object \(\mu \in \mathcal{K}_m(k)\). It remains to prove that the left actions are equivalent.
First, let \(X \in \Phi^\circ (\mathcal{K}_m)\). Let us construct a left action as in 25 . As we know from Proposition 10, \(\mathcal{K}_m\) can be seen both as categorical operad and a polynomial \(2\)-monad. So \(\mu \in \mathcal{K}_m(k)\) is given by an object in \(\mathtt{CG}(k)\). Let \(i \in \{1,\ldots,k\}\). For \(j \in \{1,\ldots,k\}\), let \(c_j=(-1,\mu_{ji})\) if \(j<i\), \(c_j=t\) if \(j=i\) and \(c_j=(1,\mu_{ij})\) is \(j>i\). otherwise. This gives an object \((\mu,c_1,\ldots,c_k,t) \in \mathtt{UCG}\) and we can defined the left action map as the one induced by this object. Note that if \(c'_1,\ldots,c'_k\) are such that \(c'_i=t\) and \((\mu,c'_1,\ldots,c'_k,t) \in \mathtt{UCG}\), then \(c'_j \leq c_j\) for all \(j \in \{1,\ldots,k\}\).
On the other hand, if \(X\) is an infinitesimal \(\mathcal{K}_m-\mathcal{K}_m\)-bimodule and \((\mu,c_1,\ldots,c_k,t) \in \mathtt{UCG}\), then we can take the functor 25 associated to \(\mu\).
We obtain two functors inverse of each other. The fact that they are inverse comes from the fact that \(\mathcal{K}_m \in \mathbb{B}\) is actually given by the constant object \(\mathcal{K}_m \in \mathcal{M}\), which means that for fixed \(\mu \in \mathtt{CG}\), the left action 25 is the same for any \((c_1,\ldots,c_k)\) such that \((\mu,c_1,\ldots,c_k,t) \in \mathtt{UCG}\). ◻
Lemma 15. The category \(\Psi(D^0(\mathcal{K}_m))\) is equivalent to the category of \(\mathcal{K}_m\)-bimodules \(X\) equipped with a map \(S^{m-1}(\mathcal{K}_m) \to X\) of \(\mathcal{K}_m\)-bimodules.
Proof. The category \(\mathbb{C}\) of \(\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\square}}\)-algebras in the category of covariant presheaves over \(\mathfrak{S}^{m-1}\) in the category of \(\mathcal{K}_m\)-algebras, by definition of the Boardman-Vogt tensor product. For \(c \in \mathbb{C}\) and \(Y \in M\), the structure of a \(\mathcal{K}_m^{\scriptsize\rotatebox[origin=c]{45}{\boxtimes}}\)-algebra on the pair \((c,Y)\) corresponds to a map \(\operatornamewithlimits{colim}_{\mathfrak{S}^{m-1}} c \to Y\). So \(\Psi(c)\) is equivalent to \(\operatornamewithlimits{colim}_{\mathfrak{S}^{m-1}} c / \mathrm{Bimod}_{\mathcal{K}_m}\). We have \[\operatornamewithlimits{colim}_{\mathfrak{S}^{m-1}} D^0(\mathcal{K}_m) / \mathrm{Bimod}_{\mathcal{K}_m} \simeq \operatornamewithlimits{hocolim}_{\mathfrak{S}^{m-1}} \mathcal{K}_m / \mathrm{Bimod}_{\mathcal{K}_m} \simeq S^{m-1}(\mathcal{K}_m) / \mathrm{Bimod}_{\mathcal{K}_m}\] where the second equivalence is given by Lemma 7. This concludes the proof. ◻
Let \(\mathrm{IBimod}_{\mathcal{K}_m}\) be the category of infinitesimal \(\mathcal{K}_m\)-bimodules.
Theorem 15. [4] Let \(Y\) be a \(\mathcal{K}_m\)-bimodule equipped with a map of \(\mathcal{K}_m\)-bimodules \(\mathcal{K}_m \to Y\). If \(Y_0\) is contractible, then there is a weak equivalence \[\Omega^m \mathrm{Bimod}_{\mathcal{K}_m}^\mathsf{h}(\mathcal{K}_m,Y) \xrightarrow{\sim} \mathrm{IBimod}_{\mathcal{K}_m}^\mathsf{h}(\mathcal{K}_m,Y).\]
Proof. We have the following sequence of weak equivalences \[\begin{align} \Omega^m \mathrm{Bimod}_{\mathcal{K}_m}^\mathsf{h}(\mathcal{K}_m,Y) &\xrightarrow{\sim} (S^{m-1}(\mathcal{K}_m) / \mathrm{Bimod}_{\mathcal{K}_m})^\mathsf{h} (\mathcal{K}_m,Y) && \text{Lemma \ref{lemmageneraldelooping}}\\ &\xrightarrow{\sim} \Psi(b)^\mathsf{h} (\mathcal{K}_m,Y) && \text{Lemma \ref{lemmapsikm}}\\ &\xrightarrow{\sim} \Phi(b)^\mathsf{h} (\mathcal{K}_m,Y) && \text{Lemma \ref{lemmaquillenreflection}}\\ &\xrightarrow{\sim} \Phi^\circ(b)^\mathsf{h} (\mathcal{K}_m,Y) && \text{Lemma \ref{lemmafibrationsequence}}\\ &\xrightarrow{\sim} \Phi^\circ(\mathcal{K}_m)^\mathsf{h} (\mathcal{K}_m,Y) && \text{Lemma \ref{lemmarectification}}\\ &\xrightarrow{\sim} \mathrm{IBimod}_{\mathcal{K}_m}^\mathsf{h} (\mathcal{K}_m,Y) && \text{Lemma \ref{lemmaphikm}} \end{align}\] ◻
By taking \(Y=\mathcal{K}_n\) in the previous theeorem, we get:
Corollary 1. There is a weak equivalence \[\Omega^m \mathrm{Bimod}_{\mathcal{K}_m}^\mathsf{h}(\mathcal{K}_m,\mathcal{K}_n) \to \mathrm{IBimod}_{\mathcal{K}_m}^\mathsf{h}(\mathcal{K}_m,\mathcal{K}_n).\]