Lax symmetric monoidal functors

Proof. Suppose first that the data in (1) are given. By Proposition 21.1.2, composition with the counit gives an equivalence \[ \Nat (\id _C,RL) \xrightarrow {\simeq } \Nat (L,L), \qquad \alpha \longmapsto (\epsilon L)(L\alpha ). \] Under the specified identifications over \(B\), the fact that \(\epsilon \) is a morphism in \(\Fun _{/B}(D,D)\) gives a commutative square

It therefore restricts to an equivalence on the fibers over \(\id _p\), which are the hom animae \[ \Hom _{\Fun _{/B}(C,C)}(\id _C,RL) \xrightarrow {\simeq } \Hom _{\Fun _{/B}(C,D)}(L,L). \] The preimage of \(\id _L\) gives a morphism \(\eta \colon \id _C \to RL\) over \(B\), together with an identification \((\epsilon L)(L\eta ) \simeq \id _L\) in \(\Fun _{/B}(C,D)\). After forgetting the maps to \(B\), this is the unit corresponding to \(\epsilon \) under Proposition 21.1.2, so it gives the data in (2).

Conversely, starting from (2), composition with the unit gives an equivalence \[ \Nat (LR,\id _D) \xrightarrow {\simeq } \Nat (R,R), \qquad \beta \longmapsto (R\beta )(\eta R). \] It likewise restricts to the fibers over the identity transformation of \(q\), and the preimage of \(\id _R\) supplies the counit in (1), together with the other triangle identification over \(B\). These constructions are inverse because they are the usual equivalence between unit and counit data from Proposition 21.1.2, restricted to the indicated fibers. □

Proof. The counit in Definition 14.3.3 and the corresponding unit of Lemma 14.3.4 may both be whiskered with a functor \(E \to C\) or \(E \to D\) over \(B\). They therefore give a unit and counit between the displayed functors, with triangle identifications inherited from those of \(L\dashv R\). The result follows from Proposition 21.1.2. □

Proof. (1) By the pointwise criterion for right adjoints from Lemma 21.1.4, it will suffice to show that for every object \(Y = \{y_j\}_{j \in J} \in D^{\otimes }\) there exists an object \(R^{\otimes }Y\) together with a counit map \(\epsilon _Y\colon L^{\otimes }R^{\otimes }Y \to Y\) such that for every other \(X \in C^{\otimes }\) the composite \begin {equation} \label {eq:Monoidal_Adjunction} \Hom _{C^{\otimes }}(X,R^{\otimes }Y) \xrightarrow {L^{\otimes }} \Hom _{D^{\otimes }}(L^{\otimes }X,L^{\otimes }R^{\otimes }Y) \xrightarrow {\epsilon _Y \circ -} \Hom _{D^{\otimes }}(L^{\otimes }X,Y) \end {equation} is an equivalence. We define \(R^{\otimes }Y := \{Ry_j\}_{j \in J}\). We then have \(L^{\otimes }R^{\otimes }Y \simeq \{LRy_j\}_{j \in J}\) and so the counit map \(\epsilon _Y\) may be taken to be the collection of morphisms \(\epsilon _{y_j}\colon LRy_j \to y_j\), resulting in a morphism in \(D^{\otimes }_J \simeq \prod _{j \in J} D\).

To show that the composite (14.1) is an equivalence, it suffices to show that it induces an equivalence on the fibers over any span \(\alpha \colon I \xleftarrow {f} K \xrightarrow {g} J\) in \(\Span (\Fin )\). Let \(\widetilde {\alpha }\colon X \to \alpha _!X\) denote a cocartesian lift of the span \(\alpha \). Since \(L^{\otimes }\) preserves cocartesian morphisms, also the map \(L^{\otimes }(\widetilde {\alpha }) \colon L^{\otimes }X \to L^{\otimes }\alpha _!X\) is cocartesian. Precomposition with these maps then induces a commutative diagram

Since the vertical maps induce equivalences on fibers over \(\Hom _{\Span (\Fin )}(J,J) \xrightarrow {- \circ \alpha } \Hom _{\Span (\Fin )}(I,J)\), we have thus reduced to the case of the identity span \(\alpha = \id _J\). In this case, the fibers are the hom animae in the fibers \(C^{\otimes }_J \simeq \prod _{j \in J} C\) and \(D^{\otimes }_J \simeq \prod _{j \in J} D\) respectively. The claim thus follows from the fact that for every \(j \in J\) the composite \[ \Hom _C((\alpha _!X)_j, Ry_j) \xrightarrow {L} \Hom _D(L(\alpha _!X)_j, LRy_j) \xrightarrow {\epsilon _{y_j} \circ -} \Hom _D(L(\alpha _!X)_j, y_j) \] is an equivalence due to the adjunction \(L \dashv R\).

(2) We will now equip \(R^{\otimes }\) with the structure of a functor over \(\Span (\Fin )\). We claim that it makes the following diagram commute:

To this end, we claim that the adjunction counit \(\epsilon \colon L^{\otimes } R^{\otimes } \to \id _{D^{\otimes }}\) induces an equivalence \[ p_C R^{\otimes } \simeq p_D L^{\otimes } R^{\otimes } \xrightarrow {p_D \epsilon } p_D \id _{D^{\otimes }} \simeq p_D. \] This may be checked objectwise for \(X \in D^{\otimes }\), where it is clear from the construction of \(\epsilon _X\). The displayed identification equips \(R^{\otimes }\) with the structure of a functor over \(\Span (\Fin )\) and lifts \(\epsilon \) to a morphism in \(\Fun _{/\Span (\Fin )}(D^{\otimes },D^{\otimes })\). Thus \(L^{\otimes }\dashv R^{\otimes }\) is a relative adjunction over \(\Span (\Fin )\) by Definition 14.3.3.

(3) The functor \(R^{\otimes }\) preserves finite products because it is a right adjoint. Thus \(R^{\otimes }\) is an operad map.

(4) By uniqueness of adjoints, the restriction of \(R^{\otimes }\) to \(D = D^{\otimes }_{\lra {1}}\) is \(R\), and we deduce that \(R^{\otimes }\) is a lax symmetric monoidal refinement of \(R\). □

Proof. By part (2) of Proposition 14.3.6, the adjunction \(L^{\otimes }\dashv R^{\otimes }\) is relative to \(\Span (\Fin )\). Applying Lemma 14.3.5 to \(p_{\Oo }\colon \Oo ^{\otimes } \to \Span (\Fin )\) gives an adjunction \[ \Fun _{/\Span (\Fin )}(\Oo ^{\otimes },C^{\otimes }) \rightleftarrows \Fun _{/\Span (\Fin )}(\Oo ^{\otimes },D^{\otimes }). \] Postcomposition with both functors preserves finite-product-preserving functors because \(L^{\otimes }\) and \(R^{\otimes }\) are morphisms of \(\infty \)-operads. Restricting the adjunction to these subcategories gives the asserted adjunction on \(\Alg _{\Oo }(C)\) and \(\Alg _{\Oo }(D)\). □

Proof. By Corollary 21.2.2, the constant-diagram functor fits into an adjunction \[ \const \colon C\rightleftarrows \Fun (I,C)\noloc \lim _I. \] Being given by restriction along \(I \to *\), the functor \(\const \) is symmetric monoidal for the pointwise symmetric monoidal structure, so Proposition 14.3.6 makes this a symmetric monoidal adjunction. By Lemma 14.3.9, it induces an adjunction \[ \Alg _{\Oo }(C) \rightleftarrows \Alg _{\Oo }(\Fun (I,C)) \simeq \Fun (I,\Alg _{\Oo }(C)). \] Under the equivalence of Lemma 14.2.12, the left adjoint is the constant-diagram functor. The right adjoint consequently computes limits in \(\Alg _{\Oo }(C)\), and its value at every color \(x\) is \(\lim _I A_x\). Thus all the evaluation functors preserve these limits. They jointly detect isomorphisms: every object of \(\Oo ^{\otimes }\) is a finite product of colors, and algebra maps preserve these products, so a natural transformation that is invertible at every color is invertible at every object. Consequently the evaluation functors jointly reflect limit cones and hence jointly create the limits. □

Proof. (1) As in the proof of Proposition 14.3.6, we use the pointwise criterion for left adjoints from Lemma 21.1.4. We define \(L^{\otimes }\) objectwise as \(L^{\otimes }(\{x_i\}_{i \in I}) := \{L(x_i)\}_{i \in I}\); the unit maps \(\eta _{x_i} \colon x_i \to RLx_i\) induce a candidate unit map \(\eta _X \colon X \to R^{\otimes }L^{\otimes }X\). We must show that for every object \(Y \in D^{\otimes }_J\) the composite \[ \Hom _{D^{\otimes }}(L^{\otimes }X, Y) \xrightarrow {R^{\otimes }} \Hom _{C^{\otimes }}(R^{\otimes }L^{\otimes }X, R^{\otimes }Y) \xrightarrow {- \circ \eta _X} \Hom _{C^{\otimes }}(X, R^{\otimes }Y) \] is an equivalence. This may be tested fiberwise over any span \(\alpha \colon I \xleftarrow {f} K \xrightarrow {g} J\). Let \[ X \to \alpha _!X, \qquad L^{\otimes }X \to \alpha _!L^{\otimes }X, \qquadtext { and } R^{\otimes }L^{\otimes }X \to \alpha _!R^{\otimes }L^{\otimes }X \] denote cocartesian lifts of \(\alpha \) in \(C^{\otimes }\) and \(D^{\otimes }\), and let \(\lax _R\colon \alpha _!R^{\otimes }L^{\otimes }X \to R^{\otimes }\alpha _!L^{\otimes }X\) denote the lax structure map of \(R\) associated to \(\alpha \). We then get a commutative diagram as follows:

It will thus suffice to show that the left vertical composite is an equivalence. Under the identifications \(C^{\otimes }_J \simeq \prod _{j \in J} C\) and \(D^{\otimes }_J \simeq \prod _{j \in J} D\), this composite is the product over \(j \in J\) of the analogous composites associated to the restricted spans \[ \alpha _j \colon I \xleftarrow {f|_{g^{-1}(j)}} g^{-1}(j) \longrightarrow \lra {1}. \] Indeed, \(R^{\otimes }\) is a morphism of \(\infty \)-operads and therefore respects these product decompositions; in particular, the \(j\)-th component of \(\lax _R\) is the lax structure map associated to \(\alpha _j\). The map \(\alpha _!\eta _X\) decomposes in the same way. It therefore suffices to treat the case \(J = \lra {1}\). Composing the vertical composite with the adjunction equivalence \(\Hom _C(\alpha _!X, RY) \simeq \Hom _D(L\alpha _!X, Y)\), we obtain a map \[ \Hom _D(\alpha _!L^{\otimes }X, Y) \to \Hom _D(L\alpha _!X,Y) \] which we must show is an equivalence. But unwinding definitions shows that this map is given by precomposition with the composite \[ L\alpha _!X \xrightarrow {L\alpha _!\eta _X} L\alpha _!R^{\otimes }L^{\otimes } X\xrightarrow {L(\lax _R)} LR\alpha _!L^{\otimes }X \xrightarrow {\epsilon _{\alpha _!L^{\otimes }X}} \alpha _!L^{\otimes } X. \] This is precisely the comparison map from the hypothesis for the family \(\{x_{f(k)}\}_{k\in K}\), and is therefore an isomorphism. This proves the existence of the left adjoint \(L^{\otimes }\).

(2) The unit maps \(\eta _X\colon X \to R^{\otimes }L^{\otimes }X\) used in the construction lie in the same fibers of \(C^{\otimes } \to \Span (\Fin )\) as their sources. Their images in \(\Span (\Fin )\) therefore give a natural identification \(p_D L^{\otimes } \simeq p_C\) which equips \(L^{\otimes }\) with the structure of a functor over \(\Span (\Fin )\) and lifts the unit to a morphism in \(\Fun _{/\Span (\Fin )}(C^{\otimes },C^{\otimes })\). By Lemma 14.3.4, this makes \(L^{\otimes }\dashv R^{\otimes }\) a relative adjunction over \(\Span (\Fin )\).

(3) The objectwise formula on tuples shows that \(L^{\otimes }\) preserves finite products. Thus it is a morphism of \(\infty \)-operads. Finally, the comparison isomorphism above identifies \(L^{\otimes }\alpha _!X\) with \(\alpha _!L^{\otimes }X\) for every span \(\alpha \). Hence \(L^{\otimes }\) preserves cocartesian morphisms and is strong symmetric monoidal.

(4) By its objectwise definition, the restriction of \(L^{\otimes }\) to the fiber over \(\lra {1}\) is \(L\). □