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Dimitri Chikhladze
Dimitri Chikhladze
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The answer might depend on how one defines the monoidal structure on the realizations $|-|:s\mathcal{V}_i\to \mathcal{V}_i$. In the topological and the simplicial cases the strong monoidal structures are determined by the preservation of products.

Observe that in both of these cases the canonical morphism $|X\times Y| \rightarrow |X|\times|Y|$ is induced through the diagonals $\Delta^n \rightarrow \Delta^n\times\Delta^n$ when one thinks of it as a map between the coends. One can generalize to the monoidal situation by requiring the cosimplicial object $\Delta \to \mathcal{V}$ to be a comonoid in $[\Delta, \mathcal{V}]$, so that we have $\delta_{\Delta^n} : \Delta^n \rightarrow \Delta^n\otimes\Delta^n$. This will define a colax monoidal structure on $|-|$

$$\int^n X_n\otimes Y_n \otimes \Delta^n \rightarrow \int^{n,m} X_n\otimes Y_m \otimes \Delta^n\otimes \Delta^m$$ $$\cong (\int^n X_n \otimes\Delta^n) \otimes (\int^m Y_m \otimes \Delta^m)$$

We also need $\otimes$ to commute with coends.

Now inIf one is given a lax monoidal functor $F$, then the composites $F|-|$ and $|F-|$ are neither lax nor colax, since $|-|$ is colax. However, one can still form the diagrams for $\tau$ by inverting in the questionthem the monoidality stucture maps will have the opposite direction sincefor $|-|$ are colax, not lax. If $\delta_{F\Delta^n}$ equals to $\delta_{\Delta^n}$ up to the isomorphism $F(\Delta^n) \cong \Delta^n$, then the firstthese diagram will commute because. The first diagram will commutive since (writing $.$ for the tensor, and omitting an index $n$)

$$FX.FY.F\Delta \rightarrow{} FX.FY.F\Delta.F\Delta \rightarrow F(X.\Delta).F(Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta)$$ $$=$$ $$FX.FY.F\Delta \rightarrow F(X.Y).F\Delta \rightarrow F(X.Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta),$$ which holds because of the naturality and coherence of the lax structure of $F$. The second diagram commutes because of a similar argument.

IfNow, if the colax monoidality morphisms are invertible, then $|-|$ will become strong monoidal, and the original diagrams in the question will commute. Hence $\tau$ will become a monoidal transformation. In particular thisThis is the case in the simplicial-topological example.

For the purpose of defining a lax structure on $|-|$ we could again look at our examples. In these examples the special casesinverses of the lax structure of $|-|$ are defined as follows. In the topological case the lax monoidal structure $|X|\times|Y| \rightarrow |X\times Y|$ is constructed (-> Theorem 11.5 "The geometry of iterated loops") using certain maps

$$X_n\times X_m \times \Delta^n\times \Delta^m \rightarrow X_{n+m}\times Y_{n+m} \times \Delta^{n+m}$$

which depend on certain isomorphisms $\Delta^n\times\Delta^m \cong \Delta^{n+m}$. In the bisimplicial situation $|X|\times|Y| \rightarrow |X\times Y|$ is explicitly given via

$$(X_n\times Y_m \times \Delta^n\times \Delta^m)_r \rightarrow (X_r\times Y_r \times \Delta^r)_r$$ $$(x, y, u : r \rightarrow n, v : r \rightarrow m) \mapsto (u^\ast(x), v^\ast(y), 1 : r \rightarrow r).$$

However, I can't tell how these can be generalizeddo not have straightforward generalization to athe monoidal context, or how they can be related to each other.

The answer might depend on how one defines the monoidal structure on the realizations $|-|:s\mathcal{V}_i\to \mathcal{V}_i$. In the topological and the simplicial cases the strong monoidal structures are determined by the preservation of products.

Observe that in both of these cases the canonical morphism $|X\times Y| \rightarrow |X|\times|Y|$ is induced through the diagonals $\Delta^n \rightarrow \Delta^n\times\Delta^n$ when one thinks of it as a map between the coends. One can generalize to the monoidal situation by requiring the cosimplicial object $\Delta \to \mathcal{V}$ to be a comonoid in $[\Delta, \mathcal{V}]$, so that we have $\delta_{\Delta^n} : \Delta^n \rightarrow \Delta^n\otimes\Delta^n$. This will define a colax monoidal structure on $|-|$

$$\int^n X_n\otimes Y_n \otimes \Delta^n \rightarrow \int^{n,m} X_n\otimes Y_m \otimes \Delta^n\otimes \Delta^m$$ $$\cong (\int^n X_n \otimes\Delta^n) \otimes (\int^m Y_m \otimes \Delta^m)$$

We also need $\otimes$ to commute with coends.

Now in the diagrams in the question the monoidality maps will have the opposite direction since $|-|$ are colax, not lax. If $\delta_{F\Delta^n}$ equals to $\delta_{\Delta^n}$ up to the isomorphism $F(\Delta^n) \cong \Delta^n$, then the first diagram will commute because (writing $.$ for the tensor, and omitting an index $n$)

$$FX.FY.F\Delta \rightarrow{} FX.FY.F\Delta.F\Delta \rightarrow F(X.\Delta).F(Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta)$$ $$=$$ $$FX.FY.F\Delta \rightarrow F(X.Y).F\Delta \rightarrow F(X.Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta),$$ which holds because of the naturality and coherence of the lax structure of $F$. The second diagram commutes because of a similar argument.

If the colax monoidality morphisms are invertible, then $|-|$ will become strong monoidal, and $\tau$ will become a monoidal transformation. In particular this is the case in the simplicial-topological example.

For the purpose of defining a lax structure on $|-|$ we could again look at the special cases. In the topological case the lax monoidal structure $|X|\times|Y| \rightarrow |X\times Y|$ is constructed (-> Theorem 11.5 "The geometry of iterated loops") using certain maps

$$X_n\times X_m \times \Delta^n\times \Delta^m \rightarrow X_{n+m}\times Y_{n+m} \times \Delta^{n+m}$$

which depend on certain isomorphisms $\Delta^n\times\Delta^m \cong \Delta^{n+m}$. In the bisimplicial situation $|X|\times|Y| \rightarrow |X\times Y|$ is explicitly given via

$$(X_n\times Y_m \times \Delta^n\times \Delta^m)_r \rightarrow (X_r\times Y_r \times \Delta^r)_r$$ $$(x, y, u : r \rightarrow n, v : r \rightarrow m) \mapsto (u^\ast(x), v^\ast(y), 1 : r \rightarrow r).$$

However, I can't tell how these can be generalized to a monoidal context, or how they can be related to each other.

The answer might depend on how one defines the monoidal structure on the realizations $|-|:s\mathcal{V}_i\to \mathcal{V}_i$. In the topological and the simplicial cases the strong monoidal structures are determined by the preservation of products.

Observe that in both of these cases the canonical morphism $|X\times Y| \rightarrow |X|\times|Y|$ is induced through the diagonals $\Delta^n \rightarrow \Delta^n\times\Delta^n$ when one thinks of it as a map between the coends. One can generalize to the monoidal situation by requiring the cosimplicial object $\Delta \to \mathcal{V}$ to be a comonoid in $[\Delta, \mathcal{V}]$, so that we have $\delta_{\Delta^n} : \Delta^n \rightarrow \Delta^n\otimes\Delta^n$. This will define a colax monoidal structure on $|-|$

$$\int^n X_n\otimes Y_n \otimes \Delta^n \rightarrow \int^{n,m} X_n\otimes Y_m \otimes \Delta^n\otimes \Delta^m$$ $$\cong (\int^n X_n \otimes\Delta^n) \otimes (\int^m Y_m \otimes \Delta^m)$$

We also need $\otimes$ to commute with coends.

If one is given a lax monoidal functor $F$, then the composites $F|-|$ and $|F-|$ are neither lax nor colax, since $|-|$ is colax. However, one can still form the diagrams for $\tau$ by inverting in them the monoidality stucture maps for $|-|$. If $\delta_{F\Delta^n}$ equals to $\delta_{\Delta^n}$ up to the isomorphism $F(\Delta^n) \cong \Delta^n$, then these diagram will commute. The first diagram will commutive since (writing $.$ for the tensor, and omitting an index $n$)

$$FX.FY.F\Delta \rightarrow{} FX.FY.F\Delta.F\Delta \rightarrow F(X.\Delta).F(Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta)$$ $$=$$ $$FX.FY.F\Delta \rightarrow F(X.Y).F\Delta \rightarrow F(X.Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta),$$ which holds because of the naturality and coherence of the lax structure of $F$. The second diagram commutes because of a similar argument.

Now, if the colax monoidality morphisms are invertible, then $|-|$ will become strong monoidal, and the original diagrams in the question will commute. Hence $\tau$ will become a monoidal transformation. This is the case in the simplicial-topological example.

For the purpose of defining a lax structure on $|-|$ we could again look at our examples. In these examples the inverses of the lax structure of $|-|$ are defined as follows. In the topological case the lax monoidal structure $|X|\times|Y| \rightarrow |X\times Y|$ is constructed (-> Theorem 11.5 "The geometry of iterated loops") using certain maps

$$X_n\times X_m \times \Delta^n\times \Delta^m \rightarrow X_{n+m}\times Y_{n+m} \times \Delta^{n+m}$$

which depend on certain isomorphisms $\Delta^n\times\Delta^m \cong \Delta^{n+m}$. In the bisimplicial situation $|X|\times|Y| \rightarrow |X\times Y|$ is explicitly given via

$$(X_n\times Y_m \times \Delta^n\times \Delta^m)_r \rightarrow (X_r\times Y_r \times \Delta^r)_r$$ $$(x, y, u : r \rightarrow n, v : r \rightarrow m) \mapsto (u^\ast(x), v^\ast(y), 1 : r \rightarrow r).$$

However, these do not have straightforward generalization to the monoidal context.

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Dimitri Chikhladze
Dimitri Chikhladze
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The answer might depend on how one defines the monoidal structure on the realizations $|-|:s\mathcal{V}_i\to \mathcal{V}_i$. In the topological and the simplicial cases the strong monoidal structures are determined by the preservation of products.

Observe that in both of these cases the canonical morphism $|X\times Y| \rightarrow |X|\times|Y|$ is induced through the diagonals $\Delta^n \rightarrow \Delta^n\times\Delta^n$ when one thinks of it as a map between the coends. One can generalize to the monoidal situation by requiring the cosimplicial object $\Delta \to \mathcal{V}$ to be a comonoid in $[\Delta, \mathcal{V}]$, so that we have $\delta_{\Delta^n} : \Delta^n \rightarrow \Delta^n\otimes\Delta^n$. This will define a colax monoidal structure on $|-|$

$$\int^n X_n\otimes Y_n \otimes \Delta^n \rightarrow \int^{n,m} X_n\otimes Y_m \otimes \Delta^n\otimes \Delta^m$$ $$\cong (\int^n X_n \otimes\Delta^n) \otimes (\int^m Y_m \otimes \Delta^m)$$

We also need $\otimes$ to commute with coends.

Now in the diagrams in the question the monoidality maps will have the opposite direction since $|-|$ are colax, not lax. If $\delta_{F\Delta^n}$ equals to $\delta_{\Delta^n}$ up to the isomorphism $F(\Delta^n) \cong \Delta^n$, then the first diagram will commute because (writing $.$ for the tensor, and omitting an index $n$)

$$FX.FY.F\Delta \rightarrow{} FX.FY.F\Delta.F\Delta \rightarrow F(X.\Delta).F(Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta)$$ $$=$$ $$FX.FY.F\Delta \rightarrow F(X.Y).F\Delta \rightarrow F(X.Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta),$$ which holds because of the naturality and coherence of the colaxlax structure of $F$. The second diagram commutes because of a similar argument.

If the colax monoidality morphisms are invertible, then $|-|$ will become strong monoidal, and $\tau$ will become a monoidal transformation. In particular this is the case in the simplicial-topological example.

For the purpose of defining a lax structure on $|-|$ we could again look at the special cases. In the topological case the lax monoidal structure $|X|\times|Y| \rightarrow |X\times Y|$ is constructed (-> Theorem 11.5 "The geometry of iterated loops") using certain maps

$$X_n\times X_m \times \Delta^n\times \Delta^m \rightarrow X_{n+m}\times Y_{n+m} \times \Delta^{n+m}$$

which depend on certain isomorphisms $\Delta^n\times\Delta^m \cong \Delta^{n+m}$. In the bisimplicial situation $|X|\times|Y| \rightarrow |X\times Y|$ is explicitly given via

$$(X_n\times Y_m \times \Delta^n\times \Delta^m)_r \rightarrow (X_r\times Y_r \times \Delta^r)_r$$ $$(x, y, u : r \rightarrow n, v : r \rightarrow m) \mapsto (u^\ast(x), v^\ast(y), 1 : r \rightarrow r).$$

However, I can't tell how these can be generalized to a monoidal context, or how they can be related to each other.

The answer might depend on how one defines the monoidal structure on the realizations $|-|:s\mathcal{V}_i\to \mathcal{V}_i$. In the topological and the simplicial cases the strong monoidal structures are determined by the preservation of products.

Observe that in both of these cases the canonical morphism $|X\times Y| \rightarrow |X|\times|Y|$ is induced through the diagonals $\Delta^n \rightarrow \Delta^n\times\Delta^n$ when one thinks of it as a map between the coends. One can generalize to the monoidal situation by requiring the cosimplicial object $\Delta \to \mathcal{V}$ to be a comonoid in $[\Delta, \mathcal{V}]$, so that we have $\delta_{\Delta^n} : \Delta^n \rightarrow \Delta^n\otimes\Delta^n$. This will define a colax monoidal structure on $|-|$

$$\int^n X_n\otimes Y_n \otimes \Delta^n \rightarrow \int^{n,m} X_n\otimes Y_m \otimes \Delta^n\otimes \Delta^m$$ $$\cong (\int^n X_n \otimes\Delta^n) \otimes (\int^m Y_m \otimes \Delta^m)$$

We also need $\otimes$ to commute with coends.

Now in the diagrams in the question the monoidality maps will have the opposite direction since $|-|$ are colax, not lax. If $\delta_{F\Delta^n}$ equals to $\delta_{\Delta^n}$ up to the isomorphism $F(\Delta^n) \cong \Delta^n$, then the first diagram will commute because (writing $.$ for the tensor, and omitting an index $n$)

$$FX.FY.F\Delta \rightarrow{} FX.FY.F\Delta.F\Delta \rightarrow F(X.\Delta).F(Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta)$$ $$=$$ $$FX.FY.F\Delta \rightarrow F(X.Y).F\Delta \rightarrow F(X.Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta),$$ which holds because of the naturality and coherence of the colax structure of $F$. The second diagram commutes because of a similar argument.

If the colax monoidality morphisms are invertible, then $|-|$ will become strong monoidal, and $\tau$ will become a monoidal transformation. In particular this is the case in the simplicial-topological example.

For the purpose of defining a lax structure on $|-|$ we could again look at the special cases. In the topological case the lax monoidal structure $|X|\times|Y| \rightarrow |X\times Y|$ is constructed (-> Theorem 11.5 "The geometry of iterated loops") using certain maps

$$X_n\times X_m \times \Delta^n\times \Delta^m \rightarrow X_{n+m}\times Y_{n+m} \times \Delta^{n+m}$$

which depend on certain isomorphisms $\Delta^n\times\Delta^m \cong \Delta^{n+m}$. In the bisimplicial situation $|X|\times|Y| \rightarrow |X\times Y|$ is explicitly given via

$$(X_n\times Y_m \times \Delta^n\times \Delta^m)_r \rightarrow (X_r\times Y_r \times \Delta^r)_r$$ $$(x, y, u : r \rightarrow n, v : r \rightarrow m) \mapsto (u^\ast(x), v^\ast(y), 1 : r \rightarrow r).$$

However, I can't tell how these can be generalized to a monoidal context, or how they can be related to each other.

The answer might depend on how one defines the monoidal structure on the realizations $|-|:s\mathcal{V}_i\to \mathcal{V}_i$. In the topological and the simplicial cases the strong monoidal structures are determined by the preservation of products.

Observe that in both of these cases the canonical morphism $|X\times Y| \rightarrow |X|\times|Y|$ is induced through the diagonals $\Delta^n \rightarrow \Delta^n\times\Delta^n$ when one thinks of it as a map between the coends. One can generalize to the monoidal situation by requiring the cosimplicial object $\Delta \to \mathcal{V}$ to be a comonoid in $[\Delta, \mathcal{V}]$, so that we have $\delta_{\Delta^n} : \Delta^n \rightarrow \Delta^n\otimes\Delta^n$. This will define a colax monoidal structure on $|-|$

$$\int^n X_n\otimes Y_n \otimes \Delta^n \rightarrow \int^{n,m} X_n\otimes Y_m \otimes \Delta^n\otimes \Delta^m$$ $$\cong (\int^n X_n \otimes\Delta^n) \otimes (\int^m Y_m \otimes \Delta^m)$$

We also need $\otimes$ to commute with coends.

Now in the diagrams in the question the monoidality maps will have the opposite direction since $|-|$ are colax, not lax. If $\delta_{F\Delta^n}$ equals to $\delta_{\Delta^n}$ up to the isomorphism $F(\Delta^n) \cong \Delta^n$, then the first diagram will commute because (writing $.$ for the tensor, and omitting an index $n$)

$$FX.FY.F\Delta \rightarrow{} FX.FY.F\Delta.F\Delta \rightarrow F(X.\Delta).F(Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta)$$ $$=$$ $$FX.FY.F\Delta \rightarrow F(X.Y).F\Delta \rightarrow F(X.Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta),$$ which holds because of the naturality and coherence of the lax structure of $F$. The second diagram commutes because of a similar argument.

If the colax monoidality morphisms are invertible, then $|-|$ will become strong monoidal, and $\tau$ will become a monoidal transformation. In particular this is the case in the simplicial-topological example.

For the purpose of defining a lax structure on $|-|$ we could again look at the special cases. In the topological case the lax monoidal structure $|X|\times|Y| \rightarrow |X\times Y|$ is constructed (-> Theorem 11.5 "The geometry of iterated loops") using certain maps

$$X_n\times X_m \times \Delta^n\times \Delta^m \rightarrow X_{n+m}\times Y_{n+m} \times \Delta^{n+m}$$

which depend on certain isomorphisms $\Delta^n\times\Delta^m \cong \Delta^{n+m}$. In the bisimplicial situation $|X|\times|Y| \rightarrow |X\times Y|$ is explicitly given via

$$(X_n\times Y_m \times \Delta^n\times \Delta^m)_r \rightarrow (X_r\times Y_r \times \Delta^r)_r$$ $$(x, y, u : r \rightarrow n, v : r \rightarrow m) \mapsto (u^\ast(x), v^\ast(y), 1 : r \rightarrow r).$$

However, I can't tell how these can be generalized to a monoidal context, or how they can be related to each other.

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Dimitri Chikhladze
Dimitri Chikhladze
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The answer might depend on how one defines the monoidal structure on the realizations $|-|:s\mathcal{V}_i\to \mathcal{V}_i$. In the topological and the simplicial cases the strong monoidal structures are determined by the preservation of products.

Observe that in both of these cases the canonical morphism $|X\times Y| \rightarrow |X|\times|Y|$ is induced through the diagonals $\Delta^n \rightarrow \Delta^n\times\Delta^n$ when one thinks of it as a map between the coends. One can generalize to the monoidal situation by requiring the cosimplicial object $\Delta \to \mathcal{V}$ to be a comonoid in $[\Delta, \mathcal{V}]$, so that we have $\delta_{\Delta^n} : \Delta^n \rightarrow \Delta^n\otimes\Delta^n$. This will define a colax monoidal structure on $|-|$

$$\int^n X_n\otimes Y_n \otimes \Delta^n \rightarrow \int^{n,m} X_n\otimes Y_m \otimes \Delta^n\otimes \Delta^m$$ $$\cong (\int^n X_n \otimes\Delta^n) \otimes (\int^m Y_m \otimes \Delta^m)$$

We also need $\otimes$ to commute with coends.

Now in the diagrams in the question the monoidality maps will have the opposite direction since $|-|$ are colax, not lax. If $\delta_{F\Delta^n}$ equals to $\delta_{\Delta^n}$ up to the isomorphism $F(\Delta^n) \cong \Delta^n$, then the first diagram will commute because (writing $.$ for the tensor, and omitting an index $n$)

$$FX.FY.F\Delta \rightarrow{} FX.FY.F\Delta.F\Delta \rightarrow F(X.\Delta).F(Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta)$$ $$=$$ $$FX.FY.F\Delta \rightarrow F(X.Y).F\Delta \rightarrow F(X.Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta),$$ which holds because of the naturality and coherence of the colax structure of $F$. The second diagram commutes because of a similar argument.

If the colax monoidality morphisms are invertible, then $|-|$ will become strong monoidal, and $\tau$ will become a monoidal transformation. In particular this is the case in the simplicial-topological example.

For the purpose of defining a lax structure on $|-|$ we could again look at the special cases. In the topological case the lax monoidal structure $|X|\times|Y| \rightarrow |X\times Y|$ is constructed (-> Theorem 11.5 "The geometry of iterated loops") using certain maps

$$X_n\times X_m \times \Delta^n\times \Delta^m \rightarrow X_{n+m}\times Y_{n+m} \times \Delta^{n+m}$$

which depend on certain isomorphisms $\Delta^n\times\Delta^m \cong \Delta^{n+m}$. In the bisimplicial situation $|X|\times|Y| \rightarrow |X\times Y|$ is explicitly given via

$$(X_n\times Y_m \times \Delta^n\times \Delta^m)_r \rightarrow (X_r\times Y_r \times \Delta^r)_r$$ $$(x, y, u : r \rightarrow n, v : r \rightarrow m) \mapsto (u^\ast(x), v^\ast(y), 1 : r \rightarrow r).$$

However, I can't tell how these can be generalized to a monoidal context, or how they can be related to each other.

The answer might depend on how one defines the monoidal structure on the realizations $|-|:s\mathcal{V}_i\to \mathcal{V}_i$. In the topological and the simplicial cases the strong monoidal structures are determined by the preservation of products.

Observe that in both of these cases the canonical morphism $|X\times Y| \rightarrow |X|\times|Y|$ is induced through the diagonals $\Delta^n \rightarrow \Delta^n\times\Delta^n$ when one thinks of it as a map between the coends. One can generalize to the monoidal situation by requiring the cosimplicial object $\Delta \to \mathcal{V}$ to be a comonoid in $[\Delta, \mathcal{V}]$, so that we have $\delta_{\Delta^n} : \Delta^n \rightarrow \Delta^n\otimes\Delta^n$. This will define a colax monoidal structure on $|-|$

$$\int^n X_n\otimes Y_n \otimes \Delta^n \rightarrow \int^{n,m} X_n\otimes Y_m \otimes \Delta^n\otimes \Delta^m$$ $$\cong (\int^n X_n \otimes\Delta^n) \otimes (\int^m Y_m \otimes \Delta^m)$$

We also need $\otimes$ to commute with coends.

Now in the diagrams in the question the monoidality maps will have the opposite direction since $|-|$ are colax, not lax. If $\delta_{F\Delta^n}$ equals to $\delta_{\Delta^n}$ up to the isomorphism $F(\Delta^n) \cong \Delta^n$, then the first diagram will commute because (writing $.$ for the tensor)

$$FX.FY.F\Delta \rightarrow{} FX.FY.F\Delta.F\Delta \rightarrow F(X.\Delta).F(Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta)$$ $$=$$ $$FX.FY.F\Delta \rightarrow F(X.Y).F\Delta \rightarrow F(X.Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta),$$ which holds because of the naturality and coherence of the colax structure of $F$. The second diagram commutes because of a similar argument.

If the colax monoidality morphisms are invertible, then $|-|$ will become strong monoidal, and $\tau$ will become a monoidal transformation. In particular this is the case in the simplicial-topological example.

For the purpose of defining a lax structure on $|-|$ we could again look at the special cases. In the topological case the lax monoidal structure $|X|\times|Y| \rightarrow |X\times Y|$ is constructed (-> Theorem 11.5 "The geometry of iterated loops") using certain maps

$$X_n\times X_m \times \Delta^n\times \Delta^m \rightarrow X_{n+m}\times Y_{n+m} \times \Delta^{n+m}$$

which depend on certain isomorphisms $\Delta^n\times\Delta^m \cong \Delta^{n+m}$. In the bisimplicial situation $|X|\times|Y| \rightarrow |X\times Y|$ is explicitly given via

$$(X_n\times Y_m \times \Delta^n\times \Delta^m)_r \rightarrow (X_r\times Y_r \times \Delta^r)_r$$ $$(x, y, u : r \rightarrow n, v : r \rightarrow m) \mapsto (u^\ast(x), v^\ast(y), 1 : r \rightarrow r).$$

However, I can't tell how these can be generalized to a monoidal context, or how they can be related to each other.

The answer might depend on how one defines the monoidal structure on the realizations $|-|:s\mathcal{V}_i\to \mathcal{V}_i$. In the topological and the simplicial cases the strong monoidal structures are determined by the preservation of products.

Observe that in both of these cases the canonical morphism $|X\times Y| \rightarrow |X|\times|Y|$ is induced through the diagonals $\Delta^n \rightarrow \Delta^n\times\Delta^n$ when one thinks of it as a map between the coends. One can generalize to the monoidal situation by requiring the cosimplicial object $\Delta \to \mathcal{V}$ to be a comonoid in $[\Delta, \mathcal{V}]$, so that we have $\delta_{\Delta^n} : \Delta^n \rightarrow \Delta^n\otimes\Delta^n$. This will define a colax monoidal structure on $|-|$

$$\int^n X_n\otimes Y_n \otimes \Delta^n \rightarrow \int^{n,m} X_n\otimes Y_m \otimes \Delta^n\otimes \Delta^m$$ $$\cong (\int^n X_n \otimes\Delta^n) \otimes (\int^m Y_m \otimes \Delta^m)$$

We also need $\otimes$ to commute with coends.

Now in the diagrams in the question the monoidality maps will have the opposite direction since $|-|$ are colax, not lax. If $\delta_{F\Delta^n}$ equals to $\delta_{\Delta^n}$ up to the isomorphism $F(\Delta^n) \cong \Delta^n$, then the first diagram will commute because (writing $.$ for the tensor, and omitting an index $n$)

$$FX.FY.F\Delta \rightarrow{} FX.FY.F\Delta.F\Delta \rightarrow F(X.\Delta).F(Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta)$$ $$=$$ $$FX.FY.F\Delta \rightarrow F(X.Y).F\Delta \rightarrow F(X.Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta),$$ which holds because of the naturality and coherence of the colax structure of $F$. The second diagram commutes because of a similar argument.

If the colax monoidality morphisms are invertible, then $|-|$ will become strong monoidal, and $\tau$ will become a monoidal transformation. In particular this is the case in the simplicial-topological example.

For the purpose of defining a lax structure on $|-|$ we could again look at the special cases. In the topological case the lax monoidal structure $|X|\times|Y| \rightarrow |X\times Y|$ is constructed (-> Theorem 11.5 "The geometry of iterated loops") using certain maps

$$X_n\times X_m \times \Delta^n\times \Delta^m \rightarrow X_{n+m}\times Y_{n+m} \times \Delta^{n+m}$$

which depend on certain isomorphisms $\Delta^n\times\Delta^m \cong \Delta^{n+m}$. In the bisimplicial situation $|X|\times|Y| \rightarrow |X\times Y|$ is explicitly given via

$$(X_n\times Y_m \times \Delta^n\times \Delta^m)_r \rightarrow (X_r\times Y_r \times \Delta^r)_r$$ $$(x, y, u : r \rightarrow n, v : r \rightarrow m) \mapsto (u^\ast(x), v^\ast(y), 1 : r \rightarrow r).$$

However, I can't tell how these can be generalized to a monoidal context, or how they can be related to each other.

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Dimitri Chikhladze
Dimitri Chikhladze
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