nLab external tensor product

Idea

The concept of external tensor product is a variant of that of tensor product in a monoidal category when the latter is generalized to indexed monoidal categories (dependent linear type theory).

Definition

Consider an indexed monoidal category given by a Cartesian fibration

$$\array{ Mod(-) \\ \downarrow \\ \mathbf{H} }$$

over a cartesian monoidal category $\mathbf{H}$.

Definition

(external tensor product)
Given $X_1, X_2 \in \mathbf{H}$ the external tensor product over these is the functor

$$\boxtimes \;\colon\; Mod(X_1)\times Mod(X_2) \longrightarrow Mod(X_1 \times X_2)$$

given on $A_1 \in Mod(X_1)$ with $A_2 \in Mod(X_2)$ by

$$A_1 \boxtimes A_2 \coloneqq (p_1^\ast A_1) \otimes_{X_1 \times X_2} (p_2^\ast A_2) \in Mod(X_1 \times X_2) \,,$$

where $p_1, p_2$ denote the projection maps out of the Cartesian product $X_1 \times X_2 \in \mathbf{H}$.

Remark

The external tensor product constitutes a tensor product on the total category $Mod$ of the given Grothendieck fibration $Mod(-)\to \mathbf{H}$; and with respect to this it is a monoidal fibration.

Properties

Relation to fiberwise tensor product

Proposition

The fiberwise (“internal”) tensor product over $X\in \mathbf{H}$ is recovered form the external one via a natural equivalence

$$A_1 \otimes_X A_2 \simeq \Delta_X^\ast (A_1 \boxtimes A_2)$$

for $A_1, A_2 \in Mod(X)$, where $\Delta \colon X \longrightarrow X \times X$ is the diagonal in $\mathbf{H}$ on $X$.

Generation of $Mod(X_1 \times X_2)$ from external tensor products

Under suitable conditions on compact generation of $Mod(-)$ then one may deduce that $Mod(X_1 \times X_2)$ is generated under external product from $Mod(X_1)$ and $Mod(X_2)$.

(Bondal-vdBerg 03, BFN 08, proof of prop. 3.24)

In indexed monoidal categories

Suppose an indexed monoidal category which satisfies the motivic yoga (Wirthmüller context-form) in that:

1. the corresponding pseudofunctor takes values in adjoint functors

   $$\array{ \mathllap{ \mathbf{C} \,\colon\, \; } Base &\longrightarrow& Cat \\ \mathcal{X} &\mapsto& \mathbf{C}_{\mathcal{X}} \\ \Big\downarrow\mathrlap{{}^{f}} && \mathllap{^{f_!}}\Big\downarrow \dashv \Big\uparrow\mathrlap{{}^{f^\ast}} \\ \mathcal{Y} &\mapsto& \mathbf{C}_{\mathcal{Y}} }$$

2. between closed monoidal categories

3. the base change functors $f^\ast$ are

   1. besides being strong monoidal functors

   2. also strong closed functors,

      hence the pushforward functors $f_!$ satisfy the projection formula (“Frobenius reciprocity”)

   3. preserving colimits

4. pull-push through cartesian squares in $Base$ satisfies the Beck-Chevalley condition.

Also assume that

• $Base$ is cocomplete and cartesian closed,

• $\mathbf{C}_{\mathcal{X}}$ is cocomplete for each $\mathcal{X} \in Base$.

Then:

Proposition

The external tensor product (Def. ) on the Grothendieck construction $\int \mathbf{C}$ preserves colimits in each variable.

Proof

Consider any object

$$\mathcal{W}_{\mathcal{Y}} \,\in\, \textstyle{\int} \mathbf{C}$$

and any diagram

$$\begin{array}{ccc} \mathcal{V}_{\mathcal{X}} \,\colon\, I &\longrightarrow& \int \mathbf{C} \\ i &\mapsto& \mathcal{V}(i)_{\mathcal{X}_i} \end{array}$$

in the Grothendieck construction category $\int \mathbf{C}$.

Then with

• the description of colimits in Grothendieck constructions as (see there)

  $$\underset{\underset{i \in I}{\longrightarrow}}{lim} \big( \mathcal{V}(i)_{\mathcal{X}_i} \big) \;\simeq\; \left( \underset{\underset{i \in I}{\longrightarrow}}{lim} q^{\mathcal{X}_i}_! \mathcal{V}(i) \right)_{ \underset{\underset{i \in I}{\longrightarrow}}{lim} \mathcal{X}_i }$$

  where

  $$q^{\mathcal{X}_i} \;\colon\; \mathcal{X}_i \longrightarrow \underset{\longrightarrow}{lim} \mathcal{X}$$

  denote the coprojections of the colimit of the underlying diagram in $Base$,

• the Beck-Chevalley condition for the following cartesian squares in $Base$

  $$\array{ \mathcal{X}_i \times \mathcal{Y} & \overset{\; pr_{\mathcal{X}_i} \;}{\longrightarrow} & \mathcal{X}_i \\ \mathllap{{q}^{\mathcal{X}_i} \times id_{\mathcal{Y}}} \Big\downarrow && \Big\downarrow \mathrlap{{}^{ q^{\mathcal{X}_i} }} \\ \underset{\underset{j \in I}{\longrightarrow}}{lim} \mathcal{X}_j \times \mathcal{Y} & \underset {pr_{\underset{\longrightarrow}{lim}\mathcal{X}}} {\longrightarrow} & \underset{\underset{j \in I}{\longrightarrow}}{lim} \mathcal{X}_j \mathrlap{\,,} }$$

• the fact that both $(-) \times \mathcal{Y}$ and $(-)\otimes (pr_{\mathcal{Y}})^\ast\mathscr{W}$ preserve colimits (being left adjoints)

we obtain the following sequence of natural isomorphisms:

$$\begin{array}{ll} \Big( \underset{\underset{i \in I}{\longrightarrow}}{lim} \mathscr{V}(i)_{\mathcal{X}_i} \Big) \boxtimes \mathscr{W}_{\mathcal{Y}} \\ \;\simeq\; \left( \big( \underset{\longrightarrow}{\lim} q^{\mathcal{X}}_!\mathscr{V} \big)_{\underset{\longrightarrow}{\lim}\mathcal{X}} \right) \boxtimes \mathscr{W}_{\mathcal{Y}} & \text{colimit in Groth. constr.} \\ \;\simeq\; \Big( \big( (pr_{\underset{\longrightarrow}{lim}\mathcal{X}})^\ast ( \underset{\longrightarrow}{\lim} q^{\mathcal{X}}_!\mathscr{V} ) \big) \,\otimes\, \big( (pr_{\mathcal{Y}})^\ast \mathscr{W} \big) \Big)_{ \big(\underset{\longrightarrow}{\lim}\mathcal{X}\big) \times \mathcal{Y} } & \text{def. of external tensor} \\ \;\simeq\; \Big( \big( \underset{\longrightarrow}{\lim} (pr_{\underset{\longrightarrow}{lim}\mathcal{X}})^\ast q^{\mathcal{X}}_! \mathscr{V} \big) \,\otimes\, \big( (pr_{\mathcal{Y}})^\ast \mathscr{W} \big) \Big)_{ \big(\underset{\longrightarrow}{\lim}\mathcal{X}\big) \times \mathcal{Y} } & \text{pullback preserves colimits} \\ \;\simeq\; \Big( \big( \underset{\longrightarrow}{\lim} (q^{\mathcal{X}} \times id_{\mathcal{Y}} )_! (pr_{\mathcal{X}})^\ast \mathscr{V} \big) \,\otimes\, \big( (pr_{\mathcal{Y}})^\ast \mathscr{W} \big) \Big)_{ \big(\underset{\longrightarrow}{\lim}\mathcal{X}\big) \times \mathcal{Y} } & \text{Beck-Chevalley} \\ \;\simeq\; \bigg( \underset{\longrightarrow}{\lim} \Big( \big( (q^{\mathcal{X}} \times id_{\mathcal{Y}} )_! (pr_{\mathcal{X}})^\ast \mathscr{V} \big) \,\otimes\, \big( (pr_{\mathcal{Y}})^\ast \mathscr{W} \big) \Big) \bigg)_{ \underset{\longrightarrow}{\lim} \big( \mathcal{X} \times \mathcal{Y} \big) } & \text{tensoring preserves colimits} \\ \;\simeq\; \bigg( \underset{\longrightarrow}{\lim} (q^{\mathcal{X}} \times id_{\mathcal{Y}} )_! \Big( \big( (pr_{\mathcal{X}})^\ast \mathscr{V} \big) \,\otimes\, \big( (pr_{\mathcal{Y}})^\ast \mathscr{W} \big) \Big) \bigg)_{ \underset{\longrightarrow}{\lim} \big( \mathcal{X} \times \mathcal{Y} \big) } & \text{projection formula} \\ \;\simeq\; \underset{\underset{i \in I}{\longrightarrow}}{lim} \bigg( \Big( \big( (pr_{\mathcal{X}_i})^\ast \mathscr{V}(i) \big) \,\otimes\, \big( (pr_{\mathcal{Y}})^\ast \mathscr{W} \big) \Big)_{ \mathcal{X}_i \times \mathcal{Y} } \bigg) & \text{colimit in Groth. constr.} \\ \;\simeq\; \underset{\underset{i \in I}{\longrightarrow}}{lim} \Big( \mathscr{V}(i)_{\mathcal{X}_i} \,\boxtimes\, \mathscr{W}_{\mathcal{Y}} \Big) & \text{def of external tensor.} \end{array}$$

The following proposition still assumes the “motivic yoga” above, but in fact we need to assume the Beck-Chevalley condition only for the very special squares of this form:

$$\array{ \mathcal{X} \times \mathcal{Y} &\overset{ f \times id }{\longrightarrow}& \mathcal{X}' \times \mathcal{Y} \\ \mathllap{{}^{ pr_{\mathcal{X}} }} \Big\downarrow && \Big\downarrow \\ \mathcal{X} &\underset{\;\;\; f \;\;\;}{\longrightarrow}& \mathcal{X}' }$$

Proposition

Given

$$\array{ f \,\colon\, \mathcal{X} &\longrightarrow& \mathcal{X}' \\ g \,\colon\, \mathcal{Y} &\longrightarrow& \mathcal{Y}' }$$

we have

1. for $\mathscr{V} \,\in\, \mathbf{C}_{\mathcal{X}'}$ and $\mathscr{W} \,\in\, \mathbf{C}_{\mathcal{Y}'}$ natural isomorphism of this form:

   (1)$$(f \times g)^\ast (\mathscr{V} \boxtimes \mathscr{W}) \;\simeq\; \big(f^\ast \mathscr{V}\big) \boxtimes \big(g^\ast \mathscr{W}\big)$$

2. for $\mathscr{V} \,\in\, \mathbf{C}_{\mathcal{X}}$ and $\mathscr{W} \,\in\, \mathbf{C}_{\mathcal{Y}}$ natural isomorphism of this form:

   (2)$$(f \times g)_! (\mathscr{V} \boxtimes \mathscr{W}) \;\simeq\; \big(f_! \mathscr{V}\big) \boxtimes \big(g_! \mathscr{W}\big) \mathrlap{\,.}$$

(The first statement is essentially immediate from the fact that pullback $(-)^*$ is assumed to be strong closed, but the second statement is not quite so immediate; it is discussed for the case of smash product of retractive spaces and parameterized spectra in May & Sigurdsson (2006), Rem. 2.5.8, Prop. 13.7.2, see also Malkiewich (2019), Lem. 3.4.1, Malkiewich (2023), Lem. 2.5.1, and is mentioned in generality but without proof in Shulman (2012), p. 624.)

Proof

For the first statement (1) we have the following sequence of natural isomorphisms:

$$\begin{array}{ll} (f \times g)^\ast \big( \mathscr{V} \,\boxtimes\, \mathscr{W} \big) \\ \;\simeq\; (f \times g)^\ast \Big( \big( \mathrm{pr}_{\mathbf{Y}}^\ast \mathscr{V} \big) \,\otimes_{\mathbf{Y} \times \mathbf{Y}'}\, \big( \mathrm{pr}_{\mathbf{Y}'}^\ast \mathscr{W} \big) \Big) & \text{by definition} \\ \;\simeq\; \big( (f \times g)^\ast \mathrm{pr}_{\mathbf{Y}}^\ast \mathscr{V} \big) \otimes_{\mathbf{X} \times \mathbf{X}'} \big( (f \times g)^\ast \mathrm{pr}_{\mathbf{Y}'}^\ast \mathscr{W} \big) & \text{since pullback is strong closed} \\ \;\simeq\; \big( \mathrm{pr}_{\mathbf{X}}^\ast f^\ast \mathscr{V} \big) \otimes_{\mathbf{X} \times \mathbf{X}'} \big( \mathrm{pr}_{\mathbf{X}'}^\ast g^\ast \mathscr{W} \big) & \text{by pseudo-functoriality} \\ \;\simeq\; \big(f^\ast \mathscr{V}\big) \boxtimes \big(g^\ast \mathscr{W}\big) & \text{by definition.} \end{array}$$

For the second statement (2), first notice the special case where one of the maps is an identity morphism and the corresponding external tensor factor is the tensor unit; and notice here that external tensoring with the tensor unit is just pullback to a product (since pullback along the other leg preserves tensor units, being strong monoidal):

$$\mathscr{V} \boxtimes \mathbb{1} \;\simeq\; (pr_{\mathcal{X}})^\ast \mathscr{V} \,.$$

This way we first find

$$\begin{array}{ll} (f \times id)_! \big( \mathscr{V} \boxtimes \mathbb{1} \big) \\ \;\simeq\; (f \times id)_! \big( pr_{\mathcal{X}}^\ast \mathscr{V} \big) & \text{ by the above comment } \\ \;\simeq\; pr_{\mathcal{X}'}^\ast\big(f_! \mathscr{V}\big) & \text{ using the BC-condition } \\ \;\simeq\; (f_! \mathscr{V}) \boxtimes \mathbb{1} & \text{ by the above comment } \,, \end{array}$$

from which the general case is obtained as follows:

$$\begin{array}{ll} (f \times g)_! \big( \mathscr{V} \boxtimes \mathscr{W} \big) \\ \;\simeq\; (f \times g)_! \big( (\mathscr{V} \boxtimes \mathbb{1}) \otimes (\mathbb{1} \boxtimes \mathscr{W}) \big) & \text{by the above comment} \\ \;\simeq\; (f \times g)_! \Big( \big( (id \times g)^\ast (\mathscr{V} \boxtimes \mathbb{1}) \big) \otimes (\mathbb{1} \boxtimes \mathscr{W}) \Big) & \text{by first claim and strong monoidalness} \\ \;\simeq\; (f \times id)_! (id \times g)_! \Big( \big( (id \times g)^\ast (\mathscr{V} \boxtimes \mathbb{1}) \big) \otimes \big(\mathbb{1} \boxtimes \mathscr{W}\big) \Big) & \text{by pseudo-functoriality} \\ \;\simeq\; (f \times id)_! \Big( \big(\mathscr{V} \boxtimes \mathbb{1}\big) \otimes \big( (id \times g)_! \big(\mathbb{1} \boxtimes \mathscr{W}\big) \big) \Big) & \text{by the projection formula} \\ \;\simeq\; (f \times id)_! \Big( \big(\mathscr{V} \boxtimes \mathbb{1}\big) \otimes \big( \mathbb{1} \boxtimes (g_!\mathscr{W}) \big) \Big) & \text{by the special case above} \\ \;\simeq\; (f \times id)_! \Big( \big(\mathscr{V} \boxtimes \mathbb{1}\big) \otimes (f \times id)^\ast \big( \mathbb{1} \boxtimes (g_!\mathscr{W}) \big) \Big) & \text{by first claim and strong monoidalness} \\ \;\simeq\; \Big( (f \times id)_! \big(\mathscr{V} \boxtimes \mathbb{1}\big) \Big) \otimes \big( \mathbb{1} \boxtimes (g_!\mathscr{W}) \big) & \text{by the projection formula} \\ \;\simeq\; \big( (f_!\mathscr{V}) \boxtimes \mathbb{1} \big) \otimes \big( \mathbb{1} \boxtimes (g_!\mathscr{W}) \big) & \text{by the special case above} \\ \;\simeq\; (f_! \mathscr{V}) \boxtimes (g_! \mathscr{W}) & \text{by the above comment}. \end{array}$$

Corollary

The $\big((f \times g)_! \dashv (f \times g)^\ast\big)$-adjunct $\widetilde{f \boxtimes g}$ of an external tensor product of morphisms into the separate pullbacks

$$\mathscr{V} \boxtimes \mathscr{W} \overset{ \phi \boxtimes \gamma }{\longrightarrow} (f^\ast \mathscr{V}') \boxtimes (g^\ast \mathscr{W}') \;\simeq\; (f \times g)^\ast (\mathscr{V}' \boxtimes \mathscr{W}')$$

is the external tensor product of the separate adjuncts

$$\widetilde{ f \boxtimes g } \,\colon\, (f \times g)_!( \mathscr{V} \boxtimes \mathscr{W} ) \,\simeq\, (f_! \mathscr{V}) \boxtimes (g_! \mathscr{W}) \overset{ \widetilde \phi \,\boxtimes\, \widetilde \gamma }{\longrightarrow} \mathscr{V}' \boxtimes \mathscr{W}' \,,$$

where the natural isomorphisms shown are those form Prop. .

Proof

After restriction along the (non-full) subcategory-inclusion

$$\array{ \mathbf{C}_{\mathcal{X}} \times \mathbf{C}_{\mathcal{Y}} &\longrightarrow& \mathbf{C}_{\mathcal{X} \times \mathcal{Y}} \\ \big(\mathscr{V},\, \mathscr{W}\big) &\mapsto& \mathscr{V} \boxtimes \mathscr{W} \\ \mathllap{{}^{ (\phi, \gamma) }} \Big\downarrow && \Big\downarrow \mathrlap{{}^{ \phi \boxtimes \gamma }} \\ \big(\mathscr{V}',\, \mathscr{W}'\big) &\mapsto& \mathscr{V}' \boxtimes \mathscr{W}' }$$

we can clearly make the restriction of the functors $(f \times g)_!$ and $(f \times g)^\ast$ into adjoints by declaring the adjunction counit $\epsilon$ to be the external tensor product of the $(f_! \dashv f^\ast)$- with the $(f_! \dashv g^\ast)$-adjunction counits. But by uniqueness of adjoints (here) this must be isomorphic to the actual adjunction counit restricted to external tensor products:

$$\epsilon^{ \big((f \times g)_! \dashv (f \times g)^\ast\big) }_{\mathcal{X} \boxtimes \mathcal{Y}} \;\; \simeq \;\; \epsilon^{ (f_! \dashv f^\ast) }_{\mathcal{X}} \,\boxtimes\, \epsilon^{ (g_! \dashv g^\ast) }_{\mathcal{Y}} \,.$$

Now using the expression on the right together with Prop. in the formula (here) that expresses adjuncts as functor-images composed with the (co)unit gives that the adjunct is formed external tensor-factor wise, as claimed:

$$\begin{array}{ll} \widetilde{ \phi \boxtimes \gamma } \\ \;\simeq\; \epsilon^{ (f \times g)_! \dashv (f \times g)^\ast }_{ \mathscr{V}' \boxtimes \mathscr{W}' } \circ (f \times g)_!\big(\phi \boxtimes \gamma\big) & \text{ by the formula for adjuncts } \\ \;\simeq\; \big( \epsilon^{ f_! \dashv f^\ast }_{\mathscr{V}'} \circ (f_! \phi) \big) \,\boxtimes\, \big( \epsilon^{ g_! \dashv g^\ast }_{\mathscr{W}'} \circ (g_! \gamma) \big) & \text{ by the previous discussion } \\ \;\simeq\; \widetilde \phi \,\boxtimes\, \widetilde \gamma & \text{ by the formula for adjuncts. } \end{array}$$

Proposition

(external pushout-product)
In the situation above, the $\boxtimes$-pushout-product (with respect to the external tensor product) is given by

$$\big( \phi_f \big) \,\widehat{\boxtimes}\, \big( \gamma_g \big) \;\;\; \simeq \;\;\; \Big( \big( (pr_{X'})^\ast \phi \big) \,\widehat{\otimes}\, \big( (pr_{Y'})^\ast \gamma \big) \Big)_{ f \,\widehat{\times}\, g }$$

where here we use the left-handed convention for component maps in morphisms in the Grothendieck construction:

$$\phi_f \,\colon\, \mathscr{V}_X \to \mathscr{V}'_{X'} \;\;\;\;\;\; \text{stands for} \;\;\;\;\;\; \begin{array}{rcl} X &\xrightarrow{f}& X' \\ f_!\mathscr{V} &\xrightarrow{\phi}& \mathscr{V}' \end{array}$$

Proof

By the general formula for colimits in Grothendieck constructions (here) the underlying colimit in the base category is the evident one and the component map over the dashed morphism is the colimiting cocone of the diagram obtained from pushing the separate component maps along the coprojections $q_\cdot$ to form a span in $\mathbf{C}_{f \widehat{\times} g}$. This yields:

$$\begin{array}{ll} (f \widehat{\times} g)_! \Bigg( \bigg( (q_l)_! \Big( \big( (pr_{X'})^\ast \phi \big) \otimes \big( (pr_{Y})^\ast \mathrm{id}_{\mathscr{W}} \big) \Big) \bigg) \wedge \bigg( (q_r)_! \Big( \big( (pr_X)^\ast \mathrm{id}_{\mathscr{V}} \big) \otimes \big( (pr_{Y'})^\ast \gamma \big) \Big) \bigg) \Bigg) \\ \;\simeq\; \Bigg( \bigg( (f \widehat{\times} g)_! (q_l)_! \Big( \big( (pr_{X'})^\ast \phi \big) \otimes \big( (pr_{Y})^\ast \mathrm{id}_{\mathscr{W}} \big) \Big) \bigg) \wedge \bigg( (f \widehat{\times} g)_! (q_r)_! \Big( \big( (pr_X)^\ast \mathrm{id}_{\mathscr{V}} \big) \otimes \big( (pr_{Y'})^\ast \gamma \big) \Big) \bigg) \Bigg) & \begin{array}{l} \text{since the left adjoint}\; (-)_! \\ \text{preserves pushouts} \end{array} \\ \;\simeq\; \Bigg( \bigg( (id \otimes g)_! \Big( \big( (pr_{X'})^\ast \phi \big) \otimes \big( (pr_{Y})^\ast \mathrm{id}_{\mathscr{W}} \big) \Big) \bigg) \wedge \bigg( (f \otimes id)_! \Big( \big( (pr_X)^\ast \mathrm{id}_{\mathscr{V}} \big) \otimes \big( (pr_{Y'})^\ast \gamma \big) \Big) \bigg) \Bigg) & \begin{array}{l} \text{by commutativity of} \\ \text{the above diagram} \end{array} \\ \;\simeq\; \Bigg( \bigg( \Big( \big( (pr_{X'})^\ast \phi \big) \otimes \big( (pr_{Y})^\ast \mathrm{id}_{f_!\mathscr{W}} \big) \Big) \bigg) \wedge \bigg( \Big( \big( (pr_X)^\ast \mathrm{id}_{f_!\mathscr{V}} \big) \otimes \big( (pr_{Y'})^\ast \gamma \big) \Big) \bigg) \Bigg) & \text{by the above result} \\ \;\simeq\; \big( (pr_{X'})^\ast \phi \big) \,\displaystyle{\widehat{\otimes}}\, \big( (pr_{Y'})^\ast \gamma \big) & \text{by definition} \end{array}$$

Here in the second-but-last step we used (2).

Examples

Example

(external tensor product of vector bundles)

Given a pair of topological spaces $X,\,Y \,\in\, Top$ and a respective pair of topological vector bundles $\mathscr{V}_X \,\in\,$ $Vect(X)$ and $\mathscr{V}_X \,\in\,$ $Vect(X)$, their external tensor product is formed via pullback of bundles $(-)^\ast$ and ordinary tensor product of vector bundles $(-) \otimes (-)$ by the formula

$$\mathscr{V}_X \,\boxtimes\, \mathscr{W}_Y \;\; \equiv \;\; \Big( \big( (pr_X)^\ast \mathscr{V} \big) \otimes \big( (pr_Y)^\ast \mathscr{W} \big) \Big)_{X \times Y} \;\;\; \in \; Vect(X \times Y) \,,$$

where we are denoting by

$$\array{ && X \times Y \\ & \mathllap{{}^{pr_X}}\swarrow && \searrow\mathrlap{{}^{pr_Y}} \\ X && && Y }$$

the product projection maps in Top.

This is a vector bundle over the product topological space $X \times Y$ whose fiber over a point $(x,y) \,\in\, X \times Y$ is the tensor product of vector spaces $\mathscr{V}_x \otimes \mathscr{W}_y$.

Example

(Cartesian product in a Grothendieck construction is external cartesian product)
Given a contravariant pseudofunctor

$$\array{ X \\ \Big\downarrow\mathrlap{{}^{f}} \\ Y } \;\;\;\;\;\mapsto\;\;\; \array{ \mathcal{C}_X \\ \Big\uparrow\mathrlap{{}^{f^\ast}} \\ \mathcal{C}_Y }$$

where the base category and all fiber categories $\mathcal{C}_{(-)}$ have Cartesian products and all base change morphisms $f^\ast$ preserve these products, then the Grothendieck construction $\int_X \mathcal{C}_X$ has cartesian products (as a special case of a general result on limits in Grothendieck constructions, discussed there) which is given on objects

$$\mathscr{V}_X \,\equiv\, \big( \mathscr{V} \,\in\, \mathcal{C}_X \big)$$

by the formula

(3)$$\mathscr{V}_X \times \mathscr{W}_Y \;\; \simeq \;\; \Big( \big(pr_X^\ast \mathscr{V}\big) \,\times\, \big(pr_Y^\ast \mathscr{W}\big) \Big)_{X \times Y} \,.$$

This is the general form of what might be called the “external cartesian product”; see also this Proposition at free coproduct completion.

Related concepts

• external tensor product of vector bundles, VectBund

• direct product group representations

• doubly closed monoidal category

References

The notion of the external tensor product of vector bundles originates in discussion of topological K-theory:

• Michael Atiyah, §2.6 in: K-theory, Harvard Lecture 1964 (notes by D. W. Anderson), Benjamin (1967) [pdf, pdf]

• Raoul Bott, p. 19 of: Lectures on $K(X)$, Benjamin (1969) [pdf, pdf]

• Max Karoubi, §4.9 in: K-Theory – An introduction, Grundlehren der mathematischen Wissenschaften 226, Springer (1978) [pdf, doi:10.1007/978-3-540-79890-3]

The notion of external tensor product of representations seems to be folklore, it is mentioned in most textbooks but without any attribution:

• William Fulton, Joe Harris, Exc. 2.36 in: Representation Theory: a First Course, Springer (1991) [doi:10.1007/978-1-4612-0979-9]

• Caroline Gruson, Vera Serganova, p. 3 of: From Finite Groups to Quivers via Algebras – A Journey Through Representation Theory, Springer (2018) [doi:10.1007/978-3-319-98271-7]

The external tensor product of perverse sheaves:

• Volodymyr Lyubashenko, External tensor product of categories of perverse sheaves, Ukr. Math. J. 53 3 (2001) 311-322 [arXiv:math/9911207]

The external tensor product of of quasicoherent sheaves in (derived) algebraic geometry:

• Alexei Bondal, Michel Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 1 (2003) 1-36 [arXiv:math/0204218]

• David Ben-Zvi, John Francis, David Nadler, Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry, J. Amer. Math. Soc. 23 4 (2010) 909-966 [arXiv:0805.0157, ISSN:1088-6834]

The external product on cobordism rings:

• Daniel Quillen, §1.6 in: Elementary Proofs of Some Results of Cobordism Theory Using Steenrod Operations, Advances in Mathematics 7 (1971) 29–56 [doi:10.1016/0001-8708(71)90041-7]

and on differential cobordism rings:

• Ulrich Bunke, Thomas Schick, Ingo Schroeder, Moritz Wiethaup, §3.3.10 in: Landweber exact formal group laws and smooth cohomology theories, Algebr. Geom. Topol. 9 (2009) 1751-1790 [arXiv:0711.1134, doi:10.2140/agt.2009.9.1751]

and in the Hodge-filtered version:

• Knut Bjarte Haus, §4.1.8, §4.10 in: Geometric Hodge filtered complex cobordism, PhD thesis (2022) [ntnuopen:3017489]

• Knut B. Haus, Gereon Quick, §2.11 in: Geometric Hodge filtered complex cobordism [arXiv:2210.13259]

The external smash product of retractive spaces and of parameterized spectra:

• Peter May, Johann Sigurdsson, §7.3 of: Parametrized Homotopy Theory, Mathematical Surveys and Monographs 132, AMS (2006) [arXiv:math/0411656]

• Cary Malkiewich, §3.1 of: Parametrized spectra, a low-tech approach [arXiv:1906.04773, user guide: pdf, pdf]

• Cary Malkiewich, §2.3 and §4.4 of: A convenient category of parametrized spectra [arXiv:2305.15327]

For general abstract literature dealing with the external tensor products see the references at indexed monoidal category and at dependent linear type theory, such as

• Mike Shulman, Enriched indexed categories, Theory Appl. Categ. 28 (2013) 616-695 [arXiv:1212.3914, tac:28-21]