What is the functor tensor product?

Question

I'm familiar with the tensor product of modules, but I've also come across functor tensor product (in emily riehls paper on homotopy limits), what are they, and how are they (if they are) related to traditional tensor products? (Emily shows that they can be defined as a particular coend, but that doesn't really provide any intuition for me).

Answer 1

It is easy to be explicit. Not in full generality, given a (small) closed symmetric monoidal category $\mathcal C$ with coequalizers, a covariant functor $M\colon \mathcal C\to \mathcal C$ and a contravariant functor $N\colon \mathcal C \to \mathcal C$, the tensor product $N\otimes_{\mathcal C} M$ is the coequalizer of the diagram

$$\coprod_{(c,d)} N(d) \otimes \mathcal C(c,d) \otimes M(c) \implies \coprod_{e} N(e)\otimes M(e).$$

Here $c,d,e$ range over the objects of $\mathcal C$ and $\implies$ indicates a pair of arrows; one is given by the evaluation maps $N(d) \otimes \mathcal C(c,d)\longrightarrow N(c)$ of $N$ and the other by the evaluation maps $\mathcal C(c,d)\otimes M(c) \longrightarrow M(d)$ of $M$. The similarity to Mike's special case should be clear. This is of course an example of a coend, but I prefer to use the tensor product notation in this special case to make the intuition clear.

Answer 2

The enriched version of the functor tensor product does literally generalize the tensor product of modules. A ring is the same as an $\mathbf{Ab}$-enriched category with one object, a (covariant or contravariant) $\mathbf{Ab}$-enriched functor from a ring to $\mathbf{Ab}$ is a (left or right) module, and the $\mathbf{Ab}$-enriched tensor product of such functors is exactly the classical tensor product of a left and a right module.

Answer 3

An intuition that I find useful is the following.

Recall that a presheaf $F : \mathcal{C}^\mathrm{op} \to \mathrm{Set}$ can be seen as "gluing specification": If $G : \mathcal{C} \to \mathcal{D}$ is some functor into a cocomplete category $\mathcal{D}$, this gluing specification can be realized as $\operatorname{colim}_{s \in F(X)} G(X)$. This colimit can also be written as the coend $$ \int^{X \in \mathcal{C}} F(X) \cdot G(X), $$ where ${\cdot} : \mathrm{Set} \times \mathcal{D} \to \mathcal{D}$ denotes the copower, i.e. $M \cdot Y = \coprod_{m \in M} Y$. This is precisely the functor tensor product $F \otimes G$!

Summarizing, $F \otimes G$ can be pictured as the $G(X)$'s, glued as specified by $F$. A few examples are interesting.

• Let $Y : \mathcal{C} \to \mathrm{PSh}(\mathcal{C})$ be the Yoneda embedding. Then $F \otimes Y = F$. (This result is also called the ninja Yoneda lemma in some circles.) Intuitively, $\mathrm{PSh}(\mathcal{C})$ is the category of gluing specifications, so "realizing" $F$ in this category just gives $F$.
• Let $\mathcal{C}$ be specifically the simplex category $\Delta$. Then $F$ is just a simplicial set, so the intuition of $F$ as a gluing specification is even more vivid. Let $G : \Delta \to \mathrm{Top}$ be the functor which sends $n$ to the topological $n$-simplex $\Delta^n$. Then $F \otimes G$ interprets the gluing specification $F$ in $\mathrm{Top}$; the result is the geometrical realization $|F|$.

Answer 4

A very comprehensive account on the tensor product of functors can be found in Section 2.4 of the paper "The fundamental pro-groupoid of an affine 2-scheme" by Alex Chirvasitu and Theo Johnson-Freyd (arXiv).

Answer 5

Familiar examples of modules can be expressed in terms of categories. For example:

• If $\mathcal{G}$ is a one-object groupoid, then functors $\mathcal{G}^\circ \to \mathbf{Set}$ are the same thing as right actions of the group $\hom_{\mathcal{G}}(*,*)$ on sets.
• If $\mathcal{R}$ is a one-object additive category, then additive functors $\mathcal{R} \to \mathbf{Ab}$ are the same thing as left modules over the ring $\hom_{\mathcal{R}}(*,*)$

Generalizing, we can view any functor $\mathcal{C} \to \mathcal{D}$ as a left action of $\mathcal{C}$ (on objects of $\mathcal{D}$). Similarly, functors $\mathcal{C}^\circ \to \mathcal{D}$ are right actions. The most common example is actions on sets; i.e. $\mathcal{D} = \mathbf{Set}$.

Let there be a presheaf $P : \mathcal{C}^\circ \to \mathbf{Set}$, a functor $F : \mathcal{C} \to \mathbf{Set}$, and a set $S$.

As with other sorts of modules, there is a "hom-tensor" adjunction: there is a functor $\otimes_\mathcal{C} : \mathbf{Set}^{\mathcal{C}^\circ} \times \mathbf{Set}^{\mathcal{C}} \to \mathbf{Set} $ such that there is a natural bijection

$$\hom_\mathbf{Set}(P \otimes_\mathcal{C} F, S) \cong \hom_{\mathbf{Set}^{\mathcal{C}^\circ}}(P, \hom_{\mathbf{Set}}(F-, S))$$

We can also characterize the tensor product by the facts. Let $\mathbf{y} : \mathcal{C} \to \mathbf{Set}^{\mathcal{C}^\circ}$ be the yoneda embedding. Then,

• $P \otimes_\mathcal{C} F$ preserves colimits in both variables
• There is a natural bijection $\mathbf{y}(C) \otimes_\mathcal{C} F \cong F(C)$

One of the important intuitions about presheaves are that they are formal colimit diagrams. If $P$ is written as a colimit of representables, then we can see that

$$ P \otimes_\mathcal{C} F \cong \left( \operatorname{colim}_j \mathbf{y}(C_j) \right) \otimes_\mathcal{C} F \cong \operatorname{colim}_j F(C_j) $$

This meshes with other tensor product intuitions; for example, you can imagine $P$ as a "free" colimit, and $- \otimes_\mathcal{C} F$ specializes the formal generators $\mathbf{y}(C_j)$ to the special values $F(C_j)$.

The story generalizes almost unchanged to functors $F : \mathcal{C} \to \mathcal{M}$ for a locally small, cocomplete $\mathcal{M}$. You still have a functor $\otimes_\mathcal{C} : \mathbf{Set}^{\mathcal{C}^\circ} \times \mathcal{M}^{\mathcal{C}} \to \mathbf{Set} $ and a hom-tensor adjunction

$$\hom_\mathcal{M}(P \otimes_\mathcal{C} F, M) \cong \hom_{\mathbf{Set}^{\mathcal{C}^\circ}}(P, \hom_{\mathcal{M}}(F-, M))$$

you still have the properties

• $P \otimes_\mathcal{C} F$ preserves colimits in both variables
• There is a natural bijection $\mathbf{y}(C) \otimes_\mathcal{C} F \cong F(C)$

and $P \otimes_\mathcal{C} F$ can be interpreted as a colimit computation in $\mathcal{M}$.

These stories can be generalized to the enriched setting; my impression that it's easier to see how various constructions should go when they're phrased in terms of homs and tensors rather than in other terms.