On Lax Monoidal Functors
Published:
What is the difference between
- a lax monoidal functor
- a monoid in a Day-monoidal category
- a morphism of lax-algebras for the free monoid 2-monad, and
- a codistributive law with the tensor product?
Well, None. Let’s see why.
To keep this post as concise as humanly possible I will assume knowledge of (symmetric)monoidal categories, kan extensions and enriched categories.
We show informally the following proposition.
Proposition. Let \((\mathcal{C}, \otimes_{\mathcal{C}}, I_{\mathcal{C}})\) be a small monoidal closed category enriched in a monoidal closed category \((\mathcal{D}, \otimes_{\mathcal{D}}, I_{\mathcal{V}})\) and let \(F : \mathcal{C} \to \mathcal{D}\) be a functor. The following statements for \(F\) are equivalent:
- It is a lax monoidal functor
- It is a monoid in the monoidal category \(([\mathcal{C}, \mathcal{D}], \otimes_\text{Day}, y(I_{\mathcal{C}}))\)
- It is a homomorphism of lax algebras for the free monoid 2-monad
It is a \(\mathbb{N}\)-indexed family of (co)distributive laws for a functor \(F : \mathcal{C} \to \mathcal{C}\)
\[\text{Nat}(\otimes^{n} \circ F^{n}, F \circ \otimes^{n})\]where \(\otimes^{n} : \mathcal{C}^{n} \to \mathcal{C}\)
Let us assume the hypothesis of the proposition.
Proof(Sketch). (1) \(\Leftrightarrow\) (2).
A lax monoidal functor is a functor which lax-preserves the monoidal structure of \(\mathcal{C}\) that is, there is a morphism
\[u : I_{\mathcal{D}} \to F I_{\mathcal{C}}\]and a family of morphisms
\[\circledast_{X,Y} : F X \otimes_{\mathcal{D}} F Y \to F (X \otimes_{\mathcal{C}} Y)\]indexed by \(X,Y\) and natural therein, subject to some coherence conditions.
On the other hand, the Day convolution provides a natural way to define a monoidal structure on the category of functors. In other words, the task is to turn the category of functors \([\mathcal{C}, \mathcal{D}]\) into a monoidal category by equipping it with a tensor product and a unit. Hence, for two functors \(F, G : \mathcal{C} \to \mathcal{D}\) the Day convolution \(\otimes_\text{Day}\) is defined as follows:
\[\begin{align*} (F \otimes_\text{Day} G) C & := \int^{X,Y \in \mathcal{C}} \mathcal{C}(X \otimes_{\mathcal{C}} Y, C) \otimes_{\mathcal{D}} F X \otimes_{\mathcal{D}} G Y\\ & = \text{Lan}_{\otimes_{\mathcal{C}}}(\otimes_{\mathcal{D}} \circ F \times F) \end{align*}\]while the unit of \([\mathcal{C}, \mathcal{D}]\) is given by the Yoneda embedding applied to the unit \(I_\mathcal{C}\) that is \(y(I_\mathcal{C}) = \mathcal{C}(I_\mathcal{C},-)\).
Now, a monoid in \(([\mathcal{C}, \mathcal{D}], \otimes_\text{Day}, y(I))\) is called a Day-monoid. This is a functor \(F : \mathcal{C} \to \mathcal{D}\) together with a unit and multiplication map.
- The unit map \(\eta : y(I) \to F\) is obtained from the unit of the lax monoidal functor (and viceversa) via the enriched Yoneda lemma
- The multiplication map \(\mu : F \otimes_{\text{Day}} F \to F\) is obtained from \(\circledast\) (and viceversa) by using the adjunction \(\text{Lan}_J \dashv - \circ J\) as follows
It remains to prove that the laws of the unit and multiplication of the monoid imply the lax monoidal properties of \(u\) and \(\circledast\) (left as exercise to the reader).
\((2) \Leftrightarrow (3)\).
This is a rather easy statement which generalises the free monoid construction to 2-categories.
In particular, the cheapest way of turning a set \(A\) into a monoid is to take the set of words over \(A\), namely \(A^*\). This is the free monoid over \(A\) where the empty word is the unit and concatenation is the multiplication of the monoid. The Eilenberg-Moore algebras of \(A^*\) are equivalent to the algebraic structure of the monoid \(A^*\).
In particular, the category of Eilenberg-Moore algebras over \(A^*\) is equivalent to the category of monoids
Similarly, given a category \(\mathcal{C}\), the cheapest way of turning this category into a monoid (a monoidal category) is to send \(\mathcal{C}\) to the category of finite sequences of objects \((A_1, \dots, A_n)\) and componentwise sequences of morphisms in \(\mathcal{C}\). In other words, \(T\) is the free monoid 2-monad in \(\textbf{Cat}\) defined as the \(\mathbb{N}\)-coproduct \(\mathcal{C}^n\), that is
\[T\mathcal{C} = \sum_{n : \mathbb{N}} \mathcal{C}^{n}\]Now, similarly to what happens in the 1-category case, we have the following equivalence
\[\textbf{Cat}^T \simeq 2\text{-Mon}\]where \(\textbf{Cat}^T\) is the 2-category of algebras for a 2-monad \(T\) and \(T\)-algebra homomorphisms and \(2\)-Mon is the 2-category of monoidal categories and monoidal functors (monoids in \(\textbf{Cat}\)). Hence (lax) \(T\)-algebra homomorphisms are (lax) monoidal functors.
\((3 \Leftrightarrow 4)\).
Clearly, if \(T\) is the free monoid 2-monad, an algebra for \(T\) is a map
\[a : \sum_{n : \mathbb{N}} \mathcal{C}^n \to \mathcal{C}\]The previous point states that this is a monoidal category where \(A \otimes_\mathcal{C} B := a (A,B)\) and \(I_\mathcal{C} = a()\), thus \(a\) sends \((A_1, \dots, A_n)\) to \(A_1 \otimes_\mathcal{C} \dots \otimes_\mathcal{C} A_n\).
A lax monoidal functor \(F\) is a lax \(T\)-algebra homomorphism, thus it has to satisfy
\[F(A_1) \otimes_\mathcal{D} \dots \otimes_\mathcal{D} F(A_n) \to F(A_1 \otimes_\mathcal{C} \dots \otimes_\mathcal{C} A_n)\]which is defined at all \(n\) and \(A_i\) hence it is a (co)distributive law
\[\text{Nat}(\otimes^{n}_{\mathcal{D}} \circ F^{n}, F \circ \otimes^{n}_{\mathcal{C}})\]