Other Monad Strengths

Last time, we saw the notion of strength for a monad. These were natural transformations

A \otimes \mathbb{T}(B) \rightarrow \mathbb{T}(A \otimes B)

which interact well with the monoidal and monad structure. As was mentioned last time, there are other ways we might consider commuting a functor with a monoidal structure. The obvious first choice is to consider a natural transformation of type

\mathbb{T}(A) \otimes B \rightarrow \mathbb{T}(A \otimes B)

again, interacting sensibly with the monoidal and monad structure (as before, we leave the details of what this means aside, to concentrate on the main ideas). Such a natural transformation is called a left-strength, denoted \mathsf{st}'. The strength is often referred to as a right-strength for emphasis.

Example: If \mathcal{V} is a symmetric monoidal category we can build a right-strength from a left-strength

\mathsf{st}_{A,B} : A \otimes \mathbb{T}(B) \rightarrow \mathbb{T}(A \otimes B)

as the composite:

\mathbb{T}(A) \otimes B \xrightarrow{\sigma_{\mathbb{T}(A),B}} B \otimes \mathbb{T}(A) \xrightarrow{\mathsf{st}_{B,A}} \mathbb{T}(B \otimes A) \xrightarrow{\mathbb{T}(\sigma_{B,A})} \mathbb{T}(A \otimes B)

Here \sigma is the symmetry isomorphism from the symmetric monoidal structure. Intuitively, we just swap the inputs, use the strength and swap them back again.

As we know every \mathsf{Set} monad is strong in a unique way, so it also has a unique left-strength. For example, the left-strength for the list monad acts as follows:

([a_1,\ldots,a_n],b) \mapsto [(a_1,b),\ldots,(a_n,b)]

Unsurprisingly, the left-strength for the powerset is very similar, visually we just swap square (list formation) brackets for curly (set formation) brackets.

(\{a_1,\ldots,a_n\},b) \mapsto \{(a_1,b),\ldots,(a_n,b)\}

We can consider left-strengths beyond symmetric monoidal categories, as their definition does not require the existence of a right-strength. In the cases that will be of interest to us, it will be the interplay between left and right strengths that will matter. To keep things simple, we will assume we are working in a symmetric monoidal category, so the left-strength can be derived from the right-strength (or vice-versa), as in the example above. (To move beyond this setting, we would need to assume further compatibility conditions between any choice of left and right strength, which are automatic when one is derived from the other via a monoidal symmetry.)

Once we have a right-strength \mathsf{st} and a left-strength \mathsf{st}', we can build a natural transformation of type \mathbb{T}(A) \otimes \mathbb{T}(B) \rightarrow \mathbb{T}(A \otimes B). In fact, there are two ways we can do this, either by applying the right-strength first:

\mathsf{dst} = \mathbb{T}(A) \otimes \mathbb{T}(B) \xrightarrow{\mathsf{st}_{A,B}} \mathbb{T}(\mathbb{T}(A) \otimes B) \xrightarrow{\mathbb{T}(\mathsf{st}'_{A,B})} \mathbb{T}^2(A\otimes B) \xrightarrow{\mu_{A \otimes B}} \mathbb{T}(A \otimes B)

or applying the left-strength first:

\mathsf{dst}' = \mathbb{T}(A) \otimes \mathbb{T}(B) \xrightarrow{\mathsf{st}'_{A,B}} \mathbb{T}(A \otimes \mathbb{T}(B)) \xrightarrow{\mathbb{T}(\mathsf{st}_{A,B})} \mathbb{T}^2(A \otimes B) \xrightarrow{\mu_{A \otimes B}} \mathbb{T}(A \otimes B)

Both these composites are referred to as double strengths. Whenever we see two composites of the same type in category theory, it is natural to consider if they are always the same.

Example: For the list monad, we have:

\mathsf{dst}([a,b],[c,d]) = [(a,c),(b,c),(a,d),(b,d)]

whereas

\mathsf{dst}'([a,b],[c,d]) = [(a,c),(a,d),(b,c),(b,d)]

notice the difference in the order of the middle two elements.

On the other hand, for the powerset monad, essentially because there is no ordering in subsets:

\mathsf{dst}(\{a,b\},\{c,d\}) = \mathsf{dst}'(\{a,b\},\{c,d\}) = \{(a,c),(b,c),(a,d),(b,d) \}

This is not an isolated example, for the powerset monad \mathsf{dst} = \mathsf{dst}'.

The previous examples show us that in general \mathsf{dst} \neq \mathsf{dst}', but the two composites may agree for some monads. This question, and its implications are another key topic in the theory of monads. We shall begin exploring the details in the next post.

A Note on Terminology

An earlier version of this post used the term costrength for what is termed a left-strength above. This terminology does not align well with the conventional use of “co” in category theory. Thanks to @varkor on mathstodon for persuading me that I should move to the more modern terminology.

There is also a discussion about this terminology on the NLab.

3 thoughts on “Other Monad Strengths”

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