Tag: math

A monad is just a one object enriched category

We have seen that the notion of monad can be interpreted in any bicategory. The aim of todays post is to explain that for a bicategory \mathcal{W}, a monad in \mathcal{W} is the same thing as a one-object \mathcal{W}-enriched category. Or more tersely:

A monad is the same thing as a one-object \mathcal{W}-category.

Mostly, this is just a case of understanding the definitions, with no complicated translation between the two structures required.

Monads and enrichment

Recall that for a monoidal category (\mathcal{V}, \otimes, I), a \mathcal{V}–enriched category, or \mathcal{V}-category \mathcal{A} is a generalisation of ordinary categories with:

  1. A collection of objects X,Y,\ldots.
  2. For every pair of object X,Y, a hom object \mathcal{A}(X,Y).
  3. For each object X, an identity \mathcal{V}-morphism j_X : I \rightarrow \mathcal{A}(X,X).
  4. For each triple of objects X,Y,Z, a composition \mathcal{V}-morphism m_{X,Y,Z} : \mathcal{A}(Y,Z) \otimes \mathcal{A}(X,Y) \rightarrow \mathcal{A}(X,Z).

These are subject to some natural axioms such that composition is associative, and unital with respect to the chosen identities. The motivating special case is that a \mathsf{Set}-enriched category is the same thing as an ordinary category.

It is a well-known fact of enriched category theory that a monoid in the monoidal category \mathcal{V} is the same thing as a one object \mathcal{V}-category. The even better known special case, which crops up in most introductions to category theory, is that a one object ordinary category is the same thing as a monoid.

Of course, the special case that we should be interested in as monad theorists is the monoidal category of endofunctors ([\mathcal{C}, \mathcal{C}], \circ, \mathsf{Id}_{\mathcal{C}}). Using the fundamental meme of monad theory, that a monad is just a monoid in the category of endofunctors, we can deduce that monads on \mathcal{C} are the same thing as one object [\mathcal{C},\mathcal{C}]-enriched categories. This claim works equally well if we consider monads in an arbitrary bicategory.

Example: We have seen previously that for a monoidal category \mathcal{V} with coproducts, a \mathcal{V}-enriched category is the same thing as a monad in the bicategory of \mathcal{V}-matrices, \mathbf{Mat}(\mathcal{V}). Applying the observation above, the following all describe the same data:

  1. A \mathcal{V}-enriched category with set of objects O.
  2. A monad on O in \mathbf{Mat}(\mathcal{V}).
  3. A one-object \mathbf{Mat}(\mathcal{V})(O,O)-enriched category.

Which is a bit of a funny conclusion, every multi-object enriched category is the same thing as a single-object enriched category over a different base. In particular, every ordinary (small) category with set of objects O is a one-object \mathbf{Mat}(\mathsf{Set})(O,O)-enriched category.

In the example above, it feels a bit clumsy to have to keep saying “…with set of objects O…”. To clean this up, and take this story a bit further, we are going to have to generalise our notion of enriched category, to categories enriched over a bicategory \mathcal{W}. Although this may sound a bit intimidating, it is actually only a small step beyond enrichment in a monoidal category.

A \mathcal{W}-enriched category \mathcal{A} consists of:

  1. A collection of objects X,Y,Z, each with an associated extent, given by a 0-cell \mathsf{ext}(X) in \mathcal{W}.
  2. For every pair of objects X,Y, a hom 1-cell \mathcal{A}(X,Y) : \mathsf{ext}(X) \rightarrow \mathsf{ext}(Y).
  3. For every object X, an identity 2-cell j_A : \mathsf{Id}_{\mathsf{ext}(X)} \Rightarrow \mathcal{A}(X,X).
  4. For every triple of objects X,Y,Z, a composition 2-cell m_{X,Y,Z} : \mathcal{A}(Y,Z) \circ \mathcal{A}(X,Y) \Rightarrow \mathcal{A}(X,Z).

As before, this data is subject to natural unitality and associativity axioms.

With this definition in place, naming things suggestively, a one object \mathcal{W}-category consists of:

  1. An object, with extent a 0-cell \mathcal{C} in \mathcal{W}.
  2. A single hom 1-cell \mathbb{T} : \mathcal{C} \rightarrow \mathcal{C}.
  3. An identity 2-cell \eta : \mathsf{Id} \Rightarrow \mathbb{T}.
  4. A single composition 2-cell \mu : \mathbb{T} \circ \mathbb{T} \Rightarrow \mathbb{T}.

This data satisfies exactly the axioms such that (\mathbb{T} : \mathcal{C} \rightarrow \mathcal{C}, \eta, \mu) is a monad. That is:

A monad in \mathcal{W} is the same thing as a one-object \mathcal{W}-category.

We rephrase the previous example in this more flexible setting.

Example: For a monoidal category \mathcal{V}, a \mathcal{V}-category is a one object \mathbf{Mat}(\mathcal{V})-enriched category.

A possibly more interesting example is as follows.

Example: We have seen previously that for a category \mathcal{C} with pullbacks, an internal category in \mathcal{C} is the same thing as a monad in the bicategory of spans \mathbf{Span}(\mathcal{C}). Now applying our previous observation, the following all describe the same data:

  1. An internal category in \mathcal{C}.
  2. A monad in \mathbf{Span}(\mathcal{C}).
  3. A one-object \mathbf{Span}(\mathcal{C})-enriched category.

This establishes a slightly surprising connection between internal and enriched category theory.

Conclusion

To an extent, this post is about almost trivial relationships between definitions that happen to coincide. This is not just an exercise in categorical showing-off or pointless abstraction. The relationship between monads and one-object enriched categories is really a matter of perspective. This may allow us to relate monads to other concepts, for example finitary monads and Lawvere theories can be connected in this way. Once monads are viewed as categories, we can consider categorical notions such as completion under certain limits or colimits, which again occur in connection with Lawvere theories. These ideas are used to startling effect in the work of Richard Garner and co-authors, which I highly recommend as further reading.

Algebras are regular quotients of free algebras

The aim of this post is to show that for a monad

(\mathbb{T} : \mathcal{C} \rightarrow \mathcal{C}, \eta : \mathsf{Id}_{\mathcal{C}} \Rightarrow \mathbb{T}, \mu : \mathbb{T}^2 \Rightarrow \mathbb{T}),

every Eilenberg-Moore algebra (A,\alpha : \mathbb{T}(A) \rightarrow A) is given as the coequalizer of a parallel pair of morphisms between free algebras in \mathcal{C}^{\mathbb{T}}. In this case, we say that (A,\alpha) is a regular quotient of free algebras. Notice we are not making any any assumptions about the base category \mathcal{C} such as (co)completeness, so we don’t have a lot to work with. Really the only available structure is:

  1. The unit and multiplication of the monad, and the axioms they satisfy.
  2. The two axioms that every Eilenberg-Moore algebra satisfies relating its structure map to the unit and multiplication.

So we have a handful of morphisms, and some equations that they satisfy, and that’s it. Let’s have a look at how this magic trick is performed.

A parallel pair of algebra morphisms

As a first step, we note that for an Eilenberg-Moore algebra (A,\alpha), we have a parallel pair of \mathcal{C}^{\mathbb{T}} morphisms:

\mu_A, \mathbb{T}(\alpha) : F^{\mathbb{T}}(\mathbb{T}(A)) \rightarrow F^{\mathbb{T}}(A)

Expanding the action of the free algebra functor, this is a pair of morphisms:

\mu_A, \mathbb{T}(\alpha) : (\mathbb{T}^2(A), \mu_{\mathbb{T}^2(A)}) \rightarrow (\mathbb{T}(A), \mu_A)

That \mu_A is an algebra morphism is equivalent to the Eilenberg-Moore algebra multiplication axiom. That \mathbb{T}(\alpha) is an algebra morphism is simply naturality of the monad multiplication.

By naturality of \mu, we note that there is an algebra morphism in the opposite direction:

\mathbb{T}(\eta_A) : (\mathbb{T}(A), \mu_A) \rightarrow (\mathbb{T}^2(A), \mu_{\mathbb{T}^2(A)})

Furthermore, by the monad right unitality axiom

\mu_A \cdot \mathbb{T}(\eta_A) = \mathsf{id}_{\mathbb{T}(A)}

and by the algebra unit axiom:

\mathbb{T}(\alpha) \cdot \mathbb{T}(\eta_A) = \mathsf{id}_{\mathbb{T}(A)}

Therefore these three morphisms form a reflexive pair.

A coequalizer in the base category

We now apply the forgetful functor U^{\mathbb{T}} : \mathcal{C}^{\mathbb{T}} \rightarrow \mathcal{C}, yielding a parallel pair.

U^{\mathbb{T}}(\mu_A), U^{\mathbb{T}}(\mathbb{T}(\alpha)) : U^{\mathbb{T}} \circ F^{\mathbb{T}}(\mathbb{T}(A)) \rightarrow U^{\mathbb{T}}(\alpha) : U^{\mathbb{T}} \circ F^{\mathbb{T}}(A)

which if we unpack the definitions is simply a parallel pair of \mathcal{C}-morphisms:

\mu_A, \mathbb{T}(\alpha) : \mathbb{T}^2(A) \rightarrow \mathbb{T}(A)

Our current aim is to find a coequalizer of this pair, knowing it must have codomain A. The obvious choice is to consider

\alpha : \mathbb{T}(A) \rightarrow A

as our candidate universal coequalizer morphism. By the algebra multiplication axiom, we have

\alpha \cdot \mu_A = \alpha \cdot \mathbb{T}(\alpha)

which is an encouraging first step to establishing this forms a coequalizer diagram. To establish the universal property, we are going to need a bit more. Using components of the monad unit, we get two other useful \mathcal{C}-morphisms:

  1. \eta_A : A \rightarrow \mathbb{T}(A).
  2. \eta_{\mathbb{T}(A)} : \mathbb{T}(A) \rightarrow \mathbb{T}^2(A).

By the algebra unit axiom

\alpha \cdot \eta_A = \mathsf{id}_A

and by the monad left unitality axiom

\mu_A \cdot \eta_{\mathbb{T}(A)} = \mathsf{id}_{\mathbb{T}(A)}

Finally, by naturality:

\mathbb{T}(\alpha) \cdot \eta_{\mathbb{T}(A)} = \eta_A \cdot \alpha.

We have shown the our parallel pair form a contractible coequalizer in the base category, and further that the parallel pair of algebra morphism:

\mu_A, \alpha : F^{\mathbb{T}}(\mathbb{T}(A)) \rightarrow F^{\mathbb{T}}(A)

form a reflexive U^{\mathbb{T}}-contractible pair.

An algebra coequalizer

We would now like to conclude we have a coequalizer in the Eilenberg-Moore category. Recall that the forgetful functor U^{\mathbb{T}} : \mathcal{C}^{\mathbb{T}} \rightarrow \mathcal{C} creates colimits that are preserved by \mathbb{T} and \mathbb{T}^2. As contractible coequalizers are absolute colimits, we can apply this result to lift the coequalizer to the Eilenberg-Moore category.

Finally, we note that \alpha is in fact an algebra morphism of type:

F^{\mathbb{T}}(A) \rightarrow (A,\alpha)

by the Eilenberg-Moore algebra multiplication axiom. Therefore

F^{\mathbb{T}}(\mathbb{T}(A)) \xrightarrow{ \mu_A, \alpha} F^{\mathbb{T}}(A) \xrightarrow{\alpha} (A,\alpha)

is a coequalizer diagram in the Eilenberg-Moore category, and (A,\alpha) is a regular quotient of free algebras as claimed.

Conclusion

We encountered reflexive U-contractible coequalizers in the statement of Beck’s monadicity theorem. In this post, we have seen one example of why these particular absolute colimits are important in the theory of monads, as they can be used to construct every Eilenberg-Moore algebra as a quotient of a free algebra.

Proving this result from very few assumptions, beyond some morphisms satisfying certain equations, leads us to the construction of coequalizers which are defined by by equations between morphisms. Such constructions are necessary absolute, and this at least partially explains the significance of absolute colimits in this context.