Connected monads weakly preserve products

If $F$ is a (not necessarily associative) monad on $Set$, then the natural transformation $F(A\times B)\to F(A)\times F(B)$ is surjective if and only if $F(\boldsymbol{1})=\boldsymbol{1}$. Specializing $F$ to $F_{\mathcal{V}}$, the free algebra functor for a variety $\mathcal{V}$, this result generalizes and clarifies an observation by Dent, Kearnes and Szendrei.


Introduction
A key observation in [2] by T. Dent, K. Kearnes, and Á. Szendrei is that for any variety V with idempotent operations each set theoretic product decomposition d : {x, y, z, u} ։ {a, b} × {a, b} always extends to a surjective homomorphism from the 4-generated free algebra in V to the square of the 2-generated one. This fact has an interesting geometric interpretation, which is relevant in the study of congruence modularity. The shifting lemma from [6], which is concerned with shifting a congruence γ from one side of an α−β-parallelogram to the opposite side modulo α ∧ β, can be specialized to axis-parallel rectangles inside a product of algebras where α and β are in fact kernels of the projections and γ a factor congruence.
Surjectivity of the above map implies that the projections on the image commute, and since ker δ = α ∧ β, it follows that α and β also commute in the preimage. In particular, therefore, the shifting lemma, which in [6] is the major geometrical tool for studying congruence modularity, is only needed in situations of permuting congruence relations α and β. The restriction to idempotent varieties in these studies is not severe, since a variety is congruence modular iff its idempotent reduct is modular.
Variations of the shifting lemma (e.g. in [1]) and, more recently, categorical generalizations as in [3] suggest to investigate the situation in a more general context. In this note, therefore, rather than exploring further ramifications of the above observation, we explore the abstract reasons behind the surjectivity of δ in (1.1).
I am sincerely indebted to Peter Jipsen and Andrew Moshier for inspiring discussions during my stay at Chapman University, where the main result of this note was obtained.
It turns out that we can deal with this in a framework which is more abstract than universal algebras and varieties. We are rather considering (not necessarily associative) Set-monads F , of which the functor F V , associating with a set X the free algebra F V (X) and with a map g : X → Y its homomorphic extension g : F V (X) → F V (Y ), is just an example.

Monads and main result
Monads on a category C are functors F : C → C together with two natural transformations ι : Id → F and µ : F • F → F , satisfying two unit laws and and an associative law. Our results will even hold for nonassociative monads, so skippping the associative law, we shall only state the unit laws: Equations (2.1) are usually expressed as a commutative diagram: Rather easy examples of monads on the category of Sets are obtained from collection data types in programming, such as List X , Set X or T ree X , see also [10]. In popular programming languages, List X denotes the type of lists of elements from a base type X. Given a function g : X → Y , the function map(g) : List X → List Y which sends [x 1 , ..., x n ] ∈ List X to the list [g(x 1 ), ..., g(x n )] ∈ List Y represents the action of the functor List on maps. In mathematical notation we write (List g) rather than map(g). Obviously, map(f • g) = map(f ) • map(g) and map(id X ) = id List X , so the pair List − with map indeed establishes a functor.
For List to be a monad, we need a natural transformation ι : Id → List, as well as a "multiplication" µ : List • List → List. The former can be chosen as the singleton operator with ι X : X → List X sending any x ∈ X to the one-element list [x].
The monad multiplication µ is for each type X defined as which is obvious. Not all monads arise from collection classes, and in recent years other uses of monads have all but revolutionized functional programming, see e.g. [11] or [14]. Relevant for universal algebraists is the fact that for every variety V the construction of the free Algebra F V (X) over a set X is a monadic functor. In this case, ι X : X → F V (X) is the inclusion of variables, or rather their interpretations as V−terms.
The defining property of F V (X) states that each map g : X → A for A ∈ V has a unique homomorphic extensionḡ : From a map f : X → Y , we therefore obtain the homomorphism can be considered as term composition: a term t(t 1 , ..., t n ), whose argument positions have been filled by other terms, is interpreted as an honest V-term. To make this precise, consider the diagram below, in which F V (X) appears in two roles -as an algebra and as a set of free variables for F V (F V (X)). (In the diagram we have dropped the lower indices to ι and id for the sake of readibility.) Here µ X is defined as the homomorphic extension of the equality map id FV (X) from F V (X), considered as set of free variables for F V (F V (X)), to F V (X) considered as V-algebra.
The first monad equation immediately follows from the definition of µ, and the second equation follows from the calculation demonstrating that both sides agree on the generators of F V (X), and consequently on all of F V (X).
The above mentioned examples T ree X , List X and Set X just correspond to the free groupoid, the free semigroup, and the free semilattice over the set X of generators, and are themselves instances of this scheme.
We are now ready to state our main result: It will be easy to see (lemma 12 below) that F weakly preserves the product A 1 × A 2 if and only if the canonical morphism δ = (F π 1 , F π 2 ) in the below diagram is epi: The starting point of our discussion, (1.1) from [2], is therefore seen to represent an instance of this result when setting A 1 = A 2 = {a, b} and F = F V . But before coming to its proof we need a few preparations.

Connected Functors
Put 1 = {0} and for any set X denote by ! X the unique (terminal) map from It is well known, see [13], that every Set-Functor F can be constructed as sum of connected functors: In the following we denote by c X y : X → Y or, if X is clear, simply by c y the constant map with value y ∈ Y. We shall need the following lemma: ιY (y) . Proof. For y ∈ Y, denote byȳ : 1 → Y the constant map with value y. Observe, that an arbitrary map f is constant if and only if it factors through 1, i.e. c X y =ȳ•! X . Applying F and adding the natural transformation ι into the picture, we obtain: ιY (y) .
In the above, we have seen, that connected functors preserve constant maps. It might be interesting to remark, that this very property characterizes connected functors:

Corollary 3. A functor F is connected if and only if for every constant morphism
In general, the elements of F (1) correspond uniquely to the natural transformations between the identity functor Id and F . This can be seen by instantiating the Yoneda Lemma with A = 1. Therefore we note: is connected if and only if ι is the only transformation from the identity functor to F . Definition 5. Let C 1 be the constant functor with C 1 (X) = 1 for all X and C 1 f = id 1 for all f . We say that a functor F possesses a constant, if there is a transformation from C 1 to F which is natural, except perhaps at X = ∅.
Clearly, each element of F (∅) gives rise to a constant, but not conversely, since there is nothing to stop us from changing F only on the empty set ∅ and on empty mappings ∅ X : ∅ → X by choosing any U ⊆ F (∅) and redefining F ′ (∅) := U as well as For that reason we were not requiring naturality at ∅ in the above definition.
We shall need a further observation: Lemma 6. A connected functor either possesses a constant or it has the identity functor as a subfunctor.
Proof. By the Yoneda-Lemma, there is exactly one natural transformation ι : Id → F . Assume that some ι X is not injective, then there are x 1 = x 2 ∈ X with ι X (x 1 ) = ι X (x 2 ). Given an an arbitrary Y with y 1 , y 2 ∈ Y, consider a map f : X → Y with f (x 1 ) = y 1 and f (x 2 ) = y 2 . By naturality, hence each ι Y is konstant and therefore factors through 1. This makes the upper and lower triangle inside the following naturality square commute, too. The left triangle commutes since 1 is terminal. If X = ∅, the terminal map ! X : X → 1 is epi, from which we now conclude that the right triangle commutes as well, except, possibly, when X = ∅. Thus F posseses a constant.

Preservation properties
We are concerned with the question, under which conditions the δ in equation (1.1) is epi. Therefore, we take a look at the canonical map δ = (F π 1 , F π 2 ) : where π i , resp η i , denote the canonical component projections.
The first thing to observe is: Proof. Assume f : A 1 → A ′ 1 and g : A 2 → A ′ 2 be given. We want to show that the following diagram commutes: Notice, that in order for δ to be surjective, the functor F must be connected or trivial: Lemma 8. If the canonical decomposition as in Theorem 1 is always epi, then either F (1) ∼ = 1 or F is the trivial functor with constant value ∅.
Next, recall some elementary categorical notions. Definition 9. Given objects A 1 , A 2 in a category C, a product of A 1 and A 2 is an object P together with morphisms p i : P → A i , such that for any "competitor", i.e. for any object Q with morphisms q i : Q → A i , there exists a unique morphism d : Q → P , such that q i = p i • δ for i = 1, 2. Products, if they exist, are unique up to isomorphism and are commonly written Similarly, given morphisms f 1 : X 1 → Y and f 2 : X 2 → Y with common codomain Y, their pullback is defined to be a pair of maps p 1 : P → X 1 and p 2 : P → X 2 with common domain P such that and for each "competitor", i.e. each object Q with morphisms q 1 : Q → X 1 and In both definitions, if we drop the uniqueness requirement, we obtain the definition of weak product, resp. weak pullback.
Notice that in case when there exists a terminal object 1, the product of A 1 with A 2 is the same as the pullback of the terminal morphisms ! Ai : A i → 1.
Weak products (weak pullbacks) arise from right invertible morphisms into products (pullbacks): Lemma 10. If (P, p 1 , p 2 ) is a product (resp. pullback), then (W, w 1 , w 2 ) is a weak product (resp. weak pullback) if and only if there is a right invertible w : W → P such that w i = p i • w.
Proof. If w has a right inverse e, and (Q, q 1 , q 2 ) is a competitor to W, then it is also a competitor to P, hence there is a morphism d : Q → P with q i = p i • d. Then e • d is the required morphism to W . Indeed, Conversely, assume that (W, w 1 , w 2 ) is a weak product, then both W and P are competitors to each other, yielding both a morphism w : W → P with w i = p i • w and a morphism e : P → W with p i = w i • e. Now (P, p 1 , p 2 ) is also acompetitor to itself, yet both p i • (w • e) = p i and p i • id P = p i for i = 1, 2. By uniqueness it follows, w • e = id P , so w is indeed right invertible. (The same proof works for the case of weak pullbacks).
Definition 11. Let F : C → D be a functor. We say that F weakly preserves products (pullbacks) if whenever (P, p 1 , p 2 ) is a product (pullback), then its image (F (P ), F p 1 , F p 2 ) is a weak product (weak pullback).
It is well known, that a functor weakly preserves a limit L, if and only it preserves weak limits, see e.g. [8]. By the axiom of choice, surjective maps are right invertible, so regarding (1.1) or its more general formulation (2.2), we now arrive at the following relevant observation: Lemma 12. The canonical map δ in (2.2) is epi if and only if F weakly preserves the product (A 1 × A 2 , π 1 , π 2 ).
Whereas the above mentioned result of [2], in which the monad F is the freealgebra-functor F V , served a purely universal algebraic purpose, it also has an interesting coalgebraic interpretation. It is well known, that coalgebraic properties of classes of F -coalgebras are to a large degree determined by weak pullback preservation properties of the functor F , which serves as a type or signature for a class Coalg F of coalgebras. Prominent structure theoretic properties can be derived from the assumptions that F weakly preserves pullbacks of preimages, kernel pairs or both, see e.g. [7], [8], [9], [4], [5], [12]. Here we add one more property to this list: preservation of pullbacks of constant maps.
Theorem 13. Let F be a nontrivial functor. Then the following are equivalent: (1) F has no constant and weakly preserves products (2) F is connected and weakly preserves pullbacks of constant maps.
Proof. If F is nontrivial and weakly preserves the product 1 × 1 ∼ = 1, then F is connected as a consequence of lemma 8. Since F has no constants, F (∅) = ∅ and moreover lemma 6 provides Id as a subfunctor of F . Thus we obtain a natural transformation ι : ιY (yi) for i = 1, 2. If y 1 = y 2 then the pullback of the c Xi yi is simply (X 1 × X 2 , π 1 , π 2 ). The F c Xi yi are constant maps with the same target value ι Y (y 1 ) = ι Y (y 2 ), so their pullback is the product F (X 1 ) × F (X 2 ) with canonical projections η i : F (X 1 ) × F (X 2 ) → F (X i ). By assumption, F weakly preserves products, which gives us a surjective canonical map δ : F (X 1 × X 2 ) → F (X 1 ) × F (X 2 ) with F π i = η i • δ, so lemma 10 assures that (F (X 1 × X 2 ), F π 1 , F π 2 ) is a weak pullback of the F c Xi yi . If y 1 = y 2 then the pullback of the c Xi yi is (∅, ∅ X1 , ∅ X2 ), the empty set ∅ with empty mappings ∅ Xi : ∅ → X i . Since ι Y is injective, the F c y1 are constant mappings, also with disjoint images, so their pullback is (∅, ∅ F (X1) , ∅ F (X2) ). This is the same we would obtain by applying F to the pullback of the c yi , taking into account that F (∅) = ∅.
For the reverse direction, suppose that F is connected and weakly preserves pullbacks of constant maps. The product (X 1 × X 2 , π 1 , π 2 ) is at the same time the pullback of the terminal maps ! Xi : X i → 1. Applying F and considering that F (1) ∼ = 1, we see that the F ! Xi are also terminal maps, so their pullback is (F (X 1 )×F (X 2 ), η 1 , η 2 ). Thus, if F weakly preserves the pullback of the ! Xi we must have that (F (X 1 × X 2 ), F π 1 , F π 2 ) is a weak pullback of the F ! Xi which by lemma 10 means that there exists a surjective map δ : The following example shows that the requirement that "F has no constants" is essential in theorem 13.

Example 14.
Consider the functor T with T (X) = X 2 /∆ where ∆ is the equivalence relation on X 2 identifying any two elements in the diagonal of X 2 . For x 1 , x 2 ∈ X, we denote the elements of X 2 /∆ by (x 1 , x 2 ) if x 1 = x 2 and by ⊥ otherwise. On maps f : X → Y the functor T is defined as (T f )(⊥) = ⊥ and . Then T is a functor and the projection π ∆ : X 2 → X 2 /∆ is a natural transformation. Even though T (∅) = ∅, the functor does have a constant, ⊥.
To see that T does not weakly preserve pullbacks of constant maps, consider and their pullback is T (X) × T (X). Clearly there is no way to find a surjective map from T (∅) = ∅ to T (X) × T (Y ) as would be required by lemma 10.

Proof of the main theorem
We are finally turning to the proof of theorem 1, verifying the surjectivity of δ = (F π 1 , F π 2 ) when (F, ι, µ) is a monad. Thus given (p, q) ∈ F (A 1 ) × F (A 2 ), we are required to find an element t ∈ F (A 1 × A 2 ) such that (F π 1 )(t) = p and (F π 2 )(t) = q.