Product preservation and stable units for reflections into idempotent subvarieties

We give a necessary and sufficient condition for the preservation of finite products by a reflection of a variety of universal algebras into an idempotent subvariety. It is also shown that simple and semi-left-exact reflections into subvarieties of universal algebras are the same. It then follows that a reflection of a variety of universal algebras into an idempotent subvariety has stable units if and only if it is simple and the above-mentioned condition holds.


Introduction
Throughout this paper, by a reflection we mean a reflection H I : C → M, with unit η : 1 C → HI, of a finitely complete category C into a full subcategory M of it, that is, a left-adjoint I of a full embedding H : M → C into a category with finite limits. T will always denote a terminal object of a given category C; for instance, if C is a variety of universal algebras, then T is any one-element algebra in C.
Simple reflections, semi-left-exact reflections and reflections having stable units were originally introduced in [3], as reflections preserving certain pullbacks. An additional structure on a reflection was described in [10], involving a pullback-preserving functor U : C → S (in particular, S can be the category of sets), allowing to simplify those preservation conditions by reducing them to the preservation of very special pullbacks.
The categorical version of monotone-light factorization for continuous maps of compact Hausdorff spaces was obtained in [2]. The results on the reflection of semigroups into semilattices obtained in [6] look similar to the results on the reflection of compact Hausdorff spaces into Stone spaces. In [10], it was shown that this is not similarity, but two special cases of the same 'theory'.
In the same setting of [10], the present paper provides new results concerning the preservation of finite products, in case S = Set and either C or M is an 'idempotent' category (see Definition 2.8). Then, it is possible to apply these results to the classification of reflections in the sense of [3].
In particular, we studied -with special care -reflections of varieties of universal algebras into subvarieties, where, to begin with, semi-left-exact and simple reflections are the same. Now, we will give a brief account of the contents of this work. The reader may also find it helpful to check the two tables at the end of the paper, summarizing all the presented results.
In Section 3, we state a necessary and sufficient condition (see Proposition 3.3) for the preservation of the product of two objects by a reflection into an 'idempotent' subcategory, provided there exists a functor U : C → Set which preserves finite limits and reflects isomorphisms, and such that U (η C ) is a surjection, for every unit morphism η C : C → HI(C), C ∈ C. It follows from Proposition 3.3 that: • if C is an 'idempotent' category (for instance, an idempotent variety of universal algebras), then finite products are preserved by the left-adjoint I (see Proposition 3.4); • if C is a variety of universal algebras and M is one of its 'idempotent' subvarieties (see Definition 2.7), then I preserves finite products if and only if I preserves the product F (x)×F (x), here F (x) stands for the free algebra on one generator in C (see Proposition 3.8).
In Section 4 (see Proposition 4.1), we will show that a reflection is simple if and only if it is semi-left-exact, provided its unit morphisms are effective descent morphisms (see the footnote in Section 4). In particular, this holds for a reflection of a variety of universal algebras into a subvariety. It is a consequence of Propositions 3.8 and 4.1 that a reflection of a variety of universal algebras C into an 'idempotent' subvariety M, has stable units in the sense of [3] if and only if it is simple and I preserves the product F (x) × F (x) (see Proposition 4.4). Consider a reflection and let (E I , M I ) be a prefactorization system as in [2, §3], that is,

Preliminaries
where H(morM) stands for the class of all morphisms in C which belong to the full subcategory M, and the arrows correspond to the diagonal fill-in Galois connection (see [2, §2.1

]).
A morphism e : A → B in C belongs to E I if and only if I(e) is an isomorphism. Hence, if e ∈ E I and e • f ∈ E I then f ∈ E I . In particular η C : C → HI(C) lies in E I , since M is a full subcategory of C.
Notice that every morphism in M lies in M I . Recall that the class M I is pullback stable (see [2, §2]), and that (E I , M I ) is a factorization system if, for some morphisms e ∈ E I and m ∈ M I , f = me for every morphism f in C.
where η A and η B are unit morphisms, and w is the unique morphism which makes the diagram commute.
Hence, (E I , M I ) is a factorization system if the reflection is simple, since π 1 in diagram (2.1) is a pullback of a morphism in M I , and so it is in M I .
where η C is a unit morphism and M ∈ M.
A semi-left-exact reflection is also called admissible in categorical Galois theory (see [1]). where η C is a unit morphism.
The following proposition is a well-known characterization of reflections having stable units.

The ground structure
We now define 'Ground Data'. The definition of an idempotent variety of universal algebras will be given in Definition 2.7, and then will be generalized in Definition 2.8 to an abstract category.
Definition 2.5. A reflection is said to satisfy Ground Data if there is a functor U : C → Set, from C into the category of sets, such that the following three conditions hold: (i) U preserves finite limits; (ii) U reflects isomorphisms; (iii) U (η C ) : U (C) → U HI(C) is a surjection, for every unit morphism η C , C ∈ C.
Example 2.6. A reflection of a variety of universal algebras C into a subvariety M satisfies Ground Data: • there exists a functor U : C → Set, the functor that assigns to an algebra C ∈ C its underlying set U (C), which reflects isomorphisms since these are just bijective homomorphisms in a variety of universal algebras; • U : C → Set preserves finite limits, since it is a right adjoint of F : Set → C, the functor that assigns to a set S the free algebra F (S) generated by the elements of S; • every unit morphism η C : C → HI(C) is a surjection in Set, since η C is the canonical projection of C into the quotient algebra C/ ∼ C ; here ∼ C is the congruence generated by the 'extra identities' satisfied in the subvariety M.
On the contrary, a forgetful functor U : C → Set from a category C of topological spaces, does not reflect isomorphisms in general. For instance, in the category C = Top of all topological spaces, consider the map i : (X, δ) → (X, τ ), in which X = {a, b}, δ is the discrete topology, τ is the topology {∅, X}, and U (i) is the identity function on the set X.
There still are cases for which a forgetful functor from a category of topological spaces reflects isomorphisms, such as for the category of compact Hausdorff spaces C = CompHaus. In fact, it is well known that U : CompHaus → Set is monadic.
In every case, either varieties of universal algebras or the two abovementioned categories of topological spaces, the forgetful functor U preserves finite limits, since U is always a right adjoint. 1 Definition 2.7. We say that a variety of universal algebras C is idempotent if any of the following three equivalent conditions hold: (i) every one-element subset {x} of any C ∈ C is a subalgebra; (ii) the free algebra F (x) on one generator 2 in C is a singleton; (iii) x = θ(x, ..., x), for every n-ary operation θ on any C ∈ C, with x ∈ C and n ∈ N 0 . Definition 2.8. Let C be a category with terminal object, such that there exists a functor U : C → Set, from C into sets, which preserves terminal objects. The category C is called idempotent with respect to the functor U : C → Set, when the restriction of U to the hom-set C(T, A), is a surjection for every object A ∈ C, with T a terminal object in C.
The category C will just be called idempotent when the functor U is the obvious one, as in the case in which C is a variety of universal algebras, or if it is the larger category in a reflection satisfying Ground Data. 1 If C is determined by a class of algebras of the same type closed under products and subalgebras, then (i) and (ii) in Definition 2.5 hold, and provided C is also closed under isomorphisms then U is a right adjoint, as it is well known. The following results concerning varieties could be easily adapted to this more general context (remember that a variety of universal algebras is just such a category C of algebras closed under homomorphic images).
2 A more precise notation would be F ({x}).
Example 2.9. A category of topological spaces C with terminal object is idempotent (with respect to the forgetful functor into sets), since every map from a singleton into any topological space is a continuous map in such a category.

Semi-left-exactness and stable units via connected components
In Definition 2.10 and Lemmas 2.11 and 2.12 below, the reflection satisfies Ground Data and the subcategory M is idempotent (with respect to the composite functor U H in Definition 2.5). These two Lemmas were proved in [10].

Preservation of finite products by reflections into subvarieties
In this section, we begin by recalling a known lemma in the category of sets. This lemma (Lemma 3.1) will be used in the proof of Lemma 3.2, which states a sufficient condition for the preservation of finite products by a reflection satisfying Ground Data. In Proposition 3.3, we state necessary and sufficient conditions for the preservation of finite products by a reflection into an idempotent subcategory, provided it satisfies Ground Data.
In Proposition 3.4, it is shown that finite products are preserved if the reflection satisfies Ground Data and C is idempotent. Note that just asking the full subcategory M to be idempotent would not be enough. See Example 4.6, where, in the reflection of M -sets into the idempotent category of sets, there is no preservation of finite products if the non-trivial monoid M satisfies the cancellation law.
In Lemma 3.7, we state a sufficient condition for the preservation of finite products by a reflection into a subvariety of universal algebras.
Finally, in Proposition 3.8, we state a necessary and sufficient condition for the preservation of finite products by a reflection into an idempotent subvariety of universal algebras.  Proof. Since η Q : Q → HI(Q) is universal from Q to H, it induces a morphism β : I(Q) → I(Q × R), such that the following diagram commutes: Applying the functor U to Diagram (3.1), one concludes that U (γ r ) factorizes through the surjective map U (η Q ). By analogous arguments, one can also conclude that U (γ q ) factorizes through the surjective map U (η R ). Now, consider the following: where the upper and bottom lines are both product diagrams. Applying the finite limits preserving functor U to Diagram (3.2) and Lemma 3.1, U HI(π 1 ) and U HI(π 2 ) are jointly monic. So U HI(π 1 ), U HI(π 2 ) is an injective map. On the other hand, since U (η Q ) and U (η R ) are surjective maps, U (η Q ) × U (η R ) = U HI(π 1 ), U HI(π 2 ) • U (η Q×R ) is also a surjective map. Finally, since U reflects isomorphisms, I(Q × R) ∼ = I(Q) × I(R). (ii) for each q fixed in U (Q), there exists a morphism γ q : Q → HI(Q × R), such that U (γ q )(e) = U (η Q×R )(q, e), f or all e ∈ U (R).
Proof. If the product Q × R is preserved by the reflector I, then w = HI(π 1 ), HI(π 2 ) in Diagram (3.2) is an isomorphism in M. Now consider for r ∈ U (R) fixed the following morphism in C: 3) where h r : T → HI(R) is a morphism in M, such that U (h r ) = f r : {r} → U HI(R); r → U (η R )(r). It is easy to check that U (γ r )(q) = U (η Q×R (q, r), for all q ∈ U (Q). There exists analogously a morphism γ q : R → HI(Q × R) in C, for any fixed q ∈ U (Q), such that U (γ q )(r) = U (η Q×R )(q, r), for all r ∈ U (R). The converse follows from Lemma 3.2.
Proposition 3.4. Consider a reflection of an idempotent category C, satisfying Ground Data. The left-adjoint I preserves finite products.
Proof. Let Q and R be objects of C. For every r ∈ U (R), consider the inclusion map f r of {r} into U (R). Since, by hypothesis, there exists a morphism f : T → R such that U (f ) = f r , and since C has finite products, there is a morphism 1 Q × f : , as in the following product diagram: Since T is a terminal object, there exists a unique morphism ! : Q → T such that U (!) : U (Q) → {r} is the unique map from U (Q) to {r}. Then, there exists a morphism 1 Q , ! : Q → Q × T (see Diagram 3.4). Therefore, there exists a morphism Hence, U (γ r )(a) = U (η Q×R )(a, r), for all a ∈ U (Q), with r ∈ U (R). One can construct, for every q ∈ U (Q), by analogous arguments, a morphism . Finally, by Lemma 3.2, one concludes that the left-adjoint I preserves the product Q × R. Example 3.6. Any reflection of an idempotent variety of magmas into one of its subvarieties preserves finite products. The reflection of the idempotent variety of quandles into its subvariety of sets preserves finite products (although it is not semi-left-exact; see [4] and check Corollary 4.5 below. In the following, we will see that the sufficient condition for the preservation of finite products in Lemma 3.2 holds for a reflection into a subvariety of universal algebras, provided I(F (x) × F (x)) = T . Lemma 3.7. Consider a reflection of a variety C of universal algebras into a subvariety, and let F (x) be the free algebra on one generator in C. If I(F (x) × F (x)) = T then the left-adjoint I preserves finite products.
Hence, there is a homomorphism λ r : Q → HI(Q×R), such that U (λ r )(q) = U (η Q×R )(q, r), with q ∈ Q, for every r ∈ R. By analogous arguments we would conclude that, for every q ∈ Q, there exists a homomorphism λ q : R → HI(Q × R), such that U (λ q )(r) = U (η Q×R )(q, r), r ∈ R. Finally, by Lemma 3.2, the left-adjoint I preserves the product Q × R. Proof. If I preserves finite products, then, in particular, I preserves the product F (x) × F (x). If I preserves the product F (x) × F (x), that is, ) is the free algebra on one generator in M. If I(F (x) × F (x)) = T , then I preserves finite products by Lemma 3.7.
Example 3.9. Consider the reflection of the variety of power associative magmas into its subvariety of semilattices (or, more generally, into any variety of bands). Recall that a magma is power-associative if any submagma generated by one of its elements is associative. The free power associative magma on one-element set F (x) is isomorphic to the commutative semigroup (N, +) of positive integer numbers. Therefore, F (x)×F (x) ∼ = N × N. On the other hand, N × N is semilattice indecomposable since it is archimedean (see [6] or [9]), that is, I(N × N) = T . Hence, the reflector I preserves finite products by Proposition 3.8.

Classifying reflections
In this section, we show that a reflection is semi-left-exact if and only if it is simple, provided its unit morphisms are effective descent morphisms 3 in C. This holds for every reflection of a variety of universal algebras into one of its subvarieties. Finally, we characterize the reflections into idempotent subvarieties of universal algebras having stable units.
First note that g, π 1 ∈ M I , because g ∈ M (see [2, §3]) and the class of morphisms M I is stable for pullbacks. Since the reflection is simple, the square (i) is a pullback, and so the outside square of Diagram (4.1) is a pullback. As HI(π 2 ) • η P = η H(X) • π 2 and HI(η B ) • η B = η HI(B) • η B , the outside square of Diagram (4.2) is also a pullback. According to Lemma 4.6 in [2], since the outside square of Diagram (4.2) is a pullback, (ii) is a pullback (because the reflection is simple) and η B is an effective descent morphism in C, then (iii) is a pullback, too. On the other hand, HI(η B ) is an isomorphism, because H I : C → M is a reflection into a full subcategory. Hence, HI(π 2 ) is also an isomorphism. Therefore, π 2 ∈ E I . Last Proposition 4.1 applies to any reflection of a variety of universal algebras into one of its subvarieties, as stated in the following proposition. Proof. It is well known that a variety of universal algebras is an exact category. In an exact category the effective descent morphisms are just the regular epimorphisms (see [2, §4.7]). On the other hand, it is easy to check that, in a variety of universal algebras the regular epimorphisms are just the surjective homomorphisms. Hence, the unit morphisms of a reflection of a variety of universal algebras into one of its subvarieties, are always effective descent morphisms, since they are surjective homomorphisms. Therefore, by Proposition 4.1, simple and semi-left-exact reflections of varieties of universal algebras are the same.
The following lemma will be used in Proposition 4.4, which characterizes the property of having stable units for reflections into idempotent subvarieties of universal algebras. Proof. The proof follows straightforwardly from Lemmas 2.11 and 2.12.
Proposition 4.4. Consider a reflection of a variety of universal algebras C into an idempotent subvariety. The following two conditions are equivalent, where F (x) is the free algebra on one generator in C: (i) the reflection is simple and the left-adjoint preserves the product F (x) × F (x); (ii) the reflection has stable units.   We present now an example of application of Proposition 4.4 to the reflection H I : M -Set → Set of M -sets into its idempotent subvariety of sets, associated to the congruence generated, on every S ∈ M -Set, by ms = s, for all m ∈ M , s ∈ S.
It is well known that this reflection of M -Set into Set is semi-leftexact (that is, simple; see Proposition 4.2), which follows from more general results (see [1, §6.2]). Here, we study the preservation of finite products and the stable units property, when M satisfies the cancellation law (ca = cb ⇒ a = b, for any a, b and c ∈ M ) 4 and when M has a zero element 0 M (0 M m = 0 M = m0 M , for every m ∈ M ).
A congruence on an M -set S contains R = {(s, ms) ∈ S × S | s ∈ S, m ∈ M } if and only if it contains R = {(ms, m s) ∈ S × S | s ∈ S, m, m ∈ M }, by symmetry and transitivity. Therefore, R and R * generate the same congruence.
Let C S be a subset of S × S consisting of those elements (a, b) ∈ S for which there exist z 0 , . . . , z n ∈ S where n ∈ N satisfying the following: Since C S is the transitive closure of R * , C S is contained in every congruence that contains R. In fact, C S is the congruence generated by R, because • {(a, a) | a ∈ S} ⊆ C S , • C S is obviously symmetric and transitive, and • C S respects the (unary) operations on S, since if there exists a finite sequence between a and b as in (4.3), then there exists a finite sequence between ma and mb as in (4.3), for all m ∈ M .
As usual, a ∼ S b will state that a and b are related, which is equivalent to (a, b) ∈ C S , and the class of equivalence of (a, b) in C S will be denoted by [(a, b)] ∼ S , for any elements a, b ∈ S.