2 Regular, homological and semi-abelian categories

2.2 Homological categories

When a regular category ${\mathcal C}$ is pointed (that is, it has a zero object, denoted by $0$ ) one says that it is homological [Reference Borceux and Bourn2] if the split short five lemma holds in ${\mathcal C}$ : given a commutative diagram

in ${\mathcal C}$ , where $f \circ s = 1_B$ and $f' \circ s' = 1_{B'}$ , and $\mathsf {ker}(f)$ is the kernel of f and $\mathsf {ker}(f')$ is the kernel of $f'$ , then v is an isomorphism whenever u and w are isomorphisms. It is well known that this assumption implies, in particular, that any regular epimorphism is a cokernel (so that regular epimorphisms coincide with normal epimorphisms), and then the classical short five lemma holds in a homological category. This implies, in particular, the validity of the following useful proposition.

Proposition 2.1 [Reference Borceux and Bourn2]

Given a commutative diagram of short exact sequences in a homological category ${\mathcal C}$ ,

u is an isomorphism if and only if the right-hand commutative square is a pullback.

3 Localization and factorization systems

In this section, we work with a semi-abelian category ${\mathcal C}$ as defined above, even though many of the facts that we recall are valid in a more general setting. The main references here are Bousfield [Reference Bousfield8] for the homotopy theory viewpoint and the article [Reference Cassidy, Hébert and Kelly12] by Cassidy et al. for the categorical side.

3.1 Factorization systems

A prefactorization system in ${\mathcal C}$ consists of classes of maps ${\mathcal E}$ and ${\mathcal M}$ determining each other by unique lifting properties or orthogonality properties. Thus, a morphism f belongs to ${\mathcal E}$ if and only if there is a unique filler in any commutative square

where p belongs to ${\mathcal M}$ . We write ${\mathcal E} = {}^{\perp } {\mathcal M}$ . Dually, ${\mathcal M} = {\mathcal E}^{\perp }$ . A factorization system is a prefactorization system in which every map can be factored into a morphism in ${\mathcal E}$ followed by a morphism in ${\mathcal M}$ .

3.3 Birkhoff subcategories

In our work, we are interested in the situation where ${\mathcal X}$ is a Birkhoff subcategory of a category ${\mathcal C}$

where U is the inclusion functor and F is its left adjoint. Being a Birkhoff subcategory means that ${\mathcal X}$ is a full (replete) and (regular epi)-reflective subcategory of ${\mathcal C}$ with the additional property that it is closed in ${\mathcal C}$ under regular quotients. Accordingly, each component $\eta _A \colon A \rightarrow UF(A)$ of the unit of the adjunction is a regular epimorphism and, moreover, ${\mathcal X}$ is also stable in ${\mathcal C}$ under regular quotients: if

is a regular epimorphism in ${\mathcal C}$ with A in ${\mathcal X}$ , then B also belongs to ${\mathcal X}$ .

Example 3.2. Any subvariety ${\mathcal X}$ of a variety ${\mathcal C}$ of universal algebras is a Birkhoff subcategory by the classical Birkhoff theorem. This applies to many situations: by adding any identity to the ones of a given algebraic theory one always determines a Birkhoff subcategory. This includes, of course, the classical examples of abelian groups or, more generally, of nilpotent or of solvable groups of a fixed class $\leq c$ in the category $\mathsf {Grp}$ of groups.

4 Conditional flatness

Our aim in this section is to study the notions of flatness and conditional flatness associated with a localization functor in a semi-abelian category. By analogy with the algebraic notion of flatness (tensoring by a flat ring preserves exactness), flatness for homotopy functors was defined by Farjoun and the second author in [Reference Farjoun and Scherer18] in terms of preservation of fibration sequences. The same was done in the category of groups in terms of preservation of extensions.

In a pointed category ${\mathcal C}$ one can translate this definition as follows.

The definition of conditional flatness from [Reference Farjoun and Scherer18] still makes sense in any pointed category.

4.1. Fiberwise localization

Definition 4.3. Given a functor $L \colon {\mathcal C} \rightarrow {\mathcal C}$ in a pointed category ${\mathcal C}$ , we say that an extension $0 \rightarrow K \rightarrow E \rightarrow Q \rightarrow 0$ in ${\mathcal C}$ admits a fiberwise localization if there is a commutative diagram of horizontal extensions

(4-1)

where $e : E \rightarrow \overline E$ is inverted by L. If the assignment $E \rightarrow \overline {E}$ is functorial (in the obvious sense), one says that it forms a functorial fiberwise localization for L.

Any localization in the category $\mathsf {Grp}$ of groups enjoys fiberwise localization as shown by Casacuberta and Descheemaeker [Reference Casacuberta and Descheemaeker10]. Other interesting examples will be considered at the end of this section, but there are also localization functors in certain homological categories that do not admit fiberwise versions (see [Reference Monjon, Scherer and Sterck32]).

4.2 Pullbacks along reflections

We are now going to show that, in any homological category, the existence of a functorial fiberwise localization has an interesting consequence. The following result refines and generalizes [Reference Farjoun and Scherer18, Proposition 4.1] from the category of groups to any homological category. The second, more amenable, condition describes admissible reflections in the sense of Janelidze–Kelly [Reference Janelidze and Kelly26], as we discuss in the next section.

Proposition 4.4. Let ${{\mathcal X} }$ be a full reflective subcategory of a homological category ${\mathcal C}$

that admits a functorial fiberwise localization. Then the following conditions are equivalent.

1. (1) The corresponding localization $(L=UF, \eta )$ is conditionally flat.

2. (2) The pullback of $\eta _C\colon C \rightarrow L(C)$ along any regular epimorphism in ${\mathcal C}$ between L-local objects is inverted by L.

Proof. Condition $(1)$ clearly implies $(2)$ , and we prove that $(2)$ implies conditional flatness of L. Let $0 \rightarrow K \rightarrow E \rightarrow Q \rightarrow 0$ be an L-flat extension and let $f : X \rightarrow Q$ be any arrow. First, we observe that, by applying fiberwise localization, there is no restriction in assuming that K is L-local. In order to see this, consider the right-hand pullback along f and the kernel $\kappa $ of $p_2$

By using the functorial fiberwise localization of L, one gets the following commutative diagram of short exact sequences (here we use the same notation as in Definition 4.3).

In any homological category, the right-hand square in the above diagram is then again a pullback (this follows from Proposition 2.1, since ${\mathcal C}$ is assumed to be a homological category). If we write $\pi \colon P \rightarrow \overline {P}$ for the L-equivalence in the construction (4-1) of the exact sequence $0 \rightarrow L(K) \longrightarrow \overline {P} \longrightarrow X \rightarrow 0$ by fiberwise localization, we obviously have that $\eta _P \cong \eta _{\overline {P}} \circ \pi $ , and this implies that the latter exact sequence is L-flat if and only if the sequence $0 \rightarrow K \longrightarrow P \longrightarrow X \rightarrow 0$ is so as well.

The next step is to reduce the proof to the case of an extension of L-local objects. This is done by noticing that, in the L-flat exact sequence $0 \rightarrow K \longrightarrow E \longrightarrow Q \rightarrow 0$ , the kernel K can already be assumed to be L-local, so that the square

is a pullback (we again use Proposition 2.1). Finally, we use the universal property of the localization and factor any map $X \rightarrow L(Q)$ through $\eta _X \colon X \rightarrow L(X)$ to decompose the pullback P of $L(E) \rightarrow L(Q)$ and $X \rightarrow L(Q)$ as the composite of two pullbacks:

Here $P'$ is a limit of L-local objects, and hence is L-local, and therefore yields, by regularity of ${\mathcal C}$ , another regular epimorphism $P' \rightarrow L(X)$ of L-local objects. The upper square is of the form required in order to apply assumption $(2)$ .

4.3 Existence of functorial fiberwise localization

In addition to the example of the category $\mathsf {Grp}$ of groups, there are many other examples of categories admitting functorial fiberwise localizations in certain circumstances. We focus from now on on localization functors for which the coaugmentation morphisms $\eta _X \colon X \rightarrow L(X)$ are always normal epimorphisms. In that case, we write $t_X \colon T(X) \rightarrow X$ for the kernel of $\eta _X$ and identify the latter with the quotient map $X \rightarrow X/T(X)$ .

Proposition 4.5. Let ${\mathcal C}$ be a homological category and let $L \colon {\mathcal C} \rightarrow {\mathcal C}$ be a localization functor such that any coaugmentation morphism $\eta _X \colon X \rightarrow L(X)$ is a normal epimorphism. Then ${\mathcal C}$ admits functorial fiberwise localization (with respect to L) if and only if one of the following conditions holds.

1. (1) For any normal monomorphism $k \colon K \rightarrow E$ in ${\mathcal C}$ , the pushout of k along $\eta _K$ exists

   (4-2)

   
   and the square (4-2) is a pullback.

2. (2) For any normal monomorphism $k \colon K \rightarrow E$ in ${\mathcal C}$ , the pushout (4-2) of k along $\eta _K$ exists and the morphism $\overline {k}$ is a monomorphism.

Proof. First, let us assume that fiberwise localization exists. Given any extension

a commutative diagram

exists by the assumption of fiberwise localization. The left-hand square is clearly a pullback, and the middle vertical morphism e is a regular epimorphism since the base category ${\mathcal C}$ is homological (see Proposition $8$ in [Reference Bourn5]). In this context, any pullback of a regular epimorphism along any morphism is a pushout (see [Reference Bourn, Carboni, Pedicchio and Rosolini4]), and this proves that $(1)$ holds. It is clear that $(1)$ implies $(2)$ since pullbacks reflect monomorphisms in ${\mathcal C}$ [Reference Bourn, Carboni, Pedicchio and Rosolini4].

Now assume that $(2)$ holds. In particular, one has the lower left-hand pushout in the commutative diagram

(4-3)

where $\overline {k}$ is a monomorphism, by assumption. Since $f \circ k \circ t_K = 0$ , the universal properties of the cokernel $\eta _K$ , and then of the left-hand pushout, yield a unique arrow $\overline {f} \colon \overline {E} \rightarrow B$ making the right-hand square above commute. The canonical factorization $\phi $ from $K/T(K)$ to the kernel $\mathsf {Ker}(\overline {f})$ of $\overline {f}$ such that $\mathsf {ker}(\overline {f}) \circ \phi = \overline {k}$ is then a monomorphism (since $\overline {k}$ is a monomorphism). It is also a regular epimorphism, since so is $\phi \circ \eta _K$ , the latter being the pullback of $\pi $ along $\mathsf {ker}(\overline {f})$ . It follows that $\phi $ is an isomorphism, and the lower sequence in diagram (4-3) is then exact.

The proof will be then complete if we show that $\pi $ is inverted by L. For this, consider the commutative diagram

where:

• • $T(k)$ is a normal monomorphism since so is $t_E \circ T(k) \colon T(K) \rightarrow E$ (the assumption that $\overline {k}$ is a monomorphism implies that $T(K)$ is the intersection $ T(E) \cap \mathsf {Ker}(\pi )$ of two normal monomorphisms) and $t_E$ is a monomorphism;

• • q is the quotient of $T(E)$ by $T(K)$ ;

• • j is the unique morphism such that $j \circ q = \pi \circ t_E$ ; and

• • p is the unique morphism such that $p \circ \pi = \eta _E$ .

Now, if $f \colon \overline {E} \rightarrow A$ is any morphism with A a local object, then there is a unique morphism $\psi \colon L(E) \rightarrow A$ such that $\psi \circ \eta _E = f \circ \pi $ . This morphism $\psi $ is also the unique one such that $\psi \circ p = f$ (since $\pi $ is an epimorphism), which proves that $p = \eta _{\overline {E}}$ (and $L(E) = L(\overline {E}$ )), so that $\pi $ is indeed inverted by L. One can easily check that this construction is functorial, and this completes the proof.

Remark 4.6. In any homological category, a property equivalent to properties $(1)$ and $(2)$ used in Proposition 4.5 consists of requiring that the monomorphism

$$ \begin{align*}k \circ t_K \colon T(K) \rightarrow K \rightarrow E\end{align*} $$

is normal. This was called condition $(N)$ in [Reference Evereart and Gran15]. The fact that $(N)$ is equivalent to condition $(1)$ easily follows from Proposition 2.1 by choosing the quotient $E/T(K)$ as $\overline {E}$ in diagram (4-2). We used condition $(N)$ in a previous version of the article, and we thank the referee for suggesting to also consider condition $(2)$ in the above proposition. The equivalent property $(N)$ will now be useful in the following examples.

Example 4.7. Proposition 4.5 can be applied to any homological category, and hence, in particular, to the category $\mathsf {Grp(Top)}$ of topological groups [Reference Borceux and Clementino3] which has the property that any regular epimorphism is normal. Consider a localization functor $L \colon \mathsf {Grp(Top)} \rightarrow \mathsf {Grp(Top)}$ for which each coaugmentation morphism $\eta _X$ of the localization is a normal epimorphism ( $=$ open surjective group homomorphism). Given any short exact sequence

(4-4)

in $\mathsf {Grp(Top)}$ , by taking the kernel $t_K \colon T(K) \rightarrow K$ of the unit $\eta _K$ of any localization, one obtains a characteristic subgroup $T(K)$ of K (see, for instance, Example $2.2$ in [Reference Evereart and Gran15]). Accordingly, the subgroup $T(K)$ is also normal in E, and hence condition $(N)$ in Remark 4.6 holds, as desired. The same observation also applies to the category $\mathsf {Grp(Haus)}$ of Hausdorff groups.

Example 4.8. Now let $\mathsf {Hopf}_{A, coc}$ be the category of cocommutative Hopf algebras over a field A, which was shown to be semi-abelian in [Reference Gran, Sterck and Vercruysse22]. Given an extension (4-4), by using the same notation as above, the Hopf subalgebra $T(K) \rightarrow K \rightarrow E$ induced by any localization functor $L \colon \mathsf {Hopf}_{A, coc} \rightarrow \mathsf {Hopf}_{A, coc}$ is also a normal Hopf subalgebra of E. Indeed, denote by $S \colon K \rightarrow K$ the antipode of E and denote by ${\phi _e \colon K \rightarrow K}$ the map defined, for any $e \in E$ , by $\phi _e (t) = e_1 t S(e_2)$ for any $t \in K$ . Here we use the usual Sweedler convention for Hopf algebras, so that we write $\Delta (e) = e_1 \otimes e_2$ , with $\Delta \colon E \rightarrow E \otimes E$ being the comultiplication. This map $\phi _e \colon K \rightarrow K$ is seen to be a Hopf algebra morphism. By functoriality of the natural transformation $t \colon T \Rightarrow \mathsf {id}_{{\mathcal C}}$ , it follows that $\phi _e$ restricts to $T(K)$ , which yields a morphism $T(K) \rightarrow T(K)$ . This means that, for any $t \in T(K)$ , $\phi _e (t) \in T(K)$ , and hence $T(K)$ is normal in E, and condition $(N)$ then holds. We conclude that, whenever we have a localization functor $L \colon \mathsf {Hopf}_{A, coc} \rightarrow \mathsf {Hopf}_{A, coc}$ with the property that the coaugmentation morphism $\eta _X\colon X \rightarrow L(X)$ is a normal epimorphism, the category $\mathsf {Hopf}_{A, coc}$ admits fiberwise localization. For instance, the ‘abelianization functor’ $\mathsf {ab} \colon \mathsf {Hopf}_{A, coc} \rightarrow \mathsf {Hopf}_{A, coc}^{comm}$ , as described in Section $4$ in [Reference Gran, Sterck and Vercruysse22], necessarily yields a functorial fiberwise localization, with $L = U \, \mathsf {ab} \colon \mathsf {Hopf}_{A, coc} \rightarrow \mathsf {Hopf}_{A, coc} $ (here $U \colon \mathsf {Hopf}_{A, coc}^{comm} \rightarrow \mathsf {Hopf}_{A, coc}$ is the forgetful functor).

Remark 4.9. One might hope that similar results hold whenever one is dealing with a category of internal groups in a finitely complete category, since the examples mentioned above, such as groups, topological groups, Hausdorff groups and cocommutative Hopf algebras, are of this type (the category $\mathsf {Hopf}_{A, coc}$ can also be seen as the category of internal groups in the category of cocommutative coalgebras). However, this is not the case, as follows from the results in [Reference Monjon, Scherer and Sterck32], where some counterexamples are given in the semi-abelian category $\mathsf {XMod}$ of crossed modules, which can be seen also (up to a category equivalence) as the category $\mathsf {Grp(Cat})$ of internal groups in the category $\mathsf {Cat}$ of (small) categories (see, for instance, [Reference Janelidze, Márki and Tholen27] and the references therein).

5 Admissible reflectors with respect to Galois theory

The type of pullbacks that appear in Proposition 4.4 are the ones appearing in categorical Galois theory, in the form presented in the article [Reference Janelidze and Kelly26] by Janelidze and Kelly. In the whole section, we work in a homological category ${\mathcal C}$ , where we fix a full reflective subcategory ${\mathcal X}$ as in

and the corresponding localization functor $L = UF\colon {\mathcal C} \rightarrow {\mathcal C}$ .

Definition 5.1 [Reference Janelidze and Kelly26]

The reflector $F \colon {{\mathcal C} } \rightarrow {\mathcal X}$ is admissible for the class of regular epimorphisms if it preserves any pullback of the form

(5-1)

where $x \colon E \rightarrow FX$ is a regular epimorphism in ${\mathcal X}$ .

One could also require F to preserve pullbacks as above for any morphism x in ${\mathcal X}$ , in which case we are looking at semi-left-exact reflections as introduced by Cassidy et al. in [Reference Cassidy, Hébert and Kelly12]. We come back to this in Section 7.

Definition 5.2 [Reference Cassidy, Hébert and Kelly12]

Let ${\mathcal X}$ be a (normal epi)-reflective subcategory of ${\mathcal C}$ . Then F is semi-left-exact, that is, F preserves all pullbacks of the form

where x is any morphism in ${\mathcal X}$ .

In other words, the morphism $P \rightarrow U(E)$ coincides with the P-component of the unit $\eta _P \colon P \rightarrow UF(P)$ of the adjunction (up to unique isomorphism). There are several equivalent ways to characterize admissibility, as we recall in the next proposition. They hold, in particular, for all the examples of semi-localizations of semi-abelian categories given in [Reference Gran and Lack20].

Proof. Let K be the object part of the kernel of the vertical morphism $U(x)$ in (5-1). This is the object part of a limit of a diagram lying in ${\mathcal X}$ , and hence it lies itself in ${\mathcal X}$ . The extension

is thus L-flat. If L is conditionally flat, then the (induced) pullback extension

must be L-flat as well. This means that $L=UF$ takes it to an extension

where $K \cong UF(K)$ remains unchanged since it lies in ${\mathcal X}$ . This extension comes with a natural transformation to the original extension.

We conclude by the short five lemma (see [Reference Borceux and Bourn2]) that the middle dotted arrow is an isomorphism, and the arrow $P \rightarrow U(E)$ in the pullback (5-1) is then isomorphic to the unit $\eta _P \colon P \rightarrow UF(P)$ . This means that the reflector is admissible with respect to the class of regular epimorphisms, as desired.

We can also reinterpret Proposition 4.4 as follows.

6 The case of Birkhoff subcategories

We restrict our attention to a Birkhoff subcategory ${\mathcal X}$ of a regular category ${\mathcal C}$ , as in Section 3.3. The suitable context in which to obtain the result of this section is the one of ideal determined categories, as introduced in [Reference Janelidze, Márki, Tholen and Ursini28] by Janelidze et al. These are regular categories ${\mathcal C}$ with binary coproducts such that:

1. (1) any regular epimorphism in ${\mathcal C}$ is normal (that is, a cokernel); and

2. (2) normal monomorphisms are stable under images; in any commutative square

   
   in ${\mathcal C}$ where f and $f'$ are normal epimorphisms, a is a normal monomorphism and b is a monomorphism, b is also a normal monomorphism.

As explained in [Reference Janelidze, Márki, Tholen and Ursini28] any semi-abelian category is ideal determined. In particular, all the examples mentioned before (groups, loops, rings, commutative algebras, associative algebras, cocommutative Hopf algebras, crossed modules, compact groups, ${\mathsf C}^{\star }$ -algebras, and so on) are ideal determined. There are also some examples of ideal determined varieties that are not semi-abelian, for instance, the variety of implication algebras [Reference Gumm and Ursini23].

The following theorem gives a natural condition that guarantees the conditional flatness of the pointed endofunctor L, without the toolkit of fiberwise localization.

Proof. We prove that $L \colon {\mathcal C} \rightarrow {\mathcal C}$ is conditionally flat. We consider an L-flat extension

(6-1)

and a morphism $g\colon A \rightarrow X$ in ${\mathcal C}$ . We construct the pullback of the original extension along g and need to prove that this extension $0 \rightarrow K \longrightarrow P \xrightarrow {p_2} A \rightarrow 0$ is again L-flat.

We know that the induced arrow $L(p_2) \colon L(P) \rightarrow L(A)$ is a normal epimorphism, since the arrow $\eta _A \circ p_2 \colon P \rightarrow L(A)$ is a normal epimorphism as it is a composite of two normal epimorphisms (see Section 2.1). We now prove that the arrow $L(K) \rightarrow L(P)$ is the kernel of $L(p_2) \colon L(P) \rightarrow L(A)$ . First, observe that the arrow $L(K) \rightarrow L(P)$ is a monomorphism, since the arrow

is a monomorphism (the original extension (6-1) being L-flat). Since the category ${\mathcal C}$ is ideal determined and the square

is commutative with $K \rightarrow P$ , which is a normal monomorphism, and $L(K) \rightarrow L(P)$ , which is a monomorphism, it follows that the arrow $L(K) \rightarrow L(P)$ is a normal monomorphism as well. Consequently, $L(K) \rightarrow L(P)$ is the kernel of its cokernel $q\colon L(P) \rightarrow Q$ in ${\mathcal C}$ . However, this latter is isomorphic to $L(p_2) \colon L(P) \rightarrow L(A)$ . Indeed, this follows from the fact that the functor $F \colon {\mathcal C} \rightarrow {\mathcal X}$ preserves cokernels (as it is a left adjoint) while $U \colon {\mathcal X} \rightarrow {\mathcal C}$ preserves them since ${\mathcal X}$ is closed in ${\mathcal C}$ under (regular) quotients, by the Birkhoff assumption.

7 Fiberwise localizations and stability under extensions

In this section, we show that, when ${\mathcal C}$ is homological, torsion-free reflections $F \colon {\mathcal C} \rightarrow {\mathcal X}$ can be characterized among (normal epi)-reflections admitting fiberwise localization in terms of the property of stability under extensions of ${\mathcal X}$ in ${\mathcal C}$ . We recall that a torsion-free reflection is associated to a torsion theory (see, for example, [Reference Gran and Lack20, Definition 1.1]). In particular, the only morphism from a torsion object to a local object is the zero morphism.

Recall that a full (replete) subcategory ${\mathcal X}$ of a pointed category ${\mathcal C}$ is stable under extensions (in ${\mathcal C}$ ) if, given any short exact sequence

(7-1)

in ${\mathcal C}$ with K and Y in ${\mathcal X}$ , we have that X is also in ${\mathcal X}$ .

Proof. $(1) \Rightarrow (2)$ We consider the short exact sequence

(7-2)

and the associated exact sequence $0 \rightarrow F (T(X)) \rightarrow \overline X \rightarrow F(X) \rightarrow 0$ that exists by the assumption of fiberwise localization. Since the subcategory ${\mathcal X}$ is closed in ${\mathcal C}$ under extensions, $\overline X$ is in ${\mathcal X}$ and the fiberwise morphism $X \rightarrow \overline X$ is, therefore, an F-equivalence to an object in ${\mathcal X}$ . Thus, it must be $\eta _X\colon X \rightarrow F(X)$ (up to isomorphism). This implies that the kernel $F (T(X))$ of the morphism $\overline {X} \rightarrow F(X)$ is zero.

$(2) \Rightarrow (3)$ and $(3) \Rightarrow (4)$ both follow from Theorem $4.12$ in [Reference Bourn and Gran6].

$(4) \Rightarrow (1)$ We briefly recall the known argument showing that a torsion-free subcategory ${\mathcal X}$ is closed under extensions in ${\mathcal C}$ . Given a short exact sequence (7-1) with K and Y in ${\mathcal X}$ , consider the canonical short exact sequence (7-2), where $T(X)$ is torsion and $F(X)$ is torsion-free. Clearly, $T(X) \rightarrow X \rightarrow Y$ is the zero morphism, and hence $t_X$ factors through K. Since $T(X)$ is a subobject of K, $T(X) \in {\mathcal X}$ ( ${\mathcal X}$ is closed under subobjects). Since it is also in the torsion subcategory, $T(X) \cong 0$ and $X \cong F(X) \in {\mathcal X}$ , as desired.

Unlike in the abelian case, in homological categories, the property of stability under extensions of a (normal epi)-reflective subcategory ${\mathcal X}$ is not strong enough to guarantee that $F \colon {\mathcal C} \rightarrow {\mathcal X}$ is a reflector to a torsion-free subcategory, as observed in [Reference Janelidze, Tholen, Davydov, Batanin, Johnson, Lack and Neeman29]. The lemma above shows that, under the assumption of fiberwise localization, this is indeed the case.

Example 7.3. We write $L_{ab}$ for the abelianization functor. The dihedral group $D_8$ of order eight abelianizes to ${\mathbb Z}/2{\mathbb Z} \times {\mathbb Z}/2{\mathbb Z}$ , an elementary abelian $2$ -group of rank two. Consider the following pullback in the category of groups.

The vertical morphism on the right-hand side is a homomorphism of abelian groups and the bottom morphism is the abelianization morphism of $D_8$ . Its pullback, however, is the map ${\mathbb Z}/2 {\mathbb Z} \rightarrow 0$ , which is not the abelianization morphism for ${\mathbb Z}/2 {\mathbb Z}$ . Since fiberwise localization always exists in the category $\mathsf {Grp}$ of groups, Proposition 7.1 applies and tells us that the above problem reflects the fact that the subcategory $\mathsf {Ab}$ of abelian groups is not closed under extensions in $\mathsf {Grp}$ and it is not torsion-free (in the categorical sense).

The fact that the abelianization functor is not semi-left-exact is well known. The ‘relative version’ of the Galois theory, which was developed by Janelidze in [Reference Janelidze25] and later in collaboration with Kelly in [Reference Janelidze and Kelly26], where the class of morphisms to be classified by the Galois theorem is the one of regular epimorphisms, was partly motivated by the possibility of applying their approach to any Birkhoff subcategory of a ‘sufficiently good’ algebraic category. Here ‘sufficiently good’ could mean being a semi-abelian variety of universal algebras [Reference Bourn and Janelidze7], for instance, yielding many examples of interest in algebra.

8 The case of nullifications

The results of the previous section apply to nullification functors. Let ${\mathcal C}$ be a semi-abelian category, let A be an object in ${\mathcal C}$ and define ${\mathcal X} \subset {\mathcal C}$ to be the (replete) reflective subcategory of A-null objects, that is, of those objects Z such that $\text {Hom}(A, Z) = 0$ .

When it exists, the associated localization functor is written $P_A$ and is called A-nullification (or A-periodization). The construction is due to Bousfield in a homotopical setting and can be found, for example, in Hirschhorn’s [Reference Hirschhorn24]. A reference in an algebraic context is Casacuberta et al. [Reference Casacuberta, Peschke, Pfenniger, Ray and Walker11, Theorem 1.4]. In all cases, $P_A (X)$ is constructed as a transfinite filtered colimit of iterated quotients of all morphisms from A. A cardinality argument is invoked to explain when one can stop the iteration.

In the recent preprint [Reference Monjon, Scherer and Sterck32], Monjon et al. gave, in Proposition $2.7$ , an explicit construction of the nullification functor in the (semi-abelian) category of crossed modules. By looking at the arguments in their proof, one realizes that these still apply to any semi-abelian variety of universal algebras [Reference Bourn and Janelidze7]. These are precisely those varieties ( $=$ finitary equational classes) whose algebraic theories have a unique constant $0$ , $n\ge 1$ binary terms $\alpha _i(x,y)$ and one $(n+1)$ ary term $\beta $ satisfying the identities $\alpha _i(x,x)=0$ (for $i\in \{1, \ldots , n\}$ ), $\beta (\alpha _1(x,y), \ldots , \alpha _n(x,y), y)= x$ . For example, in the case of the variety of groups, by using the multiplicative notation for the group operation, one can choose $0=1$ , $\alpha _1(x,y)= x\cdot y^{-1}$ and $\beta (x,y) = x\cdot y$ . Note that, for a variety of universal algebras, being homological or being semi-abelian are equivalent properties, since a variety is always Barr-exact and cocomplete. We work here with sets equipped with finitary operations satisfying a set of identities, so set-theoretic arguments are available. Moreover, any variety of universal algebras is cocomplete. Hence, the proof of [Reference Monjon, Scherer and Sterck32, Proposition 2.7] applies.

We now show that, in the presence of fiberwise localization, nullification functors are conditonally flat, in fact, even semi-left-exact. We write $\overline P_A (X)$ for the kernel of the A-nullification $\eta _X \colon X \rightarrow P_A(X)$ . The equivalent characterization from Proposition 7.1(2) that $P_A (\overline P_A (X)) = 0$ for any object X in ${\mathcal C}$ is an algebraic analog of Farjoun’s [Reference Farjoun17, Theorem 1.H.2].

9 A model categorical interpretation

In this article, we study how pullbacks of exact sequences behave and, in the previous sections, we relate this to semi-left-exactness, which is a stronger admissibility property (preservation of pullbacks along any morphism between local objects versus preservation of pullbacks along any regular epimorphism between local objects). From a model theoretic perspective, this corresponds to right properness, as we explain next.

Any category with finite limits and colimits admits a discrete model structure where weak equivalences are isomorphisms and all morphisms are fibrations and cofibrations. This easy observation has been previously made by Bousfield [Reference Bousfield8, Examples 2.3], who also constructed new model structures where the class of weak equivalences is ${\mathcal E}(L)$ that is all morphisms inverted by a localization functor L, that is, L-equivalences. Cofibrations do not change and the class of fibrations coincides now with ${\mathcal M}(L)$ (using the notation from Section 3). This is not immediately obvious as we require the lift to be unique in a factorization system, but not in a model category. This is because the model categorical lift is unique up to homotopy in the associated homotopy factorization system and, in the discrete setting, ‘unique up to homotopy’ means unique.

This process is called left Bousfield localization; we cite Salch [Reference Salch35, Proposition 3.5] for a statement in line with the present work. Our final propositions are just reformulations of the fact that a semi-left-exact reflection is also characterized by the property that, for the induced factorization system $({\mathcal E}, {\mathcal M})$ , the morphisms in ${\mathcal E}$ are stable under pullback along morphisms in ${\mathcal M}$ .

In a model category, it is a direct consequence of the axioms that the pullback of a fibration is a fibration. But weak equivalences need not be preserved by pullbacks, not even by homotopy pullbacks. A model category is right proper if the pullback of any weak equivalence $X \rightarrow B$ along any fibration $E \twoheadrightarrow B$ is a weak equivalence. The discrete model structure is right proper since the pullback of an isomorphism along any map is an isomorphism.

Now, given a localization functor L on ${\mathcal C}$ , it is then a natural question to ask when the left Bousfield localized model structure described in Proposition 9.1 is again right proper. Rosický and Tholen noticed in [Reference Rosický and Tholen34, 3.6] that a result by Cassidy et al. [Reference Cassidy, Hébert and Kelly12, Theorem 4.3] allows one to characterize right proper localized model structures as those corresponding to semi-left-exact reflections.

Therefore, Corollary 8.2 tells us that the $P_A$ -local model structure is right proper, which is an analog of the well-known fact that nullification functors in spaces yield a right proper left Bousfield localized model structure [Reference Berrick and Farjoun1] (see also Wendt [Reference Wendt37, Corollary 6.1] for simplicial sheaves on a site). However, in an algebraic setting, conditional flatness is different from right properness because pulling back an L-equivalence along a regular epimorphism is not as general as pulling back along an arbitrary fibration, that is, an arbitrary morphism in the localized model structure.