2.1. Three main results

The first result extends the definition of the left derived functors of an additive functor between abelian categories to a non-additive setting:

Theorem. Let $\mathscr {C}$ be a pointed regular category with binary coproducts and with enough projectives which satisfies Condition (P). Let $\mathscr {E}$ be a homological category with binary coproducts. Let $F:\; {\mathscr {C}}\to {\mathscr {E}}$ be a protoadditive functor that preserves proper morphisms and binary coproducts. Then the formula

\begin{equation*} \textrm {L}_n(F)(X)\; :\!= \textrm {H}_n(F(C(X))) \end{equation*}

where $X$ is an object of $\mathscr {C}$ and $C(X)$ is a chosen projective chain resolution of $X$ , defines a functor $\textrm {L}_n(F):\; {\mathscr {C}}\to {\mathscr {E}}$ for each $n\in \mathbb {Z}$ .

In other words, the left derived functors of $F$ are defined as in the abelian context. The proof follows steps (1)–(3) of Theorem 1.1.1, adapted for the notion of chain homotopy developed in this article. This is an application of Theorem 10.0.1.

In fact, when moreover $\mathscr {C}$ is protomodular (so that now both $\mathscr {C}$ and $\mathscr {E}$ are homological categories), the condition that $F$ is protoadditive is strong enough for the $\textrm {L}_n(F)$ to coincide with the derived functors obtained via projective simplicial resolutions, once those exist. In particular, we find the following result, which in the text occurs as Theorem 11.3.2:

The reason for this is two-fold: first of all, any two simplicially homotopic morphisms of simplicial objects induce homotopic chain morphisms between the Moore normalizations—this is Theorem 11.2.1. Note that this works despite the fact that the Dold–Kan equivalence (between the categories of simplicial objects and positively graded chain complexes) is not available here. Second, the normalization of a simplicial resolution is always a projective chain resolution, by closedness of the class of projectives under protosplit subobjects (Condition (P)) and the fact that in a homological category, exactness of a simplicial object agrees with exactness of its normalization [Reference Goedecke37, Reference Goedecke and Van der Linden38, Reference Tierney and Vogel72]. We require Barr exactness of the category $\mathscr {E}$ to guarantee that the normalization of any simplicial object is a proper chain complex, which is essential for the homology objects to detect exactness [Reference Van der Linden75].

A third key result mimics what happens for a right exact functor between abelian categories, Theorem 10.1.4 in the text:

Theorem. Let $\mathscr {C}$ be a homological category with binary coproducts and with enough projectives which satisfies Condition (P). Let $\mathscr {E}$ be a homological category with binary coproducts. Let $F:\; {\mathscr {C}}\to {\mathscr {E}}$ be a sequentially right exact functor which preserves protosplit monomorphisms and binary coproducts. Any short exact sequence

in $\mathscr {C}$ gives rise to a long exact sequence in $\mathscr {E}$ as in Figure 3 which depends naturally on the given short exact sequence.

Figure 3.

Long exact sequence involving the derived functors of $F$ .

The conditions on $\mathscr {C}$ are satisfied, for instance, by the variety of Lie algebras over any commutative unitary ring (4.2.2), by crossed modules (12.0.1) and more generally by $n$ -cat-groups.

If the derived functors $\textrm {L}_n(F)$ of a functor $F$ are not additive, like the title of this article indicates, then what properties do they have? We prove in Section 10 that they are protoadditive functors, which means that they preserve semidirect products, provided $F$ is a functor between semi-abelian categories which is itself protoadditive and preserves binary coproducts and proper morphisms (Corollary 10.0.6).

2.2. The context

We now explain the terminology in the statements of those results.

A category is pointed when it has an object that is initial and terminal at the same time. In such categories, we can define protosplit subobjects: such is a subobject $K\leq X$ which is represented by a kernel of a split epimorphism with domain $X$ .

A normal category in the sense of [Reference Janelidze47] is a pointed regular [Reference Barr, Grillet and van Osdol3] category where every regular epimorphism is a normal epimorphism, that is, a cokernel of some morphism.

We mainly work in the context of homological categories. A category $\mathscr {C}$ is called (Borceux–Bourn) homological [Reference Borceux and Bourn5] if it is pointed, regular and (Bourn) protomodular [Reference Bourn, Carboni, Pedicchio and Rosolini8]. The protomodularity condition says that the Split Short Five Lemma holds in $\mathscr {C}$ , or equivalently, that whenever we have a split epimorphism with its kernel as in

the object $X$ is “covered” or “generated” by the outer objects $K$ and $Y$ in the sense that the inclusions $k$ and $s$ (where $k=\textrm {ker}(\,f)$ and $s$ splits $f$ ) are jointly extremally epimorphic, which means that they do not both factor through a proper subobject of $X$ . It implies, for instance, that each regular epimorphism is a normal epimorphism. In other words, the category in question is normal.

A (Janelidze–Márki–Tholen) semi-abelian category [Reference Janelidze, Márki and Tholen46] is a category which is pointed, (Barr) exact and (Bourn) protomodular, and which has binary coproducts. This stricter definition ensures, for instance, good behavior of simplicial objects. (In particular, the normalization of any simplicial object is a proper chain complex, which means that the images of the differentials are normal subobjects.)

Examples include any abelian category, the category of groups $\mathsf {Gp}$ , and more generally any variety of $\Omega$ -groups [Reference Higgins40], of which by definition the signature admits a group operation and a unique constant—such as any variety $\mathsf {Lie}_{\mathbb {K}}$ of Lie algebras over a ring $\mathbb {K}$ or any variety of non-associative algebras over $\mathbb {K}$ [Reference Van der Linden77]. An important example of a homological category which is not semi-abelian is the category of topological groups. The category $\mathsf {OrdGp}$ of preordered groups is known to be normal but not homological [Reference Clementino, Martins-Ferreira and Montoli24]: a preordered group being a group equipped with a preorder (a reflexive and transitive relation) for which the group operation is monotone; arrows are monotone group homomorphisms.

Some of the above examples satisfy Condition (P). Indeed, it will be automatic for abelian categories; it will hold for $\mathsf {Gp}$ by the Nielsen–Schreier theorem; for $\mathsf {Lie}_{\mathbb {K}}$ (where $\mathbb {K}$ is a commutative ring) it is a non-trivial result (see Section 4); while in the category of associative algebras over a field, for instance, it is still an open question Figure 4, 5, 6.

A key concept when we are dealing with derived functors is the notion of a projective chain resolution. Let $\mathscr {C}$ be a pointed category with kernels and let $X$ be an object of $\mathscr {C}$ . A projective chain resolution of $X$ consists of a positively graded chain complex $(C,d)$ as in Figure 6, all of whose objects $C_i$ are projective (with respect to the class of regular epimorphisms) together with a regular epimorphism $d_0:\; C_0\to X$ such that for each $n\geq 0$ , the lifting $\bar {d}_{n+1}:\; {C_{n+1}\to \textrm {Ker}(d_n)}$ of $d_{n+1}$ over the kernel $\textrm {Ker}(d_n)$ of $d_n$ is a regular epimorphism. Once we assume that the category $\mathscr {C}$ has enough projectives, recursively constructed projective chain resolutions exist for any object of the category.

In the context of a normal category, a resolution of $X$ is an exact sequence as in Figure 1. Exactness in $C_n$ , for instance, means that the morphism $\bar {d}_{n+1}$ induced by the kernel of $d_n$ is a normal epimorphism, which is equivalent to saying that the image of $d_{n+1}$ is the kernel of $d_n$ . Here, the interpretation in terms of homology as in the figure’s caption makes sense.

In a pointed category, a morphism is proper if it can be factorized as a regular epimorphism followed by a normal monomorphism. In a normal category, this means that the monic part of the image factorization is a kernel, while the epic part is a cokernel. A functor $F:\; {{\mathscr {C}} \to {\mathscr {E}}}\ \ $ between between pointed categories with kernels and cokernels $\mathscr {C}$ and $\mathscr {E}$ will be called a sequentially right exact functor, as in [Reference Peschke and Van der Linden60], if it sends an exact sequence

so a proper morphism $k$ with its cokernel $f$ , to the exact sequence

These functors are meant to replace the right exact functors from the abelian context.

If we want to mimic additive functors outside the context of additive categories, we can use protoadditive functors [Reference Everaert and Gran31, Reference Everaert and Gran32]. A functor $F:\; {\mathscr {C}} \to {\mathscr {E}}$ between pointed categories $\mathscr {C}$ and $\mathscr {E}$ is protoadditive if it preserves kernels of split epimorphisms.

2.3. The abelian setting

Coming back to the above theorems, we see that the categories $\mathscr {C}$ and $\mathscr {E}$ satisfy conditions which are weaker than abelianness. On the other hand, we impose seemingly stronger conditions on the functors to compensate for the weaker context. In the first theorem, any functor between abelian categories which preserves binary coproducts is additive. Therefore, the assumption of being protoadditive is redundant in the abelian setting. And in the third one, in the abelian setting, we only require right exactness of the functor. Here we have to impose two non-trivial assumptions—preservation of protosplit monomorphisms and the preservation of binary coproducts—which are redundant in the abelian case, but just like protoadditivity turn out to be relatively strong in the non-additive setting.

3.1. Projective and free objects

We start by recalling what are projective objects (relative to the regular epimorphisms) in a category.

Definition 3.1.1. An object $P$ of a category $\mathscr {C}$ is said to be (regular) projective when, given a regular epimorphism $p:\; X\twoheadrightarrow Y$ and a morphism $f:\; P\to Y$ there exists a lifting $g:\; P\to X$ of $f$ over $p$ :

a morphism $g$ such that $pg=f$ .

A category has enough (regular) projectives when each object $A$ is a regular quotient of a projective object: there exists a projective object $P$ together with a regular epimorphism $p:\; P \twoheadrightarrow A$ .

In a variety of algebras, a convenient class of projective objects is the class of free objects: objects which belong to the image of the left adjoint $T$ to the forgetful functor $U$ to $\mathsf {Set}$ . This follows easily from the definition. In certain varieties—see Example 3.2.6 below—the two classes coincide, but in general, the converse does not hold. However, in a variety of algebras, an object is a projective object if and only it is a retract of a free object.

3.2. On condition (P)

The proof of the “classical” Horseshoe Lemma in abelian categories ([Reference Cartan and Eilenberg22, Proposition V.2.2], [Reference Weibel78, Lemma 2.2.8]) depends on the class of projective objects being closed under biproducts (see for instance [Reference Borceux4, Proposition 4.6.3]). In our present context (where biproducts are not available) this is replaced by closedness under coproducts combined with the following condition, which in the context of a homological category we will show to be equivalent to a non-additive version of the Horseshoe Lemma—see Section 6.

Remark 3.2.2. In the setting of a normal category (where every regular epimorphism is normal), this means there exists a split short exact sequence

(3)

whose kernel part is the representing monomorphism $k:\; K\to X$ .

We shall later see that, just as in the abelian case, if the domain of a functor $F$ is a Schreier variety, then each of its derived functors $\textrm {L}_n(F)$ for $n\geq 2$ is trivial (Corollary 10.1.6). So while Schreier varieties do satisfy Condition (P), they are not very interesting from a homological-algebraic point of view. This means we have to focus on non-Schreier varieties satisfying (P). It seems, however, that Condition (P) has not yet been studied at all in the literature, so currently examples are scarce. We present a non-trivial one in Section 4, and another one in Section 12.

Varieties of non-associative algebras over a field need not be Schreier:

3.3. Characterization in terms of free objects

We now investigate what happens with Condition (P), if we assume that the middle object $X$ in (4) is free rather than just projective. A priori, what we find is a weaker condition. However, it turns out that in any pointed Mal’tsev variety—which [Reference Mal’cev57] admits a ternary operation $p$ such that $p(x,x,z)=z$ and $p(x,z,z)=x$ —hence a fortiori in any homological variety, the two are equivalent.

Proof. Since, in a variety of algebras, any free object is projective, (i) implies (ii). The split epimorphism of sets $g$ in the statement of (ii) induces a diagram

(4)

as in Definition 3.2.1, where $X=T (A)$ and $Y=T (B)$ . Since $T (A)$ is a free object on $A$ , we have that (ii) implies (iii).

We now focus on the proof that (iii) implies (i). So consider a split epimorphism $f$ with section $s$ and kernel $k$ as in (4). Covering each object with a free object, we obtain the diagram in Figure 4, where the horizontal sequences are split epimorphisms with their kernel, $T U (X)$ and $T U (Y)$ are free objects and $\varepsilon _X$ and $\varepsilon _Y$ are regular epimorphisms.

Figure 4.

The implication (iii) $\Rightarrow$ (i).

The naturality of $\varepsilon$ implies that $f \varepsilon _X=\varepsilon _Y T U (\,f)$ and hence we have the morphism $\beta$ determined by the universal property of the kernel of $f$ . We claim that $\beta$ is a split epimorphism. Then $K$ will be a retract of $K'$ , which is a projective object by (iii) applied to the split epimorphism of sets $U(\,f):\; U(X)\to U(Y)$ .

Let us prove our claim. Since the codomains of $\varepsilon _X$ and $\varepsilon _Y$ are projective objects, they are split epimorphisms. Since $\mathscr {V}$ is a regular Mal’tsev category, the right-hand square is a regular pushout [Reference Bourn11, Proposition 3.2], which means that the comparison $(\varepsilon _X, TU(\,f))$ from $TU(X)$ to the pullback $X\times _UTU(Y)$ in Figure 5 is a regular epimorphism.

Figure 5.

Constructing a splitting for $\beta$ .

Let $j_Y:\; Y\to T U(Y)$ be a section of $\varepsilon _Y$ . By the pullback property we obtain a section $j^{\prime}_Y$ of $\varepsilon^{\prime} _Y$ such that $j_Y f = f^{\prime} j^{\prime}_Y$ .

Since $X$ is projective, we can define a section of $\varepsilon _X$ as a lifting of $j^{\prime}_Y$ over the regular epimorphism $(\varepsilon _X, TU(\,f))$ , so that $(\varepsilon _X, TU(\,f)) j_X= j^{\prime}_Y$ . This morphism $j_ X:\; X\to T U(X)$ is indeed a section of $\varepsilon _X$ , since $\varepsilon _X j_X=\varepsilon^{\prime} _Y (\varepsilon _X, TU(\,f)) j_X=\varepsilon^{\prime} _Y j^{\prime}_Y = 1_X$ . We see that

\begin{equation*}T U(\,f) j_X=f^{\prime} (\varepsilon _X, TU(\,f)) j_X=f^{\prime} j^{\prime}_Y = j_Y f.\end{equation*}

By the universal property of $\textrm {ker} (T U (\,f))$ , there exists a morphism $j_K:\; {K\to K'}$ such that $\textrm {ker} (T U(\,f)) j_K = j_X k$ . Now by definition of $\beta$ and $j_K$ we have

\begin{equation*}k 1_K = 1_X k = \varepsilon _X j_X k = \varepsilon _X k' j_K=k \beta j_K \end{equation*}

and since $k$ is monic this implies that $\beta j_K = 1_K$ , proving $\beta$ to be a split epimorphism.

Proposition 3.3.1 will help us in the next section, where we give a class of examples of semi-abelian varieties which need not be Schreier but do satisfy (P).

4.1. Free Lie algebras

It is well known—see for instance [Reference Reutenauer62, Section 0.2]—how to construct the free Lie algebra $T(X)$ for a given set $X$ : the elements of $T(X)$ are the (non-commutative, non-associative) polynomials with no constant term and variables in $X$ such that the binary product, which is simply a linear extension of the concatenation of monomials with variables in $X$ , satisfies the axioms of a Lie bracket.

Lie algebras over a field form a Schreier variety. The original proof due to Shirshov [Reference Shirshov68] uses the following strategy. First find a set which generates the given subalgebra of a given free Lie algebra; then prove that this subalgebra is freely generated—see for instance [Reference Reutenauer62, Theorem 2.5] for a proof written in English. The assumption that the coefficients ring is a field is a key assumption in the first step of the proof. Indeed, we need the modules involved in this construction to always admit a base—which is false when the ring is not a field. For this reason, the proof does not work for unital commutative rings, as was already noticed by Shirshov in [Reference Shirshov68]. Using an idea of Bahturin in [Reference Bahturin1], whenever $\mathbb {K}$ is not a field we can build an example of a subalgebra of a free $\mathbb {K}$ -Lie algebra which is no longer free.

Example 4.1.1 (An explicit construction of a non-free subalgebra). Let us consider a unital commutative ring $\mathbb {K}$ and a set $X$ with $m$ elements, where $m\gt 1$ . Consider the free Lie algebra $T(X)$ and its subalgebra $H\; :\!= \alpha T(X)$ , where $\alpha \in \mathbb {K}$ is a non-invertible element of $\mathbb {K}$ , different from $0$ . By definition of $H$ , the subalgebra admits a natural grading decomposition $H=\alpha T(X)_1\oplus \alpha T(X)_2\oplus \cdots$ inherited from the grading of $T(X)$ .

Let us assume that $H$ is a free Lie algebra on a set $Y$ . Then its abelianization $H^{\textrm {ab}}\; :\!= H/[H,H]$ is isomorphic to the free $\mathbb {K}$ -module $\bigoplus _{y\in Y}\mathbb {K} y$ , by composition of left adjoints. However, by the grading decomposition of $H$ , its abelianization $H^{\textrm {ab}}$ contains

\begin{align*} \alpha T_2(X)/[\alpha T_1(X),\alpha T_1(X)] \cong \begin{cases} T_2(X)/\alpha T_2(X) & \text {if $\alpha ^2\neq 0$,} \\ \alpha T_2(X) & \text {otherwise.} \end{cases} \end{align*}

In both situations, there are non-trivial torsion elements in the module $H^{\textrm {ab}}$ —which contradicts the fact that it is a free module. This explains why $H$ cannot be free.

4.2. Lazard’s elimination theorem

We fix a commutative unital ring $\mathbb {K}$ and work toward a proof that Condition (P) holds in the category $\mathsf {Lie}_{\mathbb {K}}$ . We can consider the split short exact sequence

where $(\,f,s)$ is any split epimorphism $f$ with a section $s$ in $\mathsf {Set}$ . We claim that, in this situation, we can prove that $K$ is a retract of a free Lie algebra and so a projective object. Thanks to Proposition 3.3.1, we may then conclude that $\mathsf {Lie}_{\mathbb {K}}$ satisfies Condition (P). The proof of this claim is based on the next result called Lazard’s elimination theorem.

Theorem 4.2.1. [51] Let $T(X)$ be the free Lie algebra over a commutative unital ring $\mathbb {K}$ and let $A$ and $B$ two disjoint sets such that $X=A \ \cdot{\kern-.9pt\!\!\!\cup} \ B$ . Then we have an isomorphism

\begin{equation*} T(X)\cong T(M(B) \times A) \rtimes _{\phi } T(B) \end{equation*}

where $M(B)$ is the free monoid over $B$ and $\phi$ is an appropriate action of $T(B)$ on $T(M(B) \times A)$ .

The action $\phi$ may be codified as a Lie algebra homomorphism

\begin{equation*} \phi :\; T(B)\to \textrm {Der}(T(M(B)\times A)) \end{equation*}

whose codomain is the Lie algebra of all derivations of the free algebra $T(M(B)\times A)$ .

The main idea behind the proof of this theorem is to consider the split short exact sequence

where $g$ is such that $a\in A\mapsto 0$ and $b\in B\mapsto b$ and $t$ its canonical splitting ( $b\in B\mapsto b$ ). We then notice that $K$ , as a kernel of $g$ , not only contains the element $a$ , as well as its image through (composites of) all inner derivations induced by elements of $B$ (so $\textrm {ad}_b(a)\; :\!=[b,a]$ and $\textrm {ad}_{b_1}\cdots \textrm {ad}_{b_n}(a) = [b_1,\ldots, [b_n,a]\cdots ]$ where $b\in B$ and $b_i\in B$ for all $1\leq i\leq n$ ); but in fact, those elements generate $K$ . It is, however, not clear at first whether $K$ is freely generated by those elements. The theorem tell us that this is indeed the case. The proof depends on finding a suitable action $\phi$ of $T(B)$ on $\textrm {Der}(T(M(B) \times A))$ , taking into account that $K$ is closed under inner derivations induced by the elements of $B$ .

We translate the semidirect product of Theorem 4.2.1 into the split short exact sequence

Proof. Let $f:\; X \to Y$ be a split epimorphism in $\mathsf {Set}$ and $s$ one of its sections. In order to use Lazard’s elimination theorem, we define

\begin{equation*} A\; :\!= \{x-sf(x)\mid x\in X\}\subseteq T(X) \qquad \text {and}\qquad B\; :\!= \{s(y)\mid y\in Y\}\subseteq T(X). \end{equation*}

In the diagram

the morphism $g$ sends generators in $A$ to zero and generators in $B$ to themselves, while $\phi$ and $\psi$ are defined as:

\begin{equation*} \phi (x)=(x-sf(x))+sf(x) \end{equation*}

and

\begin{equation*} \psi (x-sf(x))=x-sf(x),\qquad \psi (s(y))=s(y) \end{equation*}

for $x\in X$ and $y\in Y$ . It represents a vertical, upward-pointing split epimorphism with downward-pointing section. Taking kernels, via Theorem 4.2.1 we find a vertical, upward-pointing split epimorphism of short exact sequences as in

This proves that $\textrm {Ker}(T(\,f))$ is a projective object, as a retract of the free Lie algebra $T(M(B)\times A)$ . The result now follows from Proposition 3.3.1.

5.1. Graded objects

Let $\mathscr {C}$ be any category. A graded object in $\mathscr {C}$ is a sequence of objects $(C_{n})_{n\in \mathbb {Z}}$ or, equivalently, a functor $C:\; {\mathbb {Z}\to {\mathscr {C}}}$ where the set of integers $\mathbb {Z}$ is considered as a discrete category. Consider $k\in \mathbb {Z}$ . A morphism $f:\; {A\to B}$ of degree $k$ between graded objects $A$ and $B$ is a collection of morphisms $(\,f_{n}:\; A_{n}\to B_{n+k})_{n \in \mathbb {Z}}$ in $\mathscr {C}$ . That is to say, $f$ is a natural transformation from $A$ to $B s^{k}$ where the map $s^{k}:\;\mathbb {Z}\to \mathbb {Z}$ sends $n$ to $n+k$ . Of course, a morphism $f:\; {A\to B}$ of degree $k$ composes with a morphism $g:\; {B\to C}$ of degree $l$ to a morphism $g f:\; {A\to C}$ of degree $k+l$ .

5.2. Chain complexes

Now let $\mathscr {C}$ be pointed, and let $0$ denote the constant graded object $(0)_{n\in \mathbb {Z}}$ . We shall also write $0$ for any morphism (of whichever degree) that factors over $0$ .

A chain complex $(C,d)$ in $\mathscr {C}$ is a graded object $C$ together with a morphism $d :\; {C\to C}$ of degree $-1$ (its differential) such that $d d =0$ . So $d$ is a sequence

in which the composite of any two successive morphisms is trivial. A chain complex $(C,d)$ is bounded below when there exists $i\in \mathbb {Z}$ such that $C_{n}=0$ for all $n\lt i$ and positively graded when this $i$ is $0$ . Sometimes, we omit the differential $d$ in our notation and write $C$ for a chain complex $(C,d)$ .

A chain morphism $f:\; {(C,d^{C})\to (E,d^{E})}$ from a chain complex $(C,d^{C})$ to a chain complex $(E,d^{E})$ is a morphism $f:\; {C\to E}$ of degree $0$ such that $d^{E} f=f d^{C}$ . Chain complexes in $\mathscr {C}$ and chain morphisms between them form a category which we write ${\mathsf {Ch}} ({\mathscr {C}})$ .

Any zero-preserving functor $F :\; {{\mathscr {C}} \to {\mathscr {E}}}$ between pointed categories $\mathscr {C}$ and $\mathscr {E}$ induces a functor $F :\; {{\mathsf {Ch}}({\mathscr {C}}) \to {\mathsf {Ch}}({\mathscr {E}})}$ between the respective categories of chain complexes.

5.3. Exact complexes, proper complexes, and homology

We assume that $\mathscr {C}$ is pointed with kernels and cokernels. We say that a chain complex $(C,d)$ is exact at $C_{n}$ or exact in degree $n$ when the lifting $\bar {d}_{n+1}:\; {C_{n+1} \to \textrm {Z}_n(C)}$ of $d_{n+1}$ over the kernel $\textrm {Z}_n(C)$ of $d_{n}$ is a regular epimorphism.

The complex $C$ is exact when it is exact in all degrees: the lifting $\bar {d}$ of $d$ over $\textrm {ker} (d):\; {\textrm {Z}(d)\to C}$ is a regular epimorphism. In the context of a normal category, this is equivalent to saying that it consists of short exact sequences spliced together.

A chain complex $(C,d)$ is proper when $d$ admits a factorization as a regular epimorphism $e$ followed by a normal monomorphism $m$ .

This means that each $d_{n+1}$ is a proper morphism. Clearly, every exact complex is proper.

Given a proper chain complex $(C,d)$ in a normal category $\mathscr {C}$ and $n\in \mathbb {Z}$ , the $n$ th homology object $\textrm {H}_{n}(C)$ of $C$ is the cokernel of $\bar {d}_{n+1}:\; {C_{n+1} \to \textrm {Z}_n(C)}$ .

(5)

All homology objects together form a graded object $\textrm {H} (C)=\textrm {Coker}(\bar {d})$ , in fact, a chain complex with zero differentials. For each $n\in \mathbb {Z}$ , the operations $\textrm {Z}_n$ and $\textrm {H}_n$ define functors from ${\mathsf {Ch}}({\mathscr {C}})$ to $\mathscr {C}$ such that if $f :\; {C \to E}$ is chain morphism, then

\begin{equation*} \textrm {ker}(d^E_n) \textrm {Z}_n(\,f) = f_n \textrm {ker}(d^C_n) \end{equation*}

and

\begin{equation*} \textrm {H}_n(\,f) \textrm {coker}(\bar {d}^C_{n+1}) = \textrm {coker}(\bar {d}^E_{n+1}) \textrm {Z}_n(\,f) . \end{equation*}

In a normal category, proper chain complexes are precisely those in which homology may detect exactness so that a proper complex $C$ is exact if and only if $\textrm {H} (C)=0$ ; see [Reference Van der Linden75] for further details.

6.1. Projective resolutions

As recalled in Section 1.1, a particular type of chain complex which plays a central role in the construction of the additive left derived functors are the projective resolutions of an object. We first consider this notion in the context of a pointed category with kernels and then specialize to pointed regular categories and homological categories.

Figure 6.

A projective resolution of $X$ .

As in the abelian setting, the existence of projective resolutions is guaranteed when enough projective objects are available; see for instance [Reference Cartan and Eilenberg22] for a full proof. The idea is to take a projective cover $d_0:\; C_0\to X$ of $X$ : the object $C_0$ is the first object of the resolution. We then take the kernel $\textrm {ker}(d_0):\;\textrm {Ker}(d_0)\to C_0$ of $d_0$ and cover this object with a projective object $C_1$ . The induced composite is $d_1:\; C_1\to C_0$ . The procedure continues by recursion, and we find:

For this reason, we will assume that our categories have enough projectives. Since this implies regularity of the category as soon as it has finite limits and coequalizers of kernel pairs, we will assume this condition as well.

Figure 7.

A lifting of a morphism $x$ .

In order to define the left derived functors of a functor, we first of all need that any morphism $x:\; X\to Y$ in the domain category induces a morphism of projective resolutions $f:\; {C(X)\to E(Y)}$ over $x$ , also called a lifting of $x$ , which means that $x d_0^C=d_0^E f_0$ as in Figure 7. This is possible thanks to the next result, of which the proof is straightforward, as in the abelian context—see for instance [Reference Lane54].

Note that an object $X$ may admit several non-isomorphic projective resolutions. However, thanks to the above, we know that between them, we can always find a morphism of projective resolutions over $1_X$ . In the abelian case, such an $f$ happens to be a chain homotopy equivalence. In order to recover this result, we need an appropriate notion of chain homotopy, valid in a non-additive context. This will be the subject of Section 8.

In a normal category, the concept of a projective resolution considered above may be characterized in terms of homology objects as in the abelian case.

Remark 6.1.6. Consider an object $X$ in a normal category. A chain complex $C$

is a projective resolution of $X$ if it is projective in all degrees, while $\textrm {H}_0(C)=X$ and $\textrm {H}_n(C)=0$ for all $n\neq 0$ . Here, the morphism $d_0:\; C_0\to X$ is nothing but $\textrm {coker}(d_1):\; C_0\to X=\textrm {H}_0(C)$ . This is the viewpoint of Figure 1 .

We now restrict our attention to the context of a homological category, where a version of the Horseshoe Lemma is available.

Figure 8.

The diagram to be filled with resolutions of $X$ and of $Y$ .

6.2. The Horseshoe Lemma in homological categories

The remainder of the present section is dedicated to obtaining a non-additive version of the so-called Horseshoe Lemma, which is essential in the construction of the long exact sequence of derived functors (Section 10.1, [Reference Cartan and Eilenberg22, Reference Lane54]). We will see that the availability of this result is related to Condition (P). Given a diagram such as the one in Figure 8 where the vertical sequence is exact and the bottom exact sequence represents a projective resolution $E$ such that $\textrm {H}_0(E)=Z$ , we would like to find projective resolutions of $X$ and $Y$ and morphisms such as in Figure 9 where all vertical sequences are exact. In a homological category, this is indeed possible—as soon as Condition (P) holds; note that since in this context, biproducts are not available, the classical abelian proof strategy cannot work here.

We say “half” because unlike the abelian case we only start with an arbitrary resolution of $Z$ , not of $Z$ and $X$ , as one does in the abelian case.

Figure 9.

The diagram of Figure 8 filled.

Figure 10.

Take the pullback $P$ and a projective cover of it.

Proof. The proof is by recursion on $n\in \mathbb {N}$ . Let us start with the first step, $n=0$ .

We consider Figure 10 where the bottom square is a pullback. Being pullbacks of normal epimorphisms, $p_0$ and $p_1$ are normal epimorphisms as well. The morphism between the kernels $K_{p_0}$ and $X$ is an isomorphism by [Reference Bourn9, Lemma 1]. Notice that the normal epimorphism $p_0$ is split, by some section $i$ , since $E_0$ is a projective object. Since by assumption the category has enough projectives, $P$ is the quotient, via a normal epimorphism $\varepsilon _0$ , of a projective object $P_0$ . The composite $\beta _0=p_0 \varepsilon _0$ , as a normal epimorphism with codomain $E_0$ , is a split epimorphism with a section we denote $i_0$ . Taking the kernel of this split epimorphism $\beta _0$ will show us that $X$ is a quotient of a projective object, namely $K_{\beta _0}$ via $\overline {\varepsilon _0}$ .

Since the vertical sequence on the left is a split short exact sequence where $P_0$ is a projective object, Condition (P) implies that the kernel object $K_{\beta _0}$ is a projective object as well. Finally, since the top left square is a pullback, $\varepsilon _0$ being a normal epimorphism implies that the induced morphism $\overline {\varepsilon _0}$ is a normal epimorphism as well. If we set $C_0=K_{\beta _0}$ , then $X$ is a quotient of $K_{\beta _0}$ via $\eta _0^C=\overline {\varepsilon _0}$ and if we set $A_0=P_0$ , then $Y$ is a quotient of $P_0$ via $\eta _0^A=p_1\varepsilon _0$ .

Figure 11.

Construction of $d_n^C$ and $d_n^D$ .

Figure 12.

Part of Figure 11 on which the $(3\times 3)$ -lemma is applied.

For the recursion step, let us consider Figure 11 where we suppose that partial projective resolutions $X$ and $Y$ are defined respectively until the projective objects $C_n$ and $A_n$ . On the one hand, notice that we defined $d_n^C= \textrm {ker} ( \eta _{n-1}^C) \eta _n^C$ where $\eta _n^C$ is a normal epimorphism by assumption. We defined $d_n^A$ in an analogous way. On the other hand, we have $d_n^E=\textrm {ker} (\eta _{n-1}^E) \eta _n ^ E$ by exactness of the projective resolution $E$ of $Z$ at the level $n-1$ . Hence, the sequences are projective resolutions up to level $n-1$ .

Figure 12 shows a $(3\times 3)$ -diagram where we have three horizontal short exact sequences (by definition) and two vertical short exact sequences on the right (by assumption). By the $(3\times 3)$ -Lemma [Reference Bourn9, Theorem 12], the vertical sequence with the dashed arrow is a short exact sequence as well. We return to the situation of the the base step, with $K_{\eta _n^C}$ , $K_{\eta _n^A}$ and $K_{\eta _n^E}$ playing, respectively, the roles of $X$ , $Y$ , and $Z$ , and the normal epimorphism $\eta _{n+1}^E :\; E_{n+1}\to K_{\eta _{n}^E}$ playing the role of $\textrm {coker}(d_1^E)$ .

Remark 6.2.2 (Degreewise split short exact sequence). Like in the abelian context, the previous proof leads us to a short exact sequence of chain complexes

(6)

where for each $n\geq 0$ we have a split short exact sequence

However, in the current context, $A_n$ need not be a product of $C_n$ and $E_n$ .

Condition (P) plays a key role in the previous proof. Actually, it is not only sufficient, but also necessary:

Proof. Consider a split short exact sequence

where $X$ is a projective object. Then $Y$ is a projective object as well. We prove that $K$ is also projective.

Since $\mathscr {C}$ has enough projectives, by Proposition 6.1.2 we may consider a projective resolution of $Y$ . By assumption, we can construct suitable projective resolutions of $K$ and $X$ ; in particular, we obtain the diagram

whose horizontal sequences are short exact. Since $X$ and $Y$ are projective objects, the normal epimorphisms $\textrm {coker}(d_1^A)$ and $\textrm {coker}(d_1^E)$ are split. As a result, and since a homological category is always regular Mal’tsev [Reference Borceux and Bourn5], we are in a similar situation as in the proof for (ii) $\Rightarrow$ (i) of Proposition 3.3.1. We may thus choose suitable sections of $\textrm {coker}(d_1^A)$ and $\textrm {coker}(d_1^E)$ which restrict to a section of $\textrm {coker}(d_1^C)$ . This shows that $K$ is a retract of the projective object $C_0$ . Hence, $K$ is projective as well, and (P) holds in the category $\mathscr {C}$ .

7.1. Calculus of subtraction

For now, we assume that $\mathscr {C}$ is a pointed category with functorially chosen kernels and binary coproducts. This allows us to define the difference object functor $D:\; {\mathscr {C}}\to {\mathscr {C}}$ . Given a morphism $h:\; {X \to Y}$ , we let $D(h)$ be the unique induced arrow in the diagram

(8)

We will also need the natural transformation $\varsigma :\; D \Rightarrow 1_{{\mathscr {C}}}$ defined for $X$ in $\mathscr {C}$ by

\begin{equation*} \varsigma _X\; :\!= \langle 1_{X}, 0\rangle \delta _{X}:\; D(X)\to X. \end{equation*}

To see that $\varsigma$ is indeed natural, so that $f\varsigma _X=\varsigma _Y D(\,f)$ for any arrow $f:\; X\to Y$ , it suffices to take a look at the diagram

Approximate differences are compatible with composition:

Lemma 7.1.2. Under the above assumptions on $\mathscr {C}$ , we have

\begin{equation*} l (\,f-g)=lf-lg\qquad \text {and} \qquad (\,f-g)D(h)=fh-gh \end{equation*}

whenever these equalities make sense.

For the moment it is not so clear whether the definition of $f-g$ encodes a “difference” as in the abelian context, in the sense that $f=g$ if and only if $f-g=0$ . Under certain conditions on the surrounding category, this does indeed happen.

Lemma 7.1.4. Under the above assumptions on $\mathscr {C}$ , we have

\begin{equation*} f=g\quad \Rightarrow \quad f-g=0. \end{equation*}

If, in addition, $\mathscr {C}$ is a normal category, then $f-g=0$ implies $f=g$ .

Proof. For the first implication, as mentioned in [Reference Bourn and Janelidze15, Section 3.1], this follows from the fact that $\langle f, f\rangle =f\langle 1_X, 1_X\rangle$ while $\delta _X$ is the kernel of $\langle 1_X, 1_X\rangle$ , so that

\begin{equation*} f-g=\langle f, f\rangle \delta _X=f\langle 1_X, 1_X\rangle \delta _X=0. \end{equation*}

If $\mathscr {C}$ is a normal category, then as already mentioned in [Reference Bourn and Janelidze15, Introduction], the codiagonal $\langle 1_X, 1_X\rangle$ is the cokernel of $\delta _X$ . By assumption, we have that $\langle f, g\rangle \delta _X=0$ . Hence, $\langle f, g\rangle$ factors through $\langle 1_X, 1_X\rangle$ via a morphism $h$ so that $\langle f, g\rangle =h\langle 1_X, 1_X\rangle =\langle h, h\rangle$ and $f=h=g$ .

As a consequence, normality can be characterized in terms of approximate differences.

Proposition 7.1.5. A pointed regular finitely cocomplete category is normal if and only if for any two given morphisms $f$ , $g:\; {X\to Y}$ ,

\begin{equation*} f-g = 0\quad \Rightarrow \quad f=g. \end{equation*}

7.2. Subtractive categories

Next to the above property, we will also need that

\begin{equation*} f-0=g-0\qquad \Rightarrow \qquad f=g. \end{equation*}

Since $f\varsigma _X=f-0=g-0=g\varsigma _X$ by Remark 7.1.3, this happens as soon as $\varsigma _X$ is an epimorphism. We ask slightly more, because then we find ourselves in a known situation: the context of subtractive categories. We will not use the original definition of subtractivity here, but rather an equivalent formulation [Reference Bourn and Janelidze16, Theorem 5.1], valid for pointed regular categories with binary coproducts:

Definition 7.2.1. A pointed regular category with binary coproducts is called subtractive when for each object $X$ , the composite

\begin{equation*} 1_X - 0 =\langle 1_{X}, 0\rangle \delta _{X}=\varsigma _X :\; {D(X)\to X} \end{equation*}

is a regular epimorphism.

We see that the natural transformation $\varsigma$ is a (component-wise) regular epimorphism if and only if we have a subtractive category. Via this short observation it is easy to see that any abelian category is subtractive. Indeed, in this setting, $\delta _X=( 1_X, -1_X)$ , and the domain of the difference is $D(X)=X$ .

Almost by definition, subtractive categories form a context in which a difference of two morphisms may be expressed as an approximate difference. The disadvantage of Definition 7.2.1 is that it does not immediately show the relationships with other categorical-algebraic conditions. Subtractive categories were introduced in [Reference Janelidze47] as a categorical characterization of subtractive varieties of universal algebras, studied earlier in [Reference Ursini74], in a context which is not necessarily regular and where binary coproducts need not exist. A relation $r=( r_{1}, r_{2}):\; {R\to X\times Y}$ in a pointed category is said to be left (right) punctual [Reference Bourn10] if $(1_X, 0):\; {X\to X\times Y}$ (respectively $(0, 1_Y):\; {Y\to X\times Y}$ ) factors through $r$ . A pointed finitely complete category $\mathscr {C}$ is said to be subtractive, if every left punctual reflexive relation on any object $X$ in $\mathscr {C}$ is right punctual.

By [Reference Bourn and Janelidze16, Theorem 5.1], in a pointed regular category with binary coproducts, this agrees with Definition 7.2.1. The arguments in [Reference Janelidze47] further show that any pointed protomodular category is subtractive so that any additive category with finite limits, any homological category in the sense of [Reference Borceux and Bourn5], and in particular any semi-abelian category [Reference Janelidze, Márki and Tholen46] is subtractive.

It is explained in [Reference Janelidze47] that a variety of algebras is subtractive when the corresponding theory contains a unique constant $0$ and a binary operation $-$ , called a subtraction, subject to the equations $x-0 = x$ and $x-x = 0$ . So, for instance, any pointed variety of algebras whose theory contains a group operation is an example.

7.3. A stronger context for subtraction

Since we are eventually taking homology in the codomain of a functor $F:\; {\mathscr {C}} \to {\mathscr {E}}$ , we shall be working with minimal assumptions on $\mathscr {C}$ (a pointed regular category with binary coproducts, with enough projectives that satisfies Condition (P)) and use the stronger but natural context of a homological category for $\mathscr {E}$ . Homological categories are a suitable framework for our calculus of differences for four reasons:

1. (1) they admit kernels and cokernels of proper morphisms;

2. (2) they are subtractive;

3. (3) they are normal;

4. (4) the functor $D:\; {\mathscr {E}}\to {\mathscr {E}}$ preserves normal epimorphisms.

Our aim is now to prove that the fourth point does indeed hold. Here, we again use that a homological category is always regular Mal’tsev. Recall [Reference Bourn11, Reference Carboni, Kelly and Pedicchio19, Reference Everaert, Goedecke and Van der Linden30] that a regular category is Mal’tsev if and only if any split epimorphism of regular epimorphisms is a regular pushout—see the proof of Proposition 3.3.1 which recalls what this means.

Proof. If $h$ is a regular epimorphism, then $h+h$ is a regular epimorphism as well. Hence if $\mathscr {C}$ is regular Mal’tsev, then in the diagram (8) defining $D(h)$ , the square on the right is a regular pushout. Let $P$ denote the pullback of $\nabla _Y$ and $h$ and write $h^*(\nabla _Y):\; {P\to X}$ for the induced arrow. It is then easy to see that on the one hand, the kernel of $h^*(\nabla _Y)$ is $D(Y)$ , while, on the other hand, the square

is a pullback. It follows that the arrow $D(h)$ is a regular epimorphism.

Remark 7.3.2. The functor $D$ need not be exact, not even in the context of a semi-abelian category. We consider the alternating group $A_3$ as a normal subgroup of the symmetric group $S_3$ . In the notation of Example 7.0.5 , we have that for $\overline {(12)}\underline {(12)}\in D(S_3)$ and $\underline {(123)}\overline {(132)}\in D(A_3)$ the conjugate

\begin{equation*} \overline {(12)}\underline {(12)}\underline {(123)}\overline {(132)}\underline {(12)}\overline {(12)}=\overline {(12)}\underline {(13)}\overline {(132)}\underline {(12)}\overline {(12)} \end{equation*}

in $D(S_3)$ is not in $A_3+A_3$ and therefore not in $D(A_3)$ .

7.4. Differences and the functor $D$

Even though the functor $D$ is engaged in the construction of the approximate difference, outside of the additive context it need not interact well with those differences. It is indeed not true that $D(\,f-g)$ is equal to $D(\,f)-D(g)$ in general. If this were the case, then by Lemma 7.1.1 we should have $D(\varsigma _X)=D(1_X-0)=D(1_X)-D(0)=\varsigma _{D(X)}$ . However, in $\mathsf {Gp}$ ,

\begin{equation*} D(\varsigma _X)\bigl (\overline {(\underline {x}\overline {x}^{-1})} \underline {(\overline {x}\underline {x}^{-1})} \bigr )=\overline {x}^{-1}\underline {x}\neq \underline {x}\overline {x}^{-1}=\varsigma _{D(X)}\bigl (\overline {(\underline {x}\overline {x}^{-1})}\underline {(\overline {x}\underline {x}^{-1})} \bigr ) \end{equation*}

holds for all $x\in X$ by Example 7.0.5.

Remark 7.4.1 (Compositions of $\varsigma _X$ ). We know that in general, $\varsigma _{D(X)}\neq D(\varsigma _X)$ . However, thanks to the naturality of $\varsigma$ , for any object $X$ we have

\begin{equation*} \varsigma _X \varsigma _{D(X)}=\varsigma _X D(\varsigma _X):\; D^2(X)\to X. \end{equation*}

We will also write this composition $\varsigma _X^2$ since its domain is $D^2(X)$ . By induction, for all $n\geq 2$ we find the equality

(9) \begin{equation} \varsigma _X \varsigma _{D(X)}\cdots \varsigma _{D^{n-1}(X)}=\varsigma _X D(\varsigma _X)\cdots D^{n-1}(\varsigma _X):\; D^n(X)\to X \end{equation}

and we will denote this morphism $\varsigma _X^n$ .

By convention, we will extend the notation $\varsigma _X^n$ to all natural numbers by letting $\varsigma _X^0=1_X$ (since its domain will be $D^0(X)=X$ ) and $\varsigma _X^1=\varsigma _X$ (since its domain will be $D(X)$ ). Note that in a subtractive category, all the morphisms $\varsigma _X^n$ are regular epimorphisms (as composites of such).

For any $f:\; X\to Y$ and all $n\in \mathbb {N}$ , we have

(10) \begin{equation} f \varsigma ^n_X=\varsigma ^n_Y D^n(\,f) \end{equation}

so that the $\varsigma ^n_X$ form a natural transformation $\varsigma ^n:\; D^n\Rightarrow 1_{\mathscr {C}}$ .

8.1. Towards a definition

We let $\mathscr {C}$ be a normal category with binary coproducts and enough projectives which satisfies Condition (P). Let $C(X)$ and $E(Y)$ be projective resolutions of, respectively, $X$ and $Y$ , two objects of $\mathscr {C}$ . Given a morphism $x:\; X\to Y$ , we know from Lemma 6.1.5 that a morphism of projective resolutions $f:\; {C(X)\to E(Y)}$ exists such that $\textrm {H}_0(\,f)=x$ . Let us now assume that $g:\; {C(X)\to E(Y)}$ is another such morphism (Figure 14 again). We follow the example of the abelian context, to recursively construct what should amount to a chain homotopy from $f$ to $g$ . Since the standard definition does not immediately make sense here, we shall simply proceed with the construction itself and take what comes out as a starting point for a non-additive definition.

Construction of $h_0$ .

From Lemma 7.1.4 and Lemma 7.1.2, it follows that the composite $\textrm {coker}(d_1^E)(\,f_0-g_0)$ is zero. Hence, $f_0-g_0$ lifts over the image of $d_1^E$ via a morphism $\alpha _0$ as in Figure 15. By the assumption that the class of projective objects is closed under protosplit subobjects, the difference object $D(C_0)$ is projective. Hence, since the bottom chain complex is exact at each level, there exists a lifting $h_0$ of $\alpha _0$ over $\bar {d}_1^E$ . The construction ensures that

\begin{equation*} d_1^E h_0=f_0-g_0. \end{equation*}

We remark that this equality is the same in the abelian context. The only novelty here is the domain $D(C_0)$ shared by $h_0$ and the difference $f_0-g_0$ .

Figure 15.

Construction of $h_0$ .

Figure 16.

Construction of $h_1$ .

Construction of $h_1$ .

The construction of $h_1$ , the second morphism in the homotopy, depends on the same strategy and the same preliminary results about differences of morphisms: see Figure 16. Since

\begin{align*} d_1^E((\,f_1-g_1)-h_0 D(d_1^C)) & = d_1^E(\,f_1-g_1)-d_1^Eh_0 D(d_1^C) \\ & = (\,f_0-g_0)D(d_1^C) - (\,f_0-g_0)D(d_1^C) = 0, \end{align*}

there exists a unique morphism $\alpha _1$ such that

\begin{equation*} \textrm {ker}(d_1^E) \alpha _1 = (\,f_1-g_1) - h_0 D(d_1^C). \end{equation*}

Using the exactness of the bottom chain complex and the fact that $D^2(C_1)$ is projective, we obtain a lifting $h_1$ of $\alpha _1$ over $\bar {d}_2^E$ . We find

\begin{equation*} d_2^E h_1 = (\,f_1-g_1)-h_0 D(d_1 ^C). \end{equation*}

Construction of $h_2$ .

We repeat the same strategy once again: we have

\begin{align*} & d_2^E \left ((\,f_2-g_2)\varsigma _{D(C_2)}-h_1D^2(d_2^C) \right ) \\ & \overset {7.1.2}{\phantom {\quad }=\phantom {\quad }} d_2^E(\,f_2-g_2)\varsigma _{D(C_2)}-d_2^Eh_1D^2(d_2^C) \\ & \overset {\text {def. $h_{1}$}}{\phantom {\quad }=\phantom {\quad }} d_2^E(\,f_2-g_2)\varsigma _{D(C_2)}-\left ( (\,f_1-g_1)-h_0D(d_1^C)\right ) D^2(d_2^C) \\ & \overset {7.1.2}{\phantom {\quad }=\phantom {\quad }} d_2^E(\,f_2-g_2)\varsigma _{D(C_2)}-\left ( (\,f_1-g_1)D(d_2^C)-h_0D(d_1^C)D(d_2^C)\right ) \\ & \overset {}{\phantom {\quad }=\phantom {\quad }} d_2^E(\,f_2-g_2)\varsigma _{D(C_2)}-\left ( (\,f_1-g_1)D(d_2^C)-h_0D(d_1^C d_2^C)\right ) \\ & \overset {}{\phantom {\quad }=\phantom {\quad }} d_2^E(\,f_2-g_2)\varsigma _{D(C_2)}-\left ( (\,f_1-g_1)D(d_2^C)-h_0D(0)\right ) \\ & \overset {7.1.1}{\phantom {\quad }=\phantom {\quad }} d_2^E(\,f_2-g_2)\varsigma _{D(C_2)}-\left ( (\,f_1-g_1)D(d_2^C)-0\right ) \\& \overset {7.1.3}{\phantom {\quad }=\phantom {\quad }} d_2^E(\,f_2-g_2)\varsigma _{D(C_2)}-(\,f_1-g_1)D(d_2^C)\left ( 1_{D(C_2)}-0\right ) \\ \end{align*}

\begin{align*}& \overset {7.1.3}{\phantom {\quad }=\phantom {\quad }} d_2^E(\,f_2-g_2)\varsigma _{D(C_2)}-(\,f_1-g_1)D(d_2^C)\varsigma _{D(C_2)} \\[3pt]& \overset {7.1.2}{\phantom {\quad }=\phantom {\quad }} d_2^E(\,f_2-g_2)\varsigma _{D(C_2)}-(\,f_1d_2^C-g_1d_2^C)\varsigma _{D(C_2)} \\[3pt] & \overset {}{\phantom {\quad }=\phantom {\quad }} d_2^E(\,f_2-g_2)\varsigma _{D(C_2)}-(d_2^E f_2-d_2^Eg_2)\varsigma _{D(C_2)} \\[3pt]& \overset {7.1.2}{\phantom {\quad }=\phantom {\quad }} d_2^E(\,f_2-g_2)\varsigma _{D(C_2)}-d_2^E(\,f_2-g_2)\varsigma _{D(C_2)} \\[3pt] & \overset {7.1.4}{\phantom {\quad }=\phantom {\quad }}0. \end{align*}

Therefore, by the exactness of the bottom chain complex and by the fact that $D^3(C_2)$ is projective, we obtain a lifting $h_3$ of the unique morphism induced by the kernel of $d_2^E$ over $\bar {d}_3^E$ . In summary, $d_3^E h_2 = (\,f_2-g_2)\varsigma _{D(C_2)}-h_1 D^2(d_2^C)$ .

General case.

The construction of $h_2$ already gives us all the tools for defining $h_n$ when $n\geq 3$ . Following essentially the same strategy (which now involves Equation (10) as well), we find a lifting $h_n$ that satisfies

\begin{equation*} d_{n+1}^E h_n = (\,f_n-g_n)\varsigma ^{n-1}_{D(C_n)} - h_{n-1} D^n (d _n ^C). \end{equation*}

By the convention explained in Remark 7.4.1, this general formula agrees with the cases $n=1$ and $n=2$ . However, the case $n=0$ differs slightly. In summary: with the previous assumptions and notation, we have a collection of morphisms $(h_n)_{n\in \mathbb {N}}$ where $h_n:\; D^{n+1}(C_n) \to E_{n+1}$ such that

(11) \begin{equation} \begin{cases} d_1^E h_0 = (\,f_0-g_0) & \\ d_{n+1}^E h_n= (\,f_n-g_n)\varsigma ^{n-1}_{D(C_n)} - h_{n-1} D^n (d_n^C) & \text {for $n\geq 1$}. \end{cases} \end{equation}

In the context of a pointed category with kernels and binary coproducts, we can do exactly the same thing, but the notation is slightly different ( $d_0$ instead of $\textrm {coker}(d_1)$ , see Remark 6.1.6). This leads us to the following definition.

Definition 8.1.1. Let $\mathscr {C}$ be a pointed category with kernels and binary coproducts. Consider morphisms $f$ , $g:\; C\to E$ between positively graded chain complexes $C$ and $E$ . An (approximate) (chain) homotopy from $f$ to $g$ is a sequence of morphisms $(h_n:\; D^{n+1}(C_n)\to E_{n+1})_{n\in \mathbb {N}}$ for which Equations (11) hold. We write $h:\; f\simeq g$ .

Two positively graded chain complexes $C$ and $E$ are (approximately) chain homotopically equivalent ( $C\simeq D$ ) when chain morphisms $\phi :\; C\to E$ and $\psi :\; E\to C$ exist, together with approximate homotopies $1_C\simeq \psi \phi$ and $1_E\simeq \phi \psi$ .

We proved:

Theorem 8.1.2 (Homotopy unique existence of projective resolutions). Let $\mathscr {C}$ be a pointed category with kernels and binary coproducts for which Condition $\textrm {(P)}$ holds. Any lifting of a morphism $x:\; X\to Y$ in the sense of Lemma 6.1.5 is unique up to an approximate chain homotopy.

When, moreover, $\mathscr {C}$ has enough projectives, then for any object $X\in {\mathscr {C}}$ there exists a projective resolution, unique up to an approximate chain homotopy equivalence.

This result is of key importance in ensuring that the non-additive left-derived functors we consider in Section 10 are well defined.

Remark 8.1.3 (Extension to bounded below chain complexes). By renumbering, Definition 8.1.1 extends to bounded below chain complexes.

Remark 8.1.4 (Comparison to the abelian case). The equalities in (11) are expressed in a non-additive context. In an abelian category, we recover the usual: a collection of morphisms $(h_n:\; C_n \to E_{n+1})_{n\in \mathbb {N}}$ such that

\begin{align*} \begin{cases} d_1^E h_0 = (\,f_0-g_0) & \\ d_{n+1}^E h_n= (\,f_n-g_n) - h_{n-1} d_n^C & \text {for $n\geq 1$}. \end{cases} \end{align*}

8.2. Basic stability properties

First of all, it is easy to see that the approximate homotopy relation is compatible with pre- and post-composition with chain morphisms. Indeed, consider a chain homotopy $h:\; f\simeq g:\; C\to E$ . Lemma 7.1.2 implies that if $\alpha :\; A\to C$ is a chain morphism, then $(h_n D^{n+1}(\alpha _n))_n$ is an approximate homotopy from $f\alpha$ to $g\alpha$ , while if $\beta :\; E\to B$ is a chain morphism, then $(\beta _{n+1}h_n)_n$ is an approximate homotopy from $\beta f$ to  $\beta g$ .

Lemma 7.1.4 implies that $0:\; f\simeq f$ for any chain morphism $f$ , while $0:\; f\simeq g$ entails $f=g$ in any normal subtractive category. In particular, the homotopy relation is always reflexive. We next prove that is also symmetric. Here, we need the following construction.

For any object $X$ , we consider the twist morphism $\textrm {tw}_{D(X)}:\; D(X)\to D(X)$ , the restriction of $\langle \iota _2, \iota _1\rangle$ to $D(X)$ :

It is natural, in the sense that for any morphism $l:\; X\to Y$ ,

\begin{equation*} D(l)\textrm {tw}_{D(X)}=\textrm {tw}_{D(Y)} D(l). \end{equation*}

Moreover, for any arrows $f$ , $g:\; X\to Y$ , the equation

\begin{equation*} (\,f-g)\textrm {tw}_{D(X)}=\langle f, g\rangle \delta _X \textrm {tw}_{D(X)} = \langle f, g\rangle \langle \iota _2, \iota _1\rangle \delta _X = \langle g, f\rangle \delta _X = g-f \end{equation*}

holds.

Proof. Consider $h:\; f\simeq g$ . We construct an approximate homotopy from $g$ to $f$ by recursion. For the base step, we let $h^{\prime}_0\; :\!= h_0 \textrm {tw}_{D(C_0)}$ so that

\begin{equation*} d_1^Eh^{\prime}_0 =(\,f_0-g_0)\textrm {tw}_{D(C_0)}=g_0-f_0. \end{equation*}

For $n\geq 1$ , we assume that $h^{\prime}_{n-1}\; :\!= h_{n-1} D^{n-1}(\textrm {tw}_{D(C_{n-1})})$ is defined and satisfies

\begin{equation*} d_{n}^E h^{\prime}_{n-1}= (g_n-f_n)\varsigma ^{n-1}_{D(C_n)} - h^{\prime}_{n-2} D^{n-1}(d_{n-1}^C). \end{equation*}

If now $h^{\prime}_n\; :\!= h_nD^n(\textrm {tw}_{D(C_n)})$ , then

\begin{align*} d_{n+1}^E h^{\prime}_n & = \bigl ( (\,f_n-g_n)\varsigma ^{n-1}_{D(C_n)} - h_{n-1} D^n (d_n^C) \bigr )D^n(\textrm {tw}_{D(C_n)}) \\[3pt] & = (\,f_n-g_n)\varsigma ^{n-1}_{D(C_n)} D^{n-1}(\textrm {tw}_{D(C_n)})- h_{n-1} D^n (d_n^C) D^{n-1}(\textrm {tw}_{D(C_n)}) \\[3pt] & = (\,f_n-g_n) \textrm {tw}_{D(C_n)} \varsigma ^{n-1}_{D(C_n)}- h_{n-1} D^{n-1}\bigl ( D(d_n^C) \textrm {tw}_{D(C_n)}\bigr ) \\[3pt] & =(g_n-f_n)\varsigma ^{n-1}_{D(C_n)}- h_{n-1} D^{n-1}\left (\textrm {tw}_{D(C_{n-1})}D(d_n^C)\right ) \\[3pt] & =(g_n-f_n)\varsigma ^{n-1}_{D(C_n)}- h_{n-1} D^{n-1}(\textrm {tw}_{D(C_{n-1})})D^n(d_n^C) \\[3pt] & =(g_n-f_n)\varsigma ^{n-1}_{D(C_n)}- h_{n-1}'D^n(d_n^C). \end{align*}

Figure 17.

Comparison between $\textrm {Z}_n(\,f)-\textrm {Z}_n(g)$ and $\textrm {Z}_n (\,f-g)$ .

8.3. Approximate homotopy and homology

If, in an abelian category, two chain morphisms $f$ , $g:\; (C,d^C)\to (E,d^E)$ are homotopic, then $\textrm {H}_n(\,f)=\textrm {H}_n(g)$ for all $n$ (see for instance [Reference Cartan and Eilenberg22]). As recalled in Section 1.1, the construction of the left-derived functors crucially depends on this fact.

The usual (abelian) proof simplifies because the homology functors are additive. This is no longer true in a non-additive setting. The reason is that the kernel functor need not preserve differences. More precisely, in a pointed finitely complete category with binary coproducts, $\textrm {Z}_n(\,f-g)$ and $\textrm {Z}_n(\,f)-\textrm {Z}_n(g)$ need not agree. As depicted in Figure 17, there is a canonical comparison $k_n^C$ , the unique morphism for which $D(\textrm {ker} (d_n^C))= \textrm {ker} (D(d_n^C)) k_n^C$ . Hence,

\begin{equation*} \textrm {Z}_n(\,f-g)k_n^C = \textrm {Z}_n(\,f)-\textrm {Z}_n(g). \end{equation*}

This morphism $k_n^C$ need not be an isomorphism in general. For this reason, the non-additive proof of the compatibility of chain homotopy with homology is slightly more involved.

Proof. In a normal subtractive category $\mathscr {C}$ , suppose that $h$ is a chain homotopy $f\simeq g:\; (C,d^C)\to (E,d^E)$ . Consider $n\geq 1$ . We observe that, by definition of $\textrm {H}_n$ , it suffices to show

\begin{equation*} \textrm {H}_n(\,f) \textrm {coker}(\bar {d}^C_{n+1})= \textrm {H}_n(g) \textrm {coker}(\bar {d}^C_{n+1}). \end{equation*}

By the construction of the functors $\textrm {Z}_n$ and $\textrm {H}_n$ as in Diagram (5), this is equivalent to

\begin{equation*} \textrm {coker}(\bar {d}^E_{n+1}) \textrm {Z}_n(\,f) = \textrm {coker}(\bar {d}^E_{n+1}) \textrm {Z}_n(g). \end{equation*}

Since $\mathscr {C}$ is normal, by Lemma 7.1.4 and Lemma 7.1.2, it suffices that

\begin{equation*} \textrm {coker}(\bar {d}^E_{n+1}) \textrm {Z}_n(\,f) - \textrm {coker}(\bar {d}^E_{n+1}) \textrm {Z}_n(g) = \textrm {coker}(\bar {d}^E_{n+1}) (\textrm {Z}_n(\,f)- \textrm {Z}_n(g)) = 0. \end{equation*}

Since $\mathscr {C}$ is subtractive and by definition of $\varsigma ^n$ (see Remark 7.4.1), the equality

\begin{equation*} \textrm {coker}(\bar {d}^E_{n+1}) (\textrm {Z}_n(\,f)- \textrm {Z}_n(g)) \varsigma ^n_{D(\textrm {Z}_n(C))} = 0 \end{equation*}

is sufficient for this to happen.

Figure 18.

Homotopy versus homology.

We prove that $(\textrm {Z}_n(\,f)- \textrm {Z}_n(g)) \varsigma ^n_{D(\textrm {Z}_n(C))}$ lifts over $\bar {d}_{n+1}^E$ as in Figure 18:

\begin{align*} & \textrm {ker} (d_n^E) \bar {d}_{n+1}^Eh_n D^{n+1}(\textrm {ker} (d_n ^C)) \\ & \overset {}{\phantom {\quad }=\phantom {\quad }} d_n^Eh_n D^{n+1}(\textrm {ker} (d_n ^C)) \\ & \overset {\text {def. $h_{n}$}}{\phantom {\quad }=\phantom {\quad }} ((\,f_n-g_n)\varsigma ^{n-1}_{D(C_n)} - h_{n-1}D^n(d_n^C))D^{n+1}(\textrm {ker} (d_n ^C)) \\ & \overset {7.1.2}{\phantom {\quad }=\phantom {\quad }} (\,f_n-g_n)\varsigma ^{n-1}_{D(C_n)} D^n(\textrm {ker} (d_n ^C)) - h_{n-1}D^n(d_n^C)D^n(\textrm {ker} (d_n ^C)) \\ & \overset {}{\phantom {\quad }=\phantom {\quad }} (\,f_n-g_n)\varsigma ^{n-1}_{D(C_n)} D^n(\textrm {ker} (d_n ^C)) - h_{n-1}D^n(d_n^C \textrm {ker} (d_n ^C)) \\ & \overset {}{\phantom {\quad }=\phantom {\quad }} (\,f_n-g_n)\varsigma ^{n-1}_{D(C_n)} D^n(\textrm {ker} (d_n ^C)) - h_{n-1}D^n(0) \\ & \overset {7.1.1}{\phantom {\quad }=\phantom {\quad }} (\,f_n-g_n)\varsigma ^{n-1}_{D(C_n)} D^n(\textrm {ker} (d_n ^C)) - 0 \\ & \overset {7.1.3}{\phantom {\quad }=\phantom {\quad }} (\,f_n-g_n)\varsigma ^{n-1}_{D(C_n)} D^n(\textrm {ker} (d_n ^C)) (1_{D^n(\textrm {Z}_n(C))} - 0) \\ & \overset {}{\phantom {\quad }=\phantom {\quad }} (\,f_n-g_n)\varsigma ^{n-1}_{D(C_n)} D^n(\textrm {ker} (d_n ^C)) \varsigma _{D^n(\textrm {Z}_n(C))} \\ & \overset {(10)}{\phantom {\quad }=\phantom {\quad }} (\,f_n-g_n)D(\textrm {ker} (d_n ^C)) \varsigma ^{n-1}_{D(\textrm {Z}_n(C))} \varsigma _{D^n(\textrm {Z}_n(C))} \\ & \overset {7.4.1}{\phantom {\quad }=\phantom {\quad }} (\,f_n-g_n)D(\textrm {ker} (d_n ^C)) \varsigma ^{n}_{D(\textrm {Z}_n(C))} \\ & \overset {7.1.2}{\phantom {\quad }=\phantom {\quad }} (\,f_n \textrm {ker} (d_n ^C)-g_n \textrm {ker} (d_n ^C)) \varsigma ^{n}_{D(\textrm {Z}_n(C))} \\ & \overset {\text {def. $\textrm {Z}_{n}$}}{\phantom {\quad }=\phantom {\quad }} (\textrm {ker} (d_n ^E)\textrm {Z}_n(\,f)-\textrm {ker} (d_n ^E)\textrm {Z}_n(g)) \varsigma ^{n}_{D(\textrm {Z}_n(C))} \\ & \overset {7.1.2}{\phantom {\quad }=\phantom {\quad }}\textrm {ker} (d_n ^E) (\textrm {Z}_n(\,f)-\textrm {Z}_n(g)) \varsigma ^{n}_{D(\textrm {Z}_n(C))}. \end{align*}

Since $\textrm {ker} (d_n ^E)$ is monic, this implies that

\begin{equation*} \bar {d}_{n+1}^Eh_n D^{n+1}(\textrm {ker} (d_n ^C)) = (\textrm {Z}_n(\,f)-\textrm {Z}_n(g)) \varsigma ^{n}_{D(\textrm {Z}_n(C))}. \end{equation*}

To complete the proof, we still have to verify the case $n=0$ . Here, we can use essentially the same strategy as above. We have:

\begin{align*} \textrm {ker}(\eta ^E)\bar {d}_{1}^Eh_0D(\textrm {ker} (\eta ^C)) & =d_1^Eh_0D(\textrm {ker} (\eta ^C)) \\ & = (\,f_0-g_0)D(\textrm {ker} (\eta ^C)) \\ & =f_0 \textrm {ker} (\eta ^C) - g_0 \textrm {ker} (\eta ^C) \\ & =\textrm {ker} (\eta ^E) (\textrm {Z}_0(\,f)-\textrm {Z}_0(g)). \end{align*}

Since $\textrm {ker} (\eta ^E)$ is monic, we have $\bar {d}_{1}^E h_0 D(\textrm {ker}(\eta ^C)) = \textrm {Z}_0(\,f)-\textrm {Z}_0(g)$ , which proves that $\textrm {Z}_0(\,f)-\textrm {Z}_0(g)$ lifts over $\bar {d}_{1}^E$ .

9.1. Subtractive functors

A first idea is to replace the condition that

\begin{equation*} F(\,f+g)=F(\,f)+F(g) \qquad \text {by}\qquad F(\,f-g)=F(\,f)-F(g). \end{equation*}

We will call such a functor subtractive. Formally, this amounts to the following:

Remark 9.1.2 (Equivalent characterization of subtractive functors). When both $\mathscr {C}$ and $\mathscr {E}$ are normal, this is equivalent to asking that binary coproducts and split short exact sequences of the form

are preserved.

This is not quite enough, though, for our purposes, because a problem occurs which is non-existent in the abelian case: we need preservation of proper morphisms, in order to guarantee that the derived functor will detect exact in the codomain of $F$ —something which is not implied by subtractivity. Combining these ingredients, we find:

9.2. Protoadditive functors preserving coproducts

We now consider a convenient class of subtractive functors: the protoadditive functors of T. Everaert and M. Gran [Reference Everaert and Gran31, Reference Everaert and Gran32].

Remark 9.2.2. From the perspective of Definition 9.2.1 , protoadditivity of a functor appears to be a weak kind of left exactness property. When $\mathscr {C}$ and $\mathscr {E}$ are normal categories, it is equivalent to the condition that $F$ preserves split short exact sequences: for any split short exact sequence

in $\mathscr {C}$ , the image

by $F$ is a split short exact sequence in $\mathscr {E}$ .

Once $\mathscr {C}$ and $\mathscr {E}$ are abelian categories, any split short exact sequences is canonically induced by a biproduct. This implies that a functor between abelian categories is protoadditive if and only if it is additive—so that this left exactness aspect becomes invisible.

The articles [Reference Everaert and Gran31, Reference Everaert and Gran32] provide a list of examples: the reflector of commutative rings to reduced commutative rings, for instance, or any reflector from a category of topological semi-abelian algebras to Hausdorff algebras over the same theory. In Section 12, we focus on one in particular: the connected components functor in the context of crossed modules.

10.1. Syzygy Lemma and long exact sequence in homology

We proceed by proving some basic results that hold for functors which are right exact in an appropriate sense. Indeed, outside the abelian context, the concept of a right exact functor bifurcates. The following notion [Reference Peschke and Van der Linden60] is appropriate for our purposes:

Definition 10.1.1 (Sequentially right exact functor). A functor $F:\; {\mathscr {C}}\to {\mathscr {E}}$ where $\mathscr {C}$ and $\mathscr {E}$ are pointed categories is called sequentially right exact when for each short exact sequence

(13)

in $\mathscr {C}$ , the sequence

is exact.

This means, in particular, that the morphism $F(k)$ is proper. Note that in a category with kernels and cokernels, this is equivalent to the definition given in Subsection 2.2. In particular, sequentially right exact functors preserve proper morphisms. Hence, for such a functor between categories $\mathscr {C}$ and $\mathscr {E}$ as in Theorem 10.0.1, it suffices that it is, in addition, subtractive for the theorem to be applicable. Furthermore, protoadditivity of $F$ follows as soon as it preserves protosplit monomorphisms. Examples of such functors include the connected components functors of Section 12 (see [Reference Culot and Van der Linden25, Reference Everaert and Gran31] for further details).

We are now ready to prove versions of a few key fundamental results such as the existence of a long exact sequence in homology (see for instance [Reference Weibel78, Theorem 2.4.6] for the abelian case) or the Syzygy Lemma [Reference Weibel78, Exercise 2.4.3].

A direct consequence of this result is the Syzygy Lemma. Recall that an object $\Omega (X)$ is called a syzygy of an object $X$ if it fits a short exact sequence of the shape

(14)

where $P$ is a projective object. In a normal category with enough projectives, each object $X$ admits a syzygy $\Omega (X)$ .

Theorem 10.1.5 (Syzygy Lemma). Under the assumptions of Theorem 10.1.4, if $\Omega (X)$ is a syzygy of $X\in {\mathscr {C}}$ , then

\begin{equation*} \textrm {L}_{n+1}(F)(X)\cong \textrm {L}_n(F)(\Omega (X)) \end{equation*}

for all $n\geq 1$ .

The case of Schreier varieties (such as the category of groups, for instance) merits some special attention, because in such a category, a syzygy is always a projective object.

11.1. Simplicial objects and homology

When $\mathscr {C}$ is an abelian category, we have the well-known Dold–Kan–Puppe equivalence [Reference Dold26, Reference Dold and Puppe27], which asserts that the category of simplicial objects in $\mathscr {C}$ (denoted ${\mathsf {s}}({\mathscr {C}})$ ) and the category of positively graded chain complexes in $\mathscr {C}$ (denoted ${\mathsf {Ch}}^+({\mathscr {C}})$ ) are equivalent. In a semi-abelian category, this need no longer be the case. For instance, in [Reference Carrasco and Cegarra20], Carrasco and Cegarra provide an explicit description of ${\mathsf {s}}({\mathsf {Gp}})$ as being equivalent to the category of hypercrossed complexes of groups, which differs from ${\mathsf {Ch}}^+({\mathsf {Gp}})$ . This fact was generalized by Bourn [Reference Bourn12] to semi-abelian categories.

We can, however, still associate a chain complex to any simplicial object, even in a non-abelian category. Indeed, the Moore functor $\textrm {N}:\; {\mathsf {s}}({\mathscr {C}})\to {\mathsf {Ch}}^+({\mathscr {C}})$ can be defined for any pointed category with pullbacks $\mathscr {C}$ . It has been shown in [Reference Borceux and Bourn5, Reference Everaert and Van der Linden34] that if $\mathscr {C}$ is, in addition, exact and protomodular, then $\textrm {N}$ maps any simplicial object to a proper chain complex—hence, the concept of homology of a simplicial object, defined as the homology of the associated Moore chain complex.

For the sake of convenience, we recall some of the details. A semi-simplicial object $\mathbb {A}=(A_n)_{n\geq 0}$ in a category $\mathscr {C}$ is a family of objects, together with face operators $\partial _{i}^{\mathbb {A}}:\; A_n\to A_{n-1}$ for $i\in [n]=\{0,\ldots, n\}$ where $n\gt 0$ , such that