WEAK FACTORIZATIONS, FRACTIONS AND HOMOTOPIES

A. KURZ AND J. ROSICKÝ^{∗}

Abstract. We show that the homotopy category can be assigned to any category equipped with a weak factorization system. A classical example of this construction is the stable category of modules. We discuss a connection with the open map approach to bisimulations proposed by Joyal, Nielsen and Winskel.

1. Introduction

Weak factorization systems originated in homotopy theory (see [Q], [Bo], [Be] and [AHRT]). Having a weak factorization system (L, R) in a category K, we can formally invert the morphisms from R and form the category of fractions K[R^{−1}]. From the point of view of homotopy theory, we invert too few morphisms: only trivial fibrations and not all weak equivalences. Our aim is to show that this procedure is important in many situations.

For instance, the class Mono of all monomorphisms form a left part of the weak factorization system (Mono, R) in a category R-Mod of (left) modules over a ring R. Then R-Mod [R^{−1}] is the usual stable category of modules. Or, in the open map approach to bisimulations suggested in [JNW], one considers a weak factorization system (L, O_{P}), where O_{P} is the class of P-open morphisms w.r.t. a given full subcategory P of path objects. Then two objects K and L are P-bisimilar iff there is a span

\xymatrix{K & M \ar[l]_{f} \ar[r]^{g} & L}

of P-open morphisms. Any two P-bisimilar objects are isomorphic in the fraction category K[O_{P}^{−1}] but, in general, the fraction category makes more objects isomorphic than just P-bisimilar ones.

Date: October 21, 2004.

1991 Mathematics Subject Classification. 18A99, 55P10, 68Q85.

Key words and phrases. weak factorization system, homotopy, category of fractions, bisimilarity.

∗ Supported by the Grant Agency of the Czech Republic under the grant 201/02/0148.

Any weak factorization system (L, R) in a category K with finite coproducts yields a cylinder object in K and thus a relation ∼ of homotopy between morphisms of K. We will show that any two homotopic morphisms have the same image in the fraction category. Moreover, if K has finite coproducts and any morphism in R is a split epimorphism then the categories K[R^{−1}] and K/ ∼ are equivalent.

2. Weak factorization systems

Definition 2.1. Let K be a category and f : A → B, g : C → D morphisms such that in each commutative square

\xymatrix{A \ar[r]^{u} \ar[d]_{f} & C \ar[d]^{g} \\ B \ar[r]_{v} & D}

there is a diagonal d : B → C with d · f = u and g · d = v. Then we say that g has the right lifting property w.r.t. f and f has the left lifting property w.r.t. g.

For a class H of morphisms of K we put

H^{□} = {g|g has the right lifting property w.r.t. each f ∈ H} and
□H = {f|f has the left lifting property w.r.t. each g ∈ H}.

Definition 2.2. ([Be]) A weak factorization system (L, R) in a category K consists of two classes L and R of morphisms of K such that

(1) R = L^{□}, L = ^{□}R and

(2) any morphism h of K has a factorization h = g · f with f ∈ L and g ∈ R.

The category of fractions K[S^{−1}], where K is a category and S a class of morphisms in K, was introduced in [GZ] (see [Bor], [KP]). This category has the same objects as K and is equipped with a functor P : K → K[S^{−1}] sending any morphism from S to an isomorphism. Moreover, for every functor F : K → X sending any morphism from S to an isomorphism, there is a unique functor \overline{F} : K[S^{−1}] → X such that F = \overline{F} · P. It may happen that the category K[S^{−1}] is not locally small, i.e., that one can have a proper class of morphisms between two given objects in K[S^{−1}].

Observation 2.3. K[S^{−1}] is a quotient of the category of zig-zags

\xymatrix{K \ar[r]^{f_{1}} & X_{1} & X_{2} \ar[l]_{f_{2}} \ar[r] & … & L}

where all morphisms going backwards are in S (see [KP], II.2). If K has finite limits and S contains all isomorphisms, is closed under compositions and stable under pullbacks then these zig-zags can be reduced to spans

\xymatrix{K & X \ar[l]_{s} \ar[r]^{f} & L}

with s ∈ S. In fact, a zig-zag is reduced to a span by means of pullbacks as follows:

\xymatrix{K \ar[r]^{f_{1}} & X_{1} & X_{2} \ar[l]_{f_{2}} \ar[r] & … & L \\  & \overline{X_{1}} \ar[ul]^{_{\overline{f}_{2}}} \ar[ur]_{_{\overline{f}_{1}}}}

The equivalence relation on spans giving the category of fractions is easy to describe if S also has the property: if t · f = t · g with t ∈ S then f · s = g · s for some s ∈ S. One then says that S admits a right calculus of fractions (see [Bor]).

Let (L, R) be a weak factorization system in a category K having finite limits. Then R contains all isomorphisms, is closed under compositions and stable under pullbacks (see [AHRT]). Hence the fraction category K[R^{−1}] is a quotient category of the category of spans. But R rarely admits a right calculus of fractions.

Example 2.4. Let (L, R) be a factorization system in a category K having finite limits. This means that (L, R) is defined by means of a unique diagonalization property, i.e., that d in 2.1 is unique. Then (L, R) is a weak factorization system (see 14.6 (3) in [AHS]). We show that R admits a right calculus of fractions. Consider

\xymatrix{K \ar[r]^{h_{1}} \ar[r]_{h_{2}} & L \ar[r]^{t} & M}

such that t · h_{1} = t · h_{2} and t ∈ R. It suffices to show that the equalizer e of h_{1} and h_{2} belongs to R. This means that we have to show that e has the unique diagonalization property w.r.t. each morphism f ∈ L. Consider a commutative square

\xymatrix{X \ar[r]^{u} \ar[d]_{f} & N \ar[d]^{e} \\ Y \ar[r]_{v} & K}

with f ∈ S. Since

\xymatrix{X \ar[r]^{h_{1}eu} \ar[d]_{f} & L \ar[d]^{t} \\ Y \ar[r]_{th_{1}v} \ar[ur]^{h_{1}v} \ar[ur]_{h_{2}v} & M}

commutes, the unique diagonalization property yields that h_{1}·v = h_{2}·v. Thus v = w · e for some w : Y → N. Since e is a monomorphism, w is the unique diagonal in the starting square.

In the case of the factorization system (Iso(K), Mor(K)), the fraction category K[Mor(K)^{−1}] is the set of connected components of K. Here, Iso(K) consists of isomorphisms of K and Mor(K) of all morphisms of K.

Observation 2.5. The class

\overline{S} = \{f | P (f) is an isomorphism\}

is called the saturation of S. It is easy to see that \overline{S} is closed under retracts in the arrow category K^{→} and has the 2-out-of-3 property, i.e., with any two of f, g, g · f belonging to \overline{S} also the third morphism belongs to \overline{S}.

The following definition is motivated by [JNW].

Definition 2.6. Let (L, R) be a weak factorization system in a category K. Two objects K and L are called bisimilar if there is a span

K ⟵^{f} X ⟶^{g} L

with f, g ∈ R.

Observation 2.7. Any two bisimilar objects are clearly isomorphic in the fraction category K[R^{−1}]. If K has finite limits then bisimilarity is an equivalence relation. We will see later that (see 3.6), even in this case, two objects K, L may be isomorphic in K[R^{−1}] without being bisimilar.

Observation 2.8. Let (L, R) be a weak factorization system in a category K having an initial object 0. Then the following two conditions are equivalent:

(1) any morphism from R is a split epimorphism,

(2) any morphism 0 → K belongs to L.

Indeed, (1) ⇒ (2) because a diagonal in a square

\xymatrix{0 \ar[r] \ar[d] & L \ar[d]^{f} \\ K \ar[r]_{v} & M}

is t · v where t splits f ∈ R. Conversely, (2) ⇒ (1) because a diagonal in a square

\xymatrix{0 \ar[r] \ar[d] & L \ar[d]^{f} \\ K \ar[r]_{_{id_{K}}} & K}

splits f ∈ R.

Observation 2.9. Let K be a locally presentable category (cf. [AR]) and C a set of morphisms in K. Then (^{□}(C^{□}), C^{□}) is a weak factorization system and ^{□}(C^{□}) is the smallest class L containing C which is

(a) closed under retracts in K^{→},

(b) closed under compositions and contains all isomorphisms,

(c) stable under pushouts,

(d) closed under transfinite compositions, i.e., given a smooth chain of morphisms (f_{ij} : K_{i} → K_{j})_{i<j<λ} from L (i.e., λ is a limit ordinal, f_{jk} · f_{ij} = f_{ik} for i < j < k and f_{ij} : K_{i} → K_{j} is a colimit cocone for any limit ordinal j < λ), then a colimit cocone f_{i} : K_{i} → K has f_{0} ∈ L.

(see [Be] or [AHRT]). Even, ^{□}(C^{□}) consists of retracts of transfinite compositions of pushouts of morphisms from C.

3. Homotopies

Definition 3.1. Let K be a category with finite coproducts equipped with a weak factorization system (L, R). For an object K of K, we get a cylinder object \overline{K} by a factorization of the codiagonal

∇ : K + K ⟶^{c_{K}} \overline{K} ⟶^{s_{K}} K

with c_{K} ∈ L and s_{K} ∈ R. We denote by

c_{K}^{1} = c_{K} · i_{1} and c_{K}^{2} = c_{K} · i_{2}

the compositions of c_{K} with the coproduct injections

K ⟶^{i_{1}} K + K ⟵^{i_{2}} K .

There is a well-known way of getting homotopy from cylinder objects (see [KP]). Having two morphisms f, g : K → L, we say that f and g are homotopic and write f ∼ g if there is a morphism h : \overline{K} → L such that the following diagram commutes

\xymatrix{K + K \ar[rr]^{(f,g)} \ar[dr]_{c_{K}} &  & L \\  & \overline{K} \ar[ur]_{h}}

Here, (f, g)·i_{1} = f and (f, g)·i_{2} = g. The homotopy relation ∼ is clearly reflexive and symmetric. But, in general, the homotopy relation is not transitive. On the other hand, ∼ is compatible with the composition.

Lemma 3.2. Let K be a category with finite coproducts equipped with a weak factorization system (L, R). Let f, g : K → L, u : L → M and v : N → K be in K and f ∼ g. Then u · f ∼ u · g and f · v ∼ g · v.

Proof. Let h : \overline{K} → L make f and g homotopic. Then u · h makes u · f and u · g homotopic. Using a lifting property, there is a morphism t such that both squares in the following diagram are commutative

\xymatrix{N + N \ar[r]^{v+v} \ar[d]_{c_{N}} & K + K \ar[d]^{c_{K}} \\ \overline{N} \ar@{.>}[r]^{t} \ar[d]_{s_{N}} & \overline{K} \ar[d]^{s_{K}} \\ N \ar[r]_{v} & K}

Then h · t makes f · v and g · v homotopic. □

Observation 3.3. The homotopy relation does not depend on a choice of a cylinder object. The reason is that, for two cylinder objects

∇ : K + K ⟶^{c_{K}} \overline{K} ⟶^{s_{K}} K

and

∇ : K + K ⟶^{\overline{c}_{K}} \overline{\overline{K}} ⟶^{\overline{s}_{K}} K ,

we always have a diagonal t in the square

\xymatrix{K + K \ar[r]^{c_{K}} \ar[d]_{_{\overline{c}_{K}}} & \overline{K} \ar[d]^{s_{K}} \\ \overline{\overline{K}} \ar[ur]_{t} \ar[r]_{_{\overline{s}_{K}}} & K}

Let K be a Quillen model category (see [H]) and let L consist of cofibrations and R of trivial fibrations. Then (L, R) is a weak factorization system and any f, g homotopic in our sense are left homotopic in the standard sense. But the converse does not hold.

Any weak factorization system (L, R) gives rise to a Quillen model category if we take all morphisms of K as weak equivalences ([AHRT] 3.7). Then any two morphisms f, g : K → L are left homotopic because we have a model category cylinder

∇ : K + K ⟶^{id} K + K ⟶^{∇} K

(because ∇ is a weak equivalence).

Let K be a category with finite coproducts equipped with a weak factorization system (L, R). We get the quotient category

Q : K → K/∼ .

Following 3.2, Q(f) = Q(g) iff f and g are in the transitive closure of the homotopy relation ∼, i.e., iff there are f_{1}, …, f_{n} such that

f ∼ f_{1} ∼ ⋯ ∼ f_{n} ∼ g .

Lemma 3.4. Let K be a category with finite coproducts equipped with a weak factorization system (L, R). Then P (f) = P (g) for any morphisms f ∼ g.

Proof. We have

P (s_{K} · c_{K}^{1}) = P (∇ · i_{1}) = P (∇ · i_{2}) = P (s_{K} · c_{K}^{2}) .

Since P (s_{K}) is an isomorphism, we have P (c_{K}^{1}) = P (c_{K}^{2}). Thus P (f) = P (g) for f ∼ g. □

We therefore have a unique functor T such that

\xymatrix{K \ar[rr]^{P} \ar[dr]_{Q} &  & K[R^{−1}] \\  & K/ ∼ \ar[ur]_{T}}

commutes. In general, one cannot expect that T is an equivalence.

Example 3.5. Let K have finite coproducts and consider the factorization system ( Iso(K), Mor(K)). Then a cylinder object for K is K + K and thus any two parallel morphisms are homotopic. Hence Q is the posetal reflection of K. On the other hand, P is (up to equivalence), the projection of K to the set of connected components of K.

If (L, R) is a weak factorization system and ∼ the associated homotopy then a morphism f : K → L is called a homotopy equivalence if there is g : L → K with g · f ∼ id_{K} and f · g ∼ id_{L}. Every homotopy equivalence is sent by Q and, following 3.4, by P as well, to an isomorphism. Since ∼ is not transitive, Q inverts more morphisms than just homotopy equivalences. In fact, Q(f) is an isomorphism iff there is g such that both (g · f, id_{K}) and (f · g, id_{L}) are in the transitive closure of ∼.

The following gives an example of a homotopy relation that is not transitive and of homotopy equivalent objects that are not bisimilar.

Example 3.6. Let Set^{X} be the category of multigraphs with loops, i.e., X is the category

\xymatrix{E \ar[r]^{r_{1}} \ar[r]_{r_{2}} & V \ar[l]_{v}}

where r_{1}·v = r_{2}·v = id_{V}. Here, E is the object of edges, V is the object of vertices, r_{1} and r_{2} yield the initial and the final vertex of an edge and v yields loops. Let L = Mono be the class of all monomorphisms and R consist of morphisms g : K → L such that

(a) g is surjective on vertices and

(b) if vertices g(a) and g(b) are joined by an edge in L then a and

b are joined by an edge in K. In fact, this is the weak factorization system (^{□}(C^{□}), C^{□}) from 2.9 where C consists of the embedding of an empty multigraph into a vertex and of the embedding of two vertices not connected by an edge to the edge.

The cylinder object c_{K} : K + K → \overline{K} is obtained by joining the two copies i_{1}(x) and i_{2}(x) of a vertex x in K by two edges, one going from i_{1}(x) to i_{2}(x) and the other going from i_{2}(x) to i_{1}(x). Moreover, having an edge from x to y, there is an edge going from i_{1}(x) to i_{2}(y) and an edge going from i_{2}(x) to i_{1}(y). This means that \overline{K} = E′ × K where E′ is the non-oriented edge, i.e., a complete graph on two vertices.

Morphisms f, g : K → L are homotopic iff for each vertex x of K, we have an edge from f(x) to g(x) and an edge from g(x) to f(x) in L. Moreover, having an edge from x to y, there is an edge going from f(x) to g(y) and an edge going from g(x) to f(y). Thus the homotopy relation is not transitive.

The multigraphs (loops are not depicted)

[figure: ∘; ∘; a; b]  and

[figure: c; ∘; ∘; ∘; a; b]

are homotopy equivalent. In fact, the first multigraph K is a retract of the second multigraph L via u · v = id_{K} and the other composition v · u is homotopic to id_{L}. Hence K and L are isomorphic in K[R^{−1}]. But K and L are not bisimilar. Indeed, assume that there exist

K ⟵^{f} M ⟶^{g} L

with f, g ∈ R. There are x, y ∈ M with g(x) = b and g(y) = c. Since f(x) and f(y) are joined by an edge in K, x and y are joined by an edge in M and thus b and c are connected by an edge in L; a contradiction.

Lemma 3.7. Let K be a category having finite coproducts and (L, R) be a weak factorization system such that every morphism in R is a split epimorphism. Then every morphism in R is a homotopy equivalence.

Proof. Let r : K → L be in R and consider f with r · f = id_{L}. It suffices to show that f · r ∼ id_{K}. Consider the square

\xymatrix{K + K \ar[r]^{(f·r,id_{K})} \ar[d]_{c_{K}} & K \ar[d]^{r} \\ \overline{K} \ar[r]_{r·s_{K}} \ar@{-->}[ur]^{t} & L}

which commutes because r · f · r = r. Since c_{K} ∈ L and r ∈ R, there is a diagonal t, which is a homotopy from f · r to id_{K}. □

Remark 3.8. We have proved a stronger statement: if r ∈ R is a split epimorphism then r is a homotopy equivalence. Following 2.8, if 0 → L is in L then every r : K → L from R is a homotopy equivalence.

Theorem 3.9. Let K be a category having finite coproducts and (L, R) be a weak factorization system such that every morphism in R is a split epimorphism. Then

(1) the categories K[R^{−1}] and K/ ∼ are isomorphic,

(2) \overline{R} = \{f|Q(f) is an isomorphism\} and

(3) K[R^{−1}] is a locally small category.

Proof. (1) Following 3.7, Q inverts all morphism from R. Thus we get a unique functor U such that the triangle

\xymatrix{K \ar[rr]^{P} \ar[dr]_{Q} &  & K[R^{−1}] \ar[dl]^{U} \\  & K/ ∼}

commutes. Since both U · T and T · U are the identities, T is an isomorphism. It immediately yields (2) to (3). □

Observation 3.10. Assume that (L, R) is a weak factorization system in a category K such that every morphism in R is a split epimorphism. Let R′ consist of compositions r · f where r ∈ R and f splits some s ∈ R, i.e., s · f = id. Then two objects K and L are bisimilar iff there is h : K → L in R′.

(1) R′ is closed under compositions.

Consider the composition

K_{1} ⟶^{f_{1}} X_{1} ⟶^{r_{1}} K_{2} ⟶^{f_{2}} X_{2} ⟶^{r_{2}} K_{3}

where r_{1}, r_{2} ∈ R and s_{1} · f_{1} = id_{K_{1}}, s_{2} · f_{2} = id_{K_{2}} for s_{1}, s_{2} ∈ R. Let

\xymatrix{L \ar[r]^{^{\overline{s}^{2}}} \ar[d]_{_{\overline{r}_{1}}} & X_{1} \ar[d]^{r_{1}} \\ X_{2} \ar[r]_{s_{2}} & K_{2}}

be a pulback. Then \overline{r}_{1}, \overline{s}_{2} ∈ R and, since

s_{2} · f_{2} · r_{1} = r_{1} ,

there is a unique t : X_{1} → L with

\overline{r}_{1} · t = f_{2} · r_{1} and \overline{s}_{2} · t = id_{X_{1}} .

Thus

r_{2} · f_{2} · r_{1} · f_{1} = r_{2} · \overline{r}_{1} · t · f_{1}

where r_{2} · \overline{r}_{1} ∈ R and s_{1} · \overline{s}_{2} · t · f_{1} = id_{K_{1}}, s_{1} · \overline{s}_{2} ∈ R. Hence r_{2} · f_{2} · r_{1} · f_{1} ∈ R′.

(2) If g · h, g ∈ R′ then h ∈ R′. Let g = r_{1} · f_{1} and g · h = r_{2} · f_{2} where r_{1}, r_{2} ∈ R and f_{i} splits some s_{i} ∈ R for i = 1, 2.

Consider

\xymatrix{K_{1} \ar[r]^{h} \ar[drr]^{t} \ar[ddrr]_{f_{2}} & K_{2} \ar[r]^{f_{1}} & X \ar[r]^{r_{1}} & K_{3} \\  &  & L \ar[u]^{_{\overline{r}_{2}}} \ar[d]_{_{\overline{r}_{1}}} \\  &  & Y \ar[uur]_{r_{2}}}

where r_{1}, r_{2} ∈ R and s_{1} · f_{1} = id_{K_{2}}, s_{2} · f_{2} = id_{K_{1}} for s_{1}, s_{2} ∈ R. Take a pullback of r_{1} and r_{2} and consider the induced morphism t. Since s_{2} · \overline{r}_{1}, s_{1} · \overline{r}_{2} ∈ R,

(s_{2} · \overline{r}_{1}) · t = s_{2} · f_{2} = id_{K_{1}} ,

and

(s_{1} · \overline{r}_{2}) · t = s_{1} · f_{1} · h = h ,

we get h ∈ R′.

(3) Any g ∈ R′ ∩ L is a split monomorphism.

Let g = r · f where s · f = id and r, s ∈ R. Consider the square

\xymatrix{K \ar[r]^{f} \ar[d]_{r·f} & X \ar[d]^{r} \\ L \ar@{-->}[ur]_{t} \ar[r]_{_{id_{L}}} & L}

If g ∈ L we get a diagonal t and we have

s · t · (r · f) = s · f = id_{K} .

(4) R′ does not need to have the 2-out-of-3 property.

Let Set^{X} be the category of multigraphs with loops from 3.6 and consider the following multigraphs K and L (loops are not depicted):

[figure: ∘; ∘; a; b]  and

[figure: c; ∘; ∘; ∘; a; b]

Let f : K → L be the embedding (i.e., f(a) = a and f(b) = b) and s, t : L → K split f by means of s(c) = a and t(c) = b. Then s ∈ R and thus f ∈ R′. Assume that t ∈ R′. Then t = r · g where r ∈ R and s′ · g = id for some s′ ∈ R. Then there is an edge from g(b) to g(c) and thus and edge from b to c; a contradiction.

4. Stable equivalences

A full subcategory M of a category K is weakly reflective if for every object K in K there is a morphism r_{K} : K → K^{∗} with K^{∗} ∈ M such that every morphism f : K → M, M ∈ M factorizes through r_{K}, i.e., f = g · r_{K}.

Observation 4.1. Let (L, R) be a weak factorization system in a category K with finite products and L^{△} the full subcategory of K consisting of objects M injective w.r.t. any morphism from L. This means that for every f : X → Y in L and every h : X → M there is g : Y → M with g · f = h. Then L^{△} is weakly reflective in K where a weak reflection is given by a factorization

K ⟶^{r_{K}} K^{∗} ⟶^{t} 1

with r_{K} ∈ L and t ∈ R of the unique morphism into the terminal object 1. Let (L^{△})_{inj} consist of all morphism f such that each M ∈ L^{△} is injective to f. Then

(L^{△})_{inj} = {f| there exists g ∈ L with g = h · f for some h}

which means that (L^{△})_{inj} is the left cancellable closure of L.

Indeed, it is easy to see that the left cancellable closure of L is contained in L_{inj}^{△}. Conversely, having f : K → L in L, then a weak reflection r_{K} : K → K^{∗} factorizes through f. Thus f belongs to the left cancellable closure of L. Consequently, L = (L^{△})_{inj} whenever L is left cancellable.

Conversely, consider a weakly reflective full subcategory M of K. Then, following [AHRT] 1.5, (M_{inj} , (M_{inj} )^{□}) is a weak factorization system in K. A weak factorization of h : A → B is

h : K ⟶^{⟨r_{K},h⟩} K^{∗} × L ⟶^{p_{2}} L .

Therefore, weak factorization systems (L, R) in K with L left cancellable are precisely (M_{inj} , (M_{inj} )^{□}) for a weakly reflective full subcategory M of K.

Let M be a weakly reflective full subcategory of an additive category K. Morphisms f, g : K → L are called M-stably equivalent if f − g factorizes through an object M from M. It is easy to see that this is an equivalence relation compatible with the composition. One gets the stable category K/M and the projection S : K → K/M. We have the following results which are almost completely contained in [B] 4.5.

Lemma 4.2. Let M be a weakly reflective full subcategory of an additive category K. Then morphisms f, g : K → L are M-stably equivalent iff f ∼ g with respect to the weak factorization system (M_{inj} , (M_{inj} )^{□}).

Proof. Following 4.1, cylinder objects are

K ⊕ K ⟶^{c_{K}} K^{∗} ⊕ K^{∗} ⊕ K ⟶^{p_{3}} K

where p_{i} · c_{K}^{j} = r_{K} for i = 1, 2 and j = 1, 2. Of course, p_{3} · c_{K}^{j} = id_{K} for j = 1, 2. Hence c_{K}^{1} − c_{K}^{2} factorizes through K^{∗} ⊕ K^{∗} and thus c_{K}^{1} and c_{K}^{2} are M-stably equivalent. Since M-stable equivalence is compatible with the composition, f ∼ g implies that f and g are Mstably equivalent. Conversely, let f and g be M-stably equivalent, i.e., we have

f − g : K ⟶^{u} M ⟶^{v} L

with M ∈ M. Then f ∼ g via

h : K^{∗} ⊕ K^{∗} ⊕ K ⟶ L

given by h · i_{1} = 0, h · i_{2} = −v · t and h · i_{3} = f where

\xymatrix{K \ar[rr]^{u} \ar[dr]_{r_{K}} &  & M \\  & K^{∗} \ar[ur]_{t}}

□

The proof contains the description of a cylinder object using weak reflections to M, which may be useful in homotopy theory of additive categories. We will need this in the following proof.

Consequently, homotopy equivalences coincide with M-stable equivalences, i.e., with morphisms f admitting g with g · f and f · g Mstably equivalent to the identities. A monomorphism f is called an M-monomorphism if its cokernel belongs to M and an epimorphism g is called an M-epimorphism if its kernel belongs to M.

Theorem 4.3. Let (L, R) be a weak factorization system in an additive category K such that L is left cancellable. Then the following conditions are equivalent for a morphism h:

(i) h ∈ \overline{R}, (ii) h is a homotopy equivalence, (iii) h is an L^{△}-stable equivalence, and (iv) h = g · f with f a split L^{△}-monomorphism and g a split L^{△}-

epimorphism.

Proof. (i) ⇒ (iii). Consider h ∈ R and the factorization

h : K ⟶^{⟨r_{K},h⟩} K^{∗} ⊕ L ⟶^{p_{2}} L

from 4.1. We have

\xymatrix{K \ar[r]^{^{id^{K}}} \ar[d]_{⟨r_{K},h⟩} & K \ar[d]^{h} \\ K^{∗} ⊕ L \ar@{-->}[ur]^{t} \ar[r]_{p_{2}} & L}

and thus h is a retract of p_{2} in K^{→} via

\xymatrix{K \ar[r]^{h} \ar[d]_{⟨r_{K},h⟩} & L \ar[d]^{_{id_{L}}} & K \ar[r]^{h} & L \\ K^{∗} ⊕ L \ar[r]_{p_{2}} & L & K^{∗} ⊕ L \ar[u]^{t} \ar[r]_{p_{2}} & L \ar[u]_{_{id_{L}}}}

and

Since p_{2} · i_{2} = id_{L} and i_{2} · p_{2} = id_{K^{∗}⊕L} −i_{1} · p_{1}, p_{2} is an L^{△}-stable equivalence with an L^{△}-stable inverse i_{2}. Hence S : K → K/L^{△} factorizes through P : K → K[R^{−1}] and thus each morphism from \overline{R} is an L^{△}-stable equivalence. (ii) ⇔ (iii) following 4.2 (iii) ⇒ (iv). Let h = g·f be a factorization of an L^{△}-stable equivalence with f ∈ L and g ∈ R. Following the proof of (i) ⇒ (iii), g is a retract of a split L^{△}-epimorphisms p_{2}. Hence g is a split L^{△}-epimorphism.

Since (i) ⇒ (iii), g is an L^{△}-stable equivalence and thus f is an L^{△}- stable equivalence. Let t be an L^{△}-stable inverse of f. Then t · f is L^{△}-stably equivalent to the identity and thus we have a factorization

id_{K} −t · f : K ⟶^{u} M ⟶^{v} K

through M ∈ L^{△}. Since f ∈ L, we have a factorization

\xymatrix{K \ar[rr]^{f} \ar[dr]_{u} &  & L \ar[dl]^{w} \\  & M}

We have

id_{K} −t · f = v · u = v · w · f

and thus

id_{K} = (v · w + t) · f .

Thus f is a split monomorphism. Consequently, f has a cokernel g : L → P which is a split epimorphism.

Consider a morphism h such that S(h · f) = 0 in K/L^{△}. Then h · f factorizes through N ∈ L^{△}

\xymatrix{K \ar[r]^{f} \ar[d]_{r} & L \ar@{-->}[dl]^{p} \ar[d]^{h} \\ N \ar[r]_{s} & X}

Since f ∈ L, there is p : L → N with p · f = r. We have

(h − s · p) · f = h · f − s · p · f = h · f − s · r = 0

and thus there is q : P → X such that

q · g = h − s · p .

Hence S(q · g) = S(h) and thus S(g) is a weak cokernel of S(f) in K/L^{△}. Since S(g) is a split epimorphism, S(g) is a cokernel of S(f) in K/L^{△}. However, S(f) is an isomorphism and therefore P is the null object in K/L^{△}. Consequently, P ∈ L^{△}, which proves that f is a split L^{△}-monomorphism. (iv) ⇒ (i) Let f : K → L be a split L^{△}-monomorphism. Then f is an injection of a biproduct

f : K ⟶ K ⊕ M

where M ∈ L^{△}. Since the corresponding projection belongs to R (see 4.1 or the proof of 1.6 in [AHRT]), we get f ∈ \overline{R}. Any split L^{△}- epimorphism g : K → L is a projection of a biproduct

L ⊕ M ⟶ L

with M ∈ L^{△} and thus belongs to R. □

Observation 4.4. Following the proof of (iii) ⇒ (iv), any g ∈ R is a split L^{△}-epimorphism. Conversely, following (iv) ⇒ (i), any split L^{△}-epimorphism belongs to R. Thus compositions g · f from (iv) are precisely morphism from R′ (cf. 3.10). Consequently, homotopy equivalent objects are precisely bisimilar objects.

Example 4.5. Consider the category R-Mod of R-modules and the class L = Mono of all monomorphisms. Then L^{△} consists of injective R-modules and R = Mono^{□} of all epimorphisms with an injective kernel. Then (L, R) is a weak factorization system, L is left cancellable and R-Mod /L^{△} is the usual stable category of modules.

5. Preshaves over posets

An object K of a category K is called indecomposable if the homfunctor hom(K, −) : K → Set preserves binary coproducts.

Proposition 5.1. Let K be a category with finite coproducts and (L, R) a weak factorization system such that R = C^{□} where every morphism from C has an indecomposable domain. Then K/ ∼ is equivalent to a poset.

Proof. If suffices to show that f ∼ g for each f, g : K → L. Consider a commutative square

\xymatrix{X \ar[rr]^{u} \ar[dd]_{f} \ar@{-->}[dr]_{u′} &  & K + K \ar[dd]^{∇} \\  & K \ar[ur]_{i_{1}} \ar[dr]^{_{id_{K}}} \\ Y \ar[rr]_{v} \ar[ur]_{v} &  & K}

with f ∈ C. Then X is indecomposable and thus u factorizes through one of the coproduct injections, say, u = i_{1} · u′. Then

i_{1} · v · f = i_{1} · ∇ · u = i_{1} · ∇ · i_{1} · u′ = i_{1} · u′ = u

and ∇ · i_{1} · v = v. Hence ∇ ∈ R and thus the cylinder object is \overline{K} = K + K with

c_{K} = id_{K+K} : K + K ⟶ K + K .

Consequently, f ∼ g for any f, g : K → L. □

Remark 5.2. If K/ ∼ is equivalent to a poset then two objects K and L have QK ≅ QL iff there are morphisms both K → L and L → K.

Proposition 5.3. Let K have finite limits and finite coproducts and (L, R) be a weak factorization system such that K/ ∼ is equivalent to a poset. Then K[R^{−1}] is equivalent to a poset.

Proof. Following 2.3, K[R^{−1}] is a quotient of the category of spans

K ⟵^{r} X ⟶^{f} L

with r ∈ R. Consider two spans from K to L

\xymatrix{K & X_{1} \ar[l]_{r_{1}} \ar[r]^{f_{1}} & L \\ X_{2} \ar[u]^{r_{2}} \ar[d]_{f_{2}} & X \ar[l]^{_{\overline{r}_{1}}} \ar[u]_{_{\overline{r}_{2}}} \\ L}

and a pullback of r_{1} and r_{2}. Following our assumption the objects QX and QL are either isomorphic in K/ ∼ or Q(f_{1} · \overline{r}_{2}) = Q(f_{2} · \overline{r}_{1}). Following 3.4, the objects P X and P L are either isomorphic in K[R^{−1}] or P (f_{1} · \overline{r}_{2}) = P (f_{2} · \overline{r}_{1}). In the first case, P K ≅ P L. In the second case, we have

P (f_{1}) · P (r_{1})^{−1} = P (f_{1}) · P (\overline{r}_{2}) · P (\overline{r}_{2})^{−1} · P (r_{1})^{−1}
= P (f_{1}) · P (\overline{r}_{2}) · P (\overline{r}_{1})^{−1} · P (r_{2})^{−1}

= P (f_{2}) · P (\overline{r}_{1}) · P (\overline{r}_{1})^{−1} · P (r_{2})^{−1} = P (f_{2}) · P (r_{2})^{−1} .

Thus the starting spans yield the same morphism in K[R^{−1}]. □

Remark 5.4. If all morphisms in R are split epimorphisms then 5.3 immediately follows from 3.9.

Observation 5.5. Let P be a poset and consider the category K = Set^{P^{op}} of presheaves on P. Let P_{⊥} be the full subcategory of Set^{P^{op}} consisting of the image of P in the Yoneda embedding Y : P → Set^{P^{op}} with the initial presheaf 0 added. Then P_{⊥} is nothing else than P with a new initial element added. Following 2.9, we get a weak factorization system (^{□}(P_{⊥}^{□}), P_{⊥}^{□}) in Set^{P^{op}}. Since all objects in P_{⊥} are indecomposable, it follows from 5.1 and 5.3 that both categories K/ ∼ and K[(P_{⊥}^{□})^{−1}] are equivalent to posets. Bisimilarity in this situation was used in [JNW] to formalize bisimilarity of processes. P_{⊥}^{□} is the class of P_{⊥}-open maps. 5.1 and 3.9 show that inverting all P_{⊥}-open maps gives the category that has the same objects as Set^{P^{op}} and an arrow K → L iff there is some arrow (simulation) K → L in Set^{P^{op}}.

Proposition 5.6. Let P be a poset. Then the following conditions are equivalent:

(i) ^{□}(P_{⊥}^{□}) coincides with the class Mono of all monomorphisms, (ii) for every element x ∈ P, any non-empty subset of

{y ∈ P|y ≤ x} has a greatest element.

Proof. (i)⇒ (ii): Assume (i) and consider x ∈ P and a non-empty subset Z of {y ∈ P|y ≤ x}. Let K be a subfunctor of the representable functor Y (x) : P^{op} → Set given as follows

K(p) = \begin{cases}1 & if p ≤ z for some z ∈ Z \\ 0 & otherwise \end{cases}

(here 0 = ∅ and 1 = {∅}). Then the embedding K → Y (x) belongs to □(P_{⊥}^{□}) and thus, following 2.9, is a retract of a transfinite composition g_{λ} of a chain g_{ij} : K_{i} → K_{j}, i ≤ j < λ (λ is a limit ordinal) where K_{0} = K, g_{ii+1} is a pushout of a morphism in P_{⊥} and, for j limit, g_{ij} is a colimit of g_{ik}, i ≤ k < j. Let i be the smallest ordinal i ≤ λ such that g_{i} = g_{0i} factorizes through a representable functor. It makes sense because g_{λ} has this property. Since representable functors are finitely presentable (cf. [AR] 1.2.(7)), i is either an isolated ordinal or i = 0.

Assume that i = j +1. Then we have a pushout below where A, B ∈ P_{⊥} and a factorization of g_{i} through a representable functor C

\xymatrix{ & A \ar[r]^{f} \ar[d]_{u} & B \ar[d]^{v} \\ K \ar[r]^{g_{j}} \ar[dr]_{h} & K_{j} \ar[r]^{g_{ji}} & K_{i} \\  & C \ar[ur]_{t}}

Then either t = g_{ji} · p for some p or t = v · q for some q. In the first case,

g_{ji} · g_{j} = g_{i} = t · h = g_{ji} · p · h

and thus p · h = g_{j} because g_{ji} is a monomorphism. Hence g_{j} factorizes through a representable functor, which contradicts the definition of i. In the second case,

g_{ji} · g_{j} = t · h = v · q · h

and thus there is a unique w : K → A such that

u · w = g_{j} and f · w = q · h .

Since K ≠ 0, we have A ≠ 0 as well and thus A is representable. Hence g_{j} factorizes through a representable functor again; a contradiction.

Therefore i = 0, which means that g_{0} = id_{K} factorizes through a representable functor. Thus K is a retract of a representable functor and, since P is a poset, K is representable. Thus Z has the greatest element.

(ii) ⇒ (i) The condition (ii) clearly means that subfunctors of representable functors are representable or 0. Since P is a poset, quotients of representable functors are representable. Following the proof of [Be] 1.12, any monomorphism in Set^{P^{op}} belongs to ^{□}(P_{⊥}^{□}). □

Corollary 5.7. Let P be a poset such that, for every element x ∈ P, any non-empty subset of {y ∈ P|y ≤ x} has a greatest element. Then Set^{P^{op}}[(Mono^{□})^{−1}] is equivalent to a poset.

Example 5.8. Let P be a two-element chain. Then Set^{P^{op}} = Set^{→} is the category of maps and P_{⊥} has three elements o_{0} : 0 → 0, o_{1} : 0 → 1 and id_{1} : 1 → 1. Then a morphism (u_{1}, u_{2}) : f → g

\xymatrix{A_{1} \ar[r]^{f} \ar[d]_{u_{1}} & A_{2} \ar[d]^{u_{2}} \\ B_{1} \ar[r]_{g} & B_{2}}

belongs to P_{⊥}^{□} = Mono^{□} iff u_{2} is surjective and for every a ∈ A_{2} and b ∈ B_{1} with g(b) = u_{2}(a) there is c ∈ A_{1} such that f(c) = a and u_{1}(c) = b. This implies that u_{1} is surjective as well. In other words, (u_{1}, u_{2}) is a (surjective) bisimulation, that is, using the transition system notation b_{0} → b for g(b) = b_{0} and a → c for f(c) = a, it holds that for any “transition” b_{0} → b and any a with u_{2}(a) = b_{0} there is c such that a → c and u_{1}(c) = b.

Following 5.7, we have that Set^{→}[(Mono^{□})^{−1}] = Set^{→}/ ∼ is equivalent to a poset. The objects id_{1} and id_{1} +o_{1} are isomorphic in this category because there are morphisms

id_{1} → id_{1} +o_{1} and id_{1} +o_{1} → id_{1} .

But the objects id_{1} and id_{1} +o_{1} are not bisimilar. In fact, assume that there are f : A → B and morphisms

id_{1} ⟵^{(u_{1},u_{2})} f ⟶^{(v_{1},v_{2})} id_{1} +o_{1}

in Mono^{□}. Then (u_{1}, u_{2}) makes f surjective and thus id_{1} +o_{1} is surjective as well; a contradiction.

References

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[AR] J. Adámek and J. Rosický, Locally Presentable and Accessible Categories, Cambrige Univ. Press 1994.

[Be] T. Beke, Sheafifiable homotopy model categories, Math. Proc. Cambridge Philos. Soc. 129 (2000), 447–475.

[B] A. Beligiannis, Homotopy theory of modules and Gorenstein rings, Math. Scand. 88 (2001), 1–41.

[Bor] F. Borceux, Handbook of Categorical Algebra, Vol. I, Cambridge Univ. Press 1995.

[Bo] A. K. Bousfield, Constructions of factorization systems in categories, J. Pure Appl. Algebra 9 (1977), 207–220.

[GZ] P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Springer-Verlag, Berlin 1967.

[H] P. S. Hirschhorn, Model Categories and their Localizations, American Mathematical Society 2003.

[JNW] A. Joyal, M. Nielsen and G. Winskel, Bisimulation from open maps, Inform. and Comput. 127 (1996), 164–185.

[KP] K. H. Kamps and T. Porter, Abstract Homotopy and Simple Homotopy Theory, World Scientific, Singapore 1997.

[Q] D. Quillen, Homotopical Algebra, Lecture Notes in Math. 43, Springer-Verlag, Berlin 1967.

Alexander Kurz University of Leicester, Department of Computer Science Leicester, LE1 7RH, UK

Jiří Rosický Masaryk University, Department of Algebra and Geometry Janáčkovo nám. 2a, 662 95 Brno, Czech Republic