A FAMILY OF ACYCLIC FUNCTORS

ANTONIO DÍAZ RAMOS^{†}

Abstract. We describe the projectives in the category of functors from a DCC poset to abelian groups. Based on this description we define a related condition, pseudo-projectivity, and we prove that this condition is enough for the vanishing of the derived direct limits. We apply this result to deduce a generalized version of Whitehead’s theorem for the pushout. The dual results for inverse limits are also considered.

1. Introduction and summary

Vanishing results for derived functors of limit and colimit in the (abelian) functor category Ab^{C} are interesting, as these higher limits appear in many situations: obstruction theory, spectral sequences, dimension shifting, etc. Here, Ab denote the category of abelian groups and C is a small category. By the results of Słomińska in [14] these higher limits over C can be understood through higher limits over partially ordered sets (poset for short). In fact, if C is an EI-category there are further relations with posets, see [13] and [8]. Hence, enlarging the known family of colim-acyclic functors over posets is worthwhile.

In this paper we focus on the following problem

(1)  find colim-acyclic objects in Ab^{P},

where P is a poset. The functor colim : Ab^{P} → Ab is the direct limit functor and F ∈ Ab^{P} is colim-acyclic if colim_{n} F = 0 for n ≥ 1 (see [15] and [6], and the classical books of Cartan and Eilenberg [4] and of MacLane [10]). It is clear that if F is projective then it is colim-acyclic but, in the same way as not every flat module is projective (see, for example, [15, Section 3.2]), we may be missing colim-acyclic objects if we just consider projective ones.

To attack problem (1) we start giving a characterization of the projective objects in Ab^{P}. Recall that for any small category P (not necessarily a poset), the projective objects in Ab^{P} are well known to be, by the Yoneda Lemma, summands of direct sums of representable functors. Moreover, if P is a poset with the descending chain condition (DCC for short) then [5, Corollary 3] the projective objects in Ab^{P} are also direct sums of representable functors.

2 ANTONIO DÍAZ RAMOS^{†}

Our characterization of the projectives in Ab^{P} for P a poset follows: by the Yoneda description mentioned above, if F ∈ Ab^{P} is projective it is straightforward that the quotient of F (i) by the images of the non-identity morphisms arriving to i is free abelian (for any object i of P). More precisely, if

†Supported by MECD grant AP2001-2484 and EPSRC grant EP/D506484/1 and partially supported by MEC grant MTM2007-60016.

Definition 1.1. im_{F} (i) = ∑_{j→i,j≠i} im F (j → i) (or im_{F} (i) = 0 if the index set of the sum is empty) and coker_{F} (i) = F (i)/ im_{F} (i),

then coker_{F} (i) is free for any i ∈ P if F is projective. If p and q are objects of the poset P and p is smaller or equal than q we write p ≤ q or p → q. This gives information on the values a projective functor takes on objects. To study what occurs to the values of a projective functor on morphisms we need the following:

Definition 1.2. Let F : P → Ab be a functor over a poset P. We say that F is pseudo-projective if for any i ∈ P, any finite subset J ⊆ P_{≤i}, and elements x_{j} ∈ F (j) for j ∈ J, the condition

∑_{j∈J} F (j → i)(x_{j}) = 0

implies that x_{j} ∈ im_{F} (j) for every j ∈ max J.

Here, P_{≤i} = {j ∈ P|j ≤ i} (P_{<i}, P_{≥i} and P_{>i} are defined analogously). Also, if Q is a subset of P we denote by max Q the set of maximal elements of Q (and by min Q the set of minimal elements of Q). This condition, pseudo-projectiveness, is related to the conditions I, II and III in [5] (see Remark 4.3 for a discussion). A straightforward example of a pseudo-projective functor is a representable functor. Now we state our characterization of projectives:

Theorem A (Theorem 3.1). Let P be a DCC poset and let F : P → Ab be a functor. Then F is projective if and only if:

(1) coker_{F} (i) is free for any i ∈ P, and

(2) F is pseudo-projective.

The descending chain condition in the statement of the theorem can not be dropped: consider the poset P = ℤ of the integers. This is not a DCC poset and the constant functor of value ℤ_{n} satisfies both conditions in Theorem A but it is not projective.

Theorem A is the first step towards finding colim-acyclic objects in Ab^{P}. The reason is that the second of the conditions in the theorem, i.e., pseudo-projectiveness, implies colim-acyclicity:

Theorem B (Theorem 4.2). Let F : P → Ab be a pseudo-projective functor over a DCC poset P. Then F is colim-acyclic.

This is the main result of this work, and it gives a family of functors in Ab^{P} which are colim-acyclic but not necessarily projective. To show that there exist functors in this situation consider the functor

\xymatrix{ℤ & ℤ \ar[l]_{×2} \ar[r]^{×2} & ℤ} .

This is a pseudo-projective functor which, by Theorem B, is acyclic. Moreover, it does not satisfy condition (1) in Theorem A and so it is not projective (see Examples 3.2 and 4.4). On the other hand, pseudo-projective functors do not cover all colim-acyclic functors: the functor

\xymatrix{0 & ℤ \ar[l]_{0} \ar[r]^{1} & ℤ} .

is not pseudo-projective but, as a straightforward computation shows, it is acyclic.

For vector spaces the notion of pseudo-projectiveness becomes identical to projectiveness as condition (1) in Theorem A is unnecessary in the context of functors to k − mod (where k is a field). Even in this favorable case the functor

\xymatrix{0 & k \ar[l]_{0} \ar[r]^{1} & k}

shows that there are acyclic functors which are not projective.

We apply Theorem B to a problem in algebraic topology. Recall Whitehead’s theorem for the pushout [2, II.7.3]. It states that the pushout of classifying spaces for a diagram of groups is a classifying space for the pushout group. Using pseudoprojectiveness we can generalize this to rooted trees. We denote by Grp the category of (discrete) groups.

Theorem C (Theorem 5.1). Let P be a DCC poset which has an initial object and has no loops (in the non directed sense). Let G : P → Grp be a diagram of groups with injective morphisms. Then there is a homotopy equivalence

hocolim_{P} BG ≃ B(colim_{P} G).

Here hocolim stands for the (unpointed) homotopy colimit of topological spaces in the sense of Bouslfield-Kan [1] and BH, where H is a discrete group, denotes the classifying space of H. Pseudo-projectivity is needed in the proof of this theorem in its easiest form: a functor with injective morphisms over a rooted tree satisfies trivially Definition 1.2.

The paper is structured as follows: in Section 2 we study properties of pseudoprojective functors. Then we use them in Section 3 to characterize projective functors and in Section 4 to prove that pseudo-projectiveness implies colim-acyclicity. Next we give the generalization of Whitehead’s theorem in Section 5. We finish with Section 6, where the dual definitions and results for lim-acyclicity are stated without proof.

Acknowledgements: I would like to thank Prof. A. Viruel for his support. Also, thanks to Prof. C.A. Weibel for all his fruitful and valuable comments. Finally, I would like to thank the anonymous referees for their suggestions, which significantly improved the methods and the scope of the results.

2. Pseudo-projective functors

We begin introducing some notation: let P be a poset, let F : P → Ab be a functor and let i ∈ P. If {x_{j}}_{j∈P_{≤i}} is a collection of values x_{j} ∈ F (j) such that all but a finite number are zero the following sum makes sense

∑_{j} F (j → i)(x_{j}),

where j runs over P_{≤i}. It defines an element of F (i). Whenever we write such a sum, ∑_{j} F (j → i)(x_{j}), we assume that x_{j} is an element of F (j) for every j ∈ P_{≤i}, and that they are zero for all but a finite number of summands, i.e., that the set {j ∈ P_{≤i}|x_{j} ≠ 0} is finite.

Notice that Definition 1.2 is equivalent to

Definition 2.1. Let F : P → Ab be a functor over a poset P. We say that F is pseudo-projective if for any i ∈ P and

∑_{j} F (j → i)(x_{j}) = 0,

we have that x_{j} ∈ im_{F} (j) for each j ∈ max{p ∈ P_{≤i}|x_{p} ≠ 0}.

Remark 2.2. Observe that if {i} = max{p ∈ P_{≤i}|x_{p} ≠ 0}, i.e., if x_{i} ≠ 0, then the condition x_{i} ∈ im_{F} (i) in the definition is vacuous.

Suppose that F is pseudo-projective and that

∑_{j} F (j → i)(x_{j}) = 0.

Set Q := {p ∈ P_{≤i}|x_{p} ≠ 0}. By the definition of this sum the set Q is finite. Notice that the finite set max Q is a main ingredient in the definition of pseudo-projectivity (Definition 2.1). Hence, as F is pseudo-projective, x_{j} ∈ im_{F} (j) for each j ∈ max Q. This can be rewritten as

x_{j} = ∑_{k} F (k → j)(x_{k,j}).

Recall that this means that x_{k,j} ∈ F (k) for each k ∈ P_{≤j}, that x_{j,j} = 0 (by Definition 1.1) and that there is a finite number of non-zero summands.

If j ∈ P_{≤i} but j does not belong to max Q define x_{j,j} = x_{j} ∈ F (j) and x_{k,j} = 0 ∈ F (k) for k ∈ P_{<j} (notice that x_{j,j} = x_{j} = 0 if j ∉ Q). Then we have an equation

x_{j} = ∑_{k} F (k → j)(x_{k,j})

for each j ∈ P_{≤i} (maximal or not).

Now define

x′_{k} = ∑_{j∈P_{≤i}∩P_{≥k}} x_{k,j}

for each k ∈ P_{≤i}. Notice that because Q is finite this sum is well defined and the set {p ∈ P_{≤i}|x′_{p} ≠ 0} is also finite. Then we obtain:

∑_{k} F (k → i)(x′_{k}) = ∑_{k} F (k → i)( ∑_{j∈P_{≤i}∩P_{≥k}} x_{k,j}) = ∑_{k∈P_{≤i},j∈P_{≤i}∩P_{≥k}} F (k → i)(x_{k,j}).

This can be rewritten as

∑_{k∈P_{≤i},j∈P_{≤i}∩P_{≥k}} F (k → j → i)(x_{k,j})) = ∑_{k∈P_{≤i},j∈P_{≤i}∩P_{≥k}} F (j → i)(F (k → j)(x_{k,j})),

and this is finally equal to

∑_{j∈P_{≤i}} F (j → i)( ∑_{k∈P_{≤j}} F (k → j)(x_{k,j})) = ∑_{j} F (j → i)(x_{j}) = 0.

Next fix some k ∈ P_{≤i}. If x′_{k} ≠ 0 there exist j ∈ P_{≤i} ∩ P_{≥k} with x_{k,j} ≠ 0. Then either j ∈ max Q and k ∈ P_{<j} or k = j ∈ Q \ max Q. In both cases x_{j} ≠ 0 and there exists a strict inclusion in P_{≤i}, k → l, with l ∈ max Q ⊆ Q. That is, if k ∈ P_{≤i} and x′_{k} ≠ 0 then there exists an object l with k < l ≤ i and x_{l} ≠ 0.

Applying the same argument to the equation

∑_{k} F (k → i)(x′_{k}) = 0

and so on we obtain the following

Lemma 2.3. Let F : P → Ab be a pseudo-projective functor over a poset P. Suppose {x_{j}^{0}}_{j∈P_{≤i}} is a collection of values x_{j}^{0} ∈ F (j) such that all but a finite number are zero and such that

∑_{j} F (j → i)(x_{j}^{0}) = 0.

Then there is a collection of values {x_{k,j}^{n}}_{j∈P_{≤i},k∈P_{≤j}} for each n ≥ 0 with x_{k,j}^{n} ∈ F (k) and such for each n there is a finite number of non-zero values. Moreover, if we define

x_{k}^{n} = ∑_{j∈P_{≤i}∩P_{≥k}} x_{k,j}^{n−1} ∈ F (k)

for each k ∈ P_{≤i} and n ≥ 1, then, for each n ≥ 1, x_{k}^{n} is non zero for a finite number of objects k ∈ P_{≤i}. Moreover,

∑_{j} F (j → i)(x_{j}^{n}) = 0

for n ≥ 0 and

x_{j}^{n} = ∑_{k} F (k → j)(x_{k,j}^{n}).

for n ≥ 0, j ∈ P_{≤i}. Finally, if x_{k}^{n} ≠ 0 for some n ≥ 1 and k ∈ P_{≤i}, then there exists a strict inclusion in P_{≤i}, k → l, with x_{l}^{n−1} ≠ 0.

This is the main property of pseudo-projective functors. Note that

Remark 2.4. If x_{i}^{0} = 0 then x_{k,i}^{n} = 0 and x_{i}^{n} = 0 for each k ∈ P_{≤i} and each n ≥ 0.

Remark 2.5. By construction, if x_{j}^{n} = 0 for each j ∈ P_{≤i} and some n ≥ 0, then x_{k,j}^{m} = 0 for m ≥ n, j ∈ P_{≤i}, k ∈ P_{≤j} and x_{j}^{m} = 0 for m ≥ n + 1 and j ∈ P_{≤i}.

When the underlying poset P is a DCC poset then the collapse described in Remark 2.5 is eventually reached, i.e., the sequence of values given by Lemma 2.3 becomes zero after a large enough number of steps. To prove this we need first the following elementary fact about DCC posets Lemma 2.6. Let P be a DCC poset and let {Q_{n}}_{n≥0} be a sequence of finite subsets of P. Assume that every object in Q_{n+1} is strictly smaller that some object in Q_{n} for every n ≥ 0. Then there exist N with Q_{n} = ∅ for n ≥ N.

Proof. It is clear from the hypothesis that if Q_{n} = ∅ for some n then Q_{m} = ∅ for each m ≥ n. Assume that the thesis of the lemma does not hold, i.e., that for each n ≥ 0 there exist some object j_{n} ∈ Q_{n}. Using the hypothesis we can build, for each object j_{n} (n ≥ 0), a chain of strict inclusions in P

j_{n} → j_{n}^{n−1} → j_{n}^{n−2} → … → j_{n}^{1} → j_{n}^{0},

where j_{n}^{m} ∈ Q_{m} for m = n − 1, n − 2, …, 1, 0.

So we have an infinite number of these chains (one for each n ≥ 0). From these chains we will extract an infinite descending chain in P, reaching a contradiction with P being DCC: because all the chains contain an object of the finite set Q_{0} there exists at least one object in Q_{0} such that infinitely many chains contain that particular object. From these infinitely many chains we can also choose infinitely many which agree on a particular object of the finite set Q_{1}. Repeating this argument for Q_{2}, Q_{3}, etc, we can build an arbitrary length descending chain in P. Because P is DCC this cannot be the case, we obtain a contradiction and the statement of the lemma holds. □

Notice that the statement of the lemma above when the cardinalities satisfy |Q_{n}| ≤ 1 is exactly the definition of the descending chain condition for P. Now it is easy to prove the following

Lemma 2.7. Let F : P → Ab be a pseudo-projective functor over a DCC poset P. Suppose {x_{j}^{0}}_{j∈P_{≤i}} is a collection of values x_{j}^{0} ∈ F (j) such that all but a finite number are zero and such that

∑_{j} F (j → i)(x_{j}^{0}) = 0.

Let x_{j}^{n} ∈ F (j) for j ∈ P_{≤i} and n ≥ 1 be such that they satisfy the properties listed in Lemma 2.3. Then there exist N > 0 such that n ≥ N implies x_{j}^{n} = 0 for each j ∈ P_{≤i}.

Proof. For each n ≥ 0 consider the finite set

Q_{n} := {j ∈ P_{≤i}|x_{j}^{n} ≠ 0}.

By the last part of Lemma 2.3 the sequence of finite sets of objects {Q_{n}}_{n≥0} satisfies the hypothesis in Lemma 2.6. Hence Q_{n} = ∅ for n sufficiently large. □

Next we prove two deeper properties of pseudo-projective functors:

Lemma 2.8. Let

\xymatrix{0 \ar[r] & A \ar[r]^{α} & B \ar[r]^{β} & C \ar[r] & 0}

be a short exact sequence of functors in Ab^{P}, where P is a DCC poset. If C is pseudo-projective then

0 → im_{A}(i) →^{α_{i}} im_{B}(i) →^{β_{i}} im_{C}(i) → 0

is exact for every i ∈ P.

Proof. The only non-trivial inclusion to prove is im_{B}(i) ∩ ker β_{i} ⊆ α_{i}(im_{A}(i)).

So let x = ∑_{j} B(j → i)(x_{j}^{0}) ∈ im_{B}(i) with x_{j}^{0} ∈ B(j) for each j ∈ P_{≤i}, x_{i}^{0} = 0 and with a finite number of non-zero summands. Assume that β_{i}(x) = 0. Then

0 = β_{i}(x) = β_{i}(∑_{j} B(j → i)(x_{j}^{0})) = ∑_{j} C(j → i)(β_{j}(x_{j}^{0})).

Because C is pseudo-projective and by Lemma 2.3 we can find elements y_{k,j}^{n} ∈ C(k) for n ≥ 0, j ∈ P_{≤i}, k ∈ P_{≤j} satisfying the properties listed in this lemma. In particular, if we define

y_{k}^{n} = ∑_{j∈P_{≤i}∩P_{≥k}} y_{k,j}^{n−1}

for n ≥ 0 and y_{k}^{0} = β_{k}(x_{k}^{0}) for n = 0 (for each k ∈ P_{≤i}), then

∑_{j} C(j → i)(y_{j}^{n}) = 0

for n ≥ 0 and

y_{j}^{n} = ∑_{k} C(k → j)(y_{k,j}^{n})

for n ≥ 0, j ∈ P_{≤i}.

As β is surjective, we can choose elements x_{k,j}^{n} ∈ B(k) with β_{k}(x_{k,j}^{n}) = y_{k,j}^{n} for each n ≥ 0, j ∈ P_{≤i}, k ∈ P_{≤j} (and we choose x_{k,j}^{n} = 0 if y_{k,j}^{n} = 0). Define

x_{k}^{n} = ∑_{j∈P_{≤i}∩P_{≥k}} x_{k,j}^{n−1}

for n ≥ 1 and for each k ∈ P_{≤i}.

Now we claim that

x − ∑_{j} B(j → i)(x_{j}^{n}) ∈ α_{i}(im_{A}(i))

for each n ≥ 0. For n = 0 the claim x − ∑_{j} B(j → i)(x_{j}^{0}) = 0 holds by the very definition of x and the elements x_{j}^{0}’s. Assume the claim holds for n − 1 ≥ 0. We prove it for n ≥ 1. We start considering the following sum:

∑_{k} B(k → i)(x_{k}^{n}) = ∑_{k} B(k → i)( ∑_{j∈P_{≤i}∩P_{≥k}} x_{k,j}^{n−1})

= ∑_{k∈P_{≤i},j∈P_{≤i}∩P_{≥k}} B(k → i)(x_{k,j}^{n−1})

= ∑_{k∈P_{≤i},j∈P_{≤i}∩P_{≥k}} B(k → j → i)(x_{k,j}^{n−1})

= ∑_{k∈P_{≤i},j∈P_{≤i}∩P_{≥k}} B(j → i)(B(k → j)(x_{k,j}^{n−1})).

Swaping the order of the indexes we finally obtain:

(2)  ∑_{k} B(k → i)(x_{k}^{n}) = ∑_{j∈P_{≤i}} B(j → i)( ∑_{k∈P_{≤j}} B(k → j)(x_{k,j}^{n−1}))

= ∑_{j} B(j → i)(∑_{k} B(k → j)(x_{k,j}^{n−1})).

Now notice that

β_{j}(x_{j}^{n−1}) = β_{j}( ∑_{l∈P_{≤i}∩P_{≥j}} x_{j,l}^{n−2}) = ∑_{l∈P_{≤i}∩P_{≥j}} y_{j,l}^{n−2} = y_{j}^{n−1}

if n ≥ 2, and

β_{j}(x_{j}^{0}) = y_{j}^{0}

by the definition of y_{j}^{0} if n = 1. Hence,

β_{j}(∑_{k} B(k → j)(x_{k,j}^{n−1})) = ∑_{k} C(k → j)(β_{k}(x_{k,j}^{n−1}))

= ∑_{k} C(k → j)(y_{k,j}^{n−1}) = y_{j}^{n−1} = β_{j}(x_{j}^{n−1}).

This implies that for each j ∈ P_{≤i} there exist z_{j} ∈ K(j) with

α_{j}(z_{j}) = ∑_{k} B(k → j)(x_{k,j}^{n−1}) − x_{j}^{n−1}.

Then, by Equation (2), we obtain

x − ∑_{k} B(k → i)(x_{k}^{n}) = x − ∑_{j} B(j → i)(∑_{k} B(k → j)(x_{k,j}^{n−1}))

= x − ∑_{j} B(j → i)(x_{j}^{n−1} + α_{j}(z_{j})) = x − ∑_{j} B(j → i)(x_{j}^{n−1}) − ∑_{j} B(j → i)(α_{j}(z_{j})).

This is clearly in α_{i}(im_{A}(i)) by the induction hypothesis and because

∑_{j} B(j → i)(α_{j}(z_{j})) = α_{i}(∑_{j} A(j → i)(z_{j})) ∈ α_{i}(im_{A}(i)).

Notice that we can assume that z_{i} = 0 (and hence the sum above does belong to im_{A}(i)) because x_{i}^{0} = 0 and by Remark 2.4.

This finishes the proof of the claim, i.e., we know that for all n ≥ 0

x − ∑_{j} B(j → i)(x_{j}^{n}) ∈ α_{i}(im_{A}(i)).

Now by Lemma 2.7 there exist N > 0 such that n ≥ N implies y_{j}^{n} = 0 for each j ∈ P_{≤i}. By the choice of the values x_{k,j}^{m} and by Remark 2.5 also x_{j}^{n} = 0 for each j ∈ P_{≤i} for n ≥ N. Then the claim gives x ∈ α_{i}(im_{A}(i)) and the proof is finished. □

Lemma 2.9. Let

\xymatrix{0 \ar[r] & A \ar[r]^{α} & B \ar[r]^{β} & C \ar[r] & 0}

be a short exact sequence of functors in Ab^{P}, where P is a DCC poset. If C and B are pseudo-projective then so is A.

Proof. Let i ∈ P and suppose that

∑_{j} A(j → i)(x_{j}) = 0.

Applying α_{i} to this equality we obtain

∑_{j} B(j → i)(α_{j}(x_{j})) = 0.

Now fix j ∈ max{p ∈ P_{≤i}|α_{p}(x_{p}) ≠ 0}. Because B is pseudo-projective then α_{j}(x_{j}) ∈ im_{B}(j). It is clear that α_{j}(x_{j}) ∈ ker β_{j} and hence, by Lemma 2.8, there exists a_{j} ∈ im_{A}(j) such that α_{j}(x_{j}) = α_{j}(a_{j}). Because α_{j} is injective we obtain x_{j} = a_{j} ∈ im_{A}(j). To conclude that A is pseudo-projective notice that, as α is injective, {p ∈ P_{≤i}|x_{p} ≠ 0} = {p ∈ P_{≤i}|α_{p}(x_{p}) ≠ 0} and hence max{p ∈ P_{≤i}|x_{p} ≠ 0} = max{p ∈ P_{≤i}|α_{p}(x_{p}) ≠ 0}. □

3. Projective objects in Ab^{P}.

In this section we will prove Theorem A of the introduction as the following:

Theorem 3.1. Let F : P → Ab be a functor over a DCC poset P. Then F is projective if and only if

(1) coker_{F} (i) is free for any i ∈ P, and

(2) F is pseudo-projective.

Proof. That a projective functor satisfies both conditions in the statement is easily deduced from [5, Corollary 3] and from the fact that every representable functor satisfies the conditions.

It remains to prove that a functor F satisfying the conditions in the statement is projective. That is, given a diagram of functors with exact row as shown, we must find a natural transformation ρ : F → A making the diagram commutative:

\xymatrix{ & F \ar[d]^{σ} \ar@{-->}[dl]^{ρ} \\ A \ar[r]^{π} & B \ar[r] & 0.}

We start fixing for each i ∈ P a splitting F (i) = im_{F} (i) ⊕ coker_{F} (i) and a homomorphism ρ_{i}|_{coker_{F} (i)} : coker_{F} (i) → A(i) that makes commutative the diagram

\xymatrix{ & coker_{F} (i) \ar[d]^{σ_{i}} \ar@{-->}[dl]_{ρ_{i}|_{coker_{F} (i)}} \\ A(i) \ar[r]^{π_{i}} & B(i) \ar[r] & 0.}

Next we will prove that there is a unique natural transformation ρ : F → A such that π ∘ρ = σ and such that for each i ∈ P the restriction ρ_{i}|_{coker_{F} (i)} is the one chosen above. We prove it by Noetherian induction on the DCC poset P. More precisely, we claim that for each i ∈ P there is such a unique natural transformation defined in the subposet P_{≤i}.

Notice that if i is minimal then the claim holds as im_{F} (i) = 0. Assume that the claim holds for every j ∈ P_{<i}. We want to define ρ_{i} : F (i) → A(i) such that it makes commutative the diagram

\xymatrix{ & im_{F} (i) ⊕ coker_{F} (i) \ar[d]^{σ_{i}} \ar@{-->}[dl]_{ρ_{i}} \\ A(i) \ar[r]^{π_{i}} & B(i) \ar[r] & 0,}

and such that ρ is a natural transformation from F to A and π ∘ ρ = σ over P_{≤i}.

The restriction ρ_{i}|_{coker_{F} (i)} is fixed by hypothesis. To define ρ_{i} on im_{F} (i) notice that, if such a ρ_{i} existed, then for x = ∑_{j} F (j → i)(x_{j}) ∈ im_{F} (i) its value would be given by

(3)  ρ_{i}(x) = ∑_{j}(A(j → i) ∘ ρ_{j})(x_{j}).

Notice that ρ_{j}(x_{j}) is defined by the the induction hypothesis for each j ∈ P_{<i} (recall that x_{i} = 0). We take Equation (3) as the definition of ρ_{i}. To check that ρ_{i}(x) is well defined we have to prove that

∑_{j} F (j → i)(x_{j}) = 0 ⇒ ∑_{j}(A(j → i) ∘ ρ_{j})(x_{j}) = 0.

So assume ∑_{j} F (j → i)(x_{j}^{0}) = 0 for some collection of values x_{j}^{0} ∈ F (j), j ∈ P_{≤i}, x_{i}^{0} = 0. By the pseudo-projectivity of F and by Lemma 2.3 we can find elements x_{k,j}^{n} ∈ F (k) for each n ≥ 0, j ∈ P_{≤i}, k ∈ P_{≤j}, satisfying the following: if we define

x_{k}^{n} = ∑_{j∈P_{≤i}∩P_{≥k}} x_{k,j}^{n−1} ∈ F (k)

for each k ∈ P_{≤i} and n ≥ 1 then

∑_{j} F (j → i)(x_{j}^{n}) = 0

for n ≥ 0 and

x_{j}^{n} = ∑_{k} F (k → j)(x_{k,j}^{n}).

for n ≥ 0, j ∈ P_{≤i}.

Now we substitute x_{j}^{0} = ∑_{k} F (k → j)(x_{k,j}^{0}) in ∑_{j}(A(j → i) ∘ ρ_{j})(x_{j}^{0}):

∑_{j}(A(j → i) ∘ ρ_{j})(x_{j}^{0})) = ∑_{j} ∑_{k∈P_{≤j}}(A(j → i) ∘ ρ_{j} ∘ F (k → j))(x_{k,j}^{0})

= ∑_{j} ∑_{k∈P_{≤j}}(A(j → i) ∘ A(k → j) ∘ ρ_{k})(x_{k,j}^{0})

= ∑_{j∈P_{≤i}} ∑_{k∈P_{≤j}}(A(k → i) ∘ ρ_{k})(x_{k,j}^{0})

= ∑_{k∈P_{≤i}}(A(k → i) ∘ ρ_{k})( ∑_{j∈P_{≤i}∩P_{≥k}} x_{k,j}^{0})

= ∑_{k}(A(k → i) ∘ ρ_{k})(x_{k}^{1}).

Notice that the second equality is obtained applying the induction hypothesis. Substituting now x_{k}^{1} = ∑_{l} F (l → k)(x_{l,k}^{1}) in this last expression and so on we obtain that

(4)  ∑_{j}(A(j → i) ∘ ρ_{j})(x_{j}^{0}) = ∑_{j}(A(j → i) ∘ ρ_{j})(x_{j}^{n})

for each n ≥ 0. Recall that we want to prove that

∑_{j}(A(j → i) ∘ ρ_{j})(x_{j}^{0})) = 0.

By Lemma 2.7 there exist N > 0 such that x_{j}^{N} = 0 for each j ∈ P_{≤i} (notice that x_{j}^{N} = 0 for each j ∈ P_{≤i} does not imply, in general, x_{j}^{0} = 0 for any j ∈ P_{≤i}). Then Equation (4) gives

∑_{j}(A(j → i) ∘ ρ_{j})(x_{j}^{0}) = ∑_{j}(A(j → i) ∘ ρ_{j})(x_{j}^{N}) = 0.

This shows that we have a well defined map ρ_{i} : im_{F} (i) → A(i). That it is a homomorphism is straightforward.

It remains to prove that π_{i} ∘ ρ_{i} = σ_{i} when restricted to im_{F} (i). So take x = ∑_{j} F (j → i)(x_{j}) in im_{F} (i). Then

π_{i}(ρ_{i}(x)) = ∑_{j}(π_{i} ∘ A(j → i) ∘ ρ_{j})(x_{j})
= ∑_{j}(B(j → i) ∘ π_{j} ∘ ρ_{j})(x_{j}), π is a natural transformation
= ∑_{j}(B(j → i) ∘ σ_{j})(x_{j}), by the inductive hypothesis
= ∑_{j}(σ_{i} ∘ F (j → i))(x_{j}), σ is a natural transformation
=σ_{i}(x).

To complete the inductive step we have to prove that ρ restricted to P_{≤i} is a natural transformation. Choose a morphism j → i in P_{≤i} with j ≠ i and x′ ∈ F (j). Then x = F (j → i)(x′) ∈ im_{F} (i) and, by definition of ρ_{i} on im_{F} (i),

ρ_{i}(x) = ρ_{i}(F (j → i)(x′)) = (A(j → i) ∘ ρ_{i})(x′).

Hence ρ is a natural transformation. □

This theorem yields the following examples.

Example 3.2. For the ‘pushout category’ P with shape

\xymatrix{a \ar[r]^{f} \ar[d]^{g} & b \\ c}

a functor F : P → Ab is projective if and only if • F (a), F (b)/im F (f) and F (c)/im F (g) are free abelian, and

• F (f) and F (g) are monomorphisms. For the ‘telescope category’ P with shape

\xymatrix{a_{0} \ar[r]^{f_{1}} & a_{1} \ar[r]^{f_{2}} & a_{2} \ar[r]^{f_{3}} & a_{3} \ar[r]^{f_{4}} & a_{4}...}

a functor F : P → Ab is projective if and only if • F (a_{0}) and F (a_{i})/ im F (f_{i}) for i = 1, 2, 3, … are free abelian groups and

• F (f_{d} ∘ f_{d−1} ∘ .. ∘ f_{1}) is a monomorphism and ker F (f_{i+d} ∘ f_{i+d−1} ∘ .. ∘ f_{i+1}) is

contained in im F (f_{i}) for d = 1, 2, 3, … and i = 1, 2, 3, . . ..

4. Pseudo-projectivity

Consider a functor F : P → Ab over a DCC poset P. In this section we show that if F is pseudo-projective then it is colim-acyclic. We start proving the following:

Lemma 4.1. Consider the right exact functor

\xymatrix{Ψ : Ab^{P} \ar[r] & P \\ F \ar@{|->}[r] & F (i)/ im_{F} (i),}

where P is a DCC poset and i ∈ P. If F is pseudo-projective then the left derived functors of Ψ vanish on F, i.e., L_{n}Ψ(F) = 0 for n ≥ 1.

Proof. Consider a projective resolution of F in Ab^{P} (recall that there are enough projectives in Ab^{P}):

… → P_{2} → P_{1} → P_{0} → F.

Then L_{n}Ψ(F) = H_{n}(Ψ(P_{·})) for each n ≥ 0. Consider the kernels of the maps in the projective resolution above:

\xymatrix{… \ar[r] & P_{2} \ar[dr] \ar[rr] &  & P_{1} \ar[rr] \ar[dr] &  & P_{0} \ar[r] & F \\  &  & K_{1} \ar[ur] &  & K_{0} \ar[ur]}

We obtain a bunch of short exact sequences 0 → K_{0} → P_{0} → F → 0, 0 → K_{1} → P_{1} → K_{0} → 0, 0 → K_{2} → P_{2} → K_{1} → 0, etc. Notice that each P_{n} for n ≥ 0 is, being projective, pseudo-projective by Theorem 3.1. Then, as F is pseudo-projective by hypothesis, successive applications of Lemma 2.9 to these short exact sequences yield that also K_{n} is pseudo-projective for n ≥ 0.

Then Lemma 2.8 give short exact sequences which fit in a diagram:

\xymatrix{ & … & … & … \\ 0 \ar[r] & K_{2}(i)/ im_{K_{2}}(i) \ar[r] \ar[d]^{0} & P_{2}(i)/ im_{P_{2}}(i) \ar[r] \ar[d] & K_{1}(i)/ im_{K_{1}}(i) \ar[r] \ar[d]^{0} & 0 \\ 0 \ar[r] & K_{1}(i)/ im_{K_{1}}(i) \ar[r] \ar[d]^{0} & P_{1}(i)/ im_{P_{1}}(i) \ar[r] \ar[d] & K_{0}(i)/ im_{K_{0}}(i) \ar[r] \ar[d]^{0} & 0 \\ 0 \ar[r] & K_{0}(i)/ im_{K_{0}}(i) \ar[r] & P_{0}(i)/ im_{P_{0}}(i) \ar[r] \ar[d] & F (i)/ im_{F} (i) \ar[r] & 0 \\  &  & 0}

Notice that the middle column of this diagram is exactly Ψ(P_{·}). From the exactness of each row of the diagram it follows easily the exactness of the middle column from P_{1}(i)/ im_{P_{1}}(i) and upwards. Hence L_{n}Ψ(F) = H_{n}(Ψ(P_{·})) = 0 for n ≥ 1. □

Now we are ready to prove Theorem B in the introduction:

Theorem 4.2. Let F : P → Ab be a pseudo-projective functor over a DCC poset P. Then F is colim-acyclic.

Proof. By [7, Appendix II.3], colim_{·} F can be computed as the homology of a chain complex C_{·} with C_{n} = ⨁_{i_{0}→...→i_{n}} F (i_{0}). Consider a homology class

ζ ∈ colim_{P}_{n}F,

where n ≥ 1. We will see that ζ = 0.

The direct sum in the description of the chain complex C_{·} implies, in particular, that:

ζ ∈ colim_{Q}_{n}F,

where Q is a (full) subposet of P with finitely many elements. Define Q̄ as the (full) subposet of all the elements which precede some element of the subposet Q. Note that CQ̄ ; · given by CQ̄ ; n = ⨁_{i_{0}→...→i_{n},i_{·}∈Q̄} F (i_{0}) is a chain subcomplex of C_{·}. Hence it is enough to prove that the homology class

ζ ∈ colim_{Q̄}_{n}F

is zero. Notice that F |Q̄ is pseudo-projective because Q̄ is a left ideal of P (X is a left ideal of P if x ∈ X, y < x ⇒ y ∈ X, right ideal is defined similarly). Note also that Q̄ satisfies DCC as well as it has a finite number of maximal elements and all of them are in Q.

Recall now that Ab is a monoidal category with the tensor product of abelian groups. Using the coend construction [9, IX.6] we can form the tensor product T ⊗Q̄H of any two functors T : Q̄^{op} → Ab and H : Q̄ → Ab.

Now fix a maximal object i of Q̄. There is a short exact sequence of functors in

AbQ̄^{op}
(5)  0 → ℤ{Q̄ \ {i}} → ℤ{Q̄} → ℤ{i} → 0,

where

ℤ\{X\}(j) = \{ \begin{matrix}ℤ & if j ∈ X, \\ 0 & if j ∉ X, \end{matrix}

and X ⊆ Q̄ is a right or left ideal in Q̄. Tensoring with H ∈ AbQ̄ we obtain the exact sequence

ℤ{Q̄ \ {i}} ⊗Q̄ H → ℤ{Q̄} ⊗Q̄ H → ℤ{i} ⊗Q̄ H → 0.

By inspection it is easy to see that this sequence is isomorphic to the following one:

colim_{Q̄\{i}} H → colim_{Q̄} H → H(i)/ im_{H}(i) → 0.

The short exact sequence (5) gives rise, for any H ∈ AbQ̄, to a long exact sequence of derived functors

… → colim_{Q̄\{i}}_{1}H → colim_{Q̄}_{1}H → L_{1}Ψ(H) → colim_{Q̄\{i}} H → colim_{Q̄} H → H(i)/ im_{H}(i) → 0,

where Ψ : AbQ̄ → Q̄ maps H ↦ H(i)/ im_{H}(i). Recall that, as Q̄ a left ideal, F |Q̄ is pseudo-projective. Then Lemma 4.1 gives

colim_{Q̄}_{n}F ≃ colim_{Q̄\{i}}_{n}F,

as n ≥ 1. Applying this argument to each maximal object of Q̄ (recall that there is a finite number of them) we obtain that

colim_{Q̄}_{n}F ≃ colim_{Q̄\max Q}_{n}F.

Under this isomorphism the homology class ζ corresponds to some homology class

ζ_{1} ∈ colim_{Q̄\max Q}_{n}F.

By the argument used before we have that

ζ_{1} ∈ colim_{Q_{1}}_{n}F,

where Q_{1} is a finite (full) subposet of Q̄ \ max Q. Notice that every element of Q_{1} is strictly smaller than some element in max Q ⊆ Q. Define Q̄_{1} as the (full) subposet of all the elements which precede some element of the finite subposet Q_{1}. Again Q̄_{1} is DCC and has a finite number of maximal elements. Arguing as above we obtain that

ζ_{1} ∈ colim_{Q̄_{1}}_{n}F ≃ colim_{Q̄_{1}\max Q_{1}}_{n}F.

Denoting by ζ_{2} the image of ζ_{1} under this isomorphism we can repeat the same argument again. Defining Q_{0} = Q and ζ_{0} = ζ we can build in this way a sequence of homology classes ζ_{0}, ζ_{1}, ζ_{2}, … and finite posets Q_{0}, Q_{1}, Q_{2}, … with

ζ_{m} ∈ colim_{Q̄_{m}}_{n}F ≃ colim_{Q̄_{m}\max Q_{m}}_{n}F

for m ≥ 1. Moreover, every object of the finite subposet Q_{m+1} is strictly smaller than some object of Q_{m} for every m ≥ 0. Applying Lemma 2.6 we obtain that Q_{N} = ∅ for N sufficiently large. Then ζ_{N} = 0 and hence also ζ = ζ_{0} = 0 and the proof is finished. □

Remark 4.3. Conditions I, II and III in [5, p.127] are related to pseudo-projectivity. First, it is easy to show that if P is a poset and F ∈ Ab^{P} satisfies I and III then F is pseudo-projective. But these three conditions are stronger than pseudo-projectivity:

(1) ℤ ←^{×2} ℤ →^{×2} ℤ is pseudo-projective but do not satisfy I, II and III by [5, Lemma 2],

(2) … ℤ_{n^{3}} ↠ ℤ_{n^{2}} ↠ ℤ_{n} is pseudo-projective but do not satisfy condition I,

(3) \xymatrix{… \ar[r] & ℤ \ar[r] & ℤ \ar[r] & ℤ \ar[r] \ar@{-->}[dr] & ℤ \\ … \ar[r] & ℤ \ar[r] & ℤ \ar[r] & ℤ \ar[r] \ar@{-->}[ur] & ℤ} is pseudo-projective and satisfies I but not III.

The following examples come from Example 3.2. They show the weaker conditions that are needed for colim-acyclicity instead of projectiveness.

Example 4.4. For the “pushout category” P with shape

\xymatrix{a_{0} \ar[r]^{f} \ar[d]^{g} & b_{1} \\ c_{1}}

a functor F : P → Ab is colim-acyclic if F (f) and F (g) are monomorphisms. For the “telescope category” P with shape

\xymatrix{a_{0} \ar[r]^{f_{1}} & a_{1} \ar[r]^{f_{2}} & a_{2} \ar[r]^{f_{3}} & a_{3} \ar[r]^{f_{4}} & a_{4}...}

a functor F : P → Ab is colim-acyclic if F (f_{d} ∘ f_{d−1} ∘ .. ∘ f_{1}) is a monomorphism for d = 1, 2, 3, … and ker F (f_{i+d} ∘ f_{i+d−1} ∘ .. ∘ f_{i+1}) ⊆ im F (f_{i}) for d = 1, 2, 3, … and i = 1, 2, 3, . . .. Notice that for this it is enough that F (f_{i}) is a monomorphism for each i = 1, 2, 3, . . ..

5. Whitehead’s theorem

Whitehead’s theorem states that if G_{0} → G_{1} and G_{0} → G_{2} are injections between groups then the pushout diagram in the category of (discrete) groups:

\xymatrix{G_{0} \ar[r] \ar[d] & G_{1} \ar@{-->}[d] \\ G_{2} \ar@{-->}[r] & G_{1} ∗_{G_{0}} G_{2},}

gives rise to a pushout diagram in the category of topological spaces:

\xymatrix{BG_{0} \ar[r] \ar[d] & B(G_{1}) \ar@{-->}[d] \\ BG_{2} \ar@{-->}[r] & B(G_{1} ∗_{G_{0}} G_{2}).}

By means of pseudo-projectiveness we generalize this result as:

Theorem 5.1. Let P be a DCC poset which has an initial object and has no loops (in the non directed sense). Let G : P → Grp be a diagram of groups with injective morphisms. Then there is a homotopy equivalence

hocolim_{P} BG ≃ B(colim_{P} G).

Proof. Being the spaces BG_{i} (i ∈ P) connected, the pointed diagram BG : P → Top and Van Kampen’s spectral sequence [12] give π_{1}(hocolim_{∗} BG) = colim G and π_{0}(hocolim_{∗} BG) = 0. Then the fibration P → hocolim BG → hocolim_{∗} BG gives, as P is contractible (it has an initial object), π_{1}(hocolim BG) = colim G and π_{0}(hocolim BG) = 0, and the fibration F → hocolim BG → B(colim G) gives π_{1}(F) = 0 and π_{0}(F) = 0.

So if we prove that H_{n}(F) = 0 (integral coefficients) for n ≥ 2 the theorem is proven. The maps BG_{i} → B(colim G) give a diagram F_{.} : P → Top whose value on i ∈ P is the fiber F_{i} of BG_{i} → B(colim G). By [3], the fiber F is the homotopy colimit of the diagram of the fibers F = hocolim(F_{.}).

For each i ∈ P the pointed fibration F_{i} → BG_{i} → B(colim G) gives π_{n}(F_{i}) = 0 for n ≥ 2. Moreover, as by [11] we know that the cone G → colim G has injective morphisms, also π_{1}(F_{i}) = 0 and F_{i} is discrete. Then the Bousfield-Kan homology spectral sequence for hocolim(F_{·}) reduces to H_{n}(F) = colim_{n} H_{0}(F_{.}) for n ≥ 0.

Applying the functor free abelian group ℤ to the short exact sequence in Set^{P}: 0 → G_{.} → colim G → π_{0}(F_{.}) → 0, we obtain a short exact sequence in Ab^{P}: ℤ[G_{.}] → ℤ[colim G] → ℤ[π_{0}(F_{.})] → 0. Taking kernels object-wise we obtain another short exact sequence in Ab^{P}: 0 → H → ℤ[colim G] → H_{0}(F_{.}) → 0. Because P is contractible we obtain colim_{n} H_{0}(F_{.}) ≃ colim_{n−1} H for n ≥ 2.

To finish the theorem it is left to prove that lim_{n} H = 0 for n ≥ 1, i.e., that H is colim-acyclic. Here is where pseudo-projectivity plays a role: first notice that the functor H takes injective values on morphisms because so does ℤ[colim G]. Then, because P has an initial object and no loops it is easy to see from the definition that H is pseudo-projective. Hence by Theorem 4.2 H is colim-acyclic. □

6. Dual results for injective objects in Ab^{P}.

The appropriate notions to characterize the injective objects in the functor category Ab^{P}, where P is a poset, are the following:

Definition 6.1. ker_{F} (i) = ⋂_{i→j,j≠i} ker F (i → j) (or ker_{F} (i) = F (i) if the index set of the intersection is empty) and coim_{F} (i) = F (i)/ ker_{F} (i).

Definition 6.2. Let F : P → Ab be a functor over a poset P. We say that F is pseudo-injective if for any object i ∈ P, any subset J ⊆ P_{≥i}, and elements x_{j} ∈ ker_{F} (j) for j ∈ J, there exist x ∈ F (i) with F (i → j)(x) = x_{j} for every j ∈ min J.

Then we can prove the following, where ACC stands for the ascending chain condition:

Theorem 6.3. Let P be an ACC poset and F : P → Ab be a functor. Then F is injective if and only if

(1) ker_{F} (i) is injective in Ab for any object i ∈ P, and

(2) F is pseudo-injective.

Also in the dual case pseudo-injectiveness of F is enough for lim-acyclicity, i.e., for lim^{n} F = 0 for n ≥ 1.

Theorem 6.4. Let F : P → Ab be a pseudo-injective functor over an ACC poset P. Then F is lim-acyclic.

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Department of Mathematical Sciences, King’s College, University of Aberdeen, ABERDEEN AB24 3UE U.K.

E-mail address: a.diaz@maths.abdn.ac.uk