13.33 Derived colimits
In a triangulated category there is a notion of derived colimit.
Definition 13.33.1. Let \mathcal{D} be a triangulated category. Let (K_ n, f_ n) be a system of objects of \mathcal{D}. We say an object K is a derived colimit, or a homotopy colimit of the system (K_ n) if the direct sum \bigoplus K_ n exists and there is a distinguished triangle
\bigoplus K_ n \to \bigoplus K_ n \to K \to \bigoplus K_ n[1]
where the map \bigoplus K_ n \to \bigoplus K_ n is given by 1 - f_ n in degree n. If this is the case, then we sometimes indicate this by the notation K = \text{hocolim} K_ n.
By TR3 a derived colimit, if it exists, is unique up to (non-unique) isomorphism. Moreover, by TR1 a derived colimit of K_ n exists as soon as \bigoplus K_ n exists. The derived category D(\textit{Ab}) of the category of abelian groups is an example of a triangulated category where all homotopy colimits exist.
The nonuniqueness makes it hard to pin down the derived colimit. In More on Algebra, Lemma 15.86.5 the reader finds an exact sequence
0 \to R^1\mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (K_ n, L[-1]) \to \mathop{\mathrm{Hom}}\nolimits (\text{hocolim} K_ n, L) \to \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (K_ n, L) \to 0
describing the \mathop{\mathrm{Hom}}\nolimits s out of a homotopy colimit in terms of the usual \mathop{\mathrm{Hom}}\nolimits s.
Lemma 13.33.4. Let \mathcal{D} be a triangulated category. Let (K_ n, f_ n) be a system of objects of \mathcal{D}. Let n_1 < n_2 < n_3 < \ldots be a sequence of integers. Assume \bigoplus K_ n and \bigoplus K_{n_ i} exist. Then there exists an isomorphism \text{hocolim} K_{n_ i} \to \text{hocolim} K_ n such that
\xymatrix{ K_{n_ i} \ar[r] \ar[d]_{\text{id}} & \text{hocolim} K_{n_ i} \ar[d] \\ K_{n_ i} \ar[r] & \text{hocolim} K_ n }
commutes for all i.
Proof.
Let g_ i : K_{n_ i} \to K_{n_{i + 1}} be the composition f_{n_{i + 1} - 1} \circ \ldots \circ f_{n_ i}. We construct commutative diagrams
\vcenter { \xymatrix{ \bigoplus \nolimits _ i K_{n_ i} \ar[r]_{1 - g_ i} \ar[d]_ b & \bigoplus \nolimits _ i K_{n_ i} \ar[d]^ a \\ \bigoplus \nolimits _ n K_ n \ar[r]^{1 - f_ n} & \bigoplus \nolimits _ n K_ n } } \quad \text{and}\quad \vcenter { \xymatrix{ \bigoplus \nolimits _ n K_ n \ar[r]_{1 - f_ n} \ar[d]_ d & \bigoplus \nolimits _ n K_ n \ar[d]^ c \\ \bigoplus \nolimits _ i K_{n_ i} \ar[r]^{1 - g_ i} & \bigoplus \nolimits _ i K_{n_ i} } }
as follows. Let a_ i = a|_{K_{n_ i}} be the inclusion of K_{n_ i} into the direct sum. In other words, a is the natural inclusion. Let b_ i = b|_{K_{n_ i}} be the map
K_{n_ i} \xrightarrow {1,\ f_{n_ i},\ f_{n_ i + 1} \circ f_{n_ i}, \ \ldots ,\ f_{n_{i + 1} - 2} \circ \ldots \circ f_{n_ i}} K_{n_ i} \oplus K_{n_ i + 1} \oplus \ldots \oplus K_{n_{i + 1} - 1}
If n_{i - 1} < j \leq n_ i, then we let c_ j = c|_{K_ j} be the map
K_ j \xrightarrow {f_{n_ i - 1} \circ \ldots \circ f_ j} K_{n_ i}
We let d_ j = d|_{K_ j} be zero if j \not= n_ i for any i and we let d_{n_ i} be the natural inclusion of K_{n_ i} into the direct sum. In other words, d is the natural projection. By TR3 these diagrams define morphisms
\varphi : \text{hocolim} K_{n_ i} \to \text{hocolim} K_ n \quad \text{and}\quad \psi : \text{hocolim} K_ n \to \text{hocolim} K_{n_ i}
Since c \circ a and d \circ b are the identity maps we see that \varphi \circ \psi is an isomorphism by Lemma 13.4.3. The other way around we get the morphisms a \circ c and b \circ d. Consider the morphism h = (h_ j) : \bigoplus K_ n \to \bigoplus K_ n given by the rule: for n_{i - 1} < j < n_ i we set
h_ j : K_ j \xrightarrow {1,\ f_ j,\ f_{j + 1} \circ f_ j, \ \ldots ,\ f_{n_ i - 1} \circ \ldots \circ f_ j} K_ j \oplus \ldots \oplus K_{n_ i}
Then the reader verifies that (1 - f) \circ h = \text{id} - a \circ c and h \circ (1 - f) = \text{id} - b \circ d. This means that \text{id} - \psi \circ \varphi has square zero by Lemma 13.4.5 (small argument omitted). In other words, \psi \circ \varphi differs from the identity by a nilpotent endomorphism, hence is an isomorphism. Thus \varphi and \psi are isomorphisms as desired.
\square
Lemma 13.33.5. Let \mathcal{A} be an abelian category. If \mathcal{A} has exact countable direct sums, then D(\mathcal{A}) has countable direct sums. In fact given a collection of complexes K_ i^\bullet indexed by a countable index set I the termwise direct sum \bigoplus K_ i^\bullet is the direct sum of K_ i^\bullet in D(\mathcal{A}).
Proof.
Let L^\bullet be a complex. Suppose given maps \alpha _ i : K_ i^\bullet \to L^\bullet in D(\mathcal{A}). This means there exist quasi-isomorphisms s_ i : M_ i^\bullet \to K_ i^\bullet of complexes and maps of complexes f_ i : M_ i^\bullet \to L^\bullet such that \alpha _ i = f_ is_ i^{-1}. By assumption the map of complexes
s : \bigoplus M_ i^\bullet \longrightarrow \bigoplus K_ i^\bullet
is a quasi-isomorphism. Hence setting f = \bigoplus f_ i we see that \alpha = fs^{-1} is a map in D(\mathcal{A}) whose composition with the coprojection K_ i^\bullet \to \bigoplus K_ i^\bullet is \alpha _ i. We omit the verification that \alpha is unique.
\square
Lemma 13.33.6. Let \mathcal{A} be an abelian category. Assume colimits over \mathbf{N} exist and are exact. Then countable direct sums exists and are exact. Moreover, if (A_ n, f_ n) is a system over \mathbf{N}, then there is a short exact sequence
0 \to \bigoplus A_ n \to \bigoplus A_ n \to \mathop{\mathrm{colim}}\nolimits A_ n \to 0
where the first map in degree n is given by 1 - f_ n.
Proof.
The first statement follows from \bigoplus A_ n = \mathop{\mathrm{colim}}\nolimits (A_1 \oplus \ldots \oplus A_ n). For the second, note that for each n we have the short exact sequence
0 \to A_1 \oplus \ldots \oplus A_{n - 1} \to A_1 \oplus \ldots \oplus A_ n \to A_ n \to 0
where the first map is given by the maps 1 - f_ i and the second map is the sum of the transition maps. Take the colimit to get the sequence of the lemma.
\square
Lemma 13.33.7. Let \mathcal{A} be an abelian category. Let L_ n^\bullet be a system of complexes of \mathcal{A}. Assume colimits over \mathbf{N} exist and are exact in \mathcal{A}. Then the termwise colimit L^\bullet = \mathop{\mathrm{colim}}\nolimits L_ n^\bullet is a homotopy colimit of the system in D(\mathcal{A}).
Proof.
We have an exact sequence of complexes
0 \to \bigoplus L_ n^\bullet \to \bigoplus L_ n^\bullet \to L^\bullet \to 0
by Lemma 13.33.6. The direct sums are direct sums in D(\mathcal{A}) by Lemma 13.33.5. Thus the result follows from the definition of derived colimits in Definition 13.33.1 and the fact that a short exact sequence of complexes gives a distinguished triangle (Lemma 13.12.1).
\square
Lemma 13.33.8. Let \mathcal{D} be a triangulated category having countable direct sums. Let \mathcal{A} be an abelian category with exact colimits over \mathbf{N}. Let H : \mathcal{D} \to \mathcal{A} be a homological functor commuting with countable direct sums. Then H(\text{hocolim} K_ n) = \mathop{\mathrm{colim}}\nolimits H(K_ n) for any system of objects of \mathcal{D}.
Proof.
Write K = \text{hocolim} K_ n. Apply H to the defining distinguished triangle to get
\bigoplus H(K_ n) \to \bigoplus H(K_ n) \to H(K) \to \bigoplus H(K_ n[1]) \to \bigoplus H(K_ n[1])
where the first map is given by 1 - H(f_ n) and the last map is given by 1 - H(f_ n[1]). Apply Lemma 13.33.6 to see that this proves the lemma.
\square
The following lemma tells us that taking maps out of a compact object (to be defined later) commutes with derived colimits.
Lemma 13.33.9. Let \mathcal{D} be a triangulated category with countable direct sums. Let K \in \mathcal{D} be an object such that for every countable set of objects E_ n \in \mathcal{D} the canonical map
\bigoplus \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(K, E_ n) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(K, \bigoplus E_ n)
is a bijection. Then, given any system L_ n of \mathcal{D} over \mathbf{N} whose derived colimit L = \text{hocolim} L_ n exists we have that
\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(K, L_ n) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(K, L)
is a bijection.
Proof.
Consider the defining distinguished triangle
\bigoplus L_ n \to \bigoplus L_ n \to L \to \bigoplus L_ n[1]
Apply the cohomological functor \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(K, -) (see Lemma 13.4.2). By elementary considerations concerning colimits of abelian groups we get the result.
\square
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