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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