## Compact objects

An important part of establishing Morita equivalence between $R$ and $M_n(R)$ was that fact that the natural map \(\bigoplus_{i \in I} \operatorname{Hom}(U, N_i) \to \operatorname{Hom}(U, \bigoplus_{i \in I} N_i)\) was an isomorphism.

This is equivalent to saying we can factor any $f: U \to \bigoplus_{i \in I} N_i$ as

where $|I_f| < \infty$.

Objects with the property are very useful. They go be different names depending on the context. We will be focused on triangulated categories.

**Definition**. Let $\mathcal T$ be a triangulated category. An object $X$ is called *compact* if the natural map \(\bigoplus_{i \in I} \operatorname{Hom}(X,Y_i) \to \operatorname{Hom}(X, \bigoplus_{i \in I} Y_i)\) is isomorphism for any $Y_i, i \in I$.

The full subcategory of compact objects in $\mathcal T$ is denoted by $\mathcal T^c$.

**Lemma**. $\mathcal T^c$ is triangulated and closed under retracts.

## **Proof**. (Expand to view)

We have $$ \operatorname{Hom}(X[n], \bigoplus Y_i) \cong \operatorname{Hom}(X, \bigoplus Y_i[n]) $$ so $X[n] \in \mathcal T^c$ for any $n \in \mathbb{Z}$. Next, take a map $X \to X^\prime$ of compact objects and complete it to a triangle $$ X \to X^\prime \to C \to X[1] $$ We get a commutative diagram

**Definition**. We say $\mathcal T$ is *compactly generated* if it has small coproducts and there exists a set of compact objects $\mathcal C$ such that \(\operatorname{Hom}(C,Y) = 0 \ \forall C \in \mathcal C\) implies $Y = 0$. The set $\mathcal C$ is called a set of *compact generators*.

**Lemma**. A map $X \to Y$ in $\mathcal T$ with \(\operatorname{Hom}(C,X) \to \operatorname{Hom}(C,Y)\) an isomorphism for all $C$ in a set of compact generators is an isomorphism.

## **Proof**. (Expand to view)

The map $X \to Y$ is an isomorphism if and only if $\operatorname{cone}(X \to Y) = 0$. Since we have a set of compact generators, we only need to check that $$ \operatorname{Hom}(C,\operatorname{cone}(X \to Y)) = 0 $$ for all $C$. But this is equivalent to the map $$ \operatorname{Hom}(C,X) \to \operatorname{Hom}(C,Y) $$ being an isomorphism for all $C$. ■

**Example**. $D(R)$ is compactly generated. Indeed, \(\operatorname{Hom}_{D(R)}(R[n],Y) \cong H^{-n}Y\) If all of these are $0$, then $Y$ is acyclic and hence $0$ in $D(R)$.

**Definition**. Given a sequence of composable maps \(X_0 \overset{f_0}{\to} X_1 \overset{f_1}{\to} X_2 \overset{f_2}{\to} \cdots\) the *homotopy colimit* is the cone over the morphism $\bigoplus X_i \to \bigoplus X_i$ induced by the maps \(X_i \overset{1,-f_i}{\to} X_i \oplus X_{i+1}\)

**Lemma**. If $C$ is compact, then the natural map \(\operatorname{colim} \operatorname{Hom}(C,X_\bullet) \to \operatorname{Hom}(C,\operatorname{hocolim} X_\bullet)\) is an isomorphism.

## **Proof**. (Expand to view)

We have a commutative diagram

**Proposition**. If $\mathcal C$ is a compact set of generators, then the smallest subcategory of $\mathcal T$ that

- is closed under $\bigoplus$’s
- is triangulated and
- contains $\mathcal C$

is $\mathcal T$ itself.

## **Proof**. (Expand to view)

Let $$ U_0 := \bigcup_{C \in \mathcal C} \operatorname{Hom}(C,X) $$ Set $$ X_0 := \bigoplus_{(C,f) \in U_0} C $$ This comes with a natural evaluation map $$ X_0 \to X. $$ We work not by induction having constructed $\nu_i : X_i \to X$. We let $$ U_i := \bigcup_{C \in \mathcal C} \lbrace f : C \to X_i \mid \nu_i f = 0 \rbrace $$ Let $$ K_i := \bigoplus_{(C,f) \in U_i} C $$ and set $X_{i+1} = \operatorname{cone}(K_i \to X_i)$. The composition $K_i \to X_i \to X$ is $0$ so there exists a $X_{i+1} \to X$

If we have a compact generating set $\mathcal C$, we can also use it characterize *all* compact objects in $\mathcal T$. The following is due to Neeman.

**Proposition**. Let $\mathcal C$ be a set of compact generators for $\mathcal T$. The smallest triangulated subcategory of $\mathcal T$

- containing $\mathcal C$ and
- closed under retracts

is $\mathcal T^c$.

## **Proof**. (Expand to view)

Pick some $X \in \mathcal T^c$. From the proof the proposition, we know that $$ X \cong \operatorname{hocolim} X_\bullet $$ where each $X_i$ is obtained as a cone over a map $K_{i-1} \to X_{i-1}$ with $K_{i-1}$ a sum of elements of $\mathcal C$. We also know that $$ 1_X \in \operatorname{Hom}(X,X) \cong \operatorname{colim}\operatorname{Hom}(X,X_i) $$ Thus, there exists $X \to X_i$ such that

**Definition**. A complex $P$ in $D(R)$ is called *perfect* if it is quasi-isomorphic to a bounded complex of finitely generated projectives. The full subcategory of perfect objects is denoted by $\operatorname{perf} R$.

**Proposition**. We have \(D(R)^c = \operatorname{perf} R\)

## **Proof**. (Expand to view)

It is easy to see that $$ \operatorname{perf} R \subseteq D(R)^c $$ since compact objects are closed under finite direct sums, summands, and cones. But $\operatorname{perf} R$ - contains $R[n]$ for any $n \in \mathbb{Z}$ (a set of compact generators) - is triangulated and - is closed under retracts. Thus, the previous proposition says $$ D(R)^c \subseteq \operatorname{perf} R $$ ■

Finally it useful to introduce some notation for generation.

**Definition**. Given a subcategory $\mathcal S$ of a triangulated category $\mathcal T$, we let $\langle \mathcal S \rangle$ denote the smallest triangulated subcategory of $\mathcal T$ containing $\mathcal S$.

We also let $\overline{\langle \mathcal S \rangle}$ denote the smallest triangulated, closed under retracts, containing $\mathcal S$.