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8.4 Retracts and Idempotents

Let $\operatorname{\mathcal{C}}$ be a category containing an object $X$. Recall that an object $Y \in \operatorname{\mathcal{C}}$ is a retract of $X$ if there exist morphisms $i: Y \rightarrow X$ and $r: X \rightarrow Y$ satisfying $\operatorname{id}_{Y} = r \circ i$, so that we have a commutative diagram

8.24
\begin{equation} \begin{gathered}\label{equation:retraction-diagram-primordial} \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{ r } & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_ Y } & & Y. } \end{gathered} \end{equation}

In this case, we will refer to (8.24) as a retraction diagram in $\operatorname{\mathcal{C}}$.

There is a universal example of a retraction diagram:

Construction 8.4.0.1. We define a category $\operatorname{Ret}$ as follows:

  • The category $\operatorname{Ret}$ has exactly two objects, which we denote by $\widetilde{X}$ and $\widetilde{Y}$.

  • The morphisms sets in $\operatorname{Ret}$ are given by

    \[ \operatorname{Hom}_{\operatorname{Ret}}(\widetilde{X}, \widetilde{X}) = \{ \operatorname{id}_{\widetilde{X}}, \widetilde{e} \} \quad \quad \operatorname{Hom}_{\operatorname{Ret}}(\widetilde{X}, \widetilde{Y}) = \{ \widetilde{r} \} \]
    \[ \operatorname{Hom}_{\operatorname{Ret}}(\widetilde{Y}, \widetilde{X}) = \{ \widetilde{i} \} \quad \quad \operatorname{Hom}_{\operatorname{Ret}}(\widetilde{Y}, \widetilde{Y}) = \{ \operatorname{id}_{\widetilde{Y}} \} . \]
  • The composition law in $\operatorname{Ret}$ is given (on non-identity morphisms) by the formulae

    \[ \widetilde{r} \circ \widetilde{i} = \operatorname{id}_{\widetilde{Y}} \quad \quad \widetilde{i} \circ \widetilde{r} = \widetilde{e} \]
    \[ \widetilde{e} \circ \widetilde{i} = \widetilde{i} \quad \quad \widetilde{e} \circ \widetilde{e} = \widetilde{e} \quad \quad \widetilde{r} \circ \widetilde{e} = \widetilde{r}. \]

Exercise 8.4.0.2. Let $\operatorname{\mathcal{C}}$ be a category containing a retraction diagram

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{ r } & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_ Y } & & Y. } \]

Show that there is a unique functor $F: \operatorname{Ret}\rightarrow \operatorname{\mathcal{C}}$ which is given on objects by $F( \widetilde{X} ) = X$ and $F( \widetilde{Y} )$, and also satisfies $F( \widetilde{i} ) = i$ and $F( \widetilde{r} ) = r$. We therefore obtain a bijection

\[ \{ \textnormal{Functors $\operatorname{Ret}\rightarrow \operatorname{\mathcal{C}}$} \} \xrightarrow {\sim } \{ \textnormal{Retraction diagrams in $\operatorname{\mathcal{C}}$} \} . \]

In particular, an object $Y \in \operatorname{\mathcal{C}}$ is a retract of another object $X \in \operatorname{\mathcal{C}}$ if and only if there exists a functor $F: \operatorname{Ret}\rightarrow \operatorname{\mathcal{C}}$ satisfying $F( \widetilde{X} ) = X$ and $F( \widetilde{Y} ) = Y$.

Our first goal in this section is to extend the theory of retracts to the setting of higher category theory. Here there are (at least) two ways that we might choose to proceed:

$(a)$

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. We could define an object $Y \in \operatorname{\mathcal{C}}$ to be a retract of $X$ if there exist morphisms $i: Y \rightarrow X$ and $r: X \rightarrow Y$ such that the identity morphism $\operatorname{id}_{Y}$ is a composition of $r$ with $i$, in the sense of Definition 1.3.4.1.

$(b)$

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. We could define an object $Y \in \operatorname{\mathcal{C}}$ to be a retract of $X$ if there exists a functor of $\infty $-categories $F: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$ satisfying $F( \widetilde{X} ) = X$ and $F( \widetilde{Y} ) = Y$.

In §8.4.1, we show that these definitions are equivalent. Note that objects $X,Y \in \operatorname{\mathcal{C}}$ satisfy condition $(a)$ if and only if there exists a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ whose boundary is indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{ r } & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_ Y } & & Y; } \]

in this case, we will say that $\sigma $ is a retraction diagram in $\operatorname{\mathcal{C}}$ (Definition 8.4.1.16). We will establish the equivalence of $(a)$ and $(b)$ by showing that every retraction diagram in $\operatorname{\mathcal{C}}$ can be extended to a functor $F: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$ (Corollary 8.4.1.24). In contrast with Exercise 8.4.0.2, the functor $F$ is not necessarily unique; however, it is uniquely determined up to isomorphism (in fact, up to a contractible space of choices).

Our second goal in this section is to address the following:

Question 8.4.0.3. Given an $\infty $-category $\operatorname{\mathcal{C}}$ and an object $X \in \operatorname{\mathcal{C}}$, how can one classify the retracts of $X$?

In §8.4.2, we recall the answer to Question 8.4.0.3 in the situation where $\operatorname{\mathcal{C}}$ is an ordinary category. We say that a endomorphism $e: X \rightarrow X$ is idempotent if it satisfies the identity $e \circ e = e$ (Definition 8.4.2.1). Every retraction diagram

8.25
\begin{equation} \begin{gathered}\label{equation:retraction-diagram-primordial2} \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{ r } & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_ Y } & & Y. } \end{gathered} \end{equation}

determines an idempotent endomorphism of $X$, given by the composition $e = i \circ r$ (Example 8.4.2.3). We say that an idempotent endomorphism is split if it can obtained in this way (Example 8.4.2.3). In this case, we can recover the retraction diagram (8.25) up to (unique) isomorphism from $e$: for example the object $Y$ can be recovered as the equalizer of the pair of morphisms $(e, \operatorname{id}_{X}): X \rightrightarrows X$ (see Corollary 8.4.2.5 and its proof). We therefore obtain a bijection

\[ \{ \textnormal{Split idempotent endomorphisms of $X$} \} \simeq \{ \textnormal{Diagrams (\ref{equation:retraction-diagram-primordial2}) } \} / \textnormal{Isomorphism}. \]

We can therefore reformulate Question 8.4.0.3 as follows:

Question 8.4.0.4. What is the correct $\infty $-categorical counterpart for the notion of an idempotent endomorphism?

In §8.4.3, we propose an answer to Question 8.4.0.4. Let $\operatorname{Idem}$ denote the full subcategory of $\operatorname{Ret}$ spanned by the object $\widetilde{X}$ (Construction 8.4.2.7). We define an idempotent in $\operatorname{\mathcal{C}}$ to be a functor of $\infty $-categories $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ (Definition 8.4.3.1). Every retraction diagram in $\operatorname{\mathcal{C}}$ can be extended to a functor $\overline{F}: \operatorname{N}_{\bullet }(\operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$, which we can restrict to obtain an idempotent $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$. We will say that an idempotent in $\operatorname{\mathcal{C}}$ is split if it can be obtained in this way. In this case, we will show that $\overline{F}$ can be recovered from the idempotent $F$ up to isomorphism (Corollary 8.4.3.10). Consequently, we obtain a bijection

\[ \{ \textnormal{Split idempotents in $\operatorname{\mathcal{C}}$} \} / \textnormal{Isomorphism} \simeq \{ \textnormal{Retraction diagrams in $\operatorname{\mathcal{C}}$} \} / \textnormal{Isomorphism}. \]

To fully address Question 8.4.0.3, we also need to provide a criterion to determine when an idempotent splits. Here we have several closely related results:

  • If $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, then an idempotent endomorphism $e: X \rightarrow X$ is split if and only if the pair of morphisms $(e, \operatorname{id}_{X} ): X \rightrightarrows X$ admit an equalizer (or a coequalizer); see Corollary 8.4.2.5. If this condition is satisfied, then the (co)equalizer is the associated retract of $X$.

  • If $\operatorname{\mathcal{C}}$ is an $\infty $-category, then an idempotent $F: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ is split if and only if it admits a limit (or a colimit); see Corollary 8.4.3.11. If this condition is satisfied, then the (co)limit is the associated retract of $X = F( \widetilde{X} )$.

  • Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ be an idempotent, carrying the morphism $\widetilde{e}: \widetilde{X} \rightarrow \widetilde{X}$ of $\operatorname{Idem}$ to a morphism $e: X \rightarrow X$ of $\operatorname{\mathcal{C}}$. Then $F$ is split if and only if the sequential diagram

    \[ \cdots \rightarrow X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \rightarrow \cdots \]

    admits a limit (or a colimit); see Proposition 8.4.4.16. If this condition is satisfied, then the (co)limit is the associated retract of $X$.

In §8.4.4, we study $\infty $-categories $\operatorname{\mathcal{C}}$ in which every idempotent is split; if this condition is satisfied, we say that $\operatorname{\mathcal{C}}$ is idempotent complete (Definition 8.4.4.1). Many $\infty $-categories which arise in practice are idempotent complete. For example, an $\infty $-category which sequential limits or colimits is automatically idempotent complete (Corollary 8.4.4.17). In §8.4.5, we show that every $\infty $-category $\operatorname{\mathcal{C}}$ admits an idempotent completion $\widehat{\operatorname{\mathcal{C}}}$ which is characterized (up to equivalence) by the existence of a functor $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ having the following properties:

  • The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is idempotent complete.

  • The functor $H$ is fully faithful.

  • Every object of $\widehat{\operatorname{\mathcal{C}}}$ is a retract of $H(X)$, for some object $X \in \operatorname{\mathcal{C}}$.

Moreover, the idempotent completion $\widehat{\operatorname{\mathcal{C}}}$ can be characterized by a universal mapping property: for every idempotent complete $\infty $-category $\operatorname{\mathcal{D}}$, composition with $H$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ (see Proposition 8.4.5.8).

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $e: X \rightarrow X$ be an endomorphism of $X$. We will say that $e$ is idempotent if there exists a functor $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ satisfying $F( \widetilde{X} ) = X$ and $F( \widetilde{e} ) = e$. If $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, then the functor $F$ is completely determined by the pair $(X,e)$. In general, this need not be true: the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Idem})$ contains a nondegenerate simplex of every dimension (Remark 8.4.3.3), so the specification of $F$ requires an infinite quantity of data. Nevertheless, we prove in §8.4.6 that $F$ is determined by the pair $(X,e)$ up to (canonical) isomorphism (Corollary 8.4.6.5). This motivates the following:

Question 8.4.0.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be an endomorphism in $\operatorname{\mathcal{C}}$. How can one determine if $e$ is idempotent?

Let us first record a necessary condition. We say that an endomorphism $e: X \rightarrow X$ is homotopy idempotent if the homotopy class $[e]$ is an idempotent endomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (Definition 8.4.7.1). It follows immediately from the definitions that every idempotent endomorphism of $\operatorname{\mathcal{C}}$ is homotopy idempotent. In §8.4.7, we show that the converse is false in general (Proposition 8.4.7.15), though it is true in some important special cases (see Corollary 8.4.7.6 and Exercise 8.4.7.8).

For every integer $n \geq 0$, let $\operatorname{N}_{\leq n}( \operatorname{Idem})$ denote the $n$-skeleton of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Idem})$ (Variant 1.2.1.5). Note that an endomorphism $e: X \rightarrow X$ in $\operatorname{\mathcal{C}}$ can be identified with a diagram $F: \operatorname{N}_{\leq 1}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ (Example 8.4.8.2), and that $e$ is homotopy idempotent if and only if $F$ admits an extension to $\operatorname{N}_{\leq 2}( \operatorname{Idem})$ (Example 8.4.8.3). In §8.4.8, we address Question 8.4.0.5 by showing that $e$ is idempotent if and only if $F$ admits an extension to $\operatorname{N}_{\leq 3}( \operatorname{Idem})$ (Corollary 8.4.8.8). As an application, we show that the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}})$ commutes with filtered colimits, up to equivalence (Corollary 8.4.8.9).

Structure

  • Subsection 8.4.1: Retracts in $\infty $-Categories
  • Subsection 8.4.2: Idempotents in Ordinary Categories
  • Subsection 8.4.3: Idempotents in $\infty $-Categories
  • Subsection 8.4.4: Idempotent Completeness
  • Subsection 8.4.5: Idempotent Completion
  • Subsection 8.4.6: Idempotent Endomorphisms
  • Subsection 8.4.7: Homotopy Idempotent Endomorphisms
  • Subsection 8.4.8: Partial Idempotents
  • Subsection 8.4.9: The Thompson Groupoid