compact object of a category



compact object of a category

Idea: attachments/Pasted%20image%2020220317230918.png

Motivations:

  • Compact objects of the derived category are typically easier to construct as we may extend them over open immersions using Thomason’s localization theorem
  • \(\mathbf{D} {{\mathsf{Coh}}(X)} _{\operatorname{qc}}\) (the unbounded derived category of \({\mathcal{O}}_X{\hbox{-}}\)modules with quasicoherent cohomology) is compactly generated when \(X\in{\mathsf{Sch}}\) is quasicompact and separated.
  • Coherence: attachments/Pasted%20image%2020220323190736.png attachments/Pasted%20image%2020220323190747.png

Definitions

  • For infinity categories
    • An object \(X \in \mathsf{C}\) is compact iff \(\operatorname{Map}_{\mathsf{C}}(X,{-})\in \mathsf{Cat}(\mathsf{C}, {\mathsf{Set}})\) preserves filtered homotopy colimits.
    • \(\mathsf{C}\) is compactly generated iff there is an equivalence \(\mathsf{C} \simeq\mathsf{{\mathsf{Ind}}}(\mathsf{D})\) for some \(\mathsf{D} \in { \underset{\infty}{ \mathsf{Cat}} }\) small.
  • For locally small categories:
    • An object \(X \in \mathsf{C}\) is compact iff \(\operatorname{Map}_{\mathsf{C}}(X,{-})\in \mathsf{Cat}(\mathsf{C}, {\mathsf{Set}})\) preserves filtered colimits.
    • \(\mathsf{C}\) is compactly generated iff the following non-degeneracy condition holds: setting \(F_X({-}) \coloneqq\mathop{\mathrm{Maps}}_{\mathsf{C}}(X, {-})\), if \(F_X(M) =0\) for all compact \(X\) then \(M=0\).
  • For triangulated categories:
    • An object \(X\in \mathsf{C}\) is compact iff homs commute with arbitrary direct sums, i.e. \begin{align*}\bigoplus_{i\in I} \mathsf{C}\qty{X, Y_i} { \, \xrightarrow{\sim}\, }\mathsf{C}\qty{X, \coprod_{i\in I} Y_i}\end{align*}
    • An object \(X\) is a generator if every object \(Y\in \mathsf{C}\) is a cokernel of the form \begin{align*} \operatorname{coker}\left( X{ {}^{ \scriptscriptstyle\oplus^{n} } } \to X{ {}^{ \scriptscriptstyle\oplus^{m} } } \right) { \, \xrightarrow{\sim}\, }Y, \qquad \iff \exists\quad X{ {}^{ \scriptscriptstyle\oplus^{n} } } \to X{ {}^{ \scriptscriptstyle\oplus^{m} } } \to Y \ \end{align*}
    • \(\mathsf{C}\) is compactly generated iff there is a set of compact objects \(S\) such that for every \(S \in \mathsf{S}\), \(\mathsf{C}{ {(S, X)}_{\scriptscriptstyle \bullet}} = 0 \implies X=0\).

Examples

  • Warning: a compactly generated triangulated category need not have a compact generator!
  • Note that finite colimits of compact objects are again compact
  • In \({\mathsf{Set}}\), compact objects are finite sets
  • In \({\mathsf{Top}}\), compact objects are finite sets with the discrete topology.
  • in ${}_{R}{\mathsf{Mod}} $, compact objects are finitely presented modules
  • \({\mathsf{Spaces}}\in { \underset{\infty}{ \mathsf{Cat}} }\) is compactly generated
  • Categories of module spectra over an \({\mathbb{E}}_1{\hbox{-}}\)rings have a compact generator, namely the free module of rank 1.
  • Proving an infinity category is compactly generated: exhibit it as a siftted-colimit completion.
  • A well-known fact: for $X\in {\mathsf{Sch}}_{/ {S}} $ quasicompact and separated, \(\mathbf{D} { {}_{{\mathcal{O}}_X}{\mathsf{Mod}} } _{\operatorname{qc}}\) the derived category of quasicoherent \({\mathcal{O}}_X{\hbox{-}}\)modules has a compact generator.
  • Full subcategories of compact objects form a thick subcategory.

Compact Generation

For infty-categories

attachments/Pasted%20image%2020220318104706.png

For triangulated categories

attachments/Pasted%20image%2020220323191712.png

Misc

attachments/Pasted%20image%2020220325164212.png attachments/Pasted%20image%2020220325164459.png

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