- Tags
- Refs:
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Links:
- Balmer spectrum
- thick subcategory
- modern category theory
- conservative functor
- Grothendieck category
compact object of a category
Idea: 
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.
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Coherence:
Definitions
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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.
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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 \(M=0\).
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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
For triangulated categories
Misc
