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Compact object (mathematics)

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In mathematics, compact objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a <a href="Category_(mathematics)" title="Category (mathematics)">category</a> satisfying a certain finiteness condition.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

An object X in a category C which admits all <a href="Filtered_colimit" class="mw-redirect" title="Filtered colimit">filtered colimits</a> (also known as <a href="Direct_limit" title="Direct limit">direct limits</a>) is called compact if the functor

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{C}(X,\cdot ):C\to \mathrm {Sets} ,Y\mapsto \operatorname {Hom} _{C}(X,Y)}</annotation>

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</math><img src="./990e0c73f062bc73075f8f0f3241cbf1460ff2c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.597ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} _{C}(X,\cdot ):C\to \mathrm {Sets} ,Y\mapsto \operatorname {Hom} _{C}(X,Y)}" loading="lazy"></dd></dl>

commutes with filtered colimits, i.e., if the natural map

<mi>colim</mi>

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<mi>colim</mi>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {colim} \operatorname {Hom} _{C}(X,Y_{i})\to \operatorname {Hom} _{C}(X,\operatorname {colim} _{i}Y_{i})}</annotation>

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</math><img src="./f88d4e8be85284e82d24be4cb09eea2d32425fb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.629ex; height:2.843ex;" alt="{\displaystyle \operatorname {colim} \operatorname {Hom} _{C}(X,Y_{i})\to \operatorname {Hom} _{C}(X,\operatorname {colim} _{i}Y_{i})}" loading="lazy"></dd></dl>

is a bijection for any filtered system of objects <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle Y_{i}}</annotation>

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</math><img src="./d57be496fff95ee2a97ee43c7f7fe244b4dbf8ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.15ex; height:2.509ex;" alt="{\displaystyle Y_{i}}" loading="lazy"> in C.<a href="#cite_note-1">[1]</a> Since elements in the filtered colimit at the left are represented by maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to Y_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle X\to Y_{i}}</annotation>

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</math><img src="./f14fb9ebdfaba9b7f3b993adbe514e5b292af431.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.744ex; height:2.509ex;" alt="{\displaystyle X\to Y_{i}}" loading="lazy">, for some i, the surjectivity of the above map amounts to requiring that a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to \operatorname {colim} _{i}Y_{i}}">

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<mi>colim</mi>

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<annotation encoding="application/x-tex">{\displaystyle X\to \operatorname {colim} _{i}Y_{i}}</annotation>

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</math><img src="./906a15d1524a93eb4e098f9d6ccb5392e90310f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.356ex; height:2.509ex;" alt="{\displaystyle X\to \operatorname {colim} _{i}Y_{i}}" loading="lazy"> factors over some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle Y_{i}}</annotation>

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</math><img src="./d57be496fff95ee2a97ee43c7f7fe244b4dbf8ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.15ex; height:2.509ex;" alt="{\displaystyle Y_{i}}" loading="lazy">.

The terminology is motivated by an example arising from topology mentioned below. Several authors also use a terminology which is more closely related to algebraic categories: <a href="#CITEREFAdámekRosický1994">Adámek & Rosický (1994)</a> use the terminology finitely presented object instead of compact object. <a href="#CITEREFKashiwaraSchapira2006">Kashiwara & Schapira (2006)</a> call these the objects of finite presentation.

<div class="mw-heading mw-heading3"><h3 id="Compactness_in_∞-categories">Compactness in ∞-categories</h3></div>

The same definition also applies if C is an <a href="%E2%88%9E-category" class="mw-redirect" title="∞-category">∞-category</a>, provided that the above set of morphisms gets replaced by the mapping space in C (and the filtered colimits are understood in the ∞-categorical sense, sometimes also referred to as filtered homotopy colimits).

<div class="mw-heading mw-heading3"><h3 id="Compactness_in_triangulated_categories">Compactness in triangulated categories</h3></div>

For a <a href="Triangulated_category" title="Triangulated category">triangulated category</a> C which admits all <a href="Coproduct" title="Coproduct">coproducts</a>, <a href="#CITEREFNeeman2001">Neeman (2001)</a> defines an object to be compact if

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{C}(X,\cdot ):C\to \mathrm {Ab} ,Y\mapsto \operatorname {Hom} _{C}(X,Y)}</annotation>

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</math><img src="./e948b18eae90b9fb8d105e8f6402942f5e3d4b94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.486ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} _{C}(X,\cdot ):C\to \mathrm {Ab} ,Y\mapsto \operatorname {Hom} _{C}(X,Y)}" loading="lazy"></dd></dl>

commutes with coproducts. The relation of this notion and the above is as follows: suppose C arises as the <a href="Homotopy_category" title="Homotopy category">homotopy category</a> of a <a href="Stable_%E2%88%9E-category" title="Stable ∞-category">stable ∞-category</a> admitting all filtered colimits. (This condition is widely satisfied, but not automatic.) Then an object in C is compact in Neeman's sense if and only if it is compact in the ∞-categorical sense. The reason is that in a stable ∞-category, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} _{C}(X,-)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{C}(X,-)}</annotation>

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</math><img src="./9cc887c5447b4a55fd491b83f70a37e29ed2d2a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.954ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} _{C}(X,-)}" loading="lazy"> always commutes with finite colimits since these are limits. Then, one uses a presentation of filtered colimits as a coequalizer (which is a finite colimit) of an infinite coproduct.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

The compact objects in the <a href="Category_of_sets" title="Category of sets">category of sets</a> are precisely the finite sets.

For a ring R, the compact objects in the <a href="Category_of_modules" title="Category of modules">category of R-modules</a> are precisely the <a href="Finitely_presented_module" class="mw-redirect" title="Finitely presented module">finitely presented</a> R-modules. In particular, if R is a field, then compact objects are finite-dimensional vector spaces.

Similar results hold for any category of algebraic structures given by operations on a set obeying equational laws. Such categories, called <a href="Variety_(universal_algebra)" title="Variety (universal algebra)">varieties</a>, can be studied systematically using <a href="Lawvere_theories" class="mw-redirect" title="Lawvere theories">Lawvere theories</a>. For any Lawvere theory T, there is a category Mod(T) of models of T, and the compact objects in Mod(T) are precisely the finitely presented models. For example: suppose T is the theory of groups. Then Mod(T) is the category of groups, and the compact objects in Mod(T) are the finitely presented groups.

The compact objects in the <a href="Derived_category" title="Derived category">derived category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(R-{\text{Mod}})}">

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<annotation encoding="application/x-tex">{\displaystyle D(R-{\text{Mod}})}</annotation>

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</math><img src="./7c5d8c8bd347f9ca502f3049a12a70b6f5aed40d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.924ex; height:2.843ex;" alt="{\displaystyle D(R-{\text{Mod}})}" loading="lazy"> of R-modules are precisely the <a href="Perfect_complex" title="Perfect complex">perfect complexes</a>.

<a href="Compact_topological_space" class="mw-redirect" title="Compact topological space">Compact topological spaces</a> are not the compact objects in the <a href="Category_of_topological_spaces" title="Category of topological spaces">category of topological spaces</a>. Instead these are precisely the finite sets endowed with the <a href="Discrete_topology" class="mw-redirect" title="Discrete topology">discrete topology</a>.<a href="#cite_note-2">[2]</a> The link between compactness in topology and the above categorical notion of compactness is as follows: for a fixed topological space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, there is the category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Open}}(X)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Open}}(X)}</annotation>

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</math><img src="./cc5d120a39eff86f4bbe57c0cdecca91e3792a75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.215ex; height:2.843ex;" alt="{\displaystyle {\text{Open}}(X)}" loading="lazy"> whose objects are the open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (and inclusions as morphisms). Then, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a compact topological space if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is compact as an object in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Open}}(X)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Open}}(X)}</annotation>

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</math><img src="./cc5d120a39eff86f4bbe57c0cdecca91e3792a75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.215ex; height:2.843ex;" alt="{\displaystyle {\text{Open}}(X)}" loading="lazy">.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is any category, the category of <a href="Presheaf" class="mw-redirect" title="Presheaf">presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{PreShv}}(C)}">

<mtext>PreShv</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\text{PreShv}}(C)}</annotation>

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</math><img src="./8836b09c3fa4b35687a31a496ac4536494e4ea1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.915ex; height:2.843ex;" alt="{\displaystyle {\text{PreShv}}(C)}" loading="lazy"> (i.e., the category of functors from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{op}}">

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<annotation encoding="application/x-tex">{\displaystyle C^{op}}</annotation>

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</math><img src="./aae59a2052b797e7ccdb679cdd93a1e632f6012c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.654ex; height:2.343ex;" alt="{\displaystyle C^{op}}" loading="lazy"> to sets) has all colimits. The original category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is connected to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{PreShv}}(C)}">

<mtext>PreShv</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\text{PreShv}}(C)}</annotation>

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</math><img src="./8836b09c3fa4b35687a31a496ac4536494e4ea1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.915ex; height:2.843ex;" alt="{\displaystyle {\text{PreShv}}(C)}" loading="lazy"> by the <a href="Yoneda_embedding" class="mw-redirect" title="Yoneda embedding">Yoneda embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{(-)}:C\to {\text{PreShv}}(C),X\mapsto h_{X}:=\operatorname {Hom} (-,X)}">

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<mtext>PreShv</mtext>

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<annotation encoding="application/x-tex">{\displaystyle h_{(-)}:C\to {\text{PreShv}}(C),X\mapsto h_{X}:=\operatorname {Hom} (-,X)}</annotation>

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</math><img src="./8270ab461a118939610442c2cd3e607067a0b648.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:47.179ex; height:3.176ex;" alt="{\displaystyle h_{(-)}:C\to {\text{PreShv}}(C),X\mapsto h_{X}:=\operatorname {Hom} (-,X)}" loading="lazy">. For any object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle h_{X}}</annotation>

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</math><img src="./56faf1054608a618e4afcc6ca4ae3186098e9bb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.971ex; height:2.509ex;" alt="{\displaystyle h_{X}}" loading="lazy"> is a compact object (of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{PreShv}}(C)}">

<mtext>PreShv</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\text{PreShv}}(C)}</annotation>

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</math><img src="./8836b09c3fa4b35687a31a496ac4536494e4ea1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.915ex; height:2.843ex;" alt="{\displaystyle {\text{PreShv}}(C)}" loading="lazy">).

In a similar vein, any category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> can be regarded as a full subcategory of the category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Ind}}(C)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Ind}}(C)}</annotation>

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</math><img src="./eff110b4c2427c56c310c3d7524d344ef6526150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7ex; height:2.843ex;" alt="{\displaystyle {\text{Ind}}(C)}" loading="lazy"> of <a href="Ind-object" class="mw-redirect" title="Ind-object">ind-objects</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">. Regarded as an object of this larger category, any object of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is compact. In fact, the compact objects of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Ind}}(C)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Ind}}(C)}</annotation>

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</math><img src="./eff110b4c2427c56c310c3d7524d344ef6526150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7ex; height:2.843ex;" alt="{\displaystyle {\text{Ind}}(C)}" loading="lazy"> are precisely the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> (or, more precisely, their images in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Ind}}(C)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Ind}}(C)}</annotation>

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</math><img src="./eff110b4c2427c56c310c3d7524d344ef6526150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7ex; height:2.843ex;" alt="{\displaystyle {\text{Ind}}(C)}" loading="lazy">).

<div class="mw-heading mw-heading3"><h3 id="Non-examples">Non-examples</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Derived_category_of_sheaves_of_Abelian_groups_on_a_noncompact_X">Derived category of sheaves of Abelian groups on a noncompact X</h4></div>

In the unbounded <a href="Derived_category" title="Derived category">derived category</a> of sheaves of Abelian groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D({\text{Sh}}(X;{\text{Ab}}))}">

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<annotation encoding="application/x-tex">{\displaystyle D({\text{Sh}}(X;{\text{Ab}}))}</annotation>

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</math><img src="./53e11d1715804aae5dfb447add25268101079890.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.177ex; height:2.843ex;" alt="{\displaystyle D({\text{Sh}}(X;{\text{Ab}}))}" loading="lazy"> for a non-compact topological space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, it is generally not a compactly generated category. Some evidence for this can be found by considering an <a href="Cover_(topology)" title="Cover (topology)">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}</annotation>

</semantics>

</math><img src="./ccf91339c1648ed6bcadc0f8e06fbde97a884bbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.038ex; width:12.17ex; height:2.843ex;" alt="{\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}" loading="lazy"> (which can never be refined to a finite subcover using the non-compactness of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">) and taking a map<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \in {\text{Hom}}({\mathcal {F}}^{\bullet },{\underset {i\in I}{\text{colim}}}\mathbb {Z} _{U_{i}})}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<mtext>colim</mtext>

<mrow>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi \in {\text{Hom}}({\mathcal {F}}^{\bullet },{\underset {i\in I}{\text{colim}}}\mathbb {Z} _{U_{i}})}</annotation>

</semantics>

</math><img src="./9f9a43f2c62e3839629737c69ce335c770183254.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.923ex; height:4.176ex;" alt="{\displaystyle \phi \in {\text{Hom}}({\mathcal {F}}^{\bullet },{\underset {i\in I}{\text{colim}}}\mathbb {Z} _{U_{i}})}" loading="lazy"></blockquote>for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}^{\bullet }\in {\text{Ob}}(D({\text{Sh}}(X;{\text{Ab}})))}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}^{\bullet }\in {\text{Ob}}(D({\text{Sh}}(X;{\text{Ab}})))}</annotation>

</semantics>

</math><img src="./04372719833dd06c52cb2556d776d76107fb0628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.986ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}^{\bullet }\in {\text{Ob}}(D({\text{Sh}}(X;{\text{Ab}})))}" loading="lazy">. Then, for this map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

</semantics>

</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> to lift to an element<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \in {\underset {i\in I}{\text{colim}}}{\text{ Hom}}({\mathcal {F}}^{\bullet },\mathbb {Z} _{U_{i}})}">

<mtext>colim</mtext>

<mrow>

</mrow>

</munder>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi \in {\underset {i\in I}{\text{colim}}}{\text{ Hom}}({\mathcal {F}}^{\bullet },\mathbb {Z} _{U_{i}})}</annotation>

</semantics>

</math><img src="./079e103ad12fc3881b5cba6c4931bfa57dd4cb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.631ex; height:4.176ex;" alt="{\displaystyle \psi \in {\underset {i\in I}{\text{colim}}}{\text{ Hom}}({\mathcal {F}}^{\bullet },\mathbb {Z} _{U_{i}})}" loading="lazy"></blockquote>it would have to factor through some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{U_{i}}}">

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{U_{i}}}</annotation>

</semantics>

</math><img src="./7225b016d46a8920c1d29bb32b6e49217a1fa122.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.53ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{U_{i}}}" loading="lazy">, which is not guaranteed. Proving this requires showing that any compact object has support in some compact subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, and then showing this subset must be empty.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading4"><h4 id="Derived_category_of_quasi-coherent_sheaves_on_an_Artin_stack">Derived category of quasi-coherent sheaves on an Artin stack</h4></div>

For <a href="Algebraic_stack" title="Algebraic stack">algebraic stacks</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {X}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {X}}}</annotation>

</semantics>

</math><img src="./073868db564a5e806f61f8b2d526ba975cb5c077.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.671ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {X}}}" loading="lazy"> over positive characteristic, the unbounded derived category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{qc}({\mathfrak {X}})}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{qc}({\mathfrak {X}})}</annotation>

</semantics>

</math><img src="./ef82d4ca833f9228c1c598f5b0678b8fd75cca60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.105ex; height:3.009ex;" alt="{\displaystyle D_{qc}({\mathfrak {X}})}" loading="lazy"> of <a href="Quasi-coherent_sheaves" class="mw-redirect" title="Quasi-coherent sheaves">quasi-coherent sheaves</a> is in general not compactly generated, even if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {X}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {X}}}</annotation>

</semantics>

</math><img src="./073868db564a5e806f61f8b2d526ba975cb5c077.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.671ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {X}}}" loading="lazy"> is <a href="Quasi-compact" class="mw-redirect" title="Quasi-compact">quasi-compact</a> and <a href="Quasi-separated_morphism" title="Quasi-separated morphism">quasi-separated</a>.<a href="#cite_note-4">[4]</a> In fact, for the algebraic stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\mathbb {G} _{a}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\mathbb {G} _{a}}</annotation>

</semantics>

</math><img src="./4c42556d5dce12ff815dee1277a5a12ce5b95b34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.674ex; height:2.509ex;" alt="{\displaystyle B\mathbb {G} _{a}}" loading="lazy">, there are no compact objects other than the zero object. This observation can be generalized to the following theorem: if the stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {X}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {X}}}</annotation>

</semantics>

</math><img src="./073868db564a5e806f61f8b2d526ba975cb5c077.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.671ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {X}}}" loading="lazy"> has a stabilizer group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> such that

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is defined over a field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> of positive characteristic</li>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {G}}=G\otimes _{k}{\overline {k}}}</annotation>

</semantics>

</math><img src="./63211d9be2efa74cb1ae07472f9aa2e5859fd481.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.122ex; height:3.343ex;" alt="{\displaystyle {\overline {G}}=G\otimes _{k}{\overline {k}}}" loading="lazy"> has a subgroup isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{a}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{a}}</annotation>

</semantics>

</math><img src="./0a3a118277795dec18b43d8e6b839e6af5856180.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.91ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{a}}" loading="lazy"></li></ol>

then the only compact object in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{qc}({\mathfrak {X}})}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{qc}({\mathfrak {X}})}</annotation>

</semantics>

</math><img src="./ef82d4ca833f9228c1c598f5b0678b8fd75cca60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.105ex; height:3.009ex;" alt="{\displaystyle D_{qc}({\mathfrak {X}})}" loading="lazy"> is the zero object. In particular, the category is not compactly generated.

This theorem applies, for example, to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=GL_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=GL_{n}}</annotation>

</semantics>

</math><img src="./d1e1c3a4c8d62db922ba1db8c4322c7ffec6353f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.553ex; height:2.509ex;" alt="{\displaystyle G=GL_{n}}" loading="lazy"> by means of the embedding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{a}\to GL_{n}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{a}\to GL_{n}}</annotation>

</semantics>

</math><img src="./200e5e0b353d6c69bcffbe0112d6333dc388f2b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.152ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{a}\to GL_{n}}" loading="lazy"> sending a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {G} _{a}(S)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {G} _{a}(S)}</annotation>

</semantics>

</math><img src="./1f0c4cf8c3374a452bc3d6c3f5b9720437504b80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.389ex; height:2.843ex;" alt="{\displaystyle x\in \mathbb {G} _{a}(S)}" loading="lazy"> to the identity matrix plus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> at the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-th column in the first row.

<div class="mw-heading mw-heading2"><h2 id="Compactly_generated_categories">Compactly generated categories</h2></div>

In most categories, the condition of being compact is quite strong, so that most objects are not compact. A category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is compactly generated if any object can be expressed as a filtered colimit of compact objects in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">. For example, any vector space V is the filtered colimit of its finite-dimensional (i.e., compact) subspaces. Hence the category of vector spaces (over a fixed field) is compactly generated.

Categories which are compactly generated and also admit all colimits are called <a href="Accessible_category" title="Accessible category">accessible categories</a>.

<div class="mw-heading mw-heading2"><h2 id="Relation_to_dualizable_objects">Relation to dualizable objects</h2></div>

For categories C with a well-behaved tensor product (more formally, C is required to be a <a href="Monoidal_category" title="Monoidal category">monoidal category</a>), there is another condition imposing some kind of finiteness, namely the condition that an object is <a href="Dualizable_object" class="mw-redirect" title="Dualizable object">dualizable</a>. If the monoidal unit in C is compact, then any dualizable object is compact as well. For example, R is compact as an R-module, so this observation can be applied. Indeed, in the category of R-modules the dualizable objects are the finitely presented <a href="Projective_module" title="Projective module">projective modules</a>, which are in particular compact. In the context of ∞-categories, dualizable and compact objects tend to be more closely linked, for example in the ∞-category of complexes of R-modules, compact and dualizable objects agree. This and more general example where dualizable and compact objects agree are discussed in <a href="#CITEREFBen-ZviFrancisNadler2010">Ben-Zvi, Francis & Nadler (2010)</a>.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFLurie2009">Lurie (2009</a>, §5.3.4)

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFAdámekRosický1994">Adámek & Rosický (1994</a>, Chapter 1.A)

</li>

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</style><cite id="CITEREFNeeman" class="citation journal cs1">Neeman, Amnon. <a rel="nofollow" class="external text" href="https://eudml.org/doc/123048">"On the derived category of sheaves on a manifold"</a>. Documenta Mathematica. 6: 483–488.</cite>

</li>

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