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<title>Projective object</title>

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Projective object

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Category_theory" title="Category theory">category theory</a>, the notion of a projective object generalizes the notion of a <a href="Projective_module" title="Projective module">projective module</a>. Projective objects in <a href="Abelian_category" title="Abelian category">abelian</a> <a href="Category_(mathematics)" title="Category (mathematics)">categories</a> are used in <a href="Homological_algebra" title="Homological algebra">homological algebra</a>. The <a href="Duality_(mathematics)" title="Duality (mathematics)">dual</a> notion of a projective object is that of an <a href="Injective_object" title="Injective object">injective object</a>.

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

An <a href="Object_(category_theory)" class="mw-redirect" title="Object (category theory)">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> in a category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

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

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</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> is projective if for any <a href="Epimorphism" title="Epimorphism">epimorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e:E\twoheadrightarrow X}">

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

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</math><img src="./8d98bc7734abd55c28bd37b2a518adacc272f7e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.39ex; height:2.176ex;" alt="{\displaystyle e:E\twoheadrightarrow X}" loading="lazy"> and <a href="Morphism" title="Morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:P\to X}">

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

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</math><img src="./c7045d3acf95a616d7629c8c27378f1105a1373f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.555ex; height:2.509ex;" alt="{\displaystyle f:P\to X}" loading="lazy">, there is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {f}}:P\to E}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\overline {f}}:P\to E}</annotation>

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</math><img src="./d2eef6f49add060cedc2efdb98ce530cb9fda455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.556ex; height:3.343ex;" alt="{\displaystyle {\overline {f}}:P\to E}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\circ {\overline {f}}=f}">

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<annotation encoding="application/x-tex">{\displaystyle e\circ {\overline {f}}=f}</annotation>

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</math><img src="./08a9c5e7ac9cf59c619ec584611ef17cda9afc1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.139ex; height:3.343ex;" alt="{\displaystyle e\circ {\overline {f}}=f}" loading="lazy">, i.e. the following diagram <a href="Commutative_diagram" title="Commutative diagram">commutes</a>:

That is, every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to X}">

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

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</math><img src="./706af59e3c62c6fa253d5f87c0810d24703b99ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.34ex; height:2.176ex;" alt="{\displaystyle P\to X}" loading="lazy"> <a href="Mathematical_jargon" class="mw-redirect" title="Mathematical jargon">factors through</a> every epimorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\twoheadrightarrow X}">

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

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</math><img src="./47d55f89f438dec08139bdeb86a96d608a645169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.37ex; height:2.176ex;" alt="{\displaystyle E\twoheadrightarrow X}" loading="lazy">.<a href="#cite_note-1">[1]</a>

If C is <a href="Category_(mathematics)#Small_and_large_categories" title="Category (mathematics)">locally small</a>, i.e., in particular <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} _{C}(P,X)}">

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

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</math><img src="./77c8a67546949cd6552859278282535de6690a9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.891ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} _{C}(P,X)}" loading="lazy"> is a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> for any object X in C, this definition is equivalent to the condition that the <a href="Hom_functor" title="Hom functor">hom functor</a> (also known as <a href="Representable_functor" title="Representable functor">corepresentable functor</a>)

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (P,-)\colon {\mathcal {C}}\to \mathbf {Set} }</annotation>

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</math><img src="./7f704ace3eb5d0b0c33e7c65b218ed50a7e4b731.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.875ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (P,-)\colon {\mathcal {C}}\to \mathbf {Set} }" loading="lazy"></dd></dl>

preserves <a href="Epimorphism" title="Epimorphism">epimorphisms</a>.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading3"><h3 id="Projective_objects_in_abelian_categories">Projective objects in abelian categories</h3></div>

If the category C is an abelian category such as, for example, the <a href="Category_of_abelian_groups" title="Category of abelian groups">category of abelian groups</a>, then P is projective if and only if

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (P,-)\colon {\mathcal {C}}\to \mathbf {Ab} }</annotation>

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</math><img src="./f7301a6fc2eb4da6c38225d28f406df347bcabe3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.63ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (P,-)\colon {\mathcal {C}}\to \mathbf {Ab} }" loading="lazy"></dd></dl>

is an <a href="Exact_functor" title="Exact functor">exact functor</a>, where Ab is the category of <a href="Abelian_group" title="Abelian group">abelian groups</a>.

An abelian category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">

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

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</math><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"> is said to have enough projectives if, for every object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">

</mrow>

</mstyle>

</mrow>

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

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</math><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy">, there is a projective object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">

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</mstyle>

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

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</math><img src="./280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"> and an epimorphism from P to A or, equivalently, a <a href="Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a>

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<annotation encoding="application/x-tex">{\displaystyle 0\to K\to P\to A\to 0.}</annotation>

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</math><img src="./ed9c1aa694f55bc05561f90b816a6065052808d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.983ex; height:2.176ex;" alt="{\displaystyle 0\to K\to P\to A\to 0.}" loading="lazy"></dd></dl>

The purpose of this definition is to ensure that any object A admits a <a href="Projective_resolution" class="mw-redirect" title="Projective resolution">projective resolution</a>, i.e., a (long) exact sequence

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<annotation encoding="application/x-tex">{\displaystyle \dots P_{2}\to P_{1}\to P_{0}\to A\to 0}</annotation>

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</math><img src="./64746efe81dfe8430184116d69a99c70f2d3f0d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.112ex; height:2.509ex;" alt="{\displaystyle \dots P_{2}\to P_{1}\to P_{0}\to A\to 0}" loading="lazy"></dd></dl>

where the objects <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0},P_{1},\dots }">

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<annotation encoding="application/x-tex">{\displaystyle P_{0},P_{1},\dots }</annotation>

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</math><img src="./fec506790bae8086a055af2561aa0ceca82dd78a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.884ex; height:2.509ex;" alt="{\displaystyle P_{0},P_{1},\dots }" loading="lazy"> are projective.

<div class="mw-heading mw-heading3"><h3 id="Projectivity_with_respect_to_restricted_classes">Projectivity with respect to restricted classes</h3></div>

<a href="#CITEREFSemadeni1963">Semadeni (1963)</a> discusses the notion of projective (and dually injective) objects relative to a so-called bicategory, which consists of a pair of subcategories of "injections" and "surjections" in the given category C. These subcategories are subject to certain formal properties including the requirement that any surjection is an epimorphism. A projective object (relative to the fixed class of surjections) is then an object P so that Hom(P, −) turns the fixed class of surjections (as opposed to all epimorphisms) into surjections of sets (in the usual sense).

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

<ul><li>The <a href="Coproduct" title="Coproduct">coproduct</a> of two projective objects is projective.<a href="#cite_note-3">[3]</a></li>

<li>The <a href="Section_(category_theory)" title="Section (category theory)">retract</a> of a projective object is projective.<a href="#cite_note-4">[4]</a></li></ul>

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

The statement that all sets are projective is equivalent to the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a>.

The projective objects in the category of abelian groups are the <a href="Free_abelian_group" title="Free abelian group">free abelian groups</a>.

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

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

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> be a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> with identity. Consider the (abelian) category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">-Mod of left <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">-modules. The projective objects in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">-Mod are precisely the <a href="Projective_module" title="Projective module">projective left R-modules</a>. Consequently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

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

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is itself a projective object in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">-Mod. Dually, the injective objects in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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The category of left (right) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">-modules also has enough projectives. This is true since, for every left (right) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">-module <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, we can take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> to be the free (and hence projective) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">-module generated by a generating set <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"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> (for example we can take <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"> to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">). Then the <a href="Canonical_projection" class="mw-redirect" title="Canonical projection">canonical projection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon F\to M}">

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<annotation encoding="application/x-tex">{\displaystyle \pi \colon F\to M}</annotation>

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</math><img src="./f8a95e44dd43ed49e5068df5cf352abab5bff1ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.163ex; height:2.176ex;" alt="{\displaystyle \pi \colon F\to M}" loading="lazy"> is the required <a href="Surjection" class="mw-redirect" title="Surjection">surjection</a>.

The projective objects in the category of <a href="Compact_Hausdorff_space" class="mw-redirect" title="Compact Hausdorff space">compact Hausdorff spaces</a> are precisely the <a href="Extremally_disconnected_space" title="Extremally disconnected space">extremally disconnected spaces</a>. This result is due to <a href="#CITEREFGleason1958">Gleason (1958)</a>, with a simplified proof given by <a href="#CITEREFRainwater1959">Rainwater (1959)</a>.

In the category of <a href="Banach_spaces" class="mw-redirect" title="Banach spaces">Banach spaces</a> and contractions (i.e., functionals whose norm is at most 1), the epimorphisms are precisely the maps with dense <a href="Image_(mathematics)" title="Image (mathematics)">image</a>. <a href="#CITEREFWiweger1969">Wiweger (1969)</a> harvtxt error: no target: CITEREFWiweger1969 (help) shows that the <a href="Examples_of_vector_spaces#Trivial_or_zero_vector_space" title="Examples of vector spaces">zero space</a> is the only projective object in this category. There are non-trivial spaces, though, which are projective with respect to the class of surjective contractions. In the category of <a href="Normed_vector_space" title="Normed vector space">normed vector spaces</a> with contractions (and surjective maps as "surjections"), the projective objects are precisely the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle l^{1}}</annotation>

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</math><img src="./a611054cf766f3612a9a9700d0699780661fd8a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.748ex; height:2.676ex;" alt="{\displaystyle l^{1}}" loading="lazy">-spaces.<a href="#cite_note-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle l^{1}(S)=\{(x_{s})_{s\in S},\sum _{s\in S}||x_{s}||<\infty \}.}</annotation>

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</math><img src="./f2e8f179b8c2827861b487839a5813a8be8aa04f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.547ex; height:5.676ex;" alt="{\displaystyle l^{1}(S)=\{(x_{s})_{s\in S},\sum _{s\in S}||x_{s}||<\infty \}.}" loading="lazy"></dd></dl>

<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="#CITEREFAwodey2010">Awodey (2010</a>, §2.1)

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFMac_Lane1978">Mac Lane (1978</a>, p. 118)

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFAwodey2010">Awodey (2010</a>, p. 72)

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFAwodey2010">Awodey (2010</a>, p. 33)

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFSemadeni1963">Semadeni (1963)</a>

</li>

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</style><cite id="CITEREFAwodey2010" class="citation cs2">Awodey, Steve (2010), Category theory (2nd ed.), Oxford: Oxford University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780199237180</bdi>, <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/740446073">740446073</a></cite></li>

<li><cite id="CITEREFGleason1958" class="citation cs2">Gleason, Andrew M. (1958), "Projective topological spaces", Illinois Journal of Mathematics, 2 (4A): 482–489, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fijm%2F1255454110">10.1215/ijm/1255454110</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0121775">0121775</a></cite></li>

<li><cite id="CITEREFMac_Lane1978" class="citation cs2">Mac Lane, Saunders (1978), Categories for the Working Mathematician (Second ed.), New York, NY: Springer New York, p. 114, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1441931236</bdi>, <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/851741862">851741862</a></cite></li>

<li><cite id="CITEREFMitchell1965" class="citation book cs1">Mitchell, Barry (1965). Theory of categories. Pure and applied mathematics. Vol. 17. Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-124-99250-4</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0202787">0202787</a>.</cite></li>

<li><cite id="CITEREFPothoven1969" class="citation cs2">Pothoven, Kenneth (1969), "Projective and Injective Objects in the Category of Banach Spaces", Proceedings of the American Mathematical Society, 22 (2): 437–438, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2037073">10.2307/2037073</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2037073">2037073</a></cite></li>

<li><cite id="CITEREFRainwater1959" class="citation cs2">Rainwater, John (1959), "A Note on Projective Resolutions", Proceedings of the American Mathematical Society, 10 (5): 734–735, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2033466">10.2307/2033466</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2033466">2033466</a></cite></li>

<li><cite id="CITEREFSemadeni1963" class="citation cs2">Semadeni, Z. (1963), <a rel="nofollow" class="external text" href="http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.desklight-2c5c2dc5-5782-45c0-b156-c7853b8977b3">"Projectivity, injectivity and duality"</a>, Rozprawy Mat., 35, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0154832">0154832</a></cite></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/projective+object">projective object</a> at the <a href="NLab" title="NLab">nLab</a>

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