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There are various variations of what exactly this means, applicable in various contexts, that tend to agree when they jointly apply.</p> <h3 id="of_spaces">Of spaces</h3> <p>There are many notions of <em>dimension</em> of <a class="existingWikiWord" href="/nlab/show/space">spaces</a>. What they all have in common, is that the <a class="existingWikiWord" href="/nlab/show/cartesian+space">cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> has dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>.</p> <ul> <li> <p>The <strong>dimension</strong> of a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/cardinality">cardinality</a> of any <a class="existingWikiWord" href="/nlab/show/basis+of+a+vector+space">linear basis</a>, hence of any <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-indexed <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> (<a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> in <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>) of the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a>. (The <em><a class="existingWikiWord" href="/nlab/show/basis+theorem">basis theorem</a></em>, equivalent to the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>, says that every vector space has a unique dimension.) For <a class="existingWikiWord" href="/nlab/show/modules">modules</a> over <a class="existingWikiWord" href="/nlab/show/rings">rings</a> that are not <a class="existingWikiWord" href="/nlab/show/fields">fields</a> (for which the theorem above does not hold, neither existence nor uniqueness) the term used is <a class="existingWikiWord" href="/nlab/show/rank">rank</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/trace">trace</a> of a the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> on a vector space coincides with its categorical trace in the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a>.</p> </li> <li> <p>Generalizing from this example, the <a class="existingWikiWord" href="/nlab/show/trace">trace</a> of an object in a <a class="existingWikiWord" href="/nlab/show/spherical+category">spherical</a> <a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a> is called its <em><a class="existingWikiWord" href="/nlab/show/quantum+dimension">quantum dimension</a></em>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> has <a class="existingWikiWord" href="/nlab/show/dimension+of+a+manifold">dimension of a manifold</a> equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> if it is locally isomorphic to the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>. A <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> is of <strong>complex dimension</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> if it is locally isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}^n</annotation></semantics></math>, hence has (real) dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">2 n</annotation></semantics></math>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> has (Lebesgue) <strong><a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a></strong> less than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> if every <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/refinement">refinement</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> such that every element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> belongs to fewer than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n + 1</annotation></semantics></math> elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>. (Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> has dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> if it has dimension less than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n + 1</annotation></semantics></math> but not less than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>.) By <a class="existingWikiWord" href="/nlab/show/negative+thinking">negative thinking</a>, this makes sense for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq -1</annotation></semantics></math>; precisely the <a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a> has dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math>, and precisely the <a class="existingWikiWord" href="/nlab/show/point">point</a> (of course) has dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> has <a class="existingWikiWord" href="/nlab/show/dimension+of+a+CW-complex">dimension of a CW-complex</a>, this being the largest <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> for which there are nontrivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cells">cells</a>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a> has a <strong><a class="existingWikiWord" href="/nlab/show/Hausdorff+dimension">Hausdorff dimension</a></strong> which may be any non-negative real number.</p> </li> <li> <p>A space in <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a> (<a class="existingWikiWord" href="/nlab/show/spectral+geometry">spectral geometry</a> via <a class="existingWikiWord" href="/nlab/show/spectral+triples">spectral triples</a>) may have a notion of <a class="existingWikiWord" href="/nlab/show/KO-dimension">KO-dimension</a>.</p> </li> </ul> <p>See also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+dimension">cohomology dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a></p> </li> </ul> <h3 id="of_objects_in_an_abelian_category_length">Of objects in an abelian category: Length</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/length+of+an+object">length of an object</a></em>.</p> <h3 id="of_objects_in_a_symmetric_monoidal_category_euler_characteristic">Of objects in a symmetric monoidal category: Euler characteristic</h3> <p>The dimension of a (finite dimensional) <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> over a <a class="existingWikiWord" href="/nlab/show/field">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is equivalently the <a class="existingWikiWord" href="/nlab/show/trace">trace</a> of the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> in the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> (which is a linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">k \to k</annotation></semantics></math>, canonically identified with an element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>tr</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>tr</mi><mo stretchy="false">(</mo><msub><mi>Id</mi> <mi>V</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>k</mi><mo>→</mo><mi>V</mi><mo>⊗</mo><msup><mi>V</mi> <mo>*</mo></msup><mover><mo>→</mo><mo>≃</mo></mover><msup><mi>V</mi> <mo>*</mo></msup><mo>⊗</mo><mi>V</mi><mo>→</mo><mi>k</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> tr(V) := tr(Id_V) : k \to V \otimes V^* \stackrel{\simeq}{\to} V^* \otimes V \to k \,. </annotation></semantics></math></div> <p>Therefore it makes sense for any <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and every <a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> to call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>tr</mi><mo stretchy="false">(</mo><msub><mi>Id</mi> <mi>V</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mn>1</mn><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">tr(Id_V) : 1 \to 1</annotation></semantics></math> the <em>categorical trace of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>.</em></p> <p>This definition subsumes standard notions of <em><a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a></em> and hence may also be thought of as generalizing that notion.</p> <h3 id="of_objects_in_an_topos">Of objects in an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos</h3> <p>The following notions of dimension capture aspects of the concept for objects in a <a class="existingWikiWord" href="/nlab/show/topos">topos</a> or more generally an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+dimension">cohomological dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a></p> </li> </ul> <h3 id="frobeniusperron_dimension">Frobenius-Perron dimension</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Frobenius-Perron+dimension">Frobenius-Perron dimension</a></li> </ul> <h2 id="properties">Properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+dimension+theorem">global dimension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <div> <p><strong>notion of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+dimension">cohomology dimension</a> (<a class="existingWikiWord" href="/nlab/show/virtual+cohomological+dimension">virtual</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension+of+a+manifold">dimension of a manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension+of+a+cell+complex">dimension of a cell complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension+of+a+separable+metric+space">dimension of a separable metric space</a></p> </li> <li> <p>small <a class="existingWikiWord" href="/nlab/show/inductive+dimension">inductive dimension</a> of a <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a></p> </li> <li> <p>large <a class="existingWikiWord" href="/nlab/show/inductive+dimension">inductive dimension</a> of a <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> </ul> </div> <h2 id="references">References</h2> <p>For the dimension in symmetric monoidal categories see the references at <a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a>.</p> <p>A general abstract <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theoretic</a> discussion of notions of homotopy/cohomology/covering <em>dimension</em> is in section 7.2 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 21, 2024 at 07:32:16. 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