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For the concept of <a class='existingWikiWord' href='/nlab/show/Picard+2-group'>Picard groupoid of a monoidal category</a>, see there.</p> </blockquote> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#introduction'>Introduction</a></li><li><a href='#2category_of_picard_groupoids'>2-category of Picard groupoids</a></li><li><a href='#additivity_of_the_homotopy_category_of_picard_groupoids'>Additivity of the homotopy category of Picard groupoids</a></li><li><a href='#model_for_stable_homotopy_1types'>Model for stable homotopy 1-types</a></li><li><a href='#related_concepts'>Related concepts</a></li></ul></div> <h2 id='introduction'>Introduction</h2> <div class='num_defn'> <h6 id='definition'>Definition</h6> <p>A <em>Picard groupoid</em> is a <a class='existingWikiWord' href='/nlab/show/symmetric+monoidal+category'>strict symmetric monoidal</a> <a class='existingWikiWord' href='/nlab/show/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_54214061e8c7afb6c3d3941ed37e5af2266ec988_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>𝒜</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{A}, \otimes, 1)</annotation></semantics></math> in which every <a class='existingWikiWord' href='/nlab/show/object'>object</a> and every <a class='existingWikiWord' href='/nlab/show/morphism'>morphism</a> is strictly <a class='existingWikiWord' href='/nlab/show/inverse'>invertible</a> with respect to <math class='maruku-mathml' display='inline' id='mathml_54214061e8c7afb6c3d3941ed37e5af2266ec988_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊗</mo></mrow><annotation encoding='application/x-tex'>\otimes</annotation></semantics></math>; that is to say: there is, for every object <math class='maruku-mathml' display='inline' id='mathml_54214061e8c7afb6c3d3941ed37e5af2266ec988_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_54214061e8c7afb6c3d3941ed37e5af2266ec988_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math>, an object <math class='maruku-mathml' display='inline' id='mathml_54214061e8c7afb6c3d3941ed37e5af2266ec988_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>a</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>a^{-1}</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_54214061e8c7afb6c3d3941ed37e5af2266ec988_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_54214061e8c7afb6c3d3941ed37e5af2266ec988_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>⊗</mo><msup><mi>a</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>a \otimes a^{-1} = 1</annotation></semantics></math>, and similarly for morphisms.</p> </div> <div class='num_defn'> <h6 id='remark'>Remark</h6> <p>The notion of a Picard groupoid <a class='existingWikiWord' href='/nlab/show/vertical+categorification'>categorifies</a> that of an <a class='existingWikiWord' href='/nlab/show/abelian+group'>abelian group</a>. Note in particular that <math class='maruku-mathml' display='inline' id='mathml_54214061e8c7afb6c3d3941ed37e5af2266ec988_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊕</mo></mrow><annotation encoding='application/x-tex'>\oplus</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_54214061e8c7afb6c3d3941ed37e5af2266ec988_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math> equip the set of objects of <math class='maruku-mathml' display='inline' id='mathml_54214061e8c7afb6c3d3941ed37e5af2266ec988_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> with the structure of an abelian group.</p> </div> <div class='num_defn'> <h6 id='remark_2'>Remark</h6> <p>The notion of a Picard groupoid can be weakened in two (or three) directions: the monoidal structure can be weak instead of strict (or just the symmetry part), and the invertibility criterion can be asked to hold only up to isomorphism. On this page, we shall, however, work with the fully strict notion.</p> </div> <h2 id='2category_of_picard_groupoids'>2-category of Picard groupoids</h2> <p>Picard groupoids assemble into a <a class='existingWikiWord' href='/nlab/show/strict+2-category'>strict 2-category</a>. The objects are Picard groupoids, the 1-arrows are strict <a class='existingWikiWord' href='/nlab/show/monoidal+functor'>monoidal functors</a> (these necessarily preserve both the symmetry and the object inverses), and the 2-arrows are <a class='existingWikiWord' href='/nlab/show/monoidal+natural+transformation'>monoidal natural transformations</a>.</p> <p>This strict 2-category admits a <a class='existingWikiWord' href='/nlab/show/closed+monoidal+category'>closed</a> <a class='existingWikiWord' href='/nlab/show/symmetric+monoidal+2-category'>(fully) strict symmetric monoidal structure</a>, which categorifies the usual closed monoidal structure of the category of abelian groups.</p> <h2 id='additivity_of_the_homotopy_category_of_picard_groupoids'>Additivity of the homotopy category of Picard groupoids</h2> <p>The <em>homotopy category</em> of the category of Picard groupoids consists, roughly speaking, of Picard groupoids up to equivalence. Formally, we can obtain it by <a class='existingWikiWord' href='/nlab/show/decategorification'>decategorifying</a> the 2-category of Picard groupoids, namely identifying all 1-arrows which are 2-isomorphic, and throwing away the 2-arrows.</p> <p>The closed monoidal structure of the 2-category of Picard groupoids, together with the fact that the objects of a Picard groupoid define an abelian group under <math class='maruku-mathml' display='inline' id='mathml_54214061e8c7afb6c3d3941ed37e5af2266ec988_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊕</mo></mrow><annotation encoding='application/x-tex'>\oplus</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_54214061e8c7afb6c3d3941ed37e5af2266ec988_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>, provides immediately an <a class='existingWikiWord' href='/nlab/show/enriched+category'>enrichment</a> of the homotopy category of Picard groupoids over abelian groups. Moreover, it is obvious that this category has all coproducts. Thus it is an additive category.</p> <h2 id='model_for_stable_homotopy_1types'>Model for stable homotopy 1-types</h2> <p>Picard groupoids are well-known to model stable homotopy 1-types, at least if one adopts the weak version of the invertibility condition. This is a stable version of the 1-truncated <a class='existingWikiWord' href='/nlab/show/homotopy+hypothesis'>homotopy hypothesis</a>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/monoidal+groupoid'>monoidal groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/2-group'>2-group</a></p> </li> </ul> <p> </p> <p> </p> </div>  </div>  </body> </html>