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href='/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of (∞,1)-sheaves</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Bousfield+localization'>Bousfield localization</a>, <a class='existingWikiWord' href='/nlab/show/simplicial+localization'>simplicial localization</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/descent'>descent</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/cover'>cover</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/descent+object'>descent object</a>, <a class='existingWikiWord' href='/nlab/show/descent+morphism'>descent morphism</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/matching+family'>matching family</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/sheaf'>sheaf</a>, <a class='existingWikiWord' href='/nlab/show/%282%2C1%29-sheaf'>(2,1)-sheaf</a>/<a class='existingWikiWord' href='/nlab/show/stack'>stack</a>, <a class='existingWikiWord' href='/nlab/show/2-sheaf'>2-sheaf</a>,<a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-sheaf'>(∞,1)-sheaf</a>/<a class='existingWikiWord' href='/nlab/show/infinity-stack'>∞-stack</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/cohomological+descent'>cohomological descent</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/monadic+descent'>monadic descent</a>, <a class='existingWikiWord' href='/nlab/show/higher+monadic+descent'>higher monadic descent</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Sweedler+coring'>Sweedler coring</a>, <a class='existingWikiWord' href='/nlab/show/descent+in+noncommutative+algebraic+geometry'>descent in noncommutative algebraic geometry</a></li> </ul> </li> </ul> <div> <p> <a href='/nlab/edit/descent+and+locality+-+contents'>Edit this sidebar</a> </p> </div></div> <h4 id='topos_theory'><math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,2)</annotation></semantics></math>-Topos theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/%28infinity%2C2%29-topos'>(∞,2)-topos</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/%28infinity%2Cn%29-presheaf'>(∞,2)-presheaf</a>, <a class='existingWikiWord' href='/nlab/show/%28infinity%2C2%29-site'>(∞,2)-site</a>,</li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/%28infinity%2C2%29-sheaf'>(∞,2)-sheaf</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/codomain+fibration'>codomain fibration</a>, <a class='existingWikiWord' href='/nlab/show/tangent+%28infinity%2C1%29-category'>tangent (∞,1)-category</a>, <a class='existingWikiWord' href='/nlab/show/quasicoherent+infinity-stack'>quasicoherent ∞-stack</a></li> </ul> <h2 id='truncations'>Truncations</h2> <p><strong><a class='existingWikiWord' href='/nlab/show/2-topos'>2-topos</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/2-pretopos'>2-pretopos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/2-presheaf'>2-presheaf</a>, <a class='existingWikiWord' href='/nlab/show/2-site'>2-site</a>, <a class='existingWikiWord' href='/nlab/show/2-sheaf'>2-sheaf</a>, <a class='existingWikiWord' href='/nlab/show/stack'>stack</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/n-localic+2-topos'>n-localic 2-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/2-Giraud+theorem'>2-Giraud theorem</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/2-topos'>(2,1)-topos</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/%282%2C1%29-presheaf'>(2,1)-presheaf</a>, <a class='existingWikiWord' href='/nlab/show/%282%2C1%29-site'>(2,1)-site</a>, <a class='existingWikiWord' href='/nlab/show/%282%2C1%29-sheaf'>(2,1)-sheaf</a></li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-topos'>(∞,1)-topos</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-presheaf'>(∞,1)-presheaf</a>, <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-site'>(∞,1)-site</a>, <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-sheaf'>(∞,1)-sheaf</a></li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/1-topos'>1-topos</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/presheaf'>presheaf</a>, <a class='existingWikiWord' href='/nlab/show/site'>site</a>, <a class='existingWikiWord' href='/nlab/show/sheaf'>sheaf</a></li> </ul> </div> <h4 id='2category_theory'>2-Category theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/2-category+theory'>2-category theory</a></strong></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/2-category'>2-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/strict+2-category'>strict 2-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/bicategory'>bicategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/enriched+bicategory'>enriched bicategory</a></p> </li> </ul> <p><strong>Transfors between 2-categories</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/2-functor'>2-functor</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/pseudofunctor'>pseudofunctor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/lax+functor'>lax functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/equivalence+of+2-categories'>equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/strict+2-natural+transformation'>2-natural transformation</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/lax+natural+transformation'>lax natural transformation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/icon'>icon</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/modification'>modification</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Yoneda+lemma+for+bicategories'>Yoneda lemma for bicategories</a></p> </li> </ul> <p><strong>Morphisms in 2-categories</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/fully+faithful+morphism'>fully faithful morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/faithful+morphism'>faithful morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/conservative+morphism'>conservative morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/pseudomonic+morphism'>pseudomonic morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/discrete+morphism'>discrete morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/eso+morphism'>eso morphism</a></p> </li> </ul> <p><strong>Structures in 2-categories</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/adjunction'>adjunction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/mate'>mate</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/monad'>monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/cartesian+object'>cartesian object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/fibration+in+a+2-category'>fibration in a 2-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/codiscrete+cofibration'>codiscrete cofibration</a></p> </li> </ul> <p><strong>Limits in 2-categories</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/2-limit'>2-limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/2-pullback'>2-pullback</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/comma+object'>comma object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/inserter'>inserter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/inverter'>inverter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/equifier'>equifier</a></p> </li> </ul> <p><strong>Structures on 2-categories</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/2-monad'>2-monad</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/lax-idempotent+2-monad'>lax-idempotent 2-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/2-monad'>pseudomonad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/pseudoalgebra+for+a+2-monad'>pseudoalgebra for a 2-monad</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/monoidal+bicategory'>monoidal 2-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/cartesian+bicategory'>cartesian bicategory</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Gray+tensor+product'>Gray tensor product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/2-category+equipped+with+proarrows'>proarrow equipment</a></p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#characterization_of_over_sites'>Characterization of over <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n,r)</annotation></semantics></math>-sites</a><ul><li><a href='#over_a_1site'>Over a 1-site</a></li><li><a href='#over_a_site__as_internal_categories'>Over a <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(2,1)</annotation></semantics></math>-site – As internal categories</a></li></ul></li></ul></li><li><a href='#examples'>Examples</a><ul><li><a href='#codomain_fibrations__sheaves_of_modules'>Codomain fibrations / sheaves of modules</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><ul><li><a href='#ReferencesInTermsOfInternalCategories'>In terms of categories internal to sheaf toposes</a></li><li><a href='#InTermsOfFiberedCategories'>In terms of fibered categories</a></li><li><a href='#2sites'>2-Sites</a></li></ul></li></ul></div> <h2 id='idea'>Idea</h2> <p>The notion of <em>2-sheaf</em> is the generalization of the notion of <a class='existingWikiWord' href='/nlab/show/sheaf'>sheaf</a> to the <a class='existingWikiWord' href='/nlab/show/higher+category+theory'>higher category theory</a> of <a class='existingWikiWord' href='/nlab/show/2-category'>2-categories</a>/<a class='existingWikiWord' href='/nlab/show/bicategory'>bicategories</a>. A 2-category of 2-sheaves forms a <a class='existingWikiWord' href='/nlab/show/2-topos'>2-topos</a>.</p> <div class='num_remark'> <h6 id='remark_on_terminology'>Remark on terminology</h6> <p>A <em>2-sheaf</em> is a higher sheaf of <a class='existingWikiWord' href='/nlab/show/category'>categories</a>. More restrictive than this is a higher sheaf with values in <a class='existingWikiWord' href='/nlab/show/groupoid'>groupoids</a>, which would be a <em><a class='existingWikiWord' href='/nlab/show/%282%2C1%29-sheaf'>(2,1)-sheaf</a></em>. Both these notions are often referred to as <strong><a class='existingWikiWord' href='/nlab/show/stack'>stack</a></strong>, or sometimes “stack of groupoids” and “stack of categories” for definiteness. But moreover, traditionally a <a class='existingWikiWord' href='/nlab/show/stack'>stack</a> (in either flavor) is considered only over a <a class='existingWikiWord' href='/nlab/show/site'>1-site</a>, whereas it makes sense to consider <a class='existingWikiWord' href='/nlab/show/%282%2C1%29-sheaf'>(2,1)-sheaves</a> more generally over <a class='existingWikiWord' href='/nlab/show/%282%2C1%29-site'>(2,1)-sites</a> and 2-sheaves over <a class='existingWikiWord' href='/nlab/show/2-site'>2-sites</a>.</p> <p>Therefore, saying “2-sheaf” serves to indicate the full generality of the notion of higher sheaves in <a class='existingWikiWord' href='/nlab/show/2-category+theory'>2-category theory</a>, as opposed to various special cases of this general notion which have traditionally been considered.</p> </div> <h2 id='definition'>Definition</h2> <p>Let <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/2-site'>2-site</a> having finite <a class='existingWikiWord' href='/nlab/show/2-limit'>2-limit</a>s (for convenience). For a covering family <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><msub><mo stretchy='false'>)</mo> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>(f_i:U_i\to U)_i</annotation></semantics></math> we have the comma objects</p> <div style='text-align: center;'><svg height='10em' viewBox='-30 -20 180 150' width='10em' xmlns='http://www.w3.org/2000/svg'> <desc>Comma Square</desc> <defs> <marker id='svg295arrowhead' markerHeight='5' markerUnits='strokeWidth' markerWidth='8' orient='auto' refX='0' refY='5' viewBox='0 0 10 10'> <path d='M 0 0 L 10 5 L 0 10 z'></path> </marker> </defs> <g font-size='16'> <foreignObject height='25' width='45' x='-10' y='0'><math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_1' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <mo stretchy='false'>(</mo> <msub><mi>f</mi> <mi>i</mi></msub> <mo stretchy='false'>/</mo> <msub><mi>f</mi> <mi>j</mi></msub> <mo stretchy='false'>)</mo> </mrow> <annotation encoding='application/x-tex'>(f_i/f_j)</annotation> </semantics> </math></foreignObject> <foreignObject height='20' width='20' x='100' y='0'><math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_2' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub><mi>U</mi> <mi>i</mi></msub> </mrow> <annotation encoding='application/x-tex'>U_i</annotation> </semantics> </math></foreignObject> <foreignObject height='20' width='20' x='0' y='100'><math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_3' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub><mi>U</mi> <mi>j</mi></msub> </mrow> <annotation encoding='application/x-tex'>U_j</annotation> </semantics> </math></foreignObject> <foreignObject height='20' width='20' x='100' y='100'><math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_4' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <mi>U</mi> </mrow> <annotation encoding='application/x-tex'>U</annotation> </semantics> </math></foreignObject> <foreignObject height='25' width='20' x='110' y='50'><math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_5' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub><mi>f</mi> <mi>i</mi></msub> </mrow> <annotation encoding='application/x-tex'>f_i</annotation> </semantics> </math></foreignObject> <foreignObject height='25' width='20' x='50' y='110'><math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_6' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub><mi>f</mi> <mi>j</mi></msub> </mrow> <annotation encoding='application/x-tex'>f_j</annotation> </semantics> </math></foreignObject> <foreignObject height='30' width='20' x='-20' y='50'><math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_7' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub><mi>q</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub> </mrow> <annotation encoding='application/x-tex'>q_{i j}</annotation> </semantics> </math></foreignObject> <foreignObject height='30' width='20' x='50' y='-20'><math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_8' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub><mi>p</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub> </mrow> <annotation encoding='application/x-tex'>p_{i j}</annotation> </semantics> </math></foreignObject> <foreignObject height='25' width='20' x='60' y='60'><math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_9' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub><mi>μ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub> </mrow> <annotation encoding='application/x-tex'>\mu_{i j}</annotation> </semantics> </math></foreignObject> </g> <g fill='none' marker-end='url(#svg295arrowhead)' stroke='#000' stroke-width='1.5'> <line x1='40' x2='90' y1='10' y2='10'></line> <line x1='30' x2='90' y1='110' y2='110'></line> <line x1='10' x2='10' y1='30' y2='90'></line> <line x1='110' x2='110' y1='30' y2='90'></line> <line x1='65' x2='55' y1='55' y2='65'></line> </g> </svg></div> <p>We also have the <a class='existingWikiWord' href='/nlab/show/double+comma+object'>double comma objects</a> <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><msub><mo>×</mo> <mrow><msub><mi>U</mi> <mi>j</mi></msub></mrow></msub><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f_i/f_j/f_k) = (f_i/f_j)\times_{U_j} (f_j/f_k)</annotation></semantics></math> with projections <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>r</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>:</mo><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo>→</mo><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>r_{i j k}:(f_i/f_j/f_k)\to (f_i/f_j)</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>s</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>:</mo><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo>→</mo><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>s_{i j k}:(f_i/f_j/f_k)\to (f_j/f_k)</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>t</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>:</mo><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo>→</mo><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>t_{i j k}:(f_i/f_j/f_k)\to (f_i/f_k)</annotation></semantics></math>.</p> <p>Now, a functor <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>X:C^{op} \to Cat</annotation></semantics></math> is called a <strong>2-presheaf</strong>. It is <strong>1-separated</strong> if</p> <ul> <li>For any covering family <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><msub><mo stretchy='false'>)</mo> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>(f_i:U_i\to U)_i</annotation></semantics></math> and any <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x,y\in X(U)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>a,b: x\to y</annotation></semantics></math>, if <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X(f_i)(a) = X(f_i)(b)</annotation></semantics></math> for all <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>=</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>a=b</annotation></semantics></math>.</li> </ul> <p>It is <strong>2-separated</strong> if it is 1-separated and</p> <ul> <li>For any covering family <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><msub><mo stretchy='false'>)</mo> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>(f_i:U_i\to U)_i</annotation></semantics></math> and any <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x,y\in X(U)</annotation></semantics></math>, given <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>b</mi> <mi>i</mi></msub><mo>:</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>b_i:X(f_i)(x) \to X(f_i)(y)</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>μ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>(</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>∘</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>p</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>b</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>=</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>q</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>b</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>∘</mo><msub><mi>μ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mu_{i j}(y) \circ X(p_{i j})(b_i) = X(q_{i j})(b_i) \circ \mu_{i j}(x)</annotation></semantics></math>, there exists a (necessarily unique) <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>b:x\to y</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>b</mi> <mi>i</mi></msub><mo>=</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>b_i = X(f_i)(b)</annotation></semantics></math>.</li> </ul> <p>It is a <strong>2-sheaf</strong> if it is 2-separated and</p> <ul> <li>For any covering family <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><msub><mo stretchy='false'>)</mo> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>(f_i:U_i\to U)_i</annotation></semantics></math> and any <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub><mo>∈</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x_i\in X(U_i)</annotation></semantics></math> together with morphisms <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ζ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>:</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>p</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>→</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>q</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\zeta_{i j}:X(p_{i j})(x_i) \to X(q_{i j})(x_j)</annotation></semantics></math> such that the following diagram commutes:<div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>r</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>p</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>r</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>ζ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>r</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>q</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>→</mo><mo>≅</mo></mover></mtd> <mtd><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>s</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>p</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><msup><mo></mo><mo>≅</mo></msup><mo stretchy='false'>↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy='false'>↓</mo> <mrow><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>s</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>ζ</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo></mrow></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>t</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>p</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo></mtd> <mtd><munder><mo>→</mo><mrow><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>t</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>ζ</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo></mrow></munder></mtd> <mtd><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>t</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>q</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo></mtd> <mtd><munder><mo>→</mo><mo>≅</mo></munder></mtd> <mtd><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>s</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>q</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{X(r_{i j k})X(p_{i j})(x_i) & \overset{X(r_{i j k})(\zeta_{i j})}{\to} & X(r_{i j k})X(q_{i j})(x_j) & \overset{\cong}{\to} & X(s_{i j k})X(p_{j k})(x_j)\\ ^\cong \downarrow && && \downarrow ^{X(s_{i j k})(\zeta_{j k})}\\ X(t_{i j k}) X(p_{i k})(x_i) & \underset{X(t_{i j k})(\zeta_{i k})}{\to} & X(t_{i j k}) X(q_{i k})(x_k) & \underset{\cong}{\to} & X(s_{i j k}) X(q_{j k})(x_k)} </annotation></semantics></math></div> <p>there exists an object <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x\in X(U)</annotation></semantics></math> and isomorphisms <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>≅</mo><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>X(f_i)(x)\cong x_i</annotation></semantics></math> such that for all <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><annotation encoding='application/x-tex'>i,j</annotation></semantics></math> the following square commutes:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>p</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>)</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>→</mo><mo>≅</mo></mover></mtd> <mtd><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>p</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><msup><mo></mo><mrow><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>μ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>)</mo></mrow></msup><mo stretchy='false'>↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy='false'>↓</mo> <mrow><msub><mi>ζ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>q</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>)</mo><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mtd> <mtd><munder><mo>→</mo><mo>≅</mo></munder></mtd> <mtd><mi>X</mi><mo stretchy='false'>(</mo><msub><mi>q</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{ X(p_{i j})X(f_i)(X) & \overset{\cong}{\to} & X(p_{i j})(x_i)\\ ^{X(\mu_{i j})}\downarrow && \downarrow^{\zeta_{i j}}\\ X(q_{i j})X(f_j)(x) & \underset{\cong}{\to} & X(q_{i j})(x_j).} </annotation></semantics></math></div></li> </ul> <p>A 2-sheaf, especially on a 1-site, is frequently called a <strong><a class='existingWikiWord' href='/nlab/show/stack'>stack</a></strong>. However, this has the unfortunate consequence that a 3-sheaf is then called a 2-stack, and so on with the numbering all offset by one. Also, it can be helpful to use a new term because of the notable differences between 2-sheaves on 2-sites and 2-sheaves on 1-sites. The main novelty is that <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>μ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\mu_{i j}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ζ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\zeta_{i j}</annotation></semantics></math> <em>need not be invertible</em>.</p> <p>Note, though, they must be invertible as soon as <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is (2,1)-site: <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>μ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\mu_{i j}</annotation></semantics></math> by definition and <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ζ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\zeta_{i j}</annotation></semantics></math> since an inverse is provided by <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>ι</mi> <mrow><mi>i</mi><mi>j</mi></mrow> <mo>*</mo></msubsup><mo stretchy='false'>(</mo><msub><mi>ζ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\iota_{i j}^*(\zeta_{i j})</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ι</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>↦</mo><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><mo>→</mo><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\iota_{i j}\mapsto (f_i/f_j) \to (f_j/f_i)</annotation></semantics></math> is the symmetry equivalence.</p> <p>If <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> lacks finite limits, then in the definitions of “2-separated” and “2-sheaf” instead of the comma objects <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>/</mo><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f_i/f_j)</annotation></semantics></math>, we need to use arbitrary objects <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> equipped with maps <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>V</mi><mo>→</mo><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>p:V\to U_i</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>q</mi><mo>:</mo><mi>V</mi><mo>→</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding='application/x-tex'>q:V\to U_j</annotation></semantics></math>, and a 2-cell <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mi>p</mi><mo>→</mo><msub><mi>f</mi> <mi>j</mi></msub><mi>q</mi></mrow><annotation encoding='application/x-tex'>f_i p \to f_j q</annotation></semantics></math>. We leave the precise definition to the reader.</p> <p>A 2-site is said to be <strong>subcanonical</strong> if for any <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>U\in C</annotation></semantics></math>, the representable functor <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(-,U)</annotation></semantics></math> is a 2-sheaf. When <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> has finite limits, it is easy to verify that this is true precisely when every covering family is a (necessarily pullback-stable) quotient of its kernel <span class='newWikiWord'>2-polycongruence<a href='/nlab/new/2-polycongruence'>?</a></span>. In particular, the regular coverage on a regular 2-category is subcanonical, as is the coherent coverage on a coherent 2-category.</p> <p>The 2-category <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mi>Sh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>2Sh(C)</annotation></semantics></math> of 2-sheaves on a small 2-site <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is, by definition, a <a class='existingWikiWord' href='/nlab/show/2-topos'>Grothendieck 2-topos</a>.</p> <h2 id='properties'>Properties</h2> <h3 id='characterization_of_over_sites'>Characterization of over <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n,r)</annotation></semantics></math>-sites</h3> <p>If the underlying <a class='existingWikiWord' href='/nlab/show/2-site'>2-site</a> happens to be an <a class='existingWikiWord' href='/nlab/show/%28n%2Cr%29-site'>(n,r)-site</a> for <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> and/or <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>r</mi></mrow><annotation encoding='application/x-tex'>r</annotation></semantics></math> lower than 2, there may be other equivalent ways to think of 2-sheaves.</p> <p>A <a class='existingWikiWord' href='/nlab/show/2-topos'>2-topos</a> with a <a class='existingWikiWord' href='/nlab/show/2-site'>2-site</a> of definition that happens to be just a 1-site or <a class='existingWikiWord' href='/nlab/show/%282%2C1%29-site'>(2,1)-site</a> is <em>1-localic</em> or <em>(2,1)-localic</em>.</p> <h4 id='over_a_1site'>Over a 1-site</h4> <p>Over a 1-site, the <a class='existingWikiWord' href='/nlab/show/Grothendieck+construction'>Grothendieck construction</a> says that <a class='existingWikiWord' href='/nlab/show/2-functor'>2-functors</a> on the site are equivalent to <a class='existingWikiWord' href='/nlab/show/Grothendieck+fibration'>fibered categories</a> over the site. Hence in this case the theory of 2-sheaves can be entirely formulated in terms of fibered categories. See <em><a href='#InTermsOfFiberedCategories'>References – In terms of fibered categories</a></em>.</p> <p>Also, over a 1-site a 2-sheaf is essentially a <em><a class='existingWikiWord' href='/nlab/show/indexed+category'>indexed category</a></em>. Therefore stacks over 1-sites can also be discussed in this language, see notably the work (<a href='#BungePare'>Bunge-Pare</a>).</p> <p>In particular, if the 1-site <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/topos'>topos</a>, then every topos <em>over</em> <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> as its <a class='existingWikiWord' href='/nlab/show/base+topos'>base topos</a> (a <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>-topos) induces an <a class='existingWikiWord' href='/nlab/show/indexed+category'>indexed category</a>.</p> <div class='num_prop'> <h6 id='proposition'>Proposition</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is a topos and <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> is a <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>-topos, then (the <a class='existingWikiWord' href='/nlab/show/indexed+category'>indexed category</a> corresponding to) <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> is a 2-sheaf on <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> with respect to the <a class='existingWikiWord' href='/nlab/show/canonical+topology'>canonical topology</a>.</p> </div> <p>This appears as (<a href='#BungePare'>Bunge-Pare, corollary 2.6</a>).</p> <p>Moreover, over a <a class='existingWikiWord' href='/nlab/show/site'>1-site</a> the <a class='existingWikiWord' href='/nlab/show/2-topos'>2-topos</a> of 2-sheaves ought to be equivalent to the (suitably defined) <a class='existingWikiWord' href='/nlab/show/2-category'>2-category</a> of <a class='existingWikiWord' href='/nlab/show/internal+category'>internal categories</a> in the underlying <a class='existingWikiWord' href='/nlab/show/1-topos'>1-topos</a>. See <em><a href='#ReferencesInTermsOfInternalCategories'>References – In terms of internal categories</a></em>.</p> <h4 id='over_a_site__as_internal_categories'>Over a <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(2,1)</annotation></semantics></math>-site – As internal categories</h4> <p>Over a <a class='existingWikiWord' href='/nlab/show/%282%2C1%29-site'>(2,1)-site</a> the <a class='existingWikiWord' href='/nlab/show/2-topos'>2-topos</a> of 2-sheaves ought to be equivalent to the <a class='existingWikiWord' href='/nlab/show/2-category'>2-category</a> of <a class='existingWikiWord' href='/nlab/show/category+object+in+an+%28infinity%2C1%29-category'>internal (infinity,1)-categories</a> in the corresponding <a class='existingWikiWord' href='/nlab/show/2-topos'>(2,1)-topos</a>.</p> <p>This is discussed at <em><a href='2-topos#InTermsOfInternalCategories'>2-Topos – In terms of internal categories</a></em>.</p> <h2 id='examples'>Examples</h2> <h3 id='codomain_fibrations__sheaves_of_modules'>Codomain fibrations / sheaves of modules</h3> <p>A classical class of examples for 2-sheaves are <a class='existingWikiWord' href='/nlab/show/codomain+fibration'>codomain fibrations</a> over suitable sites, or rather their <a class='existingWikiWord' href='/nlab/show/tangent+category'>tangent categories</a>. As discussed there, this includes the case of sheaves of categories of <a class='existingWikiWord' href='/nlab/show/module'>modules</a> over sites of <a class='existingWikiWord' href='/nlab/show/algebra+over+a+Lawvere+theory'>algebras</a>.</p> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/exact+category'>exact category</a> with <a class='existingWikiWord' href='/nlab/show/finite+limit'>finite limits</a>, the <a class='existingWikiWord' href='/nlab/show/codomain+fibration'>codomain fibration</a> <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cod</mi><mo>:</mo><msup><mi>C</mi> <mi>I</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>Cod : C^I \to C</annotation></semantics></math> or equivalently (under the <a class='existingWikiWord' href='/nlab/show/Grothendieck+construction'>Grothendieck construction</a>), the self-<a class='existingWikiWord' href='/nlab/show/indexed+category'>indexing</a> of <math class='maruku-mathml' display='inline' id='mathml_71f0e94cb1da0becd34b5f402b36e7f931da8161_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is a 2-sheaf with respect to the <a class='existingWikiWord' href='/nlab/show/canonical+topology'>canonical topology</a>.</p> </div> <p>This is for instance (<a href='#BungePare'>Bunge-Pare, corollary 2.4</a>).</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/presheaf'>presheaf</a> / <a class='existingWikiWord' href='/nlab/show/sheaf'>sheaf</a> / <a class='existingWikiWord' href='/nlab/show/cosheaf'>cosheaf</a></p> </li> <li> <p><strong>2-sheaf</strong> / <a class='existingWikiWord' href='/nlab/show/stack'>stack</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-sheaf'>(∞,1)-sheaf</a> / <a class='existingWikiWord' href='/nlab/show/infinity-stack'>∞-stack</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C2%29-sheaf'>(∞,2)-sheaf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2Cn%29-sheaf'>(∞,n)-sheaf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/descent'>descent</a></p> </li> </ul> <h2 id='references'>References</h2> <p>Historically, the original definition of <em><a class='existingWikiWord' href='/nlab/show/stack'>stack</a></em> included the case of category-valued functors, hence of 2-sheaves, in:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/Jean+Giraud'>Jean Giraud</a>, <em>Cohomologie non abélienne</em> Grundlehren <strong>179</strong>, Springer (1971) [[doi:10.1007/978-3-662-62103-5](https://www.springer.com/gp/book/9783540053071)]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Jean+Giraud'>Jean Giraud</a>, <em>Classifying topos</em>, in: <a class='existingWikiWord' href='/nlab/show/William+Lawvere'>William Lawvere</a> (ed.) <em>Toposes, Algebraic Geometry and Logic</em>, Lecture Notes in Mathematics <strong>274</strong>, Springer (1972) [[doi:10.1007/BFb0073964](https://doi.org/10.1007/BFb0073964)]</p> </li> </ul> <h3 id='ReferencesInTermsOfInternalCategories'>In terms of categories internal to sheaf toposes</h3> <p>Category-valued stacks as <a class='existingWikiWord' href='/nlab/show/internal+category'>internal categories</a> in the underlying <a class='existingWikiWord' href='/nlab/show/Grothendieck+topos'>sheaf topos</a>:</p> <ul> <li id='BungePare'> <p><a class='existingWikiWord' href='/nlab/show/Marta+Bunge'>Marta Bunge</a>, <a class='existingWikiWord' href='/nlab/show/Robert+Par%C3%A9'>Robert Paré</a>, <em>Stacks and equivalence of indexed categories</em>, <a class='existingWikiWord' href='/nlab/show/Cahiers'>Cahiers de Top. et Géom. Diff. Catég</a> <strong>20</strong> 4 (1979) 373-399 [[numdam:CTGDC_1979__20_4_373_0](http://www.numdam.org/item?id=CTGDC_1979__20_4_373_0)]</p> </li> <li id='Bunge'> <p><a class='existingWikiWord' href='/nlab/show/Marta+Bunge'>Marta Bunge</a>, <em>Stack completions and Morita equivalence for categories in a topos</em>, <a class='existingWikiWord' href='/nlab/show/Cahiers'>Cahiers de Top. et Géom. Diff. Catég</a> <strong>20</strong> 4, (1979) 401-436 [[numdam](http://www.numdam.org/item?id=CTGDC_1979__20_4_401_0), <a href='http://www.ams.org/mathscinet-getitem?mr=558106'>MR558106</a>]</p> </li> <li id='JoyalTierney'> <p><a class='existingWikiWord' href='/nlab/show/Andr%C3%A9+Joyal'>André Joyal</a>, <a class='existingWikiWord' href='/nlab/show/Myles+Tierney'>Myles Tierney</a>, section 3 of: <em>Strong stacks and classifying spaces</em>, in: <em>Category Theory</em> (<a class='existingWikiWord' href='/nlab/show/Como'>Como</a>, 1990), Lecture Notes in Mathematics <strong>1488</strong>, Springer (1991) 213-236 [[doi:10.1007/BFb0084222](https://doi.org/10.1007/BFb0084222)]</p> <blockquote> <p>(establishing the <a class='existingWikiWord' href='/nlab/show/canonical+model+structure+on+Cat'>canonical model structure on Cat</a> in the internal generality)</p> </blockquote> </li> </ul> <h3 id='InTermsOfFiberedCategories'>In terms of fibered categories</h3> <p>A discussion of stacks over <a class='existingWikiWord' href='/nlab/show/site'>1-sites</a> in terms of their <a class='existingWikiWord' href='/nlab/show/Grothendieck+construction'>associated</a> <a class='existingWikiWord' href='/nlab/show/Grothendieck+fibration'>fibered categories</a> is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Angelo+Vistoli'>Angelo Vistoli</a>, <em>Notes on Grothendieck topologies, fibered categories and descent theory</em> (<a href='http://homepage.sns.it/vistoli/descent.pdf'>pdf</a>)</li> </ul> <h3 id='2sites'>2-Sites</h3> <p>The above text involves content transferred from</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Mike+Shulman'>Michael Shulman</a>, <em><a class='existingWikiWord' href='/michaelshulman/show/2-site' title='michaelshulman'>2-site</a></em></li> </ul> <p>2-sites were earlier considered in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Ross+Street'>Ross Street</a>, <em><a class='existingWikiWord' href='/nlab/show/StreetCBS'>StreetCBS</a></em></li> </ul> <p> </p> </div>  </div>  </body> </html>