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}); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">Michael Shulman</span> exactness hypothesis </h1> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#particular_cases'>Particular cases</a><ul><li><a href='#'><math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>k=0,n=1</annotation></semantics></math></a></li><li><a href='#_2'><math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>k=1,n=1</annotation></semantics></math></a></li><li><a href='#_3'><math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>=</mo><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>0</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>k=0,n=(\infty,0)</annotation></semantics></math></a></li><li><a href='#_4'><math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>k=0,n=2</annotation></semantics></math></a></li></ul></li><li><a href='#grothendieck_toposes'>Grothendieck <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-toposes</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>I propose the <strong>exactness hypothesis</strong> as one of the “guiding hypotheses of higher category theory,” alongside the <a class='existingWikiWord' href='/nlab/show/homotopy+hypothesis' title='nlab'>homotopy</a>, <a class='existingWikiWord' href='/nlab/show/delooping+hypothesis' title='nlab'>delooping</a>, and <a class='existingWikiWord' href='/nlab/show/stabilization+hypothesis' title='nlab'>stabilization</a> hypotheses. The exactness hypothesis states that</p> <ul> <li>The <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n+1)</annotation></semantics></math>-category <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mi>Mon</mi><mi>n</mi><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>k Mon n Cat</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/k-tuply+monoidal+n-category' title='nlab'>k-tuply monoidal n-categories</a> is an exact <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n+1)</annotation></semantics></math>-category.</li> </ul> <p>This is closely related to the <a class='existingWikiWord' href='/nlab/show/delooping+hypothesis' title='nlab'>delooping hypothesis</a>, and at least in low dimensions the latter is a special case of it. To get some intuition for how to get from delooping to exactness (and understand the meaning of “exact”), let <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_9' 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 category with one object <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo></mrow><annotation encoding='application/x-tex'>*</annotation></semantics></math> and consider how we might construct the corresponding <a class='existingWikiWord' href='/nlab/show/monoid' title='nlab'>monoid</a> <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mo>*</mo><mo>,</mo><mo>*</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(*,*)</annotation></semantics></math> by 2-categorical methods in <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math>. A little bit of thought shows that this monoid is given by the <a class='existingWikiWord' href='/nlab/show/2-limit' title='nlab'>comma object</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi><mo stretchy='false'>(</mo><mo>*</mo><mo>,</mo><mo>*</mo><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mo stretchy='false'>↓</mo><mo>*</mo></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd><mover><mo>→</mo><mo>*</mo></mover></mtd> <mtd><mi>C</mi><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{C(*,*) & \to & 1\\ \downarrow & \Downarrow & \downarrow * \\ 1 & \overset{*}{\to} & C.}</annotation></semantics></math></div> <p>This gives <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mo>*</mo><mo>,</mo><mo>*</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(*,*)</annotation></semantics></math> as a discrete object in <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math>, which moreover has the structure of a monoid. Note that this comma object can also be described as the <a class='existingWikiWord' href='/nlab/show/loop+space+object' title='nlab'>based loop object</a> of <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> when <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math> is equipped with the “walking arrow” <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mn>2</mn></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{2}</annotation></semantics></math> as its <a class='existingWikiWord' href='/nlab/show/interval+object' title='nlab'>interval object</a>.</p> <p>Conversely, given a monoid <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>, we can construct a category <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>B M</annotation></semantics></math> with one object and <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mi>M</mi><mo stretchy='false'>(</mo><mo>*</mo><mo>,</mo><mo>*</mo><mo stretchy='false'>)</mo><mo>=</mo><mi>M</mi></mrow><annotation encoding='application/x-tex'>B M (*,*)=M</annotation></semantics></math> as the <a class='existingWikiWord' href='/nlab/show/descent+object' title='nlab'>lax codescent object</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd></mtr> <mtr><mtd><mi>M</mi><mo>×</mo><mi>M</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>M</mi></mtd> <mtd><mo>←</mo></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{ & \to && \to \\ M\times M & \to & M & \leftarrow & 1\\ & \to && \to } </annotation></semantics></math></div> <p>In less fancy words, this means that <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>B M</annotation></semantics></math> is the universal category equipped with a functor <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>→</mo><mi>M</mi></mrow><annotation encoding='application/x-tex'>1\to M</annotation></semantics></math> and a 2-cell</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>M</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>B</mi><mi>M</mi><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{M & \to & 1\\ \downarrow & \Downarrow & \downarrow \\ 1 & \to & B M.}</annotation></semantics></math></div> <p>which is compatible with the multiplication and unit of <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>. Thus, the statement that “monoids can be identified with one-object categories” can be interpreted as a statement about an <a class='existingWikiWord' href='/nlab/show/idempotent+adjunction' title='nlab'>idempotent adjunction</a>.</p> <p>This is strikingly reminiscent of the definition of an <a class='existingWikiWord' href='/nlab/show/exact+category' title='nlab'>exact category</a>. Now the monoid <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>, which we can regard as an internal category in <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math> via <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi><mspace width='thickmathspace'></mspace><mo>⇉</mo><mspace width='thickmathspace'></mspace><mn>1</mn></mrow><annotation encoding='application/x-tex'>M \;\rightrightarrows\; 1</annotation></semantics></math>, plays the role of an <a class='existingWikiWord' href='/nlab/show/equivalence+relation' title='nlab'>equivalence relation</a>, and the comma square above plays the role of the <a class='existingWikiWord' href='/nlab/show/kernel+pair' title='nlab'>kernel pair</a> of the map <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>1\to C</annotation></semantics></math>. Clearly, then, a natural generalization of the delooping hypothesis for <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math> would involve replacing <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> by some more general category. We can define the <strong>kernel</strong> of an arbitrary functor <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>f:A\to B</annotation></semantics></math> to be the comma category</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>/</mo><mi>f</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mo stretchy='false'>↓</mo><mi>f</mi></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>B</mi><mo>;</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{(f/f) & \to & A\\ \downarrow & \Downarrow & \downarrow f \\ A & \overset{f}{\to} & B;}</annotation></semantics></math></div> <p>an appropriate statement of exactness for a 2-category should then say that certain structures that “behave like <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>/</mo><mi>f</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>⇉</mo><mspace width='thickmathspace'></mspace><mi>A</mi></mrow><annotation encoding='application/x-tex'>(f/f) \;\rightrightarrows\; A</annotation></semantics></math>”, in the same way that equivalence relations behave like kernel pairs, all arise as the kernel of some functor.</p> <p>Possibly some more general notion of exactness could be formulated in any category equipped with a (possibly directed) <a class='existingWikiWord' href='/nlab/show/interval+object' title='nlab'>interval object</a>.</p> <h2 id='particular_cases'>Particular cases</h2> <h3 id=''><math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>k=0,n=1</annotation></semantics></math></h3> <p>One formal definition of “behave like a kernel” in the case of 2-categories (due essentially to Ross Street) can be found <a class='existingWikiWord' href='/michaelshulman/show/2-congruence'>here</a>, and a corresponding definition of “exact 2-category” <a class='existingWikiWord' href='/michaelshulman/show/exact+2-category'>here</a>. In the <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>k=0,n=1</annotation></semantics></math> case of <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math>, one way to state exactness is that</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/double+category' title='nlab'>double categories</a> <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>D</mi> <mn>1</mn></msub><mspace width='thickmathspace'></mspace><mo>⇉</mo><mspace width='thickmathspace'></mspace><msub><mi>D</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>D_1 \;\rightrightarrows\; D_0</annotation></semantics></math> whose horizontal 2-categories are homwise discrete and which have a <span class='newWikiWord'>thin structure<a href='/nlab/new/thin+structure'>?</a></span> can be identified with <a class='existingWikiWord' href='/nlab/show/essentially+surjective+functor' title='nlab'>essentially surjective functors</a> <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>D</mi> <mn>0</mn></msub><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>D_0\to C</annotation></semantics></math>.</li> </ul> <p>In the case <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>D</mi> <mn>0</mn></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>D_0=1</annotation></semantics></math>, such a double category is precisely a discrete monoid in <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math> (the thin structure is automatic in this case), while an essentially surjective functor <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>1\to C</annotation></semantics></math> just makes <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> a pointed connected category.</p> <h3 id='_2'><math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>k=1,n=1</annotation></semantics></math></h3> <p>In the <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>k=1,n=1</annotation></semantics></math> case of <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>MonCat</mi></mrow><annotation encoding='application/x-tex'>MonCat</annotation></semantics></math>, one way to state exactness is that</p> <ul> <li><em>monoidal</em> double categories <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>D</mi> <mn>1</mn></msub><mspace width='thickmathspace'></mspace><mo>⇉</mo><mspace width='thickmathspace'></mspace><msub><mi>D</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>D_1 \;\rightrightarrows\; D_0</annotation></semantics></math> whose horizontal 2-categories are homwise discrete and which have a thin structure can be identified with essentially surjective (strong) monoidal functors <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>D</mi> <mn>0</mn></msub><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>D_0\to C</annotation></semantics></math>.</li> </ul> <p>Again, in the case <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>D</mi> <mn>0</mn></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>D_0=1</annotation></semantics></math> such a double category is precisely a discrete commutative monoid, and an essentially surjective monoidal functor <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>1\to C</annotation></semantics></math> makes <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> a pointed connected monoidal category.</p> <h3 id='_3'><math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>=</mo><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>0</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>k=0,n=(\infty,0)</annotation></semantics></math></h3> <p>In this case an exactness statement has been proven by Lurie:</p> <ul> <li>internal groupoids <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub><mspace width='thickmathspace'></mspace><mo>⇉</mo><mspace width='thickmathspace'></mspace><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>G_1 \;\rightrightarrows\; G_0</annotation></semantics></math> in the <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_55' xmlns='http://www.w3.org/1998/Math/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>-category of spaces can be identified with maps <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>G</mi> <mn>0</mn></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>G_0\to X</annotation></semantics></math> which are surjective on <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>\pi_0</annotation></semantics></math>.</li> </ul> <p>Taking <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>G</mi> <mn>0</mn></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>G_0=1</annotation></semantics></math>, this includes delooping in classical homotopy theory. When all 1-cells are invertible, the comma object in the definition of a kernel reduces to a (homotopy) pullback. Thus we recover the <a class='existingWikiWord' href='/nlab/show/generalized+universal+bundle' title='nlab'>observation</a> that the based loop space <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ω</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>\Omega X</annotation></semantics></math> of a pointed space <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is the homotopy pullback of the basepoint <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>1\to X</annotation></semantics></math> along itself.</p> <h3 id='_4'><math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>k=0,n=2</annotation></semantics></math></h3> <p>When <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-categories contain noninvertible <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi></mrow><annotation encoding='application/x-tex'>j</annotation></semantics></math>-morphisms for <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mo>></mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>j\gt 1</annotation></semantics></math>, there is an extra subtlety. If <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_66' 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 pointed 2-category, then the comma object</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy='false'>(</mo><mo>*</mo><mo stretchy='false'>/</mo><mo>*</mo><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mo stretchy='false'>↓</mo><mo>*</mo></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd><mover><mo>→</mo><mo>*</mo></mover></mtd> <mtd><mi>C</mi><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{(*/*) & \to & 1\\ \downarrow & \Downarrow & \downarrow * \\ 1 & \overset{*}{\to} & C.}</annotation></semantics></math></div> <p>(in the 3-category <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>2Cat</annotation></semantics></math>) is the monoidal category consists of endomorphisms of <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>*\in C</annotation></semantics></math> and <em>isomorphisms</em> between them.</p> <p>To recover information about the noninvertible 2-cells in <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mo>*</mo><mo>,</mo><mo>*</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(*,*)</annotation></semantics></math>, we can consider, in addition to the comma object, the “2-comma object,” which is the 3-limit weighted by <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>→</mo><mi>T</mi><mo>←</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>1\to T \leftarrow 1</annotation></semantics></math> where <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> is the “walking 2-cell.” With this approach, the appropriate notion of “kernel” starts to look more like the “<math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Θ</mi></mrow><annotation encoding='application/x-tex'>\Theta</annotation></semantics></math>-categories” considered by Joyal, Berger, Rezk, and others. The connection with monoidal objects also becomes less direct.</p> <p>A different approach is to consider a version of the comma object that <em>does</em> include information about the noninvertible 2-cells. This is not a 3-limit in the sense of a 2Cat-enriched limit, but it is a <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><msub><mi>Cat</mi> <mi>l</mi></msub></mrow><annotation encoding='application/x-tex'>2Cat_l</annotation></semantics></math>-enriched limit, where <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><msub><mi>Cat</mi> <mi>l</mi></msub></mrow><annotation encoding='application/x-tex'>2Cat_l</annotation></semantics></math> denotes 2Cat with the <em>lax</em> version of the <a class='existingWikiWord' href='/nlab/show/Gray+tensor+product' title='nlab'>Gray tensor product</a>. This approach might maintain the strong connection with monoidal objects and keep the notion of “kernel” looking simplicial, but it means that instead of the 3-category <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>2Cat</annotation></semantics></math> we have to be working in <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><msub><mi>Cat</mi> <mi>l</mi></msub></mrow><annotation encoding='application/x-tex'>2Cat_l</annotation></semantics></math>, which is enriched over itself but is not a 3-category (its interchange law holds only laxly).</p> <div class='query'> <p>What I wanted to say is that in the square above, the 2-cell should be a lax natural transformation (i.e. we take lax natural transformations as 2-cells in <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>2Cat</annotation></semantics></math>). I’m not sure that it’s the same as working with <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><msub><mi>Cat</mi> <mi>l</mi></msub></mrow><annotation encoding='application/x-tex'>2Cat_l</annotation></semantics></math>-enriched limits. —Mathieu</p> <p>I’m pretty sure that it is the same. Saying that you take the 2-cells in 2Cat to be lax transformations really means the same thing as saying that you’re working in <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><msub><mi>Cat</mi> <mi>l</mi></msub></mrow><annotation encoding='application/x-tex'>2Cat_l</annotation></semantics></math> rather than <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>2Cat</annotation></semantics></math>. Note that <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><msub><mi>Cat</mi> <mi>l</mi></msub></mrow><annotation encoding='application/x-tex'>2Cat_l</annotation></semantics></math> is not a 3-category. Although it is 3-category-like since it is closed monoidal and thereby enriched over itself, the interchange law only holds laxly. -Mike</p> </div> <h2 id='grothendieck_toposes'>Grothendieck <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-toposes</h2> <p>For 1-categories, exactness is one of the conditions in Giraud’s theorem characterizing <a class='existingWikiWord' href='/nlab/show/Grothendieck+topos' title='nlab'>Grothendieck toposes</a>. This is likewise true for Street’s theorem characterizing <a class='existingWikiWord' href='/michaelshulman/show/Grothendieck+2-topos'>Grothendieck 2-toposes</a> and Lurie’s theorem characterizing Grothendieck <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_84' xmlns='http://www.w3.org/1998/Math/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>-<a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-topos' title='nlab'>toposes</a>. In all three cases the other conditions are <a class='existingWikiWord' href='/nlab/show/extensive+category' title='nlab'>infinitary extensivity</a> (see <a class='existingWikiWord' href='/michaelshulman/show/extensive+2-category'>here</a> for a 2-categorical version) and the existence of a small <a class='existingWikiWord' href='/nlab/show/separator' title='nlab'>generating set</a>. Since <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>n Cat</annotation></semantics></math> should certainly satisfy these two hypotheses as well, a “corollary” of the extensivity hypothesis is the following “topos hypothesis:”</p> <ul> <li><math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>n Cat</annotation></semantics></math> is a Grothendieck <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n+1)</annotation></semantics></math>-topos.</li> </ul> <p>(On the other hand, <math class='maruku-mathml' display='inline' id='mathml_47f84f57e1455bdbc98c548680e0d5321ce91239_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mi>Mon</mi><mi>n</mi><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>k Mon n Cat</annotation></semantics></math> will not, in general, be extensive.)</p> </div>  </div>  </body> </html>