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concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>, <a class="existingWikiWord" href="/nlab/show/coherence">coherence</a></li> <li><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></li> <li><a class="existingWikiWord" href="/nlab/show/stabilization">looping and suspension</a></li> </ul> <h2 id="basic_theorems">Basic theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/periodic+table">periodic table</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization+hypothesis">stabilization hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/michaelshulman/show/exactness+hypothesis">exactness hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a></p> </li> </ul> <h2 id="models">Models</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Theta-space">Theta-space</a></li> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-fold+complete+Segal+space">n-fold complete Segal space</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+quasi-category">algebraic quasi-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a></li> <li><a class="existingWikiWord" href="/nlab/show/complete+Segal+space">complete Segal space</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-category">(∞,0)-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complex">algebraic Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+T-complex">simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2CZ%29-category">(∞,Z)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> = (n,n)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/1-category">1-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/0-category">0-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-1%29-category">(-1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-2%29-category">(-2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-poset">n-poset</a> = <a class="existingWikiWord" href="/nlab/show/n-poset">(n-1,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/poset">poset</a> = <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a> = <a class="existingWikiWord" href="/nlab/show/%281%2C2%29-category">(1,2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a> = (n,0)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>/<a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a></li> <li><a class="existingWikiWord" href="/nlab/show/geometric+definition+of+higher+category">geometric definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+model+for+weak+%E2%88%9E-categories">simplicial model for weak ∞-categories</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/complicial+set">complicial set</a></li> <li><a class="existingWikiWord" href="/nlab/show/weak+complicial+set">weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraic definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></li> <li><a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/tetracategory">tetracategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Batanin+%E2%88%9E-category">Batanin ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck-Maltsiniotis+%E2%88%9E-categories">Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></li> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+category">A-∞ category</a></li> <li><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id="morphisms">Morphisms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/transfor">transfor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></li> <li><a class="existingWikiWord" href="/nlab/show/modification">modification</a></li> </ul> </li> </ul> <h2 id="functors">Functors</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/functor">functor</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></li> <li><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></li> <li><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">(∞,1)-adjunction</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Kan+extension">(∞,1)-Kan extension</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">(∞,1)-limit</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></li> </ul> <h2 id="extra_properties_and_structure">Extra properties and structure</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/cosmic+cube">cosmic cube</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a>, <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></li> </ul> <h2 id="1categorical_presentations">1-categorical presentations</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#notions_of_pasting_diagrams'>Notions of pasting diagrams</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Given an ordinary <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, a pasting diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a sequence of composable morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>1</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover><msub><mi>a</mi> <mn>2</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mi>n</mi></msub></mrow></mover><msub><mi>a</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> a_1 \stackrel{f_1}{\to} a_2 \stackrel{f_2}{\to} \cdots \stackrel{f_n}{\to} a_{n+1} \,. </annotation></semantics></math></div> <p>We think of these arrows as not yet composed, but <em>pasted</em> together at their objects, such as to form a composable sequence, and then say the <em>value</em> of the sequence is the composite morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. Or, we could say that a pasting diagram is a specified decomposition of whatever morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> it evaluates to, thus breaking down <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> into morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">f_i</annotation></semantics></math> which, in practice, are usually “more basic” than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> relative to some type of structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">f_i</annotation></semantics></math> might be instances of associativity isomorphisms.</p> <p>Pasting decompositions become more elaborate in higher categories. An example of a pasting diagram in a (let’s say strict) 2-category is a pasting of two squares</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>a</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>b</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>f</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ a &\to& b &\to& c \\ \downarrow &\swArrow& \downarrow &\swArrow& \downarrow \\ d &\to& e &\to& f } \,. </annotation></semantics></math></div> <p>To evaluate this diagram as a single 2-morphism, we read this diagram explicitly as the vertical composite of the 2-morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>a</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>b</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>e</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>f</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ a &\to& b &\to& c \\ && \downarrow &\swArrow& \downarrow \\ && e &\to& f } </annotation></semantics></math></div> <p>with the 2-morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>a</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>b</mi></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>f</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ a &\to& b && \\ \downarrow &\swArrow& \downarrow && \\ d &\to& e &\to& f } \,. </annotation></semantics></math></div> <p>(Notice that the two halves of the boundary of each of these two 2-morphisms are themselves 1-dimensional pasting diagrams.) Similarly, there can be situations where one pastes together other shapes (triangles, pentagons, etc.), and there may be multiple paths on the way to resolving the diagram into a single 2-morphism, but the idea is that all such paths evaluate to the same 2-morphism, at least in a strict 2-category. General theorems which refer to the uniqueness of pastings are called <em>pasting theorems</em>.</p> <p>On the other hand, the following diagram is <em>not</em> a pasting diagram:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>a</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>b</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>f</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ a &\to& b &\to& c \\ \downarrow &\neArrow& \downarrow &\swArrow& \downarrow \\ d &\to& e &\to& f } \,. </annotation></semantics></math></div> <p>Thus, formal definitions of pasting diagram include conditions which impose a consistent “directionality” of the cells so they can be pasted together.</p> <p>In an <a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> a pasting diagram is similarly a collection of <a class="existingWikiWord" href="/nlab/show/k-morphism">n-morphisms</a> with a prescribed way how they are to fit together at their boundaries. There are a number of ways of formalizing pasting diagrams, depending partly on the constituent shapes one allows, and also on the technical hypotheses that allow proofs of pasting theorems, which include directionality conditions but usually also “loop-freeness” conditions to ensure there exists a unique way to resolve or evaluate the diagram. (N.B.: usually a completely unambiguous evaluation is only possible in a strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-category; in a weak <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-category, one allows uniqueness up to a “contractible space of choices”.)</p> <p>Despite some technical differences among the formalizations, the core idea throughout is that the overall geometric shapes of pasting diagrams should be (contractible) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional polyhedra, broken down into smaller polyhedral cells which come equipped with directionality or orientations, that can be sensibly pasted together as in the above descriptions once the cells have been assigned values in an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-category.</p> <h2 id="notions_of_pasting_diagrams">Notions of pasting diagrams</h2> <p>Various formalisms for pasting diagrams have been proposed. They include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pasting+scheme">pasting schemes</a> (Johnson)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parity+complex">parity complexes</a> (Street)</p> </li> <li> <p><span class="newWikiWord">directed complexes<a href="/nlab/new/directed+complex">?</a></span> (Steiner)</p> </li> <li> <p><span class="newWikiWord">generalized parity complexes<a href="/nlab/new/generalized+parity+complex">?</a></span> (Forest)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+directed+complex">regular directed complexes</a> (Hadzihasanovic)</p> </li> </ul> <p>Each of these formalisms involve graded sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>C</mi> <mi>n</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\{C_n\}_{n \geq 0}</annotation></semantics></math> together with maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo>∂</mo> <mi>n</mi> <mo>+</mo></msubsup><mo>:</mo><msub><mi>C</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>→</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\partial^+_n: C_{n+1} \to P(C_n)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo>∂</mo> <mi>n</mi> <mo>−</mo></msubsup><mo>:</mo><msub><mi>C</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>→</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\partial^-_n: C_{n+1} \to P(C_n)</annotation></semantics></math>. This goes under various names; here we call it a <strong>parity structure</strong>. It should be thought of as assigning to each “cell” of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math> a collection of positive boundary cells and negative boundary cells in dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>. The formalisms above are distinguished by the choice of axioms on parity structures, but there is definite kinship among them.</p> <p>Also related are various notions of categories of shapes, including</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/test+category">test category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+category">direct category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compositions+in+cubical+sets">compositions in cubical sets</a></p> </li> </ul> <h2 id="examples">Examples</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Toda+bracket">Toda bracket</a></li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> for <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a></li> </ul> <h2 id="references">References</h2> <p>The notion of pasting in a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> was introduced in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean+Benabou">Jean Benabou</a>, <em>Introduction to bicategories</em>, Reports of the Midwest Category Seminar, Lecture Notes in Mathematics, <strong>47</strong>, 1967. (<a href="https://doi.org/10.1007/BFb0074299">doi:10.1007/BFb0074299</a>)</li> </ul> <p>A reasonably self-contained, formal, and illustrated seven-page introduction to pasting diagrams in the context of weak 2-categories, and their relation to <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> is given in Section A.4 of:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chris+Schommer-Pries">Chris Schommer-Pries</a>, <em>The Classification of Two-Dimensional Extended Topological Field Theories</em>, 2011. (<a href="https://arxiv.org/abs/1112.1000v2">arXiv:1112.1000v2</a>)</li> </ul> <p>A survey discussion of pasting in <a class="existingWikiWord" href="/nlab/show/2-categories">2-categories</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Power">John Power</a>, <em>2-Categories</em> (<a href="http://www.brics.dk/NS/98/7/BRICS-NS-98-7.pdf">pdf</a>)</li> </ul> <p>from definition 2.10 on. Details are in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Power">John Power</a>, <em>A 2-categorical pasting theorem</em> , Journal of Algebra, <strong>129</strong>, 1990. [<a href="https://doi.org/10.1016/0021-8693(90)90229-H">doi:10.1016/0021-8693(90)90229-H</a>]</li> </ul> <p>Dominic Verity gave a bicategorical pasting theorem in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dominic+Verity">Dominic Verity</a>, <em>Enriched categories, internal categories, and change of base</em>, PhD thesis, Cambridge University, 1992. Reprinted in TAC (<a href="http://www.tac.mta.ca/tac/reprints/articles/20/tr20abs.html">link</a>).</li> </ul> <p>A definition and discussion of pasting diagrams in <a class="existingWikiWord" href="/nlab/show/strict+omega-categories">strict omega-categories</a> is in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Sjoerd+Crans">Sjoerd Crans</a>, <em>Pasting presentation for Omega-categories</em> (<a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.48.4342">web</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sjoerd+Crans">Sjoerd Crans</a>, <em>Pasting schemes for the monoidal biclosed structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Cat</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Cat}</annotation></semantics></math></em> (<a href="http://home.tiscali.nl/secrans/papers/thten.html">web</a>, <a href="http://home.tiscali.nl/secrans/papers/thten.ps.gz">ps</a>, <a class="existingWikiWord" href="/nlab/files/thten.pdf" title="">pdf</a>)</p> </li> </ul> <p>The notion of pasting scheme used by Crans was introduced by Johnson,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Johnson">Michael Johnson</a>, <em>The combinatorics of n-categorical pasting</em>, Journal of Pure and Applied Algebra, <strong>62</strong>, 1989. [<a href="https://doi.org/10.1016/0022-4049(89)90136-9">doi:10.1016/0022-4049(89)90136-9</a>]</li> </ul> <p>Other notions of pasting presentations have been given by Street and by Steiner:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Parity complexes</em>, Cahiers de Topologie et Géométrie Différentielle Catégoriques, <strong>32</strong>, 1991. (<a href="http://www.numdam.org/item/?id=CTGDC_1991__32_4_315_0">link</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Parity complexes : corrigenda</em>, Cahiers de Topologie et Géométrie Différentielle Catégoriques, <strong>35</strong>, 1994. (<a href="http://www.numdam.org/item/?id=CTGDC_1994__35_4_359_0">link</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Richard+Steiner">Richard Steiner</a>, <em>The algebra of directed complexes</em>, Applied Categorical Structures, <strong>1</strong>, 1993. (<a href="https://doi.org/10.1007/BF00873990">doi:0.1007/BF00873990</a>)</p> </li> </ul> <p>These notions were compared, and a new generalized one introduced, in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Simon+Forest">Simon Forest</a>, <em>Unifying notions of pasting diagrams</em>, Higher Structures, <strong>6</strong>, 2022 (<a href="https://arxiv.org/abs/1903.00282">arxiv:1903.00282</a>, <a href="https://doi.org/10.21136/HS.2022.01">doi:10.21136/HS.2022.01</a>)</li> </ul> <p>Another recent approach is:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Amar+Hadzihasanovic">Amar Hadzihasanovic</a>, <em>Combinatorics of higher-categorical diagrams</em>, 2024 (<a href="https://arxiv.org/abs/2404.07273">link</a>)</li> </ul> <p>For an online link to the notion of directed complex, see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Sjoerd+Crans">Sjoerd Crans</a> and <a class="existingWikiWord" href="/nlab/show/Richard+Steiner">Richard Steiner</a>, <em>Presentations of omega-categories by directed complexes</em>, Journal of the Australian Mathematical Society, <strong>63</strong>, 1997. (<a href="https://doi.org/10.1017/S1446788700000318">doi:10.1017/S1446788700000318</a>)</li> </ul> <p>There is also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Makkai">Michael Makkai</a>, <em>Computads and 2 dimensional pasting diagrams</em> (<a href="http://www.math.mcgill.ca/makkai/2dcomputads/2DCOINS1.pdf">pdf</a>)</li> </ul> <p>For a cubical approach to multiple compositions and other references see the paper</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Philip+J.+Higgins">Philip J. Higgins</a>, <em>Thin elements and commutative shells in cubical omega-categories</em>, Theory and Applications of Categories, <strong>14</strong>, 2005. (<a href="http://www.tac.mta.ca/tac/volumes/14/4/14-04abs.html">link</a>)</li> </ul> <p>A formal approach to <a class="existingWikiWord" href="/nlab/show/pasting+diagrams">pasting diagrams</a> in <a class="existingWikiWord" href="/nlab/show/Gray+categories">Gray categories</a>:</p> <ul> <li id="DiVittorio2023"><a class="existingWikiWord" href="/nlab/show/Nicola+Di+Vittorio">Nicola Di Vittorio</a>, <em>A Gray-categorical pasting theorem</em>, Theory and Applications of Categories <strong>39</strong> 5 (2023) 150-171 [<a href="http://www.tac.mta.ca/tac/volumes/39/5/39-05abs.html">tac:39-05</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 10, 2024 at 11:56:18. 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