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2 : Generalized Morphism Category by Cospa
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>"mathjaxlink" class="pcenter" ><a href="chap2.html" >[MathJax off]</a></p>"X87EB52217E1FF49C" name="X87EB52217E1FF49C" ></a></p>div class="ChapSects" ><a href="chap2_mj.html#X87EB52217E1FF49C" >2 <span class="Heading" >Generalized Morphism Category by Cospans</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X7D03633A7D98026B" >2 .1 <span class="Heading" >GAP Categories</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X8187BB8479575083" >2 .1 -1 IsGeneralizedMorphismCategoryByCospans</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X86C1877D7EC9F075" >2 .1 -2 IsGeneralizedMorphismCategoryByCospansObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X82D8F9C38330F81E" >2 .1 -3 IsGeneralizedMorphismByCospan</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X871597447BB998A1" >2 .2 <span class="Heading" >Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X815D6FE986CC3F99" >2 .2 -1 HasIdentityAsReversedArrow</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X7C701DBF7BAE649A" >2 .3 <span class="Heading" >Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7D7E0AB87D8C6BE4" >2 .3 -1 UnderlyingHonestObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7863825F85197D30" >2 .3 -2 Arrow</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X871D3BBC7E6B544B" >2 .3 -3 ReversedArrow</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7DCA5A8278F4C569" >2 .3 -4 NormalizedCospanTuple</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7F4AC0FE847140A4" >2 .3 -5 PseudoInverse</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X87E4921884F9F1FF" >2 .3 -6 GeneralizedInverseByCospan</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7CFE9A12793AF805" >2 .3 -7 IdempotentDefinedBySubobjectByCospan</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X8330AB9480BC933A" >2 .3 -8 IdempotentDefinedByFactorobjectByCospan</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7B636E0D7A9FA235" >2 .3 -9 NormalizedCospan</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X7DE8E16C7C2D387B" >2 .4 <span class="Heading" >Operations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7988FD538374DC5C" >2 .4 -1 GeneralizedMorphismFromFactorToSubobjectByCospan</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X86EC0F0A78ECBC10" >2 .5 <span class="Heading" >Constructors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X8313697F833CA064" >2 .5 -1 GeneralizedMorphismByCospan</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X865028F47CB5A9FA" >2 .5 -2 GeneralizedMorphismByCospan</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7B744FE28475F25B" >2 .5 -3 GeneralizedMorphismByCospanWithSourceAid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X79F31F9E80A5BFAD" >2 .5 -4 AsGeneralizedMorphismByCospan</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X879C3A4F82EC9C13" >2 .5 -5 GeneralizedMorphismCategoryByCospans</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X81ADB9C17D822C05" >2 .5 -6 GeneralizedMorphismByCospansObject</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X83F3C1E6877018D8" >2 .6 <span class="Heading" >Constructors of lifts of exact functors and natrual (iso)morphisms</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7A094900868E02F1" >2 .6 -1 AsGeneralizedMorphismByCospan</a></span >div ></div >div >2 <span class="Heading" >Generalized Morphism Category by Cospans</span ></h3>"X7D03633A7D98026B" name="X7D03633A7D98026B" ></a></p>2 .1 <span class="Heading" >GAP Categories</span ></h4>"X8187BB8479575083" name="X8187BB8479575083" ></a></p>2 .1 -1 IsGeneralizedMorphismCategoryByCospans</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsGeneralizedMorphismCategoryByCospans</code >( <var class="Arg" >object </var > )</td ><td class="tdright" >( filter )</td ></tr ></table ></div >code class="keyw" >true</code > or <code class="keyw" >false</code ></p>"X86C1877D7EC9F075" name="X86C1877D7EC9F075" ></a></p>2 .1 -2 IsGeneralizedMorphismCategoryByCospansObject</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsGeneralizedMorphismCategoryByCospansObject</code >( <var class="Arg" >object </var > )</td ><td class="tdright" >( filter )</td ></tr ></table ></div >code class="keyw" >true</code > or <code class="keyw" >false</code ></p>"X82D8F9C38330F81E" name="X82D8F9C38330F81E" ></a></p>2 .1 -3 IsGeneralizedMorphismByCospan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsGeneralizedMorphismByCospan</code >( <var class="Arg" >object </var > )</td ><td class="tdright" >( filter )</td ></tr ></table ></div >code class="keyw" >true</code > or <code class="keyw" >false</code ></p>"X871597447BB998A1" name="X871597447BB998A1" ></a></p>2 .2 <span class="Heading" >Properties</span ></h4>"X815D6FE986CC3F99" name="X815D6FE986CC3F99" ></a></p>2 .2 -1 HasIdentityAsReversedArrow</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; HasIdentityAsReversedArrow</code >( <var class="Arg" >alpha</var td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="keyw" >true</code > or <code class="keyw" >false</code ></p>span class="SimpleMath" >\(\alpha\)</span > by a cospan <span "SimpleMath" >\(a \rightarrow b \leftarrow c\)</span >. The output is <code class="code" >true</code > if <span class="SimpleMath" >\(b \leftarrow c\)</span > is congruent to an identity morphism, <code class="code" >false</code > otherwise.</p>"X7C701DBF7BAE649A" name="X7C701DBF7BAE649A" ></a></p>2 .3 <span class="Heading" >Attributes</span ></h4>"X7D7E0AB87D8C6BE4" name="X7D7E0AB87D8C6BE4" ></a></p>2 .3 -1 UnderlyingHonestObject</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; UnderlyingHonestObject</code >( <var class="Arg" >a</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >object in <span class="SimpleMath" >\(\mathbf{A}\)</span ></p>object <span class="SimpleMath" >\(a\)</span > in the generalized morphism category by cospans. The output is its underlying honest object .</p>"X7863825F85197D30" name="X7863825F85197D30" ></a></p>2 .3 -2 Arrow</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; Arrow</code >( <var class="Arg" >alpha</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="SimpleMath" >\(\mathrm{Hom}_{\mathbf{A}}(a,c)\)</span ></p>span class="SimpleMath" >\(\alpha\)</span > by a cospan <span "SimpleMath" >\(a \rightarrow b \leftarrow c\)</span >. The output is its arrow <span class="SimpleMath" >\(a \rightarrow b\)</span >.</p>"X871D3BBC7E6B544B" name="X871D3BBC7E6B544B" ></a></p>2 .3 -3 ReversedArrow</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; ReversedArrow</code >( <var class="Arg" >alpha</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="SimpleMath" >\(\mathrm{Hom}_{\mathbf{A}}(c,b)\)</span ></p>span class="SimpleMath" >\(\alpha\)</span > by a cospan <span "SimpleMath" >\(a \rightarrow b \leftarrow c\)</span >. The output is its reversed arrow <span class="SimpleMath" >\(b \leftarrow c\)</span >.</p>"X7DCA5A8278F4C569" name="X7DCA5A8278F4C569" ></a></p>2 .3 -4 NormalizedCospanTuple</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; NormalizedCospanTuple</code >( <var class="Arg" >alpha</var > )</td td class="tdright" >( attribute )</td ></tr ></table ></div >span class="SimpleMath" >\(\mathbf{A}\)</span >.</p>span class="SimpleMath" >\(\alpha: a \rightarrow b\)</span > by a cospan. The output is its normalized cospan pair <span class="SimpleMath" >\((a \rightarrow d, d \leftarrow b)\)</span >.</p>"X7F4AC0FE847140A4" name="X7F4AC0FE847140A4" ></a></p>2 .3 -5 PseudoInverse</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; PseudoInverse</code >( <var class="Arg" >alpha</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="SimpleMath" >\(\mathrm{Hom}_{\mathbf{G(A)}}(b,a)\)</span span class="SimpleMath" >\(\alpha: a \rightarrow b\)</span > by a cospan. The output is its pseudo inverse <span class="SimpleMath" >\(b \rightarrow a\)</span >.</p>"X87E4921884F9F1FF" name="X87E4921884F9F1FF" ></a></p>2 .3 -6 GeneralizedInverseByCospan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; GeneralizedInverseByCospan</code >( <var class="Arg" >alpha</var td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="SimpleMath" >\(\mathrm{Hom}_{\mathbf{G(A)}}(b,a)\)</span span class="SimpleMath" >\(\alpha: a \rightarrow b \in \mathbf{A}\)</span >. The output is its generalized inverse <span class="SimpleMath" >\(b \rightarrow a\)</span > by cospan.</p>"X7CFE9A12793AF805" name="X7CFE9A12793AF805" ></a></p>2 .3 -7 IdempotentDefinedBySubobjectByCospan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IdempotentDefinedBySubobjectByCospan</code >( <var class="Arg" >alpha</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="SimpleMath" >\(\mathrm{Hom}_{\mathbf{G(A)}}(b,b)\)</span span class="SimpleMath" >\(\alpha: a \hookrightarrow b \in \mathbf{A}\)</span >. The output is the idempotent <span class="SimpleMath" >\(b \rightarrow b \in \mathbf{G(A)}\)</span > by cospan defined by <span class="SimpleMath" >\(\alpha\)</span >.</p>"X8330AB9480BC933A" name="X8330AB9480BC933A" ></a></p>2 .3 -8 IdempotentDefinedByFactorobjectByCospan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IdempotentDefinedByFactorobjectByCospan</code >( <var class="Arg" >alpha</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="SimpleMath" >\(\mathrm{Hom}_{\mathbf{G(A)}}(b,b)\)</span span class="SimpleMath" >\(\alpha: b \twoheadrightarrow a \in \mathbf{A}\)</span >. The output is the idempotent <span class="SimpleMath" >\(b \rightarrow b \in \mathbf{G(A)}\)</span > by cospan defined by <span class="SimpleMath" >\(\alpha\)</span >.</p>"X7B636E0D7A9FA235" name="X7B636E0D7A9FA235" ></a></p>2 .3 -9 NormalizedCospan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; NormalizedCospan</code >( <var class="Arg" >alpha</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="SimpleMath" >\(\mathrm{Hom}_{\mathbf{G(A)}}(a,b)\)</span span class="SimpleMath" >\(\alpha: a \rightarrow b\)</span > by a cospan. The output is its normalization by cospan.</p>"X7DE8E16C7C2D387B" name="X7DE8E16C7C2D387B" ></a></p>2 .4 <span class="Heading" >Operations</span ></h4>"X7988FD538374DC5C" name="X7988FD538374DC5C" ></a></p>2 .4 -1 GeneralizedMorphismFromFactorToSubobjectByCospan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; GeneralizedMorphismFromFactorToSubobjectByCospan</code >( <var class="Arg" >beta</var >, <var class="Arg" >alpha</var > )</td ><td class="tdright" >( operation )</td tr ></table ></div >span class="SimpleMath" >\(\mathrm{Hom}_{\mathbf{G(A)}}(c,a)\)</span span class="SimpleMath" >\(\beta: b \twoheadrightarrow c\)</span >, and a subobject <span class="SimpleMath" >\(\alpha: a \hookrightarrow b\)</span >. The output is the generalized morphism by cospan from the factorobject to the subobject.</p>"X86EC0F0A78ECBC10" name="X86EC0F0A78ECBC10" ></a></p>2 .5 <span class="Heading" >Constructors</span ></h4>"X8313697F833CA064" name="X8313697F833CA064" ></a></p>2 .5 -1 GeneralizedMorphismByCospan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; GeneralizedMorphismByCospan</code >( <var class="Arg" >alpha</var >, <var class="Arg" >beta</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >span class="SimpleMath" >\(\mathrm{Hom}_{\mathbf{G(A)}}(a,c)\)</span span class="SimpleMath" >\(\alpha: a \rightarrow b\)</span > and <span class="SimpleMath" >\(\beta: c \rightarrow b\)</span > in <span class="SimpleMath" >\(\mathbf{A}\)</span >. The output is a generalized morphism by cospan with arrow <span class="SimpleMath" >\(\alpha\)</span > and reversed arrow <span class="SimpleMath" >\(\beta\)</span >.</p>"X865028F47CB5A9FA" name="X865028F47CB5A9FA" ></a></p>2 .5 -2 GeneralizedMorphismByCospan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; GeneralizedMorphismByCospan</code >( <var class="Arg" >alpha</var >, <var class="Arg" >beta</var >, <var class="Arg" >gamma</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >span class="SimpleMath" >\(\mathrm{Hom}_{\mathbf{G(A)}}(a,d)\)</span span class="SimpleMath" >\(\alpha: a \leftarrow b\)</span >, <span class="SimpleMath" >\(\beta: b \rightarrow c\)</span >, and <span class="SimpleMath" >\(\gamma: c \leftarrow d\)</span > in <span class="SimpleMath" >\(\mathbf{A}\)</span >. The output is a generalized morphism by cospan defined by the composition of the given three arrows regarded as generalized morphisms.</p>"X7B744FE28475F25B" name="X7B744FE28475F25B" ></a></p>2 .5 -3 GeneralizedMorphismByCospanWithSourceAid</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; GeneralizedMorphismByCospanWithSourceAid</code >( <var class="Arg" >alpha</var >, <var class="Arg" >beta</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >span class="SimpleMath" >\(\mathrm{Hom}_{\mathbf{G(A)}}(a,c)\)</span span class="SimpleMath" >\(\alpha: a \leftarrow b\)</span >, and <span class="SimpleMath" >\(\beta: b \rightarrow c\)</span > in <span class="SimpleMath" >\(\mathbf{A}\)</span >. The output is a generalized morphism by cospan defined by the composition of the given two arrows regarded as generalized morphisms.</p>"X79F31F9E80A5BFAD" name="X79F31F9E80A5BFAD" ></a></p>2 .5 -4 AsGeneralizedMorphismByCospan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; AsGeneralizedMorphismByCospan</code >( <var class="Arg" >alpha</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="SimpleMath" >\(\mathrm{Hom}_{\mathbf{G(A)}}(a,b)\)</span span class="SimpleMath" >\(\alpha: a \rightarrow b\)</span > in <span class="SimpleMath" >\(\mathbf{A}\)</span >. The output is the honest generalized morphism by cospan defined by <span class="SimpleMath" >\(\alpha\)</span >.</p>"X879C3A4F82EC9C13" name="X879C3A4F82EC9C13" ></a></p>2 .5 -5 GeneralizedMorphismCategoryByCospans</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; GeneralizedMorphismCategoryByCospans</code >( <var class="Arg" >A</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="SimpleMath" >\(\mathbf{A}\)</span >. The output is its generalized morphism category <span class="SimpleMath" >\(\mathbf{G(A)}\)</span > by cospans.</p>"X81ADB9C17D822C05" name="X81ADB9C17D822C05" ></a></p>2 .5 -6 GeneralizedMorphismByCospansObject</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; GeneralizedMorphismByCospansObject</code >( <var class="Arg" >a</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >object in <span class="SimpleMath" >\(\mathbf{G(A)}\)</span ></p>object <span class="SimpleMath" >\(a\)</span > in an abelian category <span class="SimpleMath" >\(\mathbf{A}\)</span >. The output is the object in the generalized morphism category by cospans whose underlying honest object is <span class="SimpleMath" >\(a\)</span >.</p>"X83F3C1E6877018D8" name="X83F3C1E6877018D8" ></a></p>2 .6 <span class="Heading" >Constructors of lifts of exact functors and natrual (iso)morphisms</span ></h4>"X7A094900868E02F1" name="X7A094900868E02F1" ></a></p>2 .6 -1 AsGeneralizedMorphismByCospan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; AsGeneralizedMorphismByCospan</code >( <var class="Arg" >F</var >, <var class="Arg" >name</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >em >exact</em > functor <var class="Arg" >F</var > to a functor <span class="SimpleMath" >\(A \to B\)</span >, where <span class="SimpleMath" >\(A := \)</span > <code class="code" >GeneralizedMorphismCategoryByCospans( SourceOfFunctor( </code ><var class="Arg" >F</var ><code class="code" > ) )</code span class="SimpleMath" >\(B := \)</span > <code class="code" >GeneralizedMorphismCategoryByCospans( RangeOfFunctor( </code ><var class="Arg" >F</var ><code class="code" > ) )</code >.</p>div class="chlinkprevnextbot" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap1_mj.html" >[Previous Chapter]</a> <a href="chap3_mj.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0_mj.html" >Top</a> <a href="chap1_mj.html" >1 </a> <a href="chap2_mj.html" >2 </a> <a href="chap3_mj.html" >3 </a> <a href="chap4_mj.html" >4 </a> <a href="chap5_mj.html" >5 </a> <a href="chap6_mj.html" >6 </a> <a href="chap7_mj.html" >7 </a> <a href="chap8_mj.html" >8 </a> <a href="chap9_mj.html" >9 </a> <a href="chapInd_mj.html" >Ind</a> </div >hr />"foot" >generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc " >GAPDoc2HTML</a></p>body >html > Messung V0.5 in Prozent C=93 H=100 G=96
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