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3 : Generalized Morphism Category by Spans<
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href="chap3_mj.html" >[MathJax on]</a></p>"X7D54782279124748" name="X7D54782279124748" ></a></p>div class="ChapSects" ><a href="chap3.html#X7D54782279124748" >3 <span class="Heading" >Generalized Morphism Category by Spans</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7D03633A7D98026B" >3 .1 <span class="Heading" >GAP Categories</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7B3EA2C279A4F9E4" >3 .1 -1 IsGeneralizedMorphismCategoryBySpans</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X78F2B8E882ECF61E" >3 .1 -2 IsGeneralizedMorphismCategoryBySpansObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7A5BDCEE78549DFB" >3 .1 -3 IsGeneralizedMorphismBySpan</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X871597447BB998A1" >3 .2 <span class="Heading" >Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7A058F5C7F7116DB" >3 .2 -1 HasIdentityAsReversedArrow</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7C701DBF7BAE649A" >3 .3 <span class="Heading" >Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7BDCF7FD7EAE01C3" >3 .3 -1 UnderlyingHonestObject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X83C847FC86AB508B" >3 .3 -2 Arrow</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X826CC0B57D6F7F3B" >3 .3 -3 ReversedArrow</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X8082EEE27FB449E5" >3 .3 -4 NormalizedSpanTuple</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X878EE24487756BD4" >3 .3 -5 PseudoInverse</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X800786037C19DEF5" >3 .3 -6 GeneralizedInverseBySpan</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7B8C00217DD3A64C" >3 .3 -7 IdempotentDefinedBySubobjectBySpan</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X82D0B42E8413A5FE" >3 .3 -8 IdempotentDefinedByFactorobjectBySpan</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X86499E0D83A9D090" >3 .3 -9 NormalizedSpan</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7DE8E16C7C2D387B" >3 .4 <span class="Heading" >Operations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7D3675B0866191EC" >3 .4 -1 GeneralizedMorphismFromFactorToSubobjectBySpan</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X86EC0F0A78ECBC10" >3 .5 <span class="Heading" >Constructors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X83FA5A4C787261DB" >3 .5 -1 GeneralizedMorphismBySpan</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7EAF87447D2A08EE" >3 .5 -2 GeneralizedMorphismBySpan</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X79A8B50C7EB5ADE2" >3 .5 -3 GeneralizedMorphismBySpanWithRangeAid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X79C48BB782BD799F" >3 .5 -4 AsGeneralizedMorphismBySpan</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X812CBF7580098CCF" >3 .5 -5 GeneralizedMorphismCategoryBySpans</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X817694F38145FD9F" >3 .5 -6 GeneralizedMorphismBySpansObject</a></span >div ></div >div >3 <span class="Heading" >Generalized Morphism Category by Spans</span ></h3>"X7D03633A7D98026B" name="X7D03633A7D98026B" ></a></p>3 .1 <span class="Heading" >GAP Categories</span ></h4>"X7B3EA2C279A4F9E4" name="X7B3EA2C279A4F9E4" ></a></p>3 .1 -1 IsGeneralizedMorphismCategoryBySpans</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsGeneralizedMorphismCategoryBySpans</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>"X78F2B8E882ECF61E" name="X78F2B8E882ECF61E" ></a></p>3 .1 -2 IsGeneralizedMorphismCategoryBySpansObject</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsGeneralizedMorphismCategoryBySpansObject</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>"X7A5BDCEE78549DFB" name="X7A5BDCEE78549DFB" ></a></p>3 .1 -3 IsGeneralizedMorphismBySpan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsGeneralizedMorphismBySpan</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>3 .2 <span class="Heading" >Properties</span ></h4>"X7A058F5C7F7116DB" name="X7A058F5C7F7116DB" ></a></p>3 .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="Math" >\alpha</span > by a span <span class="Math" >a \leftarrow b \rightarrow c</span >. The output is <code class="code" >true</code > if <span class="Math" >a \leftarrow b</span > is congruent to an identity morphism, <code class="code" >false</code > otherwise.</p>"X7C701DBF7BAE649A" name="X7C701DBF7BAE649A" ></a></p>3 .3 <span class="Heading" >Attributes</span ></h4>"X7BDCF7FD7EAE01C3" name="X7BDCF7FD7EAE01C3" ></a></p>3 .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="Math" >\mathbf{A}</span ></p>object <span class="Math" >a</span > in the generalized morphism category by spans. The output is its underlying honest object .</p>"X83C847FC86AB508B" name="X83C847FC86AB508B" ></a></p>3 .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="Math" >\mathrm{Hom}_{\mathbf{A}}(b,c)</span ></p>span class="Math" >\alpha</span > by a span <span class="Math" >a \leftarrow b \rightarrow c</span >. The output is its arrow <span class="Math" >b \rightarrow c</span >.</p>"X826CC0B57D6F7F3B" name="X826CC0B57D6F7F3B" ></a></p>3 .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="Math" >\mathrm{Hom}_{\mathbf{A}}(b,a)</span ></p>span class="Math" >\alpha</span > by a span <span class="Math" >a \leftarrow b \rightarrow c</span >. The output is its reversed arrow <span class="Math" >a \leftarrow b</span >.</p>"X8082EEE27FB449E5" name="X8082EEE27FB449E5" ></a></p>3 .3 -4 NormalizedSpanTuple</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; NormalizedSpanTuple</code >( <var class="Arg" >alpha</var > )</td ><td "tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathbf{A}</span >.</p>span class="Math" >\alpha: a \rightarrow b</span > by a span . The output is its normalized span pair <span class="Math" >(a \leftarrow d, d \rightarrow b)</span >.</p>"X878EE24487756BD4" name="X878EE24487756BD4" ></a></p>3 .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="Math" >\mathrm{Hom}_{\mathbf{G(A)}}(b,a)</span ></p>span class="Math" >\alpha: a \rightarrow b</span > by a span . The output is its pseudo inverse <span class="Math" >b \rightarrow a</span >.</p>"X800786037C19DEF5" name="X800786037C19DEF5" ></a></p>3 .3 -6 GeneralizedInverseBySpan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; GeneralizedInverseBySpan</code >( <var class="Arg" >alpha</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Hom}_{\mathbf{G(A)}}(b,a)</span ></p>span class="Math" >\alpha: a \rightarrow b \in \mathbf{A}</span >. The output is its generalized inverse <span class="Math" >b \rightarrow a</span > by span .</p>"X7B8C00217DD3A64C" name="X7B8C00217DD3A64C" ></a></p>3 .3 -7 IdempotentDefinedBySubobjectBySpan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IdempotentDefinedBySubobjectBySpan</code >( <var class="Arg" >alpha</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Hom}_{\mathbf{G(A)}}(b,b)</span ></p>span class="Math" >\alpha: a \hookrightarrow b \in \mathbf{A}</span >. The output is the idempotent <span class="Math" >b \rightarrow b \in \mathbf{G(A)}</span > by span defined by <span class="Math" >\alpha</span >.</p>"X82D0B42E8413A5FE" name="X82D0B42E8413A5FE" ></a></p>3 .3 -8 IdempotentDefinedByFactorobjectBySpan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IdempotentDefinedByFactorobjectBySpan</code >( <var class="Arg" var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Hom}_{\mathbf{G(A)}}(b,b)</span ></p>span class="Math" >\alpha: b \twoheadrightarrow a \in \mathbf{A}</span >. The output is the idempotent <span class="Math" >b \rightarrow b \in \mathbf{G(A)}</span > by span defined by <span class="Math" >\alpha</span >.</p>"X86499E0D83A9D090" name="X86499E0D83A9D090" ></a></p>3 .3 -9 NormalizedSpan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; NormalizedSpan</code >( <var class="Arg" >alpha</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Hom}_{\mathbf{G(A)}}(a,b)</span ></p>span class="Math" >\alpha: a \rightarrow b</span > by a span . The output is its normalization by span .</p>"X7DE8E16C7C2D387B" name="X7DE8E16C7C2D387B" ></a></p>3 .4 <span class="Heading" >Operations</span ></h4>"X7D3675B0866191EC" name="X7D3675B0866191EC" ></a></p>3 .4 -1 GeneralizedMorphismFromFactorToSubobjectBySpan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; GeneralizedMorphismFromFactorToSubobjectBySpan</code >( <var "Arg" >beta</var >, <var class="Arg" >alpha</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >span class="Math" >\mathrm{Hom}_{\mathbf{G(A)}}(c,a)</span ></p>span class="Math" >\beta: b \twoheadrightarrow c</span >, and a subobject <span class="Math" >\alpha: a \hookrightarrow b</span >. The output is the generalized morphism by span from the factorobject to the subobject.</p>"X86EC0F0A78ECBC10" name="X86EC0F0A78ECBC10" ></a></p>3 .5 <span class="Heading" >Constructors</span ></h4>"X83FA5A4C787261DB" name="X83FA5A4C787261DB" ></a></p>3 .5 -1 GeneralizedMorphismBySpan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; GeneralizedMorphismBySpan</code >( <var class="Arg" >alpha</var >, <var class="Arg" >beta</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >span class="Math" >\mathrm{Hom}_{\mathbf{G(A)}}(a,b)</span ></p>span class="Math" >\alpha: a \leftarrow c</span > and <span class="Math" >\beta: c \rightarrow b</span > in <span class="Math" >\mathbf{A}</span >. The output is a generalized morphism by span with arrow <span class="Math" >\beta</span > and reversed arrow <span class="Math" >\alpha</span >.</p>"X7EAF87447D2A08EE" name="X7EAF87447D2A08EE" ></a></p>3 .5 -2 GeneralizedMorphismBySpan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; GeneralizedMorphismBySpan</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="Math" >\mathrm{Hom}_{\mathbf{G(A)}}(a,d)</span ></p>span class="Math" >\alpha: a \leftarrow b</span >, <span class="Math" >\beta: b \rightarrow c</span >, and <span class="Math" >\gamma: c \leftarrow d</span > in <span class="Math" >\mathbf{A}</span >. The output is a generalized morphism by span defined by the composition of the given three arrows regarded as generalized morphisms.</p>"X79A8B50C7EB5ADE2" name="X79A8B50C7EB5ADE2" ></a></p>3 .5 -3 GeneralizedMorphismBySpanWithRangeAid</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; GeneralizedMorphismBySpanWithRangeAid</code >( <var class="Arg" var >, <var class="Arg" >beta</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >span class="Math" >\mathrm{Hom}_{\mathbf{G(A)}}(a,c)</span ></p>span class="Math" >\alpha: a \rightarrow b</span >, and <span class="Math" span > in <span class="Math" >\mathbf{A}</span >. The output is a generalized morphism by span defined by the composition of the given two arrows regarded as generalized morphisms.</p>"X79C48BB782BD799F" name="X79C48BB782BD799F" ></a></p>3 .5 -4 AsGeneralizedMorphismBySpan</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; AsGeneralizedMorphismBySpan</code >( <var class="Arg" >alpha</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Hom}_{\mathbf{G(A)}}(a,b)</span ></p>span class="Math" >\alpha: a \rightarrow b</span > in <span class="Math" >\mathbf{A}</span >. The output is the honest generalized morphism by span defined by <span class="Math" >\alpha</span >.</p>"X812CBF7580098CCF" name="X812CBF7580098CCF" ></a></p>3 .5 -5 GeneralizedMorphismCategoryBySpans</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; GeneralizedMorphismCategoryBySpans</code >( <var class="Arg" >A</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathbf{A}</span >. The output is its generalized morphism category <span class="Math" >\mathbf{G(A)}</span > by spans.</p>"X817694F38145FD9F" name="X817694F38145FD9F" ></a></p>3 .5 -6 GeneralizedMorphismBySpansObject</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; GeneralizedMorphismBySpansObject</code >( <var class="Arg" >a</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >object in <span class="Math" >\mathbf{G(A)}</span ></p>object <span class="Math" >a</span > in an abelian category <span class="Math" >\mathbf{A}</span >. The output is the object in the generalized morphism category by spans whose underlying honest object is <span class="Math" >a</span >.</p>div class="chlinkprevnextbot" > <a href="chap0.html" >[Top of Book]</a> <a href="chap0.html#contents" >[Contents]</a> <a href="chap2.html" >[Previous Chapter]</a> <a href="chap4.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0.html" >Top</a> <a href="chap1.html" >1 </a> <a href="chap2.html" >2 </a> <a href="chap3.html" >3 </a> <a href="chap4.html" >4 </a> <a href="chap5.html" >5 </a> <a href="chap6.html" >6 </a> <a href="chap7.html" >7 </a> <a href="chap8.html" >8 </a> <a href="chap9.html" >9 </a> <a href="chapInd.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=95 H=100 G=97
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