<p>Any GAP object which is IsCapCategoryObject can be added to a category and then becomes an object in this category. Any object can belong to one or no category. After a GAP object is added to the category, it knows which things can be computed in its category and - EpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObjecta>/span
<h5divjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
<div class="func"><tablejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Adds java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X87BD8BB88206D940" name="X87BD8BB88206D940"></java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>The arguments are two objects <span class="Math">a</span> and <span class="Math">b</span>. The output is <code class="code">true</code> if <span class="Math">a = b</span>, otherwise the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="
<h5>
<div="unc"tableclass="width=10%><>
tdleft> class="func">&82 java.lang.StringIndexOutOfBoundsException: Range [124, 123) out of bounds for length 256
<p>Returns: a boolean</p>
<divh5>2-1IsEqualForObjects</h5
<p>Returnsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>The arguments are two isomorphic objects <span class="Math">a</span> and <span class="Math">b</span>. The output is an isomorphism <spanp>eturns: a </p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBijectiveObject</code>( <var class="Arg"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<Returns:a boolean<p>
<p>The argument is an object <span class="Math">a</span>. The output is <code class="code">true</code> if <span class="Math">a</span> is a bijective object, otherwiseReturns:ajava.lang.StringIndexOutOfBoundsException: Range [21, 15) out of bounds for length 25
<p>The argument is an object <span class="Math">a</span>. The output is <code class="code">true</code> if <span class="Math">a</span> is an injective object, otherwise the output is <code class="code">false</code>.</
<div class="func"><table class="func" width="100%"><tr>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Returns: a boolean/pjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
<p>The argument>The is <span class=><span> a categoryspanclass="\mathbf{C}codetruecode> if <spanclass=atha<> is theinitial ofspan classMath\{}/span>, otherwise < class">false.
<ul>
<><p><em integers/>: integer simply parameter that beusedto createa random object.p>
</li>
<><><m>y lists<em>:The list usedwhen a random objectwould more oneparameter offer flexibilityatthe ofthegenericityof themethods.This happens liststhat validasinput somecategory maybenot for othercategories.Hence, hese operationsarenot used in categoricalalgorithms./>
<div >The isa <varclass"rg>objobj, that is, IsEqualForObjects( obj, ObjectConstructor( CapCategory( obj ), ObjectDatum(
/ /java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class
<p>eturns: anobject/>
<p>The argument is an object <span class=
<p>92SetOfObjects>
ect>
<div: list <trongpkg</>objects
<p>
<p>The argument is an object <span class
<p><a id1 =HeadingProjectives</span>
<h5>2.10-3 EpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject object <span classMath/span> an having projectives the allow someprojectiveobject < class="P anepimorphism ><table
p: in <<pan"">\athrm{Hom}(P,A)<><p>
<p> argumentsare <class="ath">A<span> and object<span =""></span> that theoutput <panclass">mathrm{SomeProjectiveObject}(A The output is an epimorphism \pi: P \rightarrow A.
<p>For a given object <span class="Math">A</span<>The argument is an object <span class"Math"A<span>.Theoutputisanepimorphism<span class"Math>\pii: P \ A >/><p>
<h5>2.11-1 SomeInjectiveObject</h5>
<div class="func"><table class="
0%"
P/ )
td=""> &)/divjava.lang.StringIndexOutOfBoundsException: Index 298 out of bounds for length 298
<p>:an/>
<p>The argument is an object <span class="Math">A</span>. The output is
<p>The argument is an object <span class="Math">A</span>. The output is a monomorphism <span class="Math">\iota: A \rightarrow I</span> with <span class="Math">I</span> an injective object that equals the output of <span class="Math">\mathrm{SomeInjectiveObject}(A)</span>.</p>
<p>The arguments are an object>211-SomeInjectiveObject</h5
<p<aid="" name=7"/a>/p>
<h5>2.11-4 InjectiveColiftReturns anobject</pjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
<div class="func"><table class="java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 0
<p>Returns: a morphism>12MonomorphismIntoSomeInjectiveObject>
<p>The arguments are a monomorphism <span p>Returns morphismin spanclass">{Hom}(A/>/>
<p><ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</i>
</ul
<p>The idea is that the p> arguments are amonomorphism spanclass="Math"\ B \rightarrow</span> anda orphism< ="ath">\betaB\rightarrowI<span wherespan =Math>/>i an injective object Theoutputis a morphism < class"">lambdaA rightarrowI<pan>uchthat<span class"Math"\lambda\ \=/span<>
<h5>2.12-1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code
<pReturns an<pjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
<p>The argument is an object <span class="Math">A</span>. The output is a java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width="100%"><trh5>1-1Simplify/>
<p>Returns: an object</p>
<p>The arguments are an object <span class="Math">A</span> and a positive integer <spanp>: an object</java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
<h5>2.12-java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>: a morphism <panclass"Math">mathrmHom(A_i,A</span</>
<p>The arguments are an object <spandivclass=""> class="width=10">trtdclass="">code =func>#27 SimplifyObject_IsoFromInputObject/> < class=Arg>/ <varclassArg">i
<div="unc>< class=func"width0%><><dclass"">odeclass="">R ProjectiveDimensioncode(<arclassArg>/td>td=tdrightn; td<><></iv
<p>Returns: a nonnegative integer or infinity</p>
<p>The argument is an object <span class="Math">A</span>. The output is a the projective dimension of <span class="Math">A</span>.</p>
<div class=""><pan=chlink1>GotoChapter: spana hrefchap0html>Top<> <ahrefchap1.html><a<href=hap2.html2<a <ahref="chap3.html>3/a> ></ ajava.lang.StringIndexOutOfBoundsException: Range [243, 240) out of bounds for length 656
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