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Returns: an objectp>Returns an object in ><table class="func" width="100%"><tr
<p>Returns: true or>2.5 </h5

<p>Byclass"func>< classfunc"width=0""tdleft">code class="&827RandomObject<> </>,var="rg"<v><d>td=tdright> operationnbsp;</d>tr><table<>

<p><a id="X824C0CC579E8DD7F" name="X824C0CC579E8DD7F"></a></p>

<h4>2.7 <span class

<p><a id="X825D050A8674D377" name="X825D050A8674D377"></a></p>

<h5>2.7-1 ObjectConstructor</h5>

<div="func"><table=func =10%">tr>< class="tdleft<codeclassfunc">27 ("C<> =Arg>/ >nbspnbsp;)<//d><tr></table></div>
<p>Returns: an object</p>

<p>The arguments are a category <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X7C6745F27FBA23F7" name="X7C6745F27FBA23F7"></a></p>

<h5>2.7-2 /</h5>

<div class="func"><table class=>< id"" nameX7DD668627932185C>/</>
<>Returnsanobject/p>

<p> for< classcode>ObjectConstructor( , a )</ode</>

<p><a id="X8521A36F7BBFF6BD" nameh5>.-1IsEqualForCacheForObjects</h5

<h527- ObjectDatum<h5java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26

<div class>Bydefault CAPuses tostore values Categoricaloperations.To geta valueoutof the cache one needsto compare input abasicoperationwith itspreviousinput. compareobjects  category  is.BydefaultIsEqualForCacheForObjects fallsback toIsEqualForCache ( ToolsForHomalg,whichin  defaultsto  comparisonfor listsand <code class""><>inall adda functioncode=code>AddIsEqualForCacheForObjects/code>,thatfunction isusedinstead A function<span class="">F:a,b\ bool<s> sexpectedthere. Theoutput has betrue  . Fail  notallowedin thiscontext.<p>
p>: depends the categoryp>

<p

<p><a id="X7820B09C781A90A8" name="X7820B09C781A90A8"></a></p>

<h4>2.8 <span class="Heading">Well-Definedness of Objects</span></h4>

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<>2-IsWellDefinedForObjects

< =func =funcwidth"0%>tdleft<code="func">&87;IsWellDefinedForObjectscode><var"Arg>a dright">(nbsp; 
p:abooleanpjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

<p>The argument

<p><a id="X866BE10883381E75" name="X866BE10883381E75

<h4>2.9 <span class="

tdleft>< class"(varclass=Arg"</varvarclass"Arg"C<> </td<d class"">nbspoperationnbsp;<td/></able/div

<p><a id"X8047E157804C680E" nameX8047E157804C680E"

<h5pa id="52A36F7BBFF6BD"name=2A36F7BBFF6BD"><><>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SetOfObjectsOfCategory</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr
<p>Returns a  of<strong="pkg>CAP

<p>Return a duplicate free list of objects of the category <var class="Arg">C</var>. The ordering of the returned list must always be the same.</p>

<p><a id="X7B77708881BFE682" name="X7B77708881BFE682"></a></p>

<h5>2.9-2 SetOfObjects</h5>

<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

<p>Return a duplicate free list

<p><a id="X7EFAF5CD83C3298F" 

<h4>2.10 <span class="Heading">Projectives</span></h4>

<p>For

<p><a id="X7FED5E898784C314" name="X7FED5E898784C314"></a></p>

<h5>2.10-1 SomeProjectiveObject</h5>

<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><a  ="""><

<h5>2.10-4 ProjectiveLift</h5>

<div class="func
<p>Returns: morphismin<pan="ath"mathrmHom(B)/pan>/>

<p>The arguments are a morphism <span class="Math">\pi:

<p><a id="X87100AFB7A202D2F" name="X87100AFB7A202D2F"></a></p>

<h4>2.11 <span class="Heading">Injectives</span></h4>

<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%"

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>a idX8502893C82A38864 =X8502893C82A38864><a>/>

<h5>2.11-2 MonomorphismIntoSomeInjectiveObject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MonomorphismIntoSomeInjectiveObject</code>( <var class="Arg">A</var> )</td><td class<>2.10-4 ProjectiveLift/h5>
<p>Returns: a morphism  <span="ath"\mathrm{}(IA)span>p>

<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><a id="X7A7C030382397A2B" name=>Thearguments areamorphism< class=ath\:  \ A</pan>  <spanclass"Math>Math circ =pi/></>

<h5>2.11-3 <="70AFB7A202D2F"=X87100AFB7A202D2Fajava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<
<pp>a =X7C06E1807D8775AD=X7C06E1807D8775AD</a</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

<h4

<p>Let <span: amorphism inclass"">athrmHomI,</span></>


<ul>
<li><p>an p>he argumentsare  objectspanclass"">A<s> an object< class""><spanthattheoutputof<spanMath>\SomeInjectiveObject}A<span> output amonomorphism< =Mathiota A rightarrowI<span>p>

</li>
<li< isomorphism <span="Math">^i:A\rightarrowA_ispan</>

</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<>

<p><a id="X7BA160B382F8D742" name="X7BA160B382F8D742"></a></p>

<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

<p><a id="X85B54852792DEDD7" name="X85B54852792DEDD7"></a></p>

<><id" name">/>/>

<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

<p><a id="X78C0D2CB867C5C90" name="X78C0D2CB867C5C90"></a></p>

<h5>2.12-3 SimplifyObject_IsoFromInputObject

<div class="func"><table class="func" width="100%"><tr
<p>Returns: a morphism in <span class="Math">\mathrm{Hom}(A,A_i)</span<h5212 SimplifyObject</5

<p>The arguments p>>:anobject<>

<p><a id="X835832A685F651C8" name="X835832A685F651C8"></a></p>

<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

<p><a id="X814E7CE2792896ED" name="X814E7CE2792896ED"></a></p>

<h4>2.13 <java.lang.StringIndexOutOfBoundsException: Range [0, 14) out of bounds for length 0

<p><a id="X824604E9876DDDC5" name="X824604E9876DDDC5"></a></p>

<h5>2.13-1 ProjectiveDimension</h5>

<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>

p>=X823062CF792E3359 name="X823062CF792E3359"</a>/p>

<h5>2.13-2 InjectiveDimension</h5>

<div class="<>21 < class"eading>Dimensionsspan><h4
<p>Returns

<p>The argument is an object <span class="java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 0


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