<p>For a given morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>, a kernel of <span class="SimpleMath">\(\alpha\)</span> consists of three parts:</p>
</li>
<li><p>a morphism <span class="SimpleMath">\(\iota: K \rightarrow A\)</span> such that <span class="SimpleMath">\(\alpha \circ \iota \sim_{K,B} 0\)</span>,</p>
</li>
<li><p>a dependent function <span class="SimpleMath">\(u\)</span> mapping each morphism <span class="SimpleMath">\(\tau: T \rightarrow A\)</span> satisfying <span class="SimpleMath">\(\alpha \circ \tau \sim_{T,B} 0\)</span> to a morphism <span class="SimpleMath">\(u(\tau): T \rightarrow K\)</span> such that <span class="SimpleMath">\(\iota \circ u( \tau ) \sim_{T,A} \tau\)</span>.</p>
</li>
</ul>
<p>The triple <span class="SimpleMath">\(( K, \iota, u )\)</span> is called a <em>kernel</em> of <span class="SimpleMath">\(\alpha\)</span> if the morphisms <span class="SimpleMath">\(u( \tau )\)</span> are uniquely determined up to congruence of morphisms. We denote the object <span class="SimpleMath">\(K\)</span> of such a triple by <span class="SimpleMath">\(\mathrm{KernelObject}(\alpha)\)</span>. We say that the morphism <span class="SimpleMath">\(u(\tau)\)</span> is induced by the <em>universal property of the kernel</em>. <span class="SimpleMath">\(\\ \)</span> <span class="SimpleMath">\(\mathrm{KernelObject}\)</span> is a functorial operation. This means: for <spanclass="SimpleMath">\(\mu: A \rightarrow A'\), \(\nu: B \rightarrow B'\)</span>, <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>, <span class="SimpleMath">\(\alpha': A' \rightarrow B'\) such that \(\nu \circ \alpha \sim_{A,B'} \alpha' \circ \mu\), we obtain a morphism \(\mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' )\)</span>.</p>
<p>The argument is a morphism <span class="SimpleMath">\(\alpha\)</span>. The output is the kernel <span class="SimpleMath">\(K\)</span> of <span class="SimpleMath">\(\alpha\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KernelEmbedding</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(\mathrm{KernelObject}(\alpha),A)\)</span></p>
<p>The argument is a morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>. The output is the kernel embedding <span class="SimpleMath">\(\iota: \mathrm{KernelObject}(\alpha) \rightarrow A\)</span>.</p>
<p>The arguments are a morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span> and an object <span class="SimpleMath">\(K = \mathrm{KernelObject}(\alpha)\)</span>. The output is the kernel embedding <span class="SimpleMath">\(\iota: K \rightarrow A\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MorphismFromKernelObjectToSink</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the zero morphism in <span class="SimpleMath">\(\mathrm{Hom}( \mathrm{KernelObject}(\alpha), B )\)</span></p>
<p>The argument is a morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>. The output is the zero morphism <span class="SimpleMath">\(0: \mathrm{KernelObject}(\alpha) \rightarrow B\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MorphismFromKernelObjectToSinkWithGivenKernelObject</code>( <var class="Arg">alpha</var>, <var class="Arg">K</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the zero morphism in <span class="SimpleMath">\(\mathrm{Hom}( K, B )\)</span></p>
<p>The arguments are a morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span> and an object <span class="SimpleMath">\(K = \mathrm{KernelObject}(\alpha)\)</span>. The output is the zero morphism <span class="SimpleMath">\(0: K \rightarrow B\)</span>.</p>
<p>The arguments are a morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>, a test object <span class="SimpleMath">\(T\)</span>, and a test morphism <span class="SimpleMath">\(\tau: T \rightarrow A\)</span> satisfying <span class="SimpleMath">\(\alpha \circ \tau \sim_{T,B} 0\)</span>. For convenience, the test object <var class="Arg">T</var> can be omitted and is automatically derived from <var class="Arg">tau</var> in that case. The output is the morphism <span class="SimpleMath">\(u(\tau): T \rightarrow \mathrm{KernelObject}(\alpha)\)</span> given by the universal property of the kernel.</p>
<p>The arguments are a morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>, a test object <span class="SimpleMath">\(T\)</span>, a test morphism <span class="SimpleMath">\(\tau: T \rightarrow A\)</span> satisfying <span class="SimpleMath">\(\alpha \circ \tau \sim_{T,B} 0\)</span>, and an object <span class="SimpleMath">\(K = \mathrm{KernelObject}(\alpha)\)</span>. For convenience, the test object <var class="Arg">T</var> can be omitted and is automatically derived from <var class="Arg">tau</var> in that case. The output is the morphism <span class="SimpleMath">\(u(\tau): T \rightarrow K\)</span> given by the universal property of the kernel.</p>
<p>The argument is a list <span class="SimpleMath">\(L = [ \alpha: A \rightarrow B, [ \mu: A \rightarrow A', \nu: B \rightarrow B' ], \alpha': A' \rightarrow B' ]\) of morphisms. The output is the morphism \(\mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' )\)</span> given by the functoriality of the kernel.</p>
<p>The arguments are three morphisms <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>, <span class="SimpleMath">\(\mu: A \rightarrow A'\), \(\alpha': A' \rightarrow B'\)</span>. The output is the morphism <span class="SimpleMath">\(\mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' )\) given by the functoriality of the kernel.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KernelObjectFunctorialWithGivenKernelObjects</code>( <var class="Arg">s</var>, <var class="Arg">alpha</var>, <var class="Arg">mu</var>, <var class="Arg">alpha_prime</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( s, r )\)</span></p>
<p>The arguments are an object <span class="SimpleMath">\(s = \mathrm{KernelObject}( \alpha )\)</span>, three morphisms <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>, <span class="SimpleMath">\(\mu: A \rightarrow A'\), \(\alpha': A' \rightarrow B'\)</span>, and an object <span class="SimpleMath">\(r = \mathrm{KernelObject}( \alpha' )\). The output is the morphism \(\mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' )\)</span> given by the functoriality of the kernel.</p>
<p>The arguments are an object <span class="SimpleMath">\(s = \mathrm{KernelObject}( \alpha )\)</span>, four morphisms <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>, <span class="SimpleMath">\(\mu: A \rightarrow A'\), \(\nu: B \rightarrow B'\)</span>, <span class="SimpleMath">\(\alpha': A' \rightarrow B'\), and an object \(r = \mathrm{KernelObject}( \alpha' )\)</span>. The output is the morphism <span class="SimpleMath">\(\mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' )\) given by the functoriality of the kernel.
<p>For a given morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>, a cokernel of <span class="SimpleMath">\(\alpha\)</span> consists of three parts:</p>
</li>
<li><p>a morphism <span class="SimpleMath">\(\epsilon: B \rightarrow K\)</span> such that <span class="SimpleMath">\(\epsilon \circ \alpha \sim_{A,K} 0\)</span>,</p>
</li>
<li><p>a dependent function <span class="SimpleMath">\(u\)</span> mapping each <span class="SimpleMath">\(\tau: B \rightarrow T\)</span> satisfying <span class="SimpleMath">\(\tau \circ \alpha \sim_{A, T} 0\)</span> to a morphism <span class="SimpleMath">\(u(\tau):K \rightarrow T\)</span> such that <span class="SimpleMath">\(u(\tau) \circ \epsilon \sim_{B,T} \tau\)</span>.</p>
</li>
</ul>
<p>The triple <span class="SimpleMath">\(( K, \epsilon, u )\)</span> is called a <em>cokernel</em> of <span class="SimpleMath">\(\alpha\)</span> if the morphisms <span class="SimpleMath">\(u( \tau )\)</span> are uniquely determined up to congruence of morphisms. We denote the object <span class="SimpleMath">\(K\)</span> of such a triple by <span class="SimpleMath">\(\mathrm{CokernelObject}(\alpha)\)</span>. We say that the morphism <span class="SimpleMath">\(u(\tau)\)</span> is induced by the <em>universal property of the cokernel</em>. <span class="SimpleMath">\(\\ \)</span> <span class="SimpleMath">\(\mathrm{CokernelObject}\)</span> is a functorial operation. This means: for <span class="SimpleMath">\(\mu: A \rightarrow A'\), \(\nu: B \rightarrow B'\)</span>, <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>, <span class="SimpleMath">\(\alpha': A' \rightarrow B'\) such that \(\nu \circ \alpha \sim_{A,B'} \alpha' \circ \mu\), we obtain a morphism \(\mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' )\)</span>.</p>
<p>The argument is a morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>. The output is the cokernel <span class="SimpleMath">\(K\)</span> of <span class="SimpleMath">\(\alpha\)</span>.</p>
<p>The argument is a morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>. The output is the cokernel projection <span class="SimpleMath">\(\epsilon: B \rightarrow \mathrm{CokernelObject}( \alpha )\)</span>.</p>
<p>The arguments are a morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span> and an object <span class="SimpleMath">\(K = \mathrm{CokernelObject}(\alpha)\)</span>. The outputis the cokernel projection <span class="SimpleMath">\(\epsilon: B \rightarrow \mathrm{CokernelObject}( \alpha )\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MorphismFromSourceToCokernelObject</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the zero morphism in <span class="SimpleMath">\(\mathrm{Hom}( A, \mathrm{CokernelObject}( \alpha ) )\)</span>.</p>
<p>The argument is a morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>. The output is the zero morphism <span class="SimpleMath">\(0: A \rightarrow \mathrm{CokernelObject}(\alpha)\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MorphismFromSourceToCokernelObjectWithGivenCokernelObject</code>( <var class="Arg">alpha</var>, <var class="Arg">K</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the zero morphism in <span class="SimpleMath">\(\mathrm{Hom}( A, K )\)</span>.</p>
<p>The argument is a morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span> and an object <span class="SimpleMath">\(K = \mathrm{CokernelObject}(\alpha)\)</span>. The output is the zero morphism <span class="SimpleMath">\(0: A \rightarrow K\)</span>.</p>
<p>The arguments are a morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>, a test object <span class="SimpleMath">\(T\)</span>, and a test morphism <span class="SimpleMath">\(\tau: B \rightarrow T\)</span> satisfying <span class="SimpleMath">\(\tau \circ \alpha \sim_{A, T} 0\)</span>. For convenience, the test object <var class="Arg">T</var> can be omitted and is automatically derived from <var class="Arg">tau</var> in that case. The output is the morphism <span class="SimpleMath">\(u(\tau): \mathrm{CokernelObject}(\alpha) \rightarrow T\)</span> given by the universal property of the cokernel.</p>
<p>The arguments are a morphism <span class="SimpleMath">\(\alpha: A \rightarrow B\)</span>, a test object <span class="SimpleMath">\(T\)</span>, a test morphism <span class="SimpleMath">\(\tau: B \rightarrow T\)</span> satisfying <span class="SimpleMath">\(\tau \circ \alpha \sim_{A, T} 0\)</span>, and an object <span class="SimpleMath">\(K = \mathrm{CokernelObject}(\alpha)\)</span>. For convenience, the test object <var class="Arg">T</var> can be omitted and is automatically derived from <var class="Arg">tau</var> in that case. The output is the morphism <span class="SimpleMath">\(u(\tau): K \rightarrow T\)</span> given by the universal property of the cokernel.</p>
<p>The argument is a list <span class="SimpleMath">\(L = [ \alpha: A \rightarrow B, [ \mu:A \rightarrow A', \nu: B \rightarrow B' ], \alpha': A' \rightarrow B' ]\). The output is the morphism \(\mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' )\)</span> given by the functoriality of the cokernel.</p>
<p>The arguments are three morphisms <span class="SimpleMath">\(\alpha: A \rightarrow B, \nu: B \rightarrow B', \alpha': A' \rightarrow B'\)</span>. The output is the morphism <span class="SimpleMath">\(\mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' )\) given by the functoriality of the cokernel.
<p>The arguments are an object <span class="SimpleMath">\(s = \mathrm{CokernelObject}( \alpha )\)</span>, three morphisms <span class="SimpleMath">\(\alpha: A \rightarrow B, \nu: B \rightarrow B', \alpha': A' \rightarrow B'\)</span>, and an object <span class="SimpleMath">\(r = \mathrm{CokernelObject}( \alpha' )\). The output is the morphism \(\mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' )\)</span> given by the functoriality of the cokernel.</p>
<p>The arguments are an object <span class="SimpleMath">\(s = \mathrm{CokernelObject}( \alpha )\)</span>, four morphisms <span class="SimpleMath">\(\alpha: A \rightarrow B, \mu: A \rightarrow A', \nu: B \rightarrow B', \alpha': A' \rightarrow B'\), and an object \(r = \mathrm{CokernelObject}( \alpha' )\)</span>. The output is the morphism <span class="SimpleMath">\(\mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' )\) given by the functoriality of the cokernel.
</li>
<li><p>a function <span class="SimpleMath">\(u_{\mathrm{in}}\)</span> mapping each object <span class="SimpleMath">\(A\)</span> to a morphism <span class="SimpleMath">\(u_{\mathrm{in}}(A): A \rightarrow Z\)</span>,</p>
</li>
<li><p>a function <span class="SimpleMath">\(u_{\mathrm{out}}\)</span> mapping each object <spanclass="SimpleMath">\(A\)</span> to a morphism <span class="SimpleMath">\(u_{\mathrm{out}}(A): Z \rightarrow A\)</span>.</p>
</li>
</ul>
<p>The triple <span class="SimpleMath">\((Z, u_{\mathrm{in}}, u_{\mathrm{out}})\)</span> is called a <em>zero object</em> if the morphisms <span class="SimpleMath">\(u_{\mathrm{in}}(A)\)</span>, <span class="SimpleMath">\(u_{\mathrm{out}}(A)\)</span> are uniquely determined up to congruence of morphisms. We denote the object <span class="SimpleMath">\(Z\)</span> of such a triple by <span class="SimpleMath">\(\mathrm{ZeroObject}\)</span>. We say that the morphisms <span class="SimpleMath">\(u_{\mathrm{in}}(A)\)</span> and <span class="SimpleMath">\(u_{\mathrm{out}}(A)\)</span> are induced by the <em>universal property of the zero object</em>.</p>
<p>The argument is a category <span class="SimpleMath">\(C\)</span>. The output is a zero object <span class="SimpleMath">\(Z\)</span> of <span class="SimpleMath">\(C\)</span>.</p>
<p>This is a convenience method. The argument is a cell <span class="SimpleMath">\(c\)</span>. The output is a zero object <span class="SimpleMath">\(Z\)</span> of the category <span class="SimpleMath">\(C\)</span> for which <span class="SimpleMath">\(c \in C\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UniversalMorphismFromZeroObject</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(\mathrm{ZeroObject}, A)\)</span></p>
<p>The argument is an object <span class="SimpleMath">\(A\)</span>. The output is the universal morphism <span class="SimpleMath">\(u_{\mathrm{out}}: \mathrm{ZeroObject} \rightarrow A\)</span>.</p>
<p>The arguments are an object <span class="SimpleMath">\(A\)</span>, and a zero object <span class="SimpleMath">\(Z = \mathrm{ZeroObject}\)</span>. The output is the universal morphism <span class="SimpleMath">\(u_{\mathrm{out}}: Z \rightarrow A\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UniversalMorphismIntoZeroObject</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(A, \mathrm{ZeroObject})\)</span></p>
<p>The argument is an object <span class="SimpleMath">\(A\)</span>. The output is the universal morphism <span class="SimpleMath">\(u_{\mathrm{in}}: A \rightarrow \mathrm{ZeroObject}\)</span>.</p>
<p>The arguments are an object <span class="SimpleMath">\(A\)</span>, and a zero object <span class="SimpleMath">\(Z = \mathrm{ZeroObject}\)</span>. The output is the universal morphism <span class="SimpleMath">\(u_{\mathrm{in}}: A \rightarrow Z\)</span>.</p>
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