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Epistemology - m a t h 4 w i s d o m - g m a i l
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Introduction E9F5FC Questions FFFFC0 Software |
Read p. 71-84. Exercises 2, 4.
Let {$\Delta :\mathbf{C}\Rightarrow \mathbf{C}\times\mathbf{C}$} bet the diagonal functor, which sends an object {$C$} of {$\mathbf{C}$} to the ordered pair {$(C,C)$} and a morphism {$f:C\rightarrow C'$} to the morphism {$(f,f):(C,C)\rightarrow (C',C')$}. Show that we can view the product as couniversal for this functor. The statement that the object {$(S,v:FS\rightarrow D)$} is terminal in the comma category {$(F\rightarrow D)$} is the following:
{$f=v\circ F_{\mu_j}$} In this case, the pair {$\nu =(S,v :FS\rightarrow C)$} is a For a fixed {$D\in\mathbf{D}$}, the map {$\mu_{C,D}:\textrm{hom}_D(FC,D)\rightarrow\textrm{hom}\mathbf{C}(C,S)$} that sends each {$f:FC\rightarrow D$} to its unique mediating morphism {$\mu_f:\mathbf{C}\rightarrow S$} is called the
Let {$G$} be a group and let {$R$} be a commutative ring with identity. The group ring {$RG$} is the {$R$}-algebra of formal finite sums {$\sum r_i g_i$}, where {$r_i\in R$} and {$g_i\in G$}. Multiplication is done using the product in {$G$} and linearity over {$R$}. Show that {$RG$} is a universal object. |

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