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14.4 Simplicial objects as presheaves

Another observation is that we may think of a simplicial object of $\mathcal{C}$ as a presheaf with values in $\mathcal{C}$ over $\Delta $. See Sites, Definition 7.2.2. And in fact, if $U$, $U'$ are simplicial objects of $\mathcal{C}$, then we have

14.4.0.1
\begin{equation} \label{simplicial-equation-simplicial-set-presheaf} \mathop{\mathrm{Mor}}\nolimits (U, U') = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\Delta )}(U, U'). \end{equation}

Some of the material below could be replaced by the more general constructions in the chapter on sites. However, it seems a clearer picture arises from the arguments specific to simplicial objects.

Comments (2)

Comment #969 by jojo on

"could be replace" should be "could be replaced"

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