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Let $\Delta$ be the well-known simplicial category, and denote with $\widehat\Delta_\bf C$ the category of simplicial objects in $\bf C$ ($\widehat\Delta$ for short).

Given $\mathcal X\in\widehat\Delta$ denote by $X_n:= \mathcal X([n])$ the image of $[n]\in \text{Ob}\Delta$ under the functor $\mathcal X$.

For any $X\in \bf C$ let $\mathcal K(0,X)$ be the constant functor in in $X$: it is obviously a simplicial object. Notice that there exists a functor $\widehat\Delta\times \Delta \to \bf C$ defined as evaluation, $(\mathcal Y,[n])\mapsto \text{ev}_{[n]}(\mathcal Y)=Y_n$. The functor $\mathcal K(0,-)\colon X\mapsto \mathcal K(0,X)$ is left adjoint to $\mathcal Y\mapsto \text{ev}_{[0]}(\mathcal Y)=Y_0$.

Open question: Do there exist functors $\big\{ \mathcal K(n,-) \big\}_n$ in such a way that for any $[n]\in\Delta$ one has the adjunction $\mathcal K(n,-)\dashv \text{ev}_{[n]}$?

It seems possible that $\text{ev}_{[n]}$ admits a left adjoint, just because it commutes with limits...

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In what sense are these Eilenberg–MacLane spaces? I don't see any notion of homotopy anywhere. Your question is basically asking for conditions for the (global) left Kan extension of a functor $\mathbf{1} \to \mathbf{C}$ along the inclusion $[n] : \mathbf{1} \to \mathbf{\Delta}^\textrm{op}$ to exist. When $\mathbf{C}$ has enough colimits this is possible. – Zhen Lin Aug 2 '12 at 1:57

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