## Construction of fibrant cofiber in the category of simplicial sets

A cofibration $i:X\to Y$ of simplicial sets is a monomorphism. Is there a fibrant construction of the cofiber $Ci$, along with the simplicial map $Y\to Ci$?

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What about just taking the usual mapping cone and applying the $\mathrm{Ex}^\infty$ functor? – Akhil Mathew Dec 13 2011 at 14:27