Definition [efr-003C]

Let \mathcal {C} be a model category, and consider a simplicial object X: \Delta ^\mathrm {op} \to \mathcal {C}.

The latching object of X at stage r, denoted L_rX \in \mathcal {C}, is the colimit \operatorname {colim}_{s: [r] \to [k]} X[k] where s: [r] \to [k] runs over all surjections out of [r] except the identity

The matching object M_r is the limit \lim _{d: [k] \to [r]} X[k] runs over all injections into [r] except the identity.

Note that there are canonical maps L_rX \to X[r] \to M_rX. A simplicial object is Reedy cofibrant if L_rX \to X is always a cofibration, and Reedy fibrant if X \to M_rX is always a fibration.