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We introduce the class of cointerval simplicial complexes, which generalize the classes of cointerval hypergraphs, introducedby Dochterman and Engstr\"{o}m, and shifted simplicial complexes, introduced by Erd\H{o}s-Ko-Rado (combinatorial shifting) and Kalai (algebraic shifting). We will discuss some geometric properties of cointerval complexes, and introduce a (polyhedral) complex of order-preserving homomorphisms, $OHOM (\Gamma, \Delta)$, between simplicial complexes $\Gamma$ and $\Delta$. We will show that the complex $OHOM (\Gamma,\Delta)$ supports a minimal free resolution of an associated monomial ideal when $\Delta$ is a cointerval simplicial complex. This work is joint with Jonathan Browder and Benjamin Braun.