We introduce a formal definition of a pattern poset which encompasses several
previously studied posets in the literature. Using this definition we present
some general results on the M\"obius function and topology of such pattern
posets. We prove our results using a poset fibration based on the embeddings of
the poset, where embeddings are representations of occurrences. We show that
the M\"obius function of these posets is intrinsically linked to the number of
embeddings, and in particular to so called normal embeddings. We present
results on when topological properties such as Cohen-Macaulayness and
shellability are preserved by this fibration. Furthermore, we apply these
results to some pattern posets and derive alternative proofs of existing
results, such as Bj\"orner's results on subword order.Comment: 28 Page
