  We define Euler characteristic of a category enriched by a monoidal model
category. If a monoidal model category V is equipped with Euler characteristic
that is compatible with weak equivalences and fibrations in V, then our Euler
characteristic is also compatible with weak equivalences and fibrations in the
model structure induced by that of V. In particular, we focus on the case of
topological categories; that is, categories enriched by the category of
topological spaces. As its application, we obtain the ordinary Euler
characteristic of a cellular stratified space X by computing the Euler
characteristic of the face category C(X) induced from X.
