Two factors having the same set of levels are said to be homologous. This p
aper aims to extend the domain of factorial models to designs that include
homologous factors. In doing so, it is necessary first to identify the char
acteristic property of those vector spaces that constitute the standard fac
torial models. We argue here that essentially every interesting statistical
model specified by a vector space is necessarily a representation of some
algebraic category. Logical consistency of the sort associated with the sta
ndard marginality conditions is guaranteed by category representations, but
not by group representations. Marginality is thus interpreted as invarianc
e under selection of factor levels (I-representations), and invariance unde
r replication of levels (S-representations). For designs in which each fact
or occurs once, the representations of the product category coincide with t
he standard factorial models. For designs that include homologous factors,
the set of S-representations is a subset of the I-representations. It is sh
own that symmetry and quasi-symmetry are representations in both senses, bu
t that not all representations include the constant functions (intercept).
The beginnings of an extended algebra for constructing general I-representa
tions is described and illustrated by a diallel cross design.