| Summary: | In forming a functor from category of poset to category of algebra, a
relation of object and morphism on these categories is needed. Category of
incidence algebra is a part of category of algebra. From the poset and
commutative ring can be formed an incidence algebra. In general, a morphism on
category of poset does not induce morphism on incidence algebra. This implies
that the functor formed is not well-generated. There are two methods that have
been done to make it running well, first is to define a bimodule on incidence
algebra that functioned as the morphism, and second to restrict the poset so that it
works on simplicial complexes. In this disertation a new functor is generated: a
functor from category of poset to category of simplicial complexes. This is made
in order to get a functor from the category of Poset to the category of algebra by
using functor composition.
In general incidence algebra is defined on locally finite poset so that the
multiplicative operation is well-defined. In this disertation, a new definition of
incidence algebra is defined on an abritarary poset, but with an additional
requirement, that is the finite of its sub poset, so that the incidence algebra can be
defined, called finitary incidence algebra. With this new definition, it is found that
a covariant functor from category of poset to incidence algebra is obtained, and
some isomorphism problems on finitary incidence algebra can be solved by
applying this definition. In particular, for potentially isomorphic problem, there
must be some additional requirement on one of the poset.
Covariant functor can be found from the category of simplicial complexes
to the category of abelian groups. The functor is simplicial chain group, which is
the sequence of them will form a simplicial chain complex. Morphism which
assign simplicial chain complex is simplicial chain map, and finally the new
category called category of simplicial chain complex can be obtained.
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