  A general definition of mathematical morphology has been defined within the
algebraic framework of complete lattice theory. In this framework, dealing with
deterministic and increasing operators, a dilation (respectively an erosion) is
an operation which is distributive over supremum (respectively infimum). From
this simple definition of dilation and erosion, we cannot say much about the
properties of them. However, when they form an adjunction, many important
properties can be derived such as monotonicity, idempotence, and extensivity or
anti-extensivity of their composition, preservation of infimum and supremum,
etc. Mathematical morphology has been first developed in the setting of sets,
and then extended to other algebraic structures such as graphs, hypergraphs or
simplicial complexes. For all these algebraic structures, erosion and dilation
are usually based on structuring elements. The goal is then to match these
structuring elements on given objects either to dilate or erode them. One of
the advantages of defining erosion and dilation based on structuring elements
is that these operations are adjoint. Based on this observation, this paper
proposes to define, at the abstract level of category theory, erosion and
dilation based on structuring elements. We then define the notion of
morpho-category on which erosion and dilation are defined. We then show that
topos and more precisely topos of presheaves are good candidates to generate
morpho-categories. However, topos do not allow taking into account the notion
of inclusion between substructures but rather are defined by monics up to
domain isomorphism. Therefore we define the notion of morpholizable category
which allows generating morpho-categories where substructures are defined along
inclusion morphisms. {A direct application of this framework is to generalize
modal morpho-logic to other algebraic structures than simple sets.
