In the study of categories whose morphisms display a behaviour similar to that of partial
functions, the concept of morphism domain is, obviously, central. In this paper an operation defined on
morphisms describes those properties which are related to morphisms being regarded as abstractions of
partial functions. This operation allows us to characterise the morphism domains directly, and gives rise to
an algebra defined by a simple set of identities. No product-like categorical structures are needed therefore.
We also develop the construction of topologies together with the notion of continuous morphism, in order
to test the effectiveness of this approach. It is interesting to see how much of the computational character
of the morphisms is translated into continuity.