A lot of category theory work is consisting of taking definition of sone structure ( set theory, topological space, ordered set) and constructing category where object internal structure is completely agnostic ( Set, Top, Ord) . As for every object a in every category there is an morphism id_a on that object, and there is an isomorphism between object and this identity morphism, in fact someone may forget about object completely and work with morphisms only.
Suppose we want to construct homotopy theory. Natural starting point is to define topological space. After that step next well known operations are constructed: points, paths, deformations etc. At the end of construction one gets group of paths deformations with all of the consequences and relationships with topology at the base of it. One may say, by analogue to categorical constructions above, topological properties of base space of homotopy theory is somehow like "internal structure of objects" for category theory of certain family of objects.
Question: is there any generalisation if homotopy theory which completely ignores topological nature of space it is constructed on, and focus only on abstract relationships between objects without internal structure? If so, give it possibility to define homotopy theory on more general, than topological space, structures?
( Kind of motivation explanation: by topological space I mean space for which open subsets and well known relationships between intersections and sums are defined, so it is very general structure. But in other way it is rigid, as a lot of various and interesting in practical applications sets may have interesting properties, allows path definitions but are not topological spaces because for example intersctions of subsets are not subsets or elements of the base space any more. For example - for fun - we may consider a Linux programs composable by pipes. You may connect pipes just as morphisms, and perform computations. We may think of several programs connected by pipes as a stream and it resembles paths in homotopy theory. One may consider deformations of such streams such that input data and results of the computation remains the same, which gives us equivalence classes of streams. Definitely stream ( pipe composition of programs) is a program itself. So sum of streams, at least when outputs and inputs are compatible, is defined. But intersections of a streams is not a stream ( and even it is not a program, as common elements may be "separated" for example first and the last programs in the stream ) so base space is not topological one. )