Victor Porton's Math Weblog

This short article introduces two modified definitions of category with multiple sources and destinations (from category theory from abstract mathematics).

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Standard Definition of Category

I recall that in category theory category is traditionally defined as the system of:

• a set of objects (denoted Ob);
• a set of morphisms or arrows (denoted Mor or Hom);
• a mapping from every morphism f to its source or domain (denoted Src f or dom f) and destination or range (denoted Dst f or ran f), source and range are objects;
• binary operation of composition of pairs of morphisms f and g such that Dst f = Src g, composition of f and g is denoted as gf and is a morphism (if it is defined that is Dst f = Src g) such that Dst(gf) = Dst g and Src(gf) = Src f;
• identity morphism 1_{A} for every object A which is both left and right identity for composition of morphisms operation (when it is defined).

The Problem with Standard Definition

However in applications:

1. Two morphisms of a category often differ only by their source and/or destination.
2. One object may serve as morphisms of several different categories, where source and destination are defined differently.

So we are forced to distinguish between several morphisms which differ only by their source and destination by artificial adding to their values source and destination, what is inconvenient.

So for convenience reasons we need a modified definition of category where one morphism may have several sources and destinations.

I suggest two variants of modified definition of category for consideration.

First Modified Variant of Definition

The simplest thing which comes to mind is Src f and Dst f to be (normally non-empty) sets of objects instead of single object.

Then composition of morphisms f and g will be defined when and only when the set Dst f intersects Src g. The equality Dst(gf) = Dst g and Src(gf) = Src f are left unchanged in this modified definition.

Note that identity morphisms may be common for several objects in the case of modified definition of category with multiple source and destination.

Second Modified Variant of Definition

Second, more sophisticated variant is:

Let instead of Src f and Dst f to each morphism f correspond a relation on the set of objects (Ob).

Let also composition of every two morphisms is defined and the relation of objects for the composition of two morphisms will be the composition of the relations corresponding to these two morphisms.

Note that so we may have morphisms to which correspond empty relation of objects. (These morphism are purely formal meaningless thing.)

Several different objects may again have a common identity morphisms in this modified definition.

Remaining Issues

For me the question about functors between categories with multiple sources and destination and other subtle aspects of the theory is yet open. I however do not warrant that nobody has researched categories with multiple sources and destinations before.

Also it may be interesting the question about natural mappings between categories in traditional sense (with single source and destination) and modified categories with multiple sources and destination.

I don't exclude that research of this looking trivial questions may indeed produce an interesting and important theory.

If you will research this, you may publish it in comments to this message.

Final Notes on Terminology

Finally, I'd suggest to leave the term arrow for morphisms with single source and destination and to use (with appropriate note that it is a modified definition) the term morphism for morphisms from modified definition of category where a morphism has multiple sources and destinations.

References

See my Theory of Dependencies, a research of some particular (important) categories where I use a modified definition of category with multiple sources and destinations.