No Properties/No Structure – But, Subject Identity

Jack Park has prodded me into following some category theory and data integration papers. More on that to follow but as part of that, I have been watching Bartosz Milewski’s lectures on category theory, reading his blog, etc.

In Category Theory 1.2, Mileski goes to great lengths to emphasize:

Objects are primitives with no properties/structure – a point

Morphism are primitives with no properties/structure, but do have a start and end point

Late in that lecture, Milewski says categories are the “ultimate in data hiding” (read abstraction).

Despite their lack of properties and structure, both objects and morphisms have subject identity.

Yes?

I think that is more than clever use of language and here’s why:

If I want to talk about objects in category theory as a group subject, what can I say about them? (assuming a scope of category theory)

  1. Objects have no properties
  2. Objects have no structure
  3. Objects mark the start and end of morphisms (distinguishes them from morphisms)
  4. Every object has an identity morphism
  5. Every pair of objects may have 0, 1, or many morphisms between them
  6. Morphisms may go in both directions, between a pair of morphisms
  7. An object can have multiple morphisms that start and end at it

Incomplete and yet a lot of things to say about something that has no properties and no structure. 😉

Bearing in mind, that’s just objects in general.

I can also talk about a specific object at a particular time point in the lecture and screen location, which itself is a subject.

Or an object in a paper or monograph.

We can declare primitives, like objects and morphisms, but we should always bear in mind they are declared to be primitives.

For other purposes, we can declare them to be otherwise.

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