What is category theory

Category theory is a branch of mathematics for helping make good Definitions. It guides us in defining structures in ways that are fruitful, which is to say, tap into themes that recur for all manner of interesting mathematical objects.

At the heart of categories is the associative property. This property expresses a deep duality in composition. For example, top-down is always matched by bottom-up. Likewise, avoid evaluation goes hand in hand with evaluate as soon as possible. And abstraction goes hand in hand with immersion.

Category Theory encourages us to think loosely as we make definitions, so that we might be sure to make fruitful ones, before we clamp them down tight.

Apparently, what makes a definition fruitful (for example, fruitful instructions) is that you can apply it in two different ways - knowing what you're doing, what your goal is, and specifying that ahead of time, and understanding intuitively each step, why it's necessary - or simply going through the motions, the instructions, without having to understand ultimately why. The whole point of algebra is that it lets us manipulate symbols successfully without having to intuitively understand their meaning at each step. And the whole point of understanding instructions is that our mind can then with confidence instruct our body to mechanically, obediently execute them.

So, for a definition to be fruitful, it must support thoughtful action, which involves these two directions - the "thoughtful" ordering of instructions (backwards from the goal) - and the "active" execution of that ordering (forwards from where we start). We see this, for example, when we use both "bottom up" (inductive, empirical) and "top down" (deductive, theoretical) approaches.

How is category theory useful?

Category Theory is a branch of mathematics which JosephGoguen uses in his study, What is a concept?

He also uses it in his work on Institutions. So I share my basic understanding below and note how it relates to Concepts. But then I share my own thoughts which you and category theory have inspired regarding Views.

QuiverMatrix

In order to understand category, I am thinking of it in terms of an QuiverMatrix. This lets me think in terms of matrices, which I find to be more concrete.

Duality

What is especially powerful is that - at any point - we can switch all the directions around (this is called "duality") - so that all of a sudden the real goal may be your body wants to "go for a drive", and the justification may be "get something at the store", until we realize "hmm, we're short on milk". And so you find yourself craving milk (or cigarettes) and visualizing the milk aisle until finally you get in that car, which is perhaps where your body wanted you, its goal. This means that in these chains of reasoning it's always hard to say who is in charge. And what is "bottom-up" and "top-down". But what is fruitful is that both directions can play off each other.

I think this is also the relationship between structure and activity. Structural patterns, ever finer, channel activity - and patterns of activity, ever broader, evoke structure.

Key ideas

In his paper A Categorical Manifesto, Joseph Goguen lists seven ideas for the use of category theory:

• Category: The structure which morphisms preserve. Isomorphic: Objects are equivalent if they are isomorphic, hence there should be no isomorphisms except for the identity. And there should be no inverses, for otherwise there is an isomorphism.
• Functor: A construction from one species of object to another species of object should preserve essential relationships amongst the objects such as the compositions and identities for their morphisms.
• Natural transformation: Morphisms of functors, which is to say, if there is a natural way to translate from one construction to another, then likewise there is a natural transformation for the corresponding functors.
• Limit: A diagram may be seen as a system of constraints, and its limit as all possible solutions of that system.
• Adjoint: Apparently, a relationship for relating projections and expansions, which is to say, a dual functor for a functor.
• Colimit: Limits and colimits are dual in the sense that solutions and interconnections are, or the empty diagram and an initial object (a generic object).
• Comma category: Corresponds to enriching a structure

Matrix

Note that a category may be thought of as a deductive system, a directive graph, and hence a Matrix. My thesis was on the combinatorics of the general matrix, which apparently is all generated by the symmetric functions of the eigenvalues of a matrix. So I am interested to see if that might be of value here. I want to show how a category arises from first principles. I imagine that Perspectives may be thought of as morphisms (or functors).

Pushouts

Consider push-outs (related to pullbacks): W shape and the defining of a composition of relations, as Joseph suggested. A W with an upside down W on top of it, and then an upside down V on top of that. Two projection functions yielding composition of relations in the category of sets.

AlgebraOfViews

I think that we may have different category types if we consider the subcategories in which the morphisms all assume an initial perspective (for example, Tina's perspectives on ... - would be one subcategory, and God's perspectives on ... - would be another subcategory.) Then we can have functors from one kind of subcategory to another. We can also have subcategories that all end in the same final perspective. So that some are based on immersing ourselves as deeply as we can from an initial perspective, and others are based on reframing farther and farther out from a final perspective. And these two kinds of categories are perhaps related by limits and colimits. And the relationship helps us reverse direction in some sense so that we could escape our own perspective and, for example, take up God's perspective instead.

Resources

Andrius: I find Joseph's article very inspiring and thought provoking, especially the general idea that categories guide us in making fruitful Definitions. For me, the Jean-Pierre Marquis article is helpful in understanding the basic concepts.

Stanford Encyclopedia of Philosophy: Category Theory by Jean-Pierre Marquis

There is also a helpful book online.

I received from Amazon a couple of books on category theory: "Conceptual Mathematics" by F.William Lawrere and Stephen H. Schanuel (thank you to Joseph Goguen for recommending this), and "Basic Category Theory for Computer Scientists" by Benjamin C. Pierce.

Discuss

JosephGoguen: dear andrius, i think you may be misleading yourself with these quivers, since that kind of combinatorial thinking is very far from categorical thinking, which is usually aimed at defining very general concepts that apply to many different kinds of structure. so the main examples of categories are categories of those structures, such as sets, groups, rings, fields, vector spaces, relations, matrices, topological spaces, banach spaces, partially ordered sets, ..... a categorical approach to combinatorics would possibly start with the sorts of structures that rota liked, such as mobius algebras. another idea would be a category with polynomials as morphisms and substitution as composition, for studying generating functions, etc. one would look for functors with nice properties, such as preserving limits. but this kind of application is probably more advanced than where you are now, so i would suggest that you wait a bit before trying to think about combinatorics. cheers,

JosephGoguen: dear andrius im thinking that your algebra of views could be a monoid, i.e., an associative binary operation with identity. for example, A's view of B's view would be "A of B" where "of" is the binary operation. perhaps the identity I is your "God"? since I of A = A and A of I = A. it is interesting to think about when relative idempotence holds, i.e., A of A of B = A of B, and when it doesnt - "Alex thinks his view of Betty is wrong"). relative commutativity is also interesting. note that all his is similar to Levi-Strauss' work on kinship role in various cultures, e.g., brother of father = uncle (though i think he used semigroups since the identity doesnt seem to make much sense here); you can easily find stuff on this with google. hope this helps,

AndriusKulikauskas: Joseph, Thank you for your letters. I do find them helpful. I am making some progress learning about category theory. Surprisingly, the Benjamin Pierce book is helpful because it is very terse. I'm trying to understand adjoints. They seem to pair "expansions" (of a domain) with "restrictions" (back into the original domain). Perhaps a bit like decompressions and compressions. The matrix approach is a little bit helpful to me because it simply gives a concrete notation for these very abstract ideas. One of the things that I've realized is that categories express the constraints on composition. I mean that when the category has size 1, then the arrows can all be composed. But if the category is bigger, that means that we have arrows which can't be composed. (In this case, the arrows of a category are not a monoid.) So the category splits the matter of composition into that which must be handled by the arrows, and that which can be handled by the objects. And I suppose if a particular arrow is unique, then that constrains compostion even more. So categories may be thought of as seeking the constraints on composition. Also, I think that the value of the category theory ideas is that we can look at expressions that may be understood as equivalent at some point, in some context, but are not as of yet. So it may be that in certain contexts the views form a monoid, but still there may be times when they are not even associative. So we want to be able to show how that unfolds. In that sense the quiver approach is a bit helpful because it suggests an order in which the constraints unfold. If we are forced from the start to commit to a particular labeling of our objects, then perhaps we can see where that breaks down. Ultimately, it seems that math has to do with the collapse of structure - for example, the equating of an expansion with some single object - so it is interesting to see at what point that collapse is natural or necessary. Thank you also for alerting me to ideas of relative idempotence and relative commutativity.

I note the work of Douglas R. White, his papers on the mathematics of social networks, and his interest in structural analysis and cultural kinetics.