William James answered his own question when he asked: Does category theory, being mathematics, have no associated philosophy? Yes, category theory is mathematics. Therefore its associated philosophy is whatever philosophy one chooses to associate with mathematics. As for the latter, Dusko provided a pretty good answer. I would modify it only to reflect that mathematics (which, as always in the absence of a qualifier, means pure mathematics) is a subject matter. Most mathematicians are sufficiently confident about their subject matter that they feel no need for a semantics, much less a stated philosophy. (Yes, it has been notoriously difficult to define intensionally, but that's not special to mathematics. What's the subject matter of physics? If either mathematicians or physicists were -- using Dusko's language -- to spend a lot of time defining their subject, there never would have been much mathematics or physics.) Steve thinks that the existence of a philosophy of category theory is an "of course". In one of the public meanings of the word "philosophy" he's certainly correct but not, I think, in the sense that would include something like constructivism. (The public meaning in question has even less than type-checking to do with either love or wisdom. Well, maybe it has something to do with love.) May I suggest that the applied mathematician may have a very different understanding of category theory from the mathematician. Steve says that category theory is "all things are connected". But that's an article of faith for almost any mathematician. He goes on to say, "you cannot fully describe anything purely in itself but only by the way it connects with others." This assertion about what "you cannot" do sounds like it could be a good way of describing *applied* mathematicians. To begin with, Eilenberg and Mac Lane defined categories in order to define functors and they defined functors in order to define natural transformations. Immediately it was noted that a new tool existed to pin down -- in a formal way -- how it is that some of the all things are connected. It should be noted that categories -- and more to the point, functors -- have always been considered tools for studying the subject matter of mathematics. Tools, not the subject matter itself. I am on record that the language of categories began to become respectable when Frank Adams was able to count the number of independent vector fields on each sphere using a construction that quantified over functors: it produced an n-ary transformation on the K-functor for every n-ary endofunctor on the category of finite dimensional vector spaces (which he assembled into what are now known as the Adams operations). One of the better successes since then has been the use of categories in finding connections between various foundational systems. Because some of these systems are constructivist it has apparently caused some to think that categories are intrinsically constructivist. Strange. There's another important aspect of category theory. Most categories, in the beginning at least, were categories that naturally arose from existing branches of mathematics. Some of these categories, though, had never been lived in before they were invented as categories. Joel Cohen named one of these the "Freyd Category" (named not after Peter but Jennifer): its an abelian category whose full subcategory of projectives is the stable-homotopy category; all the other objects have no easy description; the category can be described as the target of the universal homology theory. But a much better example is in Serre's dissertation. This work, hailed by many as the single most substantive dissertation ever written, contends with the two abelian categories that result when one starts with the category of abelian groups and identifies with the zero group all finite groups in one case, or all finitely generated groups in the second case. Most remarkably, Serre did all this without using category theory. (The fact that the first non-trivial construction of a category occured without benefit of category theory must be reckoned an embarrassment for category theorists.) But in recent applications I think a very different type of question is being asked: "Is it possible that there is a category in which ... can take place?" These questions are at the heart of many approaches to programming semantics. And they are at the heart of many of the uses of categories in theoretical physics. But the first serious example came, in fact, a long time ago. In the late 60's Lawvere's approach to differential geometry asked just this type of question. Elementary topoi made their first appearance as just a preliminary part of the answer. So what's this all have to do with philosophy? Not much, of course.