Category Theory

As I work through Badiou's Logiques des mondes, I've decided that it's necessary to acquaint myself with category theory. We'll see how far I'm get. Currently I'm reading Robert Goldblatt's Topoi: The Categorial Analysis of Logic, which is billed as an introductory text accessible to those who have little or no background in formal mathematics. I have not yet gotten far enough in Logiques des mondes to know precisely how Badiou plans to put category theory to use in articulating his logic of appearances or situations, but already I can be seen just how much category theory differs from set theory. As Goldblatt writes at the very beginning of his text,

The study of topoi arises within category theory, itself a relatively new branch of mathematical inquiry. One of the primary perspectives offered by category theory is that the concept of arrow, abstracted from that of function or mapping, may be used instead of set membership relation as the basic building block for developing mathematical constructions, and expressing properties of mathematical entities. Instead of defining properties of a collection by reference to its members, i.e. internal structure, one can proceed by reference to its external relationships with other collections. The links between collections are provided by functions, and the axioms for a category derive from the properties of functions under composition.

A category may be thought of in the first instance as a universe for a particular kind of mathematical discourse. Such a universe is determined by specifying a certain kind of "object", and a certain kind of "arrow" that links different objects. (1)

The founding gesture of set theory is organized around the idea of set membership, where we think elements in terms of their membership to a particular set. From the beginning I've had problems with Badiou's proposal to found ontology on set theory, as I've always had the conviction that beings can only be understood in terms of their relations, whereas relations are irrelevant to thinking elements of a set. For instance, given a set {x, y, z}, the axiom of extension tells me that there's no difference between this set and the set {y, z, x} as they both contain the same elements.

The problem is that anyone who comes from a structuralist background knows that how elements are ordered makes a profound difference in the nature of those elements. That is, elements of a structure cannot be said to exist independently of their relations, but are what they are precisely through their relations. I cannot order phonemes of a language in any old fashion, nor can I order kinship relations in any old fashions (interestingly Goldblatt uses a quote from Levi-Strauss as an epigraph to chapter 2 of his book), nor can social relations or the Lacanian subject (as a particular way of relating to the Other) be ordered in any old fashion. Take, for instance, the following anagram: "conservation" versus "conversation". From a set theoretical point of view, both of these words are identical as they contain exactly the same elements. Yet at the level of language, the position of the elements makes a deep difference in the nature of these "sets". Something important is thus lost in set theory, which, no doubt, is precisely why Badiou had such difficulties theorizing the structure of situations in Being and Event. No doubt Badiou would respond by arguing that entities such as my anagram above pose no problem for his ontology as ontology discusses only what can be said of being qua being, not specific situations. An anagram is a situation or world appearing in language, not being qua being. Fair enough. However, can we not speak of the being of the anagram? Badiou, of course, will try to demonstrate how category theory is founded on set theory, so as to respond to precisely this sort of elementary criticism.

Interestingly, Lacan's mathemes could be thought as undertaking a sort of vulgar or very basic form of category theory, in thinking various arrows relating one object to another in a structure. What would Lacanian psychoanalysis look like when viewed through the lense of a more rigorous formulation in terms of category theory?

It seems to me that a fundamental ontological decision lies in the choice between set theory and category theory. This does not entail that the choice of one excludes the choice of the other. Both claim to be foundational, and Goldblatt spends a good deal of time showing how set theory can be founded on category theory. Rather, the ontological decision in question revolves around the question of whether membership or relation are foundational in the order of being.