Among other things, category theory allows Badiou to reformulate his theory of the subject, conceived as a group in the mathematical sense (Court traité d'ontologie transitorie, p.165-77). In categorial terms "a group is a category that has one single object, and whose arrows are all isomorphisms" (p.172). This sole object provides the name of the group, its "instantiating letter". Call it "G". Its arrows all go from G and end in G, and these provide its "operative substance." they emphasise the fact that what makes a group a group, what makes G, G, is "the set of different ways in which the object-letter G is identical to itself." Any such "group-subject is infinite" that is, the number of ways it has of being identical to itself is inexhaustible" (Court traité d'ontologie transitorie, p.176).
Badiou: A Subject to Truth (p.418, n.32)
What we can see here are definite vestigial traces, albeit in highly mathematised forms, of Badiou's Sartrean legacy at work. If, as I understand it, a subject is here defined as a group through its manner of self-identity, expressed via the "sole object", then there are parallels with Sartre's non-mathematical group's identity being forged via the relations with the third party, seen by each member as "the common individual". Obviously, I think this may well be somewhat of a simplification, but it would be an intriguing argument to trace elements of Badiou's underlying mathematical ontology back towards a CDR-era Sartrean schema. Adding to these complications however is the fact that here we are clearly dealing with topos/category theory, rather than the earlier and slightly more transparent set theory (used as the basis of Badiou's discussion of being-qua-being in Being and Event) and hence why this begins to stray into the territories explored by Badiou in Logics of Worlds. Referring again to Hallward on the distinctions between these methodologies:
Category theory and set theory offer opposing approaches to "all the decisive questions of the thought of being (acts of thought, forms of immanence, identity and difference, logical framework, admissible rationality etc...)" (Topos, ou logiques de'lonto-logique: Une Introduction pour philosophes, tome 1 p.5) They are, in short, different ways of conditioning philosophy. If the first volume of l'Etre et l'envenement was written under the sole mathematical condition of set theory, the second volume will be written under the double condition of both theories... Where set theory directly articulates being-as-being, category theory is "the science of appearing, the science of what signifies that every truth of being is irredemably a local truth" (Court traité d'ontologie transitorie, p.199)... in categorial logic "relation precedes being" (Court traité d'ontologie transitorie, p.168)... The theory offers "a highly formalized language especially suited for stating the abstract property of structures." ... the theory is designed to allow for the most general possible description of logical relations or operations between such structures and entities... Methodologically, the way these structures are represented is essentially geometric: isolated manageable parts of a category are expressed in diagrams... These diagrams are made up of objects, on the one hand, and of arrows or "morphisms" on the other. Arrows are "oriented correlations between objects; an arrow 'goes' from object a (its source) to object b (its target)." ... The single most important principle of category theory is that all individuating power or action belongs to these dynamic arrows alone...
Badiou: A Subject to Truth (pp. 303-306)
All this certainly goes someway to explaining how a category theory "group" can be constructed. The type of category known as a group is specifically designated by a structure consisting of only one object, but an infinite number of isomorphic arrows (each beginning and ending with the object). Diagrammatically perhaps (!):
What remains to be seen however is the manner in which Badiou applies this: is he getting at a kind of internal structure to the group-subject? Certainly it seems his deployment of category theory (specifically the world-building device of the sub-species of category theory known as topos) is designed to enable him to think through the local logical implications of decisions set in motion by truth process sequences. Further this brings us dangerously near towards the vexed question of relationality, (for that is what the logic of appearance in Badiou's system ultimately enables him to do, to think relations rather than merely bare being-qua-being as in the ontological discourse of set theory). This idea of relationality in many ways can be seen as operating at the core of dialectical reason and transversality also, and hence will have to be a central theme of this work.