NOTE: The following is technically incorrect - I’ll post a follow-up when I’ve worked out the wrinkles.

In Logics of Worlds, Badiou defines an object as a pair (A, Id), where Id is a function from pairs of elements (p, q) of A to values in an algebra ?. A is called the “support set” of the object, and Id is its “transcendental indexing”: the algebra to which elements of A are indexed is known as the “transcendental” of the world in which the object (A, Id) appears. The value given by Id(p, q) is said to be the “degree of identity” between p and q.

I was initially rather confused by the definition of Id as a function on pairs of elements: in Goldblatt’s book on Topoi, the arrow between an object of a Topos and the subobject classifier (Badiou’s Transcendental) simply assigns a “truth value” to each discernible constituent of the object. In Badiou’s scheme, this value is said to be the value of Id(p, p), which gives the degree of identity between p and itself.

Because the values in the transcendental are arranged in a Heyting algebra, we can take two such truth values and find another value which represents the degree of “truth” on which they concur. So it’s reasonably straightforward to convert between a function that takes individual elements of A to truth values, and a function that gives the conjoined value of the truth values of pairs of elements:

Here we have an “index” set, I, that indexes the pairs (p, q) of the product A x A of A with itself, and two projections, l and r from this product set to the “left” and “right” members of each pair. ? is some “truth-valuing” arrow that takes individual elements of A to values in ?.

We can easily construct a mapping from A x A to ? x ?, by composing l and r with ? to get the unique product arrow <? . l, ? . r>. All that remains is to compose the arrow ? from ? x ? to ? with this product arrow, and we then have an arrow Id from A xA to ?.

In other words, Id is constructible on the basis of a function ? from individual elements of A to values in ?, and a function ? from pairs of truth values in ? x ? to single values in ?. All we need is the basic categorical machinery needed to set up products and map between them. Note that ? is not just any arrow - it has to be such that Id(p, p) = ?(p).

Why does Badiou put it this way around, defining Id as operating on pairs of elements of A, rather than defining an object as (A, ?), a set with a truth function on its individual elements? My guess is that it simplifies the presentation, underlining the algebraic character of ? from the outset. It also allows Badiou to define existence as intrinsically relational, a relation of self-identity mediated by the transcendental of a world (cf the mirror stage in Lacan). But nothing really obliges us to take things in this order: there is a strict mathematical equivalence between the object as (A, Id) and the object as (A, ?), and each is immediately derivable from the other.