TITLE: Lawvere's "Some thoughts on the future of category theory."
QUESTION [51 upvotes]: In Lecture Notes in Mathematics 1488, Lawvere writes the introduction to the Proceedings for a 1990 conference in Como.  
In this article, Lawvere, the inventor of Toposes and Algebraic Theories, discusses two ancient philosophical "categories": that of BEING and that of BECOMING.  And he's serious.  While some of the motivation for this article is to understand these two ancient even mystical topics, the actual content is almost purely mathematical.  Lawvere makes definitions and claims in the manner of a serious mathematician engaged in deliberate but casual explanation of ideas.
I want to understand this article, but it is difficult.  The definitions seem to be written for someone with a bit more background or expertise on topos theory and its application.
Q: I am writing to ask whether anyone here has read or understood this article (or parts of it).  I'm interested in your thoughts on it.  Have the ideas been written formally?

REPLY [2 votes]: As a prelude to the answer above:  When Lawvere wanted a mathematics of Becoming and not just Being, one thing he wanted is to work in categories that have exponentials.
For example, in the category of manifolds, we can express the motion of a body as a map $T \times B \rightarrow E$, where $T$ is an object of times, $B$ is a body, and $E$ is Euclidean space.  Lawvere wants to work in bigger categories where we can also express this as $B \rightarrow E^T$ (the possible paths of an individual part of the body) or $T \rightarrow E^B$ (the collection of instantaneous descriptions of a body's location).
A category which works for these purposes is $\mathbf{Sets^{Z^{\large Op}}}$, where $Z$ is a category of loci that extends the category of manifolds.  This leads to topoi.