### Robin Cockett (Calgary, Canada), What is a Differential PCA?

#### Schedule

- Feb. 24, 2012, 15:20 - 16:20

#### Abstract

(joint work with Jonathan Gallagher)

There is a tight correspondence between partial combinatory algebras (PCAs) and Turing categories. This correspondence can be used to help develop an answer to the question of the title: for one would hope to be able to maintain this correspondence as one adds differential structure. The talk will consider this correspondence at three different levels, namely, for ordinary Turing categories, left additive Turing categories, and differential Turing categories.

For ordinary Turing categories the ability to split idempotents is a crucial aspect of their description. In particular, the "recognition theorem" for these categories relies on this ability. An important example of an idempotent, which must be present in all PCAs and must split in the associated Turing category, is the pairing combinator. A complication, as one moves to left-additive or differential Turing categories, is that the ability to split -- while retaining the property of being left-additive or differential category -- arbitrary idempotents is lost. The structure of these settings is, therefore, linked to the question of which idempotents can still be split.

In fact, in the movement from left-additive to differential Turing categories, the question of which idempotents can be split becomes significantly more delicate. The talk presents a solution to these issues and, thus, (albeit implicitly) an answer to the question in the title.