## Restriction colimits and gluings

### Daniel Lin – 10 April 2019

A restriction category is an abstraction of sets and partial functions. In a restriction category, each map is associated to an idempotent on its domain object; these idempotents are called restriction idempotents, which we may think of as partial identity maps expressing domains of definition. In 2016, Garner and Lin proposed a notion of cocomplete restriction category; i.e., a restriction category whose restriction idempotents split, whose subcategory of total maps is cocomplete (in the ordinary sense), and whose colimits in this subcategory are preserved by its inclusion.

However, just as (ordinary) cocomplete categories may be characterised as having all (ordinary) colimits, it would be nice to have an analogous characterisation of cocomplete restriction categories. Indeed, this is what we will discuss in this talk. We will present a notion of restriction colimit, and give an application of this within the context of gluings and atlases.

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