|Title||Presheaves over join restriction categories|
|Presenter||Daniel Lin (Macquarie University)|
|Author(s)||Mr Daniel Lin|
|Session||Category Theory, Algebraic Topology, K-Theory|
Restriction categories were first introduced in the early 1990s and later studied in the early 2000s as a means of generalising the notion of a partial map category; the idea being to capture the partiality of each map through a corresponding idempotent on its domain, called a restriction idempotent. As it turns out, the hom-sets of any restriction category has a natural partial ordering, and this, together with the notion of compatibility, gave rise to the notion of a join restriction category. In this talk, we shall see that every join restriction category may be freely completed to a cocomplete join restriction category, given by the partial map category of sheaves on some site. However, we shall also see that every join restriction category has a free cocompletion equivalent to this partial map category of sheaves. This equivalent category takes a simpler form, with join restriction presheaves as its objects.