  The smooth development of large parts of mathematics hinges on the idea that
some sets are `small' or `negligible' and can therefore be ignored for a given
purpose. The perhaps most famous smallness notion, namely `measure zero',
originated with Lebesgue, while a second smallness notion, namely `meagre' or
`first category', originated with Baire around the same time. The associated
Baire category theorem is a central result governing the properties of meagre
(and related) sets, while the same holds for Tao's pigeonhole principle for
measure spaes and measure zero sets. In this paper, we study these theorems in
Kohlenbach's higher-order Reverse Mathematics, identifying a considerable
number of equivalent theorems. The latter involve most basic properties of
semi-continuous and pointwise discontinuous functions, Blumberg's theorem,
Riemann integration, and Volterra's early work circa 1881. All the
aforementioned theorems fall (far) outside of the Big Five of Reverse
Mathematics, and we investigate natural restrictions like Baire 1 and
quasi-continuity that make these theorems provable again in the Big Five (or
similar). Finally, despite the fundamental differences between measure and
category, the proofs of our equivalences turn out to be similar.
