Sheaves and sheafification on Q-sites

Introducing sheaves over commutative semicartesian quantales

“…We present sheaves on quantales as a functor that satisfies gluing properties, which are formally expressed by an equalizer diagram. This is similar to the definition of sheaves on idempotent quantales proposed in [2], but it is a completely different case since we are interested in semicartesian quantales: if the quantale was idempotent and semicartesian we would obtain a locale, by Proposition 2.8, and the theory of sheaves on locales is already well established.…”

“…In [17], the sheaf definition preserves an intimate relation with Q-sets, an object introduced in the paper as a proposal to generalize Ω-sets, defined in [9], for Ω a complete Heyting algebra 1 . More recently, in [2], sheaves are functors that make a certain diagram an equalizer. In the three formulations above, right-sided and idempotent quantales were considered, while we will study sheaves on semicartesian quantales.…”

Sheaves on semicartesian monoidal categories and applications in the quantalic case

“…Locales admit a generalization in which we have an additional binary operation ⊙ and then is the new operation ⊙ that has to distribute over arbitrary joins. Therefore, it is natural to wonder how to define sheaves on quantales and, in fact, different authors under different approaches answer this question [BB86], [BC94], [MS98], [FS79], [ASV08], [HS12], [Res12]. We have the following remarks about the currently available notions:…”

“…In [MS98], the sheaf definition preserves an intimate relation with 𝑄-sets, an object introduced in the paper as a generalization of Ω-sets, defined in [FS79], for Ω a complete Heyting algebra 1 . More recently, in [ASV08], sheaves on idempotent quantales are functors that make a certain diagram an equalizer. Besides it, an extensive work about sheaves on involutive quantales was recently studied by Hans Heymans, Isar Stubbe [HS12], and Pedro Resende [Res12], for instance.…”

“…We will study sheaves on semicartesian quantales. Our approach is similar to the one in [ASV08], in the sense that our sheaves are also described in terms of equalizers but our sheaves and theirs are orthogonal in the sense we are dealing with orthogonal kinds of quantales: every quantale that is simultaneously idempotent and semicartesian is a locale (Proposition 3.1.6). Thus, the notions of sheaves coincide only in the localic case, which is already well known (see Definition 2.2.5).…”