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Abstract

A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We show that under mild conditions subunits endow any monoidal category with a kind of topological intuition: there are well-behaved notions of restriction, localisation, and support, even though the subunits in general only form a semilattice. We develop universal constructions completing any monoidal category to one whose subunits universally form a lattice, preframe, or frame.

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... In fact, the simple main insight underlying this work is that in cartesian categories, subterminal objects can be characterised entirely algebraically as central idempotents. This improves on earlier work [25], which focused on the special case of central idempotents called subunits. From a logical point of view, it extends the sheaf representation of (topos) categorical models of higher-order intuitionistic logic [1,43,49] to (symmetric monoidal closed) categorical models of multiplicative linear logic. ...
... Finally, in Section 14 we discuss several open questions that may be attacked using the representation theorem. Appendix A compares central idempotents with the special case of subunits [25]. ...
... For an important special case, recall that a frame is a complete lattice in which finite joins distribute over suprema [35]: a frame is a commutative quantale in which the multiplication is idempotent and whose unit is the largest element. Frames form a coreflective subcategory of commutative quantales [25,Proposition 3.5], where a morphism of frames is a function that preserves , ∧, and 1: ...
Preprint
Every small monoidal category with universal (finite) joins of central idempotents is monoidally equivalent to the category of global sections of a sheaf of (sub)local monoidal categories on a topological space. Every small stiff monoidal category monoidally embeds into such a category of global sections. These representation results are functorial and subsume the Lambek-Moerdijk-Awodey sheaf representation for toposes, the Stone representation of Boolean algebras, and the Takahashi representation of Hilbert modules as continuous fields of Hilbert spaces. Many properties of a monoidal category carry over to the stalks of its sheaf, including having a trace, having exponential objects, having dual objects, having limits of some shape, and the central idempotents forming a Boolean algebra.
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