3,350 research outputs found
On EQ-monoids
An EQ-monoid A is a monoid with distinguished subsemilattice L with 1 2 L and such that any a, b 2 A have a largest right equalizer in L. The class of all such monoids equipped with a binary operation that identifies this largest right equalizer is a variety. Examples include Heyting algebras, Cartesian products of monoids with zero, as well as monoids of relations and partial maps on sets. The variety is 0-regular (though not ideal determined and hence congruences do not permute), and we describe the normal subobjects in terms of a global semilattice structure. We give representation theorems for several natural subvarieties in terms of Boolean algebras, Cartesian products and partial maps. The case in which the EQmonoid is assumed to be an inverse semigroup with zero is given particular attention. Finally, we define the derived category associated with a monoid having a distinguished subsemilattice containing the identity (a construction generalising the idea of a monoid category), and show that those monoids for which this derived category has equalizers in the semilattice constitute a variety of EQ-monoids
On some categorical-algebraic conditions in S-protomodular categories
In the context of protomodular categories, several additional conditions have
been considered in order to obtain a closer group-like behavior. Among them are
locally algebraic cartesian closedness and algebraic coherence. The recent
notion of S-protomodular category, whose main examples are the category of
monoids and, more generally, categories of monoids with operations and
Jo\'{o}nsson-Tarski varieties, raises a similar question: how to get a
description of S-protomodular categories with a strong monoid-like behavior. In
this paper we consider relative versions of the conditions mentioned above, in
order to exhibit the parallelism with the "absolute" protomodular context and
to obtain a hierarchy among S-protomodular categories
A unified framework for generalized multicategories
Notions of generalized multicategory have been defined in numerous contexts
throughout the literature, and include such diverse examples as symmetric
multicategories, globular operads, Lawvere theories, and topological spaces. In
each case, generalized multicategories are defined as the "lax algebras" or
"Kleisli monoids" relative to a "monad" on a bicategory. However, the meanings
of these words differ from author to author, as do the specific bicategories
considered. We propose a unified framework: by working with monads on double
categories and related structures (rather than bicategories), one can define
generalized multicategories in a way that unifies all previous examples, while
at the same time simplifying and clarifying much of the theory.Comment: 76 pages; final version, to appear in TA
Parabolic sheaves on logarithmic schemes
We show how the natural context for the definition of parabolic sheaves on a
scheme is that of logarithmic geometry. The key point is a reformulation of the
concept of logarithmic structure in the language of symmetric monoidal
categories, which might be of independent interest. Our main result states that
parabolic sheaves can be interpreted as quasi-coherent sheaves on certain
stacks of roots.Comment: 37 page
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