224 research outputs found
An embedding theorem for adhesive categories
Adhesive categories are categories which have pushouts with one leg a
monomorphism, all pullbacks, and certain exactness conditions relating these
pushouts and pullbacks. We give a new proof of the fact that every topos is
adhesive. We also prove a converse: every small adhesive category has a fully
faithful functor in a topos, with the functor preserving the all the structure.
Combining these two results, we see that the exactness conditions in the
definition of adhesive category are exactly the relationship between pushouts
along monomorphisms and pullbacks which hold in any topos.Comment: 8 page
Morita contexts as lax functors
Monads are well known to be equivalent to lax functors out of the terminal
category. Morita contexts are here shown to be lax functors out of the chaotic
category with two objects. This allows various aspects in the theory of Morita
contexts to be seen as special cases of general results about lax functors. The
account we give of this could serve as an introduction to lax functors for
those familiar with the theory of monads. We also prove some very general
results along these lines relative to a given 2-comonad, with the classical
case of ordinary monad theory amounting to the case of the identity comonad on
Cat.Comment: v2 minor changes, added references; to appear in Applied Categorical
Structure
Icons
Categorical orthodoxy has it that collections of ordinary mathematical
structures such as groups, rings, or spaces, form categories (such as the
category of groups); collections of 1-dimensional categorical structures, such
as categories, monoidal categories, or categories with finite limits, form
2-categories; and collections of 2-dimensional categorical structures, such as
2-categories or bicategories, form 3-categories.
We describe a useful way in which to regard bicategories as objects of a
2-category. This is a bit surprising both for technical and for conceptual
reasons. The 2-cells of this 2-category are the crucial new ingredient; they
are the icons of the title. These can be thought of as ``the oplax natural
transformations whose components are identities'', but we shall also give a
more elementary description.
We describe some properties of these icons, and give applications to monoidal
categories, to 2-nerves of bicategories, to 2-dimensional Lawvere theories, and
to bundles of bicategories.Comment: 23 page
2-nerves for bicategories
We describe a Cat-valued nerve of bicategories, which associates to every
bicategory a simplicial object in Cat, called the 2-nerve. We define a
2-category NHom whose objects are bicategories and whose 1-cells are normal
homomorphisms of bicategories, in such a way that the 2-nerve construction
becomes a full embedding of NHom in the 2-category of simplicial objects in
Cat. This embedding has a left biadjoint, and we characterize its image. The
2-nerve of a bicategory is always a weak 2-category in the sense of Tamsamani,
and we show that NHom is biequivalent to a certain 2-category whose objects are
Tamsamani weak 2-categories.Comment: 23 page
On monads and warpings
We explain the sense in which a warping on a monoidal category is the same as
a pseudomonad on the corresponding one-object bicategory, and we describe
extensions of this to the setting of skew monoidal categories: these are a
generalization of monoidal categories in which the associativity and unit maps
are not required to be invertible. Our analysis leads us to describe a
normalization process for skew monoidal categories, which produces a universal
skew monoidal category for which the right unit map is invertible.Comment: 15 pages. Version 2: revised based on a very helpful report from the
referee. To appear in the Cahiers de Topologie and Geometrie Differentielle
Categorique
Triangulations, orientals, and skew monoidal categories
A concrete model of the free skew-monoidal category Fsk on a single
generating object is obtained. The situation is clubbable in the sense of G.M.
Kelly, so this allows a description of the free skew-monoidal category on any
category. As the objects of Fsk are meaningfully bracketed words in the skew
unit I and the generating object X, it is necessary to examine bracketings and
to find the appropriate kinds of morphisms between them. This leads us to
relationships between triangulations of polygons, the Tamari lattice, left and
right bracketing functions, and the orientals. A consequence of our description
of Fsk is a coherence theorem asserting the existence of a strictly
structure-preserving faithful functor from Fsk to the skew-monoidal category of
finite non-empty ordinals and first-element-and-order-preserving functions.
This in turn provides a complete solution to the word problem for skew monoidal
categories.Comment: 48 page
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