7 research outputs found
Model structures on the category of small double categories
In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2-monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofi brant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves and categorifications
Model Structures on the Category of Small Double Categories
In this paper we obtain several model structures on {\bf DblCat}, the
category of small double categories. Our model structures have three sources.
We first transfer across a categorification-nerve adjunction. Secondly, we view
double categories as internal categories in {\bf Cat} and take as our weak
equivalences various internal equivalences defined via Grothendieck topologies.
Thirdly, {\bf DblCat} inherits a model structure as a category of algebras over
a 2-monad. Some of these model structures coincide and the different points of
view give us further results about cofibrant replacements and cofibrant
objects. As part of this program we give explicit descriptions and discuss
properties of free double categories, quotient double categories, colimits of
double categories, several nerves, and horizontal categorification.Comment: 103 pages. Included Quillen adjunctions with Cat, improved
characterization of flexible double categories, proved 2-cocompleteness of
DblCat_v, proved horizontal nerve is 2-coskeletal, cut double
categorification for a future article, removed identity squares from double
derivation schemes, improved counterexample to Reedy transfer
Algebraic exponentiation and internal homology in general categories
Includes bibliographical references (p. 101-102).We study two categorical-algebraic concepts of exponentiation:(i) Representing objects for the so-called split extension functors in semi-abelian and more general categories, whose familiar examples are automorphism groups of groups and derivation algebras of Lie algebras. We prove that such objects exist in categories of generalized Lie algebras defined with respect to an internal commutative monoid in symmetric monoidal closed abelian category. (ii) Right adjoints for the pullback functors between D. Bourns categories of points. We introduce and study them in the situations where the ordinary pullback functors between bundles do not admit right adjoints in particular for semi-abelian, protomodular, (weakly) Maltsev, (weakly) unital, and more general categories. We present a number of examples and counterexamples for the existence of such right adjoints. We use the left and right adjoints of the pullback functors between categories of points to introduce internal homology and cohomology of objects in abstract categories
An operadic approach to internal structures
We study internal structures in the category of algebras for an operad, and show that these themselves admit an operadic description. The main case of interest is where the operad is on an abelian category, and the internal structures in question are those of internal category, internaln-category, or internal (cubical) n-tuple category. This allows an operadic treatment of crossed modules and other crossed structures
An operadic approach to internal structures
We study internal structures in the category of algebras for an operad, and show that these themselves admit an operadic description. The main case of interest is wher