Familial 2-functors and parametric right adjoints

By Mark Weber. Theory and Applications of Categories, 2007

102 citations recorded by the source corpus.

We define and study familial 2-functors primarily with a view to the devel- opment of the 2-categorical approach to operads of (Weber, 2005). Also included in this paper is a result in which the well-known characterisation of a category as a simplicial set via the Segal condition, is generalised to a result about nice monads on cocomplete categories. Instances of this general result can be found in (Leinster, 2004), (Berger, 2002) and (Moerdijk-Weiss, 2007b). Aspects of this general theory are then used to show that the composite 2-monads of (Weber, 2005) that describe symmetric and braided ana- logues of the !-operads of (Batanin, 1998), are cartesian 2-monads and their underlying endo-2-functor is familial. Intricately linked to the notion of familial 2-functor is the theory of fibrations in a finitely complete 2-category (Street, 1974) (Street, 1980a), and those aspects of that theory that we require, that weren't discussed in (Weber, 2007), are reviewed here.

Subfields: Algebra and Number Theory, Endocrinology, Diabetes and Metabolism, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/34e91e08db12b24c33d0ec725c826f72bf6986f7

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In Algebra and Number Theory

In Endocrinology, Diabetes and Metabolism

In Mathematical Physics

Citations in the corpus

Polynomial functors and polynomial monads (2009, background): “For example, the set of natural numbers can be characterised as the least solution to the polynomial equation of sets X 1 + X ,”
Polynomials in categories with pullbacks (2011, unknown): “In this context the structure formed by polynomials is a degenerate kind of tricategory, called a 2-bicategory, which roughly speaking is a bicategory whose homs are 2-categories instead of categories.”
Polynomial Functors and Trees (2008, methodology): “In both cases, the interplay between algebra, combinatorics, and homotopy theory follows the same pattern, whose general theory has been worked out by Weber [21]: categories are first defined algebraically, as algebras for a nice monad on the category of graphs; there is a canonical way to distill t”
Higher quasi-categories vs higher Rezk spaces (2012, background): “24 of [42]) show that the functor Nn is fully faithful.”
Decomposition spaces, incidence algebras and Möbius inversion I: Basic theory (2015, background): “A recommended entry point to the theory is Berger–Melliès–Weber [6].”
Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness (2015, background): “The notions of generic and free maps are general notions in category theory, introduced by Weber [34, 35], who extracted the notions from earlier work of Joyal [17]; a recommended entry point to the theory is Berger–Melliès–Weber [3].”
Monads with arities and their associated theories (2011, unknown): “C T ] 1 0 Fe b 20 11”