Related papers: Categories of Optics
We give another proof of the fact that there is a dual equivalence between the $\infty$-category of monoidal $\infty$-categories with left adjoint oplax monoidal functors and that with right adjoint lax monoidal functors by constructing a…
We continue our previous modifications of the Baez-Dolan theory of opetopes to modify the Baez-Dolan definition of universality, and thereby the category of opetopic n-categories and lax functors. For the case n=2 we exhibit an equivalence…
It is shown that every concretizable category can be fully embedded into the category of accessible set functors and natural transformations.
Categorical semantics of type theories are often characterized as structure-preserving functors. This is because in category theory both the syntax and the domain of interpretation are uniformly treated as structured categories, so that we…
We introduce a general definition for colored cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from colored cyclic operads to colored operads has both adjoints, each of…
This note is a contribution written for the second volume of the Encyclopedia of mathematical physics. We give an informal introduction to the notions of an $(\infty,n)$-category and $(\infty,n)$-functor, discussing some of the different…
We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To…
This is a mostly expository paper, intended to explain a very natural relationship between two a priori distinct notions appearing in the literature: Generic Vanishing in the context of vanishing theorems and birational geometry, and…
Arboreal categories were introduced as an axiomatic framework for game comonads, which provide a comonadic view on many model-comparison games in logic. We demonstrate the inadequacy of the axiom stating that paths are connected. We then…
In this paper we investigate the construction of bicategories of fractions originally described by D. Pronk: given any bicategory $\mathcal{C}$ together with a suitable class of morphisms $\mathbf{W}$, one can construct a bicategory…
We give a 3-categorical, purely formal argument explaining why on the category of Kleisli algebras for a lax monoidal monad, and dually on the category of Eilenberg-Moore algebras for an oplax monoidal monad, we always have a natural…
Category theory provides a collective description of many arrangements in mathematics, such as topological spaces, Banach spaces and game theory. Within this collective description, the perspective from any individual member of the…
Lenses are an important tool in applied category theory. While individual lenses have been widely used in applications, many of the mathematical properties of the corresponding categories of lenses have remained unknown. In this paper, we…
A comprehensive account of the categorical properties of the category of small categories and asymmetric delta lenses is given in the recent works of Chollet et al. and Di Meglio. An important construction for proving many of these…
We continue the program of structural differential geometry that begins with the notion of a tangent category, an axiomatization of structural aspects of the tangent functor on the category of smooth manifolds. In classical geometry, having…
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,…
In some bicategories, the 1-cells are `morphisms' between the 0-cells, such as functors between categories, but in others they are `objects' over the 0-cells, such as bimodules, spans, distributors, or parametrized spectra. Many…
In this paper, we give precise mathematical form to the idea of a structure whose data and axioms are faithfully represented by a graphical calculus; some prominent examples are operads, polycategories, properads, and PROPs. Building on the…
We show that every braiding on a monoidal bicategory induces a monoidal structure on its bicategory of monoids, such that if the former is sylleptic or symmetric then the latter is braided or symmetric, respectively. This extends a classic…
Categorical aspects of the theory of modules over trusses are studied. Tensor product of modules over trusses is defined and its existence established. In particular, it is shown that bimodules over trusses form a monoidal category. Truss…