Related papers: Adjoint functor theorems for lax-idempotent pseudo…

We introduce dicodensity monads: a generalisation of pointwise codensity monads generated by functors to monads generated by mixed-variant bifunctors. Our construction is based on the notion of strong dinaturality (also known as Barr…

Starting from a comonad G on a category A, and a functor L : B -> A with a right adjoint R : A -> B, we will give a parametrization of the functors K from B to the category of all G-coalgebras that factorize throughout L in terms of…

We define $A_{\infty}$-structures -- algebras, coalgebras, modules, and comodules -- in an arbitrary monoidal DG category or bicategory by rewriting their definitions in terms of unbounded twisted complexes. We develop new notions of strong…

Given a symmetric monoidal stable $\infty$-category $\mathcal{C}$ and a left adjoint symmetric monoidal fiber functor to $\operatorname{Mod}_A^{\otimes}$ for some $\mathbb{E}_{\infty}$-ring $A$, one can construct a derived group scheme $G$…

This paper is a contribution towards a two dimensional extension of the basic ideas and results of Janelidze-Galois theory. In the present paper, we give a suitable counterpart notion to that of \textit{absolute admissible Galois structure}…

We develop the theory of relative monads and relative adjunctions in a virtual equipment, extending the theory of monads and adjunctions in a 2-category. The theory of relative comonads and relative coadjunctions follows by duality. While…

Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…

We introduce a functor of functionals which preserve maximum of comonotone functions and addition of constants. This functor is a subfunctor of the functor of order-preserving functionals and contains the idempotent measure functor as…

We show that the restriction functor from oriented factor planar algebras to subfactor planar algebras admits a left adjoint, which we call the free oriented extension functor. We show that for any subfactor planar algebra realized as the…

For our purposes, two functors {\Lambda} and {\Gamma} are said to be respectively left and right adjoints of each other if for any digraphs G and H, there exists a homomorphism of {\Lambda}(G) to H if and only if there exists a homomorphism…

We introduce a bialgebra axiom for a pair $(c,\ell)$ of a colax-monoidal and a lax-monoidal structures on a functor $F\colon \mathscr{M}_1\to \mathscr{M}_2$ between two (strict) symmetric monoidal categories. This axiom can be regarded as a…

This article is the last of the series of articles where we reprove the foundational ideas of abstract six-functor formalisms developed by Liu-Zheng. We prove the theorem of partial adjoints, which is a simplicial technique of encoding…

We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of $\infty$-categories and we use this to prove many classical results about monads in the $\infty$-categorical framework. Amongst other…

We prove a generalization of the fundamental theorem of algebraic K-theory for Verdier-localizing functors by extending the proof for algebraic K-theory of spaces to the realm of stable $\infty$-categories. The formula behaves much better…

We show that contrary to appearances, Multimodal Type Theory (MTT) over a 2-category M can be interpreted in any M-shaped diagram of categories having, and functors preserving, M-sized limits, without the need for extra left adjoints. This…

For a subanalytic Legendrian $\Lambda \subseteq S^{*}M$, we prove that when $\Lambda$ is either swappable or a full Legendrian stop, the microlocalization at infinity $m_\Lambda: \operatorname{Sh}_\Lambda(M) \rightarrow \operatorname{\mu…

The adjoint of an ideal I in a regular local ring R is the R-ideal adj(I):=H^0(Y, I\omega_Y), where f:Y -> Spec(R) is a proper birational map with Y nonsingular and IO_Y invertible, and \omega_f is a canonical relative dualizing sheaf.…

Two adjoint functors can be seen as generalisations of the two functions within a Galois connection. If instead the adjoints are not generalised from functions, but from relations, then analogously the object of study becomes a more general…

We study right quasi-representable differential graded bimodules as quasi-functors between dg-categories. We prove that a quasi-functor has a left adjoint if and only if it is left quasi-representable.

The goal of this paper is to prove coherence results with respect to relational graphs for monoidal monads and comonads, i.e. monads and comonads in a monoidal category such that the endofunctor of the monad or comonad is a monoidal functor…