Related papers: Categories enriched over a quantaloid: Isbell adju…

By using the Dold-Kan correspondence we construct a Quillen adjunction between the model categories of non-cocommutative coassociative simplicial and differential graded coalgebras over a field. We restrict to categories of connected…

We compare two possible ways of defining a category of 1-combs, the first intensionally as coend optics and the second extensionally as a quotient by the operational behaviour of 1-combs on lower-order maps. We show that there is a full and…

For a small category K enriched over a suitable monoidal category V, the free completion of K under colimits is the presheaf category [K*,V]. If K is large, its free completion under colimits is the V-category PK of small presheaves on K,…

Even a functor without an adjoint induces a monad, namely, its codensity monad; this is subject only to the existence of certain limits. We clarify the sense in which codensity monads act as substitutes for monads induced by adjunctions. We…

We give an overview of the parts of arXiv:2004.04279 that deal with 2-categories, up to and including adjunction, and explain how the Segal-type approach to 2-categories adopted there is related to the more standard approaches. As an…

We define a notion of category enriched over an oplax monoidal category $V$, extending the usual definition of category enriched over a monoidal category. Even though oplax monoidal structures involve infinitely many functors $V^n\to V$,…

We study codensity monads $T$ induced by (mostly small, mostly dense) full subcategories $A \subset K$. These monads behave quite similarly, we show some connections with the Isbell duality, pro-finite objects and compact spaces. We prove…

Linearly distributive categories were introduced to model the tensor/par fragment of linear logic, without resorting to the use of negation. Linear bicategories are the bicategorical version of linearly distributive categories. Essentially,…

We present the notion of "cyclic double multicategory", as a structure in which to organise multivariable adjunctions and mates. The classic example of a 2-variable adjunction is the hom/tensor/cotensor trio of functors; we generalise this…

We develop a number of basic concepts in the theory of categories internal to an $\infty$-topos. We discuss adjunctions, limits and colimits as well as Kan extensions for internal categories, and we use these results to prove the universal…

We introduce the notion of Kan injectivity in 2-categories and study its properties. For an adequate 2-category $\mathcal{K}$, we show that every set of morphisms $\mathcal{H}$ induces a KZ-pseudomonad on $\mathcal{K}$ whose 2-category of…

Using the language of double categories we generalise a classical result on finite-product-preserving left Kan extensions, by Ad\'amek and Rosick\'y, to one on left Kan extensions that preserve algebraic structures defined by `suitable'…

Let $b$, $b'$ be commutative monoids in a B\'{e}nabou cosmos. Motivated by six-functor formalisms in algebraic geometry, we prove that the category of commutative monoids over $b\otimes b'$ is equivalent to the category of cocontinuous lax…

Given a monad $T$ on $\mathscr{A}$ and a functor $G \colon \mathscr{A} \to \mathscr{B}$, one can construct a monad $G_\#T$ on $\mathscr{B}$ subject to the existence of a certain Kan extension; this is the pushforward of $T$ along $G$. We…

Recently, Dotsenko and Tamaroff have shown that a morphism of $T\longrightarrow S$ of monads over a category $\mathscr C$ satisfies the PBW-property if and only if it makes $S$ into a free right $T$-module. We consider an adjunction…

We prove the theorem stated in the title. More precisely, we show the stronger statement that every symmetric monoidal left adjoint functor between presentably symmetric monoidal infinity-categories is represented by a strong symmetric…

This paper considers some extensions of the notion of filter to the quantale-valued context, including saturated prefilter, $\top$-filter and bounded saturated prefilter. The question is whether these constructions give rise to monads on…

The additivity theorem for derivateurs associated to complicial biWaldhausen categories is proved. Also, to any exact category in the sense of Quillen a K-theory space is associated. This K-theory is shown to satisfy the additivity,…

Real-enriched categories are categories with real numbers as enrichment. Precisely, a real-enriched category is a category enriched over the commutative and unital quantale composed of the unit interval and a continuous t-norm. These notes…

We study the relationship between presheaf constructions and free cocompletions in the context of formal category theory, elucidating the coincidence between the two concepts in familiar settings. We show that, in a virtual equipment…