Related papers: Connected monads weakly preserve products

Graduated locally finitely presentable categories are introduced, examples include categories of sets, vector spaces, posets, presheaves and Boolean algebras. A finitary functor between graduated locally finitely presentable categories is…

Suppose that f is a projective birational morphism with at most one-dimensional fibres between d-dimensional varieties X and Y, satisfying ${\bf R}f_* \mathcal{O}_X = \mathcal{O}_Y$. Consider the locus L in Y over which f is not an…

Let T be a triangulated category with coproducts, C the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [Adams71]: All contravariant homological functors C --> Ab are the…

Let $G$ be the Grassmannian $G(d,n)$, let $X$ and $Y$ be complete irreducible varieties, and let $X\rightarrow G$ and $Y\rightarrow G$ be morphisms. Hansen proved that $X \times_G Y$ is connected when $codim f(X) + codim g(Y) < n$. We show…

The setting is the representation theory of a simply connected, semisimple algebraic group over a field of positive characteristic. There is a natural transformation from the wall-crossing functor to the identity functor. The kernel of this…

A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act itself has a finite presentation; it is weakly right coherent if every finitely generated right ideal of $S$ has a finite…

The category Rel(Set) of sets and relations can be described as a category of spans and as the Kleisli category for the powerset monad. A set-functor can be lifted to a functor on Rel(Set) iff it preserves weak pullbacks. We show that these…

Let $F$ be the category with the set of objects $\bf N$ and morphisms being the functions between the standard finite sets of the corresponding cardinalities. Let $Jf:F\rightarrow Sets$ be the obvious functor from this category to the…

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 $G$…

After a review of the concept of "monad with arities" we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere's algebraic…

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…

A fully invarient congruence relations on the free algebra on a given type induces a variety of the given type. In contrast, a congruence relation of the free algebra provides algebra of that type. This algebra is given by a so-called…

Many algorithms in numerical analysis are affine equivariant: they are immune to changes of affine coordinates. This is because those algorithms are defined using affine invariant constructions. There is, however, a crucial ingredient…

We consider the group $\mathfrak{X}(G)$ obtained from $G\ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. Deceptively complicated finitely presented groups arise from…

We investigate correspondence functors, namely the functors from the category of finite sets and correspondences to the category of $k$-modules, where $k$ is a commutative ring.They have various specific properties which do not hold for…

This paper contains results from two areas -- formal theory of Kan extensions and concrete categories. The contribution to the former topic is based on the extension of the concept of Kan extension to the cones and we prove that limiting…

For every finitary monad $T$ on sets and every endofunctor $F$ on the category of $T$-algebras we introduce the concept of an ffg-Elgot algebra for $F$, that is, an algebra admitting coherent solutions for finite systems of recursive…

If $\mathbf{C}$ is a category with pullbacks then there is a bicategory with the same objects as $\mathbf{C}$, spans as morphisms, and maps of spans as 2-morphisms, as shown by Benabou. Fong has developed a theory of "decorated" cospans,…

Birget, Margolis, Meakin and Weil proved that a finitely generated subgroup $K$ of a free group is pure if and only if the transition monoid $M(K)$ of its Stallings automaton is aperiodic. In this paper, we establish further connections…

We give an account of the basic combinatorial structure underlying the notion of type dependency. We do so by considering the category of all dependent sequent calculi, and exhibiting it as the category of algebras for a monad on a presheaf…