Search Results

Therefore, a functor between categories of algebras for two monads (regardless of rank) corresponds to a monad morphism if and only if the functor commutes with the forgetful functors. … In fact, the corresponding statement is true for monads on any category, not just $\mathrm{Set}$. …

Yes, the Eilenberg–Moore object for a monad $T$ can be presented in terms of two equifiers of the inserter $\mathbf{Ins}(T, 1)$. Denoting by $\phi \colon TU \Rightarrow U$, we equify $1_U$ and $\phi \ …

The double category of pseudoalgebras, lax and colax morphisms is defined in §5.4 of Grandis and Paré's Multiple categories of generalised quintets. As far as I'm aware, it is the only reference for t …

Let $T$ be a monad on an accessible category (i.e. an $\mathbf{Ind}$-category). If the underlying endofunctor of $T$ is finitary (i.e. preserves filtered colimits), then the Eilenberg–Moore category o …

By general properties of lax idempotent 2-monads, the action of the 2-functor $\mathbf T$ on functors $f \colon A \to B$ is given by $\mathrm{Lan}_{\eta_A} (\eta_B \circ f)$. … Consequently, since 2-monads preserve monads, if $T : A \to A$ has the structure of a monad on $A$, then $\mathbf T(T) \colon \mathbf T(A) \to \mathbf T(A)$ has the structure of a $\Phi$-cocontinuous monad …

There is a full subcategory $\text{IdemMnd}(C) \hookrightarrow \text{Mnd}(C)$ of the category of monads on $C$ spanned by the idempotent monads. …

In The Formal Theory of Monads, Street proves that a 2-category $\mathscr C$ admits the construction of algebras when the inclusion $\mathscr C \to \mathbf{Mnd}(\mathscr C)$ has a right adjoint. … It is shown in The Formal Theory of Monads II that the constructions agree on objects and 1-cells, but they do not express any further relation between the two. …

An answer to the second question has been provided in the recent preprint What is the universal property of the 2-category of monads? by Lack–Miranda. …

Typically, limits of multisorted algebraic theories (by which I mean a pair of a set $S$ and an $S$-sorted algebraic theory $\mathbb F(S) \to L$) are most easily described in terms of their presentati …

These are known as pseudo-distributive laws and are the most common notion of distributive law of 2-dimensional monads, even when both 2-dimensional monads in question are strict (i.e. 2-monads rather …

This question is inspired by Characterization of functors whose right adjoint is monadic?. Let $F : \mathbf C \rightleftarrows \mathbf D : U$ be an adjunction, and suppose that we want to establish wh …

This is a special case of the general construction of cocompletions that preserve existing colimits. The general statement can be found as Theorem 6.23 of Kelly's Basic Concepts of Enriched Category T …

I believe the original reference for this fact is Theorem 2 of Maranda's 1966 On Fundamental Constructions and Adjoint Functors, although the terminology is not modern. A more readable reference is Th …

For the purposes of this question, let us assume that the categories involved are locally presentable, and the monads are accessible (i.e. preserve $\kappa$-filtered colimits for some cardinal $\kappa$ … Now suppose we have a lax morphism of monads $(F, \varphi) : (\mathbf C, S) \to (\mathbf D, T)$, comprising a functor $F \colon \mathbf C \to \mathbf D$ and a natural transformation $\varphi : TF \Rightarrow …

The categories of $\mathbf{Rex}(\mathbb C)$-nervous monads and $\mathbf{Rex}(\mathbb C)$-theories are equivalent by Theorem 17 of ibid. … This essentially gives a classification of finitary and sifted colimit-preserving monads on presheaf categories (at least on small categories), which was suggested by Simon Henry. …