Questions tagged [monads]
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255 questions
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When does a lax monad morphism induce a functor between categories of algebras that preserves reflexive coequalisers?
Let $S$ be a monad on a category $\mathbf C$. If $\mathbf C$ is cocomplete, and the category of algebras $\mathbf C^S$ admits reflexive coequalisers, then $\mathbf C^S$ is cocomplete. Thus, it is ...
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Reference request: Algebras over monoid objects in a monoidal category [duplicate]
Looking for a reference for the following easy-to-prove fact:
Say $T$ and $S$ are monads on $\text{Set}$ admitting a monoid homomorphism $\phi : S \to T$ (i.e., a morphism in $\text{Mon}([\text{Set},\...
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Recognition theorem for a functor (bi)category or category of monads?
$\DeclareMathOperator\Mnd{Mnd}$I have a bicategory and want to recognize if it's equivalent to $\Mnd(X)$ the bicategory of monads in some other bicategory $X$. Is there a theorem which does this ...
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What are the algebras of the powerset intersection (oplax) monad?
The assignment $X\mapsto\mathcal{P}(X)$ and $f\mapsto f_*$ (direct images) defines a functor $\mathcal{P}\colon\mathsf{Sets}\to\mathsf{Sets}$.
This functor has a monad structure whose multiplication $\...
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Double category of algebras, lax and colax morphisms of algebras
As explained on this nlab page, for a 2-monad there is a double category of (strict) algebras, horizontal morphisms are lax morphisms and vertical morphisms are colax morphisms of algebras.
However it ...
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Synthetic type theory for virtual double category and its higher categories
For some monad T on a virtual equipment, the paper A unified framework for generalized multicategories by Cruttwell and Shulman (arXiv:0907.2460) proposes the normalized T-monoid.
Another paper, by ...
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Bialgebras in 1/Kl(D)
$1/Kl(D)$ is the comma category of the one element set in the Kleisli category of the distribution monad. There is mention of it here. The objects are probability distributions called states and the ...
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Monoidal categories with canonical left-strengths of monads
It is well-known that every monad on Set is left-strong and that left-strength on a Set-monad is unique.
Is there an abstract characterization of monoidal categories $C$ for which every monad on $C$ ...
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How algebraic can the dual of a topological category be?
(I'm going to try to use definitions from Abstract and Concrete Categories: The Joy of Cats by Adámek, Herrlich, and Strecker, since both of the adjectives in the title of my question seem to have at ...
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Lax morphism classifiers via lax-idempotentification
Let $T$ be a 2-monad on a nice 2-category $\mathcal K$, so that the inclusion $T\text{-}\mathbf{Alg}_s \to T\text{-}\mathbf{Alg}_l$ of the 2-category of (strict) $T$-algebras and strict $T$-algebra ...
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Big list of comonads
The concept of a monad is very well established, and there are very many examples of monads pertaining almost all areas of mathematics.
The dual concept, a comonad, is less popular.
What are examples ...
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Two definitions of a monad on an ∞-category
In the literature on $\infty$-categories (quasi-categories) I found two different definitions of a monad on an $\infty$-category, and I don't understand the relation between them.
The first ...
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Double category of monads and pseudo monad-morphisms
We can construct bicategories of monads in a bicategory $B$, $Mnd_l(B)$/$Mnd_c(B)$ with lax and colax monad-morphisms respectively.
I am failing to find a good notion of pseudo monad-morphisms.
Is ...
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If a monad in a 2-category admits a terminal resolution, does it admit an Eilenberg–Moore object?
Let $T = (t, \mu, \eta)$ be a monad on an object $A$ of a 2-category $\mathcal K$. In The formal theory of monads, Street proves (Theorem 3) that if $l \dashv r$ is the canonical adjunction associated ...
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Adjunctions: algebras of the induced monad VS. coalgebras of the induced comonad
Given an adjunction, we get a monad on one side and a comonad on the other side. What is the connection between their algebra and coalgebra categories? Are they always equivalent?
The example i have ...
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(When) does a morphism of monad induce adjoint functors between categories of algebras?
For monads $S$ and $T$ on a fixed Abelian category $C$, a morphism of monads $\sigma: S\rightarrow T$ induces a functor between Eilenberg-Moore categories $\sigma^*:C^T\rightarrow C^S$. This functor ...
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What is the category of algebras for the finitely supported measures monad?
In this post, I was introduced to the monad of finitely supported measures.
$HX$ is the set of finitely supported measures on $X$, with monad structure defined as for the Giry monad.
I have three ...
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English Reference for the Bénabou-Roubaud theorem
The Bénabou-Roubaud theorem links fibrational descent theory with monadicity. Particularly, it says that given a bifibration satisfying the Beck-Chevalley condition w.r.t some arrow $p$ in the base ...
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Monads for proof relevance in type theory
I am just getting started with homotopy type theory. After watching an introductory lecture, I was attracted to the concept of proof relevance. In my understanding, the core idea here is to elevate ...
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When is the Eilenberg-Moore category of a relative monad between two topoi a topos?
In the non-relative case, we have a theorem, that an Eilenberg-Moore category of algebras of a Monad $T$ on a topos is itself a topos if the monad in question has a right adjoint.
Now how does this ...
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Reciprocity for algebra objects in two algebraic categories
I think this question Compact Hausdorff and C^*-algebra "objects" in a category. shows that there is no reciprocity between categories of algebra-objects of two algebraic categories.
So, ...
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Example of a pseudomonad on Cat whose pseudoalgebras are not the pseudoalgebras for a 2-monad
For every pseudomonad $T$ on the 2-category of (locally small) categories $\mathbf{Cat}$, we can consider the 2-category of $T$-pseudoalgebras and pseudomorphisms $T\text{-PsAlg}_p$, which is equipped ...
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Yetter-Drinfeld modules for Hopf monads
1. Context.
1.1. Classical Yetter-Drinfeld modules.
Let $H$ a bialgebra in a braided monoidal category $\mathcal{C}$. A left-right Yetter-Drinfeld module over $H$ is a triple $(V,\rho,\Delta)$ ...
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Hopf monads in categorical probability theory
1. Context. According to [1], probability monads are arguably the most important concept in categorical probability theory. In [2] Fritz and Perrone argue that "in order for a monad to really ...
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Finitely presentable objects in the categories of algebras of $\infty$-algebraic theories
By default, all terms are understood in the infinity sense (“category” means “$(\infty, 1)$-category”, etc.).
An object $A$ in a category is said to be finitely presentable (or compact) if the functor ...
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Algebraically-free monadicity theorem
The monadicity theorem characterises when a functor $u : \mathbf B \to \mathbf E$ is the forgetful functor from the category of algebras for some monad on $\mathbf E$ (up to an equivalence over $\...
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Beck's original formulation of the precise tripleability theorem. Reference when considering reflexive pairs?
Thanks to MO's user Varkor, we have access to Beck's original untitled manuscript where Beck first stated his precise tripleability theorem. Up to terminological isomorphism, the PTT as stated in p. 3 ...
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Higher descent cohomology
Descent cohomology for a comonad is defined at degrees 0 and 1 by Mesablishvili in his paper "On Descent Cohomology" (as well as by many other authors in many other contexts). For a comonad $\bot$ on ...
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When is a container a monad?
The category of polynomial functors on Set is equivalent to the category of containers.
We have a prescription for when a container is a comonad.
There are a few other questions that come to mind. ...
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Are there cartesian closed monads that also preserve the closed structure of the CCC
When I look for cartesian closed monads, I only find monads where the endofunctor preserves the cartesian structure of a cartesian closed category
$$
\operatorname T\ (a \times b) = (\operatorname T\ ...
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Nontrivial example of when monadic functors don't compose
It is well-known that the composite of monadic functors $U: C \to C'$ and $U': C' \to C''$ need not be monadic. One standard example is the forgetful functor $\mathrm{Cat} \to \mathrm{RefGph}$ from ...
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What should be required from a model category so that the category of algebraic objects in it has the natural model structure?
I have two reference questions
What should be required of a category with finite products so that a (multi-sorted, finitary) Lawvere theory induces a monadic adjunction on it? This should be ...
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In what algebraic categories do finitely presentable objects form a dense cogenerator?
For each $C$ locally finitely presentable category, the full subcategory of finitely presentable objects $C_{fp}$ is a dense generator, i.e. the natural functor $C \to \mathrm{PSh}(C_{fp})$ is a full ...
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Quantum scattering experiments: C-modules, N-modules and their monads
I am working on a theory of particle physics where we use monads. I have a few conjectures that I need to check.
The category of $\mathbb{C}$-modules is monadic over set
The category of $\mathbb{N}$...
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Distributive law of the non-empty list comonad over the non-empty list monad
Preliminaries
A monad is a triple $(M, \eta, \mu)$, with $M$ a functor, $\eta : \mathit{id} \Rightarrow M$ and $\mu : M^2 \Rightarrow M$ two natural transformations such that the following diagrams ...
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Is there a canonical product on the category of monads on Set?
I would like to know if there is a partial monoidal product on the category of monads on Set. I want this partial monoidal product to "handle" monad composition which we understand exists ...
CommunityBot
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What are the necessary requirements to make this composite monad rewrite work?
It is well known that if you want to take two monads and compose them and get a third monad, you need a distributive law. Let us suppose we have this. So, we have two monads
$$\mathcal{M}$$
And
$$\...
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Kleisli adjunction of the distribution monad
Let $\langle D , \mu, \eta \rangle$ be the distribution monad on $Set$ and let $Kl(D)$ be the Kleisli category on the distribution monad. I am interested in the adjunction between $Kl(D)$ and $Set$, ...
CommunityBot
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Can application in untyped lambda calculus be seen as the uncurried unit of some monad?
Simply typed lambda calculus in one type variable in a Cartesian closed category $\mathbf{C}$ can be interpreted as a family of Cartesian closed functors (described below, do they have a name?) from ...
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Conditions such that split coequalizers are a symmetric notion
Consider the notion of a split coequalizer (see the nLab for the definition). Note that the definition seems to be non-symmetric. Are there any conditions on the ambient category such that it becomes ...
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Is lambda calculus polymorphism a type of generalized monad?
Let $\mathbf{C}$ be a Cartesian closed category. Then simply typed lambda calculus in $\mathbf{C}$ in one type variable can be interpreted as a category $\mathbf{STLC}_{\mathbf C}$ where the objects ...
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Eilenberg-Moore category as a 2-dimensional limit
$\require{AMScd}$Given an endofunctor $F : C\to C$, its category of algebras is the inserter of $F$ and the identity functor. This means that there is a square
$$\begin{CD}
Alg(F) @>j>> C \\
...
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Coherence laws when composing 2-monads
To have the composition of two monads be a monad itself, we need a
distributive law natural transformation satisfying certain coherence
laws.
I'm interested in the strict 2-monad case, i.e. a strict ...
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Algebras for products or limits of monads
If a category $C$ has limits of a certain type, then the category of monads on $C$ has the same type of limits, and these limits are computed "levelwise" (i.e. are preserved by the forgetful ...
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Morphism of pseudomonads induces pullback functors between pseudoalgebras
If $S$ and $T$ are monads on a category $C$, and $\lambda:S\to T$ is a morphism of monads, it is well-known that there is a functor $\lambda^*:C^T\to C^S$ which assigns to the $T$-algebra $(A,a:TA\to ...
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Intuitive meaning of Giry monad's $\sigma$-algebra
The Giry monad $G : \textbf{Meas} \to \textbf{Meas}$ maps a measurable space $(X, \mathcal{F})$ to its set of probability measures. The $\sigma$-algebra of $G(X, \mathcal{F})$ is the smallest algebra ...
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What functors between categories of algebras are induced by morphisms of monads on $\mathrm{Set}$?
Let $M, N$ be monads of rank $\lambda$, where $\lambda$ is a regular cardinal (I'm primarily interested in the case of finitary monads). Is there a known characterization of functors $\mathrm{Alg}~N \...
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Limits and colimits in the category of algebraic theories
Let $\mathrm{AlgTh}$ be the category of one-sorted algebraic theories (synonym: Lawvere theories; morphisms are functors that are identical on objects and strictly preserve products). It is known that ...
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2-completeness of stacks
I am looking for a reference which discusses the 2-categorical properties of the 2-category $St(C,J)$ of stacks and the stackification $\dashv$ inclusion of presheaves 2-adjunction.
My stacks are ...
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Commuting filtered colimits & finite limits in infinitary theories
Filtered colimits & finite limits commute in categories that are finitary monadic over Set (i.e. algebras of finitary algebraic theories). Results such as Fred Linton's result that if categories ...
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