Questions tagged [monads]
A monad is a functor from a category to itself together with two natural transformations, commonly called μ (the "multiplication") and η (the "unit"), satisfying conditions that make μ monoidal and η an identity for it.
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Proving comonad identities related to internal category
I'm going through the Elephant but I'm having a hard time verifying a given structure satisfies the comonad conditions.
Let $\mathbb{C}$ be some internal category in $\mathcal{S}$ and $\mathbb{D}$ an $...
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Multisets don't have addition
Multisets are containers, also called bags. A multiset is a set that can have repeats:
$$ \{ a, a, a, c, b, c \} $$
Usually when researchers talk about multisets, they use this kind of presentation:
$...
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What is implied by the handling of multisets by N-modules?
Multisets are containers, also called bags. A multiset is a set that can have repeats:
$$\{ a, a, a, c, b, c \}$$
Usually when researchers talk about multisets, they use this kind of presentation:
$$ ...
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Kleisli categories and Eilenberg-Moore categories induced by a pair of subcategories
Let $\mathfrak{A}$ be a full subcategory of $\mathfrak{B}$, where $\mathfrak{A}$ and $\mathfrak{B}$ are both concrete over a given category $\mathfrak{C}$. This means that we can consider the two ...
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What is the categorical construction for a list of nested lists?
In Category Theory, the List functor is the Free Monoid over a given type (i.e. object) T. One can then consider the category of ...
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Induced Comonad of a Monad on the Eilenberg-Moore and the Kleisli category
It is well known that for every monad T, we can consider the Eilenberg-Moore-Category and get an adjunction which induces T, similar for the Kleisli-Category. But since every adjunction induces a ...
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Construction of 2-category of monoidal categories and (lax) monoidal functors as strict algebra category of a 2-monad
As motivation for 2-monads, I would like to understand an explicit construction of the 2-monad $T$ of which derived 2-category $T-\operatorname{Alg}_l$ of algebras as described in Lack's 2-categories ...
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Fundamental Description Of Algebras Over The Monad $PP^{op}:\text{Set}\to\text{Set}$
$
\newcommand{\eps}{\epsilon}
\newcommand{\op}{^\text{op}}
\newcommand{\id}{\text{id}}
\newcommand{\set}{\text{Set}}
\newcommand{\xra}{\xrightarrow}
$Can someone help me to understand how algebras ...
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Understading usage of functor composed with natural transformation in CT definitions
I'm reading about Monads on Wiki, I'm confused about these two defining diagrams.
Where $T$ is the endofunctors $\mu: T^2 \to T$ and $\eta: 1_C \to T$. So, my question is, why is the two arrow in ...
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Does the monad on V associated to an V-operad preserve reflexive coequalizers for a closed monoidal category V?
This might be a very basic question, but a non-symmetric operad $P$ in the symmetric closed monoidal category $V$ induces a monad $T_P$ on $V$ which has the same category of algebras as $P$ and which ...
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How to internalize the extension operator of a monad in a Cartesian closed category?
Let $\mathbb C$ be a Cartesian closed category and $(T, \eta, \mu)$ be a monad on $\mathbb C$.
The extension operator $\_^{\sharp} : Hom(X, TY) → Hom(TX, TY)$ is defined by:
$$f^{\sharp} = \mu_{Y}\...
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Is this Comonadic?
Question:
I will first state the question, then do explanation of terminology below.
All categories should be considered as cocomplete stable $\infty$-categories, and colimits, limits etc are the ...
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Normalizer as a functor/monad
I'll motivate my question with a neat observation: let $G$ be a group, and let $\mathcal{P}(G)$ denote the power set of $G$, regarded as a category in the usual way (i.e., as a poset converted into a ...
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Coproducts in Eilenberg–Moore categories
In Category Theory In Context, Proposition 5.6.11 Riehl says that for objects $(A_1,\alpha_1)$ and $(A_2,\alpha_2)$ in the Eilenberg–Moore category of monad $\mathcal{C} \xrightarrow{T} \mathcal{C}$ (...
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When does a monad compose with a co-monad?
We know that two monads compose if there is a distributive law. What is the law that is necessary to compose a monad and a co-monad? When you combine a monad and co-monad, you get a (co)monad that ...
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Algebras for the continuation monad?
Given a monad $(T,\mu,\eta)$, a map $\alpha : TA\to A$ commuting with $\mu$ and $\eta$ is a $T$-algebra.
Given a set $D$, the continuation monad is given by the functor $C:X\mapsto D^{(D^X)}$ (see ...
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What do you need to define a map of monads?
Suppose I have a monad $M_S = \langle S , \eta_S, \mu_S \rangle$ and a monad map given by a natural transformation $\phi: T \rightarrow S$. What are the natural transformations of the resulting monad?...
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Defining multiplication transformation for free monoid monad on monoidal category
I'm learning about monads in Riehl's Category Theory in Context, and after reading an example about the free monoid monad (also known as the list monad to computer scientists) on the monoidal category ...
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Powerset monad on other categories
The powerset monad $\langle \mathrm{Pow}, \eta, \mu \rangle$ is understood on the category $\mathbf{Set}$ of sets and functions to do the following.
$$\mathrm{Pow} \colon \mathbf{Set} \rightarrow \...
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algebras for endofunctors vs monads and free monads
I have a question concerning proposition 10.14 from Steve Awodey's lecture notes (PDF) concerning the relationship between algebras for endofunctors and algebras for monads. Specifically, the ...
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Why are monadic categories over $\mathsf{Set}$ cocomplete?
$\newcommand{\set}{\mathsf{Set}}\newcommand{\T}{\mathcal{T}}$Given any monad $(\T,\eta,\mu)$ over $\set$, it is claimed that the Eilenberg-Moore category of algebras $\set^\T$ is cocomplete. More ...
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What are the inclusion arrows in the coproducts of the category of algebras for a monad?
$\newcommand{\A}{\mathscr{A}}\newcommand{\C}{\mathsf{C}}\newcommand{\T}{\mathcal{T}}\newcommand{\id}{\operatorname{id}}$Riehl, proposition $5.6.11$, from Category Theory in Context:
Suppose $\C$ is a ...
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Characterizing categories of algebras of monads as locally presentable categories
In Accessible Categories: The Foundations of Categorical Model Theory (Makkai, Paré), it is said (Introduction, p. 3):
One sign of the “rightness” of the notion [of locally presentable category] is ...
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Mathematical motivation/intuition for monads
The internet is filled with intuitive explanations of what monads are in the context of programming. This is very frustrating as i don't have programming background and i'm trying to learn about ...
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Can we turn a strong monad into a commutative one?
Let $(\mathcal C, \otimes, e, \alpha:\_\otimes (\_\otimes \_)\to (\_\otimes\_)\otimes \_, \lambda :e\otimes \_ \to 1_\mathcal C, \rho : \_\otimes e\to 1_\mathcal C, \gamma: \_\otimes = \to =\otimes \_)...
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Are finite and finitely presented categories monadic over FinSet and Set?
Finite categories, FC, are those consisting of finite sets of objects, morphisms and equations. Finitely presented categories, FPC, are those that can be generated from finitely many morphisms subject ...
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Is the category of sets and functions Kleisli?
I think that the category of sets and functions, or perhaps just the category of finite sets and functions, is a Kleisli category for some, probably trivial, monad. Is this so?
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Bimodules over a categorical monad
In category theory a monad consists in an endofunctor $M\colon \mathcal{C}\rightarrow \mathcal{C}$ together with natural transformations $\mu \colon M \circ M \Rightarrow M$ and $\eta \colon Id_{\...
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Prove that all monads are functors
It is often written that all monads are functors, but it is quite hard to find an actual mathematical proof of it.
A functor is defined as a higher level type defining the ...
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Coalgebra after counit for a monad given by adjunction
Let $L\dashv R$ be an adjunction and $LR$ the associated comonad, with comultiplication $L\eta R\colon LR\to LRLR$ and counit $\varepsilon\colon\mathrm{id}\to LR$.
A coalgebra for this comonad is a ...
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When are presheaves models of multi-sorted Lawvere theories?
As I understand it there is a correspondence between finitary monads and single-sorted (ordinary) Lawvere theories. My first question is, for a monoid $M$ in Set, when will it be the case that the ...
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Typing context as a monad for multicategories
In some ways simple type theory matches up nicer with multicategories than with categories.
A hom
$$ f \colon o_1 ; \ldots o_n \rightarrow o'$$
Matches up nicely with a well typed term
$$ x_1 \colon ...
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Internal monomorphism in the category of functors
The codensity monad can be seen as a sort of endomorphism monoid.
The internal hom or residual with respect to functor composition as the monoid being basically kan extension.
In Set
$$(G/F)(A) = \...
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When is the image of a $T$-algebra under $T$ again a $T$-algebra?
Let $T:\mathcal{C}\to\mathcal{C}$ be a monad with unit $\eta:1_\mathcal{C}\Rightarrow T$ and multiplication $\mu:T\circ T\Rightarrow T$, and let $(A,\alpha)$ be a $T$-algebra.
When is $\big(T(A),T(\...
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Cocommutative bimonads: Why does this diagram commute?
1. Definitions
Let $(C, \otimes,I, a, l,r,c)$ be a monoidal category with braiding $c:\otimes \rightarrow\otimes ^{op}$. Let $(S,\mu,\eta,\tau,\theta)$ be a bimonad on $C$.
Following Turaev and ...
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On the multiplication of the symmetric algebra monad
In their paper "Generators and bases for algebras over a monad" (https://arxiv.org/pdf/2010.10223.pdf) Zetzsche, Silva and Sammartino introduce the Symmetric algebra monad. It is an ...
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The functor that maps a (co-)monad to its (co-)Eilenberg-Moore category
I have noticed that the function that maps a monad $T : C \to C$ to the Eilenberg-Moore category $C^T$ can easily be extended into a functor $E_C$ from the category of monads $\textbf{Mnd}_C$ to $\...
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From monads to comonads by the calculus of mates
If a comonad $D$ is left adjoint to an endofunctor $T$, then $T$ can be made into a monad: its unit and multiplication are given respectively by the mates of the counit and comultiplication of $D$.
...
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(co)algebras of the adjunction between presheaves and bundles
For a topological space $X$ and the lattice of open sets $LX$, there is an extension of the "inclusion" (don't know what to call it) functor $F:LX\to\text{Top}/X$ along the Yoneda embedding $...
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What is an algebra for an algebraic theory?
Definition 1.1 of Algebraic Theories says an algebraic theory is a small category $\mathcal{T}$ with finite products, and an algebra for theory $\mathcal{T}$ is a product preserving functor $A$ from $\...
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If monads come from adjunctions what do graded monads come from?
Basically can you generalize adjunction to "graded adjunction?"
I have a hunch you would end up with something similar to some linear logic stuff. My thoughts are you can use monads for ...
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When does transporting a monad along an adjunction "preserve" its category of modules?
If I have an adjunction $F \dashv G$ where $F \colon C \to D$, and $(N, m, u)$ is a monad on $D$, then I can define a monad $\widetilde{N}$ on $C$ via
\begin{align}
(GNF, \widetilde{m} = G m_F \...
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What are the functors that can be approximated by a monadic functor?
Let $\newcommand{\Fc}{\mathbf{Func}}\Fc$ be the category whose objects are functors between small categories and whose morphisms are commutative squares (the commutativity is given by a natural ...
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Modules over monoids vs algebra over monads
I read somewhere that the construction of algebras over monads is motivated by/ similar to the construction of modules over monoids, but I'm having difficulty seeing this.
I see that a monad "...
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What makes every strong monad on a certain category be a monoidal functor?
A concept named Monad is used a lot in functional programming. And in spite their definition is not completely same with the definition of monad in category theory, as I know, Monad on a programming ...
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From Monad construction of Instantons to ADHM data
In reference to Atiyah's book - "Geometry of Yang-Mills Fields" (1979).
In chapter 5, section 3, he describes how the monad construction for $Sp(n)$ potentials can be interpreted in terms of ...
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On the monadicity of Cat
I heard that the category of small categories is monadic over the category of small graphs, the monad being the free-graph functor. I interpreted that statement as: the Eilenberg-Moore category of the ...
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Definition of strict 2-monad on Cat
The category $\text{Cat}$ can be thought of as a $2$-category. I was hoping somebody could help by telling me the explicit definition of a strict 2-monad $(T, \eta, \mu)$ on $\text{Cat}.$ In ...
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Category of Monads on $PreOrd$
In a recent paper by Adamek ("Finitary Monads on the Category of Posets", see it on Arxiv) the author introduces the category of finitary monads on $Pos$ and analyzes it in great detail. ...
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Effectus theory and the Giry monad
Is the Kleisli category of the Giry monad a monoidal effectus with copiers, in the sense of Definition 70 from An Introduction to Effectus Theory ? The fact that this category is an effectus is ...
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