Skip to main content

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.

Filter by
Sorted by
Tagged with
1 vote
1 answer
64 views

Cosimplicial resolution associated to a monad

Let $\mathcal{C}$ be a category and $\mathbf{T}$ a monad on $\mathcal{C}$ with functor part $T : \mathcal{C} \to \mathcal{C}$ (I would actually like to consider the case where $\mathcal{C}$ is an $(\...
Brendan Murphy's user avatar
3 votes
1 answer
80 views

When is $\text{Ab}(\mathcal{C}) \to \mathcal{C}$ monadic?

Consider the free abelian group monad $T: \text{Sets} \to \text{Sets}$. Then the category $\text{Ab}$ of abelian groups is equivalent to the category of $T$-algebras, and thus we have a monadic (...
Nick Mertes's user avatar
5 votes
1 answer
138 views

The monad induced by the group multiplication

Let $G$ be any group. We can define its delooping, $\mathbf{B}G$, to be the groupoid with a single object $\bullet$ and $\mathrm{Hom}_{\mathbf BG}(\bullet, \bullet) = G$ with composition given by the ...
Paweł Czyż's user avatar
  • 3,320
3 votes
1 answer
67 views

Notion of commutativity for monads

I have read that in functional programming a monad is commutative if a >>= \x -> b >>= \y -> f x y is equivalent to ...
MB7800's user avatar
  • 83
1 vote
0 answers
21 views

How does the EM category for the multiset monad work?

I am working with the standard multiset monad defined on sets and functions and I want to start looking at algebras for the monad, the EM-category of the multiset monad. I understand that this is ...
Ben Sprott's user avatar
  • 1,281
0 votes
0 answers
26 views

Majority vote EM category

I am working with the multiset monad. I want the EM category to have a structure map that is majority vote. That is to say that the map $f: X \rightarrow M(X)$ takes the set element with the highest ...
Ben Sprott's user avatar
  • 1,281
0 votes
0 answers
29 views

Concrete example on the commutative diagram of comparison functor

I am trying to familiarize myself with the properties of monad. I discovered in nlab the following convoluted picture about the whole story. https://ncatlab.org/nlab/show/comparison+functor It is ...
Y.X.'s user avatar
  • 4,223
0 votes
0 answers
23 views

Forks of algebras over monads in proof of Beck's criterion

Consider a monad $T=(T,\mu,\eta)$ and a $T$-algebra $(X,\xi).$ Then lots of expositions of Beck's monadicity criterion refer to the following fact: there is a parallel pair of morphisms of $T$-...
Sergey Guminov's user avatar
1 vote
0 answers
33 views

Free monads without fixing the base category

Usually, free monads are defined with respect to a fixed base category. By this I mean that given an endofunctor $F: \mathcal{C} \to \mathcal{C}$, the free monad $\bar{F}$ on $F$ is defined using only ...
Monoidoid's user avatar
  • 440
6 votes
0 answers
143 views

Algebraic Structures involving 𝙽𝚊𝙽 (absorbing element).

IEEE 754 floating point numbers contain the concept of 𝙽𝚊𝙽 (not a number), which "dominates" arithmetical operations ($+,-,⋅,÷$ will return ...
Hyperplane's user avatar
  • 11.8k
3 votes
1 answer
115 views

Is there a non-symmetric monoidal monad?

Recall that a monoidal monad on a monoidal category $(\mathcal{C}, \otimes, I)$ is a monad $(M, \eta, \mu)$ on $\mathcal{C}$ such that $M$ is also equipped with the structure of a lax monoidal functor ...
Naïm Favier's user avatar
  • 1,579
0 votes
0 answers
34 views

Proving that a monad $(D,\eta,\mu)$ induces a monad $(T,\lambda, \rho)$.

I'm trying to formalize some of my code within Category Theory, and I ended up having to prove that a certain construction $(T,\lambda,\rho)$ was a monad. Here is the setup. Let $(D,\eta,\mu)$ be a ...
Davi Barreira's user avatar
0 votes
1 answer
45 views

Can a (co)monad composed with itself be a (co)monad again?

I have very little categorical knowledge, my question comes mostly from programming, but I'm interested in the categorical solution of this problem. Let's assume we have a monad $(T, \eta, \mu)$ Is $T^...
1 vote
0 answers
39 views

If $\mathsf{C}$ is a cocomplete category and $\mathsf{I}\to\mathsf{J}$ is a functor, when is $\mathsf{C}^\mathsf{J}\to\mathsf{C}^\mathsf{I}$ monadic?

$\def\C{\mathsf{C}} \def\res{\operatorname{res}} \def\I{\mathsf{I}} \def\J{\mathsf{J}} \def\A{\mathsf{A}} \def\colim{\mathop{\operatorname{colim}}}$In Riehl's Category Theory in Context, we find: ...
Elías Guisado Villalgordo's user avatar
0 votes
1 answer
33 views

Reflexive Tripleability Theorem in Riehl's Category Theory in Context

Exercise 5.5.iii in Riehl's Category Theory in Context consists in proving the following result: Proposition 5.5.8 (RTT). If $U: \mathsf{D} \rightarrow \mathsf{C}$ has a left adjoint and if $\mathsf{...
Elías Guisado Villalgordo's user avatar
2 votes
0 answers
64 views

The Eilenberg-Moore cat of a monad is isomorphic to the presheaves on its Kreisli cat that restrict to representable functors (elementary proof?)

$\def\C{\mathsf{C}}$I am trying to prove by hand the following result from Riehl's Category Theory in Context (I slightly adapted the statement)¹: Exercise 5.2.vii. Let $(T,\eta,\mu)$ be a monad on a ...
Elías Guisado Villalgordo's user avatar
2 votes
0 answers
23 views

Transporting pseudo-monad structure along a pseudo-natural equivalence

Suppose that we have a pseudo-monad $\mathbb{T}$ (with underlying pseudo-functor $T$) on a $2$-category $\mathcal{K}$. Suppose also that there is a pseudo-natural equivalence between $T$ and another ...
User7819's user avatar
  • 1,621
2 votes
0 answers
32 views

Convergence of analytic functors?

Let $\mathcal{C}$ be a symmetric monoidal cocomplete category. Let $F:\mathbb{P}^{op}\rightarrow \mathcal{C}$ be a functor, where $\mathbb{P}$ denotes the permutation category. Such a functor is ...
Margaret's user avatar
  • 1,769
3 votes
0 answers
100 views

Operads and Kontsevich' polynomial functors

I am reading Kontsevich' and Soibelman's paper here. The first chapter is about operads and seems non-standard. I know the following: Let $\mathcal{C}$ be a symmetric monoidal category which is ...
Margaret's user avatar
  • 1,769
3 votes
1 answer
142 views

Monad of possibly infinite lists

It is well-known that if $T\colon\mathrm{Sets}\to\mathrm{Sets}$ is the monad which takes a set $S$ to the set of lists of elements of $S$, i.e., $\bigsqcup_{n\ge0} S^n$, with the monad structure $\mu\...
Kenta S's user avatar
  • 16.8k
6 votes
1 answer
157 views

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 $...
interregno's user avatar
3 votes
1 answer
100 views

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: $...
Ben Sprott's user avatar
  • 1,281
1 vote
0 answers
29 views

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: $$ ...
Ben Sprott's user avatar
  • 1,281
3 votes
0 answers
44 views

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 ...
TheWanderer's user avatar
  • 5,192
6 votes
1 answer
179 views

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 ...
Davi Barreira's user avatar
1 vote
0 answers
63 views

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 ...
jakobbernd's user avatar
1 vote
0 answers
46 views

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 ...
PPP's user avatar
  • 75
3 votes
0 answers
98 views

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 ...
fweth's user avatar
  • 3,584
0 votes
1 answer
33 views

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 ...
Cathartic Encephalopathy's user avatar
3 votes
1 answer
80 views

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}\...
Bob's user avatar
  • 1,548
1 vote
0 answers
61 views

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 ...
Peng Zhou's user avatar
3 votes
1 answer
75 views

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 ...
multi_porpoise's user avatar
1 vote
1 answer
93 views

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 $\mathcal{C} \xrightarrow{T} \mathcal{C}$ (...
Richard Southwell's user avatar
1 vote
0 answers
66 views

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 ...
mathlete42's user avatar
4 votes
1 answer
203 views

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 ...
Couchy's user avatar
  • 2,730
1 vote
1 answer
195 views

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?...
mathlete42's user avatar
3 votes
2 answers
131 views

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 Example 5.1.4.ii, about the free monoid monad (also known as the list monad to computer scientists) on the monoidal ...
Itserpol's user avatar
  • 439
1 vote
0 answers
100 views

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 \...
mathlete42's user avatar
1 vote
0 answers
165 views

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 ...
cemulate's user avatar
  • 928
3 votes
0 answers
82 views

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 ...
FShrike's user avatar
  • 42.7k
4 votes
3 answers
190 views

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 ...
FShrike's user avatar
  • 42.7k
0 votes
0 answers
52 views

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 ...
Dabouliplop's user avatar
  • 2,061
7 votes
3 answers
802 views

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 ...
Carla only proves trivial prop's user avatar
5 votes
0 answers
65 views

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 \_)...
chaotic's user avatar
  • 306
1 vote
0 answers
128 views

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 ...
Ben Sprott's user avatar
  • 1,281
2 votes
1 answer
115 views

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?
Ben Sprott's user avatar
  • 1,281
4 votes
0 answers
89 views

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_{\...
N.B.'s user avatar
  • 2,109
2 votes
2 answers
290 views

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 ...
Flavien's user avatar
  • 121
1 vote
2 answers
98 views

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 ...
Nikio's user avatar
  • 1,050
1 vote
1 answer
83 views

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 ...
Richard Southwell's user avatar

1
2 3 4 5 6