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.

253 questions
Filter by
Sorted by
Tagged with
1 vote
64 views

1 vote
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: ...
33 views

• 16.8k
157 views

• 1,281
1 vote
29 views

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: $$... • 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 ... • 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 ... • 3,309 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 ... 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 ... • 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 ... • 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 ... 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}\...
• 1,548
1 vote
61 views

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 ...
75 views

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 ...
1 vote
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 $\mathcal{C} \xrightarrow{T} \mathcal{C}$ (...
• 1,659
1 vote
66 views

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 ...
• 323
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 ...
• 2,730
1 vote
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?...
• 323
131 views

• 439
1 vote
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 \...
• 323
1 vote
165 views

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 ...
• 928
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 ...
• 42.7k
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 ...
• 42.7k
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 ...
• 2,061
802 views

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 ...
65 views

• 2,109
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 ...
• 121
1 vote
98 views

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 ...
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 ...