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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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Origin of η and μ as names for monad natural transformations

I swear I once stumbled across a page explaining the origin of η and μ in the context of monads. Or at the very least, an easy way to remember which is which. But I can't find it. Is there a historic ...
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Is the finite multiset monad a finitary monad?

Here is the definition of finitary monad. The finite multiset monad has a functor that maps the category of sets to itself. It does this by taking a set to the set of all finite multisets on that ...
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Is “initial Partial Fragment” of Lists a Comonad?

Lists can be captured as Monads, where the product axiom is given as concatenation. Given a list $l = Concat(l_1, l_2)$, where $Concat$ is the concatenation of two lists, we say that $l_1$ is an ...
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Construction of a monad from an operad is in the CGWH category

If $\mathcal{C}$ is an operad and if $X\in\mathcal{J}$ then $CX\in\mathcal{J}$, where $\mathcal{J}$ is the category of compactly generated weakly Hausdorff spaces well-based. I'm studying the ...
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Monad and identity element

There are 2 natural transformation are introduced in the monad definition: $$ \mu: T^2 \to T $$ $$ \eta: I \to T $$ The following conditions have to be satisfied: $$ \mu \circ \mu T = \mu \circ T \mu$...
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Interpretation of the product axiom for the Giry Monad

The product axiom for the Giry monad is given as follows: $$ \mu_{X}: P(P(X)) \to P(X) $$ given by $$ \mu_X (M)(A) := \int_{P(X)} \tau(A) M(d\tau). \,. $$ $P(X)$ is equipped with the weakest ...
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Which functors between varieties of algebras are monadic?

I was wondering if there's any known result which classifies which functors $F : \mathcal{B} \to \mathcal{C}$, where $\mathcal{B}$ and $\mathcal{C}$ are both varieties of algebras, are monadic. I ...
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The $\mathbb{C}$ vector space monad on the 2-Category of Groupoids

In this post, I am asking about the existence of something called the "Vector Space Monad" on the 2-Category of groupoids (Grpd). In a comment, it was pointed out that the monad should exist due to ...
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Proof verification: Group of units functor is not monadic

I've been fairly confused about this question (which isn't homework or anything, just something I was wondering about for myself), and coming up with different answers: Let $U : \mathbf{Rings} \to \...
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Categories in which every monomorphism is regular / effectivie?

I'm interested in conditions on a category of modules of a monad on a topos satisfies the property that Every monomorphism is the equalizer of its cokernel pair. Every topos itself satisfies this ...
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Transformation from the List monad to the Bag monad on the 2-Category of Groupoids

Kock has shown that the Bag monad and the List monad are polynomial on the 2-Category of Groupoids. He even suggests there is a transformation between them (I think, in section 3.10 Examples) going ...
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What are the special properties of adjunctions that generate polynomial monads

The subject of polynomial monads is well trodden. We know that every monad is generated by an adjunction. What are the special properties of any adjunction that generates a polynomial monad?
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Morita theory for algebras for a monad $T$

There are convincing arguments that support the claim that universal algebra is essentially the theory of $\lambda$-accessible monads $T$ over Set. Now, given two equivalent categories of algebras ...
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A pattern of adjointness proof

Let $\mathbf{Fcd}$ and $\mathbf{Top}$ be categories (the latter's objects are topological spaces but this doesn't matter for the question). The objects of $\mathbf{Fcd}$ are called endofuncoids (what ...
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Examples of monads

Recently, as a final degree project, I have been studying basic category theory. My tutor is very experienced in monads and so I am studying them. I have been researching some curious examples (the ...
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Examples of generalisations in which $k$-ring might be, for example, a monad ? What are $\textit{scalars}$, really?

I reflect on scalar nature so my question is very simple: what are $\textit{scalars}$, really ? I read about ground ring There are also generalisations in which k might be, for example, a monad. ...
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Understanding the monadicity of groupoids over splittings

In the paper The shift functor and the comprehensive factorization for internal groupoids by Bourn, the author proves that for a fixed finitely complete category, the category of internal groupoids is ...
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Composition of monadic functors may not be monadic

Let $U\colon \textbf{AbGp}\longrightarrow \textbf{Set}$ be the forgetful functor. By the Crude Monadicity Theorem, it is monadic. Any reflection is monadic, so $I \colon \textbf{tfAbGp}\...
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How can we define the stream monad and what is it's category of algebras?

Streams are infinite lists. Lists can be modeled as monads. So can Streams. Here is a definition of the Stream monad: $F(A) = A^{\mathbb{N}}$ All functions for $\mathbb{N}$ to A. $\mathbb{N}$ ...
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The comparison functor for the adjoint $F\dashv G$ with $G$ fully faithful is an equivalence

Let $\mathcal{D} \underset{G}{\overset{F}{\leftrightarrows}}\mathcal{C}$ with $F\dashv G$ and $G$ fully faithful. Write $T=GF$. $\epsilon$ denotes the counit of the adjunction. The comparison functor $...
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Forgetful functor from category of directed graphs is monadic.

I am trying to prove that the forgetful functor $U\colon \textbf{DGph}\longrightarrow \textbf{Set}$ is monadic. The strategy for that is to show that $\textbf{DGph}$ is equivalent to the category of $...
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Computing a factorization of a monad

Given a monad, $(M, \mu, \eta)$, where $M: C \rightarrow C$ for some category $C$, there is a category of factorizations, $F\cdot G = M$ where $F: X \rightarrow C$, $G: C \rightarrow X$. Though this ...
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What is the string diagram for the Bimonad law

We have string diagrams that succinctly describe the Frobenius law. Is there a string diagram that describes the Bimonad law? Bimonads are (co)monads that have a mixed distributive law. They were ...
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the List-Permutation (C0)Monad

I want to define the List-Permuation (co)monad as follows: $List-Perm = (L, \mu, \eta, \nu, \zeta)$. $$L:Set \rightarrow Set$$ such that, L returns the set of Lists of a given set. $$\mu : L \cdot ...
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A few questions about continuations and the continuation monad

In Philip Wadler's "Monads and composable continuations" (published in Lisp and Symbolic Computation, 1994, doi.org/10.1007/BF01019944) the following continuation monad is defined: $type \hspace{0....
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Some simple operators in the Lambda calculus and their interpretation

In Phillip Wadler's "Monads and composable continuations" (published in Lisp and Symbolic Computation, 1994, doi.org/10.1007/BF01019944) the following rules (amongst others) are listed as being part ...
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Continuation-passing style and the lambda calculus

I am looking for an example of continuation passing style which does not involve any examples from programming languages; in particular, an example which involves a reduction sequence from the lambda ...
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Algebraic properties of enumeration [closed]

An enumeration of a set $A$ is (or can be seen as) a bijection $$ enum : [\alpha] \rightarrow A $$ from an ordinal $\alpha$ to $A$ itself. I'm interested in countable enumertation, i.e. $\alpha$ ...
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Permutation monad

Given a set $S$, say $S= \{a,b\}$, the set of permuations of $S$ is $\{[a,b],[b,a]\}$ (these are supposed to be lists like $ab$ and $ba$). Can we define a monad that captures this? This monad would ...
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free commutative monoid monad

I would like to see a full definition of the free commutative monoid monad. Here is what I have so far. We define it with these parts $FCMonoidMonad = (CM, \mu, \eta)$. $$ CM : Set \rightarrow Set$$...
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Description of the 2-monads for (strict) monoidal categories

The nLab article on 2-monads says: For example, ordinary (non-strict) monoidal categories are the strict algebras for a strict 2-monad $T_{MC}$ on $Cat$, but usually we care about pseudo, lax, and ...
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Difficulty understanding natural transformation notation

I'm trying to fully understand the conditions for a monad as they are written in category theory. Left and right identity are expressed as follows: $$\mu \circ \eta T = \mu \circ T \eta = id_T$$ ...
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Monad/comonad that preserves equalizers

Can someone give a nontrivial example of an endofunctor on Set that is both a Monad and a comonad and also preserves equalizers? It would be great if you could show that there is a mixed distributive ...
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Monads, monoidal categories

I'm interested in learning about monads and their relations to algebraic structures (as a generalization of universal algebra, if I understand well -correct me if not) . In the process of learning ...
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2-monad and 2-operad of monoidal categories, explicit construction

It is well-known that monoids are algebras for the free monoid monad, and can be seen as well as algebras for the associative operad. Less known is the categorified statement: for example, monoidal ...
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Different definitions of monad, again.

In Moggi, Eugenio, Computational lambda-calculus and monads (1989): page 2. A monad over a category $\textbf{C}$ is defined as a triple, $(T, η, µ)$, consisting of the natural transformations $T: \...
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Continuations in mathematics: nice examples?

I wondered whether continuations, used in computer science, occur as natural and interesting mathematical structures, perhaps as algebraic (in the theory of monoids?), model-theoretic or type ...
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Different definitions of monad

A monad is usually defined (for example http://www.cs.cornell.edu/courses/cs6110/2011sp/lectures/lecture37.pdf) as a triple $(F, η, µ)$ where • $F : C → C$ is an endofunctor on a category $C$; • $η \...
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Composition of a monad with itself.

Is there any reason (counterexample) why the composition of a monad with itself should not be a monad again? It appears to me that the distributive law is in this case just the identity (on the ...
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When is the Kleisli category equivalent to the Eilenberg-Moore?

We know that, for any monad $T$, the Kleisli category $\mathcal{C}_T$ embeds into the Eilenberg-Moore category of $T$-algebras $\mathcal{C}^T$ as the full subcategory of free $T$-algebras. In the case ...
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What's the multiplication in the continuation monad?

Given a set $S$, we get an endofunctor $X \mapsto [[X,S],S]$ on $\mathbf{Set}$. This is called the continuation monad for $S$, so I guess that means it's a monad. There's a natural map $$\eta_X : X \...
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Reference request: the category of adjunctions between posets as categories that induce a partiuclar monad

I am interested in the category $A$ of adjunctions that induce a monad $c : C \to C$ where $C$ is a poset. (The description of $A$ is in a previous math.se post.) For a general $C$, of course, $A$ ...
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Conceptual explanation of preservation and reflection of certain coequalizers in Beck's monadicity theorem

Let $F\dashv U$ and let $\mathbb T=(T,\eta,U\varepsilon F)$ be the induced monad on $\mathsf C$. The proof of Beck's monadicity theorem revolves around coequalizers of pairs $$FUFA\substack{Fa\\ \...
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Applications of Monadicity theorems

Crossposted to MO. Having carefully read the proof of Beck's monadicity theorems and some related variations, I'm now hungry for cool applications. For instance, I found these blackboard pictures of ...
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Monad $W_0$ constructed by forgetful functor $\textsf{Mon}$ to $\textsf{Set}$ and its $W_0$-algebra

Let $W_0$ be the monad in $\textsf{Set}$ defined by forgetful functor $\textsf{Mon}$ to $\textsf{Set}$. Show that a $W_0$-algebra is a set $M$ with a string $v_0,v_1,\cdots$ of $n$-ary operations $v_n$...
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Monad as a monoid: intuition

I am studying the definition of a monad from Saunders Mac Lane, Categories for the Working Mathematician, Springer Verlag, 1971. I have some difficulties getting an intuition about the relation to ...
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Elaboration for $μ ∘Tμ = μ ∘ μT$ from a Monad definition

A part of Monad definition is an endofunctor $T$, and a natural transformation $\mu : T²→T$, such that the $\mu ∘T\mu = \mu ∘\mu T$ holds. I struggled hard with both sides of the equation, so here'...
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Does every finitary monad with this propery arise as a free module monad?

Let $T:\mathbf{Set} \rightarrow \mathbf{Set}$ denote a finitary monad such that $T(\emptyset) \cong 1$ and $T(A \sqcup B) \cong T(A) \times T(B)$, naturally in $A$ and $B$. Question. Does $T$ ...
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Details in applying the Barr-Beck monadicity theorem to Tannakian reconstruction

The Barr-Beck monadicity theorem gives necessary and sufficient conditions for a category $\mathcal{C}$ to be equivalent to a category of (co)algebras over a (co)monad. A functor $F:\mathcal{C}\to\...
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Left adjoint to forgetful functor between varieties of algebras

Given algebraic theories $S$ and $T$ for which there is a forgetful functor $U : S_{mod} \to T_{mod}$ (e.g. $U : \textbf{Rng} \to \textbf{Ab}$), it is known that $U$ is monadic, and hence has a left ...