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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Parallel pair in $\mathsf{cHaus}$ admitting a split coequalizer in $\mathsf{Set}$

I am reading Emily Riehl's book Category Theory in Context. There is a chapter (Chapter 5) devoted on monads and their algebras. One of the exercise (Exercise 5.6.i) asks us to prove the fact that the ...
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A monad is a monoidal monoid in the monoidal category of endofunctors?

In Awodey's book Category Theory, 2nd ed., he remarks in passing that a monad on a category $\mathbf{C}$ is exactly the same thing as a monoidal monoid in the monoidal category $\mathbf{C}^{\mathbf{C}}...
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Reference request for fiber functor of covering spaces being monadic

Let $X$ be a nice topological space (locally path-connected and simply connected). Assume also that $X$ is connected and fix a base point $x_0 \in X$. We consider the category $Cov(X)$ of covering ...
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Are there any partial order comonads?

A comonad is a triple $(F,\mu, \eta)$. $$F: C \rightarrow C$$ F is an endofunctor on some category $C$. $\mu$ is a natural transformation such that, $$\mu: F \rightarrow F \cdot F$$ $\eta$ is a ...
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Coalgebras of a free-forgetful adjunction

An algebra for a monad $T : \mathcal{C} \to \mathcal{C}$ is an arrow $m : Tc \to c$ satisfying some relevant conditions regarding the unit $\eta : 1 \Rightarrow T$ and the multiplication $\mu : T^2 \...
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What is the adjunction that generates the multiset monad on Set?

The multiset monad is a triple $<M, \mu,\eta>$. $M: Set \rightarrow Set$ $M$ sends a set to the set of all multisets on that set. What is the adjunction that generates this monad? I am looking ...
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What is the Eilenberg-Moore Category of the List monad on Set?

The List Monad is defined as a triple $< L , \mu, \eta >$. $L: Set \rightarrow Set$ $L$ takes a set to the set of all lists on that set. $\mu : L \cdot L \rightarrow L$ $\mu$ takes a list of ...
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Pseudomonads and pseudoalgebras

Are there established definitions of pseudomonads and pseudoalgebras on a (strict) 2-category? By pseudo, I mean that unitality and associativity hold up to some coherent isomorphism. If so, are "...
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Graded monoids: What does it mean to say a monoid is graded by another monoid?

The question https://cs.stackexchange.com/questions/129236/creating-a-large-tuple-from-smaller-tuples-via-a-monad-or-applicative/129482#129482 regards the existence of a procedure which uses a monad ...
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Non-evil definition of a Kleisli object in a weak 2-category

Let $t : a \to a$ be a monad in a weak 2-category. According to nLab, the 1-dimensional universal property of a Kleisli object $(f_t : a \to a_t, \lambda : f_t t \to f_t)$ is that for any right $t$-...
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What does the multiset monad functor do to morphisms?

The multiset monad, $(M, \mu, \eta)$, on the category of sets takes a set, $A$, to $M(A)$, the set of every multiset on $A$. That is what $M$ does to the objects of SET, but what does $M$ do to the ...
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Kleisli adjunction in a (weak) 2-category

In the 2-category of 1-categories, each monad $T$ on a category $\mathcal C$ determines a Kleisli category $\mathcal{C}_T$ and the so-called Kleisli adjunction between categories $\mathcal C$ and $\...
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Does the embedding of the Kleisli category of a monad into its Eilenberg-Moore category have a right adjoint?

Let $(T,m,e)$ be a monad on a category $\mathcal{A}$. There is a full and faithful functor $J_T$ from the Kleisli category $\operatorname{Kl}(T)$ of $T$ to its Eilenberg-Moore category $\operatorname{...
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Dualizing 2-categorical results in the context of locally small categories

Monad, comonad and adjunction are 2-categorical notions. Results about them can be dualized as shown in this answer. In the second part the answer, the dualization is successfully applied to the ...
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Creation of limits and diagram chasing

Let T be a monad in X. I want to prove that the forgetful functor G from category of T-algebras on X to X creates limits. I have read that it can be done via "diagram chasing". At this point, I am ...
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1answer
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Two notions of modules over a monad

If a module over a monad $T: C \rightarrow C$ is an object $c \in C$ together with a map $Tc \rightarrow c$ satisfying associativity conditions (as in https://ncatlab.org/nlab/show/algebra+over+a+...
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36 views

Unique extension of a map from $X$ to a map from the free algebra

Suppose we are in a category $\mathcal{C}$ and we have monad on this category, written as $(P,\sigma, \mu)$. We can consider the category of $P$-algebras. Among its elements is are the free algebras, ...
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Understanding the Giry monad

I would like to understand the Giry monad, which is used to reason about probability in category theory. The issue is that I'm hitting a stumbling block understanding monads in general, in the ...
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How to dualize a theorem by Eilenberg and Moore about monad, comonad and adjunction?

Eilenberg and Moore have shown that given a monad $L$, if $L$ has right adjoint $R$, then $R$ is a comonad. I see how to dualize this result to obtain the following theorem: given a comonad $R$, if $...
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Adjunction between a comonad and a monad

Although this looks like elementary, I have trouble understanding the proof of Theorem 3.1 at page 7 of this paper. As hypotheses we are given a comonad $D$, a monad $T$ and an adjunction $D \dashv T$....
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Constructing a new category from a monad

Given a monad $T: \mathbf{C} \to \mathbf{C}$ with natural transformations $\eta: id \Rightarrow T$ and $\mu: T^2 \Rightarrow T$, I want to construct a new category $\mathbf{C}^T$ where objects are the ...
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Is the monad connection from functional programming to math useful in programming?

Haskell, for example, won't have mutable states, which makes reasoning about the program much easier than object oriented languages. Due to its inhability to change state, monads are heavily used in ...
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Is direct limit created by the forgetful functor from an Eilenberg-Moore algebra

Proposition. Let $T$ be a monad on $C$ and consider the forgetful functor $$ R^T \colon C^T \to C $$ from the category of Eilenberg-Moore algebras to $C$. This functor creates limits; creates ...
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Group freely generated by monoid

There are several ways to define the group freely generated by a monoid, all of which (necessarily) produce isomorphic groups. One way starts with a presentation of the monoid, and simply reinterprets ...
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69 views

Right adjoint of a monad is a comonad

I'm having trouble proving the following statement: If a monad $T$ has a right adjoint $K$, then $K$ is a comonad and the categories of $T$-algebras and $K$-coalgebras are isomorphic. So far I've ...
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Algebras of double dual monad

Let $k$ be a field (more genererally a commutative ring) and consider the category of vector spaces (modules). The double dual endofunctor has a structure of monad where the unit is $$ \begin{align}...
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Algebras of monad of polynomials over commutative rings

Consider the endofunctor $$ \begin{align} (-)[x] \colon \operatorname{CRing} &\to \operatorname{CRing} \\ A &\mapsto A[x]. \end{align} $$ As explained here, this defines a monad over the ...
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Intuition behind $T$-algebras

The definition of a $T$-algebra on a monad seems random to me. Can anyone shed some light on it? This is the inuition I have behind monads.
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Using the Yoneda Lemma to construct a left adjoint to the restriction functor $U : C^A \to C^{\operatorname{ob} A}$

I am working through the following exercise of Emily Riehl's Category Theory in Context, Exercise 5.5.v. Generalizing Exercise 5.5.iv, for any small category $J$ and any cocomplete category $C$ the ...
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55 views

Do monads and comonads with the same functor always compose?

Given a functor $F$, a monad $M=(F, \mu,\eta)$ and a comonad $N=(F,\delta, \xi)$, do $M, N$ always compose to form a bimonad?
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Type formation rules for a monad

In type theory there are type forming rules, e.g., for product: judging $$ \Gamma \vdash A : \text{Type} \ \ \Gamma \vdash B : \text{Type} $$ allows us to judge $$ \Gamma \vdash A \times B : \text{...
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167 views

The free monad over an endofunctor on $\mathcal{Set}$

Let $F$ be an endofunctor on the category $\mathbf{Set}$. How can I construct the free monad over $F$? Can this construction be generalized to other categories than $\mathbf{Set}$?
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Is there a non-trivial monad, which is an equivalence of categories

I assume a category $C$ and a monad $M : C → C$. Then I assume that $M$ is a (possibly adjoint) equivalence of categories. Does this mean that there exists a natural isomorphism $M \simeq id_C$? I ...
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1answer
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Verifying that an adjunction induces a monad structure

I am trying to prove that the monad induced by an adjunction is indeed a monad. Recall that for a given adjunction $F : \mathbb{C} \to \mathbb{D}$ and $G : \mathbb{D} \to \mathbb{C}$ with $F$ left ...
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Recover the monoidal structure on $\mathbb{Ab}$ from the monad over Set

The category of abelian groups $\mathbb{Ab}$ has a monoidal (closed) structure $(\otimes, \mathbb{Z})$. Moreover, it is monadic over the category of sets via the free abelian group monad $$\mathbb{Z}[\...
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1answer
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Monads which preserve equalizers

I'm looking for examples of monads that preserve equalizers, and an understanding of sufficient conditions that ensure this. This is a situation where the advice given at the end of this answer fails ...
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1answer
51 views

Not all Monads are Idempotent, a Cautionary Tale on Natural Transformations

I was scratching my head over the following for almost an hour today, and since I don't have a blog to share the resolution on, I'll post it here. Let $\mathbb{T} = (T,\eta,\mu)$ be a monad on a ...
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Iteration operator and fixed points

I am trying to understand the article "Complete Axioms for Categorical Fixed-point Operators" where they introduce what an iteration operator is. This is basically, a family of functions $$ (\cdot)^\...
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Does a free algebra over a nontrivial monad have a well-defined dimension?

Let $(T,\mu,\eta)$ be a nontrivial monad on $\mathbf{Set}$. By nontrivial here I mean there is $X$ with $|T(X)|>1$. Suppose that $TX \cong TY$ as $T$-algebras (both with the usual free $T$-algebra ...
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When does monad structure on $TS$ come from a distributive law?

There is a number of equivalent ways to define a distributive law between monads $(S, ν, ϑ)$ and $(T, μ, η)$. From the elementary defintion as a compatible transformation $λ : ST → TS$, to slicker ...
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1answer
67 views

quasi-inverse of comparison functor

According to nLab(https://ncatlab.org/nlab/show/monadic+functor), if a functor $U:\mathcal{D}\to\mathcal{C}$ is monadic, the comparison functor induces an equivalence of categories $K^\mathbb{T}:\...
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Why are the two natural transformations in the definition of monad called the unity and multiplication?

Categories for the Working Mathematician says Definition. A monad $T= \langle T, \eta, \mu\rangle $ in a category $X$ consists of a functor $T: X \to X$ and two natural transformations $$...
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1answer
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How can the two diagrams in definition of monad be written as identity equations?

Categories for the Working Mathematician says Definition. A monad $T= \langle T, \eta, \mu\rangle $ in a category $X$ consists of a functor $T: X \to X$ and two natural ...
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1answer
55 views

What does the multiplication operation of a monad mean?

In the free monoid monad $(T, \eta, \mu)$ in category $Set$: $T: Set\to Set$ is a endofunctor, is the composition of the free monoid functor $List:Set→Mon$ and the forgetful functor $U:Mon→Set$. $TA$...
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Given $\eta : I_X \Rightarrow T$, is $\eta$ just $T \circ$?

Categories for the Working Mathematician says Definition. A monad $T= \langle T, \eta, \mu\rangle $ in a category $X$ consists of a functor $T: X \to X$ and two natural ...
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What are the category and monoids given rise to by a monad?

https://en.wikipedia.org/wiki/Monad_(functional_programming)#Definition Given any well-defined, basic types T, U, a monad consists of three parts: A type constructor $M$ that builds up a ...
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Coproduct and a natural transformation

What does it mean in the Example below that $\mu_X$ merges the two copies of $1$ to a single copy; can this be elaborated and explained a little bit further?
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How are two monads with isomorphic Eilenberg-Moore categories related?

If $T$ and $S$ are two monads on the category of sets $Set$ with isomorphic algebra categories, i.e. Eilenberg-Moore categories (see https://ncatlab.org/nlab/show/Eilenberg-Moore+category), in what ...
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Exception monad categorically

I have just started to `play around' with monads and I have a question on the exceptions monad (hopefully if I understand this, it will help me with other monads too). I seem to always find examples ...
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Distributive law for $\mathbb{C}$-Modules monad and $\mathbb{N}$-Modules monad

I recently asked a question and it contains a reference to the monads $\mathcal{M}_N$ and $ \mathcal{M}_C$, which are the multiset and $\mathcal{C}$-module monad, respectively. I want to define a ...