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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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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Do equivalence relations form a monadic category?

Is the category of equivalence relations and relation-preserving functions a monadic category (i.e. isomorphic to the Eilenberg-Moore category of some monad on ${\bf Set}$)?
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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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Monadic interpretation of closure operators on set systems and binary relations

It is well-known that the category of preorders may be interpreted using lax extensions (see Hoffman et al. "Monoidal Topology" - Example III.1.6.4(1)). Substantially, such an extension ...
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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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Checking that the State Monad is a Monad.

$ T(X) = (S \to X\times S)$ $\mbox{fmap}:=\lambda f:A\to B. \lambda q\in T(A). \lambda s\in S. \langle(f\circ pr_1 \circ q)(s),(pr_2 \circ q)(s)\rangle$ I defined the following functions: $\eta_a := \...
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Why the term "equivalence" in Manes' Theorem on the ultrafilter monad

Manes'Theorem is usually stated by saying that the Eilenberg-Moore category of the Ultrafilter Monad is equivalent to the category of compact Hausdorff spaces. However, it seems to me from this proof ...
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What are pullbacks for stochastic maps?

I want to understand the nature of pullbacks in the category $\mathbb{S}$ of finitely supported stochastic maps (this category is also known as the Kleisli category of the finitely supported ...
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Definition of an algebra over a monad by using equalities between natural transformations

In the definition of a monad, there are two ways to specify the equations: equalities between natural transformations, or equalities between morphisms as is done there on Wikipedia. In the usual ...
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Eilenberg-Moore Category

We call an object $C\in\mathscr{C}$ is $T$-closed if $TC=C$. Suppose that $(T,\eta,\mu)$ is a monad on a poset category $\mathscr{C}$. Prove that the Eilenberg-Moore category $\mathscr{C}^T$ is ...
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Definition of the comparison functor in category theory

I have a problem in showing that the free-forgetful adjunction $F\dashv U: \mathbf{Mon} \to \mathbf {Set}$ (call $\eta$ the unity and $\varepsilon$ the counity) is monadic, and it seems that the ...
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Examples of transfinite constructions in algebra

The nlab has a page on transfinite construction of free algebras on the existence of a free monad for a (pointed) endofunctor (accessible in a locally presentable category). My question is if the ...
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A peculiar pullback diagram involving the Kleisli category

In Exercise (ECMP) (p.91-92) of the book Toposes, Triples, and Theories, the reader is asked to prove that the following diagram is a pullback, where $T$ is a monad on $\mathcal{C}$, $\mathcal{K}=\...
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How can I compute a monad from this adjunction?

What are the steps to computing a Monad from an adjunction? There has to be the following steps: Compute the endofunctor on the base category given the adjunction Compute the product natural ...
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Are topological spaces (and open maps) comonadic over sets (and functions)?

In the neighborhood formulation of topological spaces, the data of a topological space is a pair $(X,N)$ of a set $X$ and a function $N : X \to F X$, where $FX$ denotes the set of filters of subsets ...
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Understanding the proof of Beck's Theorem

I am trying to understand the proof of Theorem 4.19 in this paper, describing when does the comparison functor $\textbf{D}\to\textbf{C}^T$ gives an isomorphism. However, I am not quite sure what the ...
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Free completion for a 2-category with Eilenberg-Moore objects

For a $2$-category $\mathcal{K}$, the free completion of $\mathcal{K}$ under Eilenberg-Moore-objects (EM-objects) is a $2$-category which has $0$-cells the (internal) monads $(A, t, \mu^t, \eta^t)$ in ...
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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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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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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 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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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 ...