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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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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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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$T(X) = (S \to X\times S)$ $\mbox{fmap}:=\lambda f:A\to B. \lambda q\in T(A). \lambda s\in S. \langle(f\circ pr_1 \circ q)(s),(pr_2 \circ q)(s)\rangle$ I defined the following functions: $\eta_a := \... 1answer 61 views Why the term "equivalence" in Manes' Theorem on the ultrafilter monad Manes'Theorem is usually stated by saying that the Eilenberg-Moore category of the Ultrafilter Monad is equivalent to the category of compact Hausdorff spaces. However, it seems to me from this proof ... 0answers 24 views What are pullbacks for stochastic maps? I want to understand the nature of pullbacks in the category$\mathbb{S}$of finitely supported stochastic maps (this category is also known as the Kleisli category of the finitely supported ... 1answer 36 views Definition of an algebra over a monad by using equalities between natural transformations In the definition of a monad, there are two ways to specify the equations: equalities between natural transformations, or equalities between morphisms as is done there on Wikipedia. In the usual ... 1answer 88 views Eilenberg-Moore Category We call an object$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$is ... 1answer 72 views Definition of the comparison functor in category theory I have a problem in showing that the free-forgetful adjunction$F\dashv U: \mathbf{Mon} \to \mathbf {Set}$(call$\eta$the unity and$\varepsilon$the counity) is monadic, and it seems that the ... 0answers 57 views Examples of transfinite constructions in algebra The nlab has a page on transfinite construction of free algebras on the existence of a free monad for a (pointed) endofunctor (accessible in a locally presentable category). My question is if the ... 0answers 61 views A peculiar pullback diagram involving the Kleisli category In Exercise (ECMP) (p.91-92) of the book Toposes, Triples, and Theories, the reader is asked to prove that the following diagram is a pullback, where$T$is a monad on$\mathcal{C}$,$\mathcal{K}=\...
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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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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 $\mathbf{C}^{\mathbf{C}}... 0answers 69 views Reference request for fiber functor of covering spaces being monadic Let$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)$of covering ... 1answer 51 views Are there any partial order comonads? A comonad is a triple$(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$is a ... 0answers 57 views Coalgebras of a free-forgetful adjunction An algebra for a monad$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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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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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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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, ...