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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### 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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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, ...
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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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### 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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### 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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### 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 ...
### 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 ...