# Tagged Questions

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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### For which category (if any) are Lie algebras the algebras of a monad?

I was reading about monads recently, and it came to me that the purpose of the category of algebras of a monad seems to be to switch to a "representation" which is easier for computations. Soon after ...
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### Monads in monoids

This question is almost a duplicate of this one, but not quite. There the person asked about examples and intuition, I am asking about terminology and applications, and I am addressing my question ...
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### What is this structure involving a monad and a comonad?

Let $F$ be a monad on some category $\mathsf{C}$ and $G$ be a comonad on the same category. Assume further that they "commute" (see below): $FG \cong GF$. Then, for lack of a better name, one can ...
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### The left adjoint to the forgetful functor $G\colon\mathsf{Vect}_\mathbb{C}\to\mathsf{Vect}_\mathbb{R}$ and Barr-Beck

Let $G\colon\mathsf{Vect}_\mathbb{C}\to\mathsf{Vect}_\mathbb{R}$ be the forgetful functor from $\mathbb{C}$-vector spaces to $\mathbb{R}$-vector spaces. I am trying to explicitly construct the left ...
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### Monads and Monoids-as-categories

I'm trying to understand the definition of a monad as a monoid and to identify this structure in the implementation of monads in Hakell. One definition is that of a structure $(T,\eta,\mu)$ given by ...
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### Composing functors with natural transformations

So I'm doing a project in Category Theory. I fully understand natural transformations and functors, but what does it mean to compose them, for example in the monad axioms where you have something like ...
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### Non-monadic adjunction

Could someone give some examples of a non-monadic adjunctions please? Possibly explaining why they are not monadic and how they contradict the monadicity theorem? Thanks!
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### Monad as not trivial adjunctions

It is well known that a monad $(T, \mu, \eta)$ can be factorized in multiple ways as adjunctions, and that in some sense, Kleisli is the initial factorization while Eilenberg-Moore is the final ...
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### How can you actually do universal algebra with monads?

Instead of digging deep into "classical" universal algebra, it seems more interesting or fruitful to look at universal algebra categorically. This should be doable with monads, since every category of ...
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### Natural map from cokernel of a monad

If I have a monad $$U \stackrel{\alpha}{\longrightarrow} V \stackrel{\beta}{\longrightarrow}W$$ then there should be a natural map $$\text{cokernel}(\alpha) \rightarrow W$$ but I can't think of ...
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### When can we use a monic morphism to copy an algebraic structure?

Let $(T,\mu, \eta)$ be a monad over the category $\textbf A$ , let $(A,a)$ be a $T$- algebra and $m: B\rightarrow A$ be monic. Prove a morphism $b:TB\rightarrow B$ is the structure for an algebra ...
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### Can every monad give rise to a monad transformer?

Can every monad give rise to a monad transformer? In the paper Calculating monad transformers with category theory by Oleksandr Manzyuk, one finds a construction of monad transformers as ...
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### Ma -> (a -> Mb) -> Mb … vs … Ma -> (a -> b) -> b

I am a newbie to the functional programming world. Is there a reason why in all example I found so far about Monad (or should I say Monoid) is written as: ...
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### Is “polynomials in $x$” a monad?

The construction of polynomials $R \mapsto R[x]$ gives a functor $P: \mathbf{Ring} \to \mathbf{Ring}$ on the category of possibly noncommutative rings. Choosing a ring $R$ for the moment, there is a ...
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### What is a monad in a $2$-category?

The wikipedia article on monads somewhat mysteriously notes that Monads can be defined in any 2-category ${\mathfrak C}$. The monads defined above are for ${\mathfrak C}$ = Cat. where Cat is the ...
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### Simple explanation of a monad

I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are ...