# Questions tagged [monads]

The monads tag has no usage guidance.

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### Relative cocompletion of a category

$\newcommand{\k}{\mathbf k}$
$\newcommand{\A}{\mathcal A}$
$\newcommand{\B}{\mathcal B}$
$\newcommand{\C}{\mathcal C}$
I'm wondering if anyone knows a reference for the following construction: let $\k$...

**5**

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84 views

### U-split coequalizer preserved by coproduct

Let $C$ be a nice symmetric monoidal category and let $T$ be a monad on $C$ such that we have a monadic adjunction
$$F:C\leftrightarrow T\text{-}alg(C):U $$
Suppose that $a\rightrightarrows b$ is a ...

**4**

votes

**1**answer

142 views

### Monad, algebras and reflexive coequalizer

Suppose we have an adjunction of categories $F:M\leftrightarrows N:U$. We define the associated (co)monad $G=F\circ U$. For any object $x\in N$ we define the simplicial resolution of $x$ given by
$$
...

**3**

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150 views

### What category of toposes is monadic over the 2-category of groupoids?

Excuse my lack of understanding of monadicity, but I have been looking at toposes and monads. I see Lambek showed that the category of Toposes are monadic over the category of categories. I see the ...

**1**

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48 views

### Schemes for conditional distributions (monads)

(Note: This is a heuristic question. I'm trying to work out if this idea makes sense. I don't have much background in this area, so apologies if I'm wide of the mark.)
Suppose you have a monad ...

**0**

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72 views

### Is there a natural transformation from Lists to Domains of Lists

The list monad, $(L, \mu_L, \eta_L)$, on $Set$, takes a set to its set of lists, with $\mu_L : L \cdot L \rightarrow L$ being concatenation of lists. Given a set of lists, there is a natural way to ...

**8**

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172 views

### Extending monads along dense functors

Let $j: \mathsf A \to \mathsf B$ be a fully faithful and dense functor where $\mathsf A$ is a small category and $\mathsf B$ is cocomplete. Let $(T, \eta, \mu)$ be a monad over $\mathsf A$.
$\require{...

**5**

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**2**answers

276 views

### Semantics-structure adjunction

In the discussion on the nLab article for monadic adjunctions, John Baez suggests and Mike Shulman confirms that the relationship between adjunctions and monads itself constitutes an adjunction called ...

**2**

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95 views

### The Kleisli Category of the Monad of Measures of Finite Support and its composition formula

In this post, I was introduced to the monad of finitely supported measures.
$HX$ is the set of finitely supported measures on $X$, with monad structure defined as for the Giry monad. Let's call this ...

**4**

votes

**1**answer

232 views

### Does the Eilenberg Moore Construction Preserve fibrations?

Say we have a Grothendieck fibration $p : E \to B$ and a monad $T$ on $B$ and a lift $T'$ of $T$ to $E$, i.e. a monad on $E$ such that $pT' = Tp$ and $p$ preserves $\eta, \mu$.
Then because the ...

**3**

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**1**answer

184 views

### Locally presentable categories

Under category
Let $C$ be a locally presentable category, and let $c$ be an object of $C$. Lets denote by $C^{/c}$ the under category, objects are maps $c\rightarrow x$ and morphisms are the evident ...

**0**

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**1**answer

251 views

### A monad that unions sets

Suppose we have a monad that maps types of some kind to other types (see below) , and let types be sets. Let $\alpha, \beta$ be types, $\rightarrow$ denote a function between types, and let $a : \...

**1**

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81 views

### The multi-set monad and modules

I am trying to analyze the category of algebras for the finite free commutative monoid monad, aka the finite multiset monad. This monad is frequently described as having a multiplication that takes a ...

**4**

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**2**answers

267 views

### $P = [-°,Set]$ is a contravariant co/lax idempotent monad, whose multiplication is determined by the unit

A unidetermined contramonad is a 2-monad $T : {\cal C}\to \cal C$ such that
$T$ is contravariant, i.e. a contravariant endofunctor;
the multiplication $\mu_A : TTA \to TA$ is determined as $T\eta_A = ...

**3**

votes

**1**answer

155 views

### Codensity monad is idempotent?

Let $j: A \to B$ be a fully faithful functor.
When $j$ has a left adjoint $L$, the codensity monad $\text{Ran}_jj$ will coincide with the monad $jL$ and thus will be idempotent, because $A$ is ...