# Questions tagged [monads]

The monads tag has no usage guidance.

224
questions

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### Is the Cartesian product of two finitely presented objects finitely presentable?

Let $C$ be a locally finitely presented category, $A, B$ are two finitely presented (synonym: compact) objects in it. Is it true that $A \times B$ is finitely representable?
At least I have looked at ...

0
votes

0
answers

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### What are the necessary requirements to make this composite monad rewrite work?

It is well known that if you want to take two monads and compose them and get a third monad, you need a distributive law. Let us suppose we have this. So, we have two monads
$$\mathcal{M}$$
And
$$\...

2
votes

1
answer

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### Literature about the category of finitary monads

This answer states that the category of finitary monads is locally presentable and monadic over the category $\mathrm{Set}^{\mathbb{N}}$. Where can I find proof of this claim?
More generally: I've ...

5
votes

0
answers

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### Does the restriction functor $j^* $ to Zariski open preserve the limit of $j^*$-split cosimplicial diagram?

This might be a trivial question but I could not find a satisfatory answer easily.
Let $X = \mathbb{C}$ and $U = \mathbb{C}^*$, and let $j: U \to X$ denote the open embedding.
Consider $j^* : QCoh(X) \...

4
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0
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### Can a non-free monad have non-trivial "quine"?

Let $\mathbf{Poly}$ denote the category of polynomial functors on $\mathbf{Set}$, and let $\mathfrak{m}\colon\mathbf{Poly}\to\mathbf{Poly}$ be the free monad monad, i.e. the functor that sends every ...

3
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0
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### What should be required from a model category that the category of algebraic objects in it has the natural model structure?

I have two reference questions
What should be required of a category with finite products so that a (multi-sorted, finitary) Lawvere theory to induce monadic adjunction in it? This should be ...

2
votes

0
answers

105
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### Cat as a bicategory of monads over another category

Let's assume infinitely many Grothendieck universes exist. Let's call $\kappa$-Cat the bicategory of $\kappa$-small categories with anafunctors and anatural transformations. Now for any $\lambda$ and ...

5
votes

1
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199
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### Lift a monad along a generic right adjoint

$\require{AMScd}$We have a neat way to lift a monad along a monadic right adjoint, through a distributive law: in a setting like
$$
\begin{CD}
X @. X \\
@VUVV @VVUV\\
C @>>T> C
\end{CD}$$
if ...

3
votes

1
answer

129
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### Is a monad functor also known as a monad map?

Suppose I have a monad $M_S = \langle S , \eta_S, \mu_S \rangle$. I want to map this monad to another monad, $M_Q = \langle Q , \eta_Q, \mu_Q \rangle$. What is the minimum I have to define to give ...

5
votes

1
answer

218
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### Is there a canonical product on the category of monads on Set?

I would like to know if there is a partial monoidal product on the category of monads on Set. I want this partial monoidal product to "handle" monad composition which we understand exists ...

1
vote

1
answer

102
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### It's there a way to take a composite monad and a monad map to create a map of the composite?

Let us suppose you start with two monads, $M_S = \langle S , \eta_S, \mu_S \rangle$ and $M_T = \langle T , \eta_T, \mu_T \rangle$ and suppose you have a distributive law, $\lambda: ST \rightarrow TS$ ...

9
votes

0
answers

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### Cocompleteness of enriched categories of algebras

A useful result due to Linton is that for a cocomplete category $C$ and monad $T$ on $C$, if the category of algebras $C^T$ admits reflexive coequalisers, then it is cocomplete (see here for a sketch ...

8
votes

1
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### How many categories $C$ are there such that $C$ and $C^{\text{op}}$ are monadic over $\mathrm{Set}$?

In this question, bimonadic category is a category $C$ such that $C$ and $C^{\text{op}}$ are monadic over $\mathrm{Set}$.
How many bimonadic categories are there? Can we classify them all?
...

1
vote

1
answer

156
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### Kleisli adjunction of the distribution monad

Let $\langle D , \mu, \eta \rangle$ be the distribution monad on $Set$ and let $Kl(D)$ be the Kleisli category on the distribution monad. I am interested in the adjunction between $Kl(D)$ and $Set$, ...

3
votes

1
answer

118
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### Commuting filtered colimits & finite limits in infinitary theories

Filtered colimits & finite limits commute in categories that are finitary monadic over Set (i.e. algebras of finitary algebraic theories). Results such as Fred Linton's result that if categories ...

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answers

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### Why are operads sometimes better than algebraic theories?

Question 1: Are there any contexts in which replacing the category of (non-symmetric or symmetric) operads (in some monoidal category or symmetric monoidal category, respectively) with the category of ...

7
votes

1
answer

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### Free idempotent monad associated to a monad

Let $C$ be a category. There is a full subcategory $\text{IdemMnd}(C) \hookrightarrow \text{Mnd}(C)$ of the category of monads on $C$ spanned by the idempotent monads. Given a monad $T$ on $C$, ...

4
votes

1
answer

187
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### Is the category of computads for a finitary monad on $n$-globular sets cocomplete?

Context
Given a finitary monad $T:\operatorname{gSet}_n\to\operatorname{gSet}_n$ we can define categories $\operatorname{Comp}_k^T$ of $k$-computads for $T$, for any $k=0,\cdots,n+1$. This is nicely ...

16
votes

1
answer

433
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### A new (?) way of composing monads

By composition of monads, I mean given two monads $S$ and $T$, making their composite $S T$ into a monad. Or more generally, given two monoid $X$ and $Y$ in a non-symetric monoidal category, making $X ...

3
votes

1
answer

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views

### If a monad in a 2-category admits a terminal resolution, does it admit an Eilenberg–Moore object?

Let $T = (t, \mu, \eta)$ be a monad on an object $A$ of a 2-category $\mathcal K$. In The formal theory of monads, Street proves (Theorem 3) that if $l \dashv r$ is the canonical adjunction associated ...

0
votes

0
answers

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views

### Over what base is Powerset polynomial?

The functor for the polynomial monad on Set takes a set and maps it to the set of subsets on that set. Is this monad polynomial on Set? If not there, then on what base is Powerset polynomial?

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### Monad map from Distribution to Powerset

I have a conjecture:
There is a monad map from the distribution monad to the powerset monad.
The natural transformation of this monad map forgets the probabilities and just gives the sets of each ...

1
vote

0
answers

61
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### Separable monads do not induce separable monoids

Let us first recall the categorical notion of monad: if we have a category $\mathcal{C}$ then a monad on it consists in an endofunctor $\mathbb{A}\colon \mathcal{C}\rightarrow \mathcal{C}$ together ...

3
votes

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### What is the free lax-idempotent adjunction?

Let $Adj$ be the free adjunction, i.e. the 2-category such that for any 2-category $K$, the functor 2-category $2Fun(Adj, K)$ is the 2-category of adjunctions in $K$ (naturally in $K$). Note that $Adj$...

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votes

2
answers

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### Are infinitary monads monadic?

As discussed here, Are monads monadic?, in "On the monadicity of finitary monads" by Steve Lack, the following is shown, the forgetful functor from $Mnd_f(C) \rightarrow Endo_f(C)$ is ...

2
votes

1
answer

202
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### Well-behaved monad quotients

Reading through Modular specification of monads through higher-order presentations, this paper includes the following lemma within set-truncated homotopy type theory:
Given a monad $R$ (they work on ...

1
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0
answers

114
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### The S-module Ass is same as the composite of Com and Lie

It has been cited in several places (eg. https://arxiv.org/pdf/1912.05519.pdf) that the S-module Ass is isomorphic to the composite of the S-modules Com and Lie. Is there a reference which gives the ...

4
votes

1
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### What are the algebras for the laxification 2-monad?

Let $C$ be a small 2-category. Let $[C , Cat]$ denote the 2-category of strict functors to $Cat$, 2-natural transformations, and modifications. Let $[[ C, Cat ]]$ denote the 2-category with the same ...

2
votes

0
answers

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### EM functor from monads to adjunctions

What is the action on $1$-cells of the functor sending a monad to its EM adjunction? What about the Kleisli adjunction?
Let $A$ be the walking adjunction. Recall that an adjunction is the same thing ...

2
votes

0
answers

120
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### Homotopy fixed points vs coalgebras

Referring to the last part of this answer https://mathoverflow.net/a/225403/170683, I would like to understand how in the case of a Galois cover $f\colon X\to Y=X/G$ with Galois group $G$ (I guess ...

1
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0
answers

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### Are pseudomonoids weak algebras for a 2-monad?

I would like to know if whether or not the pseudomonoids in an arbitrary monoidal 2-category are (equivalent to) the weak algebras for some 2-monad (I am thinking about the free monoidal category 2-...

8
votes

1
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224
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### Algebraically-free monadicity theorem

The monadicity theorem characterises when a functor $u : \mathbf B \to \mathbf E$ is the forgetful functor from the category of algebras for some monad on $\mathbf E$ (up to an equivalence over $\...

2
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### Let $T$ be a strongly cartesian monad on a presheaf category $\hat C$. Then is $\hat C$ comonadic over $\operatorname{Alg} T$?

$\DeclareMathOperator\Alg{Alg}\newcommand{\Set}{\mathit{Set}}\newcommand{\Set}{\mathit{Set}}\newcommand{\Ab}{\mathit{Ab}}$Let $C$ be a small category, and let $T$ be a strongly cartesian monad on the ...

4
votes

1
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### When is the Eilenberg-Moore category of a monad on an ind-category itself an ind-category?

I have a monad on an ind-category (specifically, my ind-category has a monoidal structure and I have an algebra object, so the monad is tensoring with it). It would be very useful in my work if the ...

2
votes

1
answer

189
views

### Uniqueness of comparison functors

Given an adjunction $F\dashv G:\mathcal{C}\rightleftarrows\mathcal{D}$ with unit $\eta$ and counit $\epsilon$, we naturally have a monad $(G\circ F,\eta,G\epsilon_F)$ on $\mathcal{C}$ and a comparison ...

4
votes

1
answer

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### What is the universal property of algebras for the codensity monad?

Let $F : A \to B$ be a functor, and suppose that the right Kan extension $T = Ran_F F : B \to B$ exists. Then $T$ is a monad, the codensity monad of $F$. Moreover, unless I'm mistaken there is a ...

4
votes

1
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### Constructing the E-M category of a monad out of inserters and equifiers

As suggested in the answer to another MO question, it seems possible to construct the E-M category of a monad $T:\mathcal{C}\to\mathcal{C}$ as an inserter followed by two equifiers as follows (I am ...

3
votes

0
answers

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### Adjoints to the forgetful functor from the $2$-category of monads

For the purpose of this post, we will identify a monad $T$ on a category $\mathcal{C}$ with a lax $2$-functor $T:{\bf 1}\to\mathfrak{Cat}$ such that $T(*)=\mathcal{C}$.
There is an obvious forgetful ...

2
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### The Kleisli category of a monoidal monad

Let $C$ be a symmetric monoidal category equipped with diagonals $\triangle_x: x \to x \otimes x$, that is, equipped with natural transformations $e_x: x \to 1$ and $\triangle_x : x \to x \otimes x $ ...

5
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### Original reference for the correspondence between commutative algebraic theories and commutative monads

Commutative algebraic theories were introduced by Linton in the 1966 paper Autonomous Equational Categories. Commutative monads were introduced by Kock in the 1970 paper Monads on symmetric monoidal ...

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1
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### Characterisation of functors whose left adjoint is Kleisli

This question is inspired by Characterization of functors whose right adjoint is monadic?.
Let $F : \mathbf C \rightleftarrows \mathbf D : U$ be an adjunction, and suppose that we want to establish ...

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### Jon Beck's untitled manuscript containing the "tripleability theorem" (i.e. the monadicity theorem)

Many papers refer to an untitled manuscript of Jon Beck (Cornell, 1966) for the origin of the monadicity theorem (originally called a "tripleability theorem"). An early proof is in Manes's ...

4
votes

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### The bidualizing monad

Let $\mathbf{C}$ be a closed symmetric monoidal category (I probably need even less than this; the examples I have in mind are simply the category of modules over a commutative ring and the category ...

2
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### Diagrammatic model for free product in monad infinity category

$\newcommand{\C}{\mathcal{C}}$ Suppose $M$ is a monad in an $\infty$-category $\C,$ and $A, B$ are two algebras over $M$. I'm willing to assume any reasonable "niceness" conditions on $\C$, $...

3
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1
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### Examples of (co)lax idempotent pseudocomonads on Cat

A lax idempotent pseudomonad, also called a KZ doctrine or KZ monad, is a pseudomonad $(T, \mu, \eta)$ with the property that $T \eta \dashv \mu \dashv \eta T$. Lax idempotent pseudomonads were ...

1
vote

1
answer

85
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### Algebras for general transfors

Algebras for endofunctors bridge the gap between functors acting on a category and structures defined in it. An algebra for an endofunctor $F$ is instantiated by some morphism $Fa \to a$, and more ...

14
votes

1
answer

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### What are internal complete atomic boolean algebras, intuitively?

The category of complete atomic boolean algebras $\mathbf{CABA}$ is equivalent to $\mathbf{Set}^{\mathrm{op}}$ via
$$\mathbf{Set}^{\mathrm{op}} \to \mathbf{CABA}, ~ X \mapsto (P(X),\bigcup,\bigcap).$$
...

2
votes

2
answers

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### Finitary endofunctors: "Support" of elements of $TM$ where $T:\mathrm{Set}\to\mathrm{Set}$ is finitary

Consider the word a.k.a. list monad $W:\mathrm{Set}\to\mathrm{Set}$ assigning to a given set $M$ the set of words over $M$. Now, given a set $M$, we can assign to every word $w\in WM$ a subset of $M$:
...

14
votes

3
answers

641
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### Reference request for Linton's theorems on equational theories

This is a reference request for the following "well-known" theorems in category theory:
There is an equivalence of categories between finitary monads on $\mathbf{Set}$ and finitary Lawvere ...

3
votes

0
answers

482
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### Cyclic lists of multisets

I am wondering if it is appropriate to ask about having a specific algebra for an endofunctor computed. We all know about the multiset monad, and it's endofunctor $\mathcal{M}_S$, and some might know ...