# Questions tagged [monads]

The monads tag has no usage guidance.

244
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### (When) does a morphism of monad induce adjoint functors between categories of algebras?

For monads $S$ and $T$ on a fixed Abelian category $C$, a morphism of monads $\sigma: S\rightarrow T$ induces a functor between Eilenberg-Moore categories $\sigma^*:C^T\rightarrow C^S$. This functor ...

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### Monads for proof relevance in type theory

I am just getting started with homotopy type theory. After watching an introductory lecture, I was attracted to the concept of proof relevance. In my understanding, the core idea here is to elevate ...

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### When is the Eilenberg-Moore category of a relative monad between two topoi a topos?

In the non-relative case, we have a theorem, that an Eilenberg-Moore category of algebras of a Monad $T$ on a topos is itself a topos if the monad in question has a right adjoint.
Now how does this ...

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### Reciprocity for algebra objects in two algebraic categories

I think this question Compact Hausdorff and C^*-algebra "objects" in a category. shows that there is no reciprocity between categories of algebra-objects of two algebraic categories.
So, ...

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### Example of a pseudomonad on Cat whose pseudoalgebras are not the pseudoalgebras for a 2-monad

For every pseudomonad $T$ on the 2-category of (locally small) categories $\mathbf{Cat}$, we can consider the 2-category of $T$-pseudoalgebras and pseudomorphisms $T\text{-PsAlg}_p$, which is equipped ...

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### Yetter-Drinfeld modules for Hopf monads

1. Context.
1.1. Classical Yetter-Drinfeld modules.
Let $H$ a bialgebra in a braided monoidal category $\mathcal{C}$. A left-right Yetter-Drinfeld module over $H$ is a triple $(V,\rho,\Delta)$ ...

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### Hopf monads in categorical probability theory

1. Context. According to [1], probability monads are arguably the most important concept in categorical probability theory. In [2] Fritz and Perrone argue that "in order for a monad to really ...

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### Finitely presentable objects in the categories of algebras of $\infty$-algebraic theories

By default, all terms are understood in the infinity sense (“category” means “$(\infty, 1)$-category”, etc.).
An object $A$ in a category is said to be finitely presentable (or compact) if the functor ...

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### Beck's original formulation of the precise tripleability theorem. Reference when considering reflexive pairs?

Thanks to MO's user Varkor, we have access to Beck's original untitled manuscript where Beck first stated his precise tripleability theorem. Up to terminological isomorphism, the PTT as stated in p. 3 ...

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### When is a container a monad?

The category of polynomial functors on Set is equivalent to the category of containers.
We have a prescription for when a container is a comonad.
There are a few other questions that come to mind. ...

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### Are there cartesian closed monads that also preserve the closed structure of the CCC

When I look for cartesian closed monads, I only find monads where the endofunctor preserves the cartesian structure of a cartesian closed category
$$
\operatorname T\ (a \times b) = (\operatorname T\ ...

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### Nontrivial example of when monadic functors don't compose

It is well-known that the composite of monadic functors $U: C \to C'$ and $U': C' \to C''$ need not be monadic. One standard example is the forgetful functor $\mathrm{Cat} \to \mathrm{RefGph}$ from ...

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### In what algebraic categories do finitely presentable objects form a dense cogenerator?

For each $C$ locally finitely presentable category, the full subcategory of finitely presentable objects $C_{fp}$ is a dense generator, i.e. the natural functor $C \to \mathrm{PSh}(C_{fp})$ is a full ...

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### Distributive law of the non-empty list comonad over the non-empty list monad

Preliminaries
A monad is a triple $(M, \eta, \mu)$, with $M$ a functor, $\eta : \mathit{id} \Rightarrow M$ and $\mu : M^2 \Rightarrow M$ two natural transformations such that the following diagrams ...

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### Can application in untyped lambda calculus be seen as the uncurried unit of some monad?

Simply typed lambda calculus in one type variable in a Cartesian closed category $\mathbf{C}$ can be interpreted as a family of Cartesian closed functors (described below, do they have a name?) from ...

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### Is lambda calculus polymorphism a type of generalized monad?

Let $\mathbf{C}$ be a Cartesian closed category. Then simply typed lambda calculus in $\mathbf{C}$ in one type variable can be interpreted as a category $\mathbf{STLC}_{\mathbf C}$ where the objects ...

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### Eilenberg-Moore category as a 2-dimensional limit

$\require{AMScd}$Given an endofunctor $F : C\to C$, its category of algebras is the inserter of $F$ and the identity functor. This means that there is a square
$$\begin{CD}
Alg(F) @>j>> C \\
...

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### Limits and colimits in the category of algebraic theories

Let $\mathrm{AlgTh}$ be the category of one-sorted algebraic theories (synonym: Lawvere theories; morphisms are functors that are identical on objects and strictly preserve products). It is known that ...

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### What functors between categories of algebras are induced by morphisms of monads on $\mathrm{Set}$?

Let $M, N$ be monads of rank $\lambda$, where $\lambda$ is a regular cardinal (I'm primarily interested in the case of finitary monads). Is there a known characterization of functors $\mathrm{Alg}~N \...

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### Intuitive meaning of Giry monad's $\sigma$-algebra

The Giry monad $G : \textbf{Meas} \to \textbf{Meas}$ maps a measurable space $(X, \mathcal{F})$ to its set of probability measures. The $\sigma$-algebra of $G(X, \mathcal{F})$ is the smallest algebra ...

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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 ...

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### 2-completeness of stacks

I am looking for a reference which discusses the 2-categorical properties of the 2-category $St(C,J)$ of stacks and the stackification $\dashv$ inclusion of presheaves 2-adjunction.
My stacks are ...

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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
$$\...

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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 ...

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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) \...

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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 ...

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### What should be required from a model category so 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 induces a monadic adjunction on it? This should be ...

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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 ...

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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 ...

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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 ...

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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 ...

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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, $\lambda: ST \rightarrow TS$ ...

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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 ...

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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?
...

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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 $Set$, ...

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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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### 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 ...

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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 $C$, ...

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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 ...

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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 ...

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### 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 ...

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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 ...

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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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### 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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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-...