Questions tagged [monads]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
1 answer
110 views

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 ...
user avatar
  • 159
1 vote
0 answers
77 views

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 ...
user avatar
  • 101
4 votes
1 answer
126 views

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 ...
user avatar
  • 49.2k
2 votes
0 answers
91 views

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 ...
user avatar
  • 6,251
1 vote
0 answers
95 views

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 ...
user avatar
  • 101
1 vote
0 answers
43 views

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-...
user avatar
  • 11
8 votes
1 answer
178 views

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 $\...
user avatar
  • 5,035
2 votes
0 answers
51 views

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 ...
user avatar
  • 49.2k
4 votes
1 answer
165 views

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 ...
user avatar
2 votes
1 answer
166 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 ...
user avatar
  • 6,251
4 votes
1 answer
117 views

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 ...
user avatar
  • 49.2k
4 votes
1 answer
132 views

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 ...
user avatar
  • 6,251
3 votes
0 answers
58 views

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 ...
user avatar
  • 6,251
2 votes
0 answers
154 views

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 $ ...
user avatar
  • 123
5 votes
0 answers
93 views

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 ...
user avatar
  • 5,035
2 votes
1 answer
150 views

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 ...
user avatar
  • 5,035
17 votes
2 answers
1k views

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 ...
user avatar
  • 5,035
4 votes
2 answers
304 views

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 ...
user avatar
  • 21.7k
2 votes
0 answers
70 views

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 $\C$, $...
user avatar
3 votes
1 answer
108 views

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 ...
user avatar
  • 5,035
1 vote
1 answer
82 views

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 ...
user avatar
15 votes
1 answer
341 views

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 $$\mathbf{Set}^{\mathrm{op}} \to \mathbf{CABA}, ~ X \mapsto (P(X),\bigcup,\bigcap).$$ ...
user avatar
2 votes
2 answers
171 views

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 $M$: ...
user avatar
13 votes
3 answers
605 views

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 ...
user avatar
3 votes
0 answers
383 views

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 ...
user avatar
  • 1,435
3 votes
0 answers
32 views

Lax algebras for pseudomonads and monads in Kleisli bicategories for the induced pseudocomonad

In Day–Street's Lax monoids, pseudo-operads, and convolution, they remark without proof: There are general principles involved here. Suppose $(T, m, j)$ is a pseudomonad on any bicategory $\mathcal K$...
user avatar
  • 5,035
3 votes
1 answer
73 views

Does the right adjoint of a comonad induce the following comodule map?

Let $\mathcal{C}$ be a category and $\mathcal{G}=(G,\delta, \epsilon)$ be a comonad on $\mathcal{C}$. Here $G: \mathcal{C}\to \mathcal{C}$ is a functor, $\delta: G\to G^2$ and $\epsilon: G\to id_{\...
user avatar
  • 8,079
5 votes
0 answers
94 views

Monads and modules, and the bicompletion under Kleisli and Eilenberg–Moore objects

In The Formal Theory of Monads, Street proves that a 2-category $\mathscr C$ admits the construction of algebras when the inclusion $\mathscr C \to \mathbf{Mnd}(\mathscr C)$ has a right adjoint. In ...
user avatar
  • 5,035
10 votes
1 answer
367 views

Characterization of functors whose right adjoint is monadic?

Let $F: \mathcal A^\to_\leftarrow \mathcal B: U$ be an adjunction, and suppose we want to know whether the comparision functor $\mathcal B \to Alg^{UF}$ is an equivalence, where $Alg^{UF}$ is the ...
user avatar
  • 49.2k
5 votes
3 answers
417 views

Contramodule as direct limit of its finitely generated subcontramodules

$\DeclareMathOperator\Hom{Hom}$Let $K$ be a field. Let $C$ be a $K$-coalgebra. A contramodule $M$ over $C$ is a $K$-space with a $K$-linear map $\pi_M:\Hom_K(C,M)\longrightarrow M$ such that $\pi_M \...
user avatar
  • 373
3 votes
0 answers
119 views

Does the notion of a Poisson monad exist?

Starting with a monoidal category with duals $C$, one may consider the category $End(C)$ of endofunctors of $C$. A Hopf monad on $C$ is a bimonad on $C$ with (a generalised notion of the) antipode. ...
user avatar
25 votes
1 answer
1k views

Infinity-categorical analogue of compact Hausdorff

Recently I became through this mathoverflow question aware of the article Codensity and the ultrafilter monad by Tom Leinster. There he shows that the ultrafilter monad on the category $\mathrm{Set}$ ...
user avatar
3 votes
1 answer
151 views

Do (co)density (co)monadic constructions stablize?

Under good conditions [1], any functor $F: C \to D$ induces a codensity monad $T: D \to D$ as a right Kan extension of $F$ along itself. It does not say explicitly, but by considering left/right Kan ...
user avatar
  • 4,005
5 votes
2 answers
299 views

A specific property of bi-adjunction

Let $$I: C \rightleftarrows D: F$$ be biadjoint [1] functors between categories $C, D$. That is, $I$ is the left and also the right adjoint of $F$ (thus vice versa). Put in notations, it's $$ \cdots \...
user avatar
  • 4,005
5 votes
0 answers
279 views

Cohomology without comonad?

TL;DR. Many cohomologies can be unified using comonads. Question: which cohomologies cannot be? For each algebraic theory, there is an adjunction, and therefore a (co)monad (or called a (co)triple). ...
user avatar
  • 4,005
8 votes
0 answers
120 views

What is the relationship between free bicompletion and the Isbell envelope?

Given a small category $\mathbb C$, we can form the free cocompletion $\mathbf y : \mathbb C \to \mathcal P(\mathbb C)$ and the free completion $\mathbf y^\circ : \mathbb C \to \mathcal P^\circ(\...
user avatar
  • 5,035
2 votes
0 answers
114 views

Are flasque sheaves exactly the retracts of "canonically" flasque sheaves?

Let $X$ be a topological space. Let $X^\delta$ denote the space whose elements are the points of $X$, and which is equipped with the discrete topology. There is a continuous map $i : X^\delta\to X$ ...
user avatar
7 votes
0 answers
181 views

Relation between two limit presentations of Eilenberg--Moore objects

Let $\mathbb{T}=({\cal T}\colon C\to C,\mu,\eta)$ be a monad (in the $2$-category $\mathsf{Cat}$), which we view as a $2$-functor $\mathbb{T}\colon\mathsf{B}\Delta_{\mathrm{a}}\to\mathsf{Cat}$ (where $...
user avatar
7 votes
1 answer
212 views

2-monads for categories with a class of (co)limits

This question concerns the strictness of (co)completions, at various levels of generality. In Blackwell–Kelly–Power's Two-dimensional monad theory, the authors state For instance, the 2-category $\...
user avatar
  • 5,035
4 votes
1 answer
304 views

Different ways to “deloop” a (topological) $A_\infty$-algebra

Let $\varphi:A\to \mathrm{Ass}$ be an $A_\infty$-operad in topological spaces, and let $X$ be an $A$-algebra. I see three possibilities to construct a delooping out of $X$: Rectify $X$ by taking the ...
user avatar
  • 1,563
2 votes
0 answers
30 views

Morphism of pseudomonads induces pullback functors between pseudoalgebras

If $S$ and $T$ are monads on a category $C$, and $\lambda:S\to T$ is a morphism of monads, it is well-known that there is a functor $\lambda^*:C^T\to C^S$ which assigns to the $T$-algebra $(A,a:TA\to ...
user avatar
  • 1,999
4 votes
0 answers
80 views

Coherence for pseudomonads and their pseudoalgebras

Let $\mathcal K$ be a bicategory. For every pseudomonad $T : \mathcal K \to \mathcal K$, does there exist a 2-monad $S : \mathcal C \to \mathcal C$, where $\mathcal C$ is a 2-category biequivalent to $...
user avatar
  • 5,035
7 votes
0 answers
106 views

Algebras for products or limits of monads

If a category $C$ has limits of a certain type, then the category of monads on $C$ has the same type of limits, and these limits are computed "levelwise" (i.e. are preserved by the forgetful ...
user avatar
  • 32.2k
4 votes
2 answers
234 views

Is monadicity preserved by the underlying functor?

Let $\mathcal{V}$ be a monoidal closed (complete, cocomplete, reasonable...) category. Let $\mathsf{T}$ be an enriched monad over $\mathcal{V}$. The forgetful functor $\mathsf{U}: \mathsf{Alg}(\...
user avatar
0 votes
1 answer
115 views

Substitution structure on pointed sets

$\def\Fin{\text{Fin}_*} \def\Set{\text{Set}_*} \def\dd{\mathop{\diamond_\land}}$ The present question is intimately related to another question. Let $\Fin$ be the category of pointed sets. The ...
user avatar
  • 11.5k
7 votes
1 answer
272 views

Finitary monads on $\operatorname{Set}$ are substitution monoids. Finitary monads on $\operatorname{Set}_*$ are...?

$\DeclareMathOperator\Fin{Fin}\DeclareMathOperator\Lan{Lan}\DeclareMathOperator\Set{Set}$ The present question is intimately related to another question. It is well known that the category of ...
user avatar
  • 11.5k
2 votes
0 answers
113 views

A morphism of monads that doesn't preserve thunkability?

Recall that for a monad $(T,\eta,\mu)$ on a category $C$, the Kleisli category $C_T$ has as objects the objects of $C$ and as morphisms $C_T(x,y) = C(x,T y)$. A morphism $f\in C_T(x,y) = C(x,T y)$ is ...
user avatar
  • 58.8k
5 votes
0 answers
142 views

Algebras of the cone monad on Top?

Let us work in Top, the category of topological spaces - although the reader is welcome to replace this by their favorite convenient category of topological spaces. If $X,Y$ are spaces, let $X\ast Y$ ...
user avatar
1 vote
0 answers
95 views

When do objects in the image of a functor $G$ have a unique action as algebras over the codensity monad of $G$?

Let $G:\mathcal{B}\longrightarrow \mathcal{A}$ be some functor which admits a right Kan extension along itself, $(\operatorname{Ran}_G G, \eta:\operatorname{Ran}_G G \circ G \rightarrow G)$. The ...
user avatar
  • 425
0 votes
0 answers
102 views

Expressing a model transformation by using monads in the simply-typed lambda calculus

In https://link.springer.com/content/pdf/10.1007/s10670-019-00128-z.pdf , page 16, the following clause is given for a modal operator $\langle R_k \rangle$ (see definition 4.2 for the definition of a ...
user avatar
  • 617