Questions tagged [monads]

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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 $M$: ...
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2answers
463 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 ...
3
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0answers
230 views
+100

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 ...
3
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0answers
26 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 $\mathcal K$...
3
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1answer
64 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_{\...
5
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0answers
67 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 ...
10
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1answer
288 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 ...
4
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3answers
344 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 \...
3
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0answers
110 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. ...
24
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1answer
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 $\mathrm{Set}$ ...
3
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1answer
131 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 ...
5
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2answers
246 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 \...
5
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0answers
176 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. A ...
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0answers
93 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(\...
2
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0answers
107 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 $i : X^\delta\to X$ ...
6
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0answers
118 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 $...
7
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1answer
164 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 $\...
4
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1answer
241 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 ...
2
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0answers
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 ...
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0answers
77 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 $...
6
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0answers
86 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 ...
4
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2answers
222 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}(\...
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1answer
108 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 ...
7
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1answer
251 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 ...
2
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0answers
94 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 ...
5
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0answers
136 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 $X\ast Y$ ...
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0answers
51 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 ...
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0answers
95 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 ...
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0answers
101 views

When $M \otimes M$ exists in the 2-Category of Monads, is there always a map $M \otimes M \rightarrow M$?

The definition of the 2-category of monads is give in this reference by Lack and Street. In it, they define the "monad morphisms", with 1 cells $f$ and 2-cells $\phi$ which obey a given pair of ...
7
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1answer
284 views

Characterisation of essentially algebraic theories as monads

The following correspondence between algebraic theories and monads on $\mathbf{Set}$ is well-known (see, for example, Algebraic Theories: A Categorical Introduction to General Algebra). The ...
13
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4answers
1k views

Conceptual reason that monadic functors create limits?

Let $U: Alg_T \to C$ be the forgetful functor from the category of algebras of $T: C \to C$ ($T$ could be a monad; I'm happy to think about the simpler case where $T$ is just an endofunctor or pointed ...
3
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2answers
267 views

Ultrafilter comonad on the category of Stone spaces

Let $\mathsf{Stone}$ denote the category of Stone spaces (compact, totally disconnected Hausdorff spaces) and continuous maps. The forgetful functor $U : \mathsf{Stone} \to \mathsf{Set}$ has a left ...
3
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2answers
252 views

The symmetric monoidal closed structure on the category of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors

In 6.5 of the book by Kelly, Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, No. 10, 2005. the author claims that the $2$-category $\mathsf{Cat}_{\...
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1answer
319 views

What are the algebras for the ultrafilter monad on topological spaces?

Motivation: Let $(X,\tau)$ be a topological space. Then the set $\beta X$ of ultrafilters on $X$ admits a natural topology (cf. Example 5.14 in Adámek and Sousa - D-ultrafilters and their monads), ...
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2answers
284 views

when is an integer sequence the trace of a monad on FinSet?

Given $(a_n \in \mathbb{N})$, when is there a monad $T$ on $\mathrm{FinSet}$ such that $$ | T(n) | = a_n\quad\forall n\in \mathbb{N}\:? $$
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0answers
110 views

Do these monads on Rel compose?

$Rel$ is the category of sets and relations. The cyclic list monad, $\mathcal{Cy}=(Cy, \mu_c, \eta_c)$ is defined as follows: $Cy : Rel \rightarrow Rel$, such that, $Cy(X)$ is all cyclic lists on ...
1
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1answer
117 views

Internal commutative monoid gives commutative monad

Let $(C,\otimes,1)$ be a symmetric monoidal category. Let $(M,\mu,\eta)$ be an internal commutative monoid object. The functor $X\mapsto M\otimes X$ has a canonical monad structure, with unit and ...
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0answers
48 views

Distributive laws of strong and/or monoidal monads

It is well-known that a commutative strong monad is the same as a monoidal monad. Is there a notion of distributive law for commutative strong monads which is equivalently a distributive law for ...
0
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1answer
137 views

What is the measures monad for FDHilb?

I am labouring under a particular assumption that, perhaps, needs to be corrected. I believe that FDHilb, the category of Finite Dimensional Hilbert spaces and general linear maps is a category of ...
1
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0answers
187 views

Existence of free functor to Banach spaces

Is there a "non-trivial" characterization of the concrete categories admitting and adjoint pair of functors $F \dashv G$ were $G$ is defined on the category sBan of separable Banach spaces and bounded ...
9
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0answers
155 views

What are the algebras for the codensity monad of $\textbf{FP-Ring} \rightarrow \textbf{Ring}$

Let $\textbf{Fin-Set}$ denote the category of finite sets, and let $\textbf{Set}$ denote the category of sets. The inclusion functor $\textbf{Fin-Set} \rightarrow \textbf{Set}$ from the category of ...
10
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1answer
453 views

Slicing up monads on categories with pullbacks: what are these mysterious “zerosumfree” monads"

Introduction I'll describe a way of taking a monad on a category $\mathcal{E}$ with pullbacks, and obtaining a monad on each slice category. I'll show that this construction is always lax-natural in $...
1
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1answer
121 views

Trying to construct the ultrafilter 2-monad on $\mathbf{Cat}$

By which I mean, following Bôrger's paper Coproducts and Ultrafilters, the terminal monad among those that preserve finite coproducts, if such a thing can be constructed. So far, what I have is, ...
0
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0answers
129 views

What is the Eilenberg-Moore category for the cyclic list?

In this paper (Kock 2012), we see a data structure with circular symmetry. It is the cyclic list monad of Examples 3.10. The author is showing that data structures with symmetries can be cast as ...
9
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0answers
179 views

What are the (co)algebras for the $(\operatorname{Hom}(A,-), A\otimes-)$ adjunction (co)monad?

A module $A$ over a commutative ring $k$ gives a pair of adjoint endofunctors, $(A\otimes_k-)$ left adjoint to $\operatorname{Hom}_k(A,-)$. They produce a monad $T_A$ and a comonad $C_A$. Is there any ...
9
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2answers
487 views

When were triples called monads for the first time?

I am fine-tuning a short note on basic category theory; any such course must introduce monads, and I want to give a bit of history of the subject. I soon realized that I don't know the precise series ...
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0answers
196 views

Partial monoid in the category of categories of modules: The spotty nature of monad composition

It seems that I am working on a conjecture in category theory. In particular, I am curious about the spotty nature of the composition of monads on Set. I am guessing that there is a category, $\...
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0answers
93 views

Composition of monads induces tensor product in the category of modules

I have recently asked a question about the composition of two monads, namely $\mathcal{M}_P = \mathcal{M}_C \cdot \mathcal{M}_C$. I am conjecturing that the cateogory of $\mathbb{C}$-Modules and the ...
3
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0answers
133 views

Quantum Scattering Experiments: C-Modules, N-Modules and Their Monads

I am working on a theory of particle physics where we use monads. I have a few conjectures that I need to check. The cateogory of $\mathbb{C}$-Modules is monadic over set The category of $\mathbb{N}...
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195 views

Understanding a monad from its fixed points

Let $(T, \eta, \mu)$ be monad over $\mathsf{C}$. And let $\iota : \mathsf{Fix}(T) \hookrightarrow \mathsf{C}$ be the inclusion of the full subcategory of fixed points of $T$. By the universal ...