Questions tagged [monads]

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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 ...
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1answer
202 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 ...
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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 ...
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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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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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79 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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95 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 ...
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1answer
230 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 ...
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4answers
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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 ...
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2answers
236 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 ...
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232 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
287 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
276 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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101 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 ...
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1answer
111 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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44 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 ...
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1answer
122 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 ...
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181 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 ...
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137 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 ...
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1answer
401 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 $...
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1answer
116 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, ...
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121 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 ...
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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 ...
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2answers
447 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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107 views

All dcpo's of streams is a monad (or comonad)

I have been working with composition of monads for a while. By that I mean that I have been positing that various compositions of monads exist. One central composition involves DCPOs, or domains, ...
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175 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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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 ...
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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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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 ...
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Big list of comonads

The concept of a monad is very well established, and there are very many examples of monads pertaining almost all areas of mathematics. The dual concept, a comonad, is less popular. What are ...
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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 $\k$...
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1answer
177 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 $$ ...
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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 ...
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51 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 ...
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235 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{...
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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 ...
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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 ...
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1answer
262 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 ...
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1answer
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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 ...
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1answer
272 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 : \...
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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 ...
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2answers
312 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 = ...
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1answer
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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 ...
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Codensity monad preserves some colimit?

Let $j: A \to B$ be a functor. When $j$ has a left adjoint $L$, the codensity monad $\text{Ran}_jj$ will coincide with the monad $jL$. Since a left adjoint preserves all colimits, it is easy to ...
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Does each monotonic endofunctor on the category of sets and relations preserve conversion?

Consider a functor $F : \mathbf{Rel} \to \mathbf{Rel}$ that is monotonic (for all relations $R$ and $S$ with $R \subseteq S$ we have $FR \subseteq FS$). Does such a functor always preserve conversion ...
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241 views

What is the category of algebras for the finitely supported measures monad?

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. I have three ...
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0answers
129 views

Colimits of algebras for $\infty$-Monad

I would like to know in anyone has developed method for constructing colimits in the category of algebra for a monad in the $(\infty,1)$-categorical framework, using transfinite constructions. I have ...
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70 views

Define a directed-complete partial order of lists

The List monad takes a set and produces the set of lists on that set. The elements of the set become the symbols or objects of the list. I would like to define a directed-complete partial order (...
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1answer
267 views

Map from the Multiset Monad to the Giry Monad: From Data to Probabilities

The Mulitiset monad, aka the free commutative monoid monad or "Bag" monad, takes a set to the set of all Multisets for that set. A Multiset is like a set, but can have duplicates. It is used in ...
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1answer
97 views

When does a 2-functor or 2-monad of Cat lift to a psuedofunctor or pseudomonad on Prof?

I'm currently reading Richard Garner's paper Polycategories via pseudo-distributive laws, and a central construction is the lifting of the symmetric strict monoidal category 2-monad to a pseudomonad ...