Questions tagged [monads]
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255 questions
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(Co)Monads with a mixed distributive law on the 2-Category of Groupoids
I am looking for containers on the 2-Category of Groupoids. In particular, though, I would like my container to be both a monad and a comonad with a mixed distributive law. Can someone provide one ...
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Frobenius monads and groupoids
For a while, I was looking for a Frobenius monad on Set. It doesn't exist as pointed out here. I am now looking at the 2-category of groupoids. Does the 2-category of groupoids admit a Frobenius ...
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What is the “free symmetric monoidal category” 2-monad?
I have come across an n-category cafe post where someone describes a monad that generates symmetric monoidal categories. Can someone give details, like what is the base category, what exactly is the ...
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Transformation from the Bag monad to the List monad
The bag monad, sometimes called the multiset monad or free commutative monoid monad is a functor on Set that takes a set to its set of bags. These bags are like strings written in the elements of the ...
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What are the special properties of adjunctions that generate polynomial monads
The subject of polynomial monads is well trodden. We know that every monad is generated by an adjunction. What are the special properties of any adjunction that generates a polynomial monad?
Take a ...
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Computing a factorization of a monad
Given a monad, $(M, \mu, \eta)$, where $M: C \rightarrow C$ for some category $C$, there is a category of factorizations, $F\cdot G = M$ where $F: X \rightarrow C$, $G: C \rightarrow X$. Though this ...
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Multiset or Bag monad on Finite-Dimensional Hilbert Spaces
Edit: I will be happy if someone can get me the Bag monad on a 2-category of groupoids, regardless of any reference to Hilbert Spaces. (It's a fire sale!!)
I am trying to create the quantum ...
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The MultiSet (Bag) Monad on FinHilb
It was recently brought to my attention that the Bag monad, also known as the MultiSet monad, is not polynomial on Set, but is Polynomial on the category of Groupoids, 3.10 Examples. I then started ...
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Relation between monads, operads and algebraic theories (Again)
This question (as the title obviously suggests) is similar to, or a continuation of, this question that was asked years ago on MO by a different user.
The present question, though, is different from ...
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What is the polynomial functor for the Bag monad
I may be wrong, but we should be able to write the Bag monad in a polynomial form. The bag monad, is exectly the multiset monad whose category of algebras are the commutative monoids. Another name ...
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Are the string diagrams for the Frobenius Algebra an example of a Polynomial Functor?
We know that Frobenius objects in a monoidal category obey a diagrammatic string calculus. We also know that trees are polynomial functors (Kock - Polynomial functors and trees). The string calculus ...
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What, mathematically speaking, does it mean to say that the continuation monad can simulate all monads?
In various places it is stated that the continuation monad can simulate all monads in some sense (see for example http://lambda1.jimpryor.net/manipulating_trees_with_monads/))
In particular, in http://...
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When does the forgetful functor from algebras over a monad commute with homotopy geometric realizations?
Let $\mathcal{C}$ be a combinatorial model category and $\mathrm{T}$ a monad on
$\mathcal{C}.$
Assume that the model structure on $\mathcal{C}$ lifts to a model structure on
the category of $\...
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Kan condition for bar construction
Let $T$ be a monad on a concrete category $\mathcal{C}$, and $A$ an algebra over $T$. The bar construction is a simplicial object in the category $\mathcal{C}^T$ of algebras which we can think of a ...
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The Curry Howard Isomorphism and models for an intuitionistic modal logic and its bimodal translation
My question regards the Curry Howard Isomorphism and how it constrains models in the case of a particular logic.
Consider quantified Lax Logic $QLL$.
https://pdfs.semanticscholar.org/468e/...
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monadic decomposition
Let $\mathrm{F}: \mathcal{C} \rightleftarrows \mathcal{D} : \mathrm{G} $ be an adjunction with associated monad $\mathrm{T} = \mathrm{G} \mathrm{F} .$
If $\mathcal{D} $ admits coequalizers of $\...
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References requestion : Pretopos are algebras for a composed monad?
Unless I'm mistaken the "Free completion under finite limits monad" $C \mapsto C^{lex}$ and the "free co-completion monad" $C \mapsto \widehat{C}$ (the categories of small presheaves) satisfies a ...
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Is Det-Stoch a factorization of the Giry Monad?
Stoch is the category of Measurable spaces and stochastic maps. It is the Klesli category of the Giry monad. Deterministic theories form a subcategory of Stoch. Specifically, the objects are just ...
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Is the Giry Monad also a Comonad and if not, is there a probability measures (Co)monad?
The Giry monad consists of an endofunctor, $P$, on the category of measureable spaces $\mathcal{M}$, as well as two natural transformations $\mu, \eta$ known as the product and unit respectively. $P$ ...
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What are the axioms of the diagrammatic calculus for containers?
Ahman et al. wrote about when a container is a comonad. Containers can also be monads, such as List. This means that we can take all containers that are endofunctors on Set and they live in the ...
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The domain monad
$\DeclareMathOperator\Set{\mathit{Set}}\DeclareMathOperator\Dom{\mathit{Dom}}\DeclareMathOperator\Hilb{\mathit{Hilb}}$Many different kinds of data structures can be captured as Monads. Lists and ...
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What is the (Co)Monad for a Bag
A Bag is a data structure, like a list, that stores items with no concept of order. The only operations on the structure is to add an item and then iterate through the items with no guarantee as to ...
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What are the adjunctions that generate the Giry Monad?
The Giry Monad captures probability measures. What is the adjunction that generates the Giry Monad? To narrow this down, perhaps we can talk about the adjunction between the category of Polish ...
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Pseudo or lax algebras for a 2-monad, reference request
I would like to find explicit definitions of pseudo, or even lax, algebras for a 2-monad, and their lax morphisms, with all the coherence diagrams included.
Alternatively, coherent lax algebras for ...
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Is there a bimonad on the category of sets that is exact?
I am wondering if it is possible to have a bimonad on $\mathsf{Set}$ that preserves equalizers on both sides? What about a bimonad that is exact? Can you give an example?
Let me try to explain what ...
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Monads associated to Higher Categories
By Koudenburg The paper (arXiv:[1511.04070])(https://arxiv.org/pdf/1511.04070) generalizes 2-monad associated to hyper virtual double category.
Another paper (arXiv:[1310.8279]) (https://arxiv.org/...
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Is this Frobenius Monad left exact? Does it preserve equalizers?
In this paper we see a Frobenius Monad in example 5.2. Suppose we take Hilb as the underlying category. Is this functor left exact? Does it preserve equalizers?
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Synthetic type theory for virtual double category and its higher categories
For some monad T on a virtual equipment, the paper A unified framework for generalized multicategories by Cruttwell and Shulman (arXiv:0907.2460) proposes the normalized T-monoid.
Another paper, by ...
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Fong's Causal Theories: Is he also describing a Monad structure? Is the causal category also a bimonad?
Fong's paper Causal Theories: A Categorical Perspective on Bayesian Networks talks about causal theories. He describes words of random variables at the top of page 42:
For the objects of CG we ...
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Why are simplicial objects monadic over split (contractible) simplicial objects?
Given an augmented simplicial object $d_\bullet:X_\bullet \to \Delta X_{-1}$, suppose there's a simplicial map $s_\bullet :\Delta X_{-1}\to X_\bullet$ making $d_\bullet$ a deformation retract, i.e ...
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Monad induced by actegory
It seems to be folklore that if we have an actegory, i.e. a monoidal functor from a monoidal category $C$ to an endofunctor category $Cat(D,D)$, we can obtain from it a monad on $D$. This appears for ...
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Relating three viewpoints on the semidirect product
It's known that giving a semidirect product $(X,m)\rtimes G$ of a $G$-group $(X,m)$ with $G$ (as defined in wiki) is the same as giving a split pair over $G$, i.e a pair of arrows $H\overset{s}{\...
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Monads on Set with trivial algebras
In an earlier post, What is known about the category of monads on Set?
the following observation was made:
What's more, all but two monads on Set have the property that there
exists an algebra ...
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English Reference for the Bénabou-Roubaud theorem
The Bénabou-Roubaud theorem links fibrational descent theory with monadicity. Particularly, it says that given a bifibration satisfying the Beck-Chevalley condition w.r.t some arrow $p$ in the base ...
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Lawvere theory and the Maybe monad
The Maybe monad is based on the endofunctor $- + 1$ (coproduct with the singleton set). Its Lawvere theory $L$ is supposed to be generated by one nullary operation (...
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Has anybody studied strict/pseudo morphisms of monads?
There is a notion of morphism from a monad $T:\mathscr C\to \mathscr C$ to another one $T':\mathscr C'\to \mathscr C'$. It arose here on MO e. g. in "Functors between monads": what are these ...
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Categories which are both monadic and comonadic over another category
I heard a professor say that $\lambda$-rings are both monadic and comonadic over commutative rings. Remark 2.11(a) on the nlab page says the same.
What does it mean, intuitively, that a category is ...
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Monads which are monoidal and opmonoidal
Do monads which are monoidal and opmonoidal have a name? (Bimonoidal?) In case they have already been studied, who can point me to a reference?
More in detail. Let $(C,\otimes)$ be a symmetric (or ...
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Monadicity of the composite of an operad and a monad
If $T$ is a monad on a category $\mathcal C$ and $T'$ is a monad on $T$-algebras, then (if I understand the answers of this question correctly) the adjunction between $\mathcal C$ and $T'$-algebras is ...
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Closure of polynomial monads under colimits
A polynomial monad on a locally cartesian closed category $C$ is a monad whose underlying endofunctor is a polynomial functor and whose unit and multiplication are cartesian transformations. Since a ...
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Applications of Monadicity theorems
This is crosspost of this MSE question.
Having carefully read the proof of Beck's monadicity theorems and some related variations, I'm now hungry for cool applications.
For instance, I found these ...
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Checking a monad is idempotent
I have a monad $T: \mathcal{C} \to \mathcal{C}$ on a (Grothendieck) abelian category which preserves filtered colimits and direct sums (but is not exact). There is a finite collection $G$ of compact, ...
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To what kind of generalized Lawvere theory does the "free cartesian closed category" 2-monad on $\mbox{Cat}_g$ correspond?
Thinking of Cat as a mere 1-category, there is a 1-monad $\Lambda$ for the free cartesian closed category on a category. To every category X it assigns the category $\Lambda(X)$ whose objects are ...
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Do "factoradic" lists form a finitary monad?
I'm trying to understand better what it means for a monad to be finitary. I know that Lawvere theories correspond to finitary monads, but I don't really understand the definition in terms of filtered ...
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Coherence laws when composing 2-monads
To have the composition of two monads be a monad itself, we need a
distributive law natural transformation satisfying certain coherence
laws.
I'm interested in the strict 2-monad case, i.e. a strict ...
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Beck's Theorem and the category of endos
Many years ago, Lawvere showed that the forgetful functor $U: \mathbf{Endo}\to \mathbf{Set}$ has a left adjoint $F$ if and only if $\mathbf{Set}$ has a natural numbers object, where $\mathbf{Endo}$ is
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"Maybe Monad" for multi-pointed objects?
Background:
A pointed object $X$ in a category $C$ with terminal object $*$ is a map $*\rightarrow X$. Such objects with basepoint-preserving maps form their own category of pointed objects $C^{*/}$. ...
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Free commutative monoid monad
Has the monad induced by the free commutative monoid functor already been studied anywhere? Does it have any particular properties (other than not being cartesian)?
I would prefer a reference on ...
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Free monad sequence versus colimit over injections of ordered sets
Kelly describes a constructive procedure for building the algebraically free monad on a pointed endofunctor. Garner gives a concise summary, which I partially review here for convenience.
Let $V$ be ...
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Is forming the Albanese variety a monad?
I'm trying to understand the idea of an Albanese variety. It reminds me of something simpler:
Given a set $X$ with a chosen point $x \in X$, we can form the free abelian group on the pointed set $(X,...
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