Questions tagged [monads]
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255 questions
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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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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 $a : \...
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$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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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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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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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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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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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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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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Domain Monad on Density Operators Using Spectral Order
The spectral order for density operators is given in this paper Coecke Martin 2010. I won't give the full definition here. Essentially, it allows for a partial order of density matrices that forms a ...
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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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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 ...
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Does the 2 category of Groupoids Admit the Vector Space Monad?
We can see here in Jacob's 2013 paper, that he seems to state that a particular kind of multiset monad is actually a vector space monad.
3.2. Vector spaces. For a semiring S one can define the ...
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Adjunctions between Groupoids and Hilbert spaces
I am interested in any adjunctions between any of the familiar categories of Groupoids and the category of finite dimensional Hilbert spaces. Do any exist? Are there any well know monads on the ...
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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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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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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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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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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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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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Why are monadicity and descent related?
This question is probably too vague for experts, but I really don't know how to avoid it.
I've read in several places that under mild conditions, a morphism is an effective descent morphism iff the ...
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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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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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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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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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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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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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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 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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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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A List-Like Frobenius Monad
Has anyone ever seen a Monad that is very much like the List Monad but is also a co-monad, and also a Frobenius monad? In this paper they give examples of List-like monads called Containers and they ...
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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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Is there any nontrivial monad on the category of graphs?
The question is in the title, but let me specify what I mean by the category of graphs.
In the context of this question, the category of graphs is the category of symmetric irreflexive relations. ...
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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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The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?
Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to its underlying ...
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