### -1 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 ...

### 1 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 ...

### 1 (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 ...

### 5 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 ...

### 4 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 ...

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 ...

### 2 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 ...

### 4 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 ...

### 1 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 ...

### 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 ...

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 ...

### 8 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 ...

### 25 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 ...

### 6 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 ...

### 1 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 ...

Many different kinds of data structures can be captured as Monads. Lists and trees are two good examples. A domain (dcpo) is like a tree, with extra axioms. Definition. A directed subset of a ...

### 3 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. $P$ ...

### 2 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 ...

### 2 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 ...

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 ...

### 6 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 ...

### 3 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 ...

### 2 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 ...

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 $H\overset{s}{\... ### 1 Monads associated to Higher Categories 0 answers, 42 views monads By Koudenburg The paper (arXiv:[1511.04070])(https://arxiv.org/pdf/1511.04070) generalizes 2-monad associated to hyper virtual double category. Another paper (arXiv:[1310.8279]) (https://arxiv.org/... ### 2 Is this Frobenius Monad left exact? Does it preserve equalizers? In this paper we see a Frobenius Monad in example 5.2. Suppose we take Hilb as the underlying category. Is this functor left exact? Does it preserve equalizers? ### 1 Synthetic type theory for virtual double category and its higher categories For some monad T on a virtual equipment, the paper A unified framework for generalized multicategories by Cruttwell and Shulman (arXiv:0907.2460) proposes the normalized T-monoid. Another paper, by ... ### 3 Fong's Causal Theories: Is he also describing a Monad structure? Is the causal category also a bimonad? 0 answers, 187 views ct.category-theory monads Fong's paper Causal Theories: A Categorical Perspective on Bayesian Networks talks about causal theories. He describes words of random variables at the top of page 42: For the objects of CG we ... ### 5 Why are simplicial objects monadic over split (contractible) simplicial objects? Given an augmented simplicial object$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$a deformation retract, i.e ... ### 4 A List-Like Frobenius Monad 1 answers, 269 views ct.category-theory monads Has anyone ever seen a Monad that is very much like the List Monad but is also a co-monad, and also a Frobenius monad? In this paper they give examples of List-like monads called Containers and they ... ### 2 Monad induced by actegory It seems to be folklore that if we have an actegory, i.e. a monoidal functor from a monoidal category$C$to an endofunctor category$Cat(D,D)$, we can obtain from it a monad on$D$. This appears for ... ### 8 Monads on Set with trivial algebras 1 answers, 353 views monads In an earlier post, What is known about the category of monads on Set? the following observation was made: What's more, all but two monads on Set have the property that there exists an algebra ... ### 3 Is there any nontrivial monad on the category of graphs? The question is in the title, but let me specify what I mean by the category of graphs. In the context of this question, the category of graphs is the category of symmetric irreflexive relations. ... ### 2 Lawvere theory and the Maybe monad 2 answers, 207 views universal-algebra monads The Maybe monad is based on the endofunctor$- + 1$(coproduct with the singleton set). Its Lawvere theory$L$is supposed to be generated by one nullary operation (... ### 6 Has anybody studied strict/pseudo morphisms of monads? There is a notion of morphism from a monad$T:\mathscr C\to \mathscr C$to another one$T':\mathscr C'\to \mathscr C'$. It arose here on MO e. g. in "Functors between monads": what are these ... ### 10 Categories which are both monadic and comonadic over another category 2 answers, 198 views adjoint-functors monads lambda-rings I heard a professor say that$\lambda$-rings are both monadic and comonadic over commutative rings. Remark 2.11(a) on the nlab page says the same. What does it mean, intuitively, that a category is ... ### 4 Monads which are monoidal and opmonoidal Do monads which are monoidal and opmonoidal have a name? (Bimonoidal?) In case they have already been studied, who can point me to a reference? More in detail. Let$(C,\otimes)$be a symmetric (or ... ### 13 The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories? Let$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$to its underlying ... ### 20 “Functors between monads”: what are these really called? 2 answers, 1.720 views ct.category-theory monads Let$(S,\eta,\mu)$be a monad on a category$C$, and$(T,\eta,\mu)$a monad on a category$D$. The following kind of gadget is ubiquitous: a functor$F:D\to C$, together with a natural map$\sigma: ...

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 ...

### 5 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 ...

### 5 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 ...

### 4 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, ...

### 4 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 ...