Questions tagged [monads]

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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 ...
geodude's user avatar
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9 votes
0 answers
516 views

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/...
user65526's user avatar
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9 votes
2 answers
357 views

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 $\...
Hadrian Heine's user avatar
6 votes
1 answer
204 views

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 ...
Simon Henry's user avatar
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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 ...
Ben Sprott's user avatar
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5 votes
1 answer
342 views

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. $P$ ...
Ben Sprott's user avatar
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2 votes
0 answers
136 views

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 ...
Ben Sprott's user avatar
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1 vote
2 answers
420 views

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 ...
Ben Sprott's user avatar
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2 votes
1 answer
497 views

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 ...
Ben Sprott's user avatar
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5 votes
4 answers
970 views

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 ...
Ben Sprott's user avatar
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3 votes
1 answer
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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 ...
geodude's user avatar
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3 votes
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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 ...
Ben Sprott's user avatar
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1 vote
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60 views

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/...
HuiFang's user avatar
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2 votes
0 answers
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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?
Ben Sprott's user avatar
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1 vote
0 answers
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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 ...
HuiFang's user avatar
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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 ...
Ben Sprott's user avatar
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6 votes
2 answers
724 views

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 ...
Arrow's user avatar
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2 votes
1 answer
327 views

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 ...
geodude's user avatar
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8 votes
1 answer
451 views

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}{\...
Arrow's user avatar
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8 votes
1 answer
393 views

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 ...
P. Corazza's user avatar
9 votes
5 answers
945 views

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 ...
Arrow's user avatar
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3 votes
2 answers
331 views

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 (...
Bartosz Milewski's user avatar
6 votes
1 answer
303 views

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 ...
მამუკა ჯიბლაძე's user avatar
13 votes
2 answers
773 views

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 ...
Arrow's user avatar
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5 votes
0 answers
75 views

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 ...
geodude's user avatar
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3 votes
0 answers
101 views

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 ...
Maxime Lucas's user avatar
5 votes
0 answers
190 views

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 ...
Mike Shulman's user avatar
7 votes
0 answers
377 views

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 ...
Arrow's user avatar
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4 votes
0 answers
208 views

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, ...
Dylan Wilson's user avatar
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9 votes
0 answers
308 views

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 ...
Mike Stay's user avatar
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1 vote
1 answer
131 views

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 ...
Mike Stay's user avatar
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5 votes
1 answer
73 views

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 ...
Mike Stay's user avatar
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7 votes
0 answers
175 views

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 ...
P. Corazza's user avatar
2 votes
1 answer
351 views

"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 $C^{*/}$. ...
user84563's user avatar
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2 votes
0 answers
285 views

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 ...
geodude's user avatar
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7 votes
1 answer
171 views

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 ...
Gabriel C. Drummond-Cole's user avatar
25 votes
1 answer
1k views

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,...
John Baez's user avatar
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6 votes
2 answers
348 views

is shuffle a Monad?

In the list monad, your map $TT \rightarrow T$ takes a list of lists and concatenates them to form a list. There is another way to take a list of lists and create a list, which is to shuffle randomly ...
Ben Sprott's user avatar
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4 votes
1 answer
418 views

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 ...
Ben Sprott's user avatar
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9 votes
1 answer
567 views

Intuition for density comonad in relation to lifting problems

In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...
Arrow's user avatar
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7 votes
1 answer
1k views

List is a monad, but is it a comonad with these natural transformations?

List is known to be a monad. It takes a set and maps it to lists of elements of that set. The natural transformations are, singleton and flatten, whereby we map a set to a set of singleton lists ...
Ben Sprott's user avatar
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30 votes
2 answers
3k views

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 ...
Arrow's user avatar
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5 votes
2 answers
517 views

Algebras for probability monad

What is the Eilenberg-Moore category for the non-finitary probability distribution monad is, that is, the monad $D \colon \mathbf{Set} \to \mathbf{Set}$ defined by $$ DX = \left\{ p \in [0,1]^X \ \...
Paul Slevin's user avatar
0 votes
1 answer
168 views

iterated loop spaces and configuration spaces [closed]

In the lecture notes by J.P. May, The geometry of iterated loop spaces, Chapter 5, formula (1), (2) and (10), a map $$ \phi: Hom_T(X,\Omega Y)\to Hom_T(SX,Y) $$ is defined. And a map $$ \eta_n=\phi^{-...
Shiquan Ren's user avatar
  • 1,970
1 vote
1 answer
170 views

tree derived from monad is itself a monad

I have constructed a functor from a monad that appears (based on computer experiments to test the monad laws) to also have monad properties but I am having trouble proving it. Here is the idea: M[A] ...
greenTara's user avatar
8 votes
0 answers
252 views

Whiskering a monad

In "The Geometry of Iterated Loop Spaces", May shows that any monad that is coming from an operad may be "whiskered", so that the unit map becomes a closed cofibration. The ability to do this is vital ...
Matthew Sartwell's user avatar
6 votes
3 answers
1k views

opposite category

In the 2-category Cat of small categories, for each category $C$ (an object of Cat) there is also the dual category (I dare not write "dual object") $C^{op}$. Is ${op}$ the instance in Cat of a more ...
Bob's user avatar
  • 476
3 votes
2 answers
268 views

Comonads from monoids

The following construction is probably known. I think it should work in any closed symmetric monoidal category, but I will play it safe and formulate the question in the concrete, cartesian closed ...
Gejza Jenča's user avatar
2 votes
1 answer
458 views

When is an Eilenberg-Moore category or Kleisli category braided monoidal? When semisimple?

I have a braided monoidal, semisimple linear category $\mathcal{C}$. (Imagine representations of a semisimple quasitriangular Hopf algebra.) I also have a monad $(T,\mu,\eta)$ on it, however, $T$ is ...
Manuel Bärenz's user avatar
6 votes
1 answer
461 views

Universal property of module categories over monads

Let $T$ be a monad on a cocomplete category $\mathcal{C}$. Let's assume that $T$ preserves reflexive coequalizers (or something weaker?). Then the category of $T$-modules $\mathsf{Mod}(T)$ is ...
Martin Brandenburg's user avatar