Questions tagged [monads]
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242
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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 ...
9
votes
0
answers
516
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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/...
9
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2
answers
357
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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 $\...
6
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1
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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 ...
1
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0
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93
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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 ...
5
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1
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342
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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. $P$ ...
2
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0
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136
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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 ...
1
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2
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420
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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 ...
2
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1
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497
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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 ...
5
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4
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970
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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 ...
3
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1
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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 ...
3
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0
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159
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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 ...
1
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0
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60
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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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0
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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 ...
3
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0
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234
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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 ...
6
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2
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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 ...
2
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1
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327
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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 ...
8
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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}{\...
8
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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 ...
9
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5
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945
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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 ...
3
votes
2
answers
331
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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 (...
6
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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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773
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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 ...
5
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75
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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 ...
3
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0
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101
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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 ...
5
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190
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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 ...
7
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377
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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 ...
4
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208
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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, ...
9
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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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1
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131
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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 ...
5
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1
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73
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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 ...
7
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175
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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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351
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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 $C^{*/}$. ...
2
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0
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285
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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 ...
7
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1
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171
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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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2
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348
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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 ...
4
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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 ...
9
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1
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567
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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 ...
7
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1
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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 ...
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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 ...
5
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2
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517
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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
$$
DX = \left\{ p \in [0,1]^X \ \...
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1
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iterated loop spaces and configuration spaces [closed]
In the lecture notes by J.P. May, The geometry of iterated loop spaces, Chapter 5, formula (1), (2) and (10), a map
$$
\phi: Hom_T(X,\Omega Y)\to Hom_T(SX,Y)
$$
is defined. And a map
$$
\eta_n=\phi^{-...
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1
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170
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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] ...
8
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252
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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 ...
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3
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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 ...
3
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268
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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 ...
2
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1
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458
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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 ...
6
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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 ...