Questions tagged [monads]
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39 questions
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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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What is known about the category of monads on Set?
Monads on the category Set of sets and functions are somehow fundamental objects of category theory, and moreover they have important applications to computer science. We know of a good number of ...
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Jon Beck's untitled manuscript containing the "tripleability theorem" (i.e. the monadicity theorem)
Many papers refer to an untitled manuscript of Jon Beck (Cornell, 1966) for the origin of the monadicity theorem (originally called a "tripleability theorem"). An early proof is in Manes's ...
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Conceptual reason that monadic functors create limits?
Let $U: Alg_T \to C$ be the forgetful functor from the category of algebras of $T: C \to C$ ($T$ could be a monad; I'm happy to think about the simpler case where $T$ is just an endofunctor or pointed ...
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Characterization of Kleisli adjunctions
There's a well known theorem due to Beck that characterizes when an adjunction is monadic, that is, if $F$ is left adjoint to $G$, $G:D \to C$, $GF:=T$ is always a monad on $C$, and the adjunction is ...
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Correspondence between operads and monads requires tensor distribute over coproduct?
In checking the details of the correspondence between operads over a symmetric monoidal category and monads on some associated endofunctor of the category, I cannot make the obvious proof work without ...
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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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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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Big list of comonads
The concept of a monad is very well established, and there are very many examples of monads pertaining almost all areas of mathematics.
The dual concept, a comonad, is less popular.
What are examples ...
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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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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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"Functors between monads": what are these really called?
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: ...
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Relation between monads, operads and algebraic theories
I've begun to interest in algebraic theories and their categorical models: in particular monads, generalized multicategories and operads, lawvere theories and their generalization. Is there any ...
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What are the algebras for the ultrafilter monad on topological spaces?
Motivation: Let $(X,\tau)$ be a topological space. Then the set $\beta X$ of ultrafilters on $X$ admits a natural topology (cf. Example 5.14 in Adámek and Sousa - D-ultrafilters and their monads), ...
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How exactly is Hochschild homology a monad homology?
Many texts which praise the generality of the bar construction associated to a monad, say that Hochschild homology is an example of this.
What exactly is in this case the underlying endofunctor of ...
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Are monads monadic?
Is there some sort of monad whose algebras are monads? How about if we are internal to a bicategory B? Are internal monads in B monadic? Certainly not always, as otherwise free T-multicategories a la ...
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What are internal complete atomic boolean algebras, intuitively?
The category of complete atomic boolean algebras $\mathbf{CABA}$ is equivalent to $\mathbf{Set}^{\mathrm{op}}$ via
$$\mathbf{Set}^{\mathrm{op}} \to \mathbf{CABA}, ~ X \mapsto (P(X),\bigcup,\bigcap).$$
...
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Characterization of functors whose right adjoint is monadic?
Let $F: \mathcal A^\to_\leftarrow \mathcal B: U$ be an adjunction, and suppose we want to know whether the comparision functor $\mathcal B \to Alg^{UF}$ is an equivalence, where $Alg^{UF}$ is the ...
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2-monads for categories with a class of (co)limits
This question concerns the strictness of (co)completions, at various levels of generality.
In Blackwell–Kelly–Power's Two-dimensional monad theory, the authors state
For instance, the 2-category $\...
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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 ...
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9
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Re-seating a monad
Let $\mathcal C$ and $\mathcal D$ be categories with suitable limits and colimits for the following discussion. Is it possible to re-interpret, or "re-seat" a monad $T : \mathcal C \to \mathcal C$ as ...
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Finitary monads on $\operatorname{Set}$ are substitution monoids. Finitary monads on $\operatorname{Set}_*$ are...?
$\DeclareMathOperator\Fin{Fin}\DeclareMathOperator\Lan{Lan}\DeclareMathOperator\Set{Set}$
The present question is intimately related to another question.
It is well known that the category of ...
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When is the Eilenberg-Moore category of a relative monad between two topoi a topos?
In the non-relative case, we have a theorem, that an Eilenberg-Moore category of algebras of a Monad $T$ on a topos is itself a topos if the monad in question has a right adjoint.
Now how does this ...
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Adjunctions: algebras of the induced monad VS. coalgebras of the induced comonad
Given an adjunction, we get a monad on one side and a comonad on the other side. What is the connection between their algebra and coalgebra categories? Are they always equivalent?
The example i have ...
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Coherence for pseudomonads and their pseudoalgebras
Let $\mathcal K$ be a bicategory. For every pseudomonad $T : \mathcal K \to \mathcal K$, does there exist a 2-monad $S : \mathcal C \to \mathcal C$, where $\mathcal C$ is a 2-category biequivalent to $...
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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 ...
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Cohomology without comonad?
TL;DR. Many cohomologies can be unified using comonads. Question: which cohomologies cannot be?
For each algebraic theory, there is an adjunction, and therefore a (co)monad (or called a (co)triple). ...
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Monad arising from operad
It's known that from every operad arises a cartesian monad whose algebras are the algebras for the operad. Leinster proved that there are different operads from which arise the same monad, in this way ...
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Free cocommutative commutative Hopf monoids
I have some questions about generalizations of abelian groups, relative to symmetric monoidal categories.
1) Let $C$ be a cocomplete cartesian monoidal category with equalizers. I can show that the ...
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What should be required from a model category so that the category of algebraic objects in it has the natural model structure?
I have two reference questions
What should be required of a category with finite products so that a (multi-sorted, finitary) Lawvere theory induces a monadic adjunction on it? This should be ...
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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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The bidualizing monad
Let $\mathbf{C}$ be a closed symmetric monoidal category (I probably need even less than this; the examples I have in mind are simply the category of modules over a commutative ring and the category ...
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Constructing the E-M category of a monad out of inserters and equifiers
As suggested in the answer to another MO question, it seems possible to construct the E-M category of a monad $T:\mathcal{C}\to\mathcal{C}$ as an inserter followed by two equifiers as follows (I am ...
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What is the category of algebras for the finitely supported measures monad?
In this post, I was introduced to the monad of finitely supported measures.
$HX$ is the set of finitely supported measures on $X$, with monad structure defined as for the Giry monad.
I have three ...
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2
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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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Uniqueness of comparison functors
Given an adjunction $F\dashv G:\mathcal{C}\rightleftarrows\mathcal{D}$ with unit $\eta$ and counit $\epsilon$, we naturally have a monad $(G\circ F,\eta,G\epsilon_F)$ on $\mathcal{C}$ and a comparison ...
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Quantum scattering experiments: C-modules, N-modules and their monads
I am working on a theory of particle physics where we use monads. I have a few conjectures that I need to check.
The category of $\mathbb{C}$-modules is monadic over set
The category of $\mathbb{N}$...
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Substitution structure on pointed sets
$\def\Fin{\text{Fin}_*}
\def\Set{\text{Set}_*}
\def\dd{\mathop{\diamond_\land}}$
The present question is intimately related to another question.
Let $\Fin$ be the category of pointed sets.
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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