Questions tagged [monads]
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242
questions
4
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When is the Eilenberg-Moore category of a monad on an ind-category itself an ind-category?
I have a monad on an ind-category (specifically, my ind-category has a monoidal structure and I have an algebra object, so the monad is tensoring with it). It would be very useful in my work if the ...
2
votes
1
answer
217
views
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 ...
4
votes
1
answer
184
views
What is the universal property of algebras for the codensity monad?
Let $F : A \to B$ be a functor, and suppose that the right Kan extension $T = Ran_F F : B \to B$ exists. Then $T$ is a monad, the codensity monad of $F$. Moreover, unless I'm mistaken there is a ...
4
votes
1
answer
190
views
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 ...
3
votes
0
answers
71
views
Adjoints to the forgetful functor from the $2$-category of monads
For the purpose of this post, we will identify a monad $T$ on a category $\mathcal{C}$ with a lax $2$-functor $T:{\bf 1}\to\mathfrak{Cat}$ such that $T(*)=\mathcal{C}$.
There is an obvious forgetful ...
2
votes
0
answers
256
views
The Kleisli category of a monoidal monad
Let $C$ be a symmetric monoidal category equipped with diagonals $\triangle_x: x \to x \otimes x$, that is, equipped with natural transformations $e_x: x \to 1$ and $\triangle_x : x \to x \otimes x $ ...
6
votes
0
answers
114
views
Original reference for the correspondence between commutative algebraic theories and commutative monads
Commutative algebraic theories were introduced by Linton in the 1966 paper Autonomous Equational Categories. Commutative monads were introduced by Kock in the 1970 paper Monads on symmetric monoidal ...
3
votes
1
answer
204
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Characterisation of functors whose left adjoint is Kleisli
This question is inspired by Characterization of functors whose right adjoint is monadic?.
Let $F : \mathbf C \rightleftarrows \mathbf D : U$ be an adjunction, and suppose that we want to establish ...
1
vote
1
answer
110
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Conditions such that split coequalizers are a symmetric notion
Consider the notion of a split coequalizer (see the nLab for the definition). Note that the definition seems to be non-symmetric. Are there any conditions on the ambient category such that it becomes ...
17
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2
answers
1k
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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 ...
4
votes
2
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348
views
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 ...
2
votes
0
answers
73
views
Diagrammatic model for free product in monad infinity category
$\newcommand{\C}{\mathcal{C}}$ Suppose $M$ is a monad in an $\infty$-category $\C,$ and $A, B$ are two algebras over $M$. I'm willing to assume any reasonable "niceness" conditions on $\C$, $...
3
votes
1
answer
141
views
Examples of (co)lax idempotent pseudocomonads on Cat
A lax idempotent pseudomonad, also called a KZ doctrine or KZ monad, is a pseudomonad $(T, \mu, \eta)$ with the property that $T \eta \dashv \mu \dashv \eta T$. Lax idempotent pseudomonads were ...
1
vote
1
answer
90
views
Algebras for general transfors
Algebras for endofunctors bridge the gap between functors acting on a category and structures defined in it. An algebra for an endofunctor $F$ is instantiated by some morphism $Fa \to a$, and more ...
14
votes
1
answer
449
views
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
$$\mathbf{Set}^{\mathrm{op}} \to \mathbf{CABA}, ~ X \mapsto (P(X),\bigcup,\bigcap).$$
...
2
votes
2
answers
176
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Finitary endofunctors: "Support" of elements of $TM$ where $T:\mathrm{Set}\to\mathrm{Set}$ is finitary
Consider the word a.k.a. list monad $W:\mathrm{Set}\to\mathrm{Set}$ assigning to a given set $M$ the set of words over $M$. Now, given a set $M$, we can assign to every word $w\in WM$ a subset of $M$:
...
14
votes
3
answers
698
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Reference request for Linton's theorems on equational theories
This is a reference request for the following "well-known" theorems in category theory:
There is an equivalence of categories between finitary monads on $\mathbf{Set}$ and finitary Lawvere ...
2
votes
0
answers
488
views
Cyclic lists of multisets
I am wondering if it is appropriate to ask about having a specific algebra for an endofunctor computed. We all know about the multiset monad, and it's endofunctor $\mathcal{M}_S$, and some might know ...
3
votes
0
answers
67
views
Lax algebras for pseudomonads and monads in Kleisli bicategories for the induced pseudocomonad
In Day–Street's Lax monoids, pseudo-operads, and convolution, they remark without proof:
There are general principles involved here. Suppose $(T, m, j)$ is a pseudomonad on any bicategory $\mathcal K$...
3
votes
1
answer
87
views
Does the right adjoint of a comonad induce the following comodule map?
Let $\mathcal{C}$ be a category and $\mathcal{G}=(G,\delta, \epsilon)$ be a comonad on $\mathcal{C}$. Here $G: \mathcal{C}\to \mathcal{C}$ is a functor, $\delta: G\to G^2$ and $\epsilon: G\to id_{\...
4
votes
1
answer
201
views
Monads and modules, and the bicompletion under Kleisli and Eilenberg–Moore objects
In The Formal Theory of Monads, Street proves that a 2-category $\mathscr C$ admits the construction of algebras when the inclusion $\mathscr C \to \mathbf{Mnd}(\mathscr C)$ has a right adjoint. In ...
10
votes
1
answer
438
views
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 ...
6
votes
3
answers
490
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Contramodule as direct limit of its finitely generated subcontramodules
$\DeclareMathOperator\Hom{Hom}$Let $K$ be a field. Let $C$ be a $K$-coalgebra. A contramodule $M$ over $C$ is a $K$-space with a $K$-linear map $\pi_M:\Hom_K(C,M)\longrightarrow M$ such that $\pi_M \...
3
votes
0
answers
126
views
Does the notion of a Poisson monad exist?
Starting with a monoidal category with duals $C$, one may consider the category $End(C)$ of endofunctors of $C$. A Hopf monad on $C$ is a bimonad on $C$ with (a generalised notion of the) antipode. ...
25
votes
1
answer
2k
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Infinity-categorical analogue of compact Hausdorff
Recently I became through this mathoverflow question aware of the article Codensity and the ultrafilter monad by Tom Leinster. There he shows that the ultrafilter monad on the category $\mathrm{Set}$ ...
4
votes
1
answer
186
views
Do (co)density (co)monadic constructions stablize?
Under good conditions [1], any functor $F: C \to D$ induces a codensity monad $T: D \to D$ as a right Kan extension of $F$ along itself. It does not say explicitly, but by considering left/right Kan ...
5
votes
2
answers
466
views
A specific property of bi-adjunction
Let $$I: C \rightleftarrows D: F$$ be biadjoint [1] functors between categories $C, D$. That is, $I$ is the left and also the right adjoint of $F$ (thus vice versa). Put in notations, it's
$$ \cdots \...
6
votes
0
answers
339
views
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). ...
8
votes
0
answers
160
views
What is the relationship between free bicompletion and the Isbell envelope?
Given a small category $\mathbb C$, we can form the free cocompletion $\mathbf y : \mathbb C \to \mathcal P(\mathbb C)$ and the free completion $\mathbf y^\circ : \mathbb C \to \mathcal P^\circ(\...
2
votes
0
answers
141
views
Are flasque sheaves exactly the retracts of "canonically" flasque sheaves?
Let $X$ be a topological space. Let $X^\delta$ denote the space whose elements are the points of $X$, and which is equipped with the discrete topology. There is a continuous map $i : X^\delta\to X$ ...
7
votes
0
answers
255
views
Relation between two limit presentations of Eilenberg--Moore objects
Let $\mathbb{T}=({\cal T}\colon C\to C,\mu,\eta)$ be a monad (in the
$2$-category $\mathsf{Cat}$), which we view as a $2$-functor
$\mathbb{T}\colon\mathsf{B}\Delta_{\mathrm{a}}\to\mathsf{Cat}$ (where
$...
9
votes
1
answer
303
views
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 $\...
4
votes
1
answer
377
views
Different ways to “deloop” a (topological) $A_\infty$-algebra
Let $\varphi:A\to \mathrm{Ass}$ be an $A_\infty$-operad in topological spaces, and let $X$ be an $A$-algebra. I see three possibilities to construct a delooping out of $X$:
Rectify $X$ by taking the ...
3
votes
1
answer
58
views
Morphism of pseudomonads induces pullback functors between pseudoalgebras
If $S$ and $T$ are monads on a category $C$, and $\lambda:S\to T$ is a morphism of monads, it is well-known that there is a functor $\lambda^*:C^T\to C^S$ which assigns to the $T$-algebra $(A,a:TA\to ...
6
votes
0
answers
151
views
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 $...
7
votes
1
answer
188
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Algebras for products or limits of monads
If a category $C$ has limits of a certain type, then the category of monads on $C$ has the same type of limits, and these limits are computed "levelwise" (i.e. are preserved by the forgetful ...
3
votes
2
answers
248
views
Is monadicity preserved by the underlying functor?
Let $\mathcal{V}$ be a monoidal closed (complete, cocomplete, reasonable...) category.
Let $\mathsf{T}$ be an enriched monad over $\mathcal{V}$. The forgetful functor $\mathsf{U}: \mathsf{Alg}(\...
0
votes
1
answer
148
views
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 ...
8
votes
1
answer
312
views
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 ...
2
votes
0
answers
125
views
A morphism of monads that doesn't preserve thunkability?
Recall that for a monad $(T,\eta,\mu)$ on a category $C$, the Kleisli category $C_T$ has as objects the objects of $C$ and as morphisms $C_T(x,y) = C(x,T y)$. A morphism $f\in C_T(x,y) = C(x,T y)$ is ...
5
votes
0
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161
views
Algebras of the cone monad on Top?
Let us work in Top, the category of topological spaces - although the reader is welcome to replace this by their favorite convenient category of topological spaces.
If $X,Y$ are spaces, let $X\ast Y$ ...
1
vote
0
answers
102
views
When do objects in the image of a functor $G$ have a unique action as algebras over the codensity monad of $G$?
Let $G:\mathcal{B}\longrightarrow \mathcal{A}$ be some functor which admits a right Kan extension along itself, $(\operatorname{Ran}_G G, \eta:\operatorname{Ran}_G G \circ G \rightarrow G)$.
The ...
0
votes
0
answers
110
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Expressing a model transformation by using monads in the simply-typed lambda calculus
In https://link.springer.com/content/pdf/10.1007/s10670-019-00128-z.pdf , page 16, the following clause is given for a modal operator $\langle R_k \rangle$ (see definition 4.2 for the definition of a ...
7
votes
1
answer
451
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Characterisation of essentially algebraic theories as monads
The following correspondence between algebraic theories and monads on $\mathbf{Set}$ is well-known (see, for example, Algebraic Theories: A Categorical Introduction to General Algebra).
The ...
21
votes
4
answers
2k
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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 ...
3
votes
2
answers
385
views
Ultrafilter comonad on the category of Stone spaces
Let $\mathsf{Stone}$ denote the category of Stone spaces (compact, totally disconnected Hausdorff spaces) and continuous maps. The forgetful functor $U : \mathsf{Stone} \to \mathsf{Set}$ has a left ...
3
votes
2
answers
325
views
The symmetric monoidal closed structure on the category of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors
In 6.5 of the book by Kelly,
Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, No. 10, 2005.
the author claims that the $2$-category $\mathsf{Cat}_{\...
15
votes
1
answer
459
views
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), ...
12
votes
2
answers
355
views
when is an integer sequence the trace of a monad on FinSet?
Given $(a_n \in \mathbb{N})$, when is there a monad $T$ on $\mathrm{FinSet}$ such that
$$
| T(n) | = a_n\quad\forall n\in \mathbb{N}\:?
$$
1
vote
0
answers
122
views
Do these monads on Rel compose?
$Rel$ is the category of sets and relations.
The cyclic list monad, $\mathcal{Cy}=(Cy, \mu_c, \eta_c)$ is defined as follows:
$Cy : Rel \rightarrow Rel$,
such that, $Cy(X)$ is all cyclic lists on ...