Questions tagged [monads]
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255 questions
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A new (?) way of composing monads
By composition of monads, I mean given two monads $S$ and $T$, making their composite $S T$ into a monad. Or more generally, given two monoid $X$ and $Y$ in a non-symetric monoidal category, making $X ...
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2
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2
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137
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If a monad in a 2-category admits a terminal resolution, does it admit an Eilenberg–Moore object?
Let $T = (t, \mu, \eta)$ be a monad on an object $A$ of a 2-category $\mathcal K$. In The formal theory of monads, Street proves (Theorem 3) that if $l \dashv r$ is the canonical adjunction associated ...
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Separable monads do not induce separable monoids
Let us first recall the categorical notion of monad: if we have a category $\mathcal{C}$ then a monad on it consists in an endofunctor $\mathbb{A}\colon \mathcal{C}\rightarrow \mathcal{C}$ together ...
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What is the free lax-idempotent adjunction?
Let $Adj$ be the free adjunction, i.e. the 2-category such that for any 2-category $K$, the functor 2-category $2Fun(Adj, K)$ is the 2-category of adjunctions in $K$ (naturally in $K$). Note that $Adj$...
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Are infinitary monads monadic?
As discussed here, Are monads monadic?, in "On the monadicity of finitary monads" by Steve Lack, the following is shown, the forgetful functor from $Mnd_f(C) \rightarrow Endo_f(C)$ is ...
- 1,347
2
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1
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243
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Well-behaved monad quotients
Reading through Modular specification of monads through higher-order presentations, this paper includes the following lemma within set-truncated homotopy type theory:
Given a monad $R$ (they work on ...
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0
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124
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The S-module Ass is same as the composite of Com and Lie
It has been cited in several places (eg. https://arxiv.org/pdf/1912.05519.pdf) that the S-module Ass is isomorphic to the composite of the S-modules Com and Lie. Is there a reference which gives the ...
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5
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1
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178
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What are the algebras for the laxification 2-monad?
Let $C$ be a small 2-category. Let $[C , Cat]$ denote the 2-category of strict functors to $Cat$, 2-natural transformations, and modifications. Let $[[ C, Cat ]]$ denote the 2-category with the same ...
- 64k
2
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223
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EM functor from monads to adjunctions
What is the action on $1$-cells of the functor sending a monad to its EM adjunction? What about the Kleisli adjunction?
Let $A$ be the walking adjunction. Recall that an adjunction is the same thing ...
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2
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0
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137
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Homotopy fixed points vs coalgebras
Referring to the last part of this answer https://mathoverflow.net/a/225403/170683, I would like to understand how in the case of a Galois cover $f\colon X\to Y=X/G$ with Galois group $G$ (I guess ...
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Are pseudomonoids weak algebras for a 2-monad?
I would like to know if whether or not the pseudomonoids in an arbitrary monoidal 2-category are (equivalent to) the weak algebras for some 2-monad (I am thinking about the free monoidal category 2-...
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1
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351
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Algebraically-free monadicity theorem
The monadicity theorem characterises when a functor $u : \mathbf B \to \mathbf E$ is the forgetful functor from the category of algebras for some monad on $\mathbf E$ (up to an equivalence over $\...
- 10.7k
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Let $T$ be a strongly cartesian monad on a presheaf category $\hat C$. Then is $\hat C$ comonadic over $\operatorname{Alg} T$?
$\DeclareMathOperator\Alg{Alg}\newcommand{\Set}{\mathit{Set}}\newcommand{\Set}{\mathit{Set}}\newcommand{\Ab}{\mathit{Ab}}$Let $C$ be a small category, and let $T$ be a strongly cartesian monad on the ...
- 64k
4
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1
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250
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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
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1
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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 ...
- 10.1k
4
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1
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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 ...
- 64k
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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 ...
- 10.1k
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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 ...
- 10.1k
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0
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283
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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 $ ...
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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 ...
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3
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1
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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 ...
- 10.7k
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1
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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 ...
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2
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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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2
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356
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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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2
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0
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76
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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 $\C$, $...
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3
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1
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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 ...
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1
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91
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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 ...
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14
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1
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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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2
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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 $M$:
...
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15
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3
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768
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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 ...
- 63.1k
2
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0
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494
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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 ...
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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 $\mathcal K$...
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1
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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_{\...
- 9,019
4
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1
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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 ...
- 10.7k
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1
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466
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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 ...
- 64k
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3
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523
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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 \...
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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. ...
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1
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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 $\mathrm{Set}$ ...
- 12.1k
4
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1
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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 ...
- 5,230
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2
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550
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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 \...
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0
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354
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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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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(\...
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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 $i : X^\delta\to X$ ...
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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
$...
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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 $\...
- 10.7k
4
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1
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391
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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 ...
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3
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1
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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 ...
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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 $...
- 10.7k
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1
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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 ...
- 42.4k
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2
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255
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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}(\...
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