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      Questions tagged [monads]

      The tag has no usage guidance.

      7
      votes
      0answers
      118 views

      Relative cocompletion of a category

      $\newcommand{\k}{\mathbf k}$ $\newcommand{\A}{\mathcal A}$ $\newcommand{\B}{\mathcal B}$ $\newcommand{\C}{\mathcal C}$ I'm wondering if anyone knows a reference for the following construction: let $\k$...
      5
      votes
      0answers
      84 views

      U-split coequalizer preserved by coproduct

      Let $C$ be a nice symmetric monoidal category and let $T$ be a monad on $C$ such that we have a monadic adjunction $$F:C\leftrightarrow T\text{-}alg(C):U $$ Suppose that $a\rightrightarrows b$ is a ...
      4
      votes
      1answer
      142 views

      Monad, algebras and reflexive coequalizer

      Suppose we have an adjunction of categories $F:M\leftrightarrows N:U$. We define the associated (co)monad $G=F\circ U$. For any object $x\in N$ we define the simplicial resolution of $x$ given by $$ ...
      3
      votes
      0answers
      150 views

      What category of toposes is monadic over the 2-category of groupoids?

      Excuse my lack of understanding of monadicity, but I have been looking at toposes and monads. I see Lambek showed that the category of Toposes are monadic over the category of categories. I see the ...
      1
      vote
      0answers
      48 views

      Schemes for conditional distributions (monads)

      (Note: This is a heuristic question. I'm trying to work out if this idea makes sense. I don't have much background in this area, so apologies if I'm wide of the mark.) Suppose you have a monad ...
      0
      votes
      0answers
      72 views

      Is there a natural transformation from Lists to Domains of Lists

      The list monad, $(L, \mu_L, \eta_L)$, on $Set$, takes a set to its set of lists, with $\mu_L : L \cdot L \rightarrow L$ being concatenation of lists. Given a set of lists, there is a natural way to ...
      8
      votes
      0answers
      172 views

      Extending monads along dense functors

      Let $j: \mathsf A \to \mathsf B$ be a fully faithful and dense functor where $\mathsf A$ is a small category and $\mathsf B$ is cocomplete. Let $(T, \eta, \mu)$ be a monad over $\mathsf A$. $\require{...
      5
      votes
      2answers
      276 views

      Semantics-structure adjunction

      In the discussion on the nLab article for monadic adjunctions, John Baez suggests and Mike Shulman confirms that the relationship between adjunctions and monads itself constitutes an adjunction called ...
      2
      votes
      0answers
      95 views

      The Kleisli Category of the Monad of Measures of Finite Support and its composition formula

      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. Let's call this ...
      4
      votes
      1answer
      232 views

      Does the Eilenberg Moore Construction Preserve fibrations?

      Say we have a Grothendieck fibration $p : E \to B$ and a monad $T$ on $B$ and a lift $T'$ of $T$ to $E$, i.e. a monad on $E$ such that $pT' = Tp$ and $p$ preserves $\eta, \mu$. Then because the ...
      3
      votes
      1answer
      184 views

      Locally presentable categories

      Under category Let $C$ be a locally presentable category, and let $c$ be an object of $C$. Lets denote by $C^{/c}$ the under category, objects are maps $c\rightarrow x$ and morphisms are the evident ...
      0
      votes
      1answer
      251 views

      A monad that unions sets

      Suppose we have a monad that maps types of some kind to other types (see below) , and let types be sets. Let $\alpha, \beta$ be types, $\rightarrow$ denote a function between types, and let $a : \...
      1
      vote
      0answers
      81 views

      The multi-set monad and modules

      I am trying to analyze the category of algebras for the finite free commutative monoid monad, aka the finite multiset monad. This monad is frequently described as having a multiplication that takes a ...
      4
      votes
      2answers
      267 views

      $P = [-°,Set]$ 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 = ...
      3
      votes
      1answer
      155 views

      Codensity monad is idempotent?

      Let $j: A \to B$ be a fully faithful functor. When $j$ has a left adjoint $L$, the codensity monad $\text{Ran}_jj$ will coincide with the monad $jL$ and thus will be idempotent, because $A$ is ...

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