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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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### 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 ...
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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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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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### 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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### 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 $\mathrm{Set}$ ...
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### Do (co)density (co)monadic constructions stablize?

Under good conditions , 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 ...
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### 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 ...
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### Internal commutative monoid gives commutative monad

Let $(C,\otimes,1)$ be a symmetric monoidal category. Let $(M,\mu,\eta)$ be an internal commutative monoid object. The functor $X\mapsto M\otimes X$ has a canonical monad structure, with unit and ...
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### Distributive laws of strong and/or monoidal monads

It is well-known that a commutative strong monad is the same as a monoidal monad. Is there a notion of distributive law for commutative strong monads which is equivalently a distributive law for ...
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### What is the measures monad for FDHilb?

I am labouring under a particular assumption that, perhaps, needs to be corrected. I believe that FDHilb, the category of Finite Dimensional Hilbert spaces and general linear maps is a category of ...
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### Existence of free functor to Banach spaces

Is there a "non-trivial" characterization of the concrete categories admitting and adjoint pair of functors $F \dashv G$ were $G$ is defined on the category sBan of separable Banach spaces and bounded ...
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### What are the algebras for the codensity monad of $\textbf{FP-Ring} \rightarrow \textbf{Ring}$

Let $\textbf{Fin-Set}$ denote the category of finite sets, and let $\textbf{Set}$ denote the category of sets. The inclusion functor $\textbf{Fin-Set} \rightarrow \textbf{Set}$ from the category of ...
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Introduction I'll describe a way of taking a monad on a category $\mathcal{E}$ with pullbacks, and obtaining a monad on each slice category. I'll show that this construction is always lax-natural in $... 1answer 121 views ### Trying to construct the ultrafilter 2-monad on$\mathbf{Cat}$By which I mean, following Bôrger's paper Coproducts and Ultrafilters, the terminal monad among those that preserve finite coproducts, if such a thing can be constructed. So far, what I have is, ... 0answers 129 views ### What is the Eilenberg-Moore category for the cyclic list? In this paper (Kock 2012), we see a data structure with circular symmetry. It is the cyclic list monad of Examples 3.10. The author is showing that data structures with symmetries can be cast as ... 0answers 179 views ### What are the (co)algebras for the$(\operatorname{Hom}(A,-), A\otimes-)$adjunction (co)monad? A module$A$over a commutative ring$k$gives a pair of adjoint endofunctors,$(A\otimes_k-)$left adjoint to$\operatorname{Hom}_k(A,-)$. They produce a monad$T_A$and a comonad$C_A$. Is there any ... 2answers 487 views ### When were triples called monads for the first time? I am fine-tuning a short note on basic category theory; any such course must introduce monads, and I want to give a bit of history of the subject. I soon realized that I don't know the precise series ... 0answers 196 views ### Partial monoid in the category of categories of modules: The spotty nature of monad composition It seems that I am working on a conjecture in category theory. In particular, I am curious about the spotty nature of the composition of monads on Set. I am guessing that there is a category,$\...
I have recently asked a question about the composition of two monads, namely $\mathcal{M}_P = \mathcal{M}_C \cdot \mathcal{M}_C$. I am conjecturing that the cateogory of $\mathbb{C}$-Modules and the ...