# Tag Info

Here are the examples of comonads that I personally find most helpful. First from topology: The universal covering is an idempotent comonad on (suitably nice) pointed topological spaces. The functor ...
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### Jon Beck's untitled manuscript containing the "tripleability theorem" (i.e. the monadicity theorem)

After reaching out to every researcher who cited the manuscript, John Kennison was kind enough to find and scan his copy of the untitled manuscript containing the crude and precise monadicity theorems....
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### Conceptual reason that monadic functors create limits?

The heart of the diagram chase is: All the operations and equations involved in an algebra structure on X have target X. (More generally: the target of each operation/equation could be any limit-...
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### Why are operads sometimes better than algebraic theories?

First - yes, for symmetric set-operads this functor is "injective", though it is not fully faithful. It is faithful on general maps and fully faithful on isomorphisms. Its image can easily ...
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### List is a monad, but is it a comonad with these natural transformations?

Todd's comment provides an important limitation on what you can do here, but here's what I think is the most interesting way to answer your question. Define an endofunctor $L^+$ on the category of ...
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Lawvere theories can be thought of as "cartesian operads." That is, we have an analogy \text{Lawvere theories} : \text{cartesian monoidal categories} :: \text{operads} : \text{symmetric monoidal ...
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### Is the Cartesian product of two finitely presented objects finitely presentable?

No. For counterexamples, see Theorems 3.8, 3.9, and 3.10 of Finiteness properties of direct products of algebraic structures Peter Mayr, Nik Ruškuc Journal of Algebra 494 (2018) 167-187. These ...
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Given a topology on a set $X$, let $2^X$ be the poset of subsets of $X$ ordered by inclusion. Then the interior operator for the topology is a comonad on $2^X$. In fact the topologies on $X$ ...
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### What is the polynomial functor for the Bag monad

The bag monad is not polynomial. Any polynomial endofunctor must preserve pullbacks: $f^*$ and $\Pi_g$ preserve all limits since they’re right adjoints, while $\Sigma_h$, being just the forgetful ...

### Monads on Set with trivial algebras

I don't know where this observation was first made, but the proof is short. Let $M$ be a monad on $Set$ such that every $M$-algebra has at most one element. For every set $A$, the set $M(A)$ has ...
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### English Reference for the Bénabou-Roubaud theorem

The following is a reasonably literal translation (made by me from the French original) of the published article: Jean Bénabou, Jacques Roubaud. Monades et descente. C. R. Acad. Sc. Paris, t. 270 (12 ...
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These questions were studied (in the dual case of comonads, née cotriples) by Applegate and Tierney in their 1970 paper Iterated cotriples. The answer to your first question is that yes, this ...
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### Adjunctions between Groupoids and Hilbert spaces

I don't know what morphisms you intend for the category of finite-dimensional Hilbert spaces, but it doesn't actually matter. The answer is no, there are no interesting adjunctions between the ...
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The category $\mathrm{Mfd}$ of finite dimensional manifolds sits fully faithfully inside the larger category $\mathrm{LocProMfd}$ of locally pro-finite dimensional manifolds, which basically extends ...
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### When were triples called monads for the first time?

This is covered in this English.SE question. In short, people were not all very happy about the term "triple", and tried to come up with something better. Jean Bénabou suggested "monad" during lunch ...
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### Conceptual reason that monadic functors create limits?

I am not able to give the high-tech answer you are clearly hoping for. For me this statement is just a straight forward generalization of the known fact that, say, $\mathsf{Grp} \to \mathsf{Set}$ ...
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### Conceptual reason that monadic functors create limits?

From an abstract point of view, the reason is that the monad $T$ always preserves any limits that exist colaxly and colax preservation is what is required. (This answer is closely related to Peter's ...
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### 2-monads for categories with a class of (co)limits

Kelly and Lack's paper On the monadicity of categories with chosen colimits answers your questions (1),(2) and (3) affirmatively. The main theorems are Theorem 6.1, 6.2 and 7.1. Their main trick is ...
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