As of May 31, 2023, we have updated our Code of Conduct.

# Tag Info

Accepted

### Why are monadicity and descent related?

Probably the reason "monadicity" gets connected with descent (and the associated terminology of descent theory) is because of its relevance to the question of descent for rings. If you're talking ...

### Why are monadicity and descent related?

I think of monads in terms of algebraic theories. Monads are substantially more general than this intuition suggests! Here is a better intuition: monads are categorified idempotents. The point of ...
Accepted

### 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....

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 ...
Accepted

### 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 ...

### 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-...
Accepted

### 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 ...

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 ...
Accepted

### 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 ...
Accepted

### 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 ...
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 $X$ ...