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14
votes
0answers
627 views

Can we describe equivariant vector bundles of free group action in terms of descent theory (Barr-Beck theorem)?

It is well known that for a compact topological group $G$ acts (say, from the right) freely on a compact space $X$. Then the category of equivariant complex vector bundles on $X$, $\text{Vect}_G(X)$, ...
8
votes
0answers
185 views

Whiskering a monad

In "The Geometry of Iterated Loop Spaces", May shows that any monad that is coming from an operad may be "whiskered", so that the unit map becomes a closed cofibration. The ability to do this is vital ...
3
votes
0answers
27 views

Intuition for density comonad in relation to lifting problems

In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...
2
votes
0answers
126 views

Distributive law between Kleisli triples

A distributive law of a monad $S$ over a monad $T$ is a natural transformation $l : T S \to S T$ such that: $l \circ T \eta^S = \eta^S T$ $l \circ \eta^T S = S \eta^T$ $\mu^S T \circ S l \circ l S = ...
2
votes
0answers
161 views

Compatibility between strength and costrength of a monoidal monad

Let $C$ be a closed monoidal category, and let $T : C \to C$ be a monad on the underlying category. Let $\sigma$ be a tensorial strength of $T$ and let $\sigma'$ be a cotensorial strength of $T$. A ...
2
votes
0answers
359 views

Cartesian-closed categories of algebras

If the Kleisli-category of a monad is Cartesian-closed, can we say when the category of Eilenberg-Moore algebras is?
1
vote
0answers
74 views

List is a monad, but is it a comonad with these natural transformations?

List is known to be a monad. It takes a set and maps it to lists of elements of that set. The natural transformations are, singleton and flatten, whereby we map a set to a set of singleton lists ...