All Questions
Tagged with higher-category-theory monads
13 questions
3
votes
1
answer
195
views
Double category of monads and pseudo monad-morphisms
We can construct bicategories of monads in a bicategory $B$, $Mnd_l(B)$/$Mnd_c(B)$ with lax and colax monad-morphisms respectively.
I am failing to find a good notion of pseudo monad-morphisms.
Is ...
- 1,347
7
votes
1
answer
281
views
Finitely presentable objects in the categories of algebras of $\infty$-algebraic theories
By default, all terms are understood in the infinity sense (“category” means “$(\infty, 1)$-category”, etc.).
An object $A$ in a category is said to be finitely presentable (or compact) if the functor ...
- 6,962
10
votes
1
answer
405
views
2-completeness of stacks
I am looking for a reference which discusses the 2-categorical properties of the 2-category $St(C,J)$ of stacks and the stackification $\dashv$ inclusion of presheaves 2-adjunction.
My stacks are ...
- 775
2
votes
0
answers
129
views
Cat as a bicategory of monads over another category
Let's assume infinitely many Grothendieck universes exist. Let's call $\kappa$-Cat the bicategory of $\kappa$-small categories with anafunctors and anatural transformations. Now for any $\lambda$ and ...
- 1,347
3
votes
0
answers
100
views
What is the free lax-idempotent adjunction?
Let $Adj$ be the free adjunction, i.e. the 2-category such that for any 2-category $K$, the functor 2-category $2Fun(Adj, K)$ is the 2-category of adjunctions in $K$ (naturally in $K$). Note that $Adj$...
- 64k
5
votes
1
answer
178
views
What are the algebras for the laxification 2-monad?
Let $C$ be a small 2-category. Let $[C , Cat]$ denote the 2-category of strict functors to $Cat$, 2-natural transformations, and modifications. Let $[[ C, Cat ]]$ denote the 2-category with the same ...
- 64k
6
votes
0
answers
354
views
Cohomology without comonad?
TL;DR. Many cohomologies can be unified using comonads. Question: which cohomologies cannot be?
For each algebraic theory, there is an adjunction, and therefore a (co)monad (or called a (co)triple). ...
- 5,230
4
votes
1
answer
440
views
Kan condition for bar construction
Let $T$ be a monad on a concrete category $\mathcal{C}$, and $A$ an algebra over $T$. The bar construction is a simplicial object in the category $\mathcal{C}^T$ of algebras which we can think of a ...
- 2,129
3
votes
1
answer
118
views
Pseudo or lax algebras for a 2-monad, reference request
I would like to find explicit definitions of pseudo, or even lax, algebras for a 2-monad, and their lax morphisms, with all the coherence diagrams included.
Alternatively, coherent lax algebras for ...
- 2,129
2
votes
1
answer
238
views
Synthetic type theory for virtual double category and its higher categories
For some monad T on a virtual equipment, the paper A unified framework for generalized multicategories by Cruttwell and Shulman (arXiv:0907.2460) proposes the normalized T-monoid.
Another paper, by ...
- 97
5
votes
1
answer
75
views
Coherence laws when composing 2-monads
To have the composition of two monads be a monad itself, we need a
distributive law natural transformation satisfying certain coherence
laws.
I'm interested in the strict 2-monad case, i.e. a strict ...
- 1,532
14
votes
2
answers
1k
views
The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?
Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to its underlying ...
- 19.6k
5
votes
1
answer
489
views
Lax and Colax Monads
Is there much known about the theory of lax and colax monads on a bicategory? Here, I really mean lax or colax, not weak. I'm aware of some literature about weak monads. I'm interested in distributive ...
- 15.5k