All Questions
Tagged with monads 2-categories
20 questions
10
votes
1
answer
466
views
Characterization of functors whose right adjoint is monadic?
Let $F: \mathcal A^\to_\leftarrow \mathcal B: U$ be an adjunction, and suppose we want to know whether the comparision functor $\mathcal B \to Alg^{UF}$ is an equivalence, where $Alg^{UF}$ is the ...
- 64k
10
votes
1
answer
333
views
2-monads for categories with a class of (co)limits
This question concerns the strictness of (co)completions, at various levels of generality.
In Blackwell–Kelly–Power's Two-dimensional monad theory, the authors state
For instance, the 2-category $\...
- 10.7k
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
9
votes
0
answers
192
views
What is the relationship between free bicompletion and the Isbell envelope?
Given a small category $\mathbb C$, we can form the free cocompletion $\mathbf y : \mathbb C \to \mathcal P(\mathbb C)$ and the free completion $\mathbf y^\circ : \mathbb C \to \mathcal P^\circ(\...
- 10.7k
9
votes
0
answers
323
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To what kind of generalized Lawvere theory does the "free cartesian closed category" 2-monad on $\mbox{Cat}_g$ correspond?
Thinking of Cat as a mere 1-category, there is a 1-monad $\Lambda$ for the free cartesian closed category on a category. To every category X it assigns the category $\Lambda(X)$ whose objects are ...
- 1,532
7
votes
1
answer
345
views
Has anybody studied strict/pseudo morphisms of monads?
There is a notion of morphism from a monad $T:\mathscr C\to \mathscr C$ to another one $T':\mathscr C'\to \mathscr C'$. It arose here on MO e. g. in "Functors between monads": what are these ...
- 18.4k
7
votes
0
answers
266
views
Relation between two limit presentations of Eilenberg--Moore objects
Let $\mathbb{T}=({\cal T}\colon C\to C,\mu,\eta)$ be a monad (in the
$2$-category $\mathsf{Cat}$), which we view as a $2$-functor
$\mathbb{T}\colon\mathsf{B}\Delta_{\mathrm{a}}\to\mathsf{Cat}$ (where
$...
- 446
7
votes
0
answers
161
views
Coherence for pseudomonads and their pseudoalgebras
Let $\mathcal K$ be a bicategory. For every pseudomonad $T : \mathcal K \to \mathcal K$, does there exist a 2-monad $S : \mathcal C \to \mathcal C$, where $\mathcal C$ is a 2-category biequivalent to $...
- 10.7k
6
votes
0
answers
75
views
What are the algebras of the powerset intersection (oplax) monad?
The assignment $X\mapsto\mathcal{P}(X)$ and $f\mapsto f_*$ (direct images) defines a functor $\mathcal{P}\colon\mathsf{Sets}\to\mathsf{Sets}$.
This functor has a monad structure whose multiplication $\...
- 11.8k
6
votes
0
answers
98
views
Example of a pseudomonad on Cat whose pseudoalgebras are not the pseudoalgebras for a 2-monad
For every pseudomonad $T$ on the 2-category of (locally small) categories $\mathbf{Cat}$, we can consider the 2-category of $T$-pseudoalgebras and pseudomorphisms $T\text{-PsAlg}_p$, which is equipped ...
- 10.7k
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
4
votes
2
answers
401
views
Transporting algebraic structure along adjoint equivalences
I have two questions, one general and the other particular to the case I am interested in.
The 'homotopically correct' notion of equivalence of categories is an adjoint equivalence (from one point of ...
- 35.5k
4
votes
1
answer
218
views
Monads and modules, and the bicompletion under Kleisli and Eilenberg–Moore objects
In The Formal Theory of Monads, Street proves that a 2-category $\mathscr C$ admits the construction of algebras when the inclusion $\mathscr C \to \mathbf{Mnd}(\mathscr C)$ has a right adjoint. In ...
- 10.7k
3
votes
1
answer
145
views
Examples of (co)lax idempotent pseudocomonads on Cat
A lax idempotent pseudomonad, also called a KZ doctrine or KZ monad, is a pseudomonad $(T, \mu, \eta)$ with the property that $T \eta \dashv \mu \dashv \eta T$. Lax idempotent pseudomonads were ...
- 10.7k
3
votes
1
answer
66
views
Morphism of pseudomonads induces pullback functors between pseudoalgebras
If $S$ and $T$ are monads on a category $C$, and $\lambda:S\to T$ is a morphism of monads, it is well-known that there is a functor $\lambda^*:C^T\to C^S$ which assigns to the $T$-algebra $(A,a:TA\to ...
- 2,129
3
votes
0
answers
49
views
Lax morphism classifiers via lax-idempotentification
Let $T$ be a 2-monad on a nice 2-category $\mathcal K$, so that the inclusion $T\text{-}\mathbf{Alg}_s \to T\text{-}\mathbf{Alg}_l$ of the 2-category of (strict) $T$-algebras and strict $T$-algebra ...
- 10.7k
3
votes
0
answers
71
views
Adjoints to the forgetful functor from the $2$-category of monads
For the purpose of this post, we will identify a monad $T$ on a category $\mathcal{C}$ with a lax $2$-functor $T:{\bf 1}\to\mathfrak{Cat}$ such that $T(*)=\mathcal{C}$.
There is an obvious forgetful ...
- 10.1k
3
votes
0
answers
70
views
Lax algebras for pseudomonads and monads in Kleisli bicategories for the induced pseudocomonad
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 $\mathcal K$...
- 10.7k
2
votes
2
answers
137
views
If a monad in a 2-category admits a terminal resolution, does it admit an Eilenberg–Moore object?
Let $T = (t, \mu, \eta)$ be a monad on an object $A$ of a 2-category $\mathcal K$. In The formal theory of monads, Street proves (Theorem 3) that if $l \dashv r$ is the canonical adjunction associated ...
- 10.7k
2
votes
1
answer
134
views
When does a 2-functor or 2-monad of Cat lift to a psuedofunctor or pseudomonad on Prof?
I'm currently reading Richard Garner's paper Polycategories via pseudo-distributive laws, and a central construction is the lifting of the symmetric strict monoidal category 2-monad to a pseudomonad ...
- 347