Questions tagged [2-categories]
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16 questions
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"Very lax" $2$-dimensional co/limits
In the setting of $1$-categories, there are two (unweighted) variant notions of limits, namely limits and colimits. For bicategories, there are sixteen of them:
Each of these notions has an ...
- 11.8k
17
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1
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2-categories for the working algebraic geometer
I study algebraic geometry / number theory and from time to time I stumble upon 2-categorical (co)limits. I have two main examples in mind:
Example 1) In étale cohomology, the (triangulated) derived ...
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10
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1
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333
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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
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1
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466
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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
9
votes
1
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Yoneda Lemma for internal presheaves
I'm looking for a reference explaining under what conditions the internal Yoneda lemma holds; in particular, I am wondering if it is known what properties of the ($2$-)category of categories are ...
- 572
9
votes
2
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434
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Pushouts of commutative pseudomonoids
Let $(\mathcal{C},\otimes)$ be a symmetric monoidal bicategory. Assume that $\mathcal{C}$ has bicategorical coequalizers which are preserved by $\otimes$ in each variable. My question is if then the ...
- 63.1k
7
votes
2
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558
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Kan extensions in concrete 2-categories
Kan extensions make sense in any 2-category. I am interested in Kan extensions in "concrete" 2-categories consisting of actual categories with some sort of structure (e.g., finite products, finite ...
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7
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0
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428
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Left Kan extensions of "strong" monoidal functors
Consider the 2-category $\mathsf{MonCat}$ where objects are monoidal categories,
1-cells are strong monoidal functors, and 2-cells are monoidal natural transformations.
Given arrows $f: \mathsf{C} \to ...
7
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161
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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
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1
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Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first?
Let $C$ be a site and $CAT$ the 2-category of categories. Given a contravariant 2-functor $A:C\rightarrow CAT$, we can of course consider the associated stack. This is done by first considering the ...
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5
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1
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Diagonal of a diagram of codescent objects
Given the following diagram in a $2$-category, in which squares of the same "type" commute, where each column and each row is a strong codescent diagram (Edit: it should be reflexive as well), is ...
- 63.1k
5
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1
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Existence and characterisations of left Kan extensions and liftings in the bicategory of relations I
The bicategory $\mathsf{Rel}$ of sets, relations, and inclusions of relations has right Kan extensions and right Kan lifts¹, however I believe it does not have all left Kan extensions/lifts.
Is it ...
- 11.8k
4
votes
0
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207
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Is the category of cochain complexes with terms in an additive category a 2-category?
$\def\hom{\operatorname{Hom}}
\def\bbZ{\mathbb{Z}}$This question is a follow-up to this other one. There the OP asks whether "the category of chain complexes" (can be interpreted in several ...
3
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0
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85
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$2$-dimensional adjunctions via co/Cartesian fibrations $\mathcal{M}\longrightarrow[1]$
Recall [HTT, Definition 5.2.2.1]:
Definition 5.2.2.1. Let $\mathcal{C}$ and $\mathcal{D}$ be $\infty$-categories. An adjunction between $\mathcal{C}$ and $\mathcal{D}$ is a map $q\colon\mathcal{M}\to\...
- 11.8k
3
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0
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A name in literature for a certain kind of 2-categories
Let $tr_2: \mathrm{sSet} \to \mathrm{sSet}_{\le 2} $ be the 2-truncation functor.
Let $C$ be a 2-truncated simplicial set such that every horn $tr_2( \Lambda^2_1) \to C$ extends to $tr_2(\Delta_2) \...
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Existence and characterisations of left Kan extensions and liftings in the bicategory of relations II
This is the second part to a previous question regarding left Kan extensions/lifts in the bicategory $\mathsf{Rel}$ of sets, relations, and inclusions of relations, which has now been split into two ...
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