All Questions
Tagged with 2-categories reference-request
23 questions
5
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66
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Bicategories in which the composition functors $\circ_{A, B, C}$ admit right adjoints
In Bénabou's 1967 Introduction to Bicategories, he mentions that, in a forthcoming part II, he would study bicategories $\mathcal K$ in which each composition functor $$\circ_{A, B, C} \colon \mathcal ...
- 10.7k
2
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answers
26
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Reference for the biequivalence between the bicategory of distributors and the bicategory of two-sided discrete fibrations
It is well known that a distributor/profunctor $A \not\rightarrow B$, i.e. a functor $B^{\text{op}} \times A \to \mathrm{Set}$, is equivalent to a two-sided discrete fibration from $A$ to $B$. ...
- 10.7k
3
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0
answers
49
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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
2
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0
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25
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Strict 2-functoriality of lax-slices of 2-categories
$\DeclareMathOperator{\Hom}{Hom}$
I'm currently interested in the homotopy theory of categories "à la Grothendieck", as he developed it in "Pursuing Stacks". I'm trying to try and ...
- 219
6
votes
0
answers
143
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Looking for Lawvere's "Closed categories and biclosed bicategories" lecture notes
The nLab page on closed bicategories reads
Closed bicategories were introduced by Lawvere in unpublished lecture notes Closed categories and biclosed bicategories (1971).
This work has also been ...
- 11.8k
3
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0
answers
124
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$\mathbf{E}_n$-algebras in nerves of 2-categories
In Example 5.1.2.4 of Higher Algebra, Lurie explains how there is a bijection between equivalence classes braided monoidal structures on one-category $\mathcal{C}$ and $\mathbf{E}_2$-algebra ...
- 5,711
5
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0
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107
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Classifying spaces of crossed modules
Let $\mathcal{G}$ be a strict topological $2$-group, i.e. a strict $2$-category with a single object, a space of invertible $1$-morphisms, a space of invertible $2$-morphisms and continuous structure ...
- 63.9k
3
votes
1
answer
56
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Reference request for facts about bi(co)descent objects
I know the following facts are true, but I struggle to find adequate references for them:
Let $T$ be a pseudo-monad on a bicategory $\mathcal{C}$, and let $A$, $B$ be pseudo-algebras for $T$. Then, ...
- 5,429
3
votes
1
answer
66
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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 ...
- 10.7k
8
votes
1
answer
778
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J.W. Gray's monumental work notes on the formal theory of internal (2-)categories
In the book "Topos Theory" of Peter Johnstone (Topos Theory, LMS Monographs no. 10. Academic, 1977) one finds at page 41 in Chapter 2:
"For a detailed account of internal categories ...
- 12.9k
3
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0
answers
152
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Free $2$-category on a $2$-quiver
The construction of the free category on a quiver is standard in category theory.
Is there some $2$-dimensional analog of a quiver such that each $2$-quiver freely gives rise to a $2$-category? How ...
- 10.1k
3
votes
0
answers
70
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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
1
answer
110
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Reference for proof that vertical composition of pseudonatural transformations is a pseudonatural transformation
Is there a standard reference for the fact that the vertical composition of two pseudonatural transformations between pseudofunctors between bicategories is a pseudonatural transformation? Recall that ...
- 11.8k
3
votes
0
answers
136
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Reference for "taking adjuncts preserves Kan extensions"
I'm using a result similar to the one below, and I would like to know if there is a reference that I can cite. It's easily proved, by "following your nose". The cell $G\phi.\eta_A$ is often ...
- 383
5
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0
answers
120
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Simplicial matrices and the nerves of weak n-categories II, III, and IV
Duskin introduced his nerve functor (see the nLab or Kerodon) in the paper
Duskin, John W. Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories. [Link].
While three ...
- 11.8k
6
votes
0
answers
118
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Retracts in the bicategory of spans
I would like to show that the category of sets and spans between them, seen as a $(2,1)$-category, is Cauchy complete, i.e. has splitting of (homotopicaly coherent) idempotent.
Ideally I would also ...
- 42.4k
3
votes
1
answer
220
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"discrete" objects of a $2$-category
Let $\mathcal{K}$ be a $2$-category. Is there a special name of those objects $B \in \mathcal{K}$ which have the property that the category $\mathrm{Hom}_{\mathcal{K}}(B,C)$ is essentially discrete ...
- 66.8k
5
votes
1
answer
339
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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
4
votes
1
answer
370
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Weak enrichment and bicategories
I'm trying to find examples where the following perspective on bicategories is developed.
We can define a 2-category as being enriched in Cat, where Cat is treated as a monoidal category using the ...
- 3,411
13
votes
1
answer
718
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On the coherence theorem for bicategories
The coherence theorem for bicategories, as usually stated, reads
Any bicategory $B$ is biequivalent to a (strict) 2-category.
It is possible to give an explicit construction of the strictification ...
- 6,051
7
votes
1
answer
345
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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 ...
- 66.8k
1
vote
2
answers
220
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Completion under weighted limits/colimits
Is there any further reference besides "Basic Concepts of Enriched Categories" (Kelly) for completion under T-(weighted) limits/colimits?
(in which T is a set of weights)
Thank you in advance
- 875
3
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0
answers
127
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Bicategorical limits with parameters
(This question was asked in https://math.stackexchange.com/questions/741334/bicategorical-limits-with-parameters with no answer.)
Let $F(-,-)\colon \mathcal{A}\times \mathcal{B}\to \mathcal{C}$ be a ...
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