Questions tagged [2-categories]
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102
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Examples and counterexamples to Lack's coherence observation
In Lack's A 2-categories companion, he states
There are general results asserting that any bicategory is biequivalent to
a 2-category, but in fact naturally occurring bicategories tend to be ...
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4
votes
1
answer
126
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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 ...
- 49.2k
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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 in a bicategory $\mathcal{C}$, and let $A$, $B$ be pseudo-algebras for $T$. Then, ...
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100
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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 ...
- 6,241
3
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180
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A 2-category of abelian categories?
Is there a reasonably well-behaved $2$-category of abelian categories, and if so, what are its objects (perhaps one needs to restrict one's attention to finite abelian categories ?), $1$-morphisms (...
- 1,121
3
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1
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101
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Horizontal closure of 'almost $2$-categories'
Is there a reference discussing the notion of 'free horizontal closure' for an 'almost $2$-category', where all that's missing are some horizontal composites of $2$-cells?
The motivation for this ...
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58
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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 ...
- 6,241
6
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1
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211
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How do the various homotopy 2-categories compare?
There are various models of $\infty$-categories floating around, so there are as many models of the associated homotopy 1- and 2-categories. Because the relations between the former are worked out in ...
- 213
3
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1
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154
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(Co)limits in lax functor categories
Let $\mathcal I$, $\mathcal C$ be $2$-categories (or $(\infty, 2)$-categories, I'm interested in both cases) and assume that $\mathcal I$ is small, $\mathcal C$ has enough weighted (co)limits as you ...
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2
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Examples of 2-categories with multiple interesting proarrow equipment structures
Proarrow equipments (also known as framed bicategories) are identity-on-objects locally fully faithful pseudofunctors $({-})_* \colon \mathcal K \to \mathcal M$ for which every 1-cell $f_*$ in the ...
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83
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Universal property of the V-Mat construction
Internal categories and enriched categories can both be realised as monads in certain bicategories. If $\mathcal E$ is a category with pullbacks, then a monad in $\mathbf{Span}(\mathcal E)$ is a ...
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1
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Are locally fully faithful 2-functors closed under 2-pushout in 2-Cat?
It is known that fully faithful functors are closed under pushouts in Cat (e.g. Lemma 4.9 of this paper). Are locally fully faithful 2-functors closed under (strict) 2-pushouts in the 2-category 2-Cat ...
- 5,035
4
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1
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152
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Adjunctions with respect to profunctors
Let $P : W° \times Y \to \mathbf{Set}$ and $Q : X° \times V \to \mathbf{Set}$ be profunctors, and let $L : X \to W$ and $R : Y \to V$ be functors. Suppose that $$P(Lx, y) \cong Q(x, Ry)$$ natural in $...
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2-morphism between circuits in a monoidal category
We are used to seeing equations between circuits in monoidal categories like this
I am wondering about morphisms between string diagrams. I think they are 2-cells. I found an example of a 2-cell ...
- 155
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72
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Adjoining extensions in bicategories
Given a bicategory $\mathcal K$, is there a universal construction of a bicategory $\mathcal K'$ and faithful locally fully faithful pseudofunctor $\mathcal K \hookrightarrow \mathcal K'$ such that ...
- 5,035
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89
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Examples of Gray comonoids
It is well-known that comonoids in Cartesian monoidal categories are uninteresting, as every object admits a unique comonoid structure given by the diagonal and projection maps. This is why for ...
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1
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314
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Long exact sequence of cohomology from 2-groups
I am trying to understand the following from Principal Infinity Bundles - General by Nikolaus, Schreiber and Stevenson.
So following the reference there to Nikolaus-Waldorf tells us that given any (...
4
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0
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85
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Coherence for biadjunctions
There are many ways to give a definition of a biadjunction. For instance, one may say that a pseudofunctor $F:\mathcal{C}\rightarrow \mathcal{D}$ is left biadjoint to $G:\mathcal{D}\rightarrow \...
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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 ...
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86
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Strict 2-Category with Lax Tensor?
I am faced with an operation $\otimes$ on a strict 2-category $C$ which walks and talks like a tensor, except that it only satisfies a "lax" interchange law. To be precise: for any $f,g,h,k\...
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94
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Factoring a natural transformation through a functor
This seems fairly similar to density. Suppose I have three categories $A,B,C$, and a functor $L: B \to C$ so that every natural transformation $f: L.F \Rightarrow L.G$, for a parallel pair $F,G: A \to ...
- 1,213
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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 $\mathcal K$...
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"Character" theory via dualisable $2$ categories
One interesting way to describe the ordinary (over $\mathbb{C}$) character theory of finite groups is to view the categories $Rep(G)$ together in a $2$ category with bimodules as morphisms. This $2$ ...
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3
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1
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118
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Coherence $2$-cells in braided monoidal bicategories
In a braided monoidal category $(\mathcal{C},\otimes_{\mathcal{C}},\mathbf{1}_{\mathcal{C}},\alpha,\lambda,\rho,\beta)$, we have $\beta_{\mathbf{1}_{\mathcal{C}},\mathbf{1}_{\mathcal{C}}}=\mathrm{id}_{...
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89
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Strictifying monoidal 2-functors
Let $F:\mathcal{C}\rightarrow\mathcal{D}$ be a (weak) monoidal 2-functor between two strict monoidal 2-categories. Up to replacing $\mathcal{C}$ by an equivalent strict monoidal 2-category, can I ...
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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\...
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83
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Is there a bicategorical Yoneda lemma for marked lax transformations?
The bicategorical Yoneda lemma (see [Johnson–Yau, Chapter 8]) states that, given a bicategory $\mathcal{C}$ and a pseudopresheaf $\mathcal{F}\colon\mathcal{C}^{\mathsf{op}}\to\mathsf{Cats}_{\mathsf{2}}...
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Can we simplify the definition of a biadjunction using coherence for bicategories?
In Biequivalences in Tricategories,
Gurski defines a biadjunction of bicategories as a sextuple $(F,G,\eta,\epsilon,\Gamma,\Sigma)$ with
$F\colon\mathcal{C}\longrightarrow\mathcal{D}$ and $G\colon\...
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4
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0
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87
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$2$-dimensional adjunctions via pre/post-composition
Recall that in the setting of $1$-categories, given functors $L\colon\mathcal{C}\longrightarrow\mathcal{D}$ and $R\colon\mathcal{D}\longrightarrow\mathcal{C}$, the following conditions are equivalent:
...
- 6,346
12
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1
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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 fourteen of them:
Here $\mathsf{LaxCones}(\Delta_{X},D)\...
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94
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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 ...
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votes
1
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367
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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 ...
- 49.2k
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Coherence for closed bicategories
A right-closed bicategory [1] is a bicategory that has all right extensions (i.e. right adjoints to precomposition with a fixed 1-cell). A one-object right-closed bicategory is therefore a right-...
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0
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A bicategorical representation theorem
The representation theorem in $1$-category theory states that a presheaf $\mathcal{F}$ is representable iff $\int_\mathcal{C}\mathcal{F}$ has a terminal object. When one tries to formulate a ...
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1
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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 ...
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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 ...
- 126
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Coherence for monoidal 2-categories vs coherence for braided monoidal categories
Gordon, Power and Street have proven that every monoidal 2-category is equivalent to a Gray monoid. This means that the only coherence 2-isomorphisms we have to be concerned about are the ...
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289
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What is a bipullback of lax functors?
$\require{AMScd}$The following question is somewhat technical, and since I firmly believe this has a small hope to be true only using all the assumptions, I am forced to introduce them all: I don't ...
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1
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1
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Unitor identities for bicategories
In the standard definition of a bicategory, the unitors are required to satisfy the 'triangle identity'
for any composable $1$-cells $f:Y\to Z,gX\to Y$. But it seems like we also want
to commute for ...
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83
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Codescent objects in Morita 2-category?
Let $\mathbf{Bimod}$ be the 2-category of algebras, bimodules and bimodules maps over a field $\mathbb{k}$. Does this 2-category have codescent objects (see my attempt at giving a definition below)?
...
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How can we view monoids as lax functors?
Let $*_2$ be the terminal $2-$category, $V$ be a monoidal category with unit object $I$ and $\mathbf BV$ be its delooping, i.e the $2-$category with $ob(\mathbf BV) = \{*\}$ and $\text{Hom}_{\mathbf ...
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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(\...
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181
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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
$...
- 336
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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 $\...
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0
answers
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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 ...
- 1,999
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80
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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 $...
- 5,035
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128
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Proper model category for "categories with finite limits"
I'm looking for a Quillen model category which model the $2$-category of 'small category with finite limits (and functors between them preserving finite limits)':
Left proper,
right proper,
Enriched ...
- 32.2k
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217
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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 ...
1
vote
1
answer
89
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Has the covariant Hom-functor of the category of additive categories a left adjoint?
Let $\mathsf{Add}$ denote the (strict) 2-category of small additive categories and additive functors. Because categories of additive functors are itself additive, we have for each additive category $\...
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66
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Examples of strictification of a weak category obtained from a generalisation of a strict category
I have made the following observation (hopefully a correct one) when reading the paper Orbifolds as stacks:
They start with the strict $2$-category category of Lie groupoids, functors, natural ...
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