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Questions tagged [coherence]

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Coherence of the graphical language for pivotal categories

Throughout I follow A survey of graphical languages for monoidal categories, Peter Selinger, arXiv. A pivotal category is a monoidal category where each object $A$ has a dual $A^*$, together with a ...
Léo S.'s user avatar
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Predicting coherence diagrams one dimension up

Assume we have a good working knowledge of $n$-dimensional category theory for some fixed $n$. It seems like it should be possible to 'predict' what coherence diagrams we're going to encounter in the ...
Alec Rhea's user avatar
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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 ...
varkor's user avatar
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3 votes
1 answer
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Bⁿ and coherence

I understand that an internal abelian group in CW-complexes (while it is not the most extensive structure for which this can be done) produces a classifying space which itself has the structure of an ...
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12 votes
0 answers
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Does this matrix norm inequality have interesting application in other areas of mathematics?

In my new paper, one of the main theorems gives an upper bound for the spectral distance of a general real symmetric matrix to diagonal matrices: Theorem 3. ‎Let $A=[a_{ij}]$ be a real symmetric ...
Mostafa's user avatar
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10 votes
2 answers
977 views

Why are the source-target rules of composition always strictly defined?

General categorical definitions always have two variants, a strict one, in which associativity and unity hold as equalities, and a weak one, in which they hold up to equivalence. However, every ...
Alexander Praehauser's user avatar
5 votes
1 answer
199 views

Why is the category of strong braided functors from the braid category to a braided monoidal $M$ equivalent to the subcategory of *strict* functors?

This is my first, and probably my last, (for a while) posting on MO. I am very much a student and I don't claim to be a research mathematician, at all, but I have seen that sometimes "regular&...
FShrike's user avatar
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4 votes
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Full coherence for non-symmetric linearly distributive categories?

1. Context Mac Lane's coherence theorem for monoidal categories can be phrased as "every formal diagram in a monoidal category commutes.“ I am interested in this type of coherence question for ...
Max Demirdilek's user avatar
2 votes
1 answer
196 views

Strictification of $\mathcal{V}$-pseudofunctors

Let $\mathcal{B}$ be a bicategory. Section 4.10 of Gordon, Power and Street's paper "Coherence for Tricategories" states that there is a bicategory $\textbf{st}\mathcal{B}$ and a ...
Zbyszek's user avatar
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13 votes
0 answers
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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 ...
varkor's user avatar
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A recursive attempt at $n$-dimensional coherence

For the purposes of this post we will use the one hom class definition of a category. Note that a functor $F:\mathcal{C}\to\mathcal{D}$ between categories is a pair of functions $F_0:{\bf Ob}_\mathcal{...
Alec Rhea's user avatar
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9 votes
2 answers
421 views

Coherence theorem in braided monoidal categories

In MacLane's Categories for the working mathematician, the author shows that the evaluation at 1 gives an equivalence of categories $\mathrm{hom}_{\mathrm{BMC}}(B,M)\simeq M_0$ where $B$ is the braid ...
QGM's user avatar
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5 votes
0 answers
265 views

Strictification for closed monoidal categories

The strictification theorem for monoidal categories states that every monoidal categorically is monoidally equivalent to a strict monoidal category. Is there a strictification theorem for closed ...
varkor's user avatar
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4 votes
0 answers
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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-...
varkor's user avatar
  • 9,080
9 votes
1 answer
312 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 $\...
varkor's user avatar
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6 votes
0 answers
153 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 $...
varkor's user avatar
  • 9,080
7 votes
2 answers
738 views

Mac Lane's proof of coherence for symmetric monoidal categories

This question only concerns the final part of the proof, so I assume that the symmetric monoidal category is a strict monoidal category $\mathsf{C}$ with the braiding $s$. Let $X_1,...,X_n$ be ...
Jxt921's user avatar
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2 votes
0 answers
161 views

Prime ideals being finitely-generated implies coherence?

Let $R$ be a non-noetherian local domain. Suppose that the following two conditions hold for $R$$\colon$ $(*)$$~\quad$An arbitrary prime ideal ${\frak P}$ of $R$ such that ${\mathrm{ht}}({\frak P}) &...
Pierre's user avatar
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0 votes
1 answer
351 views

Strange subscheme in ${\mathrm{Spec}} R \times {\Bbb A}^1_{\Bbb C}$

Let ${\Bbb C}[X_1,\ldots,X_n]$ be a $n$-variable polynomial ring over a complex number field ${\Bbb C}$. For its maximal ideal $(X_1,\ldots,X_n)$, we define the geometric regular local ring as $R \...
Pierre's user avatar
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5 votes
1 answer
73 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 ...
Mike Stay's user avatar
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1 vote
0 answers
76 views

Coherence of subrings of K[[X,Y]]

Let $K[[X,Y]]$ be a two-variables formal power series ring over a field $K$. Consider a sub-ring $\iota \colon A \subset K[[X,Y]]$. Q. Is A coherent? $\quad$ Or is it automatic that $\iota$ is ...
Pierre MATSUMI's user avatar
5 votes
1 answer
205 views

Necessity of shapes for coherence results in category theory

The classic coherence theorems of MacLane (Natural associativity and commutativity, Rice U. studies, 1963) talked about natural transformations between functors. By 1971 (Kelly-MacLane, Coherence in ...
Matt Brin's user avatar
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7 votes
0 answers
489 views

A definition of the homotopy colimit of a coherent diagram

Suppose I am given a homotopy coherent diagram of spaces of shape $I$ (This is a simplicial functor $F:\mathfrak{C}[I] \to Top$, where $\mathfrak{C}$ is the standard cofibrant replacement functor in ...
KotelKanim's user avatar
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5 votes
0 answers
246 views

Is there a reasoned derivation of the coherence conditions for symmetric rig categories?

I know what the coherence conditions are, I can look them up in M. Laplaza, Coherence for distributivity, Lecture Notes in Mathematics 281, Springer Verlag, Berlin, 1972, pp. 29-72. In theory, ...
Jacques Carette's user avatar
3 votes
0 answers
293 views

Pseudomodules, "general coherence theorem"

A pseudomonoid is defined within a monoidal bicategory. It is like a monoid in a monoidal category except that the usual axioms hold up to coherent invertible 2-cells. Pseudomonoid is like a monoidal ...
Dimitri Chikhladze's user avatar
7 votes
1 answer
709 views

Simple-minded coherence of tricategories

Recall Mac Lane's version of coherence for monoidal categories, which one can state informally as follows: "Simple-minded" coherence for monoidal categories Let $A$, $A^\prime$ be two ...
Piotr Pstrągowski's user avatar
2 votes
1 answer
197 views

What is the suitable setting for supercoherence with value in a bicategory?

It was J.F. Jardine established the so called supercoherence theory in Journal of Pure and Applied Algebra Volume 75, Issue 2, 18 October 1991, Pages 103–194. The result can be roughly stated as ...
Ma Ming's user avatar
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