Questions tagged [coherence]

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7
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1answer
137 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 $\...
4
votes
0answers
70 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 $...
7
votes
2answers
622 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 ...
2
votes
0answers
150 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}) &...
0
votes
1answer
323 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 \...
4
votes
0answers
46 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 ...
1
vote
0answers
72 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 ...
5
votes
1answer
164 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 ...
6
votes
0answers
379 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 ...
5
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0answers
203 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, ...
2
votes
0answers
239 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 ...
6
votes
1answer
540 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 ...
2
votes
1answer
183 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 ...