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2 votes

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

Kelly, Coherence theorems for lax algebras and distributive laws, Lecture Notes in Mathematics 420, Springer Verlag, Berlin, 1974, pp. 281-375. They may be found in Example 6.11 of that paper. …
varkor's user avatar
  • 10.7k
7 votes
1 answer
337 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 monoi …
3 votes

Strictification for closed monoidal categories

I have not been able to find an explicit reference for this in the literature, so I will sketch a proof here. Steve Lack suggested that one could make use of Mac Lane's proof of strictification for mo …
varkor's user avatar
  • 10.7k
6 votes
0 answers
93 views

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 …
1 vote
Accepted

Coherence laws when composing 2-monads

These have been well studied, and there are many equivalent formulations of the coherence conditions. …
varkor's user avatar
  • 10.7k
13 votes
0 answers
207 views

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 biequiva …
4 votes
0 answers
95 views

Coherence for closed bicategories

Is there a coherence theorem in the literature for right-closed bicategories, or can it be deduced from an existing result? …
10 votes
1 answer
325 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 $\ma …
7 votes
0 answers
159 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 …