8 votes
Accepted

2-monads for categories with a class of (co)limits

Kelly and Lack's paper On the monadicity of categories with chosen colimits answers your questions (1),(2) and (3) affirmatively. The main theorems are Theorem 6.1, 6.2 and 7.1. Their main trick is ...
  • 871
7 votes

Mac Lane's proof of coherence for symmetric monoidal categories

I think that this is a great question, since it is quite common to just say at this point "each relation between the transpositions follows from the relations in the group presentation of the ...
4 votes

Coherence theorem in braided monoidal categories

One way to interpret that coherence theorem in Mac Lane's book (Theorem XII.5.2) is Corollary 2.6 in the Joyal-Street paper that you cited: In the free braided monoidal category generated by a set $A$ ...
3 votes

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

The answer to the question Q is ``no''. Take $n=m=2$, $f_1(X)=X_1X^2$, $f_2(X)=X_2$, $F(X)=X_1X$. If $\alpha\ne0$ then $F(\alpha)=\alpha X_1\in(\alpha^2X_1)\in(f_1(\alpha),f_2(\alpha))$. If $\alpha=0$ ...
3 votes
Accepted

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

It is indeed true that any strong braided monoidal functor from the braid category to a strict braided monoidal category is equivalent to a strict braided monoidal functor, however that comes out of ...
3 votes
Accepted

Strictification of $\mathcal{V}$-pseudofunctors

In section 4 of my paper Not every pseudoalgebra is equivalent to a strict one, I sketched a proof that for any monoidal 2-category $\mathcal{W}$ with small sums preserved on both sides by its tensor ...
  • 60.9k
2 votes
Accepted

Coherence theorem in braided monoidal categories

Theorem 1 says precisely that for every braided monoidal category $M$ and every object $V \in M$, any isotopy class of braid on $n$ strands induces an isomorphism $$ V^{\otimes n}\longrightarrow V^{\...
  • 7,517
2 votes

Necessity of shapes for coherence results in category theory

There is an example in Saunders Mac Lane. Topology and Logic as a Source of Algebra (Retiring Presidential Address), Bulletin of the AMS 82:1, January 1976. See Section 5 ("Coherence and canonical ...
1 vote

Mac Lane's proof of coherence for symmetric monoidal categories

I'm not sure, but it appears that it is not as hard as it seemed. Martin's answer is interesting, but I still wanted to try something elementary before fully committing to powerful machinery. Let $F$ ...
  • 1,065

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