14
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Accepted
Why are the source-target rules of composition always strictly defined?
However, every definition I've seen for higher categories assumes that the source of a composite f;g is equal to the source of f (in diagrammatic notation) and the target of f;g is equal to that of g.
...
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10
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Why are the source-target rules of composition always strictly defined?
Dmitri Pavlov’s answer points out models where these equations are indeed weakened. But at the same time, there are good reason why they’re strict in many models of weak higher categories: These ...
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9
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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 ...
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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 ...
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Is there a reasoned derivation of the coherence conditions for symmetric rig categories?
I'm stating here what I think is the correct version of the conjecture in John Baez's answer.
The 1-categorical theory of rigs has morphisms given by polynomials whose coefficients are in $\mathbb{N}$....
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4
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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 ...
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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 $A$ ...
- 466
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Is there a reasoned derivation of the coherence conditions for symmetric rig categories?
I have not seen a reasoned derivation of
Laplaza's coherence laws for what we now call a symmetric rig category, namely a category with two symmetric monoidal structures $\otimes, \oplus$ obeying the ...
- 22.3k
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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 $\alpha=0$ ...
- 331
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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 ...
- 10.6k
3
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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 ...
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Bⁿ and coherence
One natural setting where the delooping operation $\def\B{{\sf B}}\B$ can be defined very quickly is Γ-spaces, described by Segal in Categories and cohomology theories.
A Γ-space is simply a functor $\...
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Is there a reasoned derivation of the coherence conditions for symmetric rig categories?
I believe you already hint at the answer.
In theory, apparently I can find these also in
G. Kelly, Coherence theorems for lax algebras and distributive laws, Lecture Notes in Mathematics 420, ...
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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^{\...
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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 $F$ ...
- 1,115
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Accepted
Coherence laws when composing 2-monads
These are known as pseudo-distributive laws and are the most common notion of distributive law of 2-dimensional monads, even when both 2-dimensional monads in question are strict (i.e. 2-monads rather ...
- 10.6k
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