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I added to walking structure a 2-categorical theorem that implies that usually “the underlying X of the walking X is the initial X”. This fact seems like it should be well-known, but I don’t offhand know a reference for it, can anyone give a pointer?
Unsure whether this is any help, but similar in spirit seem some characterizations of indiscrete categories. Internal pointer: indiscrete category, Section 2. External pointer: e.g. page 13 in pdf of this talk: http://benedikt-ahrens.de/talks/Fields_HoTT_2016.pdf.
I don’t see any relationship.
I realized that the above claim that “the underlying X of the walking X is the initial X” was a little overbroad; the situation is a bit subtler than that. The problem is that the “underlying set” functor $C(I,-)$ of a monoidal category is in general only lax monoidal, whereas the universal property of a “walking X qua monoidal category” is relative to strong monoidal functors. The argument works much better for things like categories with products, where the relevant categorical structure is preserved by the “underlying set” functor $C(1,-)$. I think we can conclude things about some walking Xs qua monoidal category, by applying the argument to multicategories instead and using the fact that a multicategory embeds fully-faithfully in the monoidal category that it freely generates (since if X is something that makes sense in a multicategory, then the walking X qua monoidal category is the free monoidal category generate by the walking X qua multicategory). But for other walking Xs qua monoidal category the claim doesn’t even make sense. I’ve updated the page walking structure with this.
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