looking at the unique morphism 1->M in the Lawvere theory of monoids and thinking "wow the absolute unit"
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go to your room and think about what you’ve done
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also you want the free monoidal category on a monoid maybe, the lawvere theory only knows what a monoid in a cartesian monoidal category is 😛
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OKAY I'VE ACTUALLY had this joke in my brain for multiple years and not been satisfied enough to say it for EXACTLY THAT REASON
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like I could go more general but nothing feels quite right. The free monoidal category on a unital magma? prolly better but no longer carries the right emotional weight...
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nah free monoidal category on a monoid is where it’s at imo. cute demonstration of the macrocosm principle plus it should work out to be something familiar, i worked this out awhile ago and totally forgot the answer
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it's the augmented simplex category! this is really important it's what makes lots of stuff work, e.g. anytime you have a (co)monad you can turn it into a [co]simplicial endofunctor...
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okay you're right
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