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The question (under the relaxed notion of operad in the original post) will be settled negatively if we show that the category $\mathrm{Grpd}$ (internal groupoids in $\mathrm{Set}$) isn't monadic over …

The answer is no, because any Day convolution product $F \ast G$ on presheaves $F, G: \mathbb{P}^{op} \to Set$ is cocontinuous in each of the separate arguments $F, G$, and yet the substitution produc …

One significant connection to mathematical physics is through vertex operator algebras, used in conformal field theory. As algebraic objects, VOA's are fairly complicated, but the operations are conve …

(These multi-part questions are always awkward to handle, because different people may answer different parts, and then how to award credit?) For question 1, I find that the shuffle notation doesn't m …

Even for single-sorted operads, the coproduct of initially monic algebras need not be initially monic. First a general construction. Let $A$ be a commutative monoid. …

I think the answer to the question as literally stated is "the trivial group", but I think there are related inquiries which get into some deep combinatorics. One way of thinking about the rooted tr …

The statement is that the Stasheff operad is initial among non-permutative non-unital operads $M$ such that each $M_n$, for $n \geq 2$, carries a basepoint and an $I$-module map $I \times M_n \to M_n$ … involves a structure $M \circ M \to M$ where $\circ$ is the well-known substitution product on graded spaces (see Tom Leinster's book for details on such substitution products, monoids of which being unital operads …

One thing to notice is that every element in $\Delta_n$, for $n \geq 2$, is obtained by iterated composition of elements of $\Delta_2$ (these are parametrized by elements $t$ in the unit interval). In …

There are in fact infinitely many nonplanar rooted tree structures having a given nonempty set of leaves $S$, because for example such a tree can look like a linear stalk of any finite height topped b …

Monoids in the monoidal category $(V^{\mathbb{P}^{op}}, \boxtimes)$ are the same thing as operads. … The nLab article on operads provides plenty of explanation and background for this view on operads. …

Of course, there Tom Leinster's book Higher Operads, Higher Categories, and there's also a lot of stuff on the nLab. … Also see Max Kelly's seminal paper (I believe unpublished until recently), On the operads of J.P. May. …