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An important example in homotopy theory is given by operadic twisting. This was introduced by T. Willwacher in "M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra" (Inventiones …

I believe that, contrary to your claim, this is not true for what you call collections, that is for the monoidal category where the monoids are operads. Indeed, if $C$ is supported at arity 2 and is j …

To complement the answer of Maxime with a useful reference, my go-to article for this circle of questions is "Coequalizers in categories of algebras" by Fred Linton (LNM 80, Seminar on Triples and Cat …

A non-cofibrant dg operad whose homology is $\mathrm{Pois}$ and which looks like $\mathrm{Com}\circ\mathrm{Lie}_\infty$ appears in Section 4.1 of this paper of Anton Khoroshkin and Pedro Tamaroff, the …

Following the comment of varkor, I re-opened Catégories structurées of Charles Ehresmann (published 1963), and I believe that Section 4 "Catégories doubles" and Section 5 "Catégories $n$-uples" give t …

My favourite example is related to some bit of work of my own, so I apologise in advance for a bit of self-promotion. It concerns dealing with symmetric operads (algebraic ones, meaning that the $n$-t …

I am not sure for how long and where you looked, but it takes less than 30 minutes to figure much more than you mention in your post. Apparently, he continued his academic career as Fritz Ulmer : if y …

The words "natural homomorphism" and "natural isomorphism" are also used (mainly in the context related to the First Isomorphism Theorem) in Pontryagin's "Topological groups" (Russian edition 1938, En …

I am not a native speaker of English and moreover belong to the ethnic group that is known to mess up the articles, but I certainly don't feel that the sentence "Behind these and several other attrac …

Depending on whether you want it to agree with the symmetric structure or only with monoidal structure, this would be usually referred to, respectively, as twisted commutative algebras or twisted asso …

If you want to find another $\sigma'_{V,W}\colon V\otimes W\to W\otimes V$ so that $\sigma'_{U,V\otimes W}=\sigma'_{U,V}\sigma'_{U,W}$ and assume that on homogeneous elements $\sigma'_{V,W}(v\otimes w …

To prove that the Hilbert series (the generating function of the sequence of dimensions of homogeneous components) of a finitely generated commutative graded algebra is a rational function, the easies …

Composition of species is closely related to the composition of symmetric collections of vector spaces ("S-modules"), which is a remarkable example of a monoidal category everyone who had ever encount …

Let me concentrate on your first question (frankly speaking, the way you formulate your second question slightly lacks motivation). The case where there is a reasonable suggestion, assumes that you wo …

A small remark that might be helpful as well is motivated by the Birkhoff theorem - in most usual senses, algebras for the given algebraic theory are closed under products, which fields are not.