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Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
3
votes
Big list of comonads
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 …
7
votes
How general is TX⊗X≃X⊗TX?
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 …
4
votes
(When) does a morphism of monad induce adjoint functors between categories of algebras?
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 …
1
vote
Poisson and homotopy Poisson operads
A non-cofibrant dg operad whose homology is Pois and which looks like Com∘Lie∞ appears in Section 4.1 of this paper of Anton Khoroshkin and Pedro Tamaroff, the …
6
votes
Who introduced the notion of 2-categories?
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 …
3
votes
The advantage of asymmetric objects
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 …
9
votes
Do non-associative objects have a natural notion of representation?
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 …
13
votes
Accepted
Mathematical life of Friedrich Ulmer
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 …
30
votes
Accepted
What are some examples of interesting uses of the theory of combinatorial species?
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 …
2
votes
History of "natural transformations"
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 …
7
votes
Understanding a quip from Gian-Carlo Rota
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 …
8
votes
Accepted
Does this notion related to species/operads/FI-modules have a name?
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 …
4
votes
Origin of the sign convention in the Tensor product of graded vector spaces
If you want to find another σ′V,W:V⊗W→W⊗V so that σ′U,V⊗W=σ′U,Vσ′U,W and assume that on homogeneous elements $\sigma'_{V,W}(v\otimes w …
2
votes
Examples where it's useful to know that a mathematical object belongs to some family of objects
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 …
4
votes
In what sense are fields an algebraic theory?
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.