Related papers: The $\infty$-Categorical Eckmann-Hilton Argument

Let $\mathcal{O}^{\otimes}$ and $\mathcal{P}^{\otimes}$ be $k$- and $\ell$-connected unital $G$-operads subject to the condition for all $S$ that $\mathcal{O}(S) = \emptyset$ if and only if $\mathcal{P}(S) = \emptyset$. We show that the…

We prove a connectivity bound for maps of $\infty$-operads of the form $\mathbb{A}_{k_1} \otimes \cdots \otimes \mathbb{A}_{k_n} \to \mathbb{E}_n$, and as a consequence, give an inductive way to construct $\mathbb{E}_n$-algebras in…

We affirm and generalize a conjecture of Blumberg and Hill: unital weak $\mathcal{N}_\infty$-operads are closed under $\infty$-categorical Boardman-Vogt tensor products and the resulting tensor products correspond with joins of weak…

The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the…

In our recent papers [Sh1,2], we introduced a {\it twisted tensor product} of dg categories, and provided, in terms of it, {\it a contractible 2-operad $\mathcal{O}$}, acting on the category of small dg categories, in which the "natural…

Let $G$ be the Grassmannian $G(d,n)$, let $X$ and $Y$ be complete irreducible varieties, and let $X\rightarrow G$ and $Y\rightarrow G$ be morphisms. Hansen proved that $X \times_G Y$ is connected when $codim f(X) + codim g(Y) < n$. We show…

Let $\mathbb{E}_d$ denote the little discs operad for $1 \le d \le \infty$ and let $\mathcal{C}$ be an $\infty$-category all of whose mapping spaces are $n$-truncated. We prove that when considering $\mathbb{E}_d$-monoids in $\mathcal{C}$,…

Let P be a quadratic operad. We determine an associated operad ~P such that for any P-algebra A and any ~P-algebra B then the tensor product $A \otimes B$ is a P-algebra.

We present a simple extension of the classical Hilton-Eckmann argument classically used to prove that the endomorphism monoid of the unit object in a monoidal category is commutative. It allows us to recover in a uniform way well-known…

We extend the theory of d-categories, by providing an explicit description of the right mapping spaces of the d-homotopy category of an $\infty$-category. Using this description, we deduce an invariant $\infty$-categorical characterization…

Given two small dg categories $C,D$, defined over a field, we introduce their (non-symmetric) twisted tensor product $C\overset{\sim}{\otimes} D$. We show that $-\overset{\sim}{\otimes} D$ is left adjoint to the functor $Coh(D,-)$, where…

A well-known result of Chv\'{a}tal and Erd\H{o}s from 1972 states that a graph with connectivity not less than its independence number plus one is hamiltonian-connected. A graph $G$ is called an $[s,t]$-graph if any induced subgraph of $G$…

We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many…

The homotopy category of $N_\infty$ operads is equivalent to a finite lattice, and as the ambient group varies, there are various image constructions between these lattices. In this paper, we explain how to lift this algebraic structure…

Under a slightly stronger hypothesis, one improves a connectedness result of Debarre [D] for a product of two projective spaces in terms of the extension problem of formal-rational functions (see Theorems 1.3 and 1.4 of the introduction)

It is well-known that the "pre-2-category" $\mathscr{C}at_\mathrm{dg}^\mathrm{coh}(k)$ of small dg categories over a field $k$, with 1-morphisms defined as dg functors, and with 2-morphisms defined as the complexes of coherent natural…

We show that in a weak globular $\omega$-category, all composition operations are equivalent and commutative for cells with sufficiently degenerate boundary, which can be considered a higher-dimensional generalisation of the Eckmann-Hilton…

Let $A$ be a, not necessarily closed, linear relation in a Hilbert space $\sH$ with a multivalued part $\mul A$. An operator $B$ in $\sH$ with $\ran B\perp\mul A^{**}$ is said to be an operator part of $A$ when $A=B \hplus (\{0\}\times \mul…

Our main result is a theorem saying that a bounded operator $A$ on a Hilbert space belongs to a certain set associated with its self-commutator $[A^*,A]$, provided that $A-zI$ can be approximated by invertible operators for all complex…

Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise…