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Questions tagged [operads]

Operads are structures encoding the properties of algebras (in a very general sense), for example associativity, commutativity, unitality, and the relations between them. Their main uses lie in (abstract-algebra), (category-theory) or (algebraic-topology).

Operads are structures encoding the properties of algebras (in a very general sense), for example associativity, commutativity, unitality, and the relations between them. Their main uses lie in , or .

121 questions
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Why is the recognition principle important?

The recognition principle basically states that (under some conditions) a topological space $X$ has the weak homotopy type of some $\Omega^k Y$ iff it is an $E_k$-algebra (ie. an algebra over the operad of the little $k$-cubes). This principle is…
Najib Idrissi
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Are groups algebras over an operad?

I'm trying to understand a little bit about operads. I think I understand that monoids are algebras over the associative operad in sets, but can groups be realised as algebras over some operad? In other words, can we require the existence of…
Jacob Bell
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Do there exist algebras of more directions of operation than left-right?

Again I am kind of new to most things algebraic, only having learned the very basics about groups. As little I have learned about groups and their operations is that an operation has two arguments, one from the left and one from the right. $$a\circ…
mathreadler
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Symmetric tensor powers as tensors over symmetric group algebra

Let $V$ be a $k$-vector space and $V^{\otimes n}$ the $n$-fold tensor power of $V$ and let $\mathbb{S}_n$ be the symmetric group of an n-element set, with its signum representation denoted by $(-1)^\sigma$ for $\sigma\in \mathbb{S}_n$. Now on one…
Mark Neuhaus
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$C_\infty$ analog of the correspondence between $A_\infty$-alg. structures on $A$ and dg coalg. strucures on $(\bar T(sA),\Delta)$

There is a 1-1-correspondence between $A_\infty$-algebra structures on a graded vector space $A$ and dg. coalgebra structures on the bar construction $(\bar T(sA),\Delta)$. My question: Is there any analogous statement for $C_\infty$-algebras?…
Bashar Saleh
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Graded Vector Spaces (definition)

I am studying Algebraic Operads with the book Algebraic Operads, by Jean-Louis Loday and Bruno Vallette and I'm having a little problem with the definition of graded vector space. My advisor and I disagree on the definition. The book defines it this…
Pryscilla Silva
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What are types of coalgebras that are more naturally described by cooperads?

Let $\mathsf{C}$ be a symmetric monoidal category. An object $X \in \mathsf{C}$ has two operads "naturally" (the two constructions aren't functorial) associated to it: the operad of endomorphisms and the operad of coendomorphisms $$\mathtt{End}_X(r)…
Najib Idrissi
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There is no "operad of fields"

I've read the following proof-less claim: there is no operad such that the algebras over it are fields. We can make that precise by asking whether there's an operad $\mathcal{P}$ in abelian groups such that the category $\mathcal{P}Alg$ is…
Najib Idrissi
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Higher (chain) homotopies

I am aware of this question, which unfortunately doesn't help me enough. Recall that a (chain) homotopy between maps $f, g\colon X_\bullet\to Y_\bullet$ of chain complexes is a collection of maps $h_\bullet\colon X_\bullet \to Y_{\bullet+1}$ such…
Bubaya
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$\Omega X$-modules are functors from $X$

Let $X$ be a connected (nice) space, $x\in X$ and $\Omega X$ the loopspace at $x$. Then $\Omega X$ has an $E_1$-structure, and so we may consider left $\Omega X$-modules. There are a few ways to do so : 1) one is to use Lurie's general formalism of…
Maxime Ramzi
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The construction of free symmetric/non-symmetric operads

I have a couple of questions concerning basic notions in operad theory. What is the ideological difference between symmetric and non-symmetric operads? I think about the difference in the following way. Symmetric operads are collections of boxes…
Gregg
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Colored operads as finitely essentially algebraic theory.

I call a planar operad what is also called planar (multi-)coloured operad or multicategory and symmetric operad a symmetric multicategory or symmetric (multi-)colored operad. I have two questions regarding them. 1. I read that a theory of planar…
Andrea Gagna
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Showing that $\text{LMod}_{\mathbb{S}}(\text{Sp}) \simeq \text{Sp}$

$\newcommand{LMod}{\text{LMod}} \newcommand{Sp}{\text{Sp}} \newcommand{SSS}{\mathbb{S}} \newcommand{Fun}{\text{Fun}}\newcommand{Op}{\text{Op}} \newcommand{CAlg}{\text{CAlg}} \newcommand{Alg}{\text{Alg}} \newcommand{LM}{\mathcal{LM}^{\otimes}} $For…
Frusciante
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Is there a concise description of the $\infty$-category $\mathrm{Mod}_A^\mathcal{O}(\mathcal{C})$ of modules over an algebra over an $\infty$-operad?

[Cross-posted to this MO question.] In Higher Algebra, Section 3.3 Lurie constructs the $\infty$-operads $\newcommand{\Mod}{\mathrm{Mod}}\newcommand{\cO}{\mathcal{O}}\newcommand{\cC}{\mathcal{C}}\Mod^{\cO}(\cC)^\otimes$ and…
Ben Steffan
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Internal Homs of (Higher) Operads and $(\infty, 2)$-Categories

While $(\infty, 1)$-categories continue to scare me (but also bring me joy!), it is almost frightening how naturally $(\infty, 2)$-categories seem to pop up if you're interested in $(\infty, 1)$-categories. Here is one such situation where they seem…
Qi Zhu
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