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).
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Origin of action of symmetric groups in Lurie's definition of operad in Higher Algebra
I'm somewhat confused about a passage in Lurie's "Higher Algebra." Let me first give the relevant definitions.
Definition ([HA, Definition 2.1.1.1]). A colored operad $\newcommand{\cO}{\...
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The role of $\mathcal K (0)$ in the Stasheff Associahedra operad
When reading "The Geometry of Iterated Loop Spaces" by P. May, I encountered the following question. He says (Remark 3.14) that the Stasheff spaces $K_j$, $j\geq 2$, together with $K_0 = *$ ...
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Is there a notion of infinity multicategories?
I know there are infinity operads, but I could not find anything about infinity multicategories. Is there any such notion or is it just useless to even consider such an object?
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Stable and Pointed Infinity-Operads
I wanted to understand the construction of the maps $\mathsf{An}^{\times} \xrightarrow{(-)_+} \mathsf{An}_{*/}^{\wedge} \xrightarrow{\Sigma^{\infty}} \mathsf{Sp}^{\otimes}$ of commutative algebras in $...
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Homotopy in the space of linear isometries
I wanted to understand the linear isometries operad $\mathscr{L}$ as a model for $E_{\infty}$-operads for which I wanted to show that $\mathscr{L}(n) \simeq *$. This all reduces down to some ...
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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 $(\infty, 1)$-...
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Composition of Associahedra Operad
Is there a nice reference which explains in detail how composition works for the $A_\infty$-operad (associahedra operad)? It would be neat to have a visualization for low dimensions of what's ...
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Sanity Check: Monoids as Algebras over an Operad
I should probably study the classical viewpoint first. I haven't yet and I will eventually but let us stick to $\infty$-operads for this question. I'm following the lecture notes from Hebestreit-...
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On infinity-morphisms between homotopy algebras over operads
Let $P$ be a Koszul operad.
In the book of Loday-Vallette "Algebraic Operads", an $\infty$-morphism between algebras $A$, $B$ over a $P_\infty$-operad is defined as a dg-morphism $P^\textrm{¡...
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Decomposable differential of operads
To my understanding, a degree $p$ derivation (for $p\in \mathbb{N}_{>0}$) on a graded reduced (meaning $P(0)=0$) operad $P$ of vector spaces is given by a sequence of equivariant degree $p$ $k$-...
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Construction of the free operad
I am reading Fresse's "Koszul duality of operads and homology of partition posets", and trying to understand the construction of the free operad $F(M)$ on a symmetric sequence $M$ as ...
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(Why) is the category of $k$-linear operads not abelian?
For $k$ a field, a $k$-linear operad is defined to be a symmetric operad in the category of $\mathbb{Z}$-graded $k$-vector spaces (that is, a symmetric sequence of $k$-vector spaces such that certain ...
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On categorical operads
A symmetric $1$-coloured operad can be defined in any symmetric monoidal category. In particular, it can be defined for the cartesian monoidal category $\mathsf{Cat}$ of small categories and functors.
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Derivations of the free operad generated by a polynomial functor
The following section in Kontsevich' and Soibelman's paper I don't understand:
Let $F$ be a polynomial functor [this is the author's terminology for analytic functor as explained here], $P= Free_{\...
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Colored operads encode 1-colored operads
In their paper BOUSFIELD LOCALIZATION AND ALGEBRAS OVER COLORED OPERADS David White and Donald Yau, write:
Colored operads encode even more general algebraic structures, inluding the category of ...
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Symmetric coloured operads as monoids?
A symmetric operad in a symmetric monoidal category $\mathcal{C}$ can be defined as a monoid in the category of presheaves $[\mathbb{P}^{op},\mathcal{C}]$ with the substitution product $\circ$. Here $\...
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Operads and Kontsevich' polynomial functors
I am reading Kontsevich' and Soibelman's paper here. The first chapter is about operads and seems non-standard.
I know the following:
Let $\mathcal{C}$ be a symmetric monoidal category which is ...
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Non unital Hopf relation
The following problem is an exercise in Loday-Vallette's Algebraic Operads. I hope I am understanding this correctly. Any suggestions or hints would be appreciated.
Show that the restriction of the
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Subcategory of functor category is complete
Let $D$ be a complete and cocomplete symmetric monoidal closed category (a Bénabou cosmos). Let $P$ be the permutation category. Consider the substitution product $\circ$ on the functor category $[P^{...
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On the relationship of Hopf-algebras and (co)Operads
In the paper Axiomatic Homotopy Theory for Operads Berger and Moerdijk construct a cooperad out of a commutative bimonoid (I have no idea, why they call the latter a Hopf-object, since in the whole ...
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Model Structure on Constant-free Symmetric Operads
I am currently trying to find a reference for the assertion that the category of positive / constant-free (meaning $\cal{O}(0)=\emptyset$ is the initial object) symmetric operads $\operatorname{Opd}_\...
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Operad structure on the finite covers of a space
Let $(\Sigma_\bullet)$ be the collection of the symmetric groups. These have a structure of an operad in $\mathsf{Set}$ (it is in fact the operad $Ass$ encoding monoids). The collection of the ...
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Adjunction between the category of operads under $P$ and $(P,P)$-bimodules
Let $C$ be a symmetric monoidal category and $\mathcal{P}$ an operad in $C$. A morphism of operads $\mathcal{P} \to \mathcal{Q}$ gives rises to a $(\mathcal{P},\mathcal{P})$-bimodule structure on $\...
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Explicit quadratic cooperad
I'm following Loday-Vallette's book Algebraic Operads, and I'm having some trouble understanding the quadratic cooperad associated to a quadratic data $(E, R)$.
They define the quadratic cooperad $\...
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Reference for Stasheff's Associahedra Operad
I am currently reading up on operads and am more than confused about the Stasheff operad. It is completely unplausible that I am the first student, who feels this way, so I hope to find a reference ...
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Reference request: Tensoring with Barratt-Eccles is $\Sigma_* $-cofibrant replacement
In some different places (e.g. in Berger & Fresse, Combinatorial operad actions on cochains), it is stated as a classical result that the Barratt-Eccles operad $\mathcal{E}$ (in chain complexes) ...
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Anticommutative operad
Let $(C, \otimes, 1)$ be a additive symmetric monoidal category with internal hom (well, lets think about $\mathrm{Mod}_R$ with $R$ a commutative ring), I am trying to figure out what would be the ...
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Koszul Duality and Andre-Quillen homology relationship
This may be a very open ended question but what is the relationship between koszul duality and Andre-Quillen homology? For example if I consider the André-Quillen homology $AQ_k(B)$ of an associative ...
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Map between endomorphism operads
Let $A$ and $B$ be alegebras of any kind (the example that I have in mind is the underlying module of a graded operad) and consider its endomorphism operads $\mathrm{End}_A$ and $\mathrm{End}_B$, ...
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The category of graded $\mathbb S$-modules form a monoidal category
I am reading paragraph 6.2 in Algebraic Operads by Jean-Louis Loday and Bruno Vallette. Proposition 6.2.2 states:
The category of graded $\mathbb S$-modules, with
the (composite) product $\circ$ and ...
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An explicit formula for group action on an operad
I read Algebraic Operads by Loday and Vallette, namely paragraph 5.3.7 Partial definition of an operad. It's not clear to me how does $\sigma''$ act (the phrase "acting identically on ... with ...
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Cohomology of algebra over an operad as naive Ext-functor instead of with Kähler module of differentials
Let $\mathcal{P}$ be an operad and $A$ a $\mathcal{P}$-algebra. In Algebraic Operads(AO) by Loday-Vallette there are some cohomology theories defined for $A$:
Operadic cohomology = cohomology of $C_{...
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Symmetric lax monoidal functor takes operads to operads
I am looking for a reference to the following theorem.
If $F:\mathcal{C}\to\mathcal{D}$ is a symmetric lax monoidal functor between symmetric monoidal categories, then it induces a well-defined ...
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Is There a Notion of Diagram in Multicategories and/or Operads?
In ordinary category theory there is a notion of a diagram in a category $\mathsf{C}$ which is usually described as a functor $F: \mathsf{J \to C}$ where $\mathsf{J}$ is some small category. Based on ...
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Koszul dual of a quadratic operad
In Proposition 7.2.4 of Loday and Vallette's Algebraic Operads, they prove that the Koszul dual operad of a quadratic operad $\mathcal{P}(E,R)$, with generators E finite dimensional in each arity and ...
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Why does the differential of a minimal model (of an algebra, operad) need be decomposable?
In operad theory, a minimal model of a dg-operad $\mathcal{P}$ (or dga A) is given by a quasi-free resolution $(\mathcal{T}(E),\partial) \overset{f}{\longrightarrow} P$ where $f$ is a quasi-...
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Is there some practical intuition when working with a cooperad given by cogenerators and corelations?
In the case of algebras and operads, a description by generators and relations is common practice and I have a good understanding of this. A non-symmetric operad $\mathcal{P}$ given by a linear space ...
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Partial composition of Swiss-Cheese Operad
I recently came across an operad called the swiss-cheese operad. I am trying to understand its definition through partial composition. I am having trouble checking the axioms (This makes me think that ...
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When does the left adjoint of the base change functor between categories of algebras over operads preserve quasi-isomorphisms?
I have been thinking about the following question: given a morphism of coloured dg-operads $$\phi: P \longrightarrow Q$$
we derive a lax morphism between their respective monads $T= T_P \...
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Differential of the Twisted complex for algebraic operads
I have a question about the proof of lemma 6.4.12 in the book Algebraic Operads (Loday-Vallette) which I do not seem to be able to fully complete on my own. Hopefully, somebody here can point out what ...
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What is the meaning of this product notation?
Let $X,Y$ be topological spaces, both of which admit an action by a group $G$. What is the definition of the space denoted by $ X \times_G Y $?
In particular, let $C_n(k)$ denote the space of ordered $...
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Koszul dual cooperad of the associative operad
I am trying to compute the $k$-modules of $\mathcal{As}^¡$, the Koszul dual cooperad of the associative operad $\mathcal{As}$. I am using sections 7.1.4 and 7.2.1 of Algebraic Operads to try to do ...
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Wedge product of permutations in definition of pairing?
In May's definition of a pairing of operads, he states that a pairing of operads $\tau: (A,O)\rightarrow C$ consists of a collection of maps $\tau: A(i)\times O(j)\rightarrow C(ij)$ that satisfies the ...
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Free algebra over an operad is an algebra over that operad
Let us focus on operads of vector spaces and let $V$ be a vector space and $P$ an operad. The free $P$-algebra on $V$ is defined by $P(V)=\bigoplus_{r=0}^\infty (P(r)\otimes V^{\otimes r})_{\Sigma_r}$,...
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Generalized multicategories to a $\Sigma$-free operad
According to Elmendorf in his paper "Left Adjoints for Generalized Multicategories", generalized multicategories associated
to a $\Sigma$-free operad $\mathcal{D}$, which we call $\mathbb{D}$...
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How does a map of two categorical operads induce a map of categories of generalized multicategories?
I'm working on a project and I have two categorical operads with a free $\Sigma_n$ action and there exists a map of operads between them. I know via a paper by Elmendorf and others titled "Left ...
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What is the name of this operad and proof verification.
Let $P_k$ denote the $k$-th permutoassociahedron and define an operad $\Xi$ by $\Xi(k)=V(P_k)$ where the composotion is given by replacement of variables and renaming. An exanmple of replacement of ...
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Universal dg-algebra of an $A_\infty$-algebra
In this document by Keller, proposition 2.1, it is stated that for every $A_\infty$-algebra $A$ there is a universal dg-algebra $U(A)$ w.r.t. the existence of an $A_\infty$-morphism $A\to U(A)$, and ...
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Questions wrt. definition of $L_\infty$- and dg-Lie-algebras
I am trying to understand this definition from nLab (Def. 3.2) of $L_\infty$-algebras, and the following example that is supposed to boil down to dg-Lie algebras.
What is the difference between ...
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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_\...
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