symmetric monoidal (∞,1)-category of spectra
equivalences in/of $(\infty,1)$-categories
The notion of $(\infty,1)$-operad is to that of (∞,1)-category as operad is to category.
So, roughly, an $(\infty,1)$-operad is an algebraic structure that has for each given type of input and one type of output an ∞-groupoid of operations that take these inputs to that output.
There is a fairly evident notion of ∞-algebra over an (∞,1)-operads. Examples include
$(\infty,1)$-Operads form an (∞,2)-category (∞,1)Operad.
Two models for $(\infty,1)$-operads exist to date, one by Cisinski–Moerdijk–Weiss, the other by Lurie. It is expected though not yet entirely proved that the two are equivalent (Higher Algebra, draft, Remark 2.0.0.8).
The first one models $(\infty,1)$-operads as dendroidal sets in close analogy to (in fact as a generalization of how) simplicial sets model (∞,1)-categories.
The second models the (∞,1)-category version of a category of operators of an operad.
Here simplicial sets are generalized to dendroidal sets. The theory of $(\infty,1)$-operads is then formulated in terms of dendroidal sets in close analogy to how the theory of (∞,1)-categories is formulated in terms of simplicial sets.
There is a model structure on dendroidal sets whose fibrant objest are the quasi-operads in direct analogy to the notion of quasi-category.
So the model structure on dendroidal sets is a presention of the (∞,1)-category of $(\infty,1)$-operads. It is Quillen equivalent to the standard model structure on operads enriched over Top or sSet. Therefore, conversely, the traditional homotopy-theoretic constructions on topological and chain operads (such as cofibrant resolutions in order to present homtopy algebras such as A-∞ algebras, L-∞ algebras, homotopy BV-algebras and the like) are also indeed presentations of $(\infty,1)$-operads.
Every operad $A$ encodes and is encoded by its category of operators $C_A$. In the approach to $(\infty,1)$-operators described below, the notion of category of operators is generalized to an (∞,1)-category of operators.
In this approach an $(\infty,1)$-operad $C^\otimes$ is regarded as an (∞,1)-category $C$ – the unary part of the $(\infty,1)$-operad to be described– with extra structure that determines (∞,1)-functors $C^{\times n} \to C$.
This and the conditions on these are encoded in requiring that $C^\otimes$ is an $(\infty,1)$-functor $C^\otimes \to \Gamma$ over Segal's category $\Gamma$ of pointed finite sets, satisfying some conditions.
In particular, any symmetric monoidal (∞,1)-category yields an example of an $(\infty,1)$-operad in this sense. In fact, symmetric monoidal $(\infty,1)$-categories can be defined as $(\infty,1)$-operads such that the functor $C^\otimes \to \Gamma$ is a coCartesian fibration. (For the moment, see monoidal (infinity,1)-category for more comments and references on higher operads in this context.)
This is the approach described in (LurieCommutative)
We are to generalize the following construction from categories to (∞,1)-categories.
For $\mathcal{O}$ a symmetric multicategory, write $\mathcal{O}^\otimes \to FinSet^{*/}$ for its category of operators.
Here $\mathcal{O}^\otimes$ is the category whose
objects are finite sequences (tuples) of objects of $\mathcal{O}$;
morphisms $(X_1, \cdots, X_{n_1}) \to (Y_1, \cdots, Y_{n_2})$ are given by a morphism $\alpha \colon \langle n_1\rangle \to \langle n_2\rangle$ in $FinSet_*$ together with a collection of multimorphisms
The functor $p \colon \mathcal{O}^\otimes \to FinSet^{*/}$ is the evident forgetful functor.
In (Lurie) this is construction 2.1.1.7.
This motivates the following definition of the generalization of this situation to (∞,1)-category theory.
Write $FinSet^{*/}$ for the category of pointed finite set (Segal's Gamma-category).
For $n \in \mathbb{N}$ we write
for the pointed set with $n+1$ elements.
A morphism in $FinSet^{*/}$
is called an inert morphism if it is a surjection, and an injection on those elements that are not sent to the base point. That is, the preimage of every non-base point is a singleton.
called an active morphism if only the basepoint goes to the basepoint.
For $n \in \mathbb{N}$ and $1 \leq i \leq n$ write
for the inert morphism that sends all but the $i$th element to the basepoint.
Notice that for each $n \in \mathbb{N}$ there is a unique active morphism $\langle n\rangle \to \langle 1\rangle$.
The $(\infty,1)$-category of operators of an $(\infty,1)$-operad is a morphism
of quasi-categories such that the following conditions hold:
For every inert morphism in $FinSet^{*/}$ and every object over it, there is a lift to a $p$-coCartesian morphism in $\mathcal{O}^\otimes$. In particular, for $f \colon \langle n_1\rangle \to \langle n_2\rangle$ inert, there is an induced (∞,1)-functor
The coCartesian lifts of the inert projection morphisms induce an equivalence of derived hom-spaces in $\mathcal{O}^{\otimes}$ between maps into multiple objects and the products of the maps into the separete objects:
For $f \colon \langle n_1 \rangle \to \langle n_2 \rangle$ write $\mathcal{O}^\otimes_f(-,-) \hookrightarrow \mathcal{O}^\otimes(-,-)$ for the components of the derived hom-space covering $f$, then the $(\infty,1)$-functor
induced as above is an equivalence.
For every finite collection of objects $C_1, \cdots c_n \in \mathcal{O}^\otimes_{\langle 1\rangle}$ there exists a multiobject $C \in \mathcal{O}^\otimes_{\langle n\rangle}$ and a collection of $p$-coCartesian morphisms $\{C \to C_i\}$ covering $\rho^i$.
Equivalently (given the first two conditions): for all $n \in \mathbb{N}$ the $(\infty,1)$-functors $\{(\rho^i)_!\}_{1 \leq i \leq n}$ induce an equivalence of (∞,1)-categories
(Lurie, def. 2.1.1.10, remark 2.1.1.14)
The conditions in def. 3 mean that $p \colon \mathcal{O}^\otimes \to FinSet^{*/}$ encodes
an (∞,1)-category $\mathcal{O} \coloneqq \mathcal{O}^\otimes_{\langle 1\rangle}$;
for each $n \in \mathbb{N}$ an $n$-ary operation given by the $(\infty,1)$-functor
induced by the unique active morphism $\langle n\rangle \to \langle 1\rangle$
a coherently associative multicomposition of these operations.
The notion of def. 3 may also be called a symmetric $(\infty,1)$-multicategory or colored $(\infty,1)$-operad. The colors are the objects of $\mathcal{O}$.
We now turn to the definition of homomorphisms of $(\infty,1)$-operads.
Given an $(\infty,1)$-operad $p \colon \mathcal{O}^\otimes \to FinSet^{*/}$ as in def. 3, a morphism $f$ in $\mathcal{O}^\otimes$ is called an inert morphism if
$p(f)$ is an inert morphism in $FinSet^{*/}$ by def. 2;
$f$ is a $p$-coCartesian morphism.
A morphism of (∞,1)-operads is a map between their (∞,1)-categories of operators over $FinSet^{*/}$ that preserves the inert morphisms of def. 4.
Morphisms of operads $\mathcal{O}_1 \to \to \mathcal{O}_2$ can be understood equivalently as exhibiting an $\mathcal{O}_1$-algebra in $\mathcal{O}_2$. Therefore:
For $\mathcal{O}_1, \mathcal{O}_2$ to $(\infty,1)$-operads, write
for the full sub-(∞,1)-category of the (∞,1)-functor (∞,1)-category on those that are morphisms of (∞,1)-operads by def. 5.
We also have the notion of
See there for more details.
There is a model category that presents the (∞,1)-category $(\infty,1)Cat_{Oper}$ of $(\infty,1)$-categories of operations.
There exists a
model category $\mathcal{P} Op_{(\infty,1)}$
whose underlying category has
objects are marked simplicial set $S$ equipped with a morphism $S \to N(FinSet_*)$ such that marked edges map to inert morphisms in $FinSet_*$ (those for which the preimage of the marked point contains just the marked point)
morphisms are morphisms of marked simplicial sets $S \to T$ such that the triangle
commutes;
which is canonically an SSet-enriched category;
and whose model structure is given by
cofibrations are those morphisms whose underlying morphisms of simplicial sets ate cofibrations, hence monomorphisms
weak equivalences are those morphisms $S \to T$ such that for all $A \to N(FinSet_*)$ that are $(\infty,1)$-categories of operations by the above definition, the morphism of SSet-hom objects
is a homotopy equivalence of simplicial sets.
an object is fibrant if and only if it is an $(\infty,1)$-category of operations, by the above definition.
This is prop 1.8 4 in
We list some examples of $(\infty,1)$-operads incarnated as their (∞,1)-categories of operators by def. 3.
The first basic examples to follow are in fact all given by 1-categories of operators.
The identity functor on $FinSet^{*/}$ exhibits an $(\infty.1)$-operad. This is the commutative operad
The (∞,1)-algebras over an (∞,1)-operad over this $(\infty,1)$-operad are E-∞ algebras.
The associative operad has $Assoc^\otimes$ the category whose objects are the natural numbers, whose $n$-ary operations are labeled by the total orders on $n$ elements, equivalently the elements of the symmetric group $\Sigma_n$, and whose composition is given by forming consecutive total orders in the obvious way.
The (∞,1)-algebras over an (∞,1)-operad over this $(\infty,1)$-operad are A-∞ algebras
In (Lurie) this is remark 4.1.1.4.
The operad for modules over an algebra $LM$ is the colored symmetric operad whose
objects are two elements, to be denoted $\mathfrak{a}$ and $\mathfrak{n}$;
multimorphisms $(X_i)_{i = 1}^n \to Y$ form
if $Y = \mathfrak{a}$ and $X_i = \mathfrak{a}$ for all $i$ then: the set of linear orders on $n$ elements, equivalently the elements of the symmetric group $\Sigma_n$;
if $Y = \mathfrak{n}$ and exactly one of the $X_i = \mathfrak{n}$ then: the set of linear order $\{i_1 \lt \cdots \lt i_n\}$ such that $X_{i_n} = \mathfrak{n}$
otherwise: the empty set;
composition is given by composition of linear orders as for the associative operad.
The (∞,1)-algebras over an (∞,1)-operad over this $(\infty,1)$-operad are pairs consisting of A-∞ algebras with (∞,1)-modules over them.
In (Lurie) this appears as def. 4.2.1.1.
The operad for bimodules over algebras $BMod$ is the colored symmetric operad whose
objects are three elements, to be denoted $\mathfrak{a}_-, \mathfrak{a}_+$ and $\mathfrak{n}$;
multimorphisms $(X_i)_{i = 1}^n \to Y$ form
if $Y = \mathfrak{a}_-$ and all $X_i = \mathfrak{a}_-$ then: the set of linear orders of $n$ elements;
if $Y = \mathfrak{a}_*$ and all $X_i = \mathfrak{a}_*$ then again: the set of linear orders of $n$ elements;
if $Y = \mathfrak{n}$: the set of linear orders $\{i_1 \lt \cdots \lt i_n\}$ such that there is exactly one index $i_k$ with $X_{i_k} = \mathfrak{n}$ and $X_{i_j} = \mathfrak{a}_-$ for all $j \lt k$ and $X_{i_j} = \mathfrak{a}_+$ for all $k \gt k$.
composition is given by the composition of linear orders as for the associative operad.
The (∞,1)-algebras over an (∞,1)-operad over this $(\infty,1)$-operad are pairs consisting of two A-∞ algebras with an (∞,1)-bimodule over them.
At the time of this writing there is no discussion in “the literature” of the relation between the definition of $(\infty,1)$-operads in terms of dendroidal sets (Cisinski, Moerdijk, Weiss) and $(\infty,1)$-categories of operators (Lurie). The following are some tentative observations. - Urs
update: meanwhile this has been worked out by some people. Results should appear in preprint form soon.
There is an obvious way to regard a tree as an $(\infty,1)$-category of operators:
(dendroidal $(\infty,1)$-category of operators)
Let
be the dendroidal object given by the following composition:
$\Omega \hookrightarrow Op$ is the functor from the tree category $\Omega$ to the category of symmetric colored operads (over Set) that sends a tree to the operad freely generated from it;
$Op \stackrel{C_{(-)}}{\to} Cat/FinSet_*$ sends an operad to its category of operators;
$Cat/FinSet_* \stackrel{N}{\to} \mathcal{P}Op_{(\infty,1)}$ takes the nerve of this category, regarded as a marked simplicial set over $N(FinSet_*)$, whose marked edges are the inert morphisms in the category of operations.
Following the general pattern of nerve and realization, we get:
(dendroidal nerve of Lurie-$\infty$-operad)
The functor
that sends a marked simplicial set $A \to N(FinSet_*)$ to the dendroidal set which sends a tree $T$ to the set of morphisms of $\omega(T)$ into $A$
is the dendroidal nerve of $A$.
One expects that $N_d$ induces a Quillen adjunction and indeed a Quillen equivalence between the above model category structure on $\mathcal{P}Op_{(\infty,1)}$ and the model structure on dendroidal sets. The following is as far as I think I can prove aspects of this. -Urs.
The dendroidal nerve functor has the following properties:
it is the right adjoint of a SSet-enriched adjunction
it sends fibrant objects to fibrant objects
i.e. it sends $(\infty,1)$-categories of operations to $(\infty,1)$-operads in their incarnation as “quasi-operads”;
it sends objects $\pi : A \to N(FinSet_*)$ that come from grouplike symmetric monoidal ∞-groupoids to fully Kan dendroidal sets (that have the extension property with respect to all horns)
it sends objects $\pi : A \to N(FinSet_*)$ that come from symmetric monoidal (∞,1)-categories to dendroidal sets that have the extension property with respect to at least one outer horn $\Lambda_{v} T$ for $v \in T$ an $n$-corolla, for all $n \in \mathbb{N}$.
its left adjoint sends cofibrations to cofibrations and acyclic cofibrations with cofibrant domain to acyclic cofibrations.
respect for fibrant objects. If $A \to N(FinSet_*)$ is fibrant, then in particual $A$ is a weak Kan complex hence has the extension property with respect to all inner horn inclusions of simplices. We need to show that this implies that $N_d(A)$ has the extension property with respect to all inner horn inclusions of trees.
By an (at the moment unpublished) result by Moerdijk, right lifting property with respect to inner horn inclusions of trees is equivalend to right lifting property with respect to inclusions of spines of trees: the union over all the corollas in a tree.
For this the extension property means that if we find a collection $\{C_{k_i} \to N_d(A)\} = Sp(T)$ of corollas in $N_d(A)$ that match at some inputs and output, then these can be composed to an image $T \to N_d(A)$ of the corresponding tree $T$ in $N_d(A)$.
An image of $T$ in $N_d(A)$ is an image of $\omega(T)$ in $A$. In the category of operators $\omega(A)$ every tree may be represented as the composite of a sequence of morphisms each of which consists of precisely one of the corollas $C_{k_i}$ in parallel to identity morphisms. This way gluing the tree from the corollas is a matter of composing a sequence of edges in $A$. But this is guaranteed to be possible if $A$ is a weak Kan complex.
symmetric monoidal product and outer horn lifting
As described at cartesian morphism, an edge $f : \Delta^1 \to A$ in $A$ is coCartesian if for all diagrams
of 0-horn lifting problems where the first edge of the horn is $f$ itself, there exists a lift
For $f$ the parallel application of an $n$-corolla with a collection of identity morphisms this implies that any outer horn $\Lambda_v T \to N_d(A)$ for which the vertex $v : C_n \to N_d(A)$ maps to $f$, the dendroidal set $N_d(A)$ has the extension property with respect to the inclusion $\Lambda_d T \hookrightarrow T$.
the left adjoint and its respect for cofibrations
By general nonsense the left adjoint to $N_d$ is given by the coend
where in the integrand we have the tautological tensoring of $\mathcal{P}Op_{(\infty,1)}$ over Set.
Notice that $\omega : \Omega \to \mathcal{P}Op_{(\infty,1)}$ is an SSet-enriched functor for the ordinary category $\Omega$ regarded as a simplicially enriched category by the canonical embedding $Set \hookrightarrow SSet$. Therefore this adjunction $F \dashv N_d$ is defined entirely in SSet-enriched category theory and hence is a simplicial adjunction.
The model structure on dendroidal sets has a set of generating cofibrations given by the boundary inclusions of trees. $\partial \Omega[T] \hookrightarrow \Omega[T]$. Tese evidenly map to monomorphisms of underlying simplicial sets under $F$, hence to cofibrations.
For $f : P \hookrightarrow Q$ an acyclic cofibration with cofibrant domain, we need to check that $C_f : C_X \to C_Y$ is a weak equivalence in $\mathcal{P}Op_{(\infty,1)}$. This is by definition the case if for every fibrant object $A$ the morphism
is a weak equivalence in the standard model structure on simplicial sets. By the simplicial adjunction $F \dashv N_d$ this is equivalent to
being a weak equivalence. By the above $N_d(A)$ is fibrant. By section 8.4 of the lecture notes on dendroidal sets cited at model structure on dendroidal sets a morphism between cofibrant dendroidal sets is a weak equivalence precisely if homming it into any fibrant dendroidal set produces an equivalence of homotopy categories.
Since $f$ is a weak equivalence between cofibrant objects by assumption, it follows that indeed $dSet(f,N_d(A))$ is a weak equivalence for all fibrant $A$.
(AHM, or does it? there is a prob here, but I need to run now…)
Hence $C_f$ is a weak equivalence.
operad / $(\infty,1)$-operad, model structure on operads
The formulation in terms of dendroidal sets is due to
Ieke Moerdijk Ittay Weiss, Dendroidal sets (web)
Denis-Charles Cisinski, Ieke Moerdijk, Dendroidal sets as models for homotopy operads (arXiv) .
Here are two blog entries on talks on this stuff:
The formulation in terms of an $(\infty,1)$-version of the category of operators is introduced in
and further discussed in
Now in section 2 of the textbook
The equivalence between the dendroidal set-formulation and the one in terms of $(\infty,1)$-categories of operators is shown in