Algebras and modules
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The notion of -operad is to that of (∞,1)-category as operad is to category.
So, roughly, an -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
-Operads form an (∞,2)-category (∞,1)Operad.
Two models for -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 188.8.131.52).
The first one models -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.
In terms of dendroidal sets
Here simplicial sets are generalized to dendroidal sets. The theory of -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 -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 -operads.
In terms of -categories of operators
Every operad encodes and is encoded by its category of operators . In the approach to -operators described below, the notion of category of operators is generalized to an (∞,1)-category of operators.
In this approach an -operad is regarded as an (∞,1)-category – the unary part of the -operad to be described– with extra structure that determines (∞,1)-functors .
This and the conditions on these are encoded in requiring that is an -functor over Segal's category of pointed finite sets, satisfying some conditions.
In particular, any symmetric monoidal (∞,1)-category yields an example of an -operad in this sense. In fact, symmetric monoidal -categories can be defined as -operads such that the functor 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 a symmetric multicategory, write for its category of operators.
Here is the category whose
objects are finite sequences (tuples) of objects of ;
morphisms are given by a morphism in together with a collection of multimorphisms
The functor is the evident forgetful functor.
In (Lurie) this is construction 184.108.40.206.
This motivates the following definition of the generalization of this situation to (∞,1)-category theory.
Write for the category of pointed finite set (Segal's Gamma-category).
For we write
for the pointed set with elements.
A morphism in
For and write
for the inert morphism that sends all but the th element to the basepoint.
Notice that for each there is a unique active morphism .
(Lurie, def. 220.127.116.11)
The -category of operators of an -operad is a morphism
of quasi-categories such that the following conditions hold:
For every inert morphism in and every object over it, there is a lift to a -coCartesian morphism in . In particular, for inter, there is an induced (∞,1)-functor
The coCartesian lifts of the inert projection morphisms induce an equivalence of derived hom-spaces in between maps into multiple objects and the products of the maps into the separete objects:
For write for the components of the derived hom-space covering (\infty,1)$-functor
induced as above is an equivalence.
For every finite collection of objects there exists a multiobject and a collection of -coCartesian morphisms covering .
Equivalently (given the first two conditions): for all the -functors induce an equivalence of (∞,1)-categories
(Lurie, def. 18.104.22.168, remark 22.214.171.124)
We now turn to the definition of homomorphisms of -operads.
Given an -operad as in def. 3, a morphism in is called an inert morphism if
is an inert morphism in by def. 2;
is a -coCartesian morphism.
Morphisms of operads can be understood equivalently as exhibiting an -algebra in . Therefore:
For to -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.
(Lurie, def. 126.96.36.199).
We also have the notion of
See there for more details.
Model for -categories of operators
There is a model category that presents the (∞,1)-category of -categories of operations.
There exists a
whose underlying category has
objects are marked simplicial set equipped with a morphism such that marked edges map to inert morphisms in (those for which the preimage of the marked point contains just the marked point)
morphisms are morphisms of marked simplicial sets such that the triangle
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 such that for all that are -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 -category of operations, by the above definition.
This is prop 1.8 4 in
We list some examples of -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.
In (Lurie) this is remark 188.8.131.52.
In (Lurie) this appears as def. 184.108.40.206.
The operad for bimodules over algebras is the colored symmetric operad whose
objects are three elements, to be denoted and ;
if and all then: the set of linear orders of elements;
if and all then again: the set of linear orders of elements;
if : the set of linear orders such that there is exactly one index with and for all and for all .
composition is given by the composition of linear orders as for the associative operad.
The (∞,1)-algebras over an (∞,1)-operad over this -operad are pairs consisting of two A-∞ algebras with an (∞,1)-bimodule over them.
Relation between the two definitions
At the time of this writing there is no discussion in “the literature” of the relation between the definition of -operads in terms of dendroidal sets (Cisinski, Moerdijk, Weiss) and -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 -category of operators:
(dendroidal -category of operators)
be the dendroidal object given by the following composition:
is the functor from the tree category to the category of symmetric colored operads (over Set) that sends a tree to the operad freely generated from it;
sends an operad to its category of operators;
takes the nerve of this category, regarded as a marked simplicial set over , 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--operad)
that sends a marked simplicial set to the dendroidal set which sends a tree to the set of morphisms of into
is the dendroidal nerve of .
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 -categories of operations to -operads in their incarnation as “quasi-operads”;
it sends objects 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 that come from symmetric monoidal (∞,1)-categories to dendroidal sets that have the extension property with respect to at least one outer horn for an -corolla, for all .
its left adjoint sends cofibrations to cofibrations and acyclic cofibrations with cofibrant domain to acyclic cofibrations.
respect for fibrant objects. If is fibrant, then in particual 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 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 of corollas in that match at some inputs and output, then these can be composed to an image of the corresponding tree in .
An image of in is an image of in . In the category of operators every tree may be represented as the composite of a sequence of morphisms each of which consists of precisely one of the corollas in parallel to identity morphisms. This way gluing the tree from the corollas is a matter of composing a sequence of edges in . But this is guaranteed to be possible if is a weak Kan complex.
symmetric monoidal product and outer horn lifting
As described at cartesian morphism, an edge in is coCartesian if for all diagrams
of 0-horn lifting problems where the first edge of the horn is itself, there exists a lift
For the parallel application of an -corolla with a collection of identity morphisms this implies that any outer horn for which the vertex maps to , the dendroidal set has the extension property with respect to the inclusion .
the left adjoint and its respect for cofibrations
By general nonsense the left adjoint to is given by the coend
where in the integrand we have the tautological tensoring of over Set.
Notice that is an SSet-enriched functor for the ordinary category regarded as a simplicially enriched category by the canonical embedding . Therefore this adjunction 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. . Tese evidenly map to monomorphisms of underlying simplicial sets under , hence to cofibrations.
For an acyclic cofibration with cofibrant domain, we need to check that is a weak equivalence in . This is by definition the case if for every fibrant object the morphism
is a weak equivalence in the standard model structure on simplicial sets. By the simplicial adjunction this is equivalent to
being a weak equivalence. By the above 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 is a weak equivalence between cofibrant objects by assumption, it follows that indeed is a weak equivalence for all fibrant .
(AHM, or does it? there is a prob here, but I need to run now…)
Hence is a weak equivalence.
The formulation in terms of dendroidal sets is due to
Here are two blog entries on talks on this stuff:
The formulation in terms of an -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 -categories of operators is shown in