symmetric monoidal (∞,1)-category of spectra
Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
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 an ∞-algebra over an (∞,1)-operad. Examples include
-Operads form an (∞,2)-category (∞,1)Operad.
Many equivalent models for -operads exist to date. Some of these include
The -category of operators model of Lurie
The complete Segal operad model of Barwick
The equifibered model of Haugseng-Kock and Barkan-Steinebrunner
The “Symmetric sequences in a closed symmetric monoidal model category” model of Berger-Moerdijk
We focus on the first two. 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.
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 objects 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.
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
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 2.1.1.7.
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
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 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 .
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 inert, 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 , then the -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. 2.1.1.10, remark 2.1.1.14)
The conditions in def. mean that encodes
an (∞,1)-category ;
for each an -ary operation given by the -functor
induced by the unique active morphism
a coherently associative multicomposition of these operations.
The notion of def. may also be called a symmetric -multicategory or colored -operad. The colors are the objects of .
We now turn to the definition of homomorphisms of -operads.
Given an -operad as in def. , a morphism in is called an inert morphism if
is an inert morphism in by def. ;
is a -coCartesian morphism.
A morphism of (∞,1)-operads is a map between their (∞,1)-categories of operators over that preserves the inert morphisms of def. .
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. .
We also have the notion of
See there for more details.
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
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 are 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. .
The first basic examples to follow are in fact all given by 1-categories of operators.
The identity functor on exhibits an -operad. This is the commutative operad
The (∞,1)-algebras over an (∞,1)-operad over this -operad are E-∞ algebras.
The associative operad has the category whose objects are the natural numbers, whose -ary operations are labeled by the total orders on elements, equivalently the elements of the symmetric group , and whose composition is given by forming consecutive total orders in the obvious way.
The (∞,1)-algebras over an (∞,1)-operad over this -operad are A-∞ algebras
In (Lurie) this is remark 4.1.1.4.
The operad for modules over an algebra is the colored symmetric operad whose
objects are two elements, to be denoted and ;
multimorphisms form
if and for all then: the set of linear orders on elements, equivalently the elements of the symmetric group ;
if and exactly one of the then: the set of linear order such that
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 -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 is the colored symmetric operad whose
objects are three elements, to be denoted and ;
multimorphisms form
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.
The dendroidal and category of operads models are equivalent, as shown in e.g. Hinich-Moerdijk. The following is another strategy to do so.
There is an obvious way to regard a tree as an -category of operators:
(dendroidal -category of operators)
Let
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)
The functor
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 .
One expects that induces a Quillen adjunction and indeed a Quillen equivalence between the above model category structure on 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 -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.
operad / -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 -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
and made symmetric monoidal in
Further equivalence to Barwick’s complete Segal operads is discussed in
For an account in terms of analytic monads, that is, monads that are cartesian (multiplication and unit transformations are cartesian) and the underlying endofunctor preserves sifted colimits and wide pullbacks (or equivalently all weakly contractible limits), see
For an account in terms of symmetric monoidal categories and equifibrations, see
Rune Haugseng, Joachim Kock, ∞-operads as symmetric monoidal ∞-categories, (arXiv:2106.12975)
Shaul Barkan, Jan Steinebrunner, The equifibered approach to ∞-properads, (arXiv:2211.02576)
On the Eckmann-Hilton argument for (∞,1)-operads:
Last revised on October 29, 2024 at 20:32:30. See the history of this page for a list of all contributions to it.