Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
Algebras and modules
Model category presentations
Geometry on formal duals of algebras
The model structure for Segal operads is a presentation of the (∞,1)-category of (∞,1)-operads regarding these as ∞Grpd-enriched operads.
It is the operadic analog of the model structure for Segal categories: its fibrant objects are operadic analogs of Segal categories.
Write for the tree category, the site for dendroidal sets.
Write for the tree with a single edge and no vertices. Write
for the category of simplicial presheaves on the tree category – simplicial dendroidal sets or dendroidal simplicial sets (see model structure for complete dendroidal Segal spaces for more on this).
A Segal pre-operad is a simplicial dendroidal set such that is a discrete simplicial set (a plain set regarded as a simplicially constant simplicial set). Write
for the full subcategory on the Segal pre-operads.
A Segal operad is a Segal pre-operad such that for every tree the powering
of the spine inclusion dSet into is an acyclic Kan fibration. Write
for the full subcategory on the Segal operads.
A Reedy-fibrant Segal operad is a Segal operad which is moreover fibrant in the generalized Reedy model structure .
This is (Cisinski-Moerdijk, def. 7.1, def. 8.1).
The inclusion def. 1 has a left and right adjoint functors
One way to see the existence of the adjoints is to note that is a category of presheaves over the site which is the localization of at morphisms of the form , where is the tree with one edge and no vertex. Write
for the localization functor, then the inclusion of Segal pre-operads is the precomposition with this functor
Therefore the left and right adjoint to are given by left and right Kan extension along .
Explicitly, these adjoints are given as follows.
For , the Segal pre-operad sends a tree either to , if is non-linear, hence if it admits no morphism to , or else to the pushout
in sSet, where the top morphism is for the unique morphism to .
In words, is obtained from precisely by contracting the simplicial set of colors to its set of connected components.
We discuss morphisms between Segal pre-operads with special properties, which will appear in the model structure.
Say a morphism in is a normal monomorphism precisely if is a normal monomorphism (see generalized Reedy model structure), which in turn is the case if it is simplicial-degreewise a normal morphisms of dendroidal sets (see there for details).
Correspondingly, a Segal pro-operad is called normal if is a normal monomorphism.
A morphism in is called an acyclic fibration precisely if it has the right lifting property against all normal monomorphisms, def. 2.
Say a morphism in is a Segal weak equivalence precisely if is a weak equivalence in the model structure for dendroidal complete Segal spaces .
Call a morphism in
a weak equivalence precisely if it is a Segal weak equivalence, def. 4;
a cofibration precisely if it is a normal monomorphism, def. 2.
Theorem 1 below asserts that this is indeed a model category struture whose fibrant objects are the Segal operads.
Of the various classes of morphisms
If in is a normal monomorphism and is a monomorphism, then is normal in .
(Cis-Moer, lemma 7.4).
The class of normal monomorphisms in is generated (under pushout, transfinite composition and retracts) by the set
(Cis-Moer, prop 7.5).
Let be fibrant. Then is a Reedy fibrant Segal operad. If is moreover fibrant in then the counit is a weak equivalence in .
(Cis-Moer, prop 8.2).
An acyclic fibration in , def. 3, is also a weak equivalence in .
(Cis-Moer, prop 8.12).
Of the model structure itself
The structures in def. 5 make the category a model category which is
This is (Cis-Moer, theorem 8.13).
The existence of the cofibrantly generated model structure follows with Smith’s theorem: by the discussion there it is sufficient to notice that
the Segal equivalences are an accessibly embedded accessible full subcategory of the arrow category;
the acyclic cofibrations are closed under pushout and retract;
(both of these because these morphisms come from the combinatorial model category )
the morphisms with right lifting against the normal monomorphisms are weak equivalences, by lemma 2.
Relation to other model structures
We discuss the relation to various other model structures for operads. For an overview see table - models for (infinity,1)-operads.
To dendroidal complete Segal spaces
(…) model structure for dendroidal complete Segal spaces