Symmetric Monoidal Categories for Operads - Index of

11.1 Recollections: The Language of Model Categories 159
11.1.10 Example: Topological Spaces. The category of topological spaces
Top is equipped with a proper model structure so that (see [28]):
– A map f : X → Y is a weak-equivalence if π0(f) :π0(X) → π0(Y )is
bijective and πn(f) :πn(X, ∗) → πn(Y,∗) is a group isomorphism for all
n>0 and for every choice of base points;
– The fibrations are the Serre fibrations, the maps which have the homotopy
lifting property with respect to cell complexes;
– The cofibrations are the maps which have the left lifting property with
respect to acyclic fibrations.
Let Δ n be the standard model of the n-simplex in the category of simplicial
sets, defined by the morphism sets Δ n = MorΔ(−,n)inthesim-
plicial category. Form the subobjects ∂Δ n = � n
Λ n k
i=1 di(Δ n ) ⊂ Δ n and
= �
i�=k di(Δ n ) ⊂ Δ n ,0≤ k ≤ n. The category of topological spaces
has a set of generating cofibrations defined by the geometric realization of
the embeddings i : ∂Δ n → Δ n , a set of generating acyclic cofibrations defined
by the geometric realization of the embeddings j : Λ n k → Δn ,sothat
Top forms a cofibrantly generated category.
11.1.11 Example: Simplicial Sets. The category of simplicial sets S is
equipped with a proper model structure so that a map f : X → Y is a weakequivalence
if its geometric realization |f| : |X| →|Y | is a weak-equivalence
of topological spaces, a fibration if f is a fibration in Kan’s sense, a cofibration
if f is dimensionwise injective (as usual, we refer to [28]).
The category of simplicial sets S is also cofibrantly generated by the embeddings
i : ∂Δ n → Δ n as generating cofibrations and the embeddings
j : Λ n k → Δn as generating acyclic cofibrations. The definition of a Kan
fibration is actually equivalent to a map f : X → Y which has the right
lifting property with respect to the embeddings j : Λ n k → Δn .
11.1.12 Model Structures and Adjunctions. Our motivation to use cofibrantly
generated model categories is to derive new model structures by
adjunction from a well-defined model category.
Say that a functor U : A→X between model categories creates weakequivalence
if the weak-equivalences of A are exactly the morphisms f such
that U(f) forms a weak-equivalence in X , and adopt similar conventions
with respect to other classes of morphisms. In many usual examples, we
have a natural adjunction relation F : X ⇄ A : U, whereX is a reference
model category, and a model structure on A is specified by assuming that
the functor U : A→X creates weak-equivalences and fibrations. The class
of cofibrations of A is determined by the left lifting property with respect to
acyclic fibrations. The difficulty is to check the axioms of model categories
to conclude that this definition returns an actual model structure on A.
In the context of cofibrantly generated model categories, the verifications
can be reduced to simple conditions: