Categorical background for A¹-homotopy theory (simplicial model categories)

I decided to post some background needed in order to understand Morel-Voevodsky's paper "A¹-homotopy theory". I explain some notions of simplicial sets, topoi, monoidal categories, enriched categories and simplicial model categories.

I tried to give many more references I found useful.

Standard model structure on simplicial sets

Let $f : A \rightarrow B$ be a morphism of simplicial sets. $f$ is said to be a topological weak equivalence if the geometric realization $|f| : |A| \rightarrow |B|$ is a weak equivalence (that is, induces isomorphisms on all homotopy groups).

$f$ is said to be a Kan fibration if it has the right lifting property with respect to all horn inclusions. A horn inclusion is a map $\Lambda^n_k \rightarrow \Delta^n$ , where the k-th horn $\Lambda^n_k$ of the n-simplex is just the simplicial set generated by faces of the n-simplex except the k-th face (so the horn is a subcomplex of the boundary of the n-simplex).

The standard model structure on simplicial sets takes as weak equivalences the topological weak equivalences, as fibrations the Kan fibrations and as cofibrations the monomorphisms (which are just degreewise injective maps).

In the standard model structure, all simplicial sets are fibrant. A Kan complex is a simplicial set that satisfies the extension condition, which is, if you take (n+1) n-simplices $x_0,...,x_{k-1},x_{k+1},...,x_{n+1}$ that satisfy for all $i < j$ , $i,j \neq k$ that $\partial_i x_j = \partial_{j-1} x_i$ , then there exists a (n+1)-simplex $x$ whose faces are $\partial_i x = x_i$ . The cofibrant objects in the standard model structure are exactly the Kan complexes. This standard model structure is sometimes called Kan model structure on simplicial sets. It is worth noting that the singular simplicial set of a topological space is always a Kan complex.

The cofibrant-fibrant replacement for a simplicial set is therefore a functor, that turn every simplicial set into a weakly equivalent Kan complex. This is achieved by either taking the singular simplicial set of the geometric realization of a simplicial set or via Kan's $Ex^\infty$ functor.

More details can be found in May's book "simplicial objects in algebraic topology".

Monoidal categories

Monoidal categories generalize various notions of tensor-like operations in categories. They will be useful to define enriched categories, which are then used to define what a simplicial model structure is.

A (lax) monoidal category is a category $\mathcal{C}$ equiped with a bifunctor, often denoted $\otimes : \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$ , an object $I \in Ob\mathcal{C}$ called identity, and natural transformations that make this $I$ the identity of $\otimes$ and the operation $\otimes$ associative, up to isomorphism of functors. There is a coherence condition to be satisfied, so that all diagrams made out of the natural transformations corresponding to associativity, left unit and right unit, commute. It can be shown that every such lax monoidal category is equivalent to a strict one, where the natural transformations are identities. This equivalence can always be done via monoidal functors, which are those functors that respect the bifunctor $\otimes$ , the identity $I$ and the natural transformations.

Good examples are the category $Set$ of sets with cartesian product and the one-point-set as identity and the category of abelian groups with tensor product over the integers and the integers as identity. The category of small categories is a monoidal category, too, with the cartesian product of categories and the one-object-with-identity-category as identity.

The category of sets has the nice property that the functor $A \mapsto A \times B$ has a right adjoint $A \mapsto Hom(A,B)$ . If a monoidal category has this property of having a right adjoint to $A \mapsto A \otimes B$ , it is called (left-)closed and the objects in the image of this right adjoint are called mapping objects, sometimes written as $Map(A,B)$ . The bifunctor that sends $(A,B)$ to the mapping object of the right adjoint of $\otimes B$ evaluated at $A$ is called internal Hom. It is important to differentiate between left-closed and right-closed categories but in many cases the monoidal structure is braided, which means there is a transformation $A \otimes B \rightarrow B \otimes A$ (satisfying some commutative diagram), and for the category of sets this braiding is symmetric, which means it is an isomorphism, so left-closed and right-closed are equivalent notions. The category of sets and the category of small categories are examples of cartesian monoidal categories, because their monoidal product coincides with the categorical product and the identity is the final object. In cartesian closed categories, the mapping objects are written as exponentials $B^A := Map(A,B)$ .

Contravariant functors from a category to a monoidal category form a monoidal category with pointwise monoidal operation. This is the general way which makes the category of simplicial sets a monoidal category. Since the category of sets is cartesian closed, the inherited structure on simplicial sets is cartesian closed, too. It is an interesting fact, that geometric realization of simplicial sets is actually a monoidal functor, when we take the standard cartesian structure on the category of compactly generated weak Hausdorff spaces. In formula, this means in particular $|A \times B| \simeq |A| \times |B|$ for any two simplicial sets $A,B$ and the geometric realization functor $|\ \cdot\ | : Set^{\Delta^{op}} \rightarrow CGHaus$ .

Now let's turn to monoidal model categories. For these, we need the notion of a Quillen bifunctor.
Let $\mathcal A, \mathcal B, \mathcal C$ be model categories. A left Quillen bifunctor is a functor $F : \mathcal A \times \mathcal B \rightarrow \mathcal C$ that preserves small colimits in each variable (seperately) and satisfies this condition (sometimes called pushout-product axiom):
For all cofibrations $i : A \rightarrow A'$ in $\mathcal A$ and $j : B \rightarrow B'$ in $\mathcal B$ , the induced morphism $i \wedge j : F(A',B) \coprod_{F(A,B)} F(A,B') \rightarrow F(A',B')$ is a cofibration in $\mathcal C$ . If either $i$ or $j$ is, in addition, a weak equivalence, then $i \wedge j$ is required to be a weak equivalence, too.

Now a monoidal model category is a closed monoidal category $(S,\otimes,I)$ equipped with a model structure such that the unit object $I$ is cofibrant and the tensor functor $S \times S \rightarrow S$ is a left Quillen bifunctor. This definition ensures that the homotopy category will be a closed monoidal category. In some rare cases, the unit object is not cofibrant and one uses a slightly weaker condition, but this isn't necessary for our purposes here.

The category of simplicial sets, with the usual cartesian monoidal structure and the standard model structure, is a monoidal model category. The (in my personal perspective) hardest part of the proof is to see that the tensor functor preserves the trivial cofibrations (that are exactly the anodyne extensions). Hovey (see below) does a very good job at explaining this.

Enriched category theory

The definition of a category enriched over some monoidal category is a priori not directly related to the definition of a category, but a posteriori it's just "ordinary category + extra structure".

A category $\mathcal{C}$ enriched over a monoidal category $(M,\otimes,I)$ is a class of objects (as usual) and for each two objects $X,Y$ an object $Map(X,Y) \in Ob(M)$ . The analog of identities are morphisms $id_X : I \rightarrow Map(X,X)$ in the category $M$ and the composition is defined as morphism $\circ : Map(Y,Z) \otimes Map(X,Y) \rightarrow Map(X,Z)$ . Of course, associativity of composition and identity axioms are required to hold.

The usual definition of a category is included in the enriched definition if we look at categories enriched over $M = Set$ (well, depending on your definition of a category, you get only locally small categories this way).

Every enriched category has an underlying ordinary category where the Hom-sets are given by $Hom(I,Map(X,Y))$ , so one can speak of giving an ordinary category an enriched structure.

A category which is enriched over $Cat$ is usually called (strict) 2-category. Of course, $Cat$ is itself a 2-category. This is very common: every closed symmetric monoidal category is enriched over itself, since it has internal Hom-functors.

One can define enriched functors and enriched transformations in the obvious manner, so it's possible to speak of functor categories and therefore the enriched categories over a fixed monoidal category form a 2-category.

Since in a $M$ -enriched category $\mathcal{C}$ we have $\mathcal{C}_M(X,Y)$ (morphisms from object $X$ to object $Y$ ) being an object of $M$ , we can for every object $K$ of $M$ consider the morphisms $M(K,\mathcal{C}_M(X,Y))$ . If this has an adjoint, namely $M(K,\mathcal{C}_M(X,Y)) \simeq \mathcal{C}_M(K \odot X,Y)$ , then the functor $X \mapsto K \odot X$ is called the copower of $X$ by $K$ . It is actually a bifunctor. In the case where $\mathcal{C} = M$ , this is always the monoidal product functor of $M$ , and thus it's often called tensor. A category is copowered if it has copowers, that is there is a copower bifunctor satisfying the adjoint relation. The dual notion of power is sometimes called cotensor. I think speaking of tensors, in general, is not a good idea but it won't hurt us for $A^1$ -homotopy theory.

To get a better feeling for copowers, look at the category of topological spaces (which carries a natural monoidal model structure, see Quillen's Homotopical Algebra for details). The copower of a topological space $X$ by a simplicial set $K$ is just the topological space $X \times |K|$ and the power of a topological space $X$ by a simplicial set $K$ is just the topological space $X^{|K|}$ .

An enriched model category $\mathcal C$ , enriched over a monoidal model category $M$ is defined to be a category $\mathcal C$ enriched over $M$ , powered and copowered, whose underlying ordinary category has a model structure such that the copower functor is a left Quillen bifunctor.

Now a simplicially enriched model category is just an enriched model category which is enriched over the monoidal model category of simplicial sets.

Topos theory

The topoi we're talking about are Grothendieck topoi. Those are, by definition, categories equivalent to the category of sheaves on a small site. A site is a category equipped with a Grothendieck topology. A Grothendieck topology can be given by a pretopology although many different pretopologies may yield the same Grothendieck topology.

A pretopology consists of a set for each object, called the set of covering families. Each such covering family is supposed to be a set of morphisms into the object in question, such that these morphisms are stable under refinement and pullback and contain all isomorphisms into the object. Refinement is, if you have a covering family $\{U_i \rightarrow A\}$ and for each $U_i$ a covering family $\{V_{ij} \rightarrow U_i\}$ then $\{V_{ij} \rightarrow A\}$ is supposed to be a covering family as well. Pullback is, if you have a morphism $B \rightarrow A$ then the covering family obtained by pullback of each morphism of a covering family $\{U_i \rightarrow A\}$ is a covering family $\{U_i \times_A B \rightarrow B\}$ is a covering family of $B$ .

A sheaf on a category $\mathcal{S}$ equipped with a pretopology is a presheaf $F : \mathcal{S}^{op} \rightarrow Set$ that satisfies for each object $X$ and each covering family $\{X_i \rightarrow X\}$ that
$! F(X) \rightarrow \prod_{i} F(X_i) {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \prod_{i,j} F(X_i\times_X X_j)$
is an equalizer.

Topoi have many useful categorical properties. To name same of them: they have all finite limits and all finite colimits and they are cartesian closed monoidal categories (so you can do some kind of Lambda calculus inside a topos). Consider "broadening" a category by using the category of presheaves on it (via Yoneda embedding). The choice of a topology and therefore what we call a sheaf, thus object of our topos, ensures categorical properties nice enough to think about the objects in our topos as the real "spaces" to define $A^1$ -homotopy theory. Look, for analogy, at topological spaces, which can be rather ill-behaved. Topologists work instead with the category of compactly generated spaces, which behave more like CW complexes. In this category, we know some nice (classical) homotopy theory, while this is not the case with the category $Top$ of all topological spaces. For more heuristic arguments why this is the "right" way to proceed, look at Voevodsky's paper in Documenta Mathematica.

The most common examples of topoi are the category of small sets (figure out how this is a topos as an exercise!) and the sheaves on the small/big Zariski sites of schemes. However, we're interested in the sheaves on Nisnevich sites, which I will therefore describe here:

The big Nisnevich site of a scheme $S$ is the category $Sm/S$ of smooth schemes over the fixed base scheme $S$ equipped with the Nisnevich topology. The Nisnevich topology is in-between the Zariski and the étale topology, so I want to describe those three topologies at once, for comparison. Nisnevich called his topology the completely decomposed topology, or just cd-topology.

The canonical topology is the biggest topology that makes all representable presheaves actually sheaves. All topologies finer than that are called subcanonical. Now look at three examples of subcanonical pretopologies, ordered from coarsest to finest:

The Zariski topology is given by covering families that are surjective families of scheme-theoretic open immersions (by open immersion I mean a morphism that decomposes uniquely into an isomorphism and the inclusion of an open subscheme; open immersions are always étale morphisms, that means flat and unramified).

The Nisnevich topology is given by covering families that are surjective families of étale morphisms $\{X_\alpha \rightarrow X\}$ with the property that for every point $x \in X$ , there exists an $\alpha$ and a point $u \in X_\alpha$ such that the induced map of residue fields $k(x) \rightarrow k(u)$ is an isomorphism.

The étale topology is given by covering families that are surjective families of étale morphisms.