Hi there! My name is Viktor Kleen. I’m currently a PhD candidate in Mathematics at USC in Los Angeles, see my CV. My research interests include:
- -homotopy theory
- Higher category theory
- Algebraic geometry
I now have a preprint describing my work on a motivic version of Snaith’s stable splitting results.
This site is supposed to serve as a research notebook to get me to write up ideas and results. As such, it is woefully unpolished, unreliable and generally to be used at your own risk.
Still, have fun! And please send email with suggestions, corrections, criticism or general rants to firstname.lastname@example.org if you like.
Interlude on constructions with torsors
Let be a right torsor for some group object . Twisting with will be a functor from objects with a left –action to objects in .
Let be an object with a left –action. The twist is defined as the quotient of by the diagonal action of , that is, it is the coequalizer where on generalized elements. If additionally carries a right action by some group object , then evidently the twist also carries a right –action; similarly, if the torsor carries a left action by some group object , then the twist also carries a left –action.
Let’s introduce some notation for the action of the epimorphism on generalized elements. Namely, given a generalized element , I will write for the composition . Because is an epimorphism, to check some relation between generalized elements downstairs it will be enough to check it on generalized elements of the form . Furthermore, if actually comes from a generalized element via projection, then by the coequalizer property we find .
The process of twisting has some exactness properties:
The functor preserves colimits and finite products.
Regarding preservation of colimits, we can in fact construct a right adjoint to . It is given by the functor which sends to the function object whose –action is given by precomposing with the action on . This won’t be particularly important in what follows, so I’ll skip the detailed calculation.
Showing that binary products are preserved is a little more difficult. Let’s first observe that the functor is naturally isomorphic to the identity functor: Consider the commutative diagram This induces a natural transformation which is easily checked to be an isomorphism on generalized elements.
Now, choose a trivializing cover of the torsor . Because pullback in a topos commutes with colimits and limits, it will then be enough to check that, for any and , the natural morphism after pulling back to the corresponding natural morphism in becomes an isomorphism. Since there is a –equivariant isomorphism , the functors and are naturally isomorphic; in particular preserves products, which is exactly what we needed to show.
Preservation of finite products in particular implies that sends group objects in to group objects in . This allows the following construction.
When endowed with the conjugation action of on itself, becomes a group object in . In particular, is naturally a group object in .
It is enough to check that the structure maps , and are equivariant with respect to conjugation. This can be checked on generalized elements where it reduces to the corresponding statement about genuine groups. There, it is a simple calculation.
More generally, if acts on itself via automorphisms, then the twist with respect to this action becomes a new group object. Sometimes, this is called a form of . This construction let’s us refine the process of twisting a little bit. It turns out that the twisted object comes naturally equipped with a left group action by . In fact, itself carries a natural left action by . Recall that, since is a righ –torsor, the map given by on generalized elements is an isomorphism. Let’s denote the inverse map by on generalized elements. Then we can construct a commutative diagram where, on generalized elements, and . The commutativity follows from the following calculation: Hence, because colimits commute with products in toposes, we have a morphism where the axioms for a left action can be checked by direct calculation.
This action makes into a left –torsor.
Take an epimorphism which trivializes . Then it will be enough to check that there is a –equivariant isomorphism . Since everything in sight commutes with pullbacks, we might as well assume that is isomorphic to the trivial right –torsor and check that then it is –isomorphic to the trivial left –torsor.
We first need to produce an isomorphism given a –isomorphism . I claim that the composition is an isomorphism. The inverse map is defined by the commutative1 diagram where denotes the conjugation action. Hence, we get a chain of isomorphisms2 The last thing to check is that this isomorphism is in fact –equivariant, that is, we need to check that the diagram commutes. On generalized elements, the top composition gives while the bottom composition is But this is equal to which proves the required commutativity.
A consequence of this theorem is that the twisting functor may be considered as a functor . Furthermore, if is a left –torsor, then we can choose an epimorphism which trivializes both and . Then, because twisting commutes with products, we find similarly to the proof of the theorem. Hence, twisting restricts to a functor from the category of left torsors to the category of left torsors. Furthermore, if is a left –torsor and a right –torsor, a left –torsor and a right –torsor then there is a natural isomorphism which follows from the fact that colimits commute with colimits and with binary products in toposes.
If is a right –torsor and hence a left –torsor, let denote the right –torsor with action given by on generalized elements. Then is a trivial right –torsor. Furthermore, as group objects and along this identification is a trivial right –torsor. It follows that is an equivalence of categories with weak inverse
Let’s first produce an isomorphism . Let be such that on generalized elements. It is straightforward, albeit tedious, to check that descends to a group homomorphism I claim that this is an isomorphism. To verify this claim, it will be enough to pull back along an epimorphism which trivializes both the right –torsor and the right –torsor , that is, we can assume that is trivial.3 So let be a –equivariant isomorphism. We can then set to be the morphism such that on generalized elements. Computing the compositions and on generalized elements, we find and Noticing that where finishes the proof of .
Next, let’s check that is a trivial right –torsor. Let be the diagonal composed with the canonical projection. I claim that this map descends to the quotient which will provide a global section of . This will then imply the claim. The calculation shows that the diagonal is is invariant under the right action of which proves the claim. The last assertion of the theorem is proven in a completely analogous way.
In summary, we have found that twisting with a right torsor is an invertible operation up to isomorphism. This will become essential in the computation of the homotopy groups of at any basepoint.
Remember that in the on the left is the trivial right torsor and the on the right is taken with the conjugation action.↩
Note that denotes the morphism induced by the –equivariant isomorphism , using the fact that twisting is functorial with respect to –equivariant morphisms in both factors.↩
By the proof of the previous theorem, this also implies that is trivial.↩
Computing Homotopy Classes in Model Categories
Let again be some Grothendieck topos and a group object in . I eventually want to give a proof of the following theorem.
There is a functorial construction of a simplicial object in and a functorial bijection between the set of –torsors and the set of simplicial homotopy classes of morphisms .
To show this we will follow an approach which was presented in the paper/preprint “Cocyle Categories” by Rick Jardine. The point is that proving our theorem comes down to computing homsets between objects of in the homotopy category. Generally, in any model category (and embeds into the local model structure on the category of simplicial objects in ) we can “calculate” homsets from to in the homotopy category by taking a cofibrant replacement of and a fibrant replacement of and taking honest homotopy classes of maps from to in .
However, computing homsets like that tends to be quite difficult because, typically, cofibrant and fibrant replacement are hard to write down explicitly (at least in a form that one can work with). Jardine came up with the following work–around.
Let be a category with weak equivalences and let and be objects in . Define a category of cocycles from to whose objects are diagrams with a weak equivalence and whose morphisms are the obvious commuting diagrams.
Remember that any category has a nerve , its –simplices are composable chains of morphisms in . By a slight abuse of notation I will not distinguish between and . A slight remark regarding notation; the geometric realization of is often called the classifying space of and is denoted by . Sometimes I will consider “a model of” to be any convenient fibrant replacement of in . In particular, it makes sense to talk about and by another slight abuse of notation I will often just write .
It turns out that in any model category is a good model for the mapping space from to as long as is cofibrant and is fibrant. This is proven for example in a preprint of Daniel Dugger’s, “Classification Spaces of Maps in Model Categories”. I will—following Jardine—give a proof of something weaker:
In a right proper model category in which weak equivalences are preserved under products, the homset in the homotopy category can in a natural way be computed as
The map from right to left is easily described as In the other direction, the map is harder to describe. Instead, let’s consider the set of (left) homotopy classes of maps from to . Any map trivially defines a cocycle. If two such maps and are left homotopic, there is a cylinder object and a homotopy such that This means that these three cocyles lie in the same path component of . Hence, we have a well defined map .
If is cofibrant and is fibrant, then the map is a bijection.
To prove this, consider the commutative diagram in which the diagonal map is a bijection because is cofibrant and is fibrant. Hence, it suffices to show that our map is surjective, i.e. that any cocycle is in the path component of a cocycle . Factor into a trivial cofibration followed by a trivial fibration to obtain where the map exists because is assumed to be fibrant. Because is cofibrant we can find a section of and obtain a commutative diagram which proves the lemma.1
If and are weak equivalences, then the induced map is a bijection.
To prove this we will describe an inverse function. Let be a cocycle and think of it as a morphism . Factor this map into a trivial cofibration followed by a fibration and observe that by 2–out–of–3 is a weak equivalence. Form the pullback in which the bottom map is a weak equivalence by our assumption on and the top map is a weak equivalence by right properness. This then implies that is a weak equivalence by 2–out–of–3, i.e. defines a cocycle in . This construction yields a well–defined map which is inverse to the map in the lemma.
Using these two lemmas the theorem about computing homsets is easy to prove. Let and be any two objects in and let be a cofibrant replacement of and a fibrant replacement of . Then we have a commutative diagram which proves the theorem.
Note that for this first lemma we haven’t used any assumption on the model category .↩
Torsors in Grothendieck Toposes
Let be some Grothendieck topos, that is a category of sheaves on some Grothendieck topology. This text will look at some properties of torsors in and how they might be classified using cohomology and homotopy theory. For this we will first need to define group objects. Torsors will then be associated to those.
A group object in a category is an object such that the associated Yoneda functor actually takes values in the category of groups.
A trivial kind of example would be just a group considered as an object in . A more involved example would be a group scheme over some base . Such a thing is essentially defined as a group object in the category . If we have any subcanonical topology on , then defines a sheaf on the associated site and we obtain in this way a group object in the corresponding topos on .
Let’s now assume we have a group object in a Grothendieck topos . Then we can define torsors over as follows:
A trivial –torsor is an object with a left –action which is isomorphic to itself with the action given by left multiplication.
A –torsor is an object with a left –action which is locally isomorphic to a trivial torsor; that is, there is an epimorphism such that is a trivial torsor in over .
I want to show that for any –torsor according to this definition the left action of on is free and transitive, that is the map given (on generalized elements) by is an isomorphism. This is going to be some relatively elementary category theory but I think it’s worth writing it up. First a few facts about isomorphisms in toposes, they can be found for example in Sheaves in Geometry and Logic.1
Epimorphisms in a topos are stable under pullback.
In a topos every morphism has a functorial factorization with a monomorphism and and epimorphism.
A morphism is an isomorphism if and only if is both monic and epic.
Now, let be a local monomorphism, i.e. there is an epimorphism such that the pullback of to is a monomorphism. Then, since epimorphisms are stable under pullback, it follows that in the commutative square both vertical maps are epimorphisms. Now let be a pair of morphisms such that . Then, denoting by and the pullbacks to , we have . but is a monomorphism by assumption, so . So we have a commutative diagram in which the vertical maps are epimorphisms. It follows that .
Now, if is a local epimorphism, then again we have the diagram and it is immediate that is an epimorphism. In summary:
In a topos, any local epimorphism is an epimorphism, any local monomorphism is a monomorphism, and any local isomorphism is an isomorphism.
Now, let’s check that –torsors as defined above are free and transitive. Take an epimorphism such that is trivial in . The action of on itself by left multiplication is plainly free and transitive, so in we have the isomorphism and and because pullback preserves products. So, is a local isomorphism, hence an isomorphism.
Saunders Mac Lane, Ieke Moerdijk. Sheaves in geometry and logic. Springer, 1994. ISBN: 0-387-97710-4↩