Torsors and Their Classification III

Interlude on constructions with torsors

Let P be a right torsor for some group object G. Twisting with P will be a functor \objl{G}\to \ca C from objects with a left G–action to objects in \ca C.

Let X be an object with a left G–action. The twist P\times^G X is defined as the quotient of P\times X by the diagonal action of G, that is, it is the coequalizer 
G\times P\times X \arrow[r,shift left=.5ex, "\act"]
\arrow[r, shift right=.5ex, "\proj"'] & P\times X \arrow[onto,r] & P\times^G X
where \act(g, p, x) = (p g^{-1}, gx) on generalized elements. If X additionally carries a right action by some group object H, then evidently the twist P\times^G X also carries a right H–action; similarly, if the torsor P carries a left action by some group object H', then the twist also carries a left H'–action.

Let’s introduce some notation for the action of the epimorphism P\times X\onto P\times^G X on generalized elements. Namely, given a generalized element (p,x)\colon T\to P\times X, I will write [p,x] for the composition T\to P\times X\to P\times^G X. Because P\times X\onto P\times^G X is an epimorphism, to check some relation between generalized elements downstairs it will be enough to check it on generalized elements of the form [p,x]. Furthermore, if (p,x) actually comes from a generalized element (g,p,x)\colon T\to G\times P\times X via projection, then by the coequalizer property we find [p,x] = [p g^{-1}, g x].

The process of twisting has some exactness properties:

The functor P\times^G{-}\colon\objl{G}\to\ca C preserves colimits and finite products.

Regarding preservation of colimits, we can in fact construct a right adjoint to P\times^G{-}. It is given by the functor ({-})^P which sends T to the function object T^P whose G–action is given by precomposing with the action on P. 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 G\times^G{-} is naturally isomorphic to the identity functor: Consider the commutative diagram 
G\times X & X \arrow[l, "e\times{\id}"'] \\
G\times G\times X \arrow[u, shift left=.5ex, "\proj"]
\arrow[u, shift right=.5ex, "\act"'] & X \arrow[l, "e\times e\times{\id}"]
\ar[u, shift left=.5ex, "{\id}"] \ar[u, shift right=.5ex, "{\id}"']
This induces a natural transformation {\id}\to G\times^G{-} which is easily checked to be an isomorphism on generalized elements.

Now, choose a trivializing cover U\onto * of the torsor P. Because pullback in a topos commutes with colimits and limits, it will then be enough to check that, for any X and Y, the natural morphism 
P\times^G (X\times Y)\to (P\times^G X) \times (P\times^G Y)
after pulling back to the corresponding natural morphism in \ca C/U becomes an isomorphism. Since there is a G\times U–equivariant isomorphism P\times U\isom G\times U, the functors P\times U\times^{G\times U}{-} and G\times U\times^{G\times U}{-} \isom \id are naturally isomorphic; in particular P\times U\times^{G\times U}{-} preserves products, which is exactly what we needed to show.

Preservation of finite products in particular implies that P\times^G{-} sends group objects in \objl{G} to group objects in \ca C. This allows the following construction.

When endowed with the conjugation action of G on itself, G becomes a group object in \objl{G}. In particular, {}^P G\coloneqq P\times^G G is naturally a group object in \ca C.

It is enough to check that the structure maps e\colon *\to G, m\colon G\times G\to G and i\colon G\to G 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 G acts on itself via automorphisms, then the twist P\times^G G with respect to this action becomes a new group object. Sometimes, this is called a form of G. This construction let’s us refine the process of twisting a little bit. It turns out that the twisted object P\times^G X comes naturally equipped with a left group action by {}^P G. In fact, P itself carries a natural left action by {}^P G. Recall that, since P is a righ G–torsor, the map P\times G\to P\times P given by (p,g)\mapsto (p, pg) on generalized elements is an isomorphism. Let’s denote the inverse map by 
(p, q) \mapsto (p, \tdiv{p}{q})
on generalized elements. Then we can construct a commutative diagram 
P\times G \times P\arrow[r, "\varphi"] & P \arrow[d, equal] \\
G\times P\times G\times P\arrow[u, shift left=.5ex]\arrow[u, shift right=.5ex]
\arrow[r, "\psi"'] & P
where, on generalized elements, \varphi(p,g,q) = p\cdot g \tdiv{p}{q} and \psi(h,p,g,q) = \varphi(p,g,q). The commutativity follows from the following calculation: 
\varphi(ph^{-1}, hgh^{-1}, q) = ph^{-1}\cdot hgh^{-1} \tdiv{ph^{-1}}{q} = p\cdot g
h^{-1} h \tdiv{p}{q} = \varphi(p,g,q).
Hence, because colimits commute with products in toposes, we have a morphism {}^P G\times P\to P where the axioms for a left action can be checked by direct calculation.

This action makes P into a left {}^P G–torsor.

Take an epimorphism U\onto * which trivializes P. Then it will be enough to check that there is a {}^P G–equivariant isomorphism P\times U\iso {}^P G\times U. Since everything in sight commutes with pullbacks, we might as well assume that P is isomorphic to the trivial right G–torsor and check that then it is {}^P G–isomorphic to the trivial left {}^P G–torsor.

We first need to produce an isomorphism P\iso {}^P G given a G–isomorphism \varphi\colon P\iso G. I claim that the composition 
G \to[g\mapsto(e,g)] G\times G\onto G\times^G G
is an isomorphism. The inverse map is defined by the commutative1 diagram 
G\times G \arrow[r, "\conj"] & G \arrow[d, equal] \\
G\times G\times G \arrow[u, shift left=.5ex]\arrow[u, shift right=.5ex]  \arrow[r, "\conj\circ\proj"'] & G
where \conj denotes the conjugation action. Hence, we get a chain of isomorphisms2 
P \arrow[r,"\varphi"] & G \arrow[r, "{g\mapsto(e,g)}"] & G\times^G G
\arrow[r, "\varphi^{-1}_*"] & P\times^G G = {}^P G.
The last thing to check is that this isomorphism P\iso {}^P G is in fact {}^P G–equivariant, that is, we need to check that the diagram 
{}^P G\times P \arrow[r, "\act"] \arrow[d] & P \arrow[d] \\
{}^P G\times G \arrow[d] & G \arrow[d] \\
{}^P G\times (G\times^G G) \arrow[d] & G\times^G G \arrow[d] \\
{}^P G\times {}^P G \arrow[r, "m"'] & {}^P G
commutes. On generalized elements, the top composition gives 
[p, g], q \mapsto [\varphi^{-1}(e), \varphi(p) g \varphi(p)^{-1}\varphi(q)]
while the bottom composition is 
[p,q], q\mapsto
[p, g\cdot\tdiv{\varphi^{-1}(e)}{p}\varphi(q)\tdiv{p}{\varphi^{-1}(e)}] = [p, g\varphi(p)^{-1}\varphi(q)\varphi(p)]
But this is equal to 
[\varphi^{-1}(e), \varphi(p) g\varphi(p)^{-1}\varphi(q)\varphi(p)\varphi(p)^{-1}]
which proves the required commutativity.

A consequence of this theorem is that the twisting functor P\times^G{-} may be considered as a functor \objl{G}\to\objl{{}^P G}. Furthermore, if Q is a left G–torsor, then we can choose an epimorphism U\onto * which trivializes both Q and P. Then, because twisting commutes with products, we find 
(P\times^G Q)\times U\isom G\times U \times^{G\times U} G\times U\isom G\times
U\isom {}^P G\times U
similarly to the proof of the theorem. Hence, twisting restricts to a functor 
P\times^G{-}\colon \Tors[G]\to\Tors[{}^P G]
from the category of left G torsors to the category of left {}^P G torsors. Furthermore, if Q is a left K–torsor and a right H–torsor, P a left H–torsor and a right G–torsor then there is a natural isomorphism 
Q\times^H (P\times^G{-})\isom (Q\times^H P)\times^G{-}
which follows from the fact that colimits commute with colimits and with binary products in toposes.

If P is a right G–torsor and hence a left {}^P G–torsor, let \op{P} denote the right {}^P G–torsor with action given by p\cdot g = g^{-1}\cdot p on generalized elements. Then \op{P}\times^{{}^P G} P is a trivial right G–torsor. Furthermore, {}^{\op{P}}({}^P G)\isom G as group objects and along this identification P\times^G \op{P} is a trivial right {}^P G–torsor. It follows that 
P\times^G{-}\colon \Tors[G]\to\Tors[{}^P G]
is an equivalence of categories with weak inverse 
\op{P}\times^{{}^P G}{-}\colon\Tors[{}^P G]\to\Tors[G].

Let’s first produce an isomorphism {}^{\op{P}}({}^P G)\iso G. Let \varphi\colon P\times P\times G\to G be such that \varphi(p,q,g) = \tdiv{q}{p} \cdot g \cdot \tdiv{p}{q} on generalized elements. It is straightforward, albeit tedious, to check that \varphi descends to a group homomorphism 
\op{P}\times^{{}^P G} (P\times^G G) = {}^{\op{P}}({}^P G)\to G.
I claim that this is an isomorphism. To verify this claim, it will be enough to pull back along an epimorphism U\onto * which trivializes both the right G–torsor P and the right {}^P G–torsor \op{P}, that is, we can assume that P is trivial.3 So let \psi\colon G\iso P be a G–equivariant isomorphism. We can then set \widetilde\varphi\colon G\to {}^{\op{P}}({}^P G) to be the morphism such that \widetilde\varphi(g) = [\psi(e), [\psi(e), g]] on generalized elements. Computing the compositions \varphi\circ\widetilde\varphi and \widetilde\varphi\circ\varphi on generalized elements, we find 
\varphi(\widetilde\varphi(g)) = \varphi([\psi(e), [\psi(e), g]]) = g
and \widetilde\varphi(\varphi([p, [q, g]])) =
[\psi(e), [\psi(e), \tdiv{q}{p}\cdot g\cdot \tdiv{p}{q}]].
Noticing that 
[p, [q, g]] = [p_0, [p_0, \tdiv{p_0}{p}]\,
 [p_0, \tdiv{q}{p_0}\cdot g\cdot \tdiv{p_0}{q}]\,
 [p_0, \tdiv{p}{p_0}]]
where p_0 = \psi(e) finishes the proof of {}^{\op{P}}({}^P G)\isom G.

Next, let’s check that \op{P}\times^{{}^P G} P is a trivial right G–torsor. Let P\to P\times P\onto \op{P}\times^{{}^P G} P be the diagonal composed with the canonical projection. I claim that this map descends to the quotient P/G\isom * which will provide a global section of \op{P}\times^{{}^P G} P. This will then imply the claim. The calculation 
[pg, pg] = [p, [p, g^{-1}]\cdot(pg)] = [p, p g g^{-1}] = [p,p]
shows that the diagonal is P\to \op{P}\times^{{}^P G} P is invariant under the right action of G 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 P is an invertible operation up to isomorphism. This will become essential in the computation of the homotopy groups of BG at any basepoint.

  1. Remember that in G\times^G G the G on the left is the trivial right torsor and the G on the right is taken with the conjugation action.

  2. Note that \varphi^{-1}_* denotes the morphism induced by the G–equivariant isomorphism \varphi^{-1}\colon G\to P, using the fact that twisting is functorial with respect to G–equivariant morphisms in both factors.

  3. By the proof of the previous theorem, this also implies that \op{P} is trivial.