Not sure if this helps, but on the leaves to the left of the outdoors "web" photo, there appear to be two small caterpillars. As this web is similar to the ones spun by some moths (for example, ermine moth, but I'm not suggesting this is what it really is!!) it might be worth looking more closely at these for an ID.
Thankyou , I will go back and try to find any caterpillars.
It looks like it has fallen from a tree above and landed in the rose - is that correct? If so it might be "fluff" from a seeding poplar (I think you call it a catkin in English).
But also google "tent caterpillar".
With Poplar and cottonwood fluff, you can usually see the plant fibres and seeds. With caterpillar webs there is a smoother surface which adheres to and covers individual twigs of a plant. From the photos it isn't clear which it is and I accept it is late in the season for caterpillar webs, but given the weather, I wouldn't rule it out. I have seen some near where I live.
Where are the larvae and why are they spinning their web on the ground?
The "surface" can be an effect of rain which clogs the fibers together when it dries.
(BTW it is late also for poplar snow, however, no date is given. But I guess the photo was taken in May or so, or at least that the "snow" fell in May and what we see are the remains of it.)
Also note that the images are quite large - you should right-click and then open in a new window (and then enlarge) and you will see much more detail.
I agree with Episcophagus that this fluff is the seeds from a Poplar, not only that but an entire inflorescence from a poplar tree rather than just a mass of poplar seeds. The end of the central woody stem of the inflorescence can be seen in the 2nd pic above the edge of the rose leaf, where the white fluff has caught and is partially covering the leaf, and can be followed down through the mass of seeds.
The central woody stem from the inflorescence can be made out in both your photos, as can the cups which held the seeds - like these seen here - especially in the first pic taken indoors. The short fibres from the seeds are also more evident in the first pic.
One way to tell for sure (assuming you still have some to try) is that on gently teasing the mass of fibres apart the poplar seeds would come away in small bundles of short distinct fibres whereas caterpillar silk would come away in long strands, probably stretch and be fairly elastic.
I tried to write that I thought that an entire inflorescence had fallen down an landed in the rose, I thus called it a catkin. (English isn't my first language, so please excuse) so your reply and my reply(ies) is overlapping (coin...something - I cant find that word). Well, I agree with you as you agree with me .
the curled up leaf-like structures are the seed pods, the actual seeds themselves are tiny and attached to the fibres which act like mini parachutes to allow the seeds to be carried on the lightest of breezes well away from the parent tree, a very clever means of distribution.
This photo shows the pods and seeds just after the pods have split. The greenish leaf-like segments of the pods then go brown and dry as in your specimen. There is an interesting sequence of photos of the maturing pods to seed dispersal here.