SCR (thyristor) - Practical Dimmer Circuit
That circuit uses a traic (not a thyristor) triggered via a diac.

C3 charges up via R3 until it reaches about 20 V or so, the diac breakdown voltage. C3 then provides a reservoir of charge to ensure that the triac receives enough of a pulse to turn the device on properly.

Some rough caluclations with R2 at 100k:


\$ \tau = R_2C_2 = 100k \times 0.1 \mu = 10 \ \text {ms} \$.
\$ \tau = R_3C_3 = 15k \times 0.1 \mu = 1.5 \ \text {ms} \$.


So C3 is following C2 with a 1.5 ms delay roughly.

The main purpose of the diac is to give a reasonably precise trigger voltage rather than rely on the triac itself which may differ on positive and negative half-cycles.