Context. The ESA/Rosetta mission made it possible to monitor the plasma environment of a comet, from near aphelion to perihelion conditions. To understand the complex dynamics and plasma structures found at the comet, a modelling effort must be carried out in parallel. Aims. Firstly, we present a 3D hybrid model of the cometary plasma environment including photoionisation, solar wind charge exchange, and electron ionisation reactions; this model is used in stationary and dynamic conditions (mimicking the solar wind variations), and is thus especially adapted to a weakly outgassing comet such as 67P/Churyumov-Gerasimenko, the target of the ESA/Rosetta mission. Secondly, we use the model to study the respective effects of ionisation processes on the formation of the dayside macroscopic magnetic and density boundaries upstream of comet 67P in perihelion conditions at 1.3 AU. Thirdly, we explore and discuss the effects of these processes on the magnetic field line draping, ionisation rates, and composition in the context of the Rosetta mission. Methods. We used a new quasi-neutral hybrid model, originally designed for weakly magnetised planetary bodies, such as Venus, Mars, and Titan, and adapted here to comets. Ionisation processes were monitored individually and together following a probabilistic interaction scheme. Three-dimensional paraboloid fits of the bow shock surface, identified for a magnetosonic Mach number equal to 2, and of the cometopause surface, were performed for a more quantitative analysis. Results. We show that charge exchange and electron ionisation play a major role in the formation of a bow shock-like structure far upstream, while photoionisation is the main driver at and below the cometopause boundary, within 1000 km cometocentric distance. Charge exchange contributes to 42% of the total production rate in the simulation box, whereas production rates from electron ionisation and photoionisation reach 33% and 25%, respectively. We also discuss implications for Rosetta's observations, regarding the detection of the bow shock and the cometopause.