Compressors, Gas Turbines and Jet Engines
An ideal air compressor cycle without clearance on P-V diagram can be represented by following processes

Two adiabatic and two isobaric
Two adiabatic, one isobaric and one constant volume
One adiabatic, two isobaric, and one constant volume
One adiabatic, one isobaric and two constant volumes

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Compressors, Gas Turbines and Jet Engines
The assumption made in two stage compression with inter-cooler is that

The compression in both the cylinders is polytropic
The suction and delivery of air takes place at constant pressure
There is no pressure drop in the inter-cooler
All of the listed here

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