Normally, the investigation process starts the criminal process. Officially, however, one must be charged with a crime before the criminal process may take course.
Another View: While the above is somewhat correct - in day-to-day reality it is usually the arrest of a perpetrator which begins the majority of criminal cases. Investigations MAY begin a case, but usually ONLY if it is necessary for probable cause to be established before a warrant can be applied for or an indictment can be sought.
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The question is somewhat unclear - - The arresting officer is usually the first person in the criminal justice system who makes initial contact with the defendant. At arraignment the defendant will appear before a judge and may possibly be introduced to a public defender, given the opportunity to obtain their own private attorney. After arraignment, the defendant's case will be assigned to a prosecutor who will be responsible for preparing and prosecuting the case against them.
Criminal cases begin with the police, or a government agency, claiming that a person has committed a criminal act.