Head Coach Lynn Howells opted for two captains in a bid to maintain continuity when Hogg is on international duty with Scotland.
The duo will take over the role from Chris Paterson.
Howells said: "Due to the commitment of many senior players to the World Cup and Six Nations, the appointment of second captain will give us continual leadership.
"We gave Allister an opportunity as captain at the end of last season and he showed he was capable of the job, as he proved throughout age group rugby.
"He has a skill that is difficult to find in many players in that he leads from the front.
"During the periods he is with us, we will have a number of leaders on the field at the same time from whom Allister can take advice and make the final decision.
"Simon has a leadership quality which will benefit the squad off the field as well as on it. We have a lot of new and young players coming in and he will gel the group into a cohesive unit.
"It's not necessarily all about playing, as we hope to have a number of leaders on the field, the best leaders bring something to the squad off the field."
Hogg said: "It is a huge honour to be asked to be captain especially with the other great people here at the club.
"Simon is a very good person to be in charge while I'm away with Scotland to make sure the boys work hard and get up to speed, as there are so many new faces in the squad.
"I want to carry on from where Chris Paterson left off so I will still bounce ideas off him. I have already learnt so much from him and other captains I've played under and so I owe him a massive thanks."
Cross added: "Last year was a tough one for me with my spell at Waterloo so I was delighted when Lynn asked me to be captain while Hoggy is away with Scotland.
"I'm immensely proud and really looking forward to the task of helping the younger players who've joined us understand how we do things at Edinburgh.
"I have been lucky enough to play under great captains including Lawrence Dallaglio, Todd Blackadder and Chris Paterson.
"Chris has been fantastic, he's a very passionate guy and I've learnt a lot from him. Hopefully I'll be able to take things from all three of them and become a good captain myself."
It was a double celebration for Hogg, who has also signed a new deal keeping him at the club until 2010.
He added: "I am happy to be staying with Edinburgh because of the recent signings which show we are an ambitious club. Being appointed captain was also a factor in extending my contract.
"We have a young squad and I want to play a part in developing and building Edinburgh Rugby."