The history of the Popes in this period is far too complex for me to go into detail here. In short, the period began with struggles against the Hohenstaufens which led to the Pope selling the claim to the Kingdom of Sicily. Henry III of Engand bought it for his son but this resulted only in Henry getting severely into debt, helping to create the circumstances for the Barons' War.
Charles of Anjou then obtained Papal backing for a claim to Sicily and finished off the Hohenstaufens.
With Ghibelline influence on the wane, the Papacy was able to extend its power further north, gaining lands which it then held fairly continously until the 19th century. Bologna was a major acquisition.
Relations with the Angevins and the French monarchy varied over the period. The populace of Rome were not always welcoming to the Pope and the danger of riots was used as a reason for the papacy moving to Avignon, although it also allowed the French monarchy to have more control over Papal policy. Rule of Rome itself was actually in the hands of the Senator. Sometime this was the Pope, at other times a native Roman of one of the great families but for some time the post was held by Charles of Anjou. Henry of Castile was a notable Senator - more of him another time.
Various Popes are mentioned by Dante. He finds Nicholas III in the 8th circle of the Inferno, buried head down with his feet on fire as a punishment for simony. This pope forsees the arrival in hell of Boniface VIII and Clement V - both of whom being political enemies of Dante.
In DBMM, the Pope is represented by a Bge(S) element. One of the benefits of this is that it gives room for a vignette and this is my attempt. The Pope is probably the most ambitious conversion I've made - he started as a mounted Hungaraian archer from Mirliton. The figure had a fur lined cap which was easy to make into a mozetta. The rest of the papal garb was made from Milliput. The litter and bearers are fantasy figures from Black Raven Foundry. They are moulded with nothing but a loin cloth so I used Milliput to give them a tunic. The throne is plastic card.
The two guardsmen at the front are from Mirliton as are the figures carrying the flabella fans and the umbraculum. The various monks are from Donnington. My original plan was to use these and their nice priest figure with the papal litter placed on the ground.
The Papal armies were, in this period, more often controlled by the Captain General of the Church (not the Gonfaloniere, apparently). This was, at times, an allied king but I have painted mine as Guillaume Durand, Captain General and governor of various Papal territories in the last two decades of the 13th century. His arms are copied from his tomb.
The figure is Mirliton, modified with a bishop's mitre of Milliput.
The figures in the background are Papal guards. I think the DBMM list is quite generous in classifying these as Sp(S) since the picture on which they are based seems to be pretty similar to other pavise carrying spearmen of the 14th century.
The DBA list for the Papacy doesn't, in my opinion, work well for this army. Normally, allies are added to the normal 12 elements but for this particular army the allies are a far more integral part of the army so should be represented as part of the normal army.
If you have access to JSTOR articles, you can download D. Waley's "Papal Armies in the Thirteenth Century". This has a lot of detail about the way in which the tributary cities owed service.
The 19th century "History of the City of Rome" by F. Gregorovius is available as a partial preview on Google books or as a pay download from here . There isn't very much military detail but it puts a lot of the events in context.