To read what you want, you have to be free! 
The Incorruptible Republic of the Immortal Library of the Grand Architect (The Incorruptible Library, The Immortal Library, or simply The Library, for short) is, to Ardsley Wooster, "..an autonomous government which claims a large section of underground Paris off the Black Market. They lend out any book to anybody. Many people find this irresponsible." Fittingly, over is inscribed the Latin phrase "scientia est similis fluminis / te capere non possunt / in perpetuum". While this is not grammatically correct, it apparently is intended to mean "Knowledge is like a river. You cannot hold it back forever."
This sprawling collection of extreme bibliophiles and the books they acquire, protect, and provide access to was founded by the mysterious Van Rijn, and has possession of some of his secrets. The Master Spark had a secret (or at least tightly locked and cleverly concealed) inside its walls, dubbed his "Hermitorium". The site now occupied by the Library was previously one of the fortresses of the Corbettite Monks, and their trap-laden storage vaults for dangerous artifacts are still present, sealed off and mostly forgotten.
The Library also has in residence a population of over-sized cats led by a human Catmaster, which can be assembled into armor-clad clowders and dispatched on raiding expeditions against surrounding fiefdoms.
Doctor Dio Zardilev takes refuge here for a time after his time-manipulation expertise makes him the target of numerous assassination attempts.
As noted by Wooster's comment, the Library is not under the rule of the Master of Paris. At the time that Agatha is allowed to the Library, its leader, the formidable Lord High Conservator, is away on a raid aimed at extracting Tarvek from the "clutches" of now-Baron Gil. Agatha is met instead by two of its lesser but still important officials, the (evidently) married couple Jon and Melissa Solera, Grand Curators of the High Western Stacks. Agatha gives the duo Van Rijn's notebook for safekeeping, in exchange for being allowed free access to the rest of the Library's books. Melissa states that Margarella Selnikov was supposed to be bringing this book to the Library, instead of using it to ransack the vaults of the Corbettite Monks.
Or more precisely, the wrong collection of Corbettite Vaults. The key that Lady Selnikov stole along with this book is revealed to open the aforementioned Hermitorium, which contains Van Rijn's mummified corpse and the imprisoned Muse of Time. Following this, Agatha and her friends use the book's information to go into the Library's resident Vaults looking for both Zardilev and Prende's Chronometric Lantern, in the process accidentally releasing the original Storm King from some form of stasis. Library employee Aldin accompanies them, with the covert (and ultimately successful) side-mission of recovering the Heart of the Muse, the control and/or activation unit for the Muse of Vengeance, which is also stored in the Library.
A couple of scenes feature an unnamedwho evidently specializes in paranoia and carping about things.
The Library Besieged
Unfortunately, the Lord High Conservator and his entire raiding party are unit of hijacking Smoke Knights in the employ of Tarvek's Grandmother, followed shortly by the Library itself being subjected to a Revenant-assisted attack led by the King of the Silver Lands and a Geister (possibly Madame Velix), which succeeds to the point that the former can to Agatha and her friends in his usual over-enthusiastic fashion that he has taken control of the facility. A timely assist by Violetta and Agatha's Wasp-Eater, as well as Princess Larana carrying on the "ancient tradition of filial usurpation" to countermand her father's orders allows the Soleras to mount a successful counter-attack. Agatha and Co. return to the surface, while the Library is drawn into a pitched battle with the Storm King as he rampages in the same general direction; in the end, they are only able to delay his travel to the surface. When last seen, the surviving Library officials are the possible deployment of the aforementioned Muse.by a