Thanks to a quick-thinking customer who spotted him, snapped a couple photos and posted them on Sina Weibo, a Chinese microblogging site, we learned Monday that Apple (AAPL) CEO Tim Cook was in Beijing, at least for a few hours.

Leaving aside the iconography of an Apple CEO setting foot in the world’s largest market for iOS devices, something Steve Jobs apparently never bothered to do, what do we suppose Cook was doing there?

According to Apple’s local public relations representative, Cook  had “great meetings with Chinese officials.”

We can imagine any number of issues that might benefit from high-level face-to-face meetings between Cook — whom the Chinese press have nicknamed “Captain Cook” — and Chinese officials. Among them:

• The long negotiated but never consummated deal to sell iPhones through China Mobile (CHL), the world’s largest cell phone carrier with 665 million mobile subscribers.
• The slower-than anticipated roll-out of Apple Stores, which at six are nowhere near the goal of 25 that was set in 2010 and was supposed to have been reached last month.
• The continuing difficulties Apple has had getting its Chinese suppliers to conform to the company’s supplier code of conduct, particularly in regard to worker overtime.
• Apple’s ongoing legal battle with Proview over ownership of the iPad trademark in China. A Hong Kong court has sided with Apple, but the company has had a harder time getting what it sees as a fair hearing in courts on the mainland.

Cook doesn’t come to China with an empty hand. According to the company’s annual report, it has budgeted an unprecedented investment of $7.1 billion for plant and equipment for 2012, much of it presumably destined for Apple’s Asian supply chain. That should give him some leverage in those meetings with Chinese officials.

Dan Butterfield, editor emeritus of iPhonAsia.com, points to China’s Ministry of Industry and Information Technology as a key sticking point.

“One dilemma for China’s MIIT,” he writes, “has been their desire to level out the playing field for the three major wireless carriers in China — China Mobile, China Unicom and China Telecom. The (by far) dominant carrier is China Mobile, and the MIIT purposefully hamstrung China Mobile with their ‘indigenously innovated’ TD-SCDMA 3G … which IMHO is akin to the Spruce Goose. The smaller carriers — China Unicom and China Telecom — have benefited via having “world standard” 3G networks.

“The question of course is will TD-LTE [the standard the next iPhone is rumored to support] unshackle China Mobile from their TD-SCDMA bondage? It will be interesting to see which comes first, MIIT issuance of the necessary TD-LTE network license or the launch of the LTE compatible iPhone 5.”