Google and the US Justice Department each believe the other wants too much of one thing: control.

“Control is the defining characteristic of a monopolist,” DOJ counsel Julia Tarver Wood said during opening statements in the federal government’s second antitrust trial against the search giant, which kicked off on Monday in Alexandria, Virginia. To the government, Google exerts too much control over every step of the way publishers sell advertising space online and how advertisers buy it, resulting in a system that benefits Google at the expense of nearly everyone else.

To Google, the government is seeking control over a successful business by making it deal with rivals on more favorable terms, disregarding the value of its investments in technology and the unique efficiencies of its integrated tools.

By the end of the trial, which is expected to last several weeks, US District Court Judge Leonie Brinkema will be left to decide which side is exerting too much control — and ultimately, if Google has illegally monopolized the markets for advertising technology.

Markets is a key word, since one question raised on the first day is how many monopolies Google might actually have. (A federal court in DC says at least one, since it recently ruled Google a monopolist in search.) The DOJ is arguing that Google has monopoly power in three different ad-related markets: those for publisher ad servers (where websites hawk ad space); ad exchanges (which facilitate ad transactions); and advertiser ad networks (where advertisers go to buy ad space). They’re also arguing that Google illegally tied together its publisher ad server with its ad exchange to maintain its monopoly power.

“One monopoly is bad enough,” Wood said during opening statements. “But a trifecta of monopolies is what we have here.”

Google says it’s not a monopolist and that there’s only one market: a two-sided market made of buyers and sellers of online ad inventory. In opening arguments, its counsel said the government is ignoring relevant Supreme Court precedent that says this is the best way to view such a market. The company also argues regulators are carving up the field with terms like “open web display advertising,” which Google calls contrived. What the government really wants here, Google claims, is to require it to deal with its rivals, something the Supreme Court has said isn’t really the job of the judicial system.

After opening statements, the DOJ began calling its first witnesses, focusing on the tools publishers use to monetize display ads. These are the ads that typically pop up at the top or the side of the page on news websites and blogs, populating through super-quick auctions that run while the page loads. During the auction, an ad exchange helps match publishers and advertisers based on things like topic and price without active intervention by a human. The process is called programmatic advertising, and it’s used by The Verge’s parent company, Vox Media, among many other outlets. (Vox Media president of revenue and growth Ryan Pauley is on the list of potential witnesses but wasn’t called today.)

Google’s tools play an essential role in the process, with some of them holding about 90 percent of the market, according to the government. Google has a publisher ad server called Google Ad Manager (formerly DoubleClick for Publishers, or DFP), which helps publishers sell ad space. It operates an ad exchange, AdX, that facilitates transactions. And it owns an advertiser ad network, rounding out its trifecta of major products across different parts of the ad world.

Four industry players testified on Monday, representing a publisher (Tim Wolfe, SVP of revenue at Gannett), an ad exchange (Andrew Casale, president and CEO of Index Exchange), a marketer (Joshua Lowcock, president of media at Quad), and a publisher ad network (James Avery, founder and CEO of Kevel). Across the testimonies, the government tried to establish that programmatic display advertising is not something publishers can easily substitute with other types of advertising, including direct deals with advertisers or ads on social media sites. And it introduced the idea that switching from Google’s tools isn’t such an easy decision, even when there might be some reason to do so.

In testimony, for instance, Wolfe and Avery both made clear that publishers are largely unwilling to switch from Google Ad Manager. They said it’s because Google packages it with access to AdX, and losing that package deal would mean giving up large amounts of revenue, even if rivals offer to take a much smaller cut for facilitating each ad sale. Wolfe testified that when Gannett received one such offer, that reduced take rate didn’t move the needle since it wouldn’t offset the benefits of AdX.

The ad server company Kevel started by targeting traditional publishers, but Avery says competing with Google proved impossibly hard. He recalled publishers asking how his company would replace the revenue they made from AdX, something Kevel simply couldn’t manage. After trying to engage Google twice about ways to connect Kevel’s ad server with AdX, Avery testified, his efforts were rebuffed. Kevel pivoted to facilitating things like sponsored listings for retailers instead.

Speaking from the ad exchange perspective, Casale testified that switching ad servers is a big lift at the technical level, so publishers rarely do it. Building a new one is “very complex and expensive.” In the ad exchange market, Casale said competing with Google’s AdX is “very challenging,” and in experiments, reducing fees had barely a “nominal” impact on the ability to gain more business. Because of the huge network effects it takes to get an exchange off the ground as well as the fact that it only gets visibility into ad impressions it wins, “I can’t imagine anyone starting a new exchange today,” he said.

Google’s attorneys poked at the witnesses’ arguments and credibility during cross-examination, pointing out ways players like Avery would benefit if the court forced Google to share access to its tools. Google will call its own witnesses to counter the DOJ later in the trial.

This trial covers very different ground from last year’s antitrust fight in the District of Columbia. But on the first day of court, both sides alluded to their earlier battle. The Department of Justice mentioned during opening statements that another court had already adjudicated the question of Google’s search monopoly, referencing a ruling Judge Amit Mehta handed down just over a month ago. And although Mehta ruled mostly against Google, the tech giant cited a piece of the ruling that went in its favor. The topic? A DOJ argument Mehta interpreted as a requirement for Google to cut deals with competitors — and, accordingly, dismissed.