Oversimplify it for me, what does "breaking up" a company mean when used commonly among politicians? What would a broken up wells fargo look like? Google? Is there historical legislation doing this, and/or case law?
The Air Mail Act of June 12, 1934, the vertically integrated United Aircraft and Transport Corporation (UATC) was broken up into three companies: United Air Lines Transportation Company, United Aircraft Manufacturing Company, and Boeing Aircraft Company.
Bell breakup: In the 1970s AT&T had a virtual monopoly as a telephone service provider. AT&T itself provided large-distance telephony, while it also controlled a number of local phone service providers under the name Bell. By 1984 AT&T - under political and legal pressure - had split off its regional companies. These formed seven independent so-called Baby Bells. This reduced the book value of AT&T by about 70 %.
Standard Oil Company: By 1911 the Standard Oil Company was an oil producing, transporting, refining, and marketing company and monopoly. It was considered in violation of the Sherman Antitrust Act of 1890 for sustaining a monopoly and restraining interstate commerce. In consequence the company was split into 34 parts. Chevron and ExxonMobil are two of the successor companies which resulted from the breakup.
In the past, companies have been broken up by the government when that company not only had a monopoly status that affected the nation's economy, but had been proven to have used that monopoly status to stifle competition. A key aspect here is that the monopoly can impact the nation's economy.
In a breakup, the monopoly is divided up into several independent companies, none of which are in a dominant position. Stockholders in the original company get equivalent shares in all of the new smaller companies.
Two notable cases of large monopolies being broken up were Standard Oil and AT&T.
In the case of the Standard Oil breakup, John D. Rockefeller was in control of the nation's oil and gasoline supply in the decade following 1900, at a time when heating oil and gasoline were becoming critical to the nation's economy. To get that control, Rockefeller had used a number of monopolistic practices in the late 1800's to muscle out or buy out competitors. Standard then began to charge whatever the market would bear. By court order, Standard Oil was broken up into a series of smaller oil producing and retailing companies, none of which were large enough to be in a monopoly position. These smaller companies were to become Chevron, Amoco, Exxon, Mobil, and Marathon.
The irony here is that Standard Oil stockholders made out like bandits on the breakup, as the smaller companies turned out to be far more valuable than a single Standard Oil.
AT&T controlled most of the nation's interstate telephone service through the 1970's... also critical to the economy. They were found to have capitalized on monopolistic practices to maintain that position, so in 1982, AT&T agreed to a breakup whereby it was divided into the 'baby Bells', none of which controlled the nation's telephone infrastructure.
It is important to note that these two cases involved a major part of the US economic infrastructure, and the predatory pricing tactics both were engaging in were bleeding money out of the nation's economy. Smaller monopolies can and do exist, as long as they don't pose a threat to economic growth. For example, Tesla has a virtual monopoly on electric cars in the US, but that isn't so important to the US economy that it's a problem.
Recently, Senator Elizabeth Warren has been campaigning on breaking up the technology companies. That is a somewhat different situation.
A company that achieves anything close to a monopoly status becomes static and complacent, protecting their monopoly at the expense of new ideas. The oil business is fairly static in general, while the slower moving tech of the early to mid 1900's, plus the huge investment in infrastructure protected AT&T. However, the current technology industry moves very quickly, too quickly for a monopolist to maintain their hold for very long.
The two most notable examples of this are IBM and Microsoft.
By the late 1970's, IBM had achieved a near monopoly on business computers, enough so that the government pursued antitrust action and threats of breakup against them. This was never pushed through, because by the 1990's IBM had been knocked off by a more agile competitor: Microsoft. Today, IBM is but a shell of its former self, a services company.
Microsoft itself had achieved a near monopoly status by the year 2000, and there were antitrust proceedings starting up against Microsoft, who had engaged in monopolistic practices... their attempt to commandeer the newly emerging Internet.
However, long before antitrust action could be enacted, Microsoft was knocked out of its monopoly position in 2008-2012 by the mobile revolution, which it completely dropped the ball on. Microsoft thought leaning on their Windows monopoly would let them dominate with Windows Mobile. They were wrong. Today, Microsoft has moved towards a services and software tools company. Ironically, their move away from the Windows monopoly has led them to pose serious competition to another monopoly: Oracle in the database field.