With a (decent) night’s sleep behind us, we spent the morning thinking about what last night’s Republican presidential debate — sponsored by the Washington Post and Bloomberg News in case you forgot — taught us about the GOP race.

Republican presidential candidates gather in New Hampshire.

* Mitt =frontrunner: In the wake of the no-go decision by New Jersey Gov. Chris Christie last week, the key question in the race was whether former Massachusetts governor Mitt Romney could consolidate support and establish himself as the clear frontrunner for the nomination. Mission accomplished. Romney’s last 48 hours — endorsements from Christie and former House Speaker Denny Hastert , a sterling performance in the debate Tuesday — may be the best of his campaign to date and solidify him as the person to beat in the race.

As the folks at NBC’s “First Read” point out, polling still suggests that there is a desire in the GOP electorate for someone other than Romney. That could be businessman Herman Cain if he can exhibit some staying power. Or it could be Texas Gov. Rick Perry is he can show some signs of life. (More on that momentarily.) But, neither of those scenarios seem likely at the moment. And that leaves Romney sitting pretty.

* Perry’s best? Maybe yes: In the aftermath of each of the first three presidential debates in which he participated, the political world wondered when the Texas governor would kick his performance into high gear. And in the run-up to Tuesday’s debate, there was much talk of how Perry was working to get better.

But the product Perry delivered — at times non-existent, at other times seemingly disinterested — raises the question of whether Perry is capable of being better than he has been in these debates. And the answer increasingly looks to be “no”. Perry and his team seem to have decided that rather than promising that his debate performances will improve, it’s a sounder strategy to simply say that he’s not all that good a debater. “Debates aren’t my strong suit,” Perry said last night.

Fine. But, if the Republican electorate is looking to see which candidate matches up best with President Obama, debates matter. And Perry struggles then matter too.

* Cain rises: Whether Cain is simply the latest Republican flavor of the month — following in the ignominious footsteps of Donald Trump, Michele Bachmann and, to a lesser extent, Perry — or the second coming of populist outsider Ross Perot — circa 1992 — remains an open question. But Cain proved on Tuesday night that he could hold his own as the centerpiece of a debate — generally handling the questions thrown his way and turning each and every one of them back to his “9/9/9” tax reform plan and outsider credentials.

On the basis of that performance, Cain is likely to keep rising in polls — both nationally and in early states — over the next few weeks. That’s bad news for the likes of Bachmann and former Pennsylvania senator Rick Santorum who are trying to court the same conservative piece of the electorate as Cain.

* Debates matter: Are these primary debates huge ratings boons for the cable networks that televise them? Generally not. But that doesn’t mean they don’t matter. Six weeks ago the dominant storyline in the Republican race was Perry as the momentum candidate who had the look and feel of a nominee. Four debates later Perry is being sold more than bought in the political world — literally. Just 12 percent of people on InTrade now believe he will be the Republican nominee.

Six weeks ago Cain was an afterthought. But, on the strength of his debate performances — and his win in a Florida straw poll — he’s now being touted as the main alternative to Romney.

These debates have tremendous ability to shape the coverage of the campaign. And, simply because of that they matter. In a major way.