The word monopoly has a meaning. You don't get to come up with your own. Amazon doesn't have a monopoly on web hosting. Not even close! There are 4 large web hosting companies, hundreds of smaller ones, and it takes very little expertise to set up your own server. You literally have hundreds of alternatives to hosting on Amazon, and there's little Amazon can do to shut out any of those options. Calling AWS a monopoly is simply absurd. Likewise it's trivial to set up the code that runs a service like Twitter, heck they actually open source their CSS (bootstrap it's called). And if you don't want to do that, there are hundreds of services that allow you to post your opinions online, you're doing it on one of them right here. Calling Twitter a monopoly is likewise absurd. Pretty much nothing in the technology stack except perhaps the internet backbone is anything anywhere close to a monopoly, and if we need to regulate the internet backbone I'm all for it.
The technology stack is literally the structural opposite to something like building a new gas pipeline infrastructure so that each house could potentially be served by more than one gas company or multiple power grids so each home could choose from multiple electric distribution companies. But even in those cases, in many states the part that's not a natural monopoly, the generation of electricity or supplier of gas, is open to competition and not regulated. We've been trying to move toward markets when we can and away from government regulated monopolies, that used to be a conservative bedrock value! And size really has nothing to do with anything when it comes to monopolies, the availability of alternatives and a company's use of it's size to block competition are what matter. There's a whole legal theory about monopolies that's fascinating, I highly recommend you take time to dive into and learn about it.
What you're conflating in an extremely dangerous way is the first amendment, which restricts the government's ability to restrict speech, and the right of private companies to say what they want or don't want. You're asking for a world where every private company is compelled to allow any legal speech. First off that entails the government controlling speech, which runs counter to the constitution. But that little trifle aside, it also means if I wanted to spend 30 minutes on Fox or 5000 words on the NYT on why Hitler is a hero, the Jews deserved to be gassed and black people deserved to be lynched they would be forced to let me....that's legal speech here in the United States. Facebook would have to allow me to post the same, or photos of corpses, or of people defecating, or legal porn....that's all legal speech. You probably don't really understand that the first amendment means that many abhorrent things are considered legal speech (apparently you don't even think obscenities or BLM protests are legal speech, which is bizarre), as a result you haven't thought through the consequences of forcing every private company to allow all "legal" speech.
Is that really what you want, or do you just want your god emperor to be allowed to lie and encourage violence and the overthrow of our democracy relentlessly and without consequence on Twitter? If you insist on the latter using your rules you also have to accept the former. And it doesn't stop there. If your version of reality prevails, than no company would be able to refuse service to anyone. You have a high end office building and a strip club wants to rent space there.....you have to let them. You run an ad company that serves health based non-profits and the tobacco industry want to hire you to make ads, no choice in the matter. You could go on and on, but suffice it to say you devolve into a dystopian future of the type the conservatives up to now have always been warning us of.
As far as a grocer store "discriminating" based on political beliefs, again I have absolutely no worry about that because as I said, good luck making that work for the grocery store. We can allow grocery stores that hypothetical because it's exactly that, a hypothetical that would result in either failure for the grocery store or failure of the policy. In no world will I realistically ever be unable to buy groceries because of my political belief, even if we don't have a law prohibiting it, so no, I don't spend any time worrying about it.
And finally, no, you're not being "discriminated against" on Twitter because you're conservative. If you're being discriminated against by Twitter it's because you're posting patently false things attempting to overthrow our democracy or cause violence, attempts that actually came to fruition on Jan 6th as you'd like to forget. I spent more than 20 years defending that democracy and I'd like to keep it, so sorry if the whining of Trump cultists about how their leader should be able to falsely claim he should be able to overthrow presidential election results doesn't elicit much sympathy with me.
First of all you're assuming all kinds of wrong things about me. I'm not Christian, conservative or a trump supporter. And I'm not American either (thank goodness

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Big Tech companies - fair enough I did not use the word monopoly correctly.
However, these companies:
a) have huge market share
b) have a strong network effect
c) have a lot of power which they are abusing (size absolutely matters here)
d) all collude with eachother on who they ban from their platforms, which makes them collectively very much like a monopoly from the point of view of the banned, I guess you call this a cartel.
e) absolutely anti competitive, did you forget the context of this thread? They blocked Parler from competing with them. Not only took away their hosting, but also kicked them off the app stores which actually are real monopolies for their own hardware. I suppose Parler should just build their own phone and app store too? Get real please.
Regarding speech, the fact that obscenity, porn and gore is considered free speech in america is a relatively recent phenomenon. For most of the history it was not considered speech and there was no issue with censoring it. This seems perfectly reasonable to me. The most convincing arguments for why free speech is important don't even apply to those things, so calling them free speech was an extreme stretch to begin with.
The analogy to Fox or NYT doesn't work. They publish curated news, they don't provide a service where people can voice their opinions (like twitter and facebook do). If FOX was selling time slots on their channels and they refused to sell to certain people, then you might have a point. This is the same conflation you did with the BLM protest and grocery store. If it was a business of providing protest spaces, and they refused to cater to BLM, then the analogy would make sense.
So you're concerned about anti black or pro hitler speech. Well twitter and facebook explicitly allow anti white racism and anti male sexism all over their platforms. You can literally call for genocide on twitter and it's perfectly fine, as long as you're targeting the approved group. Are you concerned about that too or are you a hypocrite? What is your proposed solution?
Indeed I'm not discriminated against on twitter, because I have never had an account. However claiming that conservatives aren't discriminated against there just makes you sound completely out of touch with reality. The evidence for it is overwhelming. Even the CEO has acknowledged that they have a liberal bias. Like many SV companies they are overrun by progressive/SJW employees so it's perfectly logical and expected for them to discriminate unless they are forced not to.
Regarding your examples like the strip club and tobacco ads. So what? If those things are legal, then there is no problem. Provide the service that you claim to provide, without discrimination. If there's truly some kind of exceptional case then legislate it. (I highly doubt tech companies will get such legislated exception to discriminate against their political opponents.)
There are already real world examples of banks denying people the ability to open accounts because of their politics. It is just isolated incidents so far, but it is not that much of a stretch to grocery stores or anything else, if your kind of thinking continues to gain traction.
So I am curious now, since you seem to be so pro-discrimination. Do you also favor allowing discrimination based on race, religion, sexuality etc? If not, why? what is the fundamental difference here?