Tony Yates has a nice succinct post from a couple of years ago about the "bottom up inflation fallacy" (brought up in my Twitter feed by Nick Rowe):
This inflation is caused by the sum of its parts problem rears its head every time new inflation data gets released. Where we can read that inflation was ’caused’ by the prices that went up, and inhibited by the prices that went down.
I wouldn't necessarily attribute the forces that make this fallacy a fallacy to the central bank as Tony does — at the very least, if central banks can control inflation, why are many countries (US, Japan, Canada) persistently undershooting their stated or implicit targets? But you don't really need a mechanism to understand this fallacy, because it's actually a fallacy of general reasoning. If we look at the components of inflation for the US (data from here), we can see various components rising and falling:
While the individual components move around a lot, the distribution remains roughly stable — except for the case of the 2008-9 recession (see more here). It's a bit easier to see the stability using some data from MIT's billion price project. We can think of the "stable" distribution as representing a macroeconomic equilibrium (and the recession being a non-equilibrium process). But even without that interpretation, the fact that an individual price moves still tells us almost nothing about the other prices in the distribution if that distribution is constant. And it's definitely not a causal explanation.
It does seem to us as humans that if there is something maintaining that distribution (central banks per Tony), then an excursion by one price (oil) is being offset by another (clothing) in order to maintain that distribution. However, there does not have to be any force acting to do so.
For example, if the distribution is a maximum entropy distribution then the distribution is maintained simply by the fact that it is the most likely distribution (consistent with constraints). In the same way it is unlikely that all the air molecules in your room will move to one side of it, it is just unlikely that all the prices will move in one direction — but they easily could. For molecules, that probability is tiny because there are huge numbers of them. For prices, that probability is not as negligible. In physics, the pseudo-force "causing" the molecules to maintain their distribution is called an entropic force. Molecules that make up a smell of cooking bacon will spread around a room in a way that looks like they're being pushed away from their source, but there is no force on the individual molecules making that happen. There is a macro pseudo-force (diffusion), but there is no micro force corresponding to it.
I've speculated that this general idea is involved in so-called sticky prices in macroeconomics. Macro mechanisms like Calvo prices are in fact just effective descriptions at the macro scale, and therefore studies that look at individual prices (e.g. Eichenbaum et al 2008) will not see stick prices.
In a sense, yes, macro inflation is due to the price movements of thousands of individual prices. And it is entirely possible that you could build a model where specific prices offset each other via causal forces. But you don't have to and there exist ways of constructing a model where there isn't necessarily any way to match up the macro inflation with specific individual changes because macro inflation is about the distribution of all price changes. That's why I say the "bottom up" fallacy is a fallacy of general reasoning, not just a fallacy according to the way economists understand inflation today: it assumes a peculiar model. And as Tony tells us, that's not a standard macroeconomic model (which is based on central banks setting e.g. inflation targets).
You can even take this a bit further and argue against the position that microfoundations are necessary for a macroeconomic model. It is entirely possible for macroeconomic forces to exist for which there are no microeconomic analogs. Sticky prices are a possibility; Phillips curves are another. In fact, even rational representative agents might not exist at the scale of human beings, but could be a perfectly plausible effective degrees of freedom at the macro scale (per Becker 1962 "Irrational Behavior and Economic Theory", which I use as the central theme in my book).