The following represents what an ordinary citizen of Deshret is likely to know, believe, and has been taught. It's the shape of the public truth as the majority believe. The details may differ from floor to floor and between people, but the lawful, non-heretical summary is this:
About 400 years ago, the world as we know it ended. This is canonically called the Burning: whatever exactly happened is lost, but Sepulchre's version is that humanity became too ambitious with technology. Routine examples are free-thinking machines, towers of light reaching the clouds, and attempts at reaching beyond the boundaries of the soul. As a result, the Earth was scorched - cities melted, oceans boiled, soil turned to glass, the atmosphere became unbreathable.
The in-game calendar starts counting after this fact, so we're in the year 386AC. Thus, the core doctrine is that humanity and its relationship with technology must be kept under strict control to prevent such as a catastrophe from happening again. The ones with that duty are, of course, Sepulchre. They preach that to yearn for the above is sin, and that one should aspire to instead dig deeper (and as one may remember, the Great Houses are the furthest down in the city). The closer to the scorched sky you are, the more danger you're in - and this may very well translate to actual danger due to the hazardous environment of the surface seeping in, not just spiritual danger.
The city began when nine souls, the Founders (or for some inexplicable reason in older slum poetry called the Argonauts) saw the Burning coming and led a group underground into the Earth, which is what's now Deshret. They dug out the first galleries, established the first laws, and in the modern days they're gone. The Church venerates them as Saints still. Their leader was Marrow, also called the Prophet. He is where the main doctrine of the city comes from: that technology and greed destroyed the old world, that the surface is damned, and that the people must remain underground underneath the loving thumb of the Founders where they can be kept safe.
The city may have started smaller, but over the years it has dug ever downward. Deeper floors are more prestigious, safer, and further from the radiation above. The surface-adjacent floors are the slums: it has everything from farms to workshops to crucially, the gate to the surface. The middle floors house the commercial and guild districts. These are the bourgeoisie and middle class, having everything from guild HQs to residential blocks for the sixers (i.e., guild members). The deeper floors carry the estates of the Grand Houses (and some richer Lesser Houses) as well as the Cathedral.
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