If you want to do differential geometry without points, why not reading works about Synthetic Differential Geometry ? I think that, as usual, a good start is the nLab website : ncatlab.org/nlab/show/synthetic+differential+geometry

@მამუკაჯიბლაძე A space $X$ is compact if and only if it is a compact object in the category of open subsets of $X$: it's Proposition 2.6 of ncatlab.org/nlab/show/compact+space.

The presentable objects in the category of general topological spaces are the discrete ones (it's a very known joke). A topological space is $\lambda$-presentable if and only if it is discrete of cardinal less than $\lambda$. Your question is the case $\lambda=\aleph_0$.

I did not know this notion of trifibration and the nLab page is a bit confusing: $E^{op}$ is not the opposite category.

@Max Category theory is almost everywhere in computer science.

@FoscoLoregian Not trivial in the sense of the question. If a category and its opposite are both locally presentable, it is a complete lattice.

"it requires a large staff in Ann Arbor and elsewhere, including many mathematicians" : mathematicians who work for MathSciNet are not paid for writing reviews.

Voevodsky's motivation was to build a proof assistant for doing homotopy theory (it is actually what he wrote in one of his paper). His motivation comes from some papers he published in the beginning of his career about the Homotopy Hypothesis and weak $\omega$-groupoids and which turned out to be wrong. His notion of $\omega$-groupoid is strictly associative with weakly invertible elements and it was proved later that a strictly associative law cannot lead to a model of all homotopy types. Troubles begin in dimension $3$.

I can't find Remy Tuyeres' PhD with Google, he's also not on the Mathematics Genealogy Project. Any link would be welcome. Richard Williamson 's paper does not seem to be published. How did you find these references ? You already knew them or you have a magic keyword in your pocket ?

And also : injective object in a category with respect to a class of maps, injective set map in the sense of one-to-one set map, or the notion of injective object in homological algebra.

I meant you forgot the word trivial, you should fix your post. This hypothesis is used in the construction of the left adjoint of the forgetful functor.

Interesting. The author only assumes that all trivial cofibrations are monic.

Proposition 3.3 of "On combinatorial model categories" (arxiv.org/abs/0708.2185) can be helpful. The class of (trivial) fibrations in a combinatorial model category is accessible and accessibly embedded.

@AndrejBauer Could you define precisely what is a research-level question please ? I still have a lot of difficulty to comprehend this notion. What kind of question deserves to be called "research-level" and what kind of question is not "research-level". It's not a home work, and that does not seem to be trivial. I have voted for this question, even if it does not interest me.

I have a stupid question (it is good enough for a comment I hope): can we prove that there does not exist any model of ZFC interpreting sets as non-trivial spaces (I mean non-discrete) ?

Size means something mathematically. It is not an artificial problem created by idle mathematicians to annoy people. And it is not an issue either actually. For example, all categories have a class of generators (the class of all objects) and only some of them have a set of generators.