In the very first chapter of *Elements of* $\infty$*-category theory*, E. Riehl and D. Verity define their notion of an $\infty$-cosmos, which should axiomatise a category in which $\infty$-categories live. (So, for example, the category of quasi-categories is an example of an $\infty$-cosmos.) An $\infty$-cosmos is a category enriched over quasi-categories and equipped with a collection of maps called *isofibrations*, which should satisfy some properties.


Surely, in the $\infty$-cosmos of quasi-categories, the isofibrations correspond the usual notion of an isofibration of quasi-categories. The same holds in the $\infty$-cosmos of 1-categories.


Now, since I'm just learning quasi-categories for the first time (as is expected from a reader of this book, apparently), I have no intuition whatsoever for isofibrations. **Why is this class of functors so important as to be in the very definition of an $\infty$-category (i.e., an object of an $\infty$-cosmos)?** In particular, how should I think about isofibrations?



Intuition for isofibrations in $\infty$-categories
As has been mentioned, there's no homotopically meaningful content to the notion of an isofibration, since every map of $\infty$-categories is equivalent to an isofibration. So the point is really all in the definition of an $\infty$-cosmos: isofibrations are the kinds of maps between $\infty$-categories that you can take a strict pullbacks along, or take the strict limit of a countable tower of, and so on.


It is right at the heart of homotopical category theory that we often want a strict construction to model a more complicated, homotopy coherent construction, such as a homotopy pullback, basically because this saves us from carrying around lots of coherence isomorphisms in our arguments, replacing them with equalities. This can't be done for pullbacks along arbitrary $\infty$-functors, but for isofibrations, it works, which allows an $\infty$-cosmos to behave something like the category of fibrant objects in a simplicial model category. If you don't have any familiarity with model categorical arguments, then you'll get some as you read further in the book and see how the strict limit properties of isofibrations enable many of the arguments Riehl and Verity make. But in particular, don't worry too much about isofibrations, which are just a technical convenience–keep reading to get to the good stuff!


