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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?

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  • $\begingroup$ They are the fibrations in the Joyal model structure on simplicial sets, whose fibrant objects are quasicategories. This means in particular they achieve what is stated in the Introduction: "To help us achieve this counterintuitive strictness, each ∞-cosmos comes with a specified class of maps between ∞-categories called isofibrations. The isofibrations have no homotopy-theoretic meaning, as any functor between ∞-categories is equivalent to an isofibration with the same codomain". $\endgroup$
    – Manuel Araújo
    Commented Nov 9, 2022 at 8:56
  • $\begingroup$ Dear @ManuelAraújo, would you mind explaining a little why this fact allow them work with 2-categories instead of bicategories? (I also know very little about model categories, but I would appreciate any explanation.) $\endgroup$
    – Gabriel
    Commented Nov 9, 2022 at 9:07
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    $\begingroup$ I am also not an expert (hence the comment instead of an answer), but here is an example. Say you want to to compute a homotopy pullback of a diagram of spaces. If one of the maps involved is a fibration, you can just take the strict pullback of the diagram. $\endgroup$
    – Manuel Araújo
    Commented Nov 9, 2022 at 9:22
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    $\begingroup$ @ManuelAraújo Just to clarify, the isofibrations aren't precisely the fibrations in the Joyal model structure. But every fibration is an isofibration, and every isofibration between quasicategories is a fibration, so in the $\infty$-cosmos context (where every object is fibrant) the more mysterious Joyal fibrations can be ignored. $\endgroup$
    – Kevin Carlson
    Commented Nov 9, 2022 at 20:06
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    $\begingroup$ FWIW these comments also apply to isofibrations of ordinary categories. They are technically convenient and allow us to strictify. $\endgroup$
    – Zhen Lin
    Commented Nov 9, 2022 at 21:50

1 Answer 1

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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!

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