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As far as I understand, model categories mainly provide tools for studying the "homotopy theories" (that is, $\infty$-categories) that are ubiquitous in mathematics. From this point of view, model categories (as an independent concept, not a tool) are of no interest and behave rather strangely (starting with the zigzags of Quillen equivalences). Simplicial model categories serve the same function, only they are much more convenient than ordinary model categories. But what is the function of model categories enriched in an arbitrary (good) monoidal model category?

Are enriched model categories a representation of enriched $\infty$-categories?

If so, that would be the perfect answer to my question in the title. As far as I understand, the work Rune Haugseng - Rectification of enriched infinity-categories shows that categories enriched over a good monoidal model category represent $\infty$-categories enriched over the corresponding monoidal $\infty$-category. But this does not explain why a model structure on an enriched category is needed.

If the answer to the first question is no, then maybe there is some other important invariant concept behind them? Otherwise, why do we study such a concept? What other areas (topics, issues) is the general theory of enriched model categories related to?

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  • $\begingroup$ Maybe you just don’t need them, the same way as you’re not interested in model categories. Each person has their own interests, that’s fine. $\endgroup$ May 12, 2023 at 15:03
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    $\begingroup$ As far as I know, the particular question of (fully homotopy coherent) comparison between enriched model categories and enriched quasicategories has not been settled yet. Only recently, the equivalence between combinatorial model categories and presentable quasicategories was established in arxiv.org/abs/2110.04679, and as far as know, there were no attempts so far to prove similar results in the enriched or monoidal case. (Since model categories are homotopy (co)complete and we also want to avoid size issues, the combinatoriality assumption is completely natural here.) $\endgroup$ May 12, 2023 at 19:47
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    $\begingroup$ @FernandoMuro Yes, of course! I was hoping that I managed to put the question this way: if I (as I write in the first paragraph) have no interest in (enriched) model categories per se, what other reasons could I have to care about them? Motivated by this, I ask two more specific questions. I thought about explicitly mentioning that, of course, someone might be interested in them per se, but I considered it superfluous in the final text :) $\endgroup$ May 13, 2023 at 1:41
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    $\begingroup$ @Emily thank you, indeed (regarding the second question) it would have been better written that way. By the way, I've been noticing you for a long time and I was very interested to read many of your questions, answers and your github page. $\endgroup$ May 22, 2023 at 18:50
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    $\begingroup$ @AivazianArshak Hey, thank you, that's very sweet of you to say :) I'm really happy the things I asked/answered/wrote were interesting to someone else. Again, thank you so much :) $\endgroup$
    – Emily
    May 22, 2023 at 20:42

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To me, the interest in model categories stems from Quillen's observation that the tools of topology (e.g., CW approximation) can be applied in so many different settings, especially in algebra. But not all of those settings are simplicial model categories. Many of them are dg-model categories, i.e., enriched in the category of chain complexes over a commutative ring. There is a huge literature about dg categories, and they have real applications in representation theory. Using enriched model categories, people have been able to prove that the theory of dg-categories closely parallels the theory of spectral categories, i.e., categories enriched in (your favorite) monoidal category of spectra. This has been a fruitful way to use results in stable homotopy theory to prove things in representation theory, homological algebra, and triangulated categories. It also sets the stage for results about Fukaya categories, homological mirror symmetry, etc.

Results in higher category theory (by which I mean weak $n$-categories, not $(\infty,n)$-categories) also often require enriched categories. At its most basic level, an $n$-category is a category enriched in $(n-1)$-categories. So, many of the models for weak $n$-categories (e.g., invented by Bergner, Rezk, Barwick, Ara, Tamsamani, etc.) are cartesian model categories, so that the next level can be enriched in them. To do homotopy theory in this context, you need enriched model categories. I used them a lot in a recent paper with Batanin proving a strong version of the Baez-Dolan stabilization hypothesis.

Another important reason to care about enriched model categories is monoidal model categories, i.e., self-enriched model categories. Homotopy theory is full of situations where the monoidal structure has been crucial, e.g., in chromatic homotopy theory, K-theory, etc. You probably know there was a long (and eventually successful) search for a good monoidal category of spectra, and then tons of applications. You might be interested in the paper "Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories" by Thomas Nikolaus and Steffen Sagave, that proves every presentable monoidal $\infty$-category is modeled by a combinatorial monoidal model category. Perhaps you'd be interested to generalize this result with "monoidal" replaced by "enriched."

The MathOverflow community has already answered the question "Why do we need model categories?" and "Do we still need model categories?" The same answers tell you that we still need monoidal model categories. And, even if you only want to work in the setting of monoidal $\infty$-categories, you will quickly find that you might actually need monoidal model categories to compute things, in much the same way as described in those answers.

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    $\begingroup$ Fantastic answer. Of all the unexplained downvotes I’ve seen on MathOverflow, the one on this is one of the most baffling. $\endgroup$ May 14, 2023 at 8:56
  • $\begingroup$ @PeterLeFanuLumsdaine Indeed. I also like a lot May's answer to one of the cited questions: mathoverflow.net/a/83307/12166 $\endgroup$ May 15, 2023 at 6:10
  • $\begingroup$ Thank you so much for this list of applications! Yes, of course, I am aware of these discussions on MO and I originally noted this role of model categories in my question. But what I am saying is that (unlike model categories or monoidal model categories) I don't see what the enriched model categories "model". $\endgroup$ May 22, 2023 at 16:05
  • $\begingroup$ Apparently (as Dmitri writes in his comment) the answer to my first question is open. So I will wait a little longer and if no new answers appear, I will accept your answer to my second question. $\endgroup$ May 22, 2023 at 16:11

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