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?



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> Are enriched model categories a representation of enriched $\infty$-categories?
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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](https://arxiv.org/abs/1312.3881v4) 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.



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> 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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Why do we need enriched model categories?
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](https://www.ams.org/journals/tran/2022-375-05/S0002-9947-2022-08600-1/) 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](https://arxiv.org/abs/1506.01475)" 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?](https://mathoverflow.net/questions/287091/why-do-we-need-model-categories)" and "[Do we still need model categories?](https://mathoverflow.net/questions/78400/do-we-still-need-model-categories?rq=1)" 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.


