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 (and are actually one way of defining the concept of $\infty$-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?

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?

As far as I understand, model categories basically 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 (and are actually one way of defining the concept of $\infty$-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 areas is the general theory of enriched model categories related to?

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 (and are actually one way of defining the concept of $\infty$-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?