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Recently I got acquainted with $\infty$-algebraic theories. I expect $\infty$-algebraic objects (of ordinary Lawvere theories) in $\infty\text{-}\mathrm{Groupoid}$ to behave much better than algebraic objects in the homotopy category (called $H$-objects). In a certain sense, this is a more correct concept. So my question:

Are H-objects interesting in their own right or is it just some random/pathological extension of the 1-truncation of those objects that really play a fundamental role in modern mathematics?

Or like this:

If we initially thought ∞-categorically, would we invent H-objects? Why / in connection with what?.

The answer to my question could be a list of broad interesting contexts in which exactly $H$-objects appear, but not their higher versions. Or some justification that the concept has really lost its independent value and its role has become auxiliary.

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    $\begingroup$ "Is there not a natural model structure on the category of algebraic objects in a homotopy category?" - I think this is a great question as opposed to your titular question. $H$-objects are often more difficult to study, this does not make them "incorrect", but rather interesting. $\endgroup$
    – Connor Malin
    Commented Dec 31, 2022 at 0:12
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    $\begingroup$ In regards to this question, model categories are usually defined to admit finite (co)limits. Categories of $H$-objects have very few limits, usually only (co)products. $\endgroup$
    – Connor Malin
    Commented Dec 31, 2022 at 0:21
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    $\begingroup$ Indeed, there are almost no limits and colimits, sorry, for some reason I did not think about it. In fact, I didn't like my question at all when I finished it. But I definitely have a sense of the question, sounding along the lines of "are H-objects interesting in their own right or is it just some random/pathological extension of the 1-truncation of those objects that really play a fundamental role in modern mathematics". Or like this: "if we initially thought $\infty$-categorically, would we invent H-objects? Why / in connection with what?". $\endgroup$
    – Arshak Aivazian
    Commented Dec 31, 2022 at 2:37
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    $\begingroup$ I decided to bet that I would be lucky to be understood and asked my question. But I think that I myself will gradually figure out the answer for myself. So if my question does not meet with interest, then I will calmly relate to its closure. $\endgroup$
    – Arshak Aivazian
    Commented Dec 31, 2022 at 2:37
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    $\begingroup$ I don’t know what you mean by algebraic object or algebraic-homotopy object but H-spaces (if this is what you mean by H-objects) are not wrong at all. $\endgroup$
    – Fernando Muro
    Commented Dec 31, 2022 at 9:21

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This is not an answer, but slightly too long for a comment.

The main thing I wanted to say is that there is no "correct" or "incorrect" formalization, they just serve different purposes.

For instance (grouplike) $H$-spaces are enough for some purposes (such as proving that a space is simple) and because this is a very minimal amount of structure, this can make them very handy. On the other hand, they cannot be delooped, unlike grouplike $A_\infty$-spaces, and they don't have a very workable homotopy theory, unlike (grouplike) $A_\infty$-spaces.

For your specific subquestion, it's not easy to answer in full generality: the trivial Lawvere theory has very nice up-to-homotopy models, and some simple examples might also have a nice model category that models them. Here is one way to possibly prove that something is not the homotopy category of a suitably nice model category: if $M$ is a model category, $ho(M)$ admits weak pullbacks, that is, for every cospan $A\to B \leftarrow C$, there is a cone $P\to A,C$ which satisfies a version of the universal property of a pullback, but without uniqueness. Indeed, the homotopy pullback satisfies this property.

I want to guess that in many (most ?) cases, for a nontrivial Lawvere theory $T$, $Mod_T(\mathrm{Ho})$ does not have weak pullbacks; but I'm not sure how to prove that, as I'm not sure what the correct way to say "many cases" is.

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  • $\begingroup$ I am unfamiliar with $A_n$ and $A_\infty$, but do $H$-objects (with some choice of homotopy) lie between some $A_{n-1}$ and $A_n$? $\endgroup$
    – Z. M
    Commented Jan 9, 2023 at 18:30
  • $\begingroup$ @Z.M a choice of a binary product and a homotopy for associativity is $A_3$. $\endgroup$
    – Fernando Muro
    Commented Jan 9, 2023 at 20:28
  • $\begingroup$ But without associativity (which is often what is meant by $H$-space), it is $A_2$ (in fact, $H$-space is often the unital version of $A_2$) $\endgroup$
    – Maxime Ramzi
    Commented Jan 9, 2023 at 20:29

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