the variety corresponded to the vanishing of certain purely algebraic structures (which did have a homotopical interpretation as they gave the vanishing of the Whitehead products for the corresponding homotopy types).

Homotopical for me only if the vanishing is up to homotopy and preferably up to higher homotopy ]]>

@Mike #15. No, I think that I am just more reluctant to extend the term ’homotopical algebra’ away from the Quillen context. I understand that there is a homotopical aspect to those other areas, but equally well we could claim that there was a weak infinity category theoretic aspect to them, and whilst we now tend to think of weak infinity categorical stuff as being ’homotopical’ at least in part, that latter terminology is more specific in its connotations due to the considerable influence Quillen’s theory. I wonder if modern representation theory thinks of itself as being homological algebra, yet much of it is at the same time part of homological algebra, K-theory and representation theory. To me ‘higher algebra’ is wider than just its homotopical aspect. I also make a slight distinction between algebraic homotopy and homotopical algebra as having differing aims and objectives and thus differing paradigms.

This is not really important as they are just names, but I think the question of paradigms is also of interest.

]]>Tim, what you describe still sounds “homotopical” to me in the wider sense the phrase “homotopical algebra” as e.g. Quillen used it. Are you thinking of “homotopical” as referring only to topological spaces?

]]>As an example of what I mean, with Phil Ehlers help we proved that the category of crossed complexes corresponded to a variety in the category of simplicial groups, and the variety corresponded to the vanishing of certain purely algebraic structures (which did have a homotopical interpretation as they gave the vanishing of the Whitehead products for the corresponding homotopy types). This was in some sense higher algebra but was not specifically homotopical. Similarly there will be a representation theory of various higher algebraic structures this will have a higher algebraic side but also a homotopical side and I think these will not be identical in their likely paradigms.

]]>Sure. But is that the distinction that Tim has in mind? It seems to me Tim means something different.

But never mind. Whoever feels he has more time and energy to discuss this should consider writing corresponding paragraphs into the respective entries!

]]>Higher algebra may mean also e.g. (2,2)-algebras, what is certainly not considered a homotopical algebra.

That sounds right to me.

]]>Homotopical algebra corresponds to $(\infty,1)$-structures presented often by 1-categorical versions with additional structure. Higher algebra may mean also e.g. $(2,2)$-algebras, what is certainly not considered a homotopical algebra. Do you agree ?

]]>@Urs #6. I understand that but the point is that working with coherence is (possibly) slightly more general and near the intuition that I have.

]]>But in higher algebra, we’re still working out the easy part, so higher algebra is still mostly higher universal algebra.

I’m not sure that’s really true. Stable homotopy theorists have been doing a lot of deep and interesting higher algebra with structured ring spectra for quite a while. They do, of course, often use things like operads in order to define what it means to be a structured ring spectrum, but they’ve then gone on to actually doing algebra with those things, meaning higher versions of what algebraists do.

]]>While the category of fields is not a category of models of an algebraic theory, still every field is an algebra. And vertex operator algebras are “holomorphic” algebras over some operad. It seems to me a bit too much of hair splitting for an Idea-section of an entry to make a big deal out of issues like these.

]]>To an extent, universal algebra is the *easy* part of algebra, so algebraists today are mostly concerned with harder things. But in *higher* algebra, we’re still working out the easy part, so higher algebra is still mostly higher universal algebra.

Hi Tim,

I don’t understand your last comment. Working with algebra in terms of Quillen model structures also means working with algebra up to coherent homotopies.

See for instance the big excitement about the monoidal model categories for spectra.

]]>I have tried to add something in to that entry on higher algebra, which indicates a larger view of algebra but then (towards the bottom) suggests that the higher algebraic analogues of these parts of algebra also may exist ….. and that is where there are interesting challenges as well.

I am not sure that I agree with the idea that homotopical algebra is coextensional with higher algebra however, that seems to me to be doubtful if one interprets Homotopical algebra as the study and application of those structures detectable via machinery analogous to the Quillen model category approach. (Higher algebra is, for me, more to do with algebra up to coherence instead of up to equality.)

]]>More generally, algebra is about algebraic theories, about monads and about operads. All these have higher analogs in higher algebra.

Thank you, I will contribute, soon, in few days, when I get a bit more time (I am quite in trouble these days, and shoulld restrain from extensive $n$Labing by any means). Maybe it is useful to have a bit deeper point of view here than the overly formalistic.

Namely, IMHO, this statement defines far not dominant part of modern algebra, it is just about a small subfield called universal algebra. Maybe large and dominant for pure category theorists, but small for practical algebraists. By no means algebra should be defined as subejct as being equal to universal algebra. For example, (though repairable, still tricky issue in this approach), there is no free commutative field so fields are not algebraic theory in this sense. Vertex algebras are not either, though some pseudooperadic approaches exist. Hopf algebras need cooperations together with operations so they need at least props, but this is easy extension, as well as internal versions in other monoidal categories. Various additional conditions like finiteness, representation theoretic bounds and so on are not revealed from the point of view of algebra over a monad. Algebra over a monad or over operad just *generates* plethora of structures (far not all) while being pretty unable to formulate much of algebraic concepts and most interesting subclasses in practical mathematicians textbook. So monoid aspect is defining for “universal algebra” topics (a small part of algebra) but not for great algebra as a subject.

I have expanded it slightly. See the changes. Feel free to expand on it further.

]]>I find it strange the claims in higher algebra that algebra deals with monoids and higher algebra with monoids in higher setup. It is not only monoids…and it is not only what is called “algebraic theories” (quite a limited part of algebra).

]]>I have cross-linked the two entries homotopical algebra and higher algebra.

At homotopical algebra I moved the text that had existed there into a subsection “History”, because that’s what it is about, right? I added a section “Idea” but so far only included a link to higher algebra there. We could maybe merge the two entries.

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