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created (infinity,1)-algebraic theory.
I tried to adapt Rosicky’s and Lurie’s terminology such as to match that at algebraic theory, but Mike, Toby, Todd and whoever else feels expert should please check if I did it right.
…and linked to it from derived smooth manifold
added a section on model category prresentations for (oo,1)-calgebraic theories
The treatment of algebraic theories in the generality of quasicategories in Andre Joyal's Barcelona notes is very readable.
The treatment of algebraic theories in the generality of quasicategories in Andre Joyal’s Barcelona notes is very readable.
Right, thanks. I added that to the list of references.
Do you get an equivalent of the duality between theory and models as in the 1-category case, where the opposite of the category of free algebras is equivalent to the theory (or the variant described by Forssell reported here)?
Do you get an equivalent of the duality between theory and models as in the 1-category case, where the opposite of the category of free algebras is equivalent to the theory
I am not the expert on algebraic-theory-language to ask, but as far as I can see the statement that you are after is entirely formal in that it depends only on general abstract structure, hence holds whenever the terms “functor”, “product”, “Yoneda lemma” etc. make sense.
Given a Lawvere theory, with syntactic category $C$ whose generating object I’ll write $x$ and whose models are product-preserving functors $A : C \to Set$, the underlying set of a model is $A(x)$. The free model on a given set $S$ is therefore the functor represented by $x^{\times |S|}$, since by Yoneda
$[C,Set]( j(x^{\times |S|}), B ) \simeq B(j(x^{\times |S|})) \simeq (B(x))^{\times |S|} \simeq Set(S, B(x)) \,.$So we find that the free models are precisely the representable functors, and hence, again by Yoneda, the opposite of the category of free models is equivalent to the syntactic category itself.
This reasoning depends on nothing but the assumption that we have CategoryTheory™ at work. So it applies verbatim also to $(\infty,1)$-Lawvere theories. Because we have the (infinity,1)-Yoneda lemma.
Two days ago appeared the thesis by James Cranch, where he discusses a $(2,1)$-algebraic theory whose algebras in $(\infty,1)Cat$ are symmetric monoidal $(\infty,1)$-categories and which hence knows about $E_\infty$-algebras. This very nicely fills a gap connecting $\infty$-alebraic theories with $\infty$-operad theory,
I added the reference and a (very) brief paragraph on it in the Examples-section to (infinity,1)-algebraic theory
I added a handful of details on Cranch’s results here.
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