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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexcreated [[(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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItex...and linked to it from [[derived smooth manifold]]

   …and linked to it from derived smooth manifold

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a section on <a href="http://ncatlab.org/nlab/show/(%E2%88%9E%2C1)-algebraic+theory#Models">model category prresentations for (oo,1)-calgebraic theories</a>

   added a section on model category prresentations for (oo,1)-calgebraic theories

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeApr 19th 2010
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   Author: zskoda
   Format: MarkdownThe 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.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 19th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 19th 2010
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   Author: David_Corfield
   Format: MarkdownDo 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 <a href="http://golem.ph.utexas.edu/category/2009/09/coalgebraic_modal_logic.html">here</a>)?

   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)?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeApr 19th 2010
   • (edited Apr 19th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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]].

   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.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeNov 17th 2010
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   Author: Urs
   Format: MarkdownItexTwo 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]]

   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

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeNov 17th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI added a handful of details on Cranch's results <a href="http://ncatlab.org/nlab/show/(%E2%88%9E%2C1)-algebraic+theory#EInfty">here</a>.

   I added a handful of details on Cranch’s results here.