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1. • CommentRowNumber1.
   • CommentAuthorfpaugam
   • CommentTimeMay 14th 2012
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   Author: fpaugam
   Format: TextI would like to understand what kind of theory Gamma is, in the sense of doctrines. This is not included on the Gamma-space page, and i would like something about that to be here, so this discussion is making a proposition in this sense. I mean that Delta maclane may be seen as the algebraic theory of monoids (category with finite products opposite to that of finitely generated free monoids), and its models in sets are monoids, and in categories are monoidal categories (pseudo-functors). For Gamma, it is not an algebraic theory, since n is not the product of n times 1, but it seems to me something like a monoidal theory. I would tend to define Gamma of Segal as a kind of theory of commutative monoids in monoidal categories, obtained by adding to the operations in Delta (monoidal structure) additional symmetry data. For example, if i take a monoidal functor from Gamma to sets, i get a commutative monoid, whose underlying monoid is the model of Delta (algebraic theory), but the theory Gamma seems monoidal, not algebraic in the sense of Lawvere. These notions are important to better understand higher categorical generalizations of monoidal and symmetric monoidal categories. Am i correct here? May we add these theoretical/doctrinal considerations to the Gamma-spaces page to clarify where Gamma is coming from, from a conceptual/theoretical point of view?

   I would like to understand what kind of theory Gamma is, in the sense of doctrines.
   This is not included on the Gamma-space page, and i would like something about
   that to be here, so this discussion is making a proposition in this sense.
   
   I mean that Delta maclane may be seen as the algebraic theory of monoids (category
   with finite products opposite to that of finitely generated free monoids), and its
   models in sets are monoids, and in categories are monoidal categories (pseudo-functors).
   
   For Gamma, it is not an algebraic theory, since n is not the product of n times 1, but
   it seems to me something like a monoidal theory. I would tend to define Gamma of
   Segal as a kind of theory of commutative monoids in monoidal categories, obtained
   by adding to the operations in Delta (monoidal structure) additional symmetry data.
   
   For example, if i take a monoidal functor from Gamma to sets, i get a commutative
   monoid, whose underlying monoid is the model of Delta (algebraic theory), but
   the theory Gamma seems monoidal, not algebraic in the sense of Lawvere.
   
   These notions are important to better understand higher categorical generalizations
   of monoidal and symmetric monoidal categories.
   
   Am i correct here? May we add these theoretical/doctrinal considerations to the Gamma-spaces
   page to clarify where Gamma is coming from, from a conceptual/theoretical point of view?
2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMay 14th 2012
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   Author: Mike Shulman
   Format: MarkdownItexA good question! I'm a little confused though — if by $\Delta_{maclane}$ you mean the augmented [[simplex category]] $\Delta_a$ then it's not a Lawvere finite-product theory either, but a monoidal theory: the "addition" operation on $\Delta_a$ is not the cartesian product. I believe the answer, though, is that $\Gamma$ is the [[semicartesian monoidal category|semicartesian]] symmetric monoidal theory of commutative monoids. See [this blog post](http://golem.ph.utexas.edu/category/2009/10/generalized_operads_in_classic.html). This would be great to add to the $\Gamma$-spaces page.

   A good question! I’m a little confused though — if by $\Delta_{maclane}$ you mean the augmented simplex category $\Delta_a$ then it’s not a Lawvere finite-product theory either, but a monoidal theory: the “addition” operation on $\Delta_a$ is not the cartesian product.

   I believe the answer, though, is that $\Gamma$ is the semicartesian symmetric monoidal theory of commutative monoids. See this blog post. This would be great to add to the $\Gamma$-spaces page.

3. • CommentRowNumber3.
   • CommentAuthorTom Leinster
   • CommentTimeMay 15th 2012
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   Author: Tom Leinster
   Format: MarkdownItexI'd start from the following fact. Write $\mathbf{F}$ for the category of finite sets (or a skeleton thereof), viewed as a symmetric monoidal category via coproduct. Then for any cartesian monoidal category $E$, a functor $\Gamma^{op} \to E$ is the same thing as a colax symmetric monoidal functor $\mathbf{F} \to E$. This equivalence can be refined. The functors $\Gamma^{op} \to E$ for which the Segal maps are all isomorphisms correspond to the *strong* symmetric monoidal functors $\mathbf{F} \to E$. That is, they're the commutative monoids in $E$. The same thing works in the non-symmetric world, changing $\Gamma$ to $\Delta$ (the category of *nonempty* totally ordered sets) and $\mathbf{F}$ to the category $\mathbf{D}$ of *possibly empty* totally ordered sets (which is monoidal). So, for example, a simplicial set is the same thing as a colax monoidal functor $\mathbf{D} \to Set$. So we're very much in the world of (symmetric or not) monoidal theories. It's an interesting question as to how to relate the symmetric and non-symmetric worlds, and also how to relate them to the finite-product/Lawvere world.

   I’d start from the following fact. Write $\mathbf{F}$ for the category of finite sets (or a skeleton thereof), viewed as a symmetric monoidal category via coproduct. Then for any cartesian monoidal category $E$, a functor $\Gamma^{op} \to E$ is the same thing as a colax symmetric monoidal functor $\mathbf{F} \to E$.

   This equivalence can be refined. The functors $\Gamma^{op} \to E$ for which the Segal maps are all isomorphisms correspond to the strong symmetric monoidal functors $\mathbf{F} \to E$. That is, they’re the commutative monoids in $E$.

   The same thing works in the non-symmetric world, changing $\Gamma$ to $\Delta$ (the category of nonempty totally ordered sets) and $\mathbf{F}$ to the category $\mathbf{D}$ of possibly empty totally ordered sets (which is monoidal). So, for example, a simplicial set is the same thing as a colax monoidal functor $\mathbf{D} \to Set$.

   So we’re very much in the world of (symmetric or not) monoidal theories. It’s an interesting question as to how to relate the symmetric and non-symmetric worlds, and also how to relate them to the finite-product/Lawvere world.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMay 15th 2012
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   Author: Mike Shulman
   Format: MarkdownItexThe blog post that I linked to has some thoughts about how to move between worlds.

   The blog post that I linked to has some thoughts about how to move between worlds.

5. • CommentRowNumber5.
   • CommentAuthorfpaugam
   • CommentTimeMay 15th 2012
   • (edited May 15th 2012)
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   Author: fpaugam
   Format: TextThanks for these answers. I think that Tom may be closer to what i need (a description of monoidal theories in terms of theories). I know that you will say that i am looking for some kind of circular definition, but what i was looking for is a definition of monoidal category that does not involve the notion of monoidal category (and the same for symmetric monoidal category), but only the notion of Lawvere theory (cartesian category) and its 2-categorical version. A monoidal category is a weak model of the theory of monoids in the 2-category Cat (pseudo-functor that commutes with products from the category with finite product opposite to that of free finitely generated monoids, i.e., Lawvere theory of monoids, to Cat). I don't think saying that makes me saying something wrong, because what can a monoidal category be but a monoid in Cat? One may define similarly a symmetric monoidal category, as a model of the Lawvere theory of commutative monoids. My question would be: how do we compare these descriptions of categorified algebraic structures, defined through higher sketches (here simply higher algebraic theory) to the one used in the litterature, like the definition of symmetric monoidal infinity-category defined using Gamma. The case of Gamma-spaces correspond to the notion of symmetric monoidal infinity-groupoid: one can also define a related notion using a Lawvere theory of commutative monoids. How do you relate these two approaches? How can we interpret Gamma-spaces in terms of a (maybe two step) sketch construction with values in spaces (i.e., infinity-groupoids)? To answer Mike, i agree that Delta Maclane is the initial monoidal theory, and that's interesting to know that Gamma is the semicartesian sm theory of commutative monoids, but my question is to give a definition of say, Gamma-spaces, by thinking of them as monoids in spaces, in terms of a cartesian theory. The point is to define symmetric monoidal infinity-categories, say, using the same Lawvere theory (of symmetric monoids, i guess) instead of Gamma, and to compare the two definitions. This would give me an explanation of where Gamma is coming from. I find it strange to use Gamma to define symmetric monoidal categories, without saying why it actually gives a good notion of symmetric monoidal category. In the topological case, this is given by Segal's theorem about infinite loop spaces, but i am looking for a more ``algebraic'' explanation of the origin of Gamma.

   Thanks for these answers. I think that Tom may be closer to what i need (a description of monoidal theories in terms of theories).
   
   I know that you will say that i am looking for some kind of circular definition, but what i was looking for is a definition of monoidal category that does not involve the notion of monoidal category (and the same for symmetric monoidal category), but only the notion of Lawvere theory (cartesian category) and its 2-categorical version.
   
   A monoidal category is a weak model of the theory of monoids in the 2-category Cat (pseudo-functor that commutes with products from the category with finite product opposite to that of free finitely generated monoids, i.e., Lawvere theory of monoids, to Cat). I don't think saying that makes me saying something wrong, because what can a monoidal category be but a monoid in Cat?
   
   One may define similarly a symmetric monoidal category, as a model of the Lawvere theory of commutative monoids.
   
   My question would be: how do we compare these descriptions of categorified algebraic structures, defined through higher sketches (here simply higher algebraic theory) to the one used in the litterature, like the definition of symmetric monoidal infinity-category defined using Gamma.
   
   The case of Gamma-spaces correspond to the notion of symmetric monoidal infinity-groupoid: one can also define a related notion using a Lawvere theory of commutative monoids. How do you relate these two approaches?
   
   How can we interpret Gamma-spaces in terms of a (maybe two step) sketch construction with values in spaces (i.e., infinity-groupoids)?
   
   To answer Mike, i agree that Delta Maclane is the initial monoidal theory, and that's interesting to know that Gamma is the semicartesian sm theory of commutative monoids, but my question is to give a definition of say, Gamma-spaces, by thinking of them as monoids in spaces, in terms of a cartesian theory. The point is to define symmetric monoidal infinity-categories, say, using the same Lawvere theory (of symmetric monoids, i guess) instead of Gamma, and to compare the two definitions.
   
   This would give me an explanation of where Gamma is coming from. I find it strange to use Gamma to define symmetric monoidal categories, without saying why it actually gives a good notion of symmetric monoidal category. In the topological case, this is given by Segal's theorem about infinite loop spaces, but i am looking for a more ``algebraic'' explanation of the origin of Gamma.
6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeMay 15th 2012
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   Author: Mike Shulman
   Format: MarkdownItexDid you read my blog post? When you have a morphism of doctrines $D_1\to D_2$, the left adjoint from $D_1$-theories to $D_2$-theories takes the $D_1$-theory of an X to the $D_2$-theory of an X. For instance, the morphism of doctrines from semicartesian symmetric monoidal categories to cartesian monoidal categories takes the semicartesian theory of a commutative monoid, namely $\Gamma$, to the cartesian (Lawvere) theory of a commutative monoid. Similarly, we can start from the symmetric monoidal theory of a commutative monoid and extend it up to the semicartesian situation to obtain $\Gamma$.

   Did you read my blog post? When you have a morphism of doctrines $D_1\to D_2$, the left adjoint from $D_1$-theories to $D_2$-theories takes the $D_1$-theory of an X to the $D_2$-theory of an X. For instance, the morphism of doctrines from semicartesian symmetric monoidal categories to cartesian monoidal categories takes the semicartesian theory of a commutative monoid, namely $\Gamma$, to the cartesian (Lawvere) theory of a commutative monoid. Similarly, we can start from the symmetric monoidal theory of a commutative monoid and extend it up to the semicartesian situation to obtain $\Gamma$.

7. • CommentRowNumber7.
   • CommentAuthorfpaugam
   • CommentTimeMay 15th 2012
   • (edited May 15th 2012)
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   Author: fpaugam
   Format: TextOups! I did not read it up to the end. Thanks! Is the above relation of semicartesian with cartesian monoidal theories a semantical equivalence? I mean is a Gamma-space the same thing as a symmetric monoid in spaces, in the sense of a model of the Lawvere theory of commutative monoids in the \infty-category of spaces? So my question may be transformed in: why do we use Gamma to define monoidal infinity-groupoids, and not the Lawvere theory of commutative monoids? The same question is valid for symmetric monoidal categories. I have an inclination for Lawvere theory because they are fundamental from the categorical viewpoint, in the sense that the definition of their ambient doctrine only uses limit conditions. Monoidal theories are a bit more complicated because the definition of the ambient doctrine is, for me, given by using a categorified version of the Lawvere theory of monoids. So it's not anymore a sketch type theory. Why do we have to pass to monoidal theories? Why not using directly the Lawvere theory of monoids and commutative monoids to define Gamma spaces or symmetric monoidal infinity-categories?

   Oups! I did not read it up to the end. Thanks!
   
   Is the above relation of semicartesian with cartesian monoidal theories a semantical equivalence? I mean is a Gamma-space the same thing as a symmetric monoid in spaces, in the sense of a model of the Lawvere theory of commutative monoids in the \infty-category of spaces?
   
   So my question may be transformed in: why do we use Gamma to define monoidal infinity-groupoids, and not the Lawvere theory of commutative monoids? The same question is valid for symmetric monoidal categories. I have an inclination for Lawvere theory because they are fundamental from the categorical viewpoint, in the sense that the definition of their ambient doctrine only uses limit conditions. Monoidal theories are a bit more complicated because the definition of the ambient doctrine is, for me, given by using a categorified version of the Lawvere theory of monoids. So it's not anymore a sketch type theory.
   
   Why do we have to pass to monoidal theories? Why not using directly the Lawvere theory of monoids and commutative monoids to define Gamma spaces or symmetric monoidal infinity-categories?
8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeMay 15th 2012
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   Author: Mike Shulman
   Format: MarkdownItexBy the properties of adjunctions, a (strict) model of a semicartesian theory in a cartesian category is the same as a (strict) model of the cartesian theory that it freely generates. This ought to also be true at the homotopical level as long as the adjunctions are homotopically meaningful. I don't know to what extent that is known to be true. One might hope for them to be Quillen adjunctions, in which case it would be true as long as the theory you are looking at is sufficiently cofibrant. Probably $\Gamma$ is sufficiently cofibrant. As for why we use a semicartesian theory and not a Lawvere theory, that's a good question! The fact that people usually do it that way confused me for a long time, until I figured out what a semicartesian theory was and that $\Gamma$ is one. I believe the answer is partly tradition, but there are also reasons for that tradition. $\Gamma$-spaces worked well for Segal's infinite loop space machine, and the more general semicartesian theories associated to operads (= monoidal theories) worked well for May and Thomason's comparison of infinite loop space machines. Operads, in turn — or more precisely, semicocartesian operads — worked well for May's infinite loop space machine. I don't know to what extent any of that machinery would work with Lawvere theories; my guess would be some, but not all, at least not as easily. I do know, for instance, that if you use ordinary operads rather than semicocartesian ones, then May's infinite loop space machine fails. So even if at a fully $(\infty,1)$-categorical level there should be no difference, when working with concrete presentations one choice can be easier to work with than another.

   By the properties of adjunctions, a (strict) model of a semicartesian theory in a cartesian category is the same as a (strict) model of the cartesian theory that it freely generates. This ought to also be true at the homotopical level as long as the adjunctions are homotopically meaningful. I don’t know to what extent that is known to be true. One might hope for them to be Quillen adjunctions, in which case it would be true as long as the theory you are looking at is sufficiently cofibrant. Probably $\Gamma$ is sufficiently cofibrant.

   As for why we use a semicartesian theory and not a Lawvere theory, that’s a good question! The fact that people usually do it that way confused me for a long time, until I figured out what a semicartesian theory was and that $\Gamma$ is one. I believe the answer is partly tradition, but there are also reasons for that tradition. $\Gamma$-spaces worked well for Segal’s infinite loop space machine, and the more general semicartesian theories associated to operads (= monoidal theories) worked well for May and Thomason’s comparison of infinite loop space machines. Operads, in turn — or more precisely, semicocartesian operads — worked well for May’s infinite loop space machine. I don’t know to what extent any of that machinery would work with Lawvere theories; my guess would be some, but not all, at least not as easily. I do know, for instance, that if you use ordinary operads rather than semicocartesian ones, then May’s infinite loop space machine fails. So even if at a fully $(\infty,1)$-categorical level there should be no difference, when working with concrete presentations one choice can be easier to work with than another.

9. • CommentRowNumber9.
   • CommentAuthorfpaugam
   • CommentTimeMay 16th 2012
   • (edited May 16th 2012)
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   Author: fpaugam
   Format: TextGreat! Thanks Mike, this was exactly what I was hopping for. The point is that in concrete theories, one has to replace the theory of monoids that is Cartesian by a cofibrant replacement as such, and this makes the story complicated on the concrete side. I guess this is the reason why Gamma is used as you say, its use avoids the use of cofibrant replacement.

   Great! Thanks Mike, this was exactly what I was hopping for. The point is that in concrete theories, one has to replace the theory of monoids that is Cartesian by a cofibrant replacement as such, and this makes the story complicated on the concrete side. I guess this is the reason why Gamma is used as you say, its use avoids the use of cofibrant replacement.