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Categorifications of the real numbers
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For the purposes of this question, a categorification of the real numbers is a pair $(mathcalC,r)$ consisting of:
- a symmetric monoidal category $mathcalC$
- a function $rcolon mathrmob(mathcalC)tomathbbR$
such that:
- $r(Xotimes Y) = r(X) r(Y)$ for all objects $X$ and $Y$ of $mathcalC$
- $r(mathbb1) = 1$, where $mathbb1$ is the monoidal unit
- $Xcong X'implies r(X)=r(X')$
Some examples of categorifications of $mathbbR$ are: finite sets and cardinality; finite-dimensional vector spaces and dimension; topological spaces (with the homotopy type of a CW-complex, say) and Euler characteristic. However, in these examples the map $r$ factors through $mathbbN$ or $mathbbZ$. I am interested in examples where the values of $r$ are not so restricted.
Question: What categorifications of $mathbbR$ are there where $r$ can take all values in $mathbbR$, or perhaps all values in $(0,infty)$ or $(1,infty)$?
I am especially interested in examples that already appear somewhere in the mathematical literature. I am also especially interested in examples where $mathcalC$ is symmetric monoidal abelian and r(A)+r(C) = r(B) for every short exact sequence $0to Ato Bto Cto 0$.
ct.category-theory categorification
edited Sep 6 at 19:01
Fernando Muro
11.3k13162
asked Sep 6 at 15:31
Richard Hepworth
18115
 |Â
show 1 more comment
up vote
25
down vote
favorite
For the purposes of this question, a categorification of the real numbers is a pair $(mathcalC,r)$ consisting of:
- a symmetric monoidal category $mathcalC$
- a function $rcolon mathrmob(mathcalC)tomathbbR$
such that:
- $r(Xotimes Y) = r(X) r(Y)$ for all objects $X$ and $Y$ of $mathcalC$
- $r(mathbb1) = 1$, where $mathbb1$ is the monoidal unit
- $Xcong X'implies r(X)=r(X')$
Some examples of categorifications of $mathbbR$ are: finite sets and cardinality; finite-dimensional vector spaces and dimension; topological spaces (with the homotopy type of a CW-complex, say) and Euler characteristic. However, in these examples the map $r$ factors through $mathbbN$ or $mathbbZ$. I am interested in examples where the values of $r$ are not so restricted.
Question: What categorifications of $mathbbR$ are there where $r$ can take all values in $mathbbR$, or perhaps all values in $(0,infty)$ or $(1,infty)$?
I am especially interested in examples that already appear somewhere in the mathematical literature. I am also especially interested in examples where $mathcalC$ is symmetric monoidal abelian and r(A)+r(C) = r(B) for every short exact sequence $0to Ato Bto Cto 0$.
ct.category-theory categorification
edited Sep 6 at 19:01
Fernando Muro
11.3k13162
asked Sep 6 at 15:31
Richard Hepworth
18115
1
This page has a number of examples: http://math.ucr.edu/home/baez/counting/
– MTyson
Sep 6 at 18:49
1
en.wikipedia.org/wiki/Continuous_geometry
– Qiaochu Yuan
Sep 6 at 20:56
1
By an underappreciated result of Shannon, this is true (for nonnegative reals) for the capacity of discrete memoryless channels with the usual Kronecker product of channel matrices.
– Steve Huntsman
Sep 7 at 1:25
1
Here is a cheeky answer (sufficiently cheeky I would rather write it as a comment). The Godbillon-Vey invariant of a connected oriented closed 3-manifold equipped with a co-oriented foliation is a top cohomology class in $H^3(M;Bbb R) cong Bbb R$. One may extend this additively to a $Bbb R$-valued invariant for disconnected such manifolds (or, equivalently, say "evaluate the GV class against the fundamental class"). Then the above category (with morphisms the isomorphisms) is symmetric monoidal with disjoint union, and one may take $textexp(textgv(M, mathcal F))$. But this is silly.
– Mike Miller
Sep 7 at 15:42
1
Doesnt the Category of finitely-generated Hilbert $N(G)$-modules for $G$ some fixed infinite group ans $r$ the von Neumann dimension work ?
– Berni Waterman
Sep 7 at 17:47
 |Â
show 1 more comment
up vote
25
down vote
favorite
up vote
25
down vote
favorite
For the purposes of this question, a categorification of the real numbers is a pair $(mathcalC,r)$ consisting of:
- a symmetric monoidal category $mathcalC$
- a function $rcolon mathrmob(mathcalC)tomathbbR$
such that:
- $r(Xotimes Y) = r(X) r(Y)$ for all objects $X$ and $Y$ of $mathcalC$
- $r(mathbb1) = 1$, where $mathbb1$ is the monoidal unit
- $Xcong X'implies r(X)=r(X')$
Some examples of categorifications of $mathbbR$ are: finite sets and cardinality; finite-dimensional vector spaces and dimension; topological spaces (with the homotopy type of a CW-complex, say) and Euler characteristic. However, in these examples the map $r$ factors through $mathbbN$ or $mathbbZ$. I am interested in examples where the values of $r$ are not so restricted.
Question: What categorifications of $mathbbR$ are there where $r$ can take all values in $mathbbR$, or perhaps all values in $(0,infty)$ or $(1,infty)$?
I am especially interested in examples that already appear somewhere in the mathematical literature. I am also especially interested in examples where $mathcalC$ is symmetric monoidal abelian and r(A)+r(C) = r(B) for every short exact sequence $0to Ato Bto Cto 0$.
ct.category-theory categorification
edited Sep 6 at 19:01
Fernando Muro
11.3k13162
asked Sep 6 at 15:31
Richard Hepworth
18115
For the purposes of this question, a categorification of the real numbers is a pair $(mathcalC,r)$ consisting of:
- a symmetric monoidal category $mathcalC$
- a function $rcolon mathrmob(mathcalC)tomathbbR$
such that:
- $r(Xotimes Y) = r(X) r(Y)$ for all objects $X$ and $Y$ of $mathcalC$
- $r(mathbb1) = 1$, where $mathbb1$ is the monoidal unit
- $Xcong X'implies r(X)=r(X')$
Some examples of categorifications of $mathbbR$ are: finite sets and cardinality; finite-dimensional vector spaces and dimension; topological spaces (with the homotopy type of a CW-complex, say) and Euler characteristic. However, in these examples the map $r$ factors through $mathbbN$ or $mathbbZ$. I am interested in examples where the values of $r$ are not so restricted.
Question: What categorifications of $mathbbR$ are there where $r$ can take all values in $mathbbR$, or perhaps all values in $(0,infty)$ or $(1,infty)$?
I am especially interested in examples that already appear somewhere in the mathematical literature. I am also especially interested in examples where $mathcalC$ is symmetric monoidal abelian and r(A)+r(C) = r(B) for every short exact sequence $0to Ato Bto Cto 0$.
ct.category-theory categorification
edited Sep 6 at 19:01
Fernando Muro
11.3k13162
asked Sep 6 at 15:31
Richard Hepworth
18115
edited Sep 6 at 19:01
Fernando Muro
11.3k13162
edited Sep 6 at 19:01
Fernando Muro
11.3k13162
edited Sep 6 at 19:01
Fernando Muro
11.3k13162
11.3k13162
asked Sep 6 at 15:31
Richard Hepworth
18115
asked Sep 6 at 15:31
Richard Hepworth
18115
asked Sep 6 at 15:31
Richard Hepworth
18115
18115
1
This page has a number of examples: http://math.ucr.edu/home/baez/counting/
– MTyson
Sep 6 at 18:49
1
en.wikipedia.org/wiki/Continuous_geometry
– Qiaochu Yuan
Sep 6 at 20:56
1
By an underappreciated result of Shannon, this is true (for nonnegative reals) for the capacity of discrete memoryless channels with the usual Kronecker product of channel matrices.
– Steve Huntsman
Sep 7 at 1:25
1
Here is a cheeky answer (sufficiently cheeky I would rather write it as a comment). The Godbillon-Vey invariant of a connected oriented closed 3-manifold equipped with a co-oriented foliation is a top cohomology class in $H^3(M;Bbb R) cong Bbb R$. One may extend this additively to a $Bbb R$-valued invariant for disconnected such manifolds (or, equivalently, say "evaluate the GV class against the fundamental class"). Then the above category (with morphisms the isomorphisms) is symmetric monoidal with disjoint union, and one may take $textexp(textgv(M, mathcal F))$. But this is silly.
– Mike Miller
Sep 7 at 15:42
1
Doesnt the Category of finitely-generated Hilbert $N(G)$-modules for $G$ some fixed infinite group ans $r$ the von Neumann dimension work ?
– Berni Waterman
Sep 7 at 17:47
 |Â
show 1 more comment
1
This page has a number of examples: http://math.ucr.edu/home/baez/counting/
– MTyson
Sep 6 at 18:49
1
en.wikipedia.org/wiki/Continuous_geometry
– Qiaochu Yuan
Sep 6 at 20:56
1
By an underappreciated result of Shannon, this is true (for nonnegative reals) for the capacity of discrete memoryless channels with the usual Kronecker product of channel matrices.
– Steve Huntsman
Sep 7 at 1:25
1
Here is a cheeky answer (sufficiently cheeky I would rather write it as a comment). The Godbillon-Vey invariant of a connected oriented closed 3-manifold equipped with a co-oriented foliation is a top cohomology class in $H^3(M;Bbb R) cong Bbb R$. One may extend this additively to a $Bbb R$-valued invariant for disconnected such manifolds (or, equivalently, say "evaluate the GV class against the fundamental class"). Then the above category (with morphisms the isomorphisms) is symmetric monoidal with disjoint union, and one may take $textexp(textgv(M, mathcal F))$. But this is silly.
– Mike Miller
Sep 7 at 15:42
1
Doesnt the Category of finitely-generated Hilbert $N(G)$-modules for $G$ some fixed infinite group ans $r$ the von Neumann dimension work ?
– Berni Waterman
Sep 7 at 17:47
1
1
This page has a number of examples: http://math.ucr.edu/home/baez/counting/
– MTyson
Sep 6 at 18:49
This page has a number of examples: http://math.ucr.edu/home/baez/counting/
– MTyson
Sep 6 at 18:49
1
1
en.wikipedia.org/wiki/Continuous_geometry
– Qiaochu Yuan
Sep 6 at 20:56
en.wikipedia.org/wiki/Continuous_geometry
– Qiaochu Yuan
Sep 6 at 20:56
1
1
By an underappreciated result of Shannon, this is true (for nonnegative reals) for the capacity of discrete memoryless channels with the usual Kronecker product of channel matrices.
– Steve Huntsman
Sep 7 at 1:25
By an underappreciated result of Shannon, this is true (for nonnegative reals) for the capacity of discrete memoryless channels with the usual Kronecker product of channel matrices.
– Steve Huntsman
Sep 7 at 1:25
1
1
Here is a cheeky answer (sufficiently cheeky I would rather write it as a comment). The Godbillon-Vey invariant of a connected oriented closed 3-manifold equipped with a co-oriented foliation is a top cohomology class in $H^3(M;Bbb R) cong Bbb R$. One may extend this additively to a $Bbb R$-valued invariant for disconnected such manifolds (or, equivalently, say "evaluate the GV class against the fundamental class"). Then the above category (with morphisms the isomorphisms) is symmetric monoidal with disjoint union, and one may take $textexp(textgv(M, mathcal F))$. But this is silly.
– Mike Miller
Sep 7 at 15:42
Here is a cheeky answer (sufficiently cheeky I would rather write it as a comment). The Godbillon-Vey invariant of a connected oriented closed 3-manifold equipped with a co-oriented foliation is a top cohomology class in $H^3(M;Bbb R) cong Bbb R$. One may extend this additively to a $Bbb R$-valued invariant for disconnected such manifolds (or, equivalently, say "evaluate the GV class against the fundamental class"). Then the above category (with morphisms the isomorphisms) is symmetric monoidal with disjoint union, and one may take $textexp(textgv(M, mathcal F))$. But this is silly.
– Mike Miller
Sep 7 at 15:42
1
1
Doesnt the Category of finitely-generated Hilbert $N(G)$-modules for $G$ some fixed infinite group ans $r$ the von Neumann dimension work ?
– Berni Waterman
Sep 7 at 17:47
Doesnt the Category of finitely-generated Hilbert $N(G)$-modules for $G$ some fixed infinite group ans $r$ the von Neumann dimension work ?
– Berni Waterman
Sep 7 at 17:47
 |Â
show 1 more comment
6 Answers
6
active
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votes
up vote
15
down vote
Janelidze and Street have described such a symmetric monoidal category over the nonnegative real numbers in
- George Janelidze and Ross Street, Real sets, Tbilisi Math. J., Volume 10, Issue 3 (2017), 23-49. (link)
The arXiv version is here.
answered Sep 6 at 15:46
Todd Trimble♦
42.4k5152252
1
Any explanation for the down-vote?
– Todd Trimble♦
Sep 6 at 21:18
8
A coulpe of pointers to the relevant definitions in this paper would be useful. In particular, what is the symmetric monoidal category you are referring to (I presume it's the "series monoids")? And what is the function $r:Ob(mathit C)tomathbb R$, or where is it discussed in the article?
– André Henriques
Sep 6 at 23:14
@AndréHenriques As far as I can tell, the category is $mathrmRSet_g$ (Definition 6.8) and the functor is $ # : mathrmRSet_g rightarrow [0,infty]$ (so not to $mathbbR$ but to the extended reals). It's a "categorification" of Higgs's characterization of $[0,infty]$, as described in Theorem 4.6.
– Robert Furber
Sep 7 at 20:29
add a comment |Â
up vote
13
down vote
If you drop the condition that your category is symmetric monoidal, and content yourself with a category that's merely monoidal, then the tensor category of $R$-$R$-bimodules for $R$ a factor (factor = von Neumann algebra with trivial center) is an example .
The "quantum dimension" or "statistical dimension" of an $R$-$R$-bimodule is an $mathbb R_ge 0cupinfty$-valued invariant with all the properties that you want.
See, e.g., my paper: https://arxiv.org/abs/1110.5671, specifically Proposition 5.2.
answered Sep 6 at 23:43
André Henriques
27.1k483205
add a comment |Â
up vote
11
down vote
An example might be provided by various genera. A genus is not only multiplicative with respect to the Cartesian product of manifolds but also additive with respect to their disjoint sum, and is not only isomorphism but also cobordism invariant. Some genera do not factor through $mathbb Z$ -- for example the $hat A$-genus takes non-integer values on some non-spin manifolds.
I don't know though whether there is a genus which attains all real values. The $Gamma$-genus of Morava (arXiv:1101.1647) seems to have lots of non-rational values but I am too ignorant to say anything trustworthy about it.
edited Sep 6 at 21:30
David Roberts
16.2k460170
answered Sep 6 at 15:56
მáƒÂმუკრჯიბლáƒÂძე
7,298241108
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up vote
11
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In a somewhat similar spirit to მáƒÂმუკრჯიბლáƒÂძე's answer: generalisations of Euler characteristic. As mentioned by the OP, classical Euler characteristic associates an integer to a finite complex in a way that is additive with respect to unions and multiplicative with respect to products. There are variations that apply not just to finite complexes (some are described in The Euler characteristic of a category (arXiv:math/0610260), by Tom Leinster). The extended Euler characteristic may have values that are not integer or even rational (for example, the Euler characteristic of the symmetric groupoid is $e$). But it is still a "ring map".
edited Sep 6 at 21:34
David Roberts
16.2k460170
answered Sep 6 at 19:03
Gregory Arone
4,75412336
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5
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Can we do an example along these lines:
An appropriate collection of metric spaces, with $otimes$ the Cartesian product, and $r$ a fractal dimension? Or more precisely, $r(X) = exp(dim(X))$.
Perhaps the arrows are weakly contracting $d(f(x),f(y) le d(x,y)$. And perhaps we want the finite-dimenaional (so that $dim(X) = infty$ is disallowed) fractals in the sense of Taylor (so that $dim(X times Y) = dim(X)+dim(Y)$).
answered Sep 7 at 1:12
Gerald Edgar
27k269153
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up vote
5
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Via the notion of groupoid cardinality (generalizing ordinary set-theoretic cardinality), finite non-empty groupoids can be seen as a categorification of the positive rationals, and more general groupoids as a categorification of the non-negative reals:
- John Baez and James Dolan, From Finite Sets to Feynman Diagrams, in Mathematics Unlimited—2001 and Beyond, Springer, 2001. (arXiv link)
An example that Baez and Dolan discuss is the groupoid of finite sets and bijections, which has cardinality $e$. (The notion of Euler characteristic of a category, mentioned in Gregory Arone's answer, can be seen as a further generalization of groupoid cardinality, see Example 2.7 of Leinster's article.)
answered Sep 7 at 0:43
Noam Zeilberger
2,1731416
I am sure Richard knows this example/phenomenon and it doesn't seem to address the actual question
– Yemon Choi
Sep 7 at 1:02
Why doesn't it address the question? Here we have $mathcalC = Gpd$ a symmetric monoidal (2-)category and $r : ob(Gpd) to [0,infty]$ given by groupoid cardinality, satisfying the three criteria.
– Noam Zeilberger
Sep 7 at 1:10
moreover, this specifically is an example where $r$ can take all non-negative real values.
– Noam Zeilberger
Sep 7 at 1:16
1
@YemonChoi You can write any nonnegative real as a sum of reciprocals of powers of $2$ via its binary representation. For each $1/2^n$, adjoin $B(mathbbZ/2^nmathbbZ)$ to the groupoid.
– MTyson
Sep 7 at 3:51
1
@მáƒÂმუკáƒÂჯიბლáƒÂძე See Example 2.7 in Leinster's article
– Noam Zeilberger
Sep 7 at 7:07
 |Â
show 3 more comments
6 Answers
6
active
oldest
votes
6 Answers
6
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
15
down vote
Janelidze and Street have described such a symmetric monoidal category over the nonnegative real numbers in
- George Janelidze and Ross Street, Real sets, Tbilisi Math. J., Volume 10, Issue 3 (2017), 23-49. (link)
The arXiv version is here.
answered Sep 6 at 15:46
Todd Trimble♦
42.4k5152252
1
Any explanation for the down-vote?
– Todd Trimble♦
Sep 6 at 21:18
8
A coulpe of pointers to the relevant definitions in this paper would be useful. In particular, what is the symmetric monoidal category you are referring to (I presume it's the "series monoids")? And what is the function $r:Ob(mathit C)tomathbb R$, or where is it discussed in the article?
– André Henriques
Sep 6 at 23:14
@AndréHenriques As far as I can tell, the category is $mathrmRSet_g$ (Definition 6.8) and the functor is $ # : mathrmRSet_g rightarrow [0,infty]$ (so not to $mathbbR$ but to the extended reals). It's a "categorification" of Higgs's characterization of $[0,infty]$, as described in Theorem 4.6.
– Robert Furber
Sep 7 at 20:29
add a comment |Â
up vote
15
down vote
Janelidze and Street have described such a symmetric monoidal category over the nonnegative real numbers in
- George Janelidze and Ross Street, Real sets, Tbilisi Math. J., Volume 10, Issue 3 (2017), 23-49. (link)
The arXiv version is here.
answered Sep 6 at 15:46
Todd Trimble♦
42.4k5152252
1
Any explanation for the down-vote?
– Todd Trimble♦
Sep 6 at 21:18
8
A coulpe of pointers to the relevant definitions in this paper would be useful. In particular, what is the symmetric monoidal category you are referring to (I presume it's the "series monoids")? And what is the function $r:Ob(mathit C)tomathbb R$, or where is it discussed in the article?
– André Henriques
Sep 6 at 23:14
@AndréHenriques As far as I can tell, the category is $mathrmRSet_g$ (Definition 6.8) and the functor is $ # : mathrmRSet_g rightarrow [0,infty]$ (so not to $mathbbR$ but to the extended reals). It's a "categorification" of Higgs's characterization of $[0,infty]$, as described in Theorem 4.6.
– Robert Furber
Sep 7 at 20:29
add a comment |Â
up vote
15
down vote
up vote
15
down vote
Janelidze and Street have described such a symmetric monoidal category over the nonnegative real numbers in
- George Janelidze and Ross Street, Real sets, Tbilisi Math. J., Volume 10, Issue 3 (2017), 23-49. (link)
The arXiv version is here.
answered Sep 6 at 15:46
Todd Trimble♦
42.4k5152252
Janelidze and Street have described such a symmetric monoidal category over the nonnegative real numbers in
- George Janelidze and Ross Street, Real sets, Tbilisi Math. J., Volume 10, Issue 3 (2017), 23-49. (link)
The arXiv version is here.
answered Sep 6 at 15:46
Todd Trimble♦
42.4k5152252
answered Sep 6 at 15:46
Todd Trimble♦
42.4k5152252
answered Sep 6 at 15:46
Todd Trimble♦
42.4k5152252
answered Sep 6 at 15:46
Todd Trimble♦
42.4k5152252
42.4k5152252
1
Any explanation for the down-vote?
– Todd Trimble♦
Sep 6 at 21:18
8
A coulpe of pointers to the relevant definitions in this paper would be useful. In particular, what is the symmetric monoidal category you are referring to (I presume it's the "series monoids")? And what is the function $r:Ob(mathit C)tomathbb R$, or where is it discussed in the article?
– André Henriques
Sep 6 at 23:14
@AndréHenriques As far as I can tell, the category is $mathrmRSet_g$ (Definition 6.8) and the functor is $ # : mathrmRSet_g rightarrow [0,infty]$ (so not to $mathbbR$ but to the extended reals). It's a "categorification" of Higgs's characterization of $[0,infty]$, as described in Theorem 4.6.
– Robert Furber
Sep 7 at 20:29
add a comment |Â
1
Any explanation for the down-vote?
– Todd Trimble♦
Sep 6 at 21:18
8
A coulpe of pointers to the relevant definitions in this paper would be useful. In particular, what is the symmetric monoidal category you are referring to (I presume it's the "series monoids")? And what is the function $r:Ob(mathit C)tomathbb R$, or where is it discussed in the article?
– André Henriques
Sep 6 at 23:14
@AndréHenriques As far as I can tell, the category is $mathrmRSet_g$ (Definition 6.8) and the functor is $ # : mathrmRSet_g rightarrow [0,infty]$ (so not to $mathbbR$ but to the extended reals). It's a "categorification" of Higgs's characterization of $[0,infty]$, as described in Theorem 4.6.
– Robert Furber
Sep 7 at 20:29
1
1
Any explanation for the down-vote?
– Todd Trimble♦
Sep 6 at 21:18
Any explanation for the down-vote?
– Todd Trimble♦
Sep 6 at 21:18
8
8
A coulpe of pointers to the relevant definitions in this paper would be useful. In particular, what is the symmetric monoidal category you are referring to (I presume it's the "series monoids")? And what is the function $r:Ob(mathit C)tomathbb R$, or where is it discussed in the article?
– André Henriques
Sep 6 at 23:14
A coulpe of pointers to the relevant definitions in this paper would be useful. In particular, what is the symmetric monoidal category you are referring to (I presume it's the "series monoids")? And what is the function $r:Ob(mathit C)tomathbb R$, or where is it discussed in the article?
– André Henriques
Sep 6 at 23:14
@AndréHenriques As far as I can tell, the category is $mathrmRSet_g$ (Definition 6.8) and the functor is $ # : mathrmRSet_g rightarrow [0,infty]$ (so not to $mathbbR$ but to the extended reals). It's a "categorification" of Higgs's characterization of $[0,infty]$, as described in Theorem 4.6.
– Robert Furber
Sep 7 at 20:29
@AndréHenriques As far as I can tell, the category is $mathrmRSet_g$ (Definition 6.8) and the functor is $ # : mathrmRSet_g rightarrow [0,infty]$ (so not to $mathbbR$ but to the extended reals). It's a "categorification" of Higgs's characterization of $[0,infty]$, as described in Theorem 4.6.
– Robert Furber
Sep 7 at 20:29
add a comment |Â
up vote
13
down vote
If you drop the condition that your category is symmetric monoidal, and content yourself with a category that's merely monoidal, then the tensor category of $R$-$R$-bimodules for $R$ a factor (factor = von Neumann algebra with trivial center) is an example .
The "quantum dimension" or "statistical dimension" of an $R$-$R$-bimodule is an $mathbb R_ge 0cupinfty$-valued invariant with all the properties that you want.
See, e.g., my paper: https://arxiv.org/abs/1110.5671, specifically Proposition 5.2.
answered Sep 6 at 23:43
André Henriques
27.1k483205
add a comment |Â
up vote
13
down vote
If you drop the condition that your category is symmetric monoidal, and content yourself with a category that's merely monoidal, then the tensor category of $R$-$R$-bimodules for $R$ a factor (factor = von Neumann algebra with trivial center) is an example .
The "quantum dimension" or "statistical dimension" of an $R$-$R$-bimodule is an $mathbb R_ge 0cupinfty$-valued invariant with all the properties that you want.
See, e.g., my paper: https://arxiv.org/abs/1110.5671, specifically Proposition 5.2.
answered Sep 6 at 23:43
André Henriques
27.1k483205
add a comment |Â
up vote
13
down vote
up vote
13
down vote
If you drop the condition that your category is symmetric monoidal, and content yourself with a category that's merely monoidal, then the tensor category of $R$-$R$-bimodules for $R$ a factor (factor = von Neumann algebra with trivial center) is an example .
The "quantum dimension" or "statistical dimension" of an $R$-$R$-bimodule is an $mathbb R_ge 0cupinfty$-valued invariant with all the properties that you want.
See, e.g., my paper: https://arxiv.org/abs/1110.5671, specifically Proposition 5.2.
answered Sep 6 at 23:43
André Henriques
27.1k483205
If you drop the condition that your category is symmetric monoidal, and content yourself with a category that's merely monoidal, then the tensor category of $R$-$R$-bimodules for $R$ a factor (factor = von Neumann algebra with trivial center) is an example .
The "quantum dimension" or "statistical dimension" of an $R$-$R$-bimodule is an $mathbb R_ge 0cupinfty$-valued invariant with all the properties that you want.
See, e.g., my paper: https://arxiv.org/abs/1110.5671, specifically Proposition 5.2.
answered Sep 6 at 23:43
André Henriques
27.1k483205
edited Sep 7 at 22:18
answered Sep 6 at 23:43
André Henriques
27.1k483205
answered Sep 6 at 23:43
André Henriques
27.1k483205
answered Sep 6 at 23:43
André Henriques
27.1k483205
27.1k483205
add a comment |Â
add a comment |Â
up vote
11
down vote
An example might be provided by various genera. A genus is not only multiplicative with respect to the Cartesian product of manifolds but also additive with respect to their disjoint sum, and is not only isomorphism but also cobordism invariant. Some genera do not factor through $mathbb Z$ -- for example the $hat A$-genus takes non-integer values on some non-spin manifolds.
I don't know though whether there is a genus which attains all real values. The $Gamma$-genus of Morava (arXiv:1101.1647) seems to have lots of non-rational values but I am too ignorant to say anything trustworthy about it.
edited Sep 6 at 21:30
David Roberts
16.2k460170
answered Sep 6 at 15:56
მáƒÂმუკრჯიბლáƒÂძე
7,298241108
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11
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An example might be provided by various genera. A genus is not only multiplicative with respect to the Cartesian product of manifolds but also additive with respect to their disjoint sum, and is not only isomorphism but also cobordism invariant. Some genera do not factor through $mathbb Z$ -- for example the $hat A$-genus takes non-integer values on some non-spin manifolds.
I don't know though whether there is a genus which attains all real values. The $Gamma$-genus of Morava (arXiv:1101.1647) seems to have lots of non-rational values but I am too ignorant to say anything trustworthy about it.
edited Sep 6 at 21:30
David Roberts
16.2k460170
answered Sep 6 at 15:56
მáƒÂმუკრჯიბლáƒÂძე
7,298241108
add a comment |Â
up vote
11
down vote
up vote
11
down vote
An example might be provided by various genera. A genus is not only multiplicative with respect to the Cartesian product of manifolds but also additive with respect to their disjoint sum, and is not only isomorphism but also cobordism invariant. Some genera do not factor through $mathbb Z$ -- for example the $hat A$-genus takes non-integer values on some non-spin manifolds.
I don't know though whether there is a genus which attains all real values. The $Gamma$-genus of Morava (arXiv:1101.1647) seems to have lots of non-rational values but I am too ignorant to say anything trustworthy about it.
edited Sep 6 at 21:30
David Roberts
16.2k460170
answered Sep 6 at 15:56
მáƒÂმუკრჯიბლáƒÂძე
7,298241108
An example might be provided by various genera. A genus is not only multiplicative with respect to the Cartesian product of manifolds but also additive with respect to their disjoint sum, and is not only isomorphism but also cobordism invariant. Some genera do not factor through $mathbb Z$ -- for example the $hat A$-genus takes non-integer values on some non-spin manifolds.
I don't know though whether there is a genus which attains all real values. The $Gamma$-genus of Morava (arXiv:1101.1647) seems to have lots of non-rational values but I am too ignorant to say anything trustworthy about it.
edited Sep 6 at 21:30
David Roberts
16.2k460170
answered Sep 6 at 15:56
მáƒÂმუკრჯიბლáƒÂძე
7,298241108
edited Sep 6 at 21:30
David Roberts
16.2k460170
edited Sep 6 at 21:30
David Roberts
16.2k460170
edited Sep 6 at 21:30
David Roberts
16.2k460170
16.2k460170
answered Sep 6 at 15:56
მáƒÂმუკრჯიბლáƒÂძე
7,298241108
answered Sep 6 at 15:56
მáƒÂმუკრჯიბლáƒÂძე
7,298241108
answered Sep 6 at 15:56
მáƒÂმუკრჯიბლáƒÂძე
7,298241108
7,298241108
add a comment |Â
add a comment |Â
up vote
11
down vote
In a somewhat similar spirit to მáƒÂმუკრჯიბლáƒÂძე's answer: generalisations of Euler characteristic. As mentioned by the OP, classical Euler characteristic associates an integer to a finite complex in a way that is additive with respect to unions and multiplicative with respect to products. There are variations that apply not just to finite complexes (some are described in The Euler characteristic of a category (arXiv:math/0610260), by Tom Leinster). The extended Euler characteristic may have values that are not integer or even rational (for example, the Euler characteristic of the symmetric groupoid is $e$). But it is still a "ring map".
edited Sep 6 at 21:34
David Roberts
16.2k460170
answered Sep 6 at 19:03
Gregory Arone
4,75412336
add a comment |Â
up vote
11
down vote
In a somewhat similar spirit to მáƒÂმუკრჯიბლáƒÂძე's answer: generalisations of Euler characteristic. As mentioned by the OP, classical Euler characteristic associates an integer to a finite complex in a way that is additive with respect to unions and multiplicative with respect to products. There are variations that apply not just to finite complexes (some are described in The Euler characteristic of a category (arXiv:math/0610260), by Tom Leinster). The extended Euler characteristic may have values that are not integer or even rational (for example, the Euler characteristic of the symmetric groupoid is $e$). But it is still a "ring map".
edited Sep 6 at 21:34
David Roberts
16.2k460170
answered Sep 6 at 19:03
Gregory Arone
4,75412336
add a comment |Â
up vote
11
down vote
up vote
11
down vote
In a somewhat similar spirit to მáƒÂმუკრჯიბლáƒÂძე's answer: generalisations of Euler characteristic. As mentioned by the OP, classical Euler characteristic associates an integer to a finite complex in a way that is additive with respect to unions and multiplicative with respect to products. There are variations that apply not just to finite complexes (some are described in The Euler characteristic of a category (arXiv:math/0610260), by Tom Leinster). The extended Euler characteristic may have values that are not integer or even rational (for example, the Euler characteristic of the symmetric groupoid is $e$). But it is still a "ring map".
edited Sep 6 at 21:34
David Roberts
16.2k460170
answered Sep 6 at 19:03
Gregory Arone
4,75412336
In a somewhat similar spirit to მáƒÂმუკრჯიბლáƒÂძე's answer: generalisations of Euler characteristic. As mentioned by the OP, classical Euler characteristic associates an integer to a finite complex in a way that is additive with respect to unions and multiplicative with respect to products. There are variations that apply not just to finite complexes (some are described in The Euler characteristic of a category (arXiv:math/0610260), by Tom Leinster). The extended Euler characteristic may have values that are not integer or even rational (for example, the Euler characteristic of the symmetric groupoid is $e$). But it is still a "ring map".
edited Sep 6 at 21:34
David Roberts
16.2k460170
answered Sep 6 at 19:03
Gregory Arone
4,75412336
edited Sep 6 at 21:34
David Roberts
16.2k460170
edited Sep 6 at 21:34
David Roberts
16.2k460170
edited Sep 6 at 21:34
David Roberts
16.2k460170
16.2k460170
answered Sep 6 at 19:03
Gregory Arone
4,75412336
answered Sep 6 at 19:03
Gregory Arone
4,75412336
answered Sep 6 at 19:03
Gregory Arone
4,75412336
4,75412336
add a comment |Â
add a comment |Â
up vote
5
down vote
Can we do an example along these lines:
An appropriate collection of metric spaces, with $otimes$ the Cartesian product, and $r$ a fractal dimension? Or more precisely, $r(X) = exp(dim(X))$.
Perhaps the arrows are weakly contracting $d(f(x),f(y) le d(x,y)$. And perhaps we want the finite-dimenaional (so that $dim(X) = infty$ is disallowed) fractals in the sense of Taylor (so that $dim(X times Y) = dim(X)+dim(Y)$).
answered Sep 7 at 1:12
Gerald Edgar
27k269153
add a comment |Â
up vote
5
down vote
Can we do an example along these lines:
An appropriate collection of metric spaces, with $otimes$ the Cartesian product, and $r$ a fractal dimension? Or more precisely, $r(X) = exp(dim(X))$.
Perhaps the arrows are weakly contracting $d(f(x),f(y) le d(x,y)$. And perhaps we want the finite-dimenaional (so that $dim(X) = infty$ is disallowed) fractals in the sense of Taylor (so that $dim(X times Y) = dim(X)+dim(Y)$).
answered Sep 7 at 1:12
Gerald Edgar
27k269153
add a comment |Â
up vote
5
down vote
up vote
5
down vote
Can we do an example along these lines:
An appropriate collection of metric spaces, with $otimes$ the Cartesian product, and $r$ a fractal dimension? Or more precisely, $r(X) = exp(dim(X))$.
Perhaps the arrows are weakly contracting $d(f(x),f(y) le d(x,y)$. And perhaps we want the finite-dimenaional (so that $dim(X) = infty$ is disallowed) fractals in the sense of Taylor (so that $dim(X times Y) = dim(X)+dim(Y)$).
answered Sep 7 at 1:12
Gerald Edgar
27k269153
Can we do an example along these lines:
An appropriate collection of metric spaces, with $otimes$ the Cartesian product, and $r$ a fractal dimension? Or more precisely, $r(X) = exp(dim(X))$.
Perhaps the arrows are weakly contracting $d(f(x),f(y) le d(x,y)$. And perhaps we want the finite-dimenaional (so that $dim(X) = infty$ is disallowed) fractals in the sense of Taylor (so that $dim(X times Y) = dim(X)+dim(Y)$).
answered Sep 7 at 1:12
Gerald Edgar
27k269153
answered Sep 7 at 1:12
Gerald Edgar
27k269153
answered Sep 7 at 1:12
Gerald Edgar
27k269153
answered Sep 7 at 1:12
Gerald Edgar
27k269153
27k269153
add a comment |Â
add a comment |Â
up vote
5
down vote
Via the notion of groupoid cardinality (generalizing ordinary set-theoretic cardinality), finite non-empty groupoids can be seen as a categorification of the positive rationals, and more general groupoids as a categorification of the non-negative reals:
- John Baez and James Dolan, From Finite Sets to Feynman Diagrams, in Mathematics Unlimited—2001 and Beyond, Springer, 2001. (arXiv link)
An example that Baez and Dolan discuss is the groupoid of finite sets and bijections, which has cardinality $e$. (The notion of Euler characteristic of a category, mentioned in Gregory Arone's answer, can be seen as a further generalization of groupoid cardinality, see Example 2.7 of Leinster's article.)
answered Sep 7 at 0:43
Noam Zeilberger
2,1731416
I am sure Richard knows this example/phenomenon and it doesn't seem to address the actual question
– Yemon Choi
Sep 7 at 1:02
Why doesn't it address the question? Here we have $mathcalC = Gpd$ a symmetric monoidal (2-)category and $r : ob(Gpd) to [0,infty]$ given by groupoid cardinality, satisfying the three criteria.
– Noam Zeilberger
Sep 7 at 1:10
moreover, this specifically is an example where $r$ can take all non-negative real values.
– Noam Zeilberger
Sep 7 at 1:16
1
@YemonChoi You can write any nonnegative real as a sum of reciprocals of powers of $2$ via its binary representation. For each $1/2^n$, adjoin $B(mathbbZ/2^nmathbbZ)$ to the groupoid.
– MTyson
Sep 7 at 3:51
1
@მáƒÂმუკáƒÂჯიბლáƒÂძე See Example 2.7 in Leinster's article
– Noam Zeilberger
Sep 7 at 7:07
 |Â
show 3 more comments
up vote
5
down vote
Via the notion of groupoid cardinality (generalizing ordinary set-theoretic cardinality), finite non-empty groupoids can be seen as a categorification of the positive rationals, and more general groupoids as a categorification of the non-negative reals:
- John Baez and James Dolan, From Finite Sets to Feynman Diagrams, in Mathematics Unlimited—2001 and Beyond, Springer, 2001. (arXiv link)
An example that Baez and Dolan discuss is the groupoid of finite sets and bijections, which has cardinality $e$. (The notion of Euler characteristic of a category, mentioned in Gregory Arone's answer, can be seen as a further generalization of groupoid cardinality, see Example 2.7 of Leinster's article.)
answered Sep 7 at 0:43
Noam Zeilberger
2,1731416
I am sure Richard knows this example/phenomenon and it doesn't seem to address the actual question
– Yemon Choi
Sep 7 at 1:02
Why doesn't it address the question? Here we have $mathcalC = Gpd$ a symmetric monoidal (2-)category and $r : ob(Gpd) to [0,infty]$ given by groupoid cardinality, satisfying the three criteria.
– Noam Zeilberger
Sep 7 at 1:10
moreover, this specifically is an example where $r$ can take all non-negative real values.
– Noam Zeilberger
Sep 7 at 1:16
1
@YemonChoi You can write any nonnegative real as a sum of reciprocals of powers of $2$ via its binary representation. For each $1/2^n$, adjoin $B(mathbbZ/2^nmathbbZ)$ to the groupoid.
– MTyson
Sep 7 at 3:51
1
@მáƒÂმუკáƒÂჯიბლáƒÂძე See Example 2.7 in Leinster's article
– Noam Zeilberger
Sep 7 at 7:07
 |Â
show 3 more comments
up vote
5
down vote
up vote
5
down vote
Via the notion of groupoid cardinality (generalizing ordinary set-theoretic cardinality), finite non-empty groupoids can be seen as a categorification of the positive rationals, and more general groupoids as a categorification of the non-negative reals:
- John Baez and James Dolan, From Finite Sets to Feynman Diagrams, in Mathematics Unlimited—2001 and Beyond, Springer, 2001. (arXiv link)
An example that Baez and Dolan discuss is the groupoid of finite sets and bijections, which has cardinality $e$. (The notion of Euler characteristic of a category, mentioned in Gregory Arone's answer, can be seen as a further generalization of groupoid cardinality, see Example 2.7 of Leinster's article.)
answered Sep 7 at 0:43
Noam Zeilberger
2,1731416
Via the notion of groupoid cardinality (generalizing ordinary set-theoretic cardinality), finite non-empty groupoids can be seen as a categorification of the positive rationals, and more general groupoids as a categorification of the non-negative reals:
- John Baez and James Dolan, From Finite Sets to Feynman Diagrams, in Mathematics Unlimited—2001 and Beyond, Springer, 2001. (arXiv link)
An example that Baez and Dolan discuss is the groupoid of finite sets and bijections, which has cardinality $e$. (The notion of Euler characteristic of a category, mentioned in Gregory Arone's answer, can be seen as a further generalization of groupoid cardinality, see Example 2.7 of Leinster's article.)
answered Sep 7 at 0:43
Noam Zeilberger
2,1731416
edited Sep 7 at 7:24
answered Sep 7 at 0:43
Noam Zeilberger
2,1731416
answered Sep 7 at 0:43
Noam Zeilberger
2,1731416
answered Sep 7 at 0:43
Noam Zeilberger
2,1731416
2,1731416
I am sure Richard knows this example/phenomenon and it doesn't seem to address the actual question
– Yemon Choi
Sep 7 at 1:02
Why doesn't it address the question? Here we have $mathcalC = Gpd$ a symmetric monoidal (2-)category and $r : ob(Gpd) to [0,infty]$ given by groupoid cardinality, satisfying the three criteria.
– Noam Zeilberger
Sep 7 at 1:10
moreover, this specifically is an example where $r$ can take all non-negative real values.
– Noam Zeilberger
Sep 7 at 1:16
1
@YemonChoi You can write any nonnegative real as a sum of reciprocals of powers of $2$ via its binary representation. For each $1/2^n$, adjoin $B(mathbbZ/2^nmathbbZ)$ to the groupoid.
– MTyson
Sep 7 at 3:51
1
@მáƒÂმუკáƒÂჯიბლáƒÂძე See Example 2.7 in Leinster's article
– Noam Zeilberger
Sep 7 at 7:07
 |Â
show 3 more comments
I am sure Richard knows this example/phenomenon and it doesn't seem to address the actual question
– Yemon Choi
Sep 7 at 1:02
Why doesn't it address the question? Here we have $mathcalC = Gpd$ a symmetric monoidal (2-)category and $r : ob(Gpd) to [0,infty]$ given by groupoid cardinality, satisfying the three criteria.
– Noam Zeilberger
Sep 7 at 1:10
moreover, this specifically is an example where $r$ can take all non-negative real values.
– Noam Zeilberger
Sep 7 at 1:16
1
@YemonChoi You can write any nonnegative real as a sum of reciprocals of powers of $2$ via its binary representation. For each $1/2^n$, adjoin $B(mathbbZ/2^nmathbbZ)$ to the groupoid.
– MTyson
Sep 7 at 3:51
1
@მáƒÂმუკáƒÂჯიბლáƒÂძე See Example 2.7 in Leinster's article
– Noam Zeilberger
Sep 7 at 7:07
I am sure Richard knows this example/phenomenon and it doesn't seem to address the actual question
– Yemon Choi
Sep 7 at 1:02
I am sure Richard knows this example/phenomenon and it doesn't seem to address the actual question
– Yemon Choi
Sep 7 at 1:02
Why doesn't it address the question? Here we have $mathcalC = Gpd$ a symmetric monoidal (2-)category and $r : ob(Gpd) to [0,infty]$ given by groupoid cardinality, satisfying the three criteria.
– Noam Zeilberger
Sep 7 at 1:10
Why doesn't it address the question? Here we have $mathcalC = Gpd$ a symmetric monoidal (2-)category and $r : ob(Gpd) to [0,infty]$ given by groupoid cardinality, satisfying the three criteria.
– Noam Zeilberger
Sep 7 at 1:10
moreover, this specifically is an example where $r$ can take all non-negative real values.
– Noam Zeilberger
Sep 7 at 1:16
moreover, this specifically is an example where $r$ can take all non-negative real values.
– Noam Zeilberger
Sep 7 at 1:16
1
1
@YemonChoi You can write any nonnegative real as a sum of reciprocals of powers of $2$ via its binary representation. For each $1/2^n$, adjoin $B(mathbbZ/2^nmathbbZ)$ to the groupoid.
– MTyson
Sep 7 at 3:51
@YemonChoi You can write any nonnegative real as a sum of reciprocals of powers of $2$ via its binary representation. For each $1/2^n$, adjoin $B(mathbbZ/2^nmathbbZ)$ to the groupoid.
– MTyson
Sep 7 at 3:51
1
1
@მáƒÂმუკáƒÂჯიბლáƒÂძე See Example 2.7 in Leinster's article
– Noam Zeilberger
Sep 7 at 7:07
@მáƒÂმუკáƒÂჯიბლáƒÂძე See Example 2.7 in Leinster's article
– Noam Zeilberger
Sep 7 at 7:07
 |Â
show 3 more comments
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1
This page has a number of examples: http://math.ucr.edu/home/baez/counting/
– MTyson
Sep 6 at 18:49
1
en.wikipedia.org/wiki/Continuous_geometry
– Qiaochu Yuan
Sep 6 at 20:56
1
By an underappreciated result of Shannon, this is true (for nonnegative reals) for the capacity of discrete memoryless channels with the usual Kronecker product of channel matrices.
– Steve Huntsman
Sep 7 at 1:25
1
Here is a cheeky answer (sufficiently cheeky I would rather write it as a comment). The Godbillon-Vey invariant of a connected oriented closed 3-manifold equipped with a co-oriented foliation is a top cohomology class in $H^3(M;Bbb R) cong Bbb R$. One may extend this additively to a $Bbb R$-valued invariant for disconnected such manifolds (or, equivalently, say "evaluate the GV class against the fundamental class"). Then the above category (with morphisms the isomorphisms) is symmetric monoidal with disjoint union, and one may take $textexp(textgv(M, mathcal F))$. But this is silly.
– Mike Miller
Sep 7 at 15:42
1
Doesnt the Category of finitely-generated Hilbert $N(G)$-modules for $G$ some fixed infinite group ans $r$ the von Neumann dimension work ?
– Berni Waterman
Sep 7 at 17:47