Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction...
Transcript of Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction...
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Commutative Algebra of Categories
John D. Berman
University of Virginia
September 22, 2017
John D. Berman Commutative Algebra of Categories
![Page 2: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/2.jpg)
K Theory of categories
C is an (8-)category.
C iso is an (8-)groupoid (space).
Example 1
C Fin, C iso Finiso ²
n BΣn
C iso inherits extra structure from C.
Example 2
If C` is symmetric monoidal, C iso inherits E8-space structure.
Example 3²n BΣn inherits two E8-space structures from >,.
John D. Berman Commutative Algebra of Categories
![Page 3: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/3.jpg)
K Theory of categories
C is an (8-)category.C iso is an (8-)groupoid (space).
Example 1
C Fin, C iso Finiso ²
n BΣn
C iso inherits extra structure from C.
Example 2
If C` is symmetric monoidal, C iso inherits E8-space structure.
Example 3²n BΣn inherits two E8-space structures from >,.
John D. Berman Commutative Algebra of Categories
![Page 4: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/4.jpg)
K Theory of categories
C is an (8-)category.C iso is an (8-)groupoid (space).
Example 1
C Fin, C iso Finiso ²
n BΣn
C iso inherits extra structure from C.
Example 2
If C` is symmetric monoidal, C iso inherits E8-space structure.
Example 3²n BΣn inherits two E8-space structures from >,.
John D. Berman Commutative Algebra of Categories
![Page 5: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/5.jpg)
K Theory of categories
C is an (8-)category.C iso is an (8-)groupoid (space).
Example 1
C Fin, C iso Finiso ²
n BΣn
C iso inherits extra structure from C.
Example 2
If C` is symmetric monoidal, C iso inherits E8-space structure.
Example 3²n BΣn inherits two E8-space structures from >,.
John D. Berman Commutative Algebra of Categories
![Page 6: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/6.jpg)
K Theory of categories
C is an (8-)category.C iso is an (8-)groupoid (space).
Example 1
C Fin, C iso Finiso ²
n BΣn
C iso inherits extra structure from C.
Example 2
If C` is symmetric monoidal, C iso inherits E8-space structure.
Example 3²n BΣn inherits two E8-space structures from >,.
John D. Berman Commutative Algebra of Categories
![Page 7: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/7.jpg)
K Theory of categories
C is an (8-)category.C iso is an (8-)groupoid (space).
Example 1
C Fin, C iso Finiso ²
n BΣn
C iso inherits extra structure from C.
Example 2
If C` is symmetric monoidal, C iso inherits E8-space structure.
Example 3²n BΣn inherits two E8-space structures from >,.
John D. Berman Commutative Algebra of Categories
![Page 8: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/8.jpg)
K Theory of categories
An E8-space X is grouplike if the commutative monoid π0pX q isan abelian group.
Theorem 4
Ω8 : Sp Ñ E8Top determines an equivalence
E8Topgp Sp¥0.
K pC`q ‘group completion’ of the E8-space C iso (a spectrum).
John D. Berman Commutative Algebra of Categories
![Page 9: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/9.jpg)
K Theory of categories
An E8-space X is grouplike if the commutative monoid π0pX q isan abelian group.
Theorem 4
Ω8 : Sp Ñ E8Top determines an equivalence
E8Topgp Sp¥0.
K pC`q ‘group completion’ of the E8-space C iso (a spectrum).
John D. Berman Commutative Algebra of Categories
![Page 10: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/10.jpg)
K Theory of categories
An E8-space X is grouplike if the commutative monoid π0pX q isan abelian group.
Theorem 4
Ω8 : Sp Ñ E8Top determines an equivalence
E8Topgp Sp¥0.
K pC`q ‘group completion’ of the E8-space C iso (a spectrum).
John D. Berman Commutative Algebra of Categories
![Page 11: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/11.jpg)
Examples
Example 5
Perfect modules over a ring spectrum: C` Modperf,`R
K pC`q K pRq (definition of higher algebraic K-theory)
Example 6
Finite sets: C` Fin>
K pC`q S (Barratt-Priddy-Quillen theorem)
John D. Berman Commutative Algebra of Categories
![Page 12: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/12.jpg)
Examples
Example 5
Perfect modules over a ring spectrum: C` Modperf,`R
K pC`q K pRq (definition of higher algebraic K-theory)
Example 6
Finite sets: C` Fin>
K pC`q S (Barratt-Priddy-Quillen theorem)
John D. Berman Commutative Algebra of Categories
![Page 13: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/13.jpg)
Examples
Example 5
Perfect modules over a ring spectrum: C` Modperf,`R
K pC`q K pRq (definition of higher algebraic K-theory)
Example 6
Finite sets: C` Fin>
K pC`q S (Barratt-Priddy-Quillen theorem)
John D. Berman Commutative Algebra of Categories
![Page 14: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/14.jpg)
Examples
Example 5
Perfect modules over a ring spectrum: C` Modperf,`R
K pC`q K pRq (definition of higher algebraic K-theory)
Example 6
Finite sets: C` Fin>
K pC`q S (Barratt-Priddy-Quillen theorem)
John D. Berman Commutative Algebra of Categories
![Page 15: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/15.jpg)
Ring structure
In each case, C` is a ‘commutative semiring (8-)category’:
C has a second symmetric monoidal operation b;
b distributes over `.
K pC`q should inherit the structure of an E8-ring spectrum.
Obstacles to making this precise:
1 What is a ‘semiring (8-)category’?
2 What is ‘group completion’?
John D. Berman Commutative Algebra of Categories
![Page 16: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/16.jpg)
Ring structure
In each case, C` is a ‘commutative semiring (8-)category’:
C has a second symmetric monoidal operation b;
b distributes over `.
K pC`q should inherit the structure of an E8-ring spectrum.
Obstacles to making this precise:
1 What is a ‘semiring (8-)category’?
2 What is ‘group completion’?
John D. Berman Commutative Algebra of Categories
![Page 17: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/17.jpg)
Ring structure
In each case, C` is a ‘commutative semiring (8-)category’:
C has a second symmetric monoidal operation b;
b distributes over `.
K pC`q should inherit the structure of an E8-ring spectrum.
Obstacles to making this precise:
1 What is a ‘semiring (8-)category’?
2 What is ‘group completion’?
John D. Berman Commutative Algebra of Categories
![Page 18: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/18.jpg)
Ring structure
In each case, C` is a ‘commutative semiring (8-)category’:
C has a second symmetric monoidal operation b;
b distributes over `.
K pC`q should inherit the structure of an E8-ring spectrum.
Obstacles to making this precise:
1 What is a ‘semiring (8-)category’?
2 What is ‘group completion’?
John D. Berman Commutative Algebra of Categories
![Page 19: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/19.jpg)
Ring structure
In each case, C` is a ‘commutative semiring (8-)category’:
C has a second symmetric monoidal operation b;
b distributes over `.
K pC`q should inherit the structure of an E8-ring spectrum.
Obstacles to making this precise:
1 What is a ‘semiring (8-)category’?
2 What is ‘group completion’?
John D. Berman Commutative Algebra of Categories
![Page 20: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/20.jpg)
Ring structure
In each case, C` is a ‘commutative semiring (8-)category’:
C has a second symmetric monoidal operation b;
b distributes over `.
K pC`q should inherit the structure of an E8-ring spectrum.
Obstacles to making this precise:
1 What is a ‘semiring (8-)category’?
2 What is ‘group completion’?
John D. Berman Commutative Algebra of Categories
![Page 21: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/21.jpg)
Ring structure
In each case, C` is a ‘commutative semiring (8-)category’:
C has a second symmetric monoidal operation b;
b distributes over `.
K pC`q should inherit the structure of an E8-ring spectrum.
Obstacles to making this precise:
1 What is a ‘semiring (8-)category’?
2 What is ‘group completion’?
John D. Berman Commutative Algebra of Categories
![Page 22: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/22.jpg)
Ring structure
Obstacles to making this precise:
1 What is a ‘semiring (8-)category’?
2 What is ‘group completion’?
Traditional answers (70’s):
1 bipermutative categories
2 Quillen Q construction, Waldhausen S construction
Alternative: categorify ordinary semirings and group completion!
John D. Berman Commutative Algebra of Categories
![Page 23: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/23.jpg)
Ring structure
Obstacles to making this precise:
1 What is a ‘semiring (8-)category’?
2 What is ‘group completion’?
Traditional answers (70’s):
1 bipermutative categories
2 Quillen Q construction, Waldhausen S construction
Alternative: categorify ordinary semirings and group completion!
John D. Berman Commutative Algebra of Categories
![Page 24: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/24.jpg)
Ring structure
Obstacles to making this precise:
1 What is a ‘semiring (8-)category’?
2 What is ‘group completion’?
Traditional answers (70’s):
1 bipermutative categories
2 Quillen Q construction, Waldhausen S construction
Alternative: categorify ordinary semirings and group completion!
John D. Berman Commutative Algebra of Categories
![Page 25: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/25.jpg)
Ring structure
Obstacles to making this precise:
1 What is a ‘semiring (8-)category’?
2 What is ‘group completion’?
Traditional answers (70’s):
1 bipermutative categories
2 Quillen Q construction, Waldhausen S construction
Alternative: categorify ordinary semirings and group completion!
John D. Berman Commutative Algebra of Categories
![Page 26: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/26.jpg)
Semirings and group completion
1 Given abelian groups (commutative monoids) A,B, there is anabelian group (commutative monoid) Ab B.
2 Abb (ComMonb) is a symmetric monoidal category.
3 Monoids in Abb (ComMonb) are rings (semirings).
4 Z (N) is the free abelian group (commutative monoid) on onegenerator.
5 A commutative monoid is an abelian group if and only if it isa Z-module.
6 ZbN is group completion.
John D. Berman Commutative Algebra of Categories
![Page 27: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/27.jpg)
Semirings and group completion
1 Given abelian groups (commutative monoids) A,B, there is anabelian group (commutative monoid) Ab B.
2 Abb (ComMonb) is a symmetric monoidal category.
3 Monoids in Abb (ComMonb) are rings (semirings).
4 Z (N) is the free abelian group (commutative monoid) on onegenerator.
5 A commutative monoid is an abelian group if and only if it isa Z-module.
6 ZbN is group completion.
John D. Berman Commutative Algebra of Categories
![Page 28: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/28.jpg)
Semirings and group completion
1 Given abelian groups (commutative monoids) A,B, there is anabelian group (commutative monoid) Ab B.
2 Abb (ComMonb) is a symmetric monoidal category.
3 Monoids in Abb (ComMonb) are rings (semirings).
4 Z (N) is the free abelian group (commutative monoid) on onegenerator.
5 A commutative monoid is an abelian group if and only if it isa Z-module.
6 ZbN is group completion.
John D. Berman Commutative Algebra of Categories
![Page 29: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/29.jpg)
Semirings and group completion
1 Given abelian groups (commutative monoids) A,B, there is anabelian group (commutative monoid) Ab B.
2 Abb (ComMonb) is a symmetric monoidal category.
3 Monoids in Abb (ComMonb) are rings (semirings).
4 Z (N) is the free abelian group (commutative monoid) on onegenerator.
5 A commutative monoid is an abelian group if and only if it isa Z-module.
6 ZbN is group completion.
John D. Berman Commutative Algebra of Categories
![Page 30: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/30.jpg)
Semirings and group completion
1 Given abelian groups (commutative monoids) A,B, there is anabelian group (commutative monoid) Ab B.
2 Abb (ComMonb) is a symmetric monoidal category.
3 Monoids in Abb (ComMonb) are rings (semirings).
4 Z (N) is the free abelian group (commutative monoid) on onegenerator.
5 A commutative monoid is an abelian group if and only if it isa Z-module.
6 ZbN is group completion.
John D. Berman Commutative Algebra of Categories
![Page 31: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/31.jpg)
Semirings and group completion
1 Given abelian groups (commutative monoids) A,B, there is anabelian group (commutative monoid) Ab B.
2 Abb (ComMonb) is a symmetric monoidal category.
3 Monoids in Abb (ComMonb) are rings (semirings).
4 Z (N) is the free abelian group (commutative monoid) on onegenerator.
5 A commutative monoid is an abelian group if and only if it isa Z-module.
6 ZbN is group completion.
John D. Berman Commutative Algebra of Categories
![Page 32: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/32.jpg)
Semiring categories
(Gepner-Groth-Nikolaus)
1 Given symmetric monoidal 8-categories C,D, there is asymmetric monoidal 8-category C bD.
2 SymMonb8 is a (large) symmetric monoidal 8-category.
3 (Definition) Monoids in SymMonb8 are semiring 8-categories.
4 Finiso is the free symmetric monoidal 8-category on onegenerator.
Proofs are formal, using higher algebra of presentable 8-categories.
John D. Berman Commutative Algebra of Categories
![Page 33: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/33.jpg)
Semiring categories
(Gepner-Groth-Nikolaus)
1 Given symmetric monoidal 8-categories C,D, there is asymmetric monoidal 8-category C bD.
2 SymMonb8 is a (large) symmetric monoidal 8-category.
3 (Definition) Monoids in SymMonb8 are semiring 8-categories.
4 Finiso is the free symmetric monoidal 8-category on onegenerator.
Proofs are formal, using higher algebra of presentable 8-categories.
John D. Berman Commutative Algebra of Categories
![Page 34: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/34.jpg)
Semiring categories
(Gepner-Groth-Nikolaus)
1 Given symmetric monoidal 8-categories C,D, there is asymmetric monoidal 8-category C bD.
2 SymMonb8 is a (large) symmetric monoidal 8-category.
3 (Definition) Monoids in SymMonb8 are semiring 8-categories.
4 Finiso is the free symmetric monoidal 8-category on onegenerator.
Proofs are formal, using higher algebra of presentable 8-categories.
John D. Berman Commutative Algebra of Categories
![Page 35: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/35.jpg)
Semiring categories
(Gepner-Groth-Nikolaus)
1 Given symmetric monoidal 8-categories C,D, there is asymmetric monoidal 8-category C bD.
2 SymMonb8 is a (large) symmetric monoidal 8-category.
3 (Definition) Monoids in SymMonb8 are semiring 8-categories.
4 Finiso is the free symmetric monoidal 8-category on onegenerator.
Proofs are formal, using higher algebra of presentable 8-categories.
John D. Berman Commutative Algebra of Categories
![Page 36: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/36.jpg)
Semiring categories
(Gepner-Groth-Nikolaus)
1 Given symmetric monoidal 8-categories C,D, there is asymmetric monoidal 8-category C bD.
2 SymMonb8 is a (large) symmetric monoidal 8-category.
3 (Definition) Monoids in SymMonb8 are semiring 8-categories.
4 Finiso is the free symmetric monoidal 8-category on onegenerator.
Proofs are formal, using higher algebra of presentable 8-categories.
John D. Berman Commutative Algebra of Categories
![Page 37: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/37.jpg)
Semiring categories
Examples of commutative semiring 8-categories:
closed monoidal categories (Set, Top, Vect, SetG )
categories built via some constructions (Setop, Setiso)
connective commutative ring spectra (S, KU, HR)
John D. Berman Commutative Algebra of Categories
![Page 38: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/38.jpg)
Semiring categories
Examples of commutative semiring 8-categories:
closed monoidal categories (Set, Top, Vect, SetG )
categories built via some constructions (Setop, Setiso)
connective commutative ring spectra (S, KU, HR)
John D. Berman Commutative Algebra of Categories
![Page 39: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/39.jpg)
Semiring categories
Examples of commutative semiring 8-categories:
closed monoidal categories (Set, Top, Vect, SetG )
categories built via some constructions (Setop, Setiso)
connective commutative ring spectra (S, KU, HR)
John D. Berman Commutative Algebra of Categories
![Page 40: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/40.jpg)
Semiring categories
Examples of commutative semiring 8-categories:
closed monoidal categories (Set, Top, Vect, SetG )
categories built via some constructions (Setop, Setiso)
connective commutative ring spectra (S, KU, HR)
John D. Berman Commutative Algebra of Categories
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Group completion
There is a full subcategory inclusion Sp¥0 SymMon8.
Theorem 7 (B.)
Sp¥0 ModS (i.e., C` is an S-module iff it is a spectrum).
Sb C` K pC`q if C is a groupoid.
HompS, C`q PicpC`q.
If C`,b not a groupoid but semiadditive (ModR),
K pC`q Sb Fun`,bpBurnrCobfr1 s, Cq.
John D. Berman Commutative Algebra of Categories
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Group completion
There is a full subcategory inclusion Sp¥0 SymMon8.
Theorem 7 (B.)
Sp¥0 ModS (i.e., C` is an S-module iff it is a spectrum).
Sb C` K pC`q if C is a groupoid.
HompS, C`q PicpC`q.
If C`,b not a groupoid but semiadditive (ModR),
K pC`q Sb Fun`,bpBurnrCobfr1 s, Cq.
John D. Berman Commutative Algebra of Categories
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Group completion
There is a full subcategory inclusion Sp¥0 SymMon8.
Theorem 7 (B.)
Sp¥0 ModS (i.e., C` is an S-module iff it is a spectrum).
Sb C` K pC`q if C is a groupoid.
HompS, C`q PicpC`q.
If C`,b not a groupoid but semiadditive (ModR),
K pC`q Sb Fun`,bpBurnrCobfr1 s, Cq.
John D. Berman Commutative Algebra of Categories
![Page 44: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/44.jpg)
Group completion
There is a full subcategory inclusion Sp¥0 SymMon8.
Theorem 7 (B.)
Sp¥0 ModS (i.e., C` is an S-module iff it is a spectrum).
Sb C` K pC`q if C is a groupoid.
HompS, C`q PicpC`q.
If C`,b not a groupoid but semiadditive (ModR),
K pC`q Sb Fun`,bpBurnrCobfr1 s, Cq.
John D. Berman Commutative Algebra of Categories
![Page 45: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/45.jpg)
Group completion
There is a full subcategory inclusion Sp¥0 SymMon8.
Theorem 7 (B.)
Sp¥0 ModS (i.e., C` is an S-module iff it is a spectrum).
Sb C` K pC`q if C is a groupoid.
HompS, C`q PicpC`q.
If C`,b not a groupoid but semiadditive (ModR),
K pC`q Sb Fun`,bpBurnrCobfr1 s, Cq.
John D. Berman Commutative Algebra of Categories
![Page 46: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/46.jpg)
Cartesian monoidal categories
C` a symmetric monoidal category, or C`,b a semiring category.
Definition 8
C is cartesian monoidal if ` .
C is cocartesian monoidal if ` >.
C is semiadditive if ` >.
Example 9
Set is cocartesian monoidal.
Setop is cartesian monoidal.
Ab (or ComMon) is semiadditive.
John D. Berman Commutative Algebra of Categories
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Cartesian monoidal categories
C` a symmetric monoidal category, or C`,b a semiring category.
Definition 8
C is cartesian monoidal if ` .
C is cocartesian monoidal if ` >.
C is semiadditive if ` >.
Example 9
Set is cocartesian monoidal.
Setop is cartesian monoidal.
Ab (or ComMon) is semiadditive.
John D. Berman Commutative Algebra of Categories
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Cartesian monoidal categories
C` a symmetric monoidal category, or C`,b a semiring category.
Definition 8
C is cartesian monoidal if ` .
C is cocartesian monoidal if ` >.
C is semiadditive if ` >.
Example 9
Set is cocartesian monoidal.
Setop is cartesian monoidal.
Ab (or ComMon) is semiadditive.
John D. Berman Commutative Algebra of Categories
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Cartesian monoidal categories
C` a symmetric monoidal category, or C`,b a semiring category.
Definition 8
C is cartesian monoidal if ` .
C is cocartesian monoidal if ` >.
C is semiadditive if ` >.
Example 9
Set is cocartesian monoidal.
Setop is cartesian monoidal.
Ab (or ComMon) is semiadditive.
John D. Berman Commutative Algebra of Categories
![Page 50: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/50.jpg)
Cartesian monoidal categories
C` a symmetric monoidal category, or C`,b a semiring category.
Definition 8
C is cartesian monoidal if ` .
C is cocartesian monoidal if ` >.
C is semiadditive if ` >.
Example 9
Set is cocartesian monoidal.
Setop is cartesian monoidal.
Ab (or ComMon) is semiadditive.
John D. Berman Commutative Algebra of Categories
![Page 51: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/51.jpg)
Cartesian monoidal categories
C` a symmetric monoidal category, or C`,b a semiring category.
Definition 8
C is cartesian monoidal if ` .
C is cocartesian monoidal if ` >.
C is semiadditive if ` >.
Example 9
Set is cocartesian monoidal.
Setop is cartesian monoidal.
Ab (or ComMon) is semiadditive.
John D. Berman Commutative Algebra of Categories
![Page 52: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/52.jpg)
Cartesian monoidal categories
C` a symmetric monoidal category, or C`,b a semiring category.
Definition 8
C is cartesian monoidal if ` .
C is cocartesian monoidal if ` >.
C is semiadditive if ` >.
Example 9
Set is cocartesian monoidal.
Setop is cartesian monoidal.
Ab (or ComMon) is semiadditive.
John D. Berman Commutative Algebra of Categories
![Page 53: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/53.jpg)
Cartesian monoidal categories
Theorem 10 (B.)
ModFin CocartMonCat (C` is a Fin-module iff ` >)
ModFinop CartMonCat
ModFinbFinop SemiaddCat
Results are true for categories or 8-categories.Question: What is Finb Finop?
Theorem 11 (Glasman)
The Burnside category is the free semiadditive category on oneobject.
Finb Finop Burn
John D. Berman Commutative Algebra of Categories
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Cartesian monoidal categories
Theorem 10 (B.)
ModFin CocartMonCat (C` is a Fin-module iff ` >)
ModFinop CartMonCat
ModFinbFinop SemiaddCat
Results are true for categories or 8-categories.Question: What is Finb Finop?
Theorem 11 (Glasman)
The Burnside category is the free semiadditive category on oneobject.
Finb Finop Burn
John D. Berman Commutative Algebra of Categories
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Cartesian monoidal categories
Theorem 10 (B.)
ModFin CocartMonCat (C` is a Fin-module iff ` >)
ModFinop CartMonCat
ModFinbFinop SemiaddCat
Results are true for categories or 8-categories.Question: What is Finb Finop?
Theorem 11 (Glasman)
The Burnside category is the free semiadditive category on oneobject.
Finb Finop Burn
John D. Berman Commutative Algebra of Categories
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Cartesian monoidal categories
Theorem 10 (B.)
ModFin CocartMonCat (C` is a Fin-module iff ` >)
ModFinop CartMonCat
ModFinbFinop SemiaddCat
Results are true for categories or 8-categories.
Question: What is Finb Finop?
Theorem 11 (Glasman)
The Burnside category is the free semiadditive category on oneobject.
Finb Finop Burn
John D. Berman Commutative Algebra of Categories
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Cartesian monoidal categories
Theorem 10 (B.)
ModFin CocartMonCat (C` is a Fin-module iff ` >)
ModFinop CartMonCat
ModFinbFinop SemiaddCat
Results are true for categories or 8-categories.Question: What is Finb Finop?
Theorem 11 (Glasman)
The Burnside category is the free semiadditive category on oneobject.
Finb Finop Burn
John D. Berman Commutative Algebra of Categories
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Cartesian monoidal categories
Theorem 10 (B.)
ModFin CocartMonCat (C` is a Fin-module iff ` >)
ModFinop CartMonCat
ModFinbFinop SemiaddCat
Results are true for categories or 8-categories.Question: What is Finb Finop?
Theorem 11 (Glasman)
The Burnside category is the free semiadditive category on oneobject.
Finb Finop Burn
John D. Berman Commutative Algebra of Categories
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Cartesian monoidal categories
Theorem 10 (B.)
ModFin CocartMonCat (C` is a Fin-module iff ` >)
ModFinop CartMonCat
ModFinbFinop SemiaddCat
Results are true for categories or 8-categories.Question: What is Finb Finop?
Theorem 11 (Glasman)
The Burnside category is the free semiadditive category on oneobject.
Finb Finop Burn
John D. Berman Commutative Algebra of Categories
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Modules over semiring categories
Semiring 8-category R R-modulesS Spectra
Finiso Symmetric monoidal
Fin Cocartesian monoidal
Finop Cartesian monoidal
Fininj Symmetric monoidal with initial unit
Fininj,op Symmetric monoidal with terminal unit
Fin Cocartesian monoidal with 0 1
Finop Cartesian monoidal with 0 1
Burn Semiadditive
Burngp Additive
John D. Berman Commutative Algebra of Categories
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Algebraic theories
Definition 12
A PROP (PROduct and Permutation category) is a symmetricmonoidal category P` generated by one object under `.
Think: objects labeled by finite sets, ` >.
Definition 13
A P`-algebra in Cb is a symmetric monoidal functor
AlgPpCbq HompP`, Cbq.
John D. Berman Commutative Algebra of Categories
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Algebraic theories
Definition 12
A PROP (PROduct and Permutation category) is a symmetricmonoidal category P` generated by one object under `.
Think: objects labeled by finite sets, ` >.
Definition 13
A P`-algebra in Cb is a symmetric monoidal functor
AlgPpCbq HompP`, Cbq.
John D. Berman Commutative Algebra of Categories
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Algebraic theories
Definition 12
A PROP (PROduct and Permutation category) is a symmetricmonoidal category P` generated by one object under `.
Think: objects labeled by finite sets, ` >.
Definition 13
A P`-algebra in Cb is a symmetric monoidal functor
AlgPpCbq HompP`, Cbq.
John D. Berman Commutative Algebra of Categories
![Page 64: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/64.jpg)
Algebraic theories
Example 14
Fin is the PROP for commutative monoids;
Finop is the PROP for cocommutative comonoids;
Burn Finb Finop is the PROP forcommutative-cocommutative bimonoids;
Burngp is the PROP for Hopf algebras.
Subcategory of finitely generated free objects.
Definition 15
A Lawvere theory is a cartesian monoidal PROP L. Algebras aretaken in Set (1-categories) or Top (8-categories):
AlgL AlgLpTopq HompL,Topq.
John D. Berman Commutative Algebra of Categories
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Algebraic theories
Example 14
Fin is the PROP for commutative monoids;
Finop is the PROP for cocommutative comonoids;
Burn Finb Finop is the PROP forcommutative-cocommutative bimonoids;
Burngp is the PROP for Hopf algebras.
Subcategory of finitely generated free objects.
Definition 15
A Lawvere theory is a cartesian monoidal PROP L. Algebras aretaken in Set (1-categories) or Top (8-categories):
AlgL AlgLpTopq HompL,Topq.
John D. Berman Commutative Algebra of Categories
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Algebraic theories
Example 14
Fin is the PROP for commutative monoids;
Finop is the PROP for cocommutative comonoids;
Burn Finb Finop is the PROP forcommutative-cocommutative bimonoids;
Burngp is the PROP for Hopf algebras.
Subcategory of finitely generated free objects.
Definition 15
A Lawvere theory is a cartesian monoidal PROP L. Algebras aretaken in Set (1-categories) or Top (8-categories):
AlgL AlgLpTopq HompL,Topq.
John D. Berman Commutative Algebra of Categories
![Page 67: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/67.jpg)
Algebraic theories
Example 14
Fin is the PROP for commutative monoids;
Finop is the PROP for cocommutative comonoids;
Burn Finb Finop is the PROP forcommutative-cocommutative bimonoids;
Burngp is the PROP for Hopf algebras.
Subcategory of finitely generated free objects.
Definition 15
A Lawvere theory is a cartesian monoidal PROP L. Algebras aretaken in Set (1-categories) or Top (8-categories):
AlgL AlgLpTopq HompL,Topq.
John D. Berman Commutative Algebra of Categories
![Page 68: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/68.jpg)
Algebraic theories
Example 14
Fin is the PROP for commutative monoids;
Finop is the PROP for cocommutative comonoids;
Burn Finb Finop is the PROP forcommutative-cocommutative bimonoids;
Burngp is the PROP for Hopf algebras.
If P` is cartesian monoidal, Pop AlgPpSetq:
Subcategory of finitely generated free objects.
Definition 15
A Lawvere theory is a cartesian monoidal PROP L. Algebras aretaken in Set (1-categories) or Top (8-categories):
AlgL AlgLpTopq HompL,Topq.
John D. Berman Commutative Algebra of Categories
![Page 69: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/69.jpg)
Algebraic theories
Example 14
Fin is the PROP for commutative monoids;
Finop is the PROP for cocommutative comonoids;
Burn Finb Finop is the PROP forcommutative-cocommutative bimonoids;
Burngp is the PROP for Hopf algebras.
If P` is cartesian monoidal, Pop AlgPpTopq:
Subcategory of finitely generated free objects.
Definition 15
A Lawvere theory is a cartesian monoidal PROP L. Algebras aretaken in Set (1-categories) or Top (8-categories):
AlgL AlgLpTopq HompL,Topq.
John D. Berman Commutative Algebra of Categories
![Page 70: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/70.jpg)
Algebraic theories
Example 14
Fin is the PROP for commutative monoids;
Finop is the PROP for cocommutative comonoids;
Burn Finb Finop is the PROP forcommutative-cocommutative bimonoids;
Burngp is the PROP for Hopf algebras.
If P` is cartesian monoidal, Pop AlgPpTopq:Subcategory of finitely generated free objects.
Definition 15
A Lawvere theory is a cartesian monoidal PROP L. Algebras aretaken in Set (1-categories) or Top (8-categories):
AlgL AlgLpTopq HompL,Topq.
John D. Berman Commutative Algebra of Categories
![Page 71: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/71.jpg)
Algebraic theories
Example 14
Fin is the PROP for commutative monoids;
Finop is the PROP for cocommutative comonoids;
Burn Finb Finop is the PROP forcommutative-cocommutative bimonoids;
Burngp is the PROP for Hopf algebras.
If P` is cartesian monoidal, Pop AlgPpTopq:Subcategory of finitely generated free objects.
Definition 15
A Lawvere theory is a cartesian monoidal PROP L. Algebras aretaken in Set (1-categories) or Top (8-categories):
AlgL AlgLpTopq HompL,Topq.
John D. Berman Commutative Algebra of Categories
![Page 72: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/72.jpg)
Lawvere theories
Example 16
Lawvere theory Set-algebras Top-algebras
Finop Set Top
Burn SpanpFinq Ab Sp¥0
Poly BispanpFinq ComRing ? (ComRingSp¥0)
Theorem 17 (B.)
A PROP is a cyclic Finiso-module.
A Lawvere theory is a cyclic Finop-module.
If P,P 1 are PROPs/Lawvere theories, so is P b P 1.
If P` is a PROP, the associated Lawvere theory is P`bFinop:
AlgPpTopq AlgPbFinoppTopq.
John D. Berman Commutative Algebra of Categories
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Lawvere theories
Example 16
Lawvere theory Set-algebras Top-algebras
Finop Set Top
Burn SpanpFinq Ab Sp¥0
Poly BispanpFinq ComRing ? (ComRingSp¥0)
Theorem 17 (B.)
A PROP is a cyclic Finiso-module.
A Lawvere theory is a cyclic Finop-module.
If P,P 1 are PROPs/Lawvere theories, so is P b P 1.
If P` is a PROP, the associated Lawvere theory is P`bFinop:
AlgPpTopq AlgPbFinoppTopq.
John D. Berman Commutative Algebra of Categories
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Lawvere theories
Example 16
Lawvere theory Set-algebras Top-algebras
Finop Set Top
Burn SpanpFinq Ab Sp¥0
Poly BispanpFinq ComRing ? (ComRingSp¥0)
Theorem 17 (B.)
A PROP is a cyclic Finiso-module.
A Lawvere theory is a cyclic Finop-module.
If P,P 1 are PROPs/Lawvere theories, so is P b P 1.
If P` is a PROP, the associated Lawvere theory is P`bFinop:
AlgPpTopq AlgPbFinoppTopq.
John D. Berman Commutative Algebra of Categories
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Lawvere theories
Example 16
Lawvere theory Set-algebras Top-algebras
Finop Set Top
Burn SpanpFinq Ab Sp¥0
Poly BispanpFinq ComRing ? (ComRingSp¥0)
Theorem 17 (B.)
A PROP is a cyclic Finiso-module.
A Lawvere theory is a cyclic Finop-module.
If P,P 1 are PROPs/Lawvere theories, so is P b P 1.
If P` is a PROP, the associated Lawvere theory is P`bFinop:
AlgPpTopq AlgPbFinoppTopq.
John D. Berman Commutative Algebra of Categories
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Lawvere theories
Example 16
Lawvere theory Set-algebras Top-algebras
Finop Set Top
Burn SpanpFinq Ab Sp¥0
Poly BispanpFinq ComRing ? (ComRingSp¥0)
Theorem 17 (B.)
A PROP is a cyclic Finiso-module.
A Lawvere theory is a cyclic Finop-module.
If P,P 1 are PROPs/Lawvere theories, so is P b P 1.
If P` is a PROP, the associated Lawvere theory is P`bFinop:
AlgPpTopq AlgPbFinoppTopq.
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Lawvere theories
Example 16
Lawvere theory Set-algebras Top-algebras
Finop Set Top
Burn SpanpFinq Ab Sp¥0
Poly BispanpFinq ComRing ? (ComRingSp¥0)
Theorem 17 (B.)
A PROP is a cyclic Finiso-module.
A Lawvere theory is a cyclic Finop-module.
If P,P 1 are PROPs/Lawvere theories, so is P b P 1.
If P` is a PROP, the associated Lawvere theory is P`bFinop:
AlgPpTopq AlgPbFinoppTopq.
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Lawvere theories
Example 16
Lawvere theory Set-algebras Top-algebras
Finop Set Top
Burn SpanpFinq Ab Sp¥0
Poly BispanpFinq ComRing ? (ComRingSp¥0)
Theorem 17 (B.)
A PROP is a cyclic Finiso-module.
A Lawvere theory is a cyclic Finop-module.
If P,P 1 are PROPs/Lawvere theories, so is P b P 1.
If P` is a PROP, the associated Lawvere theory is P`bFinop:
AlgPpTopq AlgPbFinoppTopq.
John D. Berman Commutative Algebra of Categories
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Application: equivariant homotopy theory
Definition 18 (B.)
An equivariant Lawvere theory is a cyclic FinopG -module L.
AlgL HompL,Topq.
Theorem 19 (Elmendorf)
FinopG is the equivariant Lawvere theory for TopG .
Theorem 20 (Guillou-May)
BurnG SpanpFinG q is the equivariant Lawvere theory for Sp¥0G .
Conjecture
PolyG BispanpFinG q is the equivariant Lawvere theory forCRingSp¥0
G .
John D. Berman Commutative Algebra of Categories
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Application: equivariant homotopy theory
Definition 18 (B.)
An equivariant Lawvere theory is a cyclic FinopG -module L.
AlgL HompL,Topq.
Theorem 19 (Elmendorf)
FinopG is the equivariant Lawvere theory for TopG .
Theorem 20 (Guillou-May)
BurnG SpanpFinG q is the equivariant Lawvere theory for Sp¥0G .
Conjecture
PolyG BispanpFinG q is the equivariant Lawvere theory forCRingSp¥0
G .
John D. Berman Commutative Algebra of Categories
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Application: equivariant homotopy theory
Definition 18 (B.)
An equivariant Lawvere theory is a cyclic FinopG -module L.
AlgL HompL,Topq.
Theorem 19 (Elmendorf)
FinopG is the equivariant Lawvere theory for TopG .
Theorem 20 (Guillou-May)
BurnG SpanpFinG q is the equivariant Lawvere theory for Sp¥0G .
Conjecture
PolyG BispanpFinG q is the equivariant Lawvere theory forCRingSp¥0
G .
John D. Berman Commutative Algebra of Categories
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Application: equivariant homotopy theory
Definition 18 (B.)
An equivariant Lawvere theory is a cyclic FinopG -module L.
AlgL HompL,Topq.
Theorem 19 (Elmendorf)
FinopG is the equivariant Lawvere theory for TopG .
Theorem 20 (Guillou-May)
BurnG SpanpFinG q is the equivariant Lawvere theory for Sp¥0G .
Conjecture
PolyG BispanpFinG q is the equivariant Lawvere theory forCRingSp¥0
G .
John D. Berman Commutative Algebra of Categories
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Application: operads
Operad O:
given a finite set X , set OpX q of ways to multiply objects of X
composition maps
associative
Example 21
Commutative operad CommpX q .
John D. Berman Commutative Algebra of Categories
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Application: operads
Operad O:
given a finite set X , set OpX q of ways to multiply objects of X
composition maps
associative
Example 21
Commutative operad CommpX q .
John D. Berman Commutative Algebra of Categories
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Application: operads
Operad O:
given a finite set X , set OpX q of ways to multiply objects of X
composition maps
associative
Example 21
Commutative operad CommpX q .
John D. Berman Commutative Algebra of Categories
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Application: operads
Operad O:
given a finite set X , set OpX q of ways to multiply objects of X
composition maps
associative
Example 21
Commutative operad CommpX q .
John D. Berman Commutative Algebra of Categories
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Application: operads
Symmetric monoidal category EnvpOq>:
Objects are finite sets.
Morphism X Ñ Y is a way to turn X into Y using operationsin O.
Symmetric monoidal operation is >.
EnvpOq> is a PROP; algebras are O-algebras
HompEnvpOq>, Cbq AlgOpCbq.
Example 22
EnvpCommq> Fin>.
John D. Berman Commutative Algebra of Categories
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Application: operads
Symmetric monoidal category EnvpOq>:
Objects are finite sets.
Morphism X Ñ Y is a way to turn X into Y using operationsin O.
Symmetric monoidal operation is >.
EnvpOq> is a PROP; algebras are O-algebras
HompEnvpOq>, Cbq AlgOpCbq.
Example 22
EnvpCommq> Fin>.
John D. Berman Commutative Algebra of Categories
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Application: operads
Symmetric monoidal category EnvpOq>:
Objects are finite sets.
Morphism X Ñ Y is a way to turn X into Y using operationsin O.
Symmetric monoidal operation is >.
EnvpOq> is a PROP; algebras are O-algebras
HompEnvpOq>, Cbq AlgOpCbq.
Example 22
EnvpCommq> Fin>.
John D. Berman Commutative Algebra of Categories
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Application: operads
Symmetric monoidal category EnvpOq>:
Objects are finite sets.
Morphism X Ñ Y is a way to turn X into Y using operationsin O.
Symmetric monoidal operation is >.
EnvpOq> is a PROP; algebras are O-algebras
HompEnvpOq>, Cbq AlgOpCbq.
Example 22
EnvpCommq> Fin>.
John D. Berman Commutative Algebra of Categories
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Application: operads
Symmetric monoidal category EnvpOq>:
Objects are finite sets.
Morphism X Ñ Y is a way to turn X into Y using operationsin O.
Symmetric monoidal operation is >.
EnvpOq> is a PROP; algebras are O-algebras
HompEnvpOq>, Cbq AlgOpCbq.
Example 22
EnvpCommq> Fin>.
John D. Berman Commutative Algebra of Categories
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Applications: operads
Operads
O
Ñ
PROPs
EnvpOq
Ñ
Lawvere Theories
EnvpOq b Finop
Theorem 23
Given an operad O, EnvpOq b Finop is:
the Lawvere theory associated to O;
the PROP for O Commbialgebras;
an explicit span construction.
Conjecture
The PROP for O O1bialgebras can be computed via a spanconstruction.
John D. Berman Commutative Algebra of Categories
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Applications: operads
Operads
O
Ñ
PROPs
EnvpOq
Ñ
Lawvere Theories
EnvpOq b Finop
Theorem 23
Given an operad O, EnvpOq b Finop is:
the Lawvere theory associated to O;
the PROP for O Commbialgebras;
an explicit span construction.
Conjecture
The PROP for O O1bialgebras can be computed via a spanconstruction.
John D. Berman Commutative Algebra of Categories
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Applications: operads
Operads
O
Ñ
PROPs
EnvpOq
Ñ
Lawvere Theories
EnvpOq b Finop
Theorem 23
Given an operad O, EnvpOq b Finop is:
the Lawvere theory associated to O;
the PROP for O Commbialgebras;
an explicit span construction.
Conjecture
The PROP for O O1bialgebras can be computed via a spanconstruction.
John D. Berman Commutative Algebra of Categories
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Applications: operads
Operads
O
Ñ
PROPs
EnvpOq
Ñ
Lawvere Theories
EnvpOq b Finop
Theorem 23
Given an operad O, EnvpOq b Finop is:
the Lawvere theory associated to O;
the PROP for O Commbialgebras;
an explicit span construction.
Conjecture
The PROP for O O1bialgebras can be computed via a spanconstruction.
John D. Berman Commutative Algebra of Categories
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Applications: operads
Operads
O
Ñ
PROPs
EnvpOq
Ñ
Lawvere Theories
EnvpOq b Finop
Theorem 23
Given an operad O, EnvpOq b Finop is:
the Lawvere theory associated to O;
the PROP for O Commbialgebras;
an explicit span construction.
Conjecture
The PROP for O O1bialgebras can be computed via a spanconstruction.
John D. Berman Commutative Algebra of Categories
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Future work
Push/pull square of rings:
Finiso //
Fin
Finop // Burn
Descent: Can Cb P SymMon8 be reconstructed from C b Fin andC b Finop?Answer: Not always!
Example 24
Sb Fin Sb Finop 0, but S 0.
John D. Berman Commutative Algebra of Categories
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Future work
Push/pull square of rings:
SymMon8bFin//
bFinop
CocartMon8
bBurn
CartMon8bBurn
// SemiaddCat8
Descent: Can Cb P SymMon8 be reconstructed from C b Fin andC b Finop?Answer: Not always!
Example 24
Sb Fin Sb Finop 0, but S 0.
John D. Berman Commutative Algebra of Categories
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Future work
Push/pull square of rings:
Finiso //
Fin
Finop // Burn
Descent: Can Cb P SymMon8 be reconstructed from C b Fin andC b Finop?
Answer: Not always!
Example 24
Sb Fin Sb Finop 0, but S 0.
John D. Berman Commutative Algebra of Categories
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Future work
Push/pull square of rings:
Finiso //
Fin
Finop // Burn
Descent: Can Cb P SymMon8 be reconstructed from C b Fin andC b Finop?Answer: Not always!
Example 24
Sb Fin Sb Finop 0, but S 0.
John D. Berman Commutative Algebra of Categories
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Future work
Push/pull square of rings:
Finiso //
Fin
Finop // Burn
Descent: Can Cb P SymMon8 be reconstructed from C b Fin andC b Finop?Answer: Not always!
Example 24
Sb Fin Sb Finop 0, but S 0.
John D. Berman Commutative Algebra of Categories
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Future work
Example 25
Can operad O be recovered from EnvpOq b Fin andEnvpOq b Finop?
EnvpOq b Finop LO (Lawvere theory)
EnvpOq b Fin Fin
LO b Burn Burn
Conjecture
There is an equivalence of (8-)categories between unital(8-)operads and cyclic Finop-modules with trivialization over Burn.
Applications:
earlier conjecture on operadic bialgebras
equivariant 8-operads
John D. Berman Commutative Algebra of Categories
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Future work
Example 25
Can operad O be recovered from EnvpOq b Fin andEnvpOq b Finop?
EnvpOq b Finop LO (Lawvere theory)
EnvpOq b Fin Fin
LO b Burn Burn
Conjecture
There is an equivalence of (8-)categories between unital(8-)operads and cyclic Finop-modules with trivialization over Burn.
Applications:
earlier conjecture on operadic bialgebras
equivariant 8-operads
John D. Berman Commutative Algebra of Categories
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Future work
Example 25
Can operad O be recovered from EnvpOq b Fin andEnvpOq b Finop?
EnvpOq b Finop LO (Lawvere theory)
EnvpOq b Fin Fin
LO b Burn Burn
Conjecture
There is an equivalence of (8-)categories between unital(8-)operads and cyclic Finop-modules with trivialization over Burn.
Applications:
earlier conjecture on operadic bialgebras
equivariant 8-operads
John D. Berman Commutative Algebra of Categories
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Future work
Example 25
Can operad O be recovered from EnvpOq b Fin andEnvpOq b Finop?
EnvpOq b Finop LO (Lawvere theory)
EnvpOq b Fin Fin
LO b Burn Burn
Conjecture
There is an equivalence of (8-)categories between unital(8-)operads and cyclic Finop-modules with trivialization over Burn.
Applications:
earlier conjecture on operadic bialgebras
equivariant 8-operads
John D. Berman Commutative Algebra of Categories
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Future work
Example 25
Can operad O be recovered from EnvpOq b Fin andEnvpOq b Finop?
EnvpOq b Finop LO (Lawvere theory)
EnvpOq b Fin Fin
LO b Burn Burn
Conjecture
There is an equivalence of (8-)categories between unital(8-)operads and cyclic Finop-modules with trivialization over Burn.
Applications:
earlier conjecture on operadic bialgebras
equivariant 8-operads
John D. Berman Commutative Algebra of Categories
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Future work
Example 25
Can operad O be recovered from EnvpOq b Fin andEnvpOq b Finop?
EnvpOq b Finop LO (Lawvere theory)
EnvpOq b Fin Fin
LO b Burn Burn
Conjecture
There is an equivalence of (8-)categories between unital(8-)operads and cyclic Finop-modules with trivialization over Burn.
Applications:
earlier conjecture on operadic bialgebras
equivariant 8-operads
John D. Berman Commutative Algebra of Categories
![Page 108: Commutative Algebra of Categories - people.uic.edu - commutative algebra of... · construction Alternative: categorify ordinary semirings and group completion! ... are rings (semirings).](https://reader036.fdocuments.us/reader036/viewer/2022081516/60fa606ec499b611c5001ff2/html5/thumbnails/108.jpg)
Future work
Example 25
Can operad O be recovered from EnvpOq b Fin andEnvpOq b Finop?
EnvpOq b Finop LO (Lawvere theory)
EnvpOq b Fin Fin
LO b Burn Burn
Conjecture
There is an equivalence of (8-)categories between unital(8-)operads and cyclic Finop-modules with trivialization over Burn.
Applications:
earlier conjecture on operadic bialgebras
equivariant 8-operads
John D. Berman Commutative Algebra of Categories