Weights for The Object of Monoids

Transkrypt

Lukasz Sienkiewicz, Marek Zawadowski
University of Warsaw
July 23, 2013
Lukasz Sienkiewicz, Marek Zawadowski (Uw)
July 23, 2013
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Motivation
Let T be a monad in a 2-category K. There exists a notion of
Eilenberg-Moore object for T [Street,1972].
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Motivation
Question
How about ‘2-algebraic set’ of monoids for any monoidal category object?
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Motivation
Question
How about ‘2-algebraic set’ of monoids for any monoidal category object?
Example
Starting with abelian groups in the category Set. Define an abelian group
op
in the functor category SetC . Reflect by Yoneda embedding
YC : C −→ SetC
op
to derive notion of an abelian group in C
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Description of procedure
Start with a notion in Cat.
op
Generalize in the pointwise manner to 2-functor categories CatK .
Then using 2-Yoneda embedding
YK : K −→ CatK
op
one can reflect the algebraic notion in question to K.
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Objects of Monoids
Monst (Cat)
mon
u
⇒
?
|−|
?
Cat
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Objects of Monoids
Monst (K)
Monst (Cat)
monK
| − |K
uK
⇒
?
mon
?
?
K
u
⇒
|−|
?
Cat
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Objects of Monoids
Monst (K)
Monst (Cat)K
monK
| − |K
uK
⇒
?
op
monK
?
?
op
| − |K
op
?
CatK
K
uK
⇒
op
op
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Objects of Monoids
op
Monst (CatK )
Monst (K)
λK
?
Monst (Cat)K
monK
| − |K
uK
⇒
?
op
monK
?
?
op
| − |K
op
?
CatK
K
uK
⇒
op
op
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Objects of Monoids
Monst (K)
Monst (YK )
- Mon (CatKop )
st
λK
?
Monst (Cat)K
monK
| − |K
uK
⇒
?
op
monK
?
K
YK
uK
⇒
op
op
| − |K
op
?
?
- CatKop
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Objects of Monoids
Monst (K)
Monst (YK )
- Mon (CatKop )
st
λK
?
Monst (Cat)K
monK
| − |K
uK
⇒
op
monK
uK
⇒
op
op
| − |K
op
σK
⇒
?
?
K
YK
?
?
- CatKop
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Monadicity of the object of monoids
Theorem (Monadicity for the monoids)
Let (C, ⊗, I ) be a monoidal category and U : mon(C) → C be its category
of monoids together with the canonical forgetful functor U. Suppose that
U is a right adjoint. Then U is monadic i.e mon(C) is the
Eilenberg-Moore object for the corresponding monad.
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Monadicity of the object of monoids
Theorem (Monadicity for the monoids)
Let (C, ⊗, I ) be a monoidal category and U : mon(C) → C be its category
of monoids together with the canonical forgetful functor U. Suppose that
U is a right adjoint. Then U is monadic i.e mon(C) is the
Eilenberg-Moore object for the corresponding monad.
Theorem (S, Zawadowski)
Let (c, ⊗, I ) be a monoidal object in a 2-category K and u : mon(c) → c
be the object of monoids for (c, ⊗, I ). Suppose that u is a right adjoint.
Then u is monadic i.e mon(c) is the Eilenberg-Moore object for the
corresponding monad.
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Motivation
Let T be a monad in a 2-category K. Then the Eilenberg-Moore
object for T is a weighted 2-limit [Lawvere,1969].
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Motivation
Let T be a monad in a 2-category K. Then the Eilenberg-Moore
object for T is a weighted 2-limit [Lawvere,1969].
Question
Is ‘2-algebraic set’ of monoids a weighted 2-limit for a suitable diagram?
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The weights
Three 2-categories:
Operations −→ Coherences −→ Algebras
iso on category part/locally fully faithful factorizations
Operations
- Coherences
@
@
@
R
@
2Structure
- Algebras
- Weight
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The weights
2-category 2Structure will be a 2-Lawvere theory.
We have a 2-functor
2Structure −→ Weight
Weight has the terminal object 1
2Structure
- Weight
Weight(1, −)
- Cat
is the weight W.
Let
F : 2Structure −→ K
be a finite products preserving 2-functor. Pick limW F .
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The weights
To construct a sequence:
Operations −→ Coherences −→ Algebras
we will use symmetric operads.
Define categories
associated with symmetric operads:
and 2-categories
A1 B1
FA , A B ,
.
A0 B0
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The Category FA
A a symmetric operad with multiplication µA .
The objects of FA are [n] = {1, ..., n} for n ∈ N.
Morphims are
[n]
hf , ai ii∈[m]
- [m]
where ai ∈ A(f −1 (i)).
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The Category FA
[k]
(g , bj )j∈[n]
[n]
(f , ai )i∈[m]
[m]
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The Category FA
g −1 (f −1 (i))
g
f −1 (i)
f
i
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The Category FA
g −1 (f −1 (i))
g
(f −1 (i), ai )
f −1 (i)
f
i
i
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The Category FA
(g −1 (j1 ), bj1 )
g −1 (f −1 (i))
g
(f −1 (i), ai )
f −1 (i)
f
i
i
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The Category FA
(g −1 (j1 ), bj1 )
...
g −1 (f −1 (i))
g
(f −1 (i), ai )
f −1 (i)
f
i
i
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The Category FA
(g −1 (j1 ), bj1 )
...
(g −1 (jr ), bjr )
g −1 (f −1 (i))
g
(f −1 (i), ai )
f −1 (i)
f
i
i
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The Category FA
(g −1 (j1 ), bj1 )
...
(g −1 (jr ), bjr )
g −1 (f −1 (i))
g
(f −1 (i), ai )
f −1 (i)
f
i
i
Use multiplication in A to get ci = µA (bj1 , ..., bjr , ai ). The composition is
(f ◦ g , ci )
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Categories of spans from symmetric operads
A B
The objects of A B are [n].
Morphism of A B are classes
[r ]
hf , ai ii∈[n]
[n]
@
@ hg , bj ij∈[m]
R
@
[m]
with hf , ai ii∈[n] in FA and hg , bj ij∈[m] in FB .
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[r ]
A B
[s]
(f 0 , ai 0 )
(f , ai )
[n]
(g , bj )
[m]
(g 0 , bj 0 )
[k]
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A B
[l]
g 00
f 00
[r ]
[s]
(f 0 , ai 0 )
(f , ai )
[n]
(g , bj )
[m]
(g 0 , bj 0 )
[k]
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A B
[l]
(g 00 , bj 00 )
(f 00 , ai 00 )
[r ]
[s]
(f 0 , ai 0 )
(f , ai )
[n]
(g , bj )
[m]
(g 0 , bj 0 )
[k]
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Arrays of symmetric operads and higher categories of
spans
α0 : A1 → A0 , α1 : A1 → B0 , β0 : B1 → A0 , β1 : B1 → B0 morphisms
of symmetric operads.
2-category
of 2-spans defined by an array of symmetric operads
A1 B1
A0 B0
Its category part is A0 B0 .
Its 2-cells are 2-spans defined using A1 and B1 and morphisms given
above.
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Some Operads
BTr operad of binary trees i.e BTr(X) is the set of all binary trees
with leaves partially labeled by elements of X
Lo operad of linear orders i.e Lo(X) is the set of all linear orders on X
Have a morphism
BTr → Lo
that remembers order of leaves.
⊥,> the initial and the terminal symmetric operads, respectively.
Have a sequence of symmetric operads
⊥ → BTr → Lo → >
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The weights for the objects of monoids
Consider the following two 2-functors
> BTr
−→
> Lo
−→
⊥ Lo
> Lo
Take a factorization (iso on category part/locally fully faithful):
- > Lo
- ⊥ Lo
> BTr
> Lo
@
@
@
R
@
M
-
WM
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The weights for the objects of monoids
Theorem (S. Zawadowski)
The 2-category M is the 2-Lawvere theory for the monoidal category
objects. The 2-functor W, the composite of
M
- WM
WM(1, −)
- Cat
is the weight for objects monoids over a monoidal category objects i.e., a
finite product preserving 2-functor F : M → K corresponds to a monoidal
category object in K and, if it exists, the weighted limit limW F is the
object of monoids for F in K.
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The weights for other sturctures
Comonoids
Lo ⊥
> BTr −→ > Lo −→
> Lo
comonoids over a monoidal category object.
the weight for the object of
Commutative monoids
⊥ >
> BTr −→ > > −→
> >
commutative monoids over a symmetric monoidal category object.
Bi-monoids
Lo Lo
> >
bi-monoids over a symmetric monoidal category object.
> BTr
−→
> >
−→
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Remarks
To define an object of monoids we need to have a bi-monoidal
2-category.
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Remarks
2-category.
In the definition of a monoid we need to consider ’two copies’ of M to
consider M ⊗ M.
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Remarks
2-category.
consider M ⊗ M.
We need ’diagonals’ and ’projections’ i.e a bi-monoidal structure.
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Remarks
2-category.
consider M ⊗ M.
Microcosm prinicple must be formulated carefully.
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Remarks
2-category.
consider M ⊗ M.
Using colored symmetric operads one can construct the weights for
the objects of actions.
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Remarks
2-category.
consider M ⊗ M.
Using colored symmetric operads one can construct the weights for
the objects of actions.
As in Cat lax monoidal 1-cells induce 1-cells between the objects of
monoids
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The End
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Weights for The Object of Monoids

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