1 Jun 2011 16:03

### TYPES 2011 in Bergen 8 - 11 Sept: Deadline for abstracts 19 June

                         Types Meeting 2011
Bergen, 8 - 11 September 2011

http://www.types.name

The 18-th Workshop "Types for Proofs and Programs" will take place
in Bergen, Norway from 8 to 11 September 2011. CSL'11 will take place
in Bergen from 12 to 15 September: http://www.eacsl.org/csl11

The Types Meeting is a forum to present new and on-going work in all
aspects of type theory and its applications, especially in formalized
and computer assisted reasoning and computer programming.

Invited speakers:

* Georges Gonthier, Cambridge
* Dag Normann, Oslo

We encourage all researchers to contribute talks on subjects related
to the Types area of interest. These include, but are not limited to:

- Foundations of type theory and constructive mathematics;
- Applications of type theory;
- Dependently typed programming;
- Industrial uses of type theory technology;
- Meta-theoretic studies of type systems;
-Proof-assistants and proof technology;
- Automation in computer-assisted reasoning;
- Links between type theory and functional programming;


7 Jun 2011 20:10

### existence of limit and colimits in product categories

Hi,

I am looking for references about the existence of limits and colimits
in product categories. More precisely, if A and B are categories, and
AxB is the product category:
- If all pushouts exist in A and B, do the pushouts exist in AxB?
- If not all pushouts exist in A, and all pushout exist in B, do the
pushouts exist in AxB only when they exist in A ?
- If pushouts exist in A and B, do the pushouts in AxB be the product of
the pushout in A and the pushout in B?

I have some difficulties to find references about product categories.
Maybe the answer is simple but it's not evident for me.

Best regards,
Viktor Kronenbourg

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9 Jun 2011 17:02

### Hopf - Lie algebra adjunction in monoidal categories

Dear all,

It is well-known that there exists an adjunction between the category
of Lie algebras and the category of Hopf algebras, by taking the
universal algebra of a lie algebra, and the primitive elements of a
Hopf algebra.

Both the notion of a Hopf algebra and a Lie algebra make sense in the
setting of an additive symmetric monoidal category.
Is it possible to preform the above mentioned adjunction in this
general setting, probably assuming extra conditions on the monoidal
category one is working in, such as the existence of certain
(co)limits and preservation of these (co)limits by the tensor product ?
Does anyone know a reference for such a result ?

Many thanks and best wishes,
Joost Vercruysse.

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10 Jun 2011 00:25

### Codensity and the ultrafilter monad

Dear all,

Any functor from a small category A to a complete category E induces a
contravariant adjunction between E and Set^A.  This in turn induces a

(The construction of the adjunction is better known in its dual form,
starting with a functor from a small category to a COcomplete category.
For example, the usual functor from Delta into Top induces the usual
adjunction between topological spaces and simplicial sets.)

The codensity monad of the inclusion FinSet --> Set is the ultrafilter
monad.  This seems a rather basic fact, but I've been unable to find it in
the literature.  I'd be grateful if someone could tell me a reference.

(I'm aware of the 1987 paper by Reinhard Börger giving a different but
related characterization of the ultrafilter monad.)

Thanks,
Tom

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10 Jun 2011 03:04

### Re: Codensity and the ultrafilter monad

Tom Leinster <Tom.Leinster <at> glasgow.ac.uk> wrote, in part:

> The codensity monad of the inclusion FinSet --> Set is the ultrafilter
> monad.  This seems a rather basic fact, but I've been unable to find it  in
> the literature.  I'd be grateful if someone could tell me a reference.

I can't be certain, but I can easily imagine Oswald Wyler or
Ernie Manes having noticed that fact in the dim, dark, distant past.
Perhaps Ernie will chime in  .

HTH. Cheers, -- Fred

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10 Jun 2011 10:19

### Re: Codensity and the ultrafilter monad

Hi Tom,

This appears as exercise 3.2.12(e) in Manes' book (Algebraic Theories). The
references there are Lawvere's thesis and Linton's 1966, but I don't know
if this part of the exercise is solved or mentioned explicitly there.
Best regards,
Michel

On Fri, Jun 10, 2011 at 12:25 AM, Tom Leinster
<Tom.Leinster <at> glasgow.ac.uk>wrote:

> Dear all,
>
> Any functor from a small category A to a complete category E induces a
> contravariant adjunction between E and Set^A.  This in turn induces a monad
> on E, the "codensity monad" of the functor.
>
> (The construction of the adjunction is better known in its dual form,
> starting with a functor from a small category to a COcomplete category. For
> example, the usual functor from Delta into Top induces the usual adjunction
> between topological spaces and simplicial sets.)
>
> The codensity monad of the inclusion FinSet --> Set is the ultrafilter
> monad.  This seems a rather basic fact, but I've been unable to find it in
> the literature.  I'd be grateful if someone could tell me a reference.
>
> (I'm aware of the 1987 paper by Reinhard Börger giving a different but
> related characterization of the ultrafilter monad.)
>
> Thanks,


13 Jun 2011 03:28

### Re: Codensity and the ultrafilter monad

A few days ago, I asked where in the literature I could find the fact that
the codensity monad of the inclusion FinSet --> Set is the ultrafilter

> This appears as exercise 3.2.12(e) in Manes' book (Algebraic Theories).  The
> references there are Lawvere's thesis and Linton's 1966, but I don't know
> if this part of the exercise is solved or mentioned explicitly there.

Thanks very much, Michel.  The references to Lawvere's thesis (Section
III, Theorem 2) and Linton's 1966 La Jolla paper (Section 2) seem to be
general references for the structure-semantics adjunction, which is what
the earlier parts of the exercise are about.  The word "ultrafilter" does
not appear in either Lawvere or Linton.

So I currently believe that Manes was the first to publish this fact.  If
someone knows better (perhaps Bill, Fred, Anders Kock or Myles Tierney), I
hope they will let me know.

(I suspect that John Isbell would have known it, at some level, when he
wrote his 1960 paper "Adequate subcategories", even though the language of
monads wasn't available then.  But I haven't found it mentioned in his
words; Manes's exercise is the only written reference to this fact that I
know of.)

through history: it has also been called the model-induced monad/triple
Lawvere), and the algebraic completion (e.g. by Manes).

Thanks to all who replied.


13 Jun 2011 04:00

### Wikipedia on Eilenberg-Mac Lane spaces

Something in Wikipedia on E.-M. spaces I think they've got not quite right.
The article in question:

http://en.wikipedia.org/wiki/Eilenberg%E2%80%93MacLane_space .

The problem: after stating (more or less correctly) that "An important
property of K(G,n) is that, for any abelian group G, and any CW-complex X,
the set

[X, K(G,n)]

of homotopy classes of maps from X to K(G,n) is in natural bijection with
the n-th singular cohomology group

H^n(X; G)"

the article goes on to say (incorrectly) that "Since H^n(K(G,n); G) =
Hom(G,G), there is a distinguished element u {\in} H^n(K(G,n);G)
corresponding to the identity."

Seems to me all that's justified here would be that 'the set

[K(G,n), K(G,n)]

of homotopy classes of maps from K(G,n) to itself is in natural bijection
with H^n(K(G,n); G)', whence "there is a distinguished element u {\in}
H^n(K(G,n);G) corresponding to the identity."

What exact role Hom(G,G) may have to play here might be of interest in its
own right, but there's no groundwork for that laid anywhere in this Wiki


15 Jun 2011 14:00

### Preprint: From cubical to globular higher categories

The following preprint is available

M. Grandis - R. Pare
From cubical to globular higher categories
Dip. Mat. Univ. Genova, Preprint 594 (2011).

http://www.dima.unige.it/~grandis/CGlb.pdf

Abstract. 	We show that a strict symmetric (infinite-dimensional)
cubical category A
has an associated omega-category Glb(A), consisting of its 'globular
cubes'.
The procedure of globularisation generally destroys important
features of A,
like the existence of limits and colimits or the presence of symmetries.
Then we examine the much more complex weak case, up to constructing the
tricategory associated to a weak symmetric 3-cubical category.

Best regards

Marco Grandis

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16 Jun 2011 13:40

### CSASC 2011

September 25-28, 2011, there will take place the Joint Mathematical Conference
CSASC 2011 with a minisymposium Categorical Algebra, Homotopy theory,
and Applications (chaired by C. Casacuberta, J. Trlifaj and me). See:
http://www.dmg.tuwien.ac.at/OMG/OMG-Tagung/index.html