Subject: Orthogonality and toposes question.

From: categories <[email-masked]>
Date: Thu, 14 Mar 1996 17:11:10 -0400 (AST)

Date: Thu, 14 Mar 1996 18:35:00 GMT
From: Marcelo Fiore <[email-masked]>

Below I consider a notion of orthogonality with respect to cones, 
generalising that of orthogonality with respect to maps and the sheaf 
condition for a cover in a Grothendieck topology:

  1. We say that an object  K  is orthogonal to a cone  D --> C  
     whenever for every cone  D --> K  there exists a unique  
     C --> K  such that  (D --> C --> K) = (D --> K).

  2. For a category  K  and a class  J  of cones in  K  we define 
     O(K,J)  as the full subcategory of  K  consisting of all those 
     objects orthogonal to every cone in  J.

My question is:

  Let  A  be a small category and write  Psh A  for the topos of 
  presheaves on A. 

  Is there a characterisation of the classes  J  of cones in  A  for 
  which  O(Psh A,J)  is a topos?

Comments and pointers to relevant literature are welcome.  Many thanks,
Marcelo.

---

From: categories <[email-masked]>
Date: Thu, 14 Mar 1996 23:08:32 -0400 (AST)

Date: Thu, 14 Mar 1996 19:21:43 -0500 (EST)
From: Dan Christensen <[email-masked]>

| Below I consider a notion of orthogonality with respect to cones, 
| generalising that of orthogonality with respect to maps and the sheaf 
| condition for a cover in a Grothendieck topology:
| 
|   1. We say that an object  K  is orthogonal to a cone  D --> C  
|      whenever for every cone  D --> K  there exists a unique  
|      C --> K  such that  (D --> C --> K) = (D --> K).

  I've come across examples of the above in stable homotopy theory,
except that uniqueness of the map C --> K doesn't hold in general.
(In general one only gets these weakened versions of colimits in the
homotopy category, if one gets anything at all.)  It seemed to me to
be unnatural, but now I'm happy to find that someone else has come
across these partial colimits.  Could you pass on any references you
know, or any that people send to you?

Thanks,

Dan

---

From: categories <[email-masked]>
Date: Fri, 15 Mar 1996 09:36:00 -0400 (AST)

Date: Fri, 15 Mar 96 9:47:37 MET
From: Koslowski <[email-masked]>

With regard to Marcelo Fiore's question about orthogonality with
respect to cones:  together with Gabriele Castellini and George Strecker
I have studied some aspects of this phenomenon for discrete cones in

Regular Closure Operators
Applied Categorical Structures 2 (1994), 219--244
Kluwer Academic Publishers

Section 5 of this article might be of interest.

Best regards,

-- J"urgen Koslowski