zigzag category

zigzag category

[Paul Levy]

For a set A, let Cat(A) be the category whose objects are small
categories with object set A, and whose morphisms are identity-on-
objects functors.

For a small category C, let zigzag(C) be the coproduct of C and C^op
within Cat(ob C).

Explicitly, in zigzag(C), a non-identity morphism z : a ---> b is a
nonempty sequence of non-identity C-morphisms that alternately go
forwards or backwards.  Depending on the direction of the first and
last C-morphism, z can take one of four different forms.

Surely this appears in the literature?  Google gave me a zillion
categorical papers that mention zigzags, but I didn't find this
construction, although several were close (e.g. the special case where
C is the free category on a graph).

Paul




--
Paul Blain Levy
School of Computer Science, University of Birmingham
+44 (0)121 414 4792
http://www.cs.bham.ac.uk/~pbl











[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: zigzag category

[Chris Heunen]

Dear Paul,

My PhD thesis covers a construction for dagger categories in 3.1.18 and
further that is slightly different but related; at least I also called
this functor Zigzag:Cat->DagCat. In that case, it is left adjoint to the
evident forgetful functor.

best,
Chris

> For a set A, let Cat(A) be the category whose objects are small
> categories with object set A, and whose morphisms are identity-on-
> objects functors.
>
> For a small category C, let zigzag(C) be the coproduct of C and C^op
> within Cat(ob C).
>
> Explicitly, in zigzag(C), a non-identity morphism z : a ---> b is a
> nonempty sequence of non-identity C-morphisms that alternately go
> forwards or backwards.  Depending on the direction of the first and
> last C-morphism, z can take one of four different forms.
>
> Surely this appears in the literature?  Google gave me a zillion
> categorical papers that mention zigzags, but I didn't find this
> construction, although several were close (e.g. the special case where
> C is the free category on a graph).
>
> Paul
>
>
>
>
> --
> Paul Blain Levy
> School of Computer Science, University of Birmingham
> +44 (0)121 414 4792
> http://www.cs.bham.ac.uk/~pbl
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]