Dear all, I'm sure the following question has been answered to. Could anyone give me a precise answer and references to this answer. Many thanks. QUESTION Let p: S --> X be a functor. What conditions should satisfy p to be called a structure functor, i.e. such that every object s of S can be thought of as a structure on the object p(s). [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Jean, The simplest answer is: faithful. But a better one (in view of `everything up to isomorphism') is: faithful and amnestic. The latter means that p reflects identity morphisms: an isomorphism in S is an identity if its image by p is. See The Joy of Cats (free on the web). Best, Jiri
QUESTION Let p: S --> X be a functor. What conditions should satisfy p to be called a structure functor, i.e. such that every object s of S can be thought of as a structure on the object p(s).
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Jean and Jiri, As we know there is no such notion accepted by everybody. I would probably vote for faithful + amnestic + iso-fibration. Best regards, George -------------------------------------------------- From: "Jir? Ad?mek" <j.adamek@tu-bs.de> Sent: Wednesday, February 8, 2017 6:34 PM To: "categories net" <categories@mta.ca> Subject: categories: Re: Terminology
Dear Jean,
The simplest answer is: faithful. But a better one (in view of `everything up to isomorphism') is: faithful and amnestic. The latter means that p reflects identity morphisms: an isomorphism in S is an identity if its image by p is. See The Joy of Cats (free on the web).
Best, Jiri
QUESTION Let p: S --> X be a functor. What conditions should satisfy p to be called a structure functor, i.e. such that every object s of S can be thought of as a structure on the object p(s).
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Jean, unless there is a technical meaning of "structure" I'm not aware of, the answer may be "Concrete categories" in the sense of Adámek, Herrlich, and Strecker: http://katmat.math.uni-bremen.de/acc/acc.pdf. A concrete category is just a faithful functor, but a remarkable amount of theory can be build on that notion. In particular, a classification of "algebra-like" and "space-like" structures is already possible at that level. On Wed, Feb 8, 2017 at 4:56 PM Jean Benabou <jean.benabou@wanadoo.fr> wrote:
Dear all,
I'm sure the following question has been answered to. Could anyone give me a precise answer and references to this answer. Many thanks.
QUESTION Let p: S --> X be a functor. What conditions should satisfy p to be called a structure functor, i.e. such that every object s of S can be thought of as a structure on the object p(s).
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Jean, in Remark 13.18 of their book on "Algebraic Theories" Adamek, Rosicky and Vitale suggest the following conditions 1) p faithful (what they call "concrete over X") 2) p-vertical isos are identities (what they call "amnestic")) 3) p is an isofibration (what they call "transportable") These seem to be reasonable conditions validated by most examples. Does this confirm with your intuition? Thomas
I'm sure the following question has been answered to. Could anyone give me a precise answer and references to this answer. Many thanks.
QUESTION Let p: S --> X be a functor. What conditions should satisfy p to be called a structure functor, i.e. such that every object s of S can be thought of as a structure on the object p(s).
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
Carsten Führmann -
George Janelidze -
Jean Benabou -
Jirí Adámek -
Thomas Streicher