Dear Robert, Peter and all, We often turn covariant functors into contravariant ones: If C is a small category, then the category [C,Set] of covariant set valued functors on C is the topos of presheaves on C^{op}. Recall that the category \Gamma introduced by Graeme Segal is the opposite of the category Fin_\star of finite pointed sets. https://ncatlab.org/nlab/show/Segal%27s+category A Gamma-space was not defined by Segal to be a covariant functor Fin_\star --->Space but as a contravariant functor \Gamma---->Space https://ncatlab.org/nlab/show/Gamma-space -André ________________________________________ From: Peter Selinger [selinger@mathstat.dal.ca] Sent: Thursday, September 14, 2017 11:58 AM To: Categories List Subject: categories: Re: opposite category Robert Pare wrote:
He said there may come a time when we have to consider covariant functors as contravariant ones on the opposite category.
This anecdote seems to have prompted a few posts about opposite categories, but I thought the point of the original anecdote was that Fred said that *covariant* functors should be considered as contravariant functors on the opposite category, i.e., that he considered contravariant functors to be the more fundamental concept. An interesting thought, and obviously tongue-in-cheek. -- Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]