Re: opposite category

["Joyal, André" <joyal.andre@uqam.ca>]

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





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