Dear Jean, Within directed algebraic topology (dat) and for (perhaps) all researchers working in this domain, a category is by itself a 'directed structure' while a groupoid is not (or trivially so): - a topological space has a fundamental groupoid, - a directed topological space (in each of its versions) has a fundamental category, - a directed topological space (in each of its versions) has an 'opposite object', whose fundamental category is the opposite of the previous one. I hope this clarifies our use of the term 'directed' in connection with category theory. Thank you for your interest in my book, and more generally in dat - I think it is a fascinating subject. Best regards Marco PS By the way, the French term for dat is - as far as I know - "topologie algebrique dirigee", not "directe". I think it is a good choice. M On 27 Nov 2013, at 09:40, Jean Bénabou wrote:

Dear Marco,

Many thanks for your explanations, and of course for giving me the possibility to study your book, which I shall do very soon.

I have a notion which I hesitated to call directed category for fear of a conflict in using the word directed. Your non technical use of this word encourage me to use it, but I shall first read your book before I make up my mind.

Best regards, Jean

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