Interesting. And FGA is not exactly an obscure source, so it is really bizarre how the notion of an indexed category is misattributed.

]]>the first extensive usage of the indexed categories (contravariant pseudofunctors from a usual category to Cat) is from Grothendieck’s first FGA article about descent around 1958

Right, p. 3 here.

I have removed the claim that the notion is due to Lawvere and instead added this:

The notion of

indexed categorieswas introduced in

- Alexander Grothendieck, §A.1 (p. 3) of
Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats, Séminaire N. Bourbaki exp. no190 (1960) 299-327 [numdam:SB_1958-1960__5__299_0]but under the name “fibered category” (

catégorie fibrée) which later became the standard term, instead, for the (equivalent) Grothendieck construction on an indexed category, cf.

- Jean Bénabou, p. 898 (2 of 41) of:
Fibrations petites et localement petites, C. R. Acad. Sci. Paris281Série A (1975) 897-900 [gallica]

Let’s add in attribution to Lawvere if and when we have concrete referenc to support it. (Our entry *William Lawvere* claims that an “approach to indexed categories” is in the Perugia 1972 lecture notes pdf, but on browsing through it once I haven’t spotted such).

3 “Indexed categories were introduced and developed by Lawvere in early 1970s”

The *name* indexed category is indeed from that period (with major influence of the book by Pare and Schumacher), but the first extensive usage of the indexed categories (contravariant pseudofunctors from a usual category to Cat) is from Grothendieck’s first FGA article about descent around 1958. Only a year or two later Grothendieck switched to fibered categories, and the article in SGA on descent is indeed in fibered formalism (article written by Pierre Gabriel under Grothendieck’s guidance). There is often heard falacy in 1-category community, totally wrong, that Grothendieck (extensively) used (only) fibered categories and American school pioneered/used contravariant pseudofunctors/indexed categories. Benabou, who made huge contribution to the subject, himself complained to this historical falacy on category list more than once. I once wrote my own complaint to a thread in category list and my contribution has been rejected by (presumably an American school belonging) moderator. Probably, under the influence of Gabriel-Grothendieck more clean SGA approach to descent it became more standard ro use property-approach of fibered categories and everybody looked as THE approach to descent of Grothendieck school so the structure point of view embraced and further developed by American-Canadian school was making impression of being a new approach.

(One should also point out that it was Grothendieck who introduced the common notation for writng fibration-like functions/functors with drawing up to down function arrow suggesting the original point of view, including on pseudofunctors as “indexed” families.)

]]>Thanks. I am making all these author-hyperlinks work…

]]>Added:

Indexed categories were introduced and developed by Lawvere in early 1970s. Further contributions were made by Bénabou. An early account of the theory can be found in

- P. T. Johnstone, R. Paré, R. D. Rosebrugh, D. Schumacher, R. J. Wood, G. C. Wraith,
*Indexed Categories and Their Applications*, Lecture Notes in Mathematics 661 (1978). doi:10.1007/bfb0061362.

added pointer to:

- Andrzej Tarlecki, Rod M. Burstall Joseph A. Goguen,
*Some fundamental algebraic tools for the semantics of computation: Part 3. indexed categories*, Theoretical Computer Science**91**2 (1991) 239-264 [doi:10.1016/0304-3975(91)90085-G]

This article speaks as if inventing the notion/terminology of “indexed catgeories”. Is that the case?

]]>Used $\tilde \mathbb{C}$ for the name of the indexed category

Bartosz Milewski

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