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Ge-categories, i.e., categores enriched over groupoids (these are 2-categories where the set of morhisms $HOM(a,b)$ has a groupoid structure) seem to be useful in homotopy theory.

Question: What are results in homotopy theory proved using ge-categories?

I am especially interested in ge-categories where 1-morphisms are not invertible. I would also appreciate references.

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    $\begingroup$ Is someone really using "e.g." not to mean "for example"? There's not some misunderstanding here? I tried googling but of course found nothing because the search results were all in the sense of "for example." $\endgroup$
    – Noah Snyder
    Commented Jan 7, 2018 at 2:57
  • $\begingroup$ @NoahSnyder: I changed it to g.e. everywhere. If you want to know what it is, see MR3654359, for example. If you want to use google, search for "2-categories enriched over groupoids". But if I could google an answer, I would not be posting a question here. $\endgroup$
    – user6976
    Commented Jan 7, 2018 at 3:09
  • $\begingroup$ Sorry, I didn't mean to imply that googling would give an answer, just that I hadn't heard the term before, and so I googled it. $\endgroup$
    – Noah Snyder
    Commented Jan 7, 2018 at 3:27
  • $\begingroup$ I'm a little confused about the indexing here. The MR you mentioned seems to be talking about categories enriched in groupoids (i.e. it's a category where the homs form groupoids which are a special kind of category, which makes the whole thing a special kind of 2-category (or category enriched in categories)), not 2-categories enriched in groupoids (which would be a special kind of 3-category). Assuming it's the former, the more standard name these days would be (2,1)-category. $\endgroup$
    – Noah Snyder
    Commented Jan 7, 2018 at 3:32
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    $\begingroup$ @NoahSnyder: Yes, these are 2-categories which are categories enriched over (in) groupoids. I will fix the grammar. $\endgroup$
    – user6976
    Commented Jan 7, 2018 at 3:34

2 Answers 2

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There are lots of results provable in this context. In the book I wrote with Heiner Kamps (which is easily found via Google so I won't advertise here!) we looked at the problem of what results in homotopy theory could be proved with a restricted set of fillers for boxes in a cubical enrichment of a category. This applies to your question since groupoid enriched categories give rise to such cubical homotopy theories very easily.

There is an old paper: P. H. H. Fantham and E. J. Moore, Groupoid enriched categories and homotopy theory, Canad. J. Math., 35, (1983), 385 – 416, which also examines this question and of course, some of the classical book by Gabriel and Zisman is devoted to developing GE-categories in your sense.

As Noah points out, these 2-categories are nowadays more often called (strict) (2,1)-categories although that term (without the `strict') also is used for bicategories in which the homs are groupoids. Try looking up locally groupoidal 2-category in the nLab for more on that side of things.

(Edited (08-01-2018): I should have mentioned the extensive work by Hans Baues and his coworkers on what he calls 'track categories'. These are the 'ge-categories' of the question. There are many problems solved within the more calculative part of homotopy theory that are stated in terms of these track categories but which have direct interpretation in more classical approaches homotopy theory.)

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  • $\begingroup$ About "nowadays", the paper I mentioned above is published last year. My question is still open "what results in homotopy theory were proved using ge-categories?" The paper by Fantham and Moore is about reformulation of old results in the new language. Or did I miss any new results in that paper? Perhaps I had to say explicitly that I am interested in new results. $\endgroup$
    – user6976
    Commented Jan 7, 2018 at 11:12
  • $\begingroup$ The comment about the cube structure is interesting although you did not seem to mention in the book that the universal cover of the cube complex you produce is CAT(0) (at least in the ge-category case). It was noticed at the end of the 90s by Dan Farley. $\endgroup$
    – user6976
    Commented Jan 7, 2018 at 11:25
  • $\begingroup$ Mark: your question "what results in homotopy theory were proved using ge-categories?" could perhaps be refined to say : "what results in homotopy theory were first proved using ge-categories?" I answered a slightly different interpretation of it ,namely "what results in homotopy theory can be proved using ge-categories?" Which comes down to be a question about what `homotopy theory' is to mean in this setting. $\endgroup$
    – Tim Porter
    Commented Jan 8, 2018 at 7:25
  • $\begingroup$ Your comment about Cat(0) is interesting. How does this relate to more recent models for (cubical) $\infty$-categories? Any ideas? $\endgroup$
    – Tim Porter
    Commented Jan 8, 2018 at 7:27
  • $\begingroup$ I do not know what $\infty$-categories are. Yes, I would like to know what results in homotopy theory were first proved using ge-categories. I would like to see any such result where the word "category" is not in the formulation. $\endgroup$
    – user6976
    Commented Jan 8, 2018 at 13:56
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I claim that a good example of the use of Grpd-enriched categories is the Murayama-Shimakawa model for equivariant classifying spaces for equivariant principal bundles:

Such equivariant classifying spaces were abstractly characterized by the original authors (tom Dieck, Bierstone, Lashof, May) but remained somewhat impalpable until Murayama & Shimakawa 1995 presented a concrete model based on topological realization of equivariant topological mapping groupoids. This construction was only more recently highlighted by Guillou, May & Merling 2017 for its neat category-theoretic nature.

While none of these authors makes the ambient Grpd-enriched category theory fully explicit, one recognizes it behind the scenes.

(For instance, the groupoid of "crossed homomorphism" is equivalently simply the mapping groupoid of sections of the projection out of the delooping groupoid of the corresponding semidirect product group...)

We are writing up a systematic account of universal equivariant principal bundle theory as an excercise in Grpd-enriched category theory. Will add a pointer when its ready for public consumption.

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