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added a sentence to the Idea-section at Kan complex
I have uploaded a copy of the letter that was referred to there (16-06-1983.pdf).
Thanks, that looks nicer!
Tim, I have edited your pointer to make it be more in line with other references, it now reads as such:
Okay?
By the way, the entry would deserve many more (historical) references. The References-section should start out as: “Kan complexes were first considered in … then further developed in …” etc. Something like this. Do you maybe feel like adding something?
Also, as I guess you have seen, I am right now editing the entry. I want to bring it into better shape, such that it becomes suitable as a source to use in lectures.
So far I have reworked the Idea-section a bit. Check it out and further edit where you feel the need.
Here’s a question from someone who is trying to learn some homotopical algebra: Why Kan complexes? As evidenced by the vast body of results, the definition is a good one. But, even assuming that simplicial sets are a good thing to consider (and of this I have yet to find a satisfactory explanation), I am unable to see what the inspiration behind the definition was…
Kan complexes are but a model, for something else that does have a profound “why”-answer. As all models, they justify themselves only by what they allow to do easily. One should not worry about the “why” of the model too much, I think.
Zhen Lin: Good point, and this needs what Urs was suggesting. Historically simplicial sets came to the fore because of (co)homology. I believe it was Sammy Eilenberg who used the singular complex rather than simplicial complexes for this and then noted that this led to a thing that was like a simplicial complex but was floppier! The degeneracies gave some extra structure that was not there in the simplicial case and then Kan noted that to do anything nice with simplicial sets you needed filler conditions, otherwise you did not get composability of homtopies and hence things went wrong. (He probably thought of the filler conditions for cubical sets first but then applied the same idea to their simplicial cousins.
Have you looked at Kamps & Porter (shameless advertising!!!!). We tried to give a way through what you ask starting on page 159. One of our points was that some proofs in homotopy theory do not need the whole of the Kan complex condition and perhaps seeing what result needs what conditions sheds light both on that particular case (for instance a particular horn needs to be filled..) and on those conditions in general.
Why are simplicial objects a good thing? Even in homological algebra one ends up needing something like them and the resolutions one gets are usually quite naturally simplicial.
Another source you might like is my Menagerie notes that can be found via the n-Lab…. and are FREE!!!!!
At Kan complex I have expanded the remark right beneath the definition, to now read as follows:
+– {: .num_remark }
The last characterization in def. \ref{KanComplexes} is sometimes taken to induce the generalization to internal Kan complexes in ambient geometric contexts. For instance the generalization of Lie groupoids to “Lie Kan complexes” might be defined to be given by simplicial objects in the category SmoothMfd of smooth manifolds such that the morphisms
$[\Delta[n], S] \to [\Lambda^i[n],S]$While this is useful for some purposes, one should beware that this naive generalization, if taken at face value, may break the homotopy theoretic interpretation of (smooth, say) Kan complexes as models for (smooth, say) ∞-groupoids. A homotopy-good theory of Lie Kan complexes is discussed in (NSS, section 4.2). See at internal ∞-groupoid for more..
=–
@Tim Porter: A very interesting book, thank you!
In reaction to Zhen Lin’s comment above I have added the following remark to Kan complex (currently Remark 4 in the section “As models for $\infty$-groupoids”)
Of all the models for ∞-groupoids known in the literature, Kan complexes are probably the most widely used, certainly in homotopy theory and related “geometric” approaches to higher category (such as in terms of n-fold complete Segal spaces etc.), less so in “algebraic” approaches to higher category theory. To a large extent this is because the category of Kan complexes, in particular when thought of as the full sub-category of fibrant objects inside the standard model structure on simplicial sets lends itself usefully to many computations; to some extent it is maybe a historical coincidence that specifically for this model the theory was worked out in such detail. Maybe if Kan – who first tried cubical sets and then rejected them in favor of simplicial sets due to some technical issues – had tried cubical set with connection first, things would have developed differently. See at cubical set for discussion of this issue.
But in any case it seems clear that there is no “fundamental” conceptual role to prefer Kan complexes over other models for ∞-groupoids. Instead, in view of modern developments it seems right to regard the abstract concept of homotopy type (not as an equivalence class, but as a representative, though) as fundamental, and everything else to be “just a model” for this, which may or may not be useful for a particular computation. This point of view is formalized by the univalent foundations of mathematics in terms of homotopy type theory. Here the theory of homotopy types is given as an abstract foundational notion and then Kan complexes and related structures are shown to be a model. For more on this see at homotopy type theory.
I have added to Kan complex the observation that any simplicial model of a Mal’cev theory is a Kan complex. (For example, the subobject classifier is Kan because it is a model of the theory of Heyting algebras, which is a Mal’cev theory.)
I think the proof (which I have not seen; I’m guessing) is a straightforward adaptation of the proof in the case of simplicial groups, where elements of the form $x y^{-1} z$ that recur there are replaced by values $t(x, y, z)$ of the Mal’cev operation.
In fact, it is true for simplicial objects in Mal’cev categories in general. This is proved in [Carboni, Kelly, and Pedicchio, Some remarks on Maltsev and Goursat categories].
Thanks for mentioning that; I’ve now added that as a remark.
added Kan fibration as a related concept.
Cross-linked with Kan object.
added pointer to:
Added:
Originally, Kan complexes were defined in a cubical setting, where Kan introduced the extension property for cubical horns and established the elementary properties of cubical homotopy groups:
The second paper in the series proves that the cubical singular complex functor yields cubical Kan complexes:
The third paper introduces simplicial Kan complexes (as “c.s.s. complexes which satisfy the extension condition”), and defines and studies the Kan fibrant replacement functor:
The fourth paper proves that simplicial groups are Kan complexes and studies their properties, including the Kan loop group functor:
The details for Part III above appeared in the paper
John C. Moore in his review of this paper for Mathematical Reviews indicates that the term “Kan complex” was already in use at the time:
For such complexes, usually called Kan complexes, homotopy groups may be defined, and, further, the notion of homotopy between maps of an arbitrary semi-simplicial complex into a Kan complex is an equivalence relation.
The details for Part IV above are given in the following paper:
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