Alain Connes's cycle category $\Lambda$ (sometimes denoted $\mathcal{C}$), often called his cyclic category or category of cycles, is a useful source for presheaves analogous to the simplex category $\Delta$.
The cycle category may be defined as the subcategory of Cat whose objects are the categories $[n]_\Lambda$ which are freely generated by the graph $0\to 1\to 2\to\ldots\to n\to 0$, and whose morphisms $\Lambda([m],[n])\subset\mathrm{Cat}([m],[n])$ are precisely the functors of degree $1$ (seen either at the level of nerves or via the embedding $\mathrm{Ob}[n]_\Lambda\to \mathbf{R}/\mathbf{Z}\cong S^1$ given by $k\mapsto k/(n+1)\,\mathrm{mod}\,\mathbf{Z}$ on the level of objects, the rest being obvious).
The simplex category $\Delta$ can be identified with a subcategory of $\Lambda$, having the same objects but with fewer morphisms. This identification does not respect the inclusions into $Cat$, however, since $[n]$ and $[n]_\Lambda$ are different categories.
Denote the generator of $\Aut_\Lambda([n])$ by $\tau_n$; then of course $\tau_n^{n+1} = \mathrm{id}_{[n]}$. This enables more standard, and equivalent, presentation of $\Lambda$ by generators and relations. In addition to the cosimplicial identities between the coboundaries $\delta_i$ and codegeneracies $\sigma_j$ and $\tau^{n+1}_n = \mathrm{id}$ there are following identities:
Cyclic objects in a category $C$ are the contravariant functors $\Lambda^{\mathrm{op}}\to C$, cocyclic objects are the covariant functors $\Lambda\to C$. Note that $\Lambda$ itself is, via its inclusion into $Cat$, an example of a cocyclic object in $Cat$. (Thus, the common term “the cyclic category” to refer to $\Lambda$ is misleading, just like using “the simplicial category” to refer to the simplex category $\Delta$.)
If $A$ is an abelian category then the category of $A$-presheaves on $\Lambda$ is usually called (Connes's) category of cyclic modules in $A$.
$\Aut_\Lambda([n]) = \mathbf{Z}/(n+1)\mathbf{Z}$
$\Lambda([n],[m]) = \Delta([n],[m])\times \mathbf{Z}/(n+1)\mathbf{Z}$ (as a set)
Any morphism $f$ in $\Lambda([n],[m])$ can be uniquely written as a composition $f = \phi\circ g$ where $\phi\in\Delta([n],[m])$ and $g\in\Aut_\Lambda([n])$.
The generalizations of this theorem are the starting point of the theory of skew-simplicial sets of Krasausukas or equivalently crossed simplicial groups of Loday and Fiedorowicz.
The cyclic category is a generalized Reedy category, as explained here.
The cycle category is a generalized Reedy category. Hence “cyclic spaces” carry a generalized Reedy model structure.
Blog discussion
Literature:
J.-L. Loday, Cyclic homology, Grundleheren Math.Wiss. 301, Springer 2nd ed.
V. Drinfeld, On the notion of geometric realization, arXiv:math.CT/0304064
Alain Connes, Noncommutative geometry, Academic Press 1994 (also at http://www.alainconnes.org)
R. Krasauskas, Skew-simplicial groups, (Russian) Litovsk. Mat. Sb. 27 (1987), no. 1, 89–99, MR88m:18022 (English transl. R. Krasauskas, Skew-simplicial groups, Lith. Math. J. )
W. G. Dwyer, D. M. Kan, Normalizing the cyclic modules of Connes, Comment. Math. Helv. 60 (1985), no. 4, 582–600.
W. G. Dwyer, M. J. Hopkins, D. M. Kan, The homotopy theory of cyclic sets, Trans. Amer. Math. Soc. 291 (1985), no. 1, 281–289.
Z. Fiedorowicz, Jean-Louis Loday, Crossed simplicial groups and their associated homology, Trans. Amer. Math. Soc. 326 (1991), no. 1, 57–87, MR91j:18018, doi
This discussion is about getting the definition right.
Mike: Sorry, I don’t think I believe this either. The category freely generated by any of the above graphs (for $n\gt 0$) has infinitely many morphisms between any pair of objects, and therefore (since it is free) infinitely many endomorphisms. But aren’t the hom-sets of $\Lambda$ supposed to be finite?
Zoran Škoda Hom sets of $\Lambda$-yes, but we are now trying to find the concrete realization of objects of $\Lambda$: there are many realizations some involving some sort of categories with infinitely many morphisms. Each $C_n$ in presentation above have infinitely many morphisms in $\mathrm{Cat}$, but only finitely many in $\Lambda$ as I just corrected (thanks for being alert and convince me to think three times!). The infinities in models for $\Lambda$ have to do with a ‘reason’ why for example finite group model has $SL(2,Z)$ symmetry – as it is natural to be explained in terms of second iteration of inertia orbifold, which is closely related to cyclic cohomology. But it is confusing because the t-operators are finite…Drinfeld talks about $Z_+$ categories when talking about $\Lambda$, maybe I confused something, I’ll think about it, in that setup there are infinitely many morphisms for what he calls $[n]_{cyc}$ but maybe it is not the same what I intended to do.
Mike: What does it mean for a functor to be “of degree 1?” I assume that your parenthetical is meant to explain this, but to me it is not obvious.
Zoran Skoda: By degree I mean the degree of a map in the sense of homotopy theory – the class of circle to circle map, either at the level of nerves or looking at the subset of circle.
Mike: It’s not immediately obvious to me that the nerve of your category $[n]_\Lambda$ is homotopically a sphere or something else for which the notion of ‘degree’ makes sense. What does “looking at the subset of circle” mean? I would prefer if we give a more explicit combinatorial description of $\Lambda$ as its definition, although we could also include this version later on the page.
Zoran Skoda: As I wrote above, $k\mapsto k/(n+1)\,\mathrm{mod}\,\mathbf{Z}$, is THE formula for embedding $[n]_{\Lambda}$ into the circle. On the other hand, the nerve the free category on the graph $0\to 1\to ..\to n\to 0$ is homotopically the circle, isn’t it? I think the definition is cleaner than the explanation below via generators and relations.
Mike: What about “$\Lambda$ is the category of finite nonempty cyclically ordered sets?” I think that gets across the intended intuition better than either, and is cleaner than either modulo a definition of “cyclic order.”
Zoran Skoda: very good!
Mike: I think I understand what you are getting at with your definition now, although I still don’t think it’s quite right yet. I agree that the nerve of $[n]_\Lambda$ is homotopically a circle—except when $n=0$. And I think that exception means that not all the functors you want have degree 1—those that factor through $[0]_\Lambda$ have degree 0. It seems like those might be the only functors with degree 0, though so maybe it would suffice to consider all functors with degree 0 or 1.
Mike Shulman: Apparently I’m wrong: the $0$-cycle is still supposed to be a “loop” of some sort. So maybe your definition is right as long as the category $[0]_\Lambda$ is defined as freely generated by an endomorphism $0\to 0$.
This page should probably be rewritten with an “Idea” section at the beginning and then descriptions of the many different ways to define it formally.
This discussion is about the name of the category.
We might also call it the cycle category in analogy with simplex category, cube category, and globe category that we've already got here. If that's a good system. —Toby
Mike: I like that system.
Zoran Škoda I personally prefer category of cycles, even sometimes category of simplices, category of (hyper)cubes as I hear from geometers. Partly because when you translate to other languages, bahuvrihi style (which is anyway an abbreviation of the other form) is not preferred (unlike in German where it is even written as one word, and in English in which it is one word but is written as two), or sometimes impossible, hence one needs to convert the modifier back into an adjective when translating, what one does not need with saxon genitive. But I am ambivalent to that issue in other cases, but cycle category sounds too similar to cyclic category (for simplicial there is no problem as it sounds very different from simplex)…
Mike: Of course, also “category of simplices” has a different meaning: one talks about “the category of simplices of a simplicial set” to mean the comma category of $\Delta$ over it. The simplex category is then the category of simplices of the terminal simplicial set.
Regarding translation, I would be inclined to just regard that as something that happens in translation. Since English uses noun adjuncts frequently, it must be commonplace for translators to replace them with the preferred forms in other languages. There are lots of other cases in translation where you can’t just replace word-for-word; doesn’t translation really consist instead of writing new sentences in the target language with the same meaning as those in the source language?
Zoran Skoda: Look, one can translate a phrase, but not just a stack of nouns. Stack of nouns either stays stack of nouns (what is very awkward nowdays, with young people using lots of stacks of nouns literally from English semi-translated to languages which do not do massive bahuvrihi compounds) or need to be expanded/described. But how to expand cycle category then to category of cycles. Hence I have no problem to translate cycle category to Croatian, but then it will coincide in Croatian with category of cycles. Or I can make unnatural compounds with hyphen ciklus-kategorija, what sounds like water-fruit for juicy fruit or for fruit with juice. What do you, mathematics-man think of this word-compound tongue-thing ? And beware that the cost (unusualness) in many languages of such compounds is orders of magnitude more unusual than in English. Or give me another descriptive expansion (if category of cycles already used/not accepted!).
By the way, I hear around certain students of Kan talking “the category of simplices” for $\Delta$. I see no problem with the fact that there are more general “categories of simplices” in specialistic usage (non-specialists use them very little and specialists will anyway say simplices where…$\Delta$ is used by everybody, not only homotopy theorists or simplicial experts, the latter comma category is a rather specialists’ thing).
Mike: You appear to be trying to ridicule me for exactly I was saying one shouldn’t do. Of course, when you translate “cycle category” to Croatian it will come out the same as “category of cycles.” When written in language X, mathematics should be written in language X, not simply obtained from language Y by replacing things word-for-word; this is just as true when X=English and Y=Croatian as when X=Croatian and Y=English. In particular, a noun together with some noun adjuncts is a phrase in English, not just a “stack of nouns,” and should be translated to result in a grammatical phrase in whatever other language one is translating to. Don’t blame me because some people translate things from English incorrectly.
I don’t have a problem with people saying “category of simplices” for $\Delta$, but I prefer to say “simplex category” myself as it is slightly more precise.
Mike: You do, however, have a good point that when writing in English we should not try to distinguish in meaning between “cycle category” and “category of cycles,” since when translating into many other languages they will become the same. I would still prefer that we name the page “cycle category” here on the nLab, since it accords better with our existing terminology for similar categories, but I think you should feel free to use “category of cycles.”