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I've seen many different notion of $\infty$-categories, actual I've seen the operadic-globular ones of Batanin and Leinster and the opetopic too and eventually I'll see the simplicial ones too. Although there are so many notion of $\infty$-category so far I've only seen the following examples:

  • $\infty$-grupoids as fundamental groupoids topological spaces;

  • $(\infty,1)$-categories, mostly via topological example and application in algebraic geometry (in particular in derived algebraic geometry);

  • strict $(\infty,\infty)$-categories, and their $n$-dimensional versions, for instance the various categories of strict-$n$-categories (here I intend $n \in \omega+\{\infty\}$).

There are other example of $\infty$-categories, especially from algebraic topology or algebraic geometry, but also mathematical physics and computer science and logic? In particular I wondering if there's a concrete example, well known, weak $(\infty,\infty)$-category.

(Edit:) after the a discussion with Mr.Porter I think adding some specifications may help:

I'm looking for models/presentations of $\infty$-weak-categories for which is possible to give a combinatorial description, in which is possible to make manipulations and explicit calculations, but also $\infty$-categories arising in practice in various mathematical context.

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There is the $(\infty, n)$-category of bordisms. It is very interesting for algebraic topologists. –  Chris Schommer-Pries Aug 26 '11 at 14:14
Chris, do you know of an up-to-date account of this in the literature? (I know of a couple of accounts, but they might not be considered the most up to date). If you do, I encourage you to post it as an answer, because it's a very important sort of example. Jacob Lurie's name should probably be mentioned as well. –  Todd Trimble Aug 26 '11 at 16:16
Is Lurie's paper sketching the classification of topological field theories not considered up-to-date enough now? –  Jeffrey Giansiracusa Aug 26 '11 at 16:23
In James Cranch's thesis (front.math.ucdavis.edu/1011.3243) there is the $(\infty,1)$-category of spans, which is an example of a rather different flavour from the ones that you mention. –  Neil Strickland Aug 26 '11 at 17:03
Jeffrey: you're probably right -- I'm only asking. –  Todd Trimble Aug 26 '11 at 17:10
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6 Answers

As per Todd's suggestion I am posting this as an answer.

The $(\infty, n)$-category of bordisms is an important example for many reasons, the most imporant of which is its role in the Baez-Dolan cobordism hypothesis. There are several constructions of it, but one of the more modern ones, in the language of $(\infty,n)$-categories ($n$-fold complete Segal spaces), is given in Jacob Lurie's On the Classification of Topological Field Theories.

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I think it's worth mentioning the examples of weak $\infty$-groupoids constructed from type theory due to van den Berg and Garner and Lumsdaine. These examples are constructed out of syntax and so are very concrete in a certain sense (cf. Tom Leinster's answer). These models of $\infty$-groupoids arise from Martin-Löf type theory which was designed in such a way as to exhibit good computational properties. So there is a certain sense in which it is possible to compute with types (although it is a more technical sense of "compute" than you may have in mind). Of course, one expects that many of these examples (at least of $(\infty,1)$-categories) to be very closely related (as are, e.g., fundamental groupoids of spaces and Kan complexes).

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This probably isn't "well known", and you might think it's cheating. Nevertheless, it's a very concrete, maximally weak, example of an $\infty$-category that isn't an $(\infty, n)$-category for any $n < \infty$. It's simply the free $\infty$-category on one cell in each dimension.

For "free" to make sense, you have to use an appropriate definition of $\infty$-category (e.g. any one in which $\infty$-categories are defined as algebras for some particular monad or operad). For Segal-type definitions, it's not clear that it does make sense.

I say "very concrete" because the cells and operations of this free $\infty$-category can be described in an explicit combinatorial way, much as the elements of a free group admit an explicit combinatorial description -- except that this, of course, is more complex.

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@tomleinster This should be the free $\infty$-category on the terminal globular set, shouldn't it? Is there any reference in which there's the explicit combinatorial description of this weak category? –  Giorgio Mossa Sep 1 '11 at 16:06
I think the original reference should be Batanin's paper "Monoidal Globular Categories as a natural setting..." (which unfortunately doesn't seem to be freely available). Somewhere in among the various categories of pasting diagrams and so forth in Leinster's book "Higher Operads, Higher Categories" (freely available on his website) lies this particular object, but I'm not sure exactly where. Street's computads (see the nlab page) are supposed to enable explicit combinatorial, generators-and-relations treatment of higher categories. –  Tim Campion May 27 '12 at 2:57
It's hard for a non-expert to gauge just what what role computads play -- Batanin erroneously claimed (I think in the paper mentioned above?) that they form a presheaf category; this was corrected by Makkai and Zawadowski in their paper "The category of 3-computads is not cartesian closed". In Leinster's book, I think one roughly finds the idea "computads = many-in/many-out version of opetopes", and the fact that they don't form a presheaf category is taken as a sort of "no-go" theorem for using them in an "operadic" way. But Batanin has some sort of preprint saying that they're still useful –  Tim Campion May 27 '12 at 3:03
available from his webpage web.science.mq.edu.au/~mbatanin/papers.html . But that preprint is pretty old, and my impression as a non-expert is that Batanin and Weber's work over the last decade doesn't really work through stuff combinatorially... I would really love for someone to explain to me what's wrong or unclear with this or any other of my above statements. –  Tim Campion May 27 '12 at 3:07
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It's plausible that there's a translation between the notion of an "$A_\infty$ disk-like n-category" (c.f. my paper "The blob complex" with Kevin Walker) and the usual notion of an $(\infty,n)$-category. (Roughly, the only difference should be that disk-like requires lots of duality.)

In that case the blob complex of an $(n{-}k)$-manifold with coefficients in any disklike $n$-category gives an $(\infty, k)$-category.

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The obvious concrete example is any Kan complex considered as a weak infinity groupoid. If that is not concrete enough, take a space and its singular complex is a weak infinity category. If you want category as against groupoid, the homotopy coherent nerve of a simplicially enriched category $\mathcal{B}$, is another example (provided $\mathcal{B}$ is `locally Kan' i.e. fibrant.) Thus setting size issues aside, the category of topological spaces yields an infinity category. (Look up homotopy coherent nerve in the nLab if you need. It is a very neat idea.)

(Edit: I should have started by asking what `concrete' means for you.)

(Edit number 2:) I see my original answer did not address the last part of the question. For that you are requiring morphisms in all dimensions to be potentially non-(invertible up to higher cells) so there is a non-reversiblility about things. Chris's examples give some idea of this but there is a nice set of ideas that have not been fully explored as yet that may give another. The context in which this arises is that of directed homotopy. This arises is computer science when modelling concurrent and distributed computing. An action takes time and resources, so is non reversible. If you model things by a directed space (and there are various interpretations of that idea see Marco Grandis' book for instance), and then use directed $n$-simplices for all $n$ and the test spaces, you get a singular complex with quite `singular' properties! (Look at directed space in the nLab for some ideas of what is going on here.) I tried to capture some of this in a paper (Enriched categories and models for spaces of evolving states, Theoretical Computer Science, 405, (2008), pp. 88 - 100.) The structure would seem to be related to the Segal category types constructions, but an adequate description is still lacking.

The challenge is then to solidify this link and then to find out if it does give an adequate model of the sorts of situation modelled by the directed spaces in the first instance.

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By concrete in this context I mean a model in which one can actually do manipulations or carry out some calculations. An example from group theory: a concrete group is for instance the group of word over an alphabet, or the group of permutation; other examples from algebra and category theory are the polinomial rings but also the free-category over a graph. In this case what I'm looking for is some example of ∞-weak-categories in which it's easy to see what are the n-morphisms, for each n∈N, and its also easy to describe the compositions between the morphisms. I hope this answer your question. –  Giorgio Mossa Aug 26 '11 at 21:47
In that case, take any simplicial group, and look at the Moore complex. The simplicial group sits on its underlying simplical set which is a Kan complex, and there is a filler algorithm which gives a (weak) composite of a pasting scheme of faces of a horn of a simplex. Have a look at the Menagerie notes a fairly short version of which can be got from the nLab page of that name. In short you want to study the combinatorial homotopy (in J.H.C.Whitehead's sense) of weak infinity categories, a worthy aim. (I can say more if you like but I never know how much detail to give in a comment!) –  Tim Porter Aug 27 '11 at 10:36
To my mind, the most concrete example of a multiple weak category in which one can easily see the compositions is the singular cubical complex $S^\square X$ of a space $X$. (Even better in some ways is the cubical singular complex $R X_*$ of a filtered space, but that is another story.) This has the advantage one can easily formulate multiple compositions $[\alpha_{(r)}]$ where $(r)$ is a multi-index. I don't know how to do that simplicially or globularly. –  Ronnie Brown Jan 29 '12 at 11:30
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A concrete example of a weak $\infty$-category, but not well studied abstractly, is the cubical singular complex of a space, preferably with the connections introduced in our 1981 JPAA paper with Philip Higgins. The clear advantage of the cubical setup is the command of multiple compositions, using an easy matrix notation. Thus one can express for the diagram


that the big square is the composition of the little squares, by simply writing something like $\alpha=[\alpha_{ij}]$. (How does one do that simplicially, or globularly?) This and higher dimensional versions are useful in expressing algebraic inverses to subdivision, a useful tool in local-to-global problems. Because higher groupoids are nonabelian, unlike higher groups, one can also obtain nonabelian results in algebraic topology.

The book "Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids" (EMS, (2011), pdf available from here) has a large number of uses of algebraic and geometric (e.g. homotopical) conclusions derived from rewriting such multiple arrays. The main results were conjectured and eventually proved by cubical methods.

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