You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Yuri (Nov 10 2025 at 18:03):Is there a technical or historical reason why, in my limited exposure to category theory, I tend to see more references to simplicial rather than globular sets in the context of higher categories?
I’ve been reading about models of higher categories and keep encountering simplicial approaches far more often than globular ones — is this mostly a matter of historical development or are there deeper technical reasons?
Kevin Carlson (Nov 10 2025 at 18:45):There are deep reasons. A globular set contains morphisms of every dimension, together with their source and target maps. To use one as the material of a higher category, you thus have to equip it with extra structure in the form of composition operations. In higher category theory, the more operations you add, the more coherences they have to observe, and as the dimension goes to infinity, so do the coherence laws. Thus, for instance, bicategories are manageable but sufficiently harder than categories that weak double categories come in to reduce some of the coherence difficulties, tricategories are just barely possible, and tetracategories are essentially impossible for a human to work with (it's an act of great virtuosity just to write down the definition.) These are all globular definitions, but to get to -categories, you clearly need something different: you simply can't just write down a globular definition of an -category in a naive way. The solution to this is provided by the theory of operads, which algebraic topologists developed in the '60s to handle spaces with continuous families of multiplications satisfying parameterized families associativity conditions, such as spaces. Speaking all too roughly, you can think of a globular -category as a kind of -object in globular sets. What people actually do (in the option I fail least severely to understand) is define an -category as an algebra for the initial contractible [[globular operad]].
Kevin Carlson (Nov 10 2025 at 18:46):Anyway, long story short, it's really really hard to even write down the definition of an -category in this way, and nobody has ever really tried that hard to develop -category theory on the basis of such a definition. I think the objects just become too large, floppy, and inexplicit to really allow for doing explicit calculations. Let's compare to the simplicial situation.
Kevin Carlson (Nov 10 2025 at 18:48):A simplicial definition of an -category is as a quasicategory, which is a simplicial set which is injective with respect to all the maps determined by removing the th face from a simplex, except for or
Kevin Carlson (Nov 10 2025 at 18:52):That's a really short definition! It's also familiar for homotopy theorists: if I had stopped before the "except" clause, that would just be the definition of a Kan complex, the simplicial analogue of topological space that's been what lots of homotopy theorists have worked with in preference to topological spaces for generations. So you can say the definition quickly and you already know there exists a theory that uses a similar definition very productively; more abstractly you get a very nice model category of -categories, and so homotopy theorists have found it quite easy to adapt their existing skills to this case. (You might imagine they could've adapted their skills with spaces to the Batanin -categories too, but the ratio of existing literature on Kan complexes to spaces is probably something like That said, in the stable situation, it really is a common approach to study the -generalization of differential graded categories, categories enriched in chain complexes. This is what you'll find if you Google around for -categories. )
Kevin Carlson (Nov 10 2025 at 18:54):So what's the core, conceptually, of why the simplicial picture is quicker to get off the ground? It's that the simplicial set encodes, not only the sources and targets of a cell, but also the composition. To find the composite of two composable morphisms you say "give me the long edge of any 2-simplex whose two short edges are and " rather than "pick some 2-dimensional operation from the initial contractible globular operad and apply it to and " This puts compositions on the same level as sources and targets, which turns out to be pretty great.
Kevin Carlson (Nov 10 2025 at 18:55):The downside is that, if every 2-simplex is a composite, it's not so obvious how to get non-invertible 2-morphisms. So this works wonderfully for -categories, but then takes more work to upgrade further. But people can do this pretty well, nowadays.
Yuri (Nov 11 2025 at 01:26):@Kevin Carlson, thank you so much for the detailed answer! I actually follow a bit of your reasoning here—though only at a surface level.
FWIW, one of my main sources has been “Computads for Weak ω-Categories as an Inductive Type” by Dean, Finster, Markakis, Reutter, and Vicary (and related papers). I believe Vicary also gave a presentation on some of this work at the Topos Institute a few years ago.
“We give a new description of computads for weak globular ω-categories by giving an explicit inductive definition of the free words. This yields a new understanding of computads and allows a new definition of ω-category that avoids the technology of globular operads. Our framework permits direct proofs of important results via structural induction, and we use this to give new proofs that every ω-category is equivalent to a free one, and that the category of computads with generator-preserving maps is a presheaf topos, giving a direct description of the index category. We prove that our resulting definition of ω-category agrees with that of Batanin and Leinster, and that the induced notion of cofibrant replacement for ω-categories coincides with that of Garner.”
And:
“In our work we show that this quotient set does in fact admit a direct description in each dimension, as a family of inductive sets. This yields an elementary and fully explicit definition of computads for weak globular ω-categories.
Since our definition of computad is concrete and elementary, it allows us to demonstrate known results about computads in new and simple ways.”
So I ended up digging deeper here—without a solid grasp of the alternatives. Still, what mattered to me was that the research turned out to be far more accessible than I expected.