homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
Todd Trimble’s definition of weak $n$-category is an example of a notion of weak omega-category which is based on using an operad induced by all the possible compositions of an interval object with itself for describing weak composition of 1-morphisms. The higher cells and their composition are then obtained iteratively.
Trimble originally called these flabby $n$-categories, which was intended to distinguish them from “weak” $n$-categories (since at the time he wasn’t convinced that they were truly weak enough, owing to the fact that we use strict maps of algebras along the way), and also to recall the word “flab” as used by Frank Adams in his Infinite Loop Spaces book (since there is rather a lot of topological flab in the definition).
An arrow looks like an interval. So, the theory of categories and even n-categories should have a lot to do with the interval — especially when it comes to applications to topology!
In a category we can glue arrows together, ‘composing’ them. But an interval can be chopped apart or ‘decomposed’ into a bunch of intervals. So, there should be a cocategory or something like that lurking around here.
In fact the closed unit interval gives an A-infinity-cocategory: a cocategory where the laws hold up to homotopy, where the homotopies satisfy nice laws up to homotopy, ad infinitum.
The space of maps out of an $A_\infty$-cocategory into something should form an A-infinity category. So, the space of maps out of an interval into a space forms an $A_\infty$-category. And this is an important first step in how Todd Trimble constructs the fundamental n-groupoid of a space!
But the really cool part is how this construction goes hand in hand with his definition of $n$-categories, in an inductive kind of way.
In 1999, Todd Trimble introduced a definition of weak $n$-category in a lecture at Cambridge University. This definition was introduced in order to give an algebraic notion of fundamental $n$-groupoid, and has since been taken up in a number of directions by Tom Leinster and Eugenia Cheng, who compare it with globular approaches (e.g., Batanin’s) and give a weak $\omega$-category extension.
Trimble never published this material himself (in effect leaving it to others present at the Cambridge lecture to develop their own accounts, which has been a good thing), but it may be of at least historical interest to have Trimble’s original account available. We will then pick out the highlights and streamline the approach, following subsequent work by Leinster and Cheng.
The original idea was to use an $A_\infty$-cocategory structure on the topological interval $I$, in the form of an topological operad $T$ which is used to control all ways of weakly composing arrows. Then one proceeds to define weak n-categories by induction, defining at each stage a fundamental n-groupoid functor
and then defining an $(n+1)$-category as a category “weakly enriched” in $nCat$, meaning roughly speaking a structure of $\Pi_n(T)$-algebra in the category of $n Cat$-graphs.
It will be helpful to review the fundamental (1-)groupoid of a space first, doing it in such a way that we will see how to inductively define the fundamental $n$-groupoid.
Let $X$ be a topological space, and let $X_0$ denote its underlying set, viewed as a discrete category. If $V$ is a category, then a $V$-graph over $X_0$ is just a functor
(which here amounts to a function $X_0 \times X_0 \to Ob(V)$).There is an obvious category of $V$-graphs over $X_0$.
A basic example of $V$-graph for our purposes is the “graph of paths” for a space $X$: define
by letting $X^\sim(x, y)$ be the space of paths from $x$ to $y$. Two paths from $x$ to $y$ are homotopic if they lie in the same connected component of $X^\sim(x, y)$, so by applying the connected components functor,
we get the graph of homotopy classes of paths. What we want is to put a category structure on this $Set$-graph, so as to obtain the fundamental groupoid $\Pi_1(X)$.
The $Top$-graph (1) can also be thought of as a topological span from the discrete space $X_0$ to itself, and this is obviously closely related to the span
where $i_0, i_1: 1 \to I$ are the endpoint inclusions. Indeed, we get (1) from (2) by first precomposing with a span from $X_0$ to $X$,
(both arrows being identity functions), followed by postcomposing with the opposite span. This pre- and post-composition gives a functor
to which we return later, but for the time being we work with the span (2).
As a functor in $X$, the span (2) is obtained by homming out of the interval cospan
and the fundamental groupoid structure on the span (1) ultimately comes about by homming out of an H-cogroupoid structure on the interval cospan, just as the fundamental group of a based space is obtained by homming out of the pointed circle, seen as an H-cogroup.
Let us describe the H-cocategory structure on the interval cospan. A cospan from a point to itself is a bipointed space, and the composite of two such cospans $X$, $Y$ is obtained by identifying the second point of $X$ with the first point of $Y$, resulting in a new bipointed space we denote by $X \vee Y$. In particular,
and the co-composition for the H-cocategory structure on $I$, which is a cospan map $I \to I \vee I$, is essentially just multiplication by 2:
This co-operation $I \to I \vee I$ is not strictly coassociative, but is coassociative up to homotopy. As a matter of fact, it follows straight away from the convexity of the $n$-fold cospan composite $I^{\vee n} \cong [0, n]$ that the space of cospan maps $\hom(I, I^{\vee n})$ is contractible: the convexity actually gives a way of defining coherent homotopies (the pentagon, etc.) all the way up the ”$A_{\infty}$-ladder”. In other words, the cospan $I$ carries an $A_{\infty}$-cocategory structure.
But we don’t actually need the $A_{\infty}$-jargon: all we will need are two basic facts:
There is a ‘tautological’ topological operad $T$ where the component $T_n$ is the space $\hom(I, I^{\vee n})$ of cospan maps;
Each component $T_n$ is contractible.
The operad structure on $T$ is derived from pure abstract nonsense: if $A$ is an object in a monoidal category, then the objects $\hom(A, A^{\otimes n})$ are the components of an operad, just as are the objects $\hom(A^{\otimes n}, A)$ the components of the familiar tautological endomorphism operad. This applies in particular to the cospan $I$, seen as an object in the monoidal category of cospans from a point to itself; in analogy, we call $T$ ‘tautological’ as well.
Now we just hom out of the cospan co-operations encapsulated in the operad $T$ to get span operations on the topological span above. That is, we use the facts that
$\hom(-, X)$ takes the $n$-fold cospan composite $I^{\vee n}$ to the $n$-fold span composite of $X \leftarrow X^I \to X$, so
Each cospan co-operation $\theta \in T_n = \hom(I, I^{\vee n})$ induces a span operation
from the $n$-fold span composite to the span itself.
Again, we could express this by saying we derive an $A_\infty$-category structure on the topological span (2), but in less fancy terms, the composition map
gives a $T$-algebra structure on the topological span:
where $\otimes$ indicates span composite.
This change of base is (strong) monoidal. As a matter of fact, this change of base factors through the center of the monoidal category $TopSpan$, which we actually need because one needs symmetries in order to express the notion of even a nonpermutative operad internal to a monoidal category.
We get a similar $T$-algebra structure on the $Top$-graph $X^\sim: X_0 \times X_0 \to Top$ we started with. Indeed, the functor
is a lax monoidal functor, and hence takes $T$-algebras in $TopSpan(X, X)$ (as in (4)) to $T$-algebras in $TopGraph_{X_0}$. This gives a $T$-algebra structure
in the monoidal category of $Top$-graphs. Considered geometrically, this is almost pure tautology, but (5) is the crucial topological input in Trimble’s machine.
Let us make this more explicit. In general, if $V$ is a monoidal category with coproducts, such that the tensor $u \otimes v$ preserves coproducts in each of its separate arguments $u$ and $v$, then the category of $V$-graphs also carries a monoidal structure, where for two $V$-graphs $G, H$ we define
Hence, in the case $V = Top$, (5) takes the more explicit form
Now apply the functor $\Pi_0: Top \to Set$ (which preserves products and incidentally sums as well) to (6). This makes the $Set$-graph $\Pi_0 X^\sim$ an algebra over the operad $\Pi_0 T$. But a $Set$-graph equipped with a $\Pi_0 T$-algebra structure is just a category, since $\Pi_0 T_n$ consists of just a point, i.e., is the associative operad whose algebras are monoids, and a category is a monoid in a category of graphs.
This category is precisely the fundamental groupoid $\Pi_1(X)$. It is pretty clear from this description that $\Pi_1(X)$ is functorial in $X$. Indeed, if $f: X \to Y$ is a continuous map, we can pull back the $Top$-graph $Y^\sim$ over $Y_0$ to a $Top$-graph over $X_0$,
which we call $f_0^{-1} Y^\sim$. The continuous map $f: X \to Y$ induces a comparison map of $T$-algebras in $TopGraph_{X_0}$,
and again by composing with $\Pi_0: Top \to Set$, we get a map $\Pi_0 f^*: \Pi_1(X) \to \Pi_1(Y)$, i.e., a functor between the fundamental groupoids.
Thus, after a series of small tautological steps, we have bootstrapped our way up from $\Pi_0$ to $\Pi_1$. This same procedure suggests a way to bootstrap from $\Pi_1$ to a definition of $\Pi_2$, and so on up an $n$-categorical ladder. Here are the general inductive definitions:
A (Trimble) 0-category is a set. The fundamental 0-groupoid $Top \to 0 Cat$ is the connected components functor $\Pi_0: Top \to Set$.
For $n \geq 0$, a Trimble $(n+1)$-category consists of a set $X_0$ and a $n Cat$-graph
equipped with a structure of algebra over the operad $\Pi_n T$ (a priori an operad in the cartesian monoidal category $n Cat$, but made into an operad in $n Cat Graph_{X_0}$ by an obvious change of base; cf. the remark above).
A morphism of Trimble $(n+1)$-categories $(X_0, X) \to (Y_0, Y)$ consists of of a function $f_0: X_0 \to Y_0$ together with a strict $\Pi_n T$-algebra map $f: X \to f_0^{-1} Y$. The category $(n+1)Cat$ admits finite products and arbitrary sums over which products distribute.
The fundamental $(n+1)$-groupoid functor
sends a space $X$ to the pair $(X_0, \Pi_n X^\sim)$, seen as a $\Pi_n T$-algebra by applying the product-preserving functor $\Pi_n$ to the tautological $T$-algebra $X^\sim$.
Theorem: $\Pi_{n+1}$ preserves finite products and arbitrary coproducts. (This theorem is needed to push the induction through.)
And that’s it. Notice this approach is in essence globular, that is we have an underlying functor
to $n$-globular sets. This is defined by induction: the underlying $(n+1)$-globular set of an $(n+1)$-category $(X_0, X)$ is given by
using the fact that an $(n+1)$-globular set is naturally an $n$-globular graph over its set of 0-cells. In fact, Cheng has shown that Trimble $n$-categories (and related structures; see the next section) are the algebras of a Batanin operad, that is a globular operad satisfying a contractibility condition.
As Cheng put it, Trimble’s inductive definition is disarmingly straightforward, and yet it codes up all the essential structure one expects to see in weak $n$-categories. For example, an associator for 2-categories is coded up as a 1-cell in $\Pi_1 T_3$, i.e., as a homotopy class of paths between two points in $\hom(I, I \vee I \vee I)$:
where $\delta: [0, 1] \to [0, 2]$ is a co-composition. The pentagonator is similarly coded up as a 2-cell in $\Pi_2 T_4$. And so on.
To be continued
The account above was given in a traditional language of operads, but subsequent analysis by Leinster and Cheng led to the following notion, involving a hybrid between algebras over an operad and enriched categories.
Fix a symmetric monoidal category $V$, and let $P$ be an operad valued in $V$. A $(V, P)$-category consists of
A set of objects $X_0$,
A $V$-graph $X(-, -): X_0 \times X_0 \to V$;
Maps $P_n \otimes X(x_0, x_1) \otimes \ldots X(x_{n-1}, x_n) \to X(x_0, x_n)$ parametrized over all choices $x_0, x_1, \ldots, x_n$
subject to axioms similar to the usual axioms for algebras over an operad.
Trimble’s definition above amounts to an inductive definition of weak $n$-Cat (call it $V_n$) where
0-Cat = Set;
$V_{n+1} = (V_n, P_n)$-Cat
for some series of operads $P_n$ valued in $V_n$. The operads $P_n$ in his definition came from one master operad $T$, by applying a product-preserving functor
to $T$. But other operads are possible. For the purposes of $n$-category theory, we should choose operads that are contractible in some sense (which the $\Pi_n(T)$ should be), but even in the realm of contractible operads there are many possible choices which may be convenient for different purposes.
Using some theory of terminal coalgebras, Leinster and Cheng described a viable notion of Trimble $\infty$-category.
There are several equivalent ways to describe this construction.
This approach starts from the observation that the category $Str \infty Cat$ of strict $\infty$-categories is the inverse limit in $CAT$ of the sequence
where each functor is “truncation,” i.e. discard all top-dimensional cells. We would like to imitate this, but the problem is that if we discard the $(n+1)$-cells in a (Trimble) weak $(n+1)$-category, we don’t get a weak $n$-category; for that we would have to also identify $n$-cells that are connected by some $(n+1)$-cell.
We can see this going on at the operad level as well. Consider the basic shape of the fundamental $n$-groupoid $\Pi_n(X)$ as a globular set: in dimensions $j \lt n$, the $j$-cells are just continuous maps $D^j \to X$, but the dimension $j = n$ is exceptional: the putative $\infty$-groupoid $\Pi_\omega(X)$ is “truncated” to a fundamental $n$-groupoid $\Pi_n(X)$ by forcing all $j$-cells in dimensions $j \gt n$ to be identity cells, that is by taking the quotient of the set of $n$-cells $D^n \to X$ with respect to homotopy-equivalence rel boundary $\partial D^n$. Thinking of homotopy-equivalence classes in terms of applying a path-components functor $\Pi_0$ to a suitable function space, one sees after some reflection that this globular “truncation”, which is a sort of enforcement of coherence at the level of top-dimensional cells, is really the result of beginning the iterative enrichment with the path-components functor
in the first place.
In slightly different words: this form of “truncation” from a higher-dimensional category to a lower one is iterated “decategorification”, going from isomorphisms between $n$-cells to isomorphism classes of $n$-cells: again the effect of applying a connected components functor $\Pi_0$. To take weak $\omega$-categories seriously is to eliminate all vestiges of such decategorification: there are no coherence equations to impose at the top level; there are only higher and higher coherence data all the way up the dimensional chain, which one could elect to chop off past a certain point (instead of modding out by).
So in order to perform the inverse limit construction successfully, we should not perform any identifications at all, merely truncation. Precisely, if instead of starting out the iterative enrichment with $\Pi_0$, one could start by replacing it with the underlying set functor
and again set the machine in motion. In this way we obtain an “incoherent Trimble 1-category” as a graph equipped with a structure of algebra over the operad $D_0 = U T$, an “incoherent Trimble 2-category” as a graph enriched in incoherent 1-categories and with a structure of algebra over the operad $D_1 = D_0 T$, and so on. Again, one proves that $D_n$ preserves coproducts and finite products by induction, starting with the base observation that the underlying set functor $U$ preserves coproducts and finite products. Thus by induction we have a category of incoherent $n$-categories, $n iCat$.
As a result, we have actual truncation functors
where one just saws off the $(n+1)$-cells at each step, and one may define the category of Trimble $\omega$-categories as the inverse limit of this sequence.
This approach begins with the observation of Simpson that $Str$-$\omega$-$Cat$ is a fixed point of the ordinary enrichment process
taking a (complete, cocomplete) closed symmetric monoidal $V$ to the (complete, cocomplete) closed symmetric monoidal category of small $V$-categories. In fact, it is more than a fixed point; it is a terminal coalgebra for the functor $E$. This is the same as the previous observation that $Str \omega Cat$ is the inverse limit of the sequence
because of Adamek’s theorem that the limit
always produces a terminal $E$-coalgebra whenever the functor $E$ preserves limits of sequences of this form. In our case, $E^n !$ works out to be the usual globular truncation from $(n+1)$-categories to $n$-categories. We now observe that the same principle applies to the weak enrichment process, as long as we carry along the information of the fundamental $n$-groupoid functor, i.e. we consider the functor
where $\Pi: Top \to V$ preserves finite products (and coproducts), where $(V, \Pi)$-$Cat$ is the category of $(V,\Pi(T))$-categories, and where $\Pi^+$ takes a space $X$ to $\Pi(X^\sim)$ as a $(V,\Pi(T))$-category. This time $E^n !$ works out to be precisely the globular truncation
and so the terminal $E$-coalgebra is the category $Trimble(\omega)Cat$ as described above, equipped with the fundamental $\omega$-groupoid functor
It is often convenient to reason about terminal coalgebras using coinduction and corecursion. In this spirit, it is possible to reformulate the definition of Trimble $\omega$-categories coinductively. Namely, we define a category $\omega Cat = Trimble \omega Cat$ and a functor $\Pi\colon Top \to \omega Cat$, which preserves coproducts and finite products, by mutual coinduction, as follows.
This sort of coinductive definition is that used in the calculus of coinductive constructions, which when rephrased in terms of a terminal coalgebra reduces to the definitions given above.
Urs:
suppose I wanted to serious apply the Trimble $n$-category tool to some class of problems. It would be helpful to have an idea about some of the following points.
The main reason why quasi-categories and other models for (infinity,1)-categories are popular is that Joyal and Lurie showed that beyond just having a definition, one can do category theory with them. I would like to understand to which extent we know how to “do category theory” with Trimble $n$-categories.
I understand that much of this will remain to be worked out, but probably in particular Todd has many ideas about how one should go about approaching this. I’d be interested in whatever idea or tentative suggestion there is.
So:
What can be said about
43 in Trimble $n$-catgeories?
For instance for what I am doing I’d need in particular weak pullbacks and the resulting fibration sequences in a Trimble $n$-category. What are the chances to get hold of these notions in this context?
What is known about the reflection of Trimble $n$-categories on themselves
Is there a definition of Trimble $(n+1)$-categorry of all Trimble $n$-categories?
A Trimble $\infty$-category of all Trimble $\infty$-categories?
What is the relation to
simplicial models for weak omega categories,
do we have a notion of nerve of a Trimble $n$-category?
can we characterize the nerves of Trimble $n$-categories?
Probably we want to define the cosimplicial Trimble $\infty$-category that sends $[n]$ to the fundamental Trimble $\infty$-path category of the topological $n$-simplex and induce a notion of nerve of that. But the naive version of that will give just $\infty$-groupoidal nerve. What is generally needed is probably a notion of directed Trimble path $n$-category.
What can be said about (n,r)-category-cases of Trimble $n$-categories. How do we say “Trimble $n$-groupoid”?
What about monoidal structures? Is there a good guess for what it would mean to have a
monoidal Trimble $n$-category
symmetric (i.e. stably) monoidal Trimble $n$-category?
The first appearance in print of the definition was in
Later the definition was applied tentatively for the construction of weak $n$-cateories of cobordisms in
This article first reviews the original definition
and then generalizes it
A comparison of Trimble’s definition with that of a Batanin omega-category is in
Previous versions of this entry led to the following discussions:
John has just substituted “topological” for my (Todd’s) original “tautological”, which I don’t mind at all – “topological” is correct and arguably more informative – but maybe this is a good place to explain that “tautological” was not a typo, but rather an expansion of a technical usage:
One of the first examples of an operad that people are taught is the one where the n-th component consists of all n-ary operations $X^n \to X$. (Here $X^n$ denotes an n-fold tensor power in a monoidal category.) Because it’s the simplest or most obvious example of an operad, and because it’s canonically there no matter what $X$ is, it’s often in the literature called a “tautological operad”. What is less often noted is that by the same logic, $hom(X, X^n)$ is an equally tautological operad, except that it consists of all “co-operations” instead of operations. Then, I am applying this observation to the example of the monoidal category of topological cospans from a point to itself, where $X$ is the interval.
Toby: Considering that the original was ‘tautological topological’, which is more informative, I've changed it back. But perhaps this explanation has a place in the main text?
Todd: Sounds like a good idea. If you can figure out a smooth way to do that, I’d be grateful.
Toby: Actually, I just realised that most of it is already in the next paragraph below. I'll just add a sentence noting that this explains the name.
Urs: Thanks, Todd, this is great. If I may, let me try, also to check if I am following, to see how much of this we can do while abstracting away from Top and the concrete interval object in there.
It seems that we need:
a closed monoidal homotopical category $V$ (not your $V$ above, but I’ll call it $V$ nevertheless);
an object $I \in V$ fitting in an internal co-graph $pt \stackrel{\sigma, \tau}{\to} I$ such that
all the end-to-end pushouts $I^{\vee n}$ exist in $V$;
all the $V$-internal hom-objects $[I, I^{\vee n}]$ are weakly equivalent to the point (the terminal object) $[I, I^{\vee n}] \stackrel{\sim}{\to} pt$;
equipped with the structure of a co-operad internal to $V$ on $\{[I,I^{\vee n}]\}$, $n \in \mathbb{N}$.
Then
Hm, let me just let it stand this way for you to either erase all of this or maybe comment on it for the moment
Todd: Yes, that’s more or less right. But there is a god-given nonpermutative operad (not co-operad) structure on the graded $V$-object $\{[I, I^{\vee n}]\}$, and that’s what I’m using here. If you know how the “endomorphism operad” works (as in the entry operad), then the operad here works the same way, mutatis mutandis.
As for the last part: in the original definition, I didn’t know how to pass to the full-fledged $\omega$-categorical definition; I just had $n$-categories, one for each $n$. But Tom and Eugenia figured out how to make “Trimble-like” $\omega$-categories work by using coalgebraic methods. I haven’t thought about whether that allows a $\Pi_\omega$… But aside from these technicalities: yes, you seem to understand what this is all about.
There were some comments to this effect after I gave that lecture; someone asked what you really need to do this definition very generally. Martin Hyland cried, “Not much!”
Urs Schreiber: Okay, thanks. I tried now to formalize this at interval object.