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Trimble n-category

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Content

Todd Trimble’s definition of weak nn-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 nn-categories, which was intended to distinguish them from “weak” nn-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).

Basic idea

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 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 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 nn-categories, in an inductive kind of way.

History

In 1999, Todd Trimble introduced a definition of weak nn-category in a lecture at Cambridge University. This definition was introduced in order to give an algebraic notion of fundamental nn-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 ω\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

The original idea was to use an A A_\infty-cocategory structure on the topological interval II, in the form of an topological operad TT 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

Π n:TopnCat\Pi_n: Top \to n Cat

and then defining an (n+1)(n+1)-category as a category “weakly enriched” in nCatnCat, meaning roughly speaking a structure of Π n(T)\Pi_n(T)-algebra in the category of nCatn 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 nn-groupoid.

Let XX be a topological space, and let X 0X_0 denote its underlying set, viewed as a discrete category. If VV is a category, then a VV-graph over X 0X_0 is just a functor

X 0×X 0VX_0 \times X_0 \to V

(which here amounts to a function X 0×X 0Ob(V)X_0 \times X_0 \to Ob(V)).There is an obvious category of VV-graphs over X 0X_0.

  • We can also consider a more general category VV-Graph, where a morphism (C 0,C(,))(D 0,D(,))(C_0, C(-, -)) \to (D_0, D(-, -)) consists of a function f 0:C 0D 0f_0: C_0 \to D_0 together with a collection C(c,c)D(f 0c,f 0c)C(c, c') \to D(f_0 c, f_0 c') of morphisms in DD.

A basic example of VV-graph for our purposes is the “graph of paths” for a space XX: define

(1)X :X 0×X 0Top X^\sim : X_0 \times X_0 \to Top

by letting X (x,y)X^\sim(x, y) be the space of paths from xx to yy. Two paths from xx to yy are homotopic if they lie in the same connected component of X (x,y)X^\sim(x, y), so by applying the connected components functor,

X 0×X 0X TopΠ 0SetX_0 \times X_0 \stackrel{X^\sim}{\to} Top \stackrel{\Pi_0}{\to} Set

we get the graph of homotopy classes of paths. What we want is to put a category structure on this SetSet-graph, so as to obtain the fundamental groupoid Π 1(X)\Pi_1(X).

The TopTop-graph (1) can also be thought of as a topological span from the discrete space X 0X_0 to itself, and this is obviously closely related to the span

(2)X 1X i 0X IX i 1X 1 X^1 \stackrel{X^{i_0}}{\leftarrow} X^I \stackrel{X^{i_1}}{\to} X^1

where i 0,i 1:1Ii_0, i_1: 1 \to I are the endpoint inclusions. Indeed, we get (1) from (2) by first precomposing with a span from X 0X_0 to XX,

X 0X 0X,X_0 \leftarrow X_0 \to X,

(both arrows being identity functions), followed by postcomposing with the opposite span. This pre- and post-composition gives a functor

(3)TopSpan(X,X)TopSpan(X 0,X 0)=TopGraph X 0 TopSpan(X, X) \to TopSpan(X_0, X_0) = TopGraph_{X_0}

to which we return later, but for the time being we work with the span (2).

As a functor in XX, the span (2) is obtained by homming out of the interval cospan

1i 0Ii 111 \stackrel{i_0}{\to} I \stackrel{i_1}{\leftarrow} 1

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 XX, YY is obtained by identifying the second point of XX with the first point of YY, resulting in a new bipointed space we denote by XYX \vee Y. In particular,

II=[0,1][0,1][0,2]I \vee I = [0, 1] \vee [0, 1] \cong [0, 2]

and the co-composition for the H-cocategory structure on II, which is a cospan map IIII \to I \vee I, is essentially just multiplication by 2:

[0,1][0,2][0, 1] \to [0, 2]

This co-operation IIII \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 nn-fold cospan composite I n[0,n]I^{\vee n} \cong [0, n] that the space of cospan maps hom(I,I n)\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 A_{\infty}-ladder”. In other words, the cospan II carries an A A_{\infty}-cocategory structure.

But we don’t actually need the A A_{\infty}-jargon: all we will need are two basic facts:

  • There is a ‘tautological’ topological operad TT where the component T nT_n is the space hom(I,I n)\hom(I, I^{\vee n}) of cospan maps;

  • Each component T nT_n is contractible.

The operad structure on TT is derived from pure abstract nonsense: if AA is an object in a monoidal category, then the objects hom(A,A n)\hom(A, A^{\otimes n}) are the components of an operad, just as are the objects hom(A n,A)\hom(A^{\otimes n}, A) the components of the familiar tautological endomorphism operad. This applies in particular to the cospan II, seen as an object in the monoidal category of cospans from a point to itself; in analogy, we call TT ‘tautological’ as well.

Now we just hom out of the cospan co-operations encapsulated in the operad TT to get span operations on the topological span above. That is, we use the facts that

  • hom(,X)\hom(-, X) takes the nn-fold cospan composite I nI^{\vee n} to the nn-fold span composite of XX IXX \leftarrow X^I \to X, so

  • Each cospan co-operation θT n=hom(I,I n)\theta \in T_n = \hom(I, I^{\vee n}) induces a span operation

    hom(θ,X):hom(I n,X)hom(I,X)\hom(\theta, X): \hom(I^{\vee n}, X) \to \hom(I, X)

    from the nn-fold span composite to the span itself.

Again, we could express this by saying we derive an A A_\infty-category structure on the topological span (2), but in less fancy terms, the composition map

hom(I,I n)×hom(I n,X)hom(I,X)\hom(I, I^{\vee n}) \times \hom(I^{\vee n}, X) \to \hom(I, X)

gives a TT-algebra structure on the topological span:

(4)T n×(XX IX) n(XX IX) T_n \times (X \leftarrow X^I \to X)^{\otimes n} \to (X \leftarrow X^I \to X)

where \otimes indicates span composite.

  • Remark: TT is an operad in TopTop, not in the monoidal category TopSpan(X,X)TopSpan(X, X). But we can make it an operad there by applying an evident change-of-base TopTopSpan(X,X)Top \to TopSpan(X, X), sending a space YY in TopTop to the span
    XπX×YπXX \stackrel{\pi}{\leftarrow} X \times Y \stackrel{\pi}{\to} X

    This change of base is (strong) monoidal. As a matter of fact, this change of base factors through the center of the monoidal category TopSpanTopSpan, 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 TT-algebra structure on the TopTop-graph X :X 0×X 0TopX^\sim: X_0 \times X_0 \to Top we started with. Indeed, the functor

(5)TopSpan(X,X)TopSpan(X 0,X 0)TopGraph X 0 TopSpan(X, X) \to TopSpan(X_0, X_0) \cong TopGraph_{X_0}

is a lax monoidal functor, and hence takes TT-algebras in TopSpan(X,X)TopSpan(X, X) (as in (4)) to TT-algebras in TopGraph X 0TopGraph_{X_0}. This gives a TT-algebra structure

(5)T n×(X ) nX T_n \times (X^\sim)^{\otimes n} \to X^\sim

in the monoidal category of TopTop-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 VV is a monoidal category with coproducts, such that the tensor uvu \otimes v preserves coproducts in each of its separate arguments uu and vv, then the category of VV-graphs also carries a monoidal structure, where for two VV-graphs G,HG, H we define

(GH)(x,z)= yG(x,y)H(y,z)(G \otimes H)(x, z) = \sum_y G(x, y) \otimes H(y, z)

Hence, in the case V=TopV = Top, (5) takes the more explicit form

(6)T n× y 1,,y n1X (x,y 1)×X (y 1,y 2)××X (y n1,z)X (x,z) T_n \times \sum_{y_1, \ldots, y_{n-1}} X^\sim(x, y_1) \times X^\sim(y_1, y_2) \times \ldots \times X^\sim(y_{n-1}, z) \to X^{\sim}(x, z)

Now apply the functor Π 0:TopSet\Pi_0: Top \to Set (which preserves products and incidentally sums as well) to (6). This makes the SetSet-graph Π 0X \Pi_0 X^\sim an algebra over the operad Π 0T\Pi_0 T. But a SetSet-graph equipped with a Π 0T\Pi_0 T-algebra structure is just a category, since Π 0T n\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 Π 1(X)\Pi_1(X). It is pretty clear from this description that Π 1(X)\Pi_1(X) is functorial in XX. Indeed, if f:XYf: X \to Y is a continuous map, we can pull back the TopTop-graph Y Y^\sim over Y 0Y_0 to a TopTop-graph over X 0X_0,

X 0×X 0f 0×f 0Y 0×Y 0Y Top,X_0 \times X_0 \stackrel{f_0 \times f_0}{\to} Y_0 \times Y_0 \stackrel{Y^\sim}{\to} Top,

which we call f 0 1Y f_0^{-1} Y^\sim. The continuous map f:XYf: X \to Y induces a comparison map of TT-algebras in TopGraph X 0TopGraph_{X_0},

f *:X f 0 1Y :X 0×X 0Top f^*: X^\sim \to f_0^{-1} Y^\sim: X_0 \times X_0 \to Top

and again by composing with Π 0:TopSet\Pi_0: Top \to Set, we get a map Π 0f *:Π 1(X)Π 1(Y) \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 Π 0\Pi_0 to Π 1\Pi_1. This same procedure suggests a way to bootstrap from Π 1\Pi_1 to a definition of Π 2\Pi_2, and so on up an nn-categorical ladder. Here are the general inductive definitions:

  • A (Trimble) 0-category is a set. The fundamental 0-groupoid Top0CatTop \to 0 Cat is the connected components functor Π 0:TopSet\Pi_0: Top \to Set.

  • For n0n \geq 0, a Trimble (n+1)(n+1)-category consists of a set X 0X_0 and a nCatn Cat-graph

    X:X 0×X 0nCatX: X_0 \times X_0 \to n Cat

    equipped with a structure of algebra over the operad Π nT\Pi_n T (a priori an operad in the cartesian monoidal category nCatn Cat, but made into an operad in nCatGraph X 0n Cat Graph_{X_0} by an obvious change of base; cf. the remark above).

  • A morphism of Trimble (n+1)(n+1)-categories (X 0,X)(Y 0,Y)(X_0, X) \to (Y_0, Y) consists of of a function f 0:X 0Y 0f_0: X_0 \to Y_0 together with a strict Π nT\Pi_n T-algebra map f:Xf 0 1Yf: X \to f_0^{-1} Y. The category (n+1)Cat(n+1)Cat admits finite products and arbitrary sums over which products distribute.

  • The fundamental (n+1)(n+1)-groupoid functor

    Π n+1:Top(n+1)Cat\Pi_{n+1}: Top \to (n+1)Cat

    sends a space XX to the pair (X 0,Π nX )(X_0, \Pi_n X^\sim), seen as a Π nT\Pi_n T-algebra by applying the product-preserving functor Π n\Pi_n to the tautological TT-algebra X X^\sim.

  • Theorem: Π n+1\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

U n:nCat(n)GlobularSetU_n: n Cat \to (n)GlobularSet

to nn-globular sets. This is defined by induction: the underlying (n+1)(n+1)-globular set of an (n+1)(n+1)-category (X 0,X)(X_0, X) is given by

X 0×X 0(n)CatU n(n)GlobularSetX_0 \times X_0 \to (n)Cat \stackrel{U_n}{\to} (n)GlobularSet

using the fact that an (n+1)(n+1)-globular set is naturally an nn-globular graph over its set of 0-cells. In fact, Cheng has shown that Trimble nn-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 nn-categories. For example, an associator for 2-categories is coded up as a 1-cell in Π 1T 3\Pi_1 T_3, i.e., as a homotopy class of paths between two points in hom(I,III)\hom(I, I \vee I \vee I):

(δ1)δ,(1δ)δ:[0,1][0,3](\delta \vee 1)\delta, (1 \vee \delta)\delta: [0, 1] \to [0, 3]

where δ:[0,1][0,2]\delta: [0, 1] \to [0, 2] is a co-composition. The pentagonator is similarly coded up as a 2-cell in Π 2T 4\Pi_2 T_4. And so on.

To be continued

Iterative operadic enrichment

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 VV, and let PP be an operad valued in VV. A (V,P)(V, P)-category consists of

  • A set of objects X 0X_0,

  • A VV-graph X(,):X 0×X 0VX(-, -): X_0 \times X_0 \to V;

  • Maps P nX(x 0,x 1)X(x n1,x n)X(x 0,x n)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,,x nx_0, x_1, \ldots, x_n

subject to axioms similar to the usual axioms for algebras over an operad.

  • Example: For V=(Top,×)V = (Top, \times) and PP the operad T={hom cospan(I,I n)}T = \{\hom_{cospan}(I, I^{\vee n})\} discussed above, we constructed a (V,P)(V, P)-category (X 0,X )(X_0, X^\sim) out of any space XX.

Trimble’s definition above amounts to an inductive definition of weak nn-Cat (call it V nV_n) where

  • 0-Cat = Set;

  • V n+1=(V n,P n)V_{n+1} = (V_n, P_n)-Cat

for some series of operads P nP_n valued in V nV_n. The operads P nP_n in his definition came from one master operad TT, by applying a product-preserving functor

Π n:TopV n\Pi_n: Top \to V_n

to TT. But other operads are possible. For the purposes of nn-category theory, we should choose operads that are contractible in some sense (which the Π n(T)\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.

Extension to \infty-categories

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.

As an inverse limit

This approach starts from the observation that the category StrCatStr \infty Cat of strict \infty-categories is the inverse limit in CATCAT of the sequence

Str3CatStr2CatCatSet\dots \to Str3Cat \to Str2Cat \to Cat \to Set

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)(n+1)-cells in a (Trimble) weak (n+1)(n+1)-category, we don’t get a weak nn-category; for that we would have to also identify nn-cells that are connected by some (n+1)(n+1)-cell.

We can see this going on at the operad level as well. Consider the basic shape of the fundamental nn-groupoid Π n(X)\Pi_n(X) as a globular set: in dimensions j<nj \lt n, the jj-cells are just continuous maps D jXD^j \to X, but the dimension j=nj = n is exceptional: the putative \infty-groupoid Π ω(X)\Pi_\omega(X) is “truncated” to a fundamental nn-groupoid Π n(X)\Pi_n(X) by forcing all jj-cells in dimensions j>nj \gt n to be identity cells, that is by taking the quotient of the set of nn-cells D nXD^n \to X with respect to homotopy-equivalence rel boundary D n\partial D^n. Thinking of homotopy-equivalence classes in terms of applying a path-components functor Π 0\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

Π 0:TopSet\Pi_0: Top \to Set

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 nn-cells to isomorphism classes of nn-cells: again the effect of applying a connected components functor Π 0\Pi_0. To take weak ω\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 Π 0\Pi_0, one could start by replacing it with the underlying set functor

U:TopSetU: Top \to Set

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=UTD_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 0TD_1 = D_0 T, and so on. Again, one proves that D nD_n preserves coproducts and finite products by induction, starting with the base observation that the underlying set functor UU preserves coproducts and finite products. Thus by induction we have a category of incoherent nn-categories, niCatn iCat.

As a result, we have actual truncation functors

(n+1)iCatniCat...Set\ldots (n+1)iCat \to n iCat \to \ldots ... \to Set

where one just saws off the (n+1)(n+1)-cells at each step, and one may define the category of Trimble ω\omega-categories as the inverse limit of this sequence.

As a terminal coalgebra

This approach begins with the observation of Simpson that StrStr-ω\omega-CatCat is a fixed point of the ordinary enrichment process

E:VVCatE: V \mapsto V-Cat

taking a (complete, cocomplete) closed symmetric monoidal VV to the (complete, cocomplete) closed symmetric monoidal category of small VV-categories. In fact, it is more than a fixed point; it is a terminal coalgebra for the functor EE. This is the same as the previous observation that StrωCatStr \omega Cat is the inverse limit of the sequence

Strict(n+1)CatE n!Strict(n)CatSet!1\ldots \to Strict(n+1)Cat \overset{E^n !}{\to} Strict(n)Cat \to \ldots Set \overset{!}{\to} 1

because of Adamek’s theorem that the limit

E n1E 21E11 \dots \to E^n 1 \to \dots \to E^2 1 \to E 1 \to 1

always produces a terminal EE-coalgebra whenever the functor EE preserves limits of sequences of this form. In our case, E n!E^n ! works out to be the usual globular truncation from (n+1)(n+1)-categories to nn-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 nn-groupoid functor, i.e. we consider the functor

E:V,Π(V,Π)Cat,Π +E: \langle V, \Pi \rangle \mapsto \langle (V, \Pi)-Cat, \Pi^+ \rangle

where Π:TopV\Pi: Top \to V preserves finite products (and coproducts), where (V,Π)(V, \Pi)-CatCat is the category of (V,Π(T))(V,\Pi(T))-categories, and where Π +\Pi^+ takes a space XX to Π(X )\Pi(X^\sim) as a (V,Π(T))(V,\Pi(T))-category. This time E n!E^n ! works out to be precisely the globular truncation

Dec(n+1)CatDec(n)CatDec(n+1)Cat \to Dec(n)Cat

and so the terminal EE-coalgebra is the category Trimble(ω)CatTrimble(\omega)Cat as described above, equipped with the fundamental ω\omega-groupoid functor

Π ω:TopTrimble(ω)Cat\Pi_\omega: Top \to Trimble(\omega)Cat

As a coinductive definition

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 ω\omega-categories coinductively. Namely, we define a category ωCat=TrimbleωCat\omega Cat = Trimble \omega Cat and a functor Π:TopωCat\Pi\colon Top \to \omega Cat, which preserves coproducts and finite products, by mutual coinduction, as follows.

  • An object of ωCat\omega Cat (that is, a Trimble ω\omega-category) is an (ωCat,Π(T))(\omega Cat,\Pi(T))-category, i.e. an ωCat\omega Cat-enriched graph XX with composition operations Π(T) n×X(x 0,x 1)×X(x n1,x n)X(x 0,x n)\Pi(T)_n \times X(x_0, x_1) \times \ldots X(x_{n-1}, x_n) \to X(x_0, x_n) satisfying the evident axioms.
  • Similarly, a morphism in ωCat\omega Cat is a (strict) (ωCat,Π(T))(\omega Cat,\Pi(T))-functor, i.e. a map of ωCat\omega Cat-enriched graphs which respects the Π(T)\Pi(T)-structure.
  • The functor Π\Pi takes a space XX to the ω\omega-category Π(X )\Pi(X^\sim), whose objects are the points of XX, whose hom-ω\omega-category from xx to yy is Π(X (x,y))\Pi(X^\sim(x,y)), and whose Π(T)\Pi(T)-structure is given by composing paths.

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.

Category theory for Trimble nn-categories

Urs:

suppose I wanted to serious apply the Trimble nn-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 nn-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 nn-catgeories?

For instance for what I am doing I’d need in particular weak pullbacks and the resulting fibration sequences in a Trimble nn-category. What are the chances to get hold of these notions in this context?

What is known about the reflection of Trimble nn-categories on themselves

  • Is there a definition of Trimble (n+1)(n+1)-categorry of all Trimble nn-categories?

  • A Trimble \infty-category of all Trimble \infty-categories?

What is the relation to

Probably we want to define the cosimplicial Trimble \infty-category that sends [n][n] to the fundamental Trimble \infty-path category of the topological nn-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 nn-category.

What can be said about (n,r)-category-cases of Trimble nn-categories. How do we say “Trimble nn-groupoid”?

What about monoidal structures? Is there a good guess for what it would mean to have a

  • monoidal Trimble nn-category

  • symmetric (i.e. stably) monoidal Trimble nn-category?

References

The first appearance in print of the definition was in

Later the definition was applied tentatively for the construction of weak nn-cateories of cobordisms in

  • Eugenia Cheng and Nick Gurski, Towards an nn-category of cobordisms, Theory and Applications of Categories, Vol. 18, 2007, No. 10, pp 274-302. (tac)

This article first reviews the original definition

  • section 2, Trimble’s definition recalls the original definition,

and then generalizes it

  • section 4 Generalised Trimble definition proposes a generalization

A comparison of Trimble’s definition with that of a Batanin omega-category is in

Discussion

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 nXX^n \to X. (Here X nX^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 XX 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)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 XX 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 VV (not your VV above, but I’ll call it VV nevertheless);

  • an object IVI \in V fitting in an internal co-graph ptσ,τIpt \stackrel{\sigma, \tau}{\to} I such that

    • all the end-to-end pushouts I nI^{\vee n} exist in VV;

    • all the VV-internal hom-objects [I,I n][I, I^{\vee n}] are weakly equivalent to the point (the terminal object) [I,I n]pt[I, I^{\vee n}] \stackrel{\sim}{\to} pt;

    • equipped with the structure of a co-operad internal to VV on {[I,I n]}\{[I,I^{\vee n}]\}, nn \in \mathbb{N}.

Then

  • for CC any VV-enriched homotopical category (I want to use now that CC is powered over VV) we get – now let me see… – a functor
    Π ω:CTrimbleωCat.\Pi_{\omega}: C \to Trimble-\omega-Cat.

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 VV-object {[I,I n]}\{[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 ω\omega-categorical definition; I just had nn-categories, one for each nn. But Tom and Eugenia figured out how to make “Trimble-like” ω\omega-categories work by using coalgebraic methods. I haven’t thought about whether that allows a Π ω\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.

Revised on February 20, 2013 14:06:00 by Tim Porter (95.147.236.41)