Does “Quillen A with an extra direction” hold?

1. Some notation A bicategory X is a bisimplicial set such that the simplicial sets X_{i,∗} and X_{∗,j} are (nerves of) categories for all i, j ⩾ 0. A bifunctor between bicategories is simply a bisimplicial map. We write maps on the right. The face operators in a bisimplicial set are denoted by d_{i}^{(1)} in the first, and by d_{j}^{(2)} in the second direction. Analogously in a trisimplicial set. Write d_{⌊m+n+1,m+1⌋}^{(2)} := d_{m+n+1}^{(2)} ⋯ d_{m+1}^{(2)}. Etc. If X is a bisimplicial set, denote by X const_{2} the trisimplicial set that has (X const_{2})_{i,j,k} = X_{i,k}. Etc. A bisimplicial map is called a weak homotopy equivalence if its diagonalisation is a weak homotopy equivalence. Likewise for a trisimplicial map.

2. The question on “Quillen A with an extra direction” Let X and Y be bicategories, and let X ^{\overline{}^{f}} Y be a bifunctor. Form the trisimplicial set T_{f} that has

(T_{f})_{s,m,n} = {(x, y) ∈ X_{s,m} × Y_{s,m+n+1} : xf = yd_{⌊m+n+1,m+1⌋}^{(2)}}

for s, m, n ⩾ 0. To sketch a schematic picture,

(T_{f})_{s,m,n} ∋ (  ^{m}^{s}  x  , \underbrace{^{m}  ^{n}_{s}  xf}_{y}) .

The trisimplicial operation in the first (aka s-) direction on (x, y) is the operation on x and on y in the first direction. The trisimplicial operation in the second (aka m-) direction on (x, y) is the operation on x in the second direction and the operation on y in the “front part” in the second direction. So e.g. if (x, y) ∈ (T_{f})_{s,m,n}, then (x, y)d_{i}^{(2)} = (xd_{i}^{(2)}, yd_{i}^{(2)}). The trisimplicial operation in the third (aka n-) direction on (x, y) is the identical operation on x and the operation on y in the “back part” in the second direction. So e.g. if (x, y) ∈ (T_{f})_{s,m,n}, then (x, y)d_{i}^{(3)} = (x, yd_{m+1+i}^{(2)}).

We have trisimplicial “projection” maps

T_{f}  ^{\overline{}^{p_{1,f}}} X const_{3}
(x, y)  \overline{}x
T_{f}  ^{\overline{}^{p_{2,f}}} Y const_{2}
(x, y)  \overline{}yd_{⌊m,0⌋}^{(2)}

For a schematic picture of p_{2,f}, cf. §3 (upper row of picture).

Remark. Imitating Quillen’s proof of Theorem A, it is not difficult to show that f is a weak homotopy equivalence if and only if p_{2,f} is a weak homotopy equivalence.

Consider the bisimplicial map p_{2,f}|_{n=0} that is given at (s, m) by (T_{f})_{s,m,0} ^{\overline{}^{(p_{2,f})_{s,m,0}}} Y_{s,0} . For a schematic picture of p_{2,f}|_{n=0}, cf. §3 (lower row of picture).

Question. If p_{2,f}|_{n=0} is a weak homotopy equivalence, is then p_{2,f} (and thus f) a weak homotopy equivalence?

Remark. If X and Y are constant in the first (aka s-) direction (that is, if this simplicial direction is “not there”), then the answer is affirmative by Quillen A. In fact, (T_{f})_{0,∗,0} is the disjoint union of the over-categories f_{0,∗}/y, indexed by y ∈ Y_{0,0}, and p_{2,f}|_{n=0} maps an element in that disjoint union just to its indexing element.

Remark. If the simplicial subset p_{2,f}^{−1}(ỹ) of (T_{f})_{s,∗,0} is weakly contractible for all s ⩾ 0 and all ỹ ∈ Y_{s,0} , then p_{2,f} and p_{2,f}|_{n=0} are both weak homotopy equivalences. In fact, in this case it follows that p_{2,f}^{−1}(ỹ) is weakly contractible for all s, n ⩾ 0 and all ỹ ∈ Y_{s,n}, for p_{2,f}^{−1}(ỹ) ≃ p_{2,f}^{−1}(ỹd_{⌊n,1⌋}^{(2)}) (isomorphism of simplicial sets).

3. A question for a homotopy pullback

Fix n ⩾ 0. Consider the following commutative quadrangle of bisimplicial sets.

(∗)  \xymatrix{(T_{f})_{∗,˜∗,n} \ar[r]^{(p_{2,f})_{∗,˜∗,n}} \ar[d]_{d_{⌊n,1⌋}^{(3)}} & Y_{∗,n} \ar[d]^{d_{⌊n,1⌋}^{(2)}} \\ (T_{f})_{∗,˜∗,0} \ar[r]^{(p_{2,f})_{∗,˜∗,0}} & Y_{∗,0}}

To sketch a schematic picture,

[figure: (  ^{m}^{s}  x  , \underbrace{^{m}  ^{n}_{s}  xf}_{y}); (p_{2,f})_{s,m,n}; ^{^{n}^{s} yd_{⌊m,0⌋}^{(2)}}; d_{⌊n,1⌋}^{(3)}; d_{⌊n,1⌋}^{(2)}; (  ^{m}^{s}  x  ,  ^{m}^{s} xf  ); (p_{2,f})_{s,m,0}; _{s} yd_{⌊m,0⌋}^{(2)}d_{⌊n,1⌋}^{(2)}]

Question. Is (∗) a homotopy pullback?

Remark. If the answer to this question is affirmative, so is the answer to the question in §2.

Remark. If X and Y are constant in the first direction, then, as far as I can see, this is true.

Speculation. Is there a categorical model for the homotopy pullback (of categories, to begin with; then of bicategories)? With objects like in the comma category, only with an eventually bothsided constant zigzag instead of simply a morphism? Such a model could then be used to compare – one would need to get rid of the zigzag again somehow.

Matthias Künzer

Lehrstuhl D für Mathematik

RWTH Aachen Templergraben 64 D-52062 Aachen

kuenzer@math.rwth-aachen.de www.math.rwth-aachen.de/∼kuenzer