nLab Simpson's conjecture

Idea

Carlos Simpson conjectured, in the context of semi-strict infinity-categories that every weak omega-category is equivalent to one that is a strict omega-category except for the units.

Statement

In

• Carlos Simpson, Homotopy types of strict 3-groupoids (arXiv:9810059)

it was conjectured (page 27) that

Simpson’s Conjecture: Every weak $\infty$-category is equivalent to an $\infty$-category in which composition and exchange laws are strict and only the unit laws are allowed to hold weakly.

For weak unit laws in $\infty$-categories see

• Joachim Kock, Weak identity arrows in higher categories (arXiv)

In

• Andre Joyal, Kock, Weak units and homotopy 3-types (arXiv)

Simpson’s conjecture is proven up to the case of 3-categories with a single object.

Remarks

One expects several alternative such semi-strictification statements. Eugenia Cheng and Nick Gurski write the following at the end of their paper:

Finally we consider the question of eliminating the distinguished invertible elements by using a stricter form of $n$-category. Generalising from the previous sections, we see that we do not need to restrict all the way to strict $n$-categories – a semistrict version will suffice. One form of semistrictness has everything strict except interchange (cf. Gray-categories and see Crans 2000b, Crans 2000a); another has everything strict except units (Koch 2005, Simpson 1998). These have both been proposed as solutions to the coherence problem for $n$-categories.

However, there are other possible “shades” of semistrictness and the above notions do not appear to be right for the present purposes. Instead, we need a form of semistrict $n$-category in which the units and interchange for $(n-1)$-cells are strict, but everything else can be weak. This is to eliminate the constraint $n$-cells that become distinguished invertible elements in our $n$-degenerate situation; we expect that as in the case $n = 2$ the associator is automatically forced to be the identity.

Hypothesis 5.3. Semistrictness Let $n \ge 3$. Then an $n$-degenerate semistrict $n$-category in the above sense is precisely a commutative monoid.

E. Cheng and N. Gurski’s, The periodic table of $n$-categories for low dimensions I (arXiv)

History

The conjecture was initially triggered by the claim in

• Mikhail Kapranov, Vladimir Voevodsky, $\infty$-Groupoids and homotopy types, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 32 no. 1 (1991), p. 29-46 (Numdam)

that “semi-strict infinity-groupoids”, in which only the inverses could be weak and everything else strict, model all homotopy types (hence solve the homotopy hypothesis)

In (Simpson 98) it was pointed out that this seems to be wrong for the homotopy type of the 2-sphere.

The issue, however, is quite subtle, as highlighted by Voevodsky here)

(For instance Corollary 5.2 in (Simpson 98) leaves open the possibility that the geometric realization in (Kapranov-Voevodsky 91) is essentially surjective, only that the homotopy groups of the resulting space could not be read off from the groupoid in the obvious way.

Examining KV 91‘s motivation, Simpson noticed a singularity at the origin of Moore paths, which led him to conjecture that weakening invertibility and units (while keeping strict associativity and interchange) would be enough to get at all homotopy types. Specifically, he writes on p. 27

I think that the argument of (K. and V.) (which is unclear on the question of identity elements) actually serves to prove the above statement. I have called the above statement a “conjecture” because I haven’t checked this.