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Simpson's conjecture

References

In

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

it was conjectured (page 27) that

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

For weak unit laws in $\mathrm{\infty }$-categories see

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

In

• 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)

References

$n$Café discussion:

• Semistrict $\mathrm{\infty }$-categories and $\omega$-Semi-Categories