homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
A strictification theorem is a theorem of the form: every weak structure is suitably equivalent to a semi-strict structure.
There are typically two varieties of strictification theorems:
For instance, in the case of monoidal categories, the following are true.
However, one must be careful not to assume that it is always possible to strictify everything. For instance, the strictification theorem for symmetric monoidal categories states the following.
This is why we use the term “semi-strict” above: a strictification theorem expresses the extent to which a structure may be strictified, which may not be completely. Another classical example is tricategories, which may be strictified to Gray-categories, but not in general to 3-categories.
Strictification theorems are related to, but not the same as, coherence theorems.
See coherence theorem for a list of coherence theorems that also include strictification results.
There exist several approaches for proving strictification theorems.
for describing “doctrines” or algebraic structures on categories, which can be used to describe free algebras and state coherence theorems. All are closely related.
A 2-monad can be regarded as the “extensional essence” of an algebraic structure on categories (or on objects of some other 2-category). Of course, if $T$ is a 2-monad describing some structure, then $T A$ is the free such structure on $A$; thus the first sort of coherence theorem can be precisely stated as “describe $T A$ as explicitly as possible in terms of $A$.”
However, the notion of monad is so general that in practice, for proving coherence theorems it is useful to have a more explicit way of “presenting” a monad in terms of generating operations and relations; this is the purpose of the structures presented below, such as operads and clubs. Not all monads can be presented in such a more explicit way, but for those that can, it is a very useful simplification.
The notion of strict 2-monad on a strict 2-category also provides a general way to state a strictification theorem, although one must beware that this theorem is not always true, and when it is true, it doesn’t necessarily give what one was looking for. This general “coherence theorem schema” for a 2-monad $T$ would be that every pseudo $T$-algebra is equivalent to a strict one. A stronger statement, which is true in most cases when the weaker version is, would be that the inclusion
of the 2-category of strict $T$-algebras and strict $T$-morphisms into the 2-category of pseudo $T$-algebras and pseudo $T$-morphisms has a strict left 2-adjoint, usually written $(-)'$, and moreover for a pseudo algebra $A$, the unit $A \to A'$ is an equivalence in $Ps T Alg$. Thus $A$ is not only equivalent to a strict $T$-algebra, it is equivalent to a canonically defined one which is characterized by a universal property.
This “coherence theorem” is true in a lot of generality, although there are 2-monads for which it fails (see (Shulman)). However, often it leaves a lot of work to be done in order to extract what is commonly thought of as a coherence theorem. This is because the notion of pseudo $T$-algebra is always an unbiased weak structure, whereas it is more usual to study biased structures. Moreover, in most cases, proving an equivalence between biased and unbiased notions requires the full strength of the coherence theorem (in the description-of-free-algebras sense) for the biased notion!
For now, see operad.
For now, see club.
For now, see coherence theorem for bicategories with finite limits, which gives an example of this approach.
See strictification versus coherence.
Steve Lack, Codescent objects and coherence, MR
Mike Shulman, “Not every pseudoalgebra is equivalent to a strict one”, Adv. Math. 229 no. 3 (2012), 2024–2041, arXiv
Last revised on September 25, 2024 at 10:22:00. See the history of this page for a list of all contributions to it.