Coherency (homotopy theory)
In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism".
The adjectives such as "pseudo-" and "lax-" are used to refer to the fact equalities are weakened in coherent ways; e.g., pseudo-functor, pseudoalgebra.
Coherent isomorphism
In some situations, isomorphisms need to be chosen in a coherent way. Often, this can be achieved by choosing canonical isomorphisms. But in some cases, such as prestacks, there can be several canonical isomorphisms and there might not be an obvious choice among them.
In practice, coherent isomorphisms arise by weakening equalities; e.g., strict associativity may be replaced by associativity via coherent isomorphisms. For example, via this process, one gets the notion of a weak 2-category from that of a strict 2-category.
Replacing coherent isomorphisms by equalities is usually called strictification or rectification.
Coherence theorem
Mac Lane's coherence theorem states, roughly, that if diagrams of certain types commute, then diagrams of all types commute.[1] A simple proof of that theorem can be obtained using the permutoassociahedron, a polytope whose combinatorial structure appears implicitly in Mac Lane's proof.[2]
There are several generalizations of Mac Lane's coherence theorem.[3] Each of them has the rough form that "every weak structure of some sort is equivalent to a stricter one".[4]
Homotopy coherence
Homotopy Coherence refres to the concept in algebraic topology that deals with the coherence of higher homotopies. It explores the relationship between various homotopies and their interactions. Homotopy coherence is an important aspect of studying higher-dimensional algebraic structures and figures.
Example
Suppose there are two loops in a loop space, wanting to understand how they can be composed. Homotopy coherence allows definition of a higher-dimensional composition, called the loop concatenation, which combines these two loops into a single loop. This composition satisfies certain coherence conditions, ensuring the resulting loop is well-defined and independent of the chosen loops
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See also
- Coherence condition
- Canonical isomorphism
Notes
- ↑ Mac Lane 1978, Chapter VII, Section 2
- ↑ See Kapranov 1993 and Reiner & Ziegler 1994
- ↑ See, for instance coherence theorem (nlab)
- ↑ Shulman 2012, Section 1
References
- Cordier, Jean-Marc; Porter, Timothy (1997). "Homotopy coherent category theory". Transactions of the American Mathematical Society 349 (1): 1–54. doi:10.1090/S0002-9947-97-01752-2.
- § 5. of "Topology and Logic as a Source of Algebra (Retiring Presidential Address)". Bulletin of the American Mathematical Society 82 (1): 1–40. January 1976. doi:10.1090/S0002-9904-1976-13928-6.
- Categories for the working mathematician. Graduate texts in mathematics. Springer-Verlag. 1978. doi:10.1007/978-1-4757-4721-8.
- Ch. 5 of Kamps, Klaus Heiner; Porter, Timothy (April 1997). Abstract Homotopy and Simple Homotopy Theory. World Scientific. doi:10.1142/2215. ISBN 9810216025.
- Shulman, Mike (2012). "Not every pseudoalgebra is equivalent to a strict one". Advances in Mathematics 229 (3): 2024–2041. doi:10.1016/j.aim.2011.01.010.
- "The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation". Journal of Pure and Applied Algebra 85 (2): 119–142. 1993. doi:10.1016/0022-4049(93)90049-Y.
- Reiner, Victor (1994). "Coxeter-associahedra". Mathematika 41 (2): 364–393. doi:10.1112/S0025579300007452.
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