nLab hyperstructure

Idea

Nils Baas has been emphasizing for many years, in print and in private communication, the conviction that the usual notions of n-category, infinity-category omega-category in higher category theory are not naturally suited for describing

• extended cobordisms such as appearing in

  • the tangle hypothesis

  • in extended quantum field theory

• hierarchical systems such as appearing in

  • complex systems;

  • biology.

The point is essentially that the directedness of morphisms and — related to that — the binary notion of source and target in categories and higher categories are notions alien to these contexts, which in applications have to and are essentially removed again in a second step by adding extra structure and requiring further properties, such as various monoidal structures and dualities, which allow to change the direction of morphisms, to collect objects together, etc.

In contrast to that, Baas pointed out that more naturally the above situations are thought of from the beginning in terms of hierarchies of what he calls bonds, where, quite generally, a bond is an object equipped with information of how a collection of sub-bonds sits inside it, bound by the bond.

A sketch of a generic such a situation of hierarchical bonds is a diagram

$$\array{ \beta_2 &&=&& \beta_2 && \beta_4 \\ \downarrow && && \downarrow & \swarrow \\ b_1 &\to& B &\leftarrow& b_2 \\ \uparrow && \uparrow && \uparrow \\ \beta_1 && b_3 && \beta_3 \\ && \uparrow \\ && \beta_5 }$$

where a bond $B$ binds sub-bonds $b_1, b_2$ and $b_3$ which in turn bind sub-bonds $\beta_i$.

For instance $B$ might be an extended cobordism with three boundary components $b_1, b_2, b_3$ which in turn have pieces of boundary components $\beta_i$.

Remark

The notion of replacing morphisms by bonds is familiar and concretely realized at least in a hierarchy of depth 2 in the context of groupoidification and geometric function theory, where morphisms are replaced by spans. Regarding these as cospans in the opposite category produces diagrams like the above sketch of a generic bond system.

Accordingly, the notion of multispan and multi-cospan may come close to exhibiting some crucial aspects of the idea that motivated the concept of hyperstructures.

Related concepts

• cyclic operad

• $(\infty,n)$-category with adjoints

References

Nils Baas has made, in print and in private communication, suggestions for a formalization of such systems of hierarchical bonds and coined the term hyperstructures for these. Some references are

• Nils A. Baas, Higher Algebraic Structures and General Field Theories [arXiv:2111.11205]

• Nils A. Baas, A Mathematical Approach to the Hierarchical Structure of Languages [arXiv:1806.05551]

• Nils A. Baas, On the Mathematics of Higher Structures [arXiv:1805.11944]

• Nils A. Baas, On the Philosophy of Higher Structures [arXiv:1805.11943]

• Nils A. Baas, On Higher Order Structures, arXiv:1509.00403 (2015), 15 pp.

• Nils A. Baas, Higher order architecture of collections of objects, International Journal of General Systems vol. 44 (1), pp. 55-75, 2015. arXiv:1409.0344

• Nils A. Baas, N.C. Seeman and A. Stacey, Synthesising topological links, Journal of Mathematical Chemistry, vol. 53, pp. 183-199, 2015.

• Nils A. Baas, D.V. Fedorov, A.S. Jensen, K. Riisager, A.G. Volosniev and N.T. Zinner, Higher-Order Brunnian Structures and Possible Physical Realizations, Physics of Atomic Nuclei, vol. 77(3), pp. 336-343, 2014. arXiv:1205.0746

• Nils A. Baas, New states of matter suggested by new topological structures, International Journal of General Systems, vol. 42 (2), pp. 137-169, 2012. arXiv:1012.2698

• Nils A. Baas, On structure and organization: an organizing principle, International Journal of General Systems, vol. 42 (2), pp. 170-196, 2012. arXiv:1201.6228

• Nils A. Baas and N.C. Seeman, On the chemical synthesis of new topological structures, Journal of Mathematical Chemistry, vol. 50 (1), pp. 220-232, 2012.

• Nils A. Baas, New structures in complex systems, The European Physical Journal Special Topics, vol. 178 (1), pp. 25-44, 2010.

• Nils A. Baas, Hyperstructures, Topology and Datasets, Axiomathes, vol. 19 (3), pp. 281-295, 2009.

• Nils A. Baas, Hyperstructures as Abstract Matter, Advances in Complex Systems, vol. 9 (3), pp. 157-182, 2006.

• Nils A. Baas, A. Ehresmann and J.-P. Vanbremeersch, Hyperstructures and Memory Evolutive Systems, International Journal of General Systems, vol. 33 (5), pp. 553-568, 2004.

• Nils A. Baas, Higher order cognitive structures and processes. In Toward a Science of Consciousness, S.R. Hameroff, A.W.Kaszniak and A.C.Scott (editors), MIT Press, pp. 633-648, 1996.

• Nils A. Baas, Hyper-structures as a tool in nanotechnology, Nanobiology, vol. 3 (1), pp. 49-60, 1994.

For a discussion of the idea of multispans as a formalization of hyperstructures, see the thread commencing from this comment by Urs Schreiber, and its possible application to the description of extended cobordisms and the generalized tangle hypothesis in this comment.