nLab combinatorial category

Combinatorial category

Combinatorial category


A combinatorial category [Pultr (1973, Def. 1.7)] CC is a locally finite category satisfying the unnatural isomorphism property:

  • For all objects A,BCA, B \in C, if there is an unnatural isomorphism C(,A)C(,B)C(-, A) \cong C(-, B), then ABA \cong B.

Or equivalently

  • For all objects A,BCA, B \in C, |C(X,A)|=|C(X,B)|\left|C(X, A)\right| = \left|C(X, B)\right| for all objects XCX \in C.



Let CC be a combinatorial category and let A,B,XCA, B, X \in \C. If A×XB×XA \times X \cong B \times X and XX is weakly terminal, then ABA \cong B.


For all YCY \in C, we have

|C(Y,A)|×|C(Y,X)|=|C(Y,A×X)|=|C(Y,B×X)|=|C(Y,B)|×|C(Y,X)| \left|C(Y, A)\right| \times \left|\C(Y, X)\right| = \left|C(Y, A \times X)\right| = \left|C(Y, B \times X)\right| = \left|C(Y, B)\right| \times \left|C(Y, X)\right|

Hence we may divide both sides by |C(Y,X)|\left|C(Y, X)\right| (which is nonzero since XX is weakly terminal) and so |C(Y,A)|=|C(Y,B)|\left|C(Y, A)\right| = \left|C(Y, B)\right| for all YCY \in \C, from which we have ABA \cong B.


  • Věra Trnková, Unnatural isomorphisms of products in a category, Category Theory: Applications to Algebra, Logic and Topology Proceedings of the International Conference Held at Gummersbach, July 6–10, 1981. Berlin, Heidelberg: Springer Berlin Heidelberg.

  • A. Pultr, Isomorphism types of objects in categories determined by numbers of morphisms, Acta Scientiarum Mathematicarum 35 (1973) 155-160 [pdf]

  • Ehsan Momtahan, Afshin Amini, and Babak Amini, From Hom(A,X)Hom(B,X)Hom (A, X) \cong Hom(B, X) to ABA \cong B, Filomat 32.11 (2018): 4079-4087.

  • Luca Reggio?, Polyadic sets and homomorphism counting, Advances in Mathematics 410 (2022): 108712.

  • Shoma Fujino and Makoto Matsumoto, Lovász’s hom-counting theorem by inclusion-exclusion principle, arXiv:2206.01994 (2022).

Last revised on May 7, 2024 at 21:28:24. See the history of this page for a list of all contributions to it.