combinatorial species

Let $\mathbf{P}$ be the groupoid of finite cardinals. The category of combinatorial species is the copresheaf category $\mathsf{Sp}=[\mathbf{P},\mathbf{Set}]$.

• The category of species is the free symmetric monoidal closed cocomplete category on the Yoneda embedding of 1, see (Garner, Lopez Franco, 2015) and A Universal Property of the Convolution Monoidal Structure (Kelly, Im, 1985).

References.

• Une théorie combinatoire des séries formelles (Joyal, 1981) introduces the notion of combinatorial species. Fosco Loregian has an introductory course on this work (https://tetrapharmakon.github.io/species/).
• Combinatorial species and tree-like structures (Bergeron et al) is a complete book on combinatorial structures. See also (Aguiar and Mahajan, 2009).