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4. For any stuff types F : X −→ E and G : Y −→ E we have: ∞ (X , F ), (Y , G) (X , F )n · (Y , G)n · n! = n=0 Proof. 2 we have: (X , F ), (Y , G) = (X , F ), (Y , G) ∞ = (X , F ), (Y , G) , F, G n=0 ∞ = n=0 ∞ | F, G −1 n | = n! ∞ n=0 n |F −1 n | |G−1 n | · · n! n! n! (X , F )n · (Y , G)n · n! 8 Stuff operator Continuing in the spirit of thinking of stuff types as vectors, we seek to create an analogous construct on stuff types. In fact we have already seen two such examples, being the pointing (•) and the derivative (∂) operations.

Baez and James Dolan, From finite sets to Feynman diagrams, Mathematics unlimited—2001 and beyond, Springer, Berlin, 2001, pp. 29–50. [3] Andr´e Joyal, Une th´eorie combinatoire des s´eries formelles, Adv. in Math. 42 (1981), no. 1, 1–82. [4] G. M. Kelly, An abstract approach to coherence, Coherence in categories, Springer, Berlin, 1972, pp. 106–147. , Vol. 281. [5] , Many-variable functorial calculus. I, Coherence in categories, Springer, Berlin, 1972, pp. 66–105. , Vol. 281. [6] J. P. May, The geometry of iterated loop spaces, Springer-Verlag, Berlin, 1972, Lectures Notes in Mathematics, Vol.

Proof. This proof can be obtained rather easily by constructing the following: A f ff || ff || ff | | f3 | |} Φ Q C V U | | fff ff || || | | ff || || f3 }|| }|| ∆ Γ CQ Pf Xf ff ff || ff ff || f ff | | L ff F f3 }|| K 3 E ff ff ff ff 3 CQ Y || | | || |} | G E where Γ, ∆, Φ are all natural isomorphisms. 5. Considering the adjoint of the previous example, we obtain: (X , F ), (E, 1E ) dd +{∗} dd E1 F dd E dd d2 CQ E ff ff || | ff || ff | | 1E +{∗} ff | ~|| 2  ~~ ~~ ~ ~~ ~~ ~ Xf f ff ff f F fff 2 E E +{∗} So the operator E1E adds a new element to the set.

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2-transitive abstract ovals of odd order by Korchmaros G.


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