Let $\mathbb{R}P^2=e^0\cup e^1\cup e^2$ be the real projective plane given its standard CW structure. Give $\mathbb{R}P^2\times\mathbb{R}P^2$ the product CW structure.
Does anyone know an explicit formula for a cellular approximation to the diagonal map $\Delta:\mathbb{R}P^2\rightarrow\mathbb{R}P^2\times\mathbb{R}P^2$, $[x,y,z]\mapsto ([x,y,z],[x,y,z])$?
I've just uploaded a file on this Diagonals.pdf. This gives diagonal approximations for the fundamental crossed complexes $\Pi X_*$ of the standard cell structures $X_*$ of the Torus, Klein Bottle and Projective Plane. The diagonal approximatiions are maps $d:C \to C \otimes C$ where $\otimes$ is the monoidal part of the monoidal closed structure on the category of crossed complexes. See the book referenced [BHS11] in the preprint.
To give the details for the Projective Plane $P$, with cell structure $e^0 \cup e^1 \cup e^2$, and with boundary of the $2$-cell $e$ given by $\delta e= 2a$, say. Thus a diagonal in dimension $1$ is given by $\Delta(a) = a_1 + a_2$. In dimension $2$ it is given by $$\Delta(e) = (-a_1 )\otimes (-a_2) +e_1 ^{-a_2} +e_2.$$ This is fairly explicit, but is not given in terms of coordinates.