# Cellular diagonal approximation

Let $$X$$ be a Co-H space with a finite CW structure. Composing the comultiplication $$c:X \rightarrow X \vee X$$ with the inclusion $$i:X \vee X \rightarrow X \times X$$ gives a map $$i \circ c \simeq \Delta:X \rightarrow X \times X,$$ homotopic to the usual diagonal $$\Delta:X \rightarrow X \times X$$.

Let $$p_1,p_2:X \times X \rightarrow X$$ denote the projections onto the first and second factor, respectively. By definition, $$p_1 \circ (i \circ c) \simeq id_X \simeq p_2 \circ (i \circ c)$$ since $$X$$ is Co-H.

In particular, $$i \circ c$$ is homotopic to a cellular diagonal approximation which induces the following on the level of cellular chains

$$\Delta'_*:C_*(X) \rightarrow C_*(X) \otimes C_*(X), e \mapsto (e \otimes *) + (* \otimes e).$$

Having such an explicit description of a cellular diagonal approximation (on the chain level) enables us to investigate cup and cap products in $$X$$. Are there other (or wider) known classes of spaces for which an explicit cellular diagonal approximation can be written down in this way?