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?


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