A topological monoid is an object $M \in \textup{\textsf{cat}}$ together with morphisms $\mu \colon M \times M \to M$ and $\eta \colon 1 \to M$ so that the following diagrams commute:\n\(\xymatrix{ M \times M \times M \ar[d]_{\mu \times 1_M} \ar[r]^-{1_M \times \mu} & M \times M \ar[d]^\mu & M \ar[r]^-{\eta \times 1_M} \ar[dr]_{1_M} & M \times M \ar[d]^\mu & M \ar[l]_-{1_M \times \eta} \ar[dl]^{1_M} \\ M \times M \ar[r]_-\mu & M && M}\)\n\nA unital ring is an object $R \in \textup{\textsf{cat}}$ together with morphisms $\mu \colon R \otimes_\mathbb{Z} R \to R$ and $\eta \colon \mathbb{Z} \to R$ so that the following diagrams commute:\n\(\xymatrix{ R \otimes_\ZZ R \otimes_\ZZ R \ar[d]_{\mu \otimes_\ZZ 1_R} \ar[r]^-{1_R \otimes_\mathbb{Z} \mu} & R \otimes_\mathbb{Z} R \ar[d]^\mu & R \ar[r]^-{\eta \otimes_\mathbb{Z} 1_R} \ar[dr]_{1_R} & R \otimes_\mathbb{Z} R \ar[d]^\mu & R \ar[l]_-{1_R \otimes_\mathbb{Z} \eta} \ar[dl]^{1_R} \\ R \otimes_\mathbb{Z} R \ar[r]_-\mu & R && R}\)\n\nA $\mathbbe{k}$-algebra is an object $R \in \textup{\textsf{cat}}\mathbbe{k}$ together with morphisms $\mu \colon R \otimes\mathbbe{k} R \to R$ and $\eta \colon \mathbbe{k} \to R$ so that the following diagrams commute:\n\(\xymatrix{ R \otimes_\kk R \otimes_\kk R \ar[d]_{\mu \otimes_\kk 1_R} \ar[r]^-{1_R \otimes_\mathbbe{k} \mu} & R \otimes_\mathbbe{k} R \ar[d]^\mu & R \ar[r]^-{\eta \otimes_\mathbbe{k} 1_R} \ar[dr]_{1_R} & R \otimes_\mathbbe{k} R \ar[d]^\mu & R \ar[l]_-{1_R \otimes_\mathbbe{k} \eta} \ar[dl]^{1_R} \\ R \otimes_\mathbbe{k} R \ar[r]_-\mu & R && R}\)