Homomorphisms of D-Categories

After introducing the notion of a strict \(D\)-category, we seek ways to relate different \(D\)-categories and their interleavings to each other. To this end let \(\mathbf{C}\) and \(\mathbf{C}'\) be strict \(D\)-categories with smoothing functor \(\mathcal{S}\) respectively \(\mathcal{S}'\).

  • Definition (1-Homomorphism). A \(1\)-homomorphism of strict \(D\)-categories from \(\mathbf{C}\) to \(\mathbf{C}'\) is a functor \(F \colon \mathbf{C} \rightarrow \mathbf{C}'\) such that \[ \xymatrix{ \mathbf{C} \ar[r]^{\mathcal{S}(\mathbf{a})} \ar[d]_F & \mathbf{C} \ar[d]^F \\ \mathbf{C}' \ar[r]_{\mathcal{S}'(\mathbf{a})} & \mathbf{C}' } \] and \[ \xymatrix@C+=6pc@R+=5pc{ \mathbf{C} \rtwocell<6>_{\mathcal{S}(\mathbf{b})}^{\mathcal{S}(\mathbf{a})} {\qquad ~ \mathcal{S}(\mathbf{a} \preceq \mathbf{b})} \ar[d]_F & \mathbf{C} \ar[d]^F \\ \mathbf{C}' \rtwocell<6>_{\mathcal{S}'(\mathbf{b})}^{\mathcal{S}'(\mathbf{a})} {\qquad ~ \, \mathcal{S}'(\mathbf{a} \preceq \mathbf{b})} & \mathbf{C}' } \] commute for all \(\mathbf{a}, \mathbf{b} \in D\) with \(\mathbf{a} \preceq \mathbf{b}\).

  • Remark. We read the second diagram of the above definition as \(F \circ \mathcal{S} (\mathbf{a} \preceq \mathbf{b}) = \mathcal{S}' (\mathbf{a} \preceq \mathbf{b}) \circ F\).

Now let \(F \colon \mathbf{C} \rightarrow \mathbf{C}'\) be a \(1\)-homomorphism and let \(A\) and \(B\) be objects of \(\mathbf{C}\).

  • Lemma. Let \(\varphi \colon A \rightarrow \mathcal{S}(\mathbf{a})(B)\) and \(\psi \colon B \rightarrow \mathcal{S}(\mathbf{b})(A)\) be an \((\mathbf{a}, \mathbf{b})\)-interleaving for some \((\mathbf{a}, \mathbf{b}) \in \mathcal{D}\), then \(F(\varphi)\) and \(F(\psi)\) form an \((\mathbf{a}, \mathbf{b})\)-interleaving in \(\mathbf{C}'\).

  1. Corollary. We have \(M_{\mathcal{S}'} (F(A), F(B)) \leq M_{\mathcal{S}} (A, B)\) and \(\mu_{\mathcal{S}'} (F(A), F(B)) \leq \mu_{\mathcal{S}} (A, B)\).

  • Lemma. Now suppose that \(F\) is faithful as a functor from \(\mathbf{C}\) to \(\mathbf{C}'\) and that \(\varphi \colon A \rightarrow \mathcal{S}(\mathbf{a})(B)\) and \(\psi \colon B \rightarrow \mathcal{S}(\mathbf{b})(A)\) are arbitrary homomorphisms for some \((\mathbf{a}, \mathbf{b}) \in \mathcal{D}\), with \(F(\varphi)\) and \(F(\psi)\) an \((\mathbf{a}, \mathbf{b})\)-interleaving of \(F(A)\) and \(F(B)\). Then \(\varphi\) and \(\psi\) are an \((\mathbf{a}, \mathbf{b})\)-interleaving as well.

The previous two lemmata have the following

  • Corollary. If \(F\) is full and faithful, then \(F\) induces a bijection between the \((\mathbf{a}, \mathbf{b})\)-interleavings of \(A\) and \(B\) and the \((\mathbf{a}, \mathbf{b})\)-interleavings of \(F(A)\) and \(F(B)\) for any \((\mathbf{a}, \mathbf{b}) \in \mathcal{D}\).

  • Corollary. If \(F\) is full and faithful we have \(M_{\mathcal{S}'} (F(A), F(B)) = M_{\mathcal{S}} (A, B)\) and \(\mu_{\mathcal{S}'} (F(A), F(B)) = \mu_{\mathcal{S}} (A, B)\).

We also note that we can compose \(1\)-homomorphisms. To this end let \(\mathbf{C}''\) be another strict \(D\)-category.

  • Lemma. Let \(G \colon \mathbf{C}' \rightarrow \mathbf{C}''\) be another \(1\)-homomorphism, then \(G \circ F\) is a \(1\)-homomorphism from \(\mathbf{C}\) to \(\mathbf{C}''\).

Now we defined \(1\)-homomorphisms as functors with special properties. For any two functors \(F\) and \(G\) from \(\mathbf{C}\) to \(\mathbf{C}'\) we have the class of natural transformations from \(F\) to \(G\). Now suppose \(F\) and \(G\) are \(1\)-homomorphisms, then \(F\) and \(G\) are in some sense compatible with \(\mathcal{S}\) and \(\mathcal{S}'\). We name a natural transformation from \(F\) to \(G\), that is compatible with \(\mathcal{S}\) and \(\mathcal{S}'\), a \(2\)-homomorphism from \(F\) to \(G\).

  • Definition (2-Homomorphism) . A \(2\)-homomorphism of strict \(D\)-categories from \(F\) to \(G\) is a natural transformation \(\eta \colon F \Rightarrow G\) such that \[ \xymatrix@+=3pc{ \mathbf{C} \ar[r]^{\mathcal{S}(\mathbf{a})} \dtwocell<4>_F^G{^\eta} & \mathbf{C} \dtwocell<4>_F^G{^\eta} \\ \mathbf{C}' \ar[r]_{\mathcal{S}'(\mathbf{a})} & \mathbf{C}' } \] commutes for all \(\mathbf{a} \in D\).

  • Remark. We read the diagram of the previous definition as \(\eta \circ \mathcal{S} (\mathbf{a}) = \mathcal{S}' (\mathbf{a}) \circ \eta\). This definition does not include the condition \(\eta \circ \mathcal{S} (\mathbf{a} \preceq \mathbf{b}) = \mathcal{S}' (\mathbf{a} \preceq \mathbf{b}) \circ \eta\) for \(\mathbf{a} \preceq \mathbf{b}\), where \(\circ\) is the Godement product of natural transformations. The reason is, that this equation follows from the previous two definitions. If we choose one of the two formulas for the Godement product of \(\eta\) and \(\mathcal{S} (\mathbf{a} \preceq \mathbf{b})\) and rewrite the term using the previous two definitions, then we obtain the other of the two formulas for the Godement product of \(\mathcal{S}' (\mathbf{a} \preceq \mathbf{b})\) and \(\eta\).

Just like we can compose natural transformations we can also compose \(2\)-homomorphisms.

  • Lemma. Let \(\eta \colon F \Rightarrow G\) and \(\theta \colon G \Rightarrow H\) be \(2\)-homomorphisms, then \(\theta \circ \eta\) is a \(2\)-homomorphism as well.

Moreover we can compose \(1\)-homomorphisms with \(2\)-homomorphisms and vice versa.

  • Lemma. Let \(F\) and \(G\) be \(1\)-homomorphisms from \(\mathbf{C}\) to \(\mathbf{C}'\), let \(\eta \colon F \Rightarrow G\) be a \(2\)-homomorphism, and let \(H\) be a \(1\)-homomorphism from \(\mathbf{C}'\) to \(\mathbf{C}''\), then the composition \(H \circ \eta\), pictorially \[ \xymatrix{ \mathbf{C} \rtwocell^F_G{\eta} & \mathbf{C}' \ar[r]^H & \mathbf{C}'' , } \] is \(2\)-homomorphism from \(H \circ F\) to \(H \circ G\).

    Similarly if \(F\) and \(G\) are \(1\)-homomorphisms from \(\mathbf{C}'\) to \(\mathbf{C}''\) and if \(H\) is a \(1\)-homomorphism from \(\mathbf{C}\) to \(\mathbf{C}'\), then the composition \(\eta \circ H\), pictorially \[ \xymatrix{ \mathbf{C} \ar[r]^H & \mathbf{C}' \rtwocell^F_G{\eta} & \mathbf{C}'' , } \] is again a \(2\)-homomorphism.

  • Remark. Altogether we obtain the structure of a strict 2-category on the collection of all strict \(D\)-categories.