I'm writing this post to better understand the possible equivalence between Borceux and Adamek's notions of topological functor. Usually, when working with Adamek's book, one must be very careful with definitions and subtleties. Indeed, when a notion is unique up to isomorphisms, he uses the terminology "essentially unique". Otherwise, when he uses the term "unique", he says exactly one (not up to isomorphisms!)
I give below Borceux' definition (see Handbook of Categorical Algebra Vol. 2).
Let $U: \mathfrak{A} \to \mathfrak{B}$ be a functor. Consider
- a category $\mathfrak{D}$ and a functor $H: \mathfrak{D} \to \mathfrak{A}$;
- a cone $(f_D: B \to (U \circ H)(\mathfrak{D}))_{D \in \mathfrak{D}}$ on $U \circ H$ in $\mathfrak{B}$.
An initial structure for these data is a cone $(g_D: A \to H(\mathfrak{D}))_{D \in \mathfrak{D}}$ on $H$ in $\mathfrak{A}$ such that:
- $U(A)=B$ and, for all $D \in \mathfrak{D}$, $f_D=U(g_D)$;
- if $(h_D: A' \to H(\mathfrak{D}))_{D \in \mathfrak{D}}$ is a cone on $H$ in $\mathfrak{A}$ whose image $(U(h_D): U(A') \to (U \circ H)(\mathfrak{D}))_{D \in \mathfrak{D}}$ under $U$ factors via a morphism $h: U(A') \to U(A)$ through $(f_D: U(A) \to (U \circ H)(\mathfrak{D}))_{D \in \mathfrak{D}}$, there exists a unique morphism $a: A' \to A$ such that $U(a)=b$ and $h_D=g_D \circ a$ for every $D \in \mathfrak{D}$.
Given a category $\mathfrak{D}$, we say that $U$ has initial structure of shape $\mathfrak{D}$ if for every functor $H: \mathfrak{D} \to \mathfrak{A}$ and every cone on $U \circ H$ there exists a corresponding initial structure. A functor is said topological when it has initial structures of shape $\mathfrak{D}$ for every category $\mathfrak{D}$.
Now, let me give the notion of Adamek's topological functor (see http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf).
Let $U: \mathfrak{A} \to \mathfrak{B}$ be a functor. A $U$-structured arrow with domain $B \in \mathfrak{B}$ is a pair $(f,A)$ consisting of an $\mathfrak{A}$-object A and a $\mathfrak{B}$-morphism $f: B \to U(A)$.
A source $\mathcal{S}=(f_i: A \to A_i)_{i \in I}$ in $\mathfrak{A}$ is called $U$-initial provided that for each source $\mathcal{T}=(g_i: B \to A_i)_{i \in I}$ in $\mathfrak{A}$ with the same codomain as $\mathcal{S}$ and each $\mathfrak{B}$-morphism $h: U(B) \to U(A)$ with $U(\mathcal{T})=U(\mathcal{S}) \circ h$ there exists a unique $\mathfrak{A}$-morphism $\overline{h}: B \to A$ with $\mathcal{T}=\mathcal{S} \circ \overline{h}$ and $h=U(\overline{h})$.
A source $\mathcal{S}=(f_i: A \to A_i)_{i \in I}$ in $\mathfrak{A}$ is a lift of $\mathcal{T}=(g_i: B \to U(A_i))_{i \in I}$ via the functor $U$ if $U(A)=B$ and $U(f_i)=g_i$.
We say that $U$ topological if every $U$-structured source $(f_i: B \to U(A_i))_{i \in I}$ has a unique $U$-initial lift $(\overline{f}_i: A \to A_i)_{i \in I}$.
If I'm not wrong, we can relate both the terminologies as follows:
| Borceux | Adamek |
|---|---|
| Cone | Natural Source (def. 11.3) |
| Initial Structure | $U$-initial lift |
It might seem that the two definitions of a topological functor are equivalent. However, it is quite easy to see that the uniqueness of $a$ in Borceux' definition implies the uniqueness (up to isomorphism) of the initial structure. In other terms, the initial structure is essentially unique. On the other hand, Adamek requires the uniqueness (and not the essential uniqueness) of the $U$-initial lift. If my reinterpretation of the initial structure as the $U$-initial lift is correct, there would be a fundamental difference between the two definitions and, actually, we would have Adamek $\implies$ Borceux, and the converse does not hold! I wanted to know if my argument is correct or not.