# Example of Hopf categories and conatural transformation

Let $\mathcal{C}$ be a category. It is well known how to internalize the notion of category. Let $(C_0,C_1)$ be an internal category, with source $s$, target $t$, composition $c$ and unit $e$. One can define an internal functor from two internal categories $(C_0, C_1)$ and $(D_0, D_1)$. as a pair of arrows $(F_0 : C_0 \rightarrow D_0$, $F_1 : C_1 \rightarrow D_1)$ such that

1) respect of sources and targets: $F_1 \circ s = s \circ F_1$, $F_1 \circ t = t \circ F_1$,

2) preservation of unit: $F_1 \circ e = e \circ F_0$,

3) preservation of composition: $c \circ F_1 \times F_1 = F_1 \circ c$.

We take the "convention" that the composition $c$ has to be understood morally as $c(f,g) = f\circ_c g$.

Now, let $(F_0, F_1)$ and $(G_0, G_1)$ be two internal functors from $(C_0, C_1)$ to $(D_0, D_1)$. If I'm doing no mistake, an internal natural transformation $\alpha : F \Rightarrow G$ is an arrow $\alpha : C_0 \rightarrow D_1$ such that:

1) respect of sources and targets: $s \circ \alpha = F_0$, $t \circ \alpha = G_0,$

2) naturality square (from $C_0 \times C_1$ to $D_1 \times D_1$): $c \circ \sigma \circ (\alpha \times G_1) = c \circ (\alpha \times F_1)$, where $\sigma: D_1 \times D_1 \rightarrow D_1 \times D_1$ is an "internal swap" characterized by $p_1 \circ \sigma = p_2$ and $p_2 \circ \sigma = p_1$, where $p_1$, $p_2$ are the projection of $(D_0, D_1)$ that allow us to define the pullback for the composition.

With such definition, it is easily seen what a conatural transformation is, between two cocategories and two cofunctors. Still, I don't know if this concept is useful. Does anyone have any example of them? Moreover, how would the "cosource" and "cotarget" map be interpreted in a cocategory?

Finally, one might also want to formalize the "theory of cocategory" from scratch, and even makes them interact in order to produce the theory of hopf-category (that is, a "category" with (co)source, (co)target, (co)composition, (co)unit, that makes all the operations compatible in the obvious way. Is there any example of such Hopf category?

PS: one could then define the notion of Hopf-functor, Hopf-natural transformation, ...

• I think that trying to make a theory of cocategory (whose purpose is to describe the notion of cocomposition) might be interesting, but I have some doubts about the fact that one could easily make a theory of "Hopf category". Indeed, one would like to have $\Delta \circ_1= \circ_2 \Delta\times \Delta$, which suggests that such formalization would automatically require an infinite-category. It might be interesting to see if trying to build such Hopf-category theory could make a meaningful definition of higher category. – sure Mar 11 '15 at 14:45
• Well, categories are exactly the internal monoids in the bicategory of spans over $\bf Set$, so you might look at comonoids and Hopf monoids therein. – Berci Mar 13 '15 at 23:41
• I'm not sure to understand how a category is nothing else than a monoid object in Set. The product $m$ over $X \leftarrow M \rightarrow Y$ is given by three maps $m_1: X\times X \rightarrow X$, $m_2: Y \times Y \rightarrow Y$, $m_3: M\times M \rightarrow M$. associativity of such products is nothing else than the associativity of the three $m_i$ products and their compatibility, while unit is nothing else than the existence of three units and their compatibility. How should I interprete all of them? – sure Mar 14 '15 at 9:10