The one-arrow category as a weighted limit in Cat

Many categories can be defined as weighted limits or colimits in the 2-category of categories Cat. For example the category 1 (one object with its identity) is the terminal object of Cat, the category 2 (two object with their respective identities) is the coproduct 1+1. Even comma categories are weighted limits in Cat.

What about the category {0 --> 1} consisting of two objects 0 and 1 with their identities and one morphism between them? Can it be defined as some kind of limit (or composition of limits like 1+1) in Cat?

• A similar question arose regarding one of my answers. Maybe Zhen Lin's following comment can be made into a construction of limits and colimits: math.stackexchange.com/questions/782426/… Nov 1 '16 at 22:14

First, a trivial remark. Any object $c$ of a category $C$ is both the colimit and the limit of the functor $c \colon 1 \longrightarrow C$ that sends the unique object of the terminal category $1$ to $c$. So in particular, the category $\mathbf{2} = (0\to 1)$ is both the colimit and the limit of the functor $\mathbf{2} \colon 1 \longrightarrow \mathbf{\text{Cat}}$.

Now, more interesting, as you suggest in your question, is when we can build up an object as a colimit of simpler objects. We can see the category $\mathbf{2}$ as a weighted colimit in $\mathbf{Cat}$ in a number of ways, for example:

1. as the tensor $\mathbf{2}\ast 1$, i.e. the colimit of the functor $1 \colon 1 \longrightarrow \mathbf{\text{Cat}}$ weighted by the functor $\mathbf{2} \colon 1 \longrightarrow \mathbf{\text{Cat}}$;

2. as the coinserter of the parallel pair $0,1 \colon 1 \longrightarrow 2$ (here $2$ is the discrete category with objects $0$ and $1$), which is the colimit weighted by the parallel pair $0,1\colon 1 \longrightarrow \mathbf{2}$;

3. as the co-comma category $$\require{AMScd} \begin{CD} 1 @>1>> 1 \\ @V1VV \Longrightarrow @VV1V \\ 1 @>>0> \mathbf{2} \end{CD}$$ i.e. the colimit of the span $1 \longleftarrow 1 \longrightarrow 1$ weighted by the cospan $0 \colon 1 \longrightarrow \mathbf{2} \longleftarrow 1 \colon 1$. In fact this follows from the previous example, since co-comma categories can in general be constructed by first taking a coproduct ($1+1 \cong 2$) and then taking a coinserter (as above);

4. as the colimit of the arrow $1 \longrightarrow 1$ (seen as a functor $\mathbf{2} \longrightarrow \mathbf{\text{Cat}}$) weighted by the arrow $1 \colon 1 \longrightarrow \mathbf{2}$. (For reasons to be hinted at below, this kind of weighted colimit is called a "lax colimit of an arrow".)

Hence the category $\mathbf{2}$ can be seen in a number of ways as a weighted colimit of discrete category (i.e. set)-valued diagrams, and moreover of diagrams with constant value $1$. However this is in some sense not entirely satisfying, since $\mathbf{2}$ already features in the weights of all these colimits. We can remedy this concern by considering lax colimits in $\mathbf{\text{Cat}}$.

The lax colimit of a functor $F \colon A \longrightarrow \mathbf{\text{Cat}}$ is a category $\texttt{laxcolim}F$ together with a lax natural transformation $F \longrightarrow \Delta (\texttt{laxcolim}F)$, such that for each category $C$, the induced functor $$[\texttt{laxcolim}F,C] \longrightarrow \text{Lax}[A,\mathbf{\textbf{Cat}}](F,\Delta C)$$ is an isomorphism. Here the domain is the category of functors $\texttt{laxcolim}F \longrightarrow C$ and natural transformations between them, and the codomain is the category of lax natural transformations $F \longrightarrow \Delta C$ and modifications between them.

By spelling out the definition, one can see that any category $A$ is the lax colimit of the functor $\Delta 1 \colon A \longrightarrow \mathbf{\text{Cat}}$ with constant value $1$. In particular, the category $\mathbf{2}$ is the lax colimit of the constant functor $\Delta 1 \colon \mathbf{2}\longrightarrow \mathbf{\text{Cat}}$, also known as the lax colimit of the arrow $1 \longrightarrow 1$.

(Note that there is a general construction by which a lax colimit can be calculated as a weighted colimit; for our description of $\mathbf{2}$ as a lax colimit, this construction retrieves the weighted colimit of example 4.)

• In all those definitions of 2 (even the lax-colimit one), 2 appears in its own definition! Does it mean that 2 can be defined as a least (or greatest) fixpoint?
– Bob
Nov 4 '16 at 7:48
• @Bob The same is true of the colimit $1+1 \cong 2$, for this is none other than the colimit of the functor $\Delta 1 \colon 2 \longrightarrow \mathbf{\text{Set}}$. Nov 4 '16 at 8:32
• @Bob The above colimits are not intended to be the "definition" of the category $\mathbf{2}$. The fundamental universal property of the category $\mathbf{2}$ is that it represents the functor $\mathbf{\text{Cat}} \longrightarrow \mathbf{\text{Set}}$ that sends a category to its set of morphisms; this suffices to define $\mathbf{2}$ up to isomorphism. But why not just define it is as the category with two objects and one non-identity morphism? Nov 4 '16 at 8:40
• @Bob Also note that just as one can unpack the definition of the colimit of the functor $\Delta 1 \colon 2 \longrightarrow \mathbf{\text{Set}}$ to remove explicit reference to the set $2$, one can similarly unpack the definitions colimits in examples 2,3,4 and the lax colimit to remove explicit reference to the category $\mathbf{2}$. For just as we can say "the coproduct $1+1$", we can also say "the lax colimit of the arrow $1\longrightarrow 1$", as I indeed did in my answer. Nov 4 '16 at 9:12
• @Bob I strongly encourage you to unpack the universal property of the lax colimit of an arrow, you will see that this is an elementary and simple statement just using the 2-category structure of Cat. (You need only to concern yourself with the assertion that the displayed functor in the general definition of lax colimit above be an isomorphism on objects.) Nov 4 '16 at 9:29