2 and 3 pullbacks

Question

If $F:A\to C$ and $G:B\to C$ are morphisms in $Cat$, then their pseudo-pullback (I hope it's the right notion) can be calculated as the strict limit $A\times_C C^I \times_C B$, where $I$ is the category with two objects and one isomorphism between them.

If we take $I$ to be the category with two objects and one morphism between them, then the strict limit above becomes the category that has:

1. Objects are $(a,b,f)$, where $a\in Ob(A)$, $b\in Ob(B)$ and $f:F(a)\to G(b)$ is a morphism in $C$.
2. A morphism from $(a,b,f)$ to $(a',b',f')$ is a pair $(g,h)$ such that $g:a\to a'$ is a morphism in $A$, $h:b\to b'$ is a morphism in $B$ and we have: $G(h)\circ f=f'\circ F(g)$.

My question is, what did we get? Is it also some sort of a 2 pullback?

Going one dimension higher, if $F:A\to C$ and $G:B\to C$ are morphisms in $2-Cat$ we can consider the 2 category that has:

1. Objects are $(a,b,f)$, where $a\in Ob(A)$, $b\in Ob(B)$ and $f:F(a)\to G(b)$ is a 1 morphism in $C$.
2. A 1 morphism from $(a,b,f)$ to $(a',b',f')$ is a triple $(g,h,\alpha)$ such that $g:a\to a'$ is a morphism in $A$, $h:b\to b'$ is a morphism in $B$ and $\alpha$ is a 2 morphism in $C$ from $f'\circ F(g)$ to $G(h)\circ f$.
3. A 2 morphism from $(g,h,\alpha)$ to $(g',h',\alpha')$ is a pair $(k.l)$ such that $k:g\to g'$ is a 2 morphism in $A$, $l:h\to h'$ is a 2 morphism in $B$ and we have: $(G(l)\circ id_{f})\circ \alpha=\alpha'\circ (id_{f'}\circ F(k))$.

Again my question is, what do we get? Is it some sort of a 3 pullback?

What is a good reference for these sort of things?

I'm asking because such a construction arose naturally in my work.

Answer 1

What you have described is a general construction of comma objects from pullbacks and cotensors with $2$. In your first construction you will get the comma object (comma category) $F \downarrow G$. Here is the full story.

The category $0 \rightarrow 1$ consisting of two objects and one morphism between distinct objects is usually denoted by $2$. If we fix a sufficiently finitely complete 2-category $\mathbb{W}$ (you may assume $\mathbb{W} = \mathbf{Cat}$, but the situation is more general) then for every object $C \in \mathbb{W}$, there exists an object $C^2$, more commonly written as $2\pitchfork C$, and called "the cotensor of $2$ with $C$". It is a kind of a weighted limit, universally charactersiable as a 2-representation of the functor $\hom_{\mathbf{Cat}}(2, \hom_{\mathbb{W}}(-, C))$. That is: $$\hom_{\mathbb{W}}(-, 2 \pitchfork C) \approx \hom_{\mathbf{Cat}}(2, \hom_{\mathbb{W}}(-, C))$$ (in case $\mathbb{W} = \mathbf{Cat}$, we have: $\hom(2, \hom(-, C)) \approx \hom(2 \times -, C) \approx \hom(-, \hom(2, C))$, therefore $2 \pitchfork C$ is the usual exponent $C^2$)

Because pullbacks along identities do not change objects, in your construction $2 \pitchfork C$ may be also written as $\mathit{Id} \downarrow \mathit{Id}$ (there is of course a bit more to show if you start with the usual definition of comma objects). Now, the point is that comma objects are always stable under pullbacks --- i.e. if $H \colon A \rightarrow C$ then: $$A \times_C (F \downarrow G) \approx (F \circ H) \downarrow G$$ and similarly on the other side. Thus, in your case: $$A \times_C (2 \pitchfork C) \times_C B \approx A \times_C (\mathit{Id} \downarrow \mathit{Id}) \times_C B \approx F \downarrow G$$

In your second construction, you will get an object, which I believe, was called "2-comma object" by John Gray. If I recall correctly, the relevant reference is: John W. Gray, "Formal Category Theory: Adjointness for 2-Categories", Lecture Notes in Mathematics, Volume 391, 1974.

Answer 2

These limits are more or less described in the paper "limits in double categories" by Grandis and Pare. Your 3 pullbacks are their double comma categories I believe.