4
$\begingroup$

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.

$\endgroup$
2
  • 2
    $\begingroup$ In the first case you will get the comma object (comma category) $F \downarrow G$. In the second case, you will get an object, which I believe, was called "2-comma object" by John Gray. I may write more about the subject later. $\endgroup$ – Michal R. Przybylek Dec 3 '13 at 21:45
  • 2
    $\begingroup$ ncatlab.org/nlab/show/comma+category $\endgroup$ – Mike Shulman Dec 3 '13 at 22:33
7
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Yes I found it (pg.29), thanks! However he takes the 2 morphism that I called $\alpha$ (in the definition of 1 morphisms) to go from $G(h)\circ f$ to $f'\circ F(g)$, and accordingly the definition of 2 morphisms changes. In my work it arose like I wrote. However as I understand it is just a matter of choice. $\endgroup$ – Ilan Barnea Dec 4 '13 at 1:40
  • $\begingroup$ @IlanBarnea, I remember having the same problem :-) But, as you said, it is just a matter of choice --- one is just the opposite version of the other. $\endgroup$ – Michal R. Przybylek Dec 6 '13 at 13:35
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.