Post History

Unification and generalization of limit and colimit

The recent post by r~~ reminded me of my own idea to unify and generalise limit and colimit. I also only occasionally dabble in category theory, and thus I wonder

  * if the following is a sound construction, and if so,
  * if it is a known concept.

The idea is as follows: We have a category $\mathcal C$, in which the generalised limit will be defined, a category $\mathcal P$ which I'll call the pattern category, and a subcategory $\mathcal I$ of $\mathcal P$ which will serve as index category.

The construction starts with a given functor (diagram) $D: \mathcal I\to\mathcal C$. Now I consider extensions of $D$ to functors $F:\mathcal P\to\mathcal C$ (that is for any object $X$ in $\mathcal I$, $F(X)=D(X)$, and likewise for morphisms).

Now suppose there exists some such extension $L$ such that from any other extension $F$ there exists an unique natural transformation from $F$ to $L$. In that case, I call the image $F(\mathcal P)$ the $(\mathcal P,\mathcal I)$-limit of the image of $D$.

Now if I'm not mistaken, if $\mathcal P$ forms a cone over $\mathcal I$, then this construction gives the limit, and if $\mathcal P$ forms a co-cone over $\mathcal I$, it gives the colimit.

An application beyond limits and colimits would be e.g. if $\mathcal C$ is the category of sets, $\mathcal P$ is the category with three objects $A,B,C$ and unique non-identity morphisms $A\to B$, $A\to C$ and $B\to C$, and $\mathcal I$ consists only of the objects $A$ and $C$ and the morphisms $A\to C$ (and the corresponding identities, of course). Then the ($\mathcal P,\mathcal I)$-limit decomposes a function $f:X\to Y$ into an injection $i:X\to Z$ and a surjection $s:Z\to Y$ where $Z$ is (isomorphic to) the disjoint union of $X$ and $Y\setminus f(X)$. Basically, the injection adds all the points not in the image of $f$, and then the surjection combines all the points that have the same image. I would guess that this generalizes to canonical decompositions of morphisms into monomorphisms and epimorphisms in other categories.