Preserving, Reflecting, and Creating Limits

March 21, 2023 05:39 UTC (Last updated on March 21, 2023 07:26 UTC)

Tags: math, category theory

Introduction

This is a brief article about the notions of preserving, reflecting, and creating limits and, by duality, colimits. Preservation is relatively intuitive, but the distinction between reflection and creation is subtle.

Preservation of Limits

A functor, $F$, preserves limits when it takes limiting cones to limiting cones. As often happens in category theory texts, the notation focuses on the objects. You’ll often see things like $F(X\times Y)\cong FX\times FY$, but implied is that one direction of this isomorphism is the canonical morphism $⟨F{\pi }_1,F{\pi }_2⟩$. To put it yet another way, in this example we require $F(X\times Y)$ to satisfy the universal property of a product with the projections $F{\pi }_1$ and $F{\pi }_2$.

Other than that subtlety, preservation is fairly intuitive.

Reflection of Limits versus Creation of Limits

A functor, $F$, reflects limits when whenever the image of a cone is a limiting cone, then the original cone was a limiting cone. For products this would mean that if we had a wedge $A\stackrel{p}{\leftarrow }Z\stackrel{q}{\to }B$, and $FZ$ was the product of $FA$ and $FB$ with projections $Fp$ and $Fq$, then $Z$ was the product of $A$ and $B$ with projections $p$ and $q$.

A functor, $F$, creates limits when whenever the image of a diagram has a limit, then the diagram itself has a limit and $F$ preserves the limiting cones. For products this would mean if $FX$ and $FY$ had a product, $FX\times FY$, then $X$ and $Y$ have a product and $F(X\times Y)\cong FX\times FY$ via the canonical morphism.

Creation of limits implies reflection of limits since we can just ignore the apex of the cone. While creation is more powerful, often reflection is enough in practice as we usually have a candidate limit, i.e. a cone. Again, this is often not made too explicit.

Example

Consider the posets:

$$

Failure of reflection

Let $$ with $$ and $$ mapping to $$ where $$. Reflection fails because $$ maps to a meet but is not itself a meet.

Failure of creation

If we change the source to just $$, then creation fails because $$ and $$ have a meet in the image but not in the source. Reflection succeeds, though, because there are no non-trivial cones in the source, so every cone (trivially) gets mapped to a limit cone. It’s just that we don’t have any cones with both $$ and $$ in them.

In general, recasting reflection and creation of limits for posets gives us: Let $$ be a monotonic function. $$ reflects limits if every lower bound that $$ maps to a meet is already a meet. $$ creates limits if whenever $$ has a meet for $$, then $$ already had a meet and $$ sends the meet of $$ to the meet of $$.