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I would like to ask about properties that are preserved under equivalence of categories. To be more specific, is it true that equivalences preserve limits? Why?

Ying Zhou
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    An equivalence of categories can be "improved" to an adjoint equivalence, see here. But right adjoints preserve limits, see here. – Dietrich Burde Jul 02 '15 at 15:05
  • @DietrichBurde Thank you very much! What about other properties such as initial objects, final objects, products, coproducts, kernels, cokernels, monomorphisms, epimorphisms? May I ask.. – Ying Zhou Jul 02 '15 at 15:28
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    Everything in that list is preserved by either a left adjoint or a right adjoint, and equivalences are simultaneously left and right adjoints. – Zhen Lin Jul 02 '15 at 15:55
  • One could also ask whether a functor which is part of such an adjoint equivalence reflects or creates limits. – Stefan Hamcke Jul 02 '15 at 18:04
  • Related (identical?): http://math.stackexchange.com/questions/1201451 – Martin Brandenburg Jul 02 '15 at 20:34
  • @StefanHamcke: It's enough to show that equivalences reflect limits. But 1. equivalences are fully faithful, and fully faithful functors reflect limits. 2. alternatively, equivalences are conservative, and conservative functors that preserve limits reflect them too.

    In fact, as a special kind of (co)reflection, every equivalence is (co)monadic. This can either be shown directly, or (if you are feeling trigger happy) by Beck's theorem.

     – user54748 Jul 03 '15 at 01:43
  • I don't believe it's possible to give a satisfactory answer to this question other than "the properties that are preserved under equivalence". – Najib Idrissi Jul 03 '15 at 14:42
  • @user54748: Isn't it that a convervative functor reflects limits only for a diagram which has a limit which is preserved? – Stefan Hamcke Jul 03 '15 at 15:11
  • @user54748: Also, I think one can show that each diagram $D$ in $\cal X$ whose image $F(D)$ has a limit in $\cal A$, has a limit in $\cal X$, and to do this, we only need the fact that $G:\cal A\to X$ preserves limits and that $GF\cong 1_\cal X$. – Stefan Hamcke Jul 03 '15 at 15:16
  • @StefanHamcke: 1. Sure, otherwise preservation is vacuous, and you get a very wrong statement. Sorry for the mangled formulation. 2. Yes, in fact that's exactly the missing step from preservation + reflection to creation. – user54748 Jul 03 '15 at 21:05

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Basically any property that can be considered categorical in nature. Any textbook would list a warning if a property isn't preserved. Wikipedia lists some simple examples.

Here are some things that aren't necessarily preserved:

  • Number of objects
  • Number of morphisms (total)
  • Underlying graph
  • Other evil properties

Tip for Proofs:

Equivalences preserve hom-sets. This helps, for example, if you are trying to proof that a morphism is unique.

Community
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Christopher King
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  • Thanks..the problem is that I can not prove them. I have read the article on Wikipedia before but have trouble with proofs. For isomorphism of categories it is usually easy but for equivalences I need help. Would you please give me some hints on how to prove them? Thank you very much! – Ying Zhou Jul 02 '15 at 21:30
  • Thank you very much! This tip is very helpful! – Ying Zhou Jul 06 '15 at 13:59
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In Categories, Allegories, Freyd and Scedrov describe a diagrammatic form of logic. The simplest kind of proposition consists of a finite sequence of categories

$$ J_0 \xrightarrow{i_0} J_1 \xrightarrow{i_1} \ldots \xrightarrow{i_{n-1}} J_n $$

A functor $F_0 : J_0 \to C$ satisfies this statement if and only if:

  • there exists a functor $F_1 : J_1 \to C$ such that $F_1 i_0 = F_0$ such that
  • for every functor $F_2 : J_2 \to C$ such that $F_2 i_1 = F_1$ such that
  • there exists a functor $F_3 : J_3 \to C$ such that $F_3 i_2 = F_2$ such that
  • ...

For example, the proposition that a $J$-shaped diagram has a limit is given by the sequence (all of the $i$ maps are inclusions)

  • $J_0 = J$
  • $J_1$ is the diagram that includes the object, arrows, and equations describing a cone
  • $J_2$ is the diagram that includes the object, arrows, and equations describing another cone
  • $J_3$ is the diagram that includes the arrow and equations describing a morphism between the cones
  • $J_4$ is the diagram that includes the arrow and equations describing another morphism between the cones
  • $J_5$ includes the relations asserting both of those arrows are equal

The theorem they prove is

A proposition respects equivalence of categories if and only if it can be described on a blackboard

which is an amusing description whose actual content is

A proposition respects equivalence of categories if and only if every $i_n$ is injective on objects

The meat of the theorem is part of the canonical model structure:

A diagram of categories and functors $$ \require{AMScd} \begin{CD} A @>F>> B \\ @VGVV @VHVV \\ C @>K>> D \end{CD} $$ has the lifting property iff there exists a functor $L:C \to B$ that makes the diagram commute: that is, $F = LC$ and $K = HL$.

Theorem: If there is a specific functor $G$ such that every square where $H$ is a surjective equivalence has the lifting property, then $G$ is injective on objects

To see why this is relevant, suppose we have $$ \require{AMScd} \begin{CD} J_n @>F_n>> C \\ @Vi_nVV @VVUV \\ J_{n+1} & & D \end{CD} $$

where $U$ is a surjective equivalence. Define $G_n = U F_n$ and suppose $i_n$ is injective on objects.

Regarding satisfying propositions, we ask about the possible existence of functors $F_{n+1} : J_{n+1} \to C$ and $G_{n+1} : J_{n+1} \to D$ that satisfy $F_{n} = F_{n+1} i_n$ and $G_n = G_{n+1} i_n$, and are still related by $G_{n+1} = U F_{n+1}$.

  • If we're given an $F_{n+1}$, then it's clear that we can also construct $G_{n+1}$ simply by setting $G_{n+1} = U F_{n+1}$.
  • If we're given a $G_{n+1}$, then it completes the square above, and the lifting property gives us an $F_{n+1}$.

This can be used to show that $F_0 : J_0 \to C$ satisfies a sentence if and only if $U F_0 : J_0 \to D$ satisfies the sentence.