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Consider the following examples.

  1. The category whose objects are the rings with unity, and whose arrows are precisely the ring homomorphisms (so that, in particular, arrows needn't preserve unity).

  2. The category whose objects are bounded lattices, and whose arrows are precisely the lattice homomorphisms (so that, in particular, arrows needn't preserve topness or bottomness).

Are concrete categories such as these (whose arrows don't preserve all the structure) ever interesting?

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It does happen sometimes – by accident – because of a phenomenon known as "property-like structure". For instance, it is a fact that every complete meet semilattice is also a complete join semilattice (hence, a complete lattice) in a unique way, so these three classes of structures have the same objects as subcategories of $\mathbf{Poset}$, yet have different homomorphisms. –  Zhen Lin Oct 27 '13 at 11:47
    
@Zhen: Perhaps you can make this an answer. –  Martin Brandenburg Oct 27 '13 at 12:57

3 Answers 3

It does happen sometimes – by accident – because of a phenomenon known as "property-like structure". For instance, it is a fact that every complete meet semilattice is also a complete join semilattice (hence, a complete lattice) in a unique way, so these three classes of structures have the same objects as subcategories of $\mathbf{Poset}$, yet have different homomorphisms.

More generally, a functor $U : \mathcal{D} \to \mathcal{C}$ forgets property-like structure if it is faithful and fully faithful on isomorphisms (i.e. for any $A$ and $B$ in $\mathcal{D}$ and any isomorphism $g : U A \to U B$ in $\mathcal{C}$, there is a unique isomorphism $f : A \to B$ in $\mathcal{D}$ such that $U f = g$). Given such a functor, we may think of objects in $\mathcal{D}$ as objects in $\mathcal{C}$ equipped with extra structure that is uniquely determined (if it exists), and the morphisms in $\mathcal{D}$ are those morphisms in $\mathcal{C}$ that preserve this extra structure.

One can also make the following observation. Suppose $U : \mathcal{D} \to \mathcal{C}$ has a left adjoint, say $F : \mathcal{C} \to \mathcal{D}$. Then we get a comonad $\mathbb{G} = (G, \epsilon, \delta)$ on $\mathcal{D}$, where $G = F U$, $\epsilon$ is the adjunction counit, and $\delta = F \eta U$, and thence a canonical functor $\mathcal{D}^\mathbb{G} \to \mathcal{C}$, where $\mathcal{D}^\mathbb{G}$ is the Kleisli category induced by $\mathbb{G}$. It is not hard to see that this functor is fully faithful, and its image in $\mathcal{C}$ is precisely the full subcategory spanned by the image of $U : \mathcal{D} \to \mathcal{C}$. So one can think of the objects in $\mathcal{C}$ that are in the image of $U$ as being objects in $\mathcal{D}$ equipped with a certain extra coalgebraic structure.

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One could also make a joke about "forgetting structure" = "equipping costructure"... –  Zhen Lin Oct 27 '13 at 16:19
    
OMG I am totally using this one! –  Malice Vidrine Nov 5 '13 at 18:34

Well if I'm right you can consider two kind of different morphism in the category of toposes: namely geometric morphisms and logical morphisms.

Based on the link a send you it seems that such kind of morphisms do not preserve completely the whole structure of topos.

Another example is that of topological spaces and homotopy equivalence classes of maps. In this category morphisms do not preserve the topological structure.

Of course you could argue that in such example the morphisms preserve exactly all the structure that we want to consider: in the case of logical functor we want to preserve the internal logic, while with geometric morphism we preserve those properties of Grothendieck toposes and homotopy classes of continuous function preserve the homotopy structure.

To such argument I can just reply that when dealing with (concrete) categories we are interested in studying object through they morphisms, and from the categorical point of view are exactly those morphism that determinate the structure of the object we are interested in.

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I think, yes. This gives us forgetting functors, and by the lasts one can build free objects.

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