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Everyone has their own mental image of mathematical objects and the relations and connections between them. I came to think of morphisms in categories: does $\text{Mor} (A,B)$ only consist of structure preserving arrows between the objects $A$ and $B$ by definition, or does $\text{Mor} (A,B)$ contain higgledy-piggledy morphisms as well, just that we only care for the nice well-behaved ones?

I can't see that for a morphism, this worth of studying-property of it being structure preserving, for example being continuous in Top, follows from the axioms, nor have I found anything related in MacLane or in other standard texts. Is there a definite answer to this question?

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    $\begingroup$ I don't get your question. Once you work in a particular category, the notion of morphism is defined in that category. So it is exactly what you defined it to be. There are weird categories though. In some categories the morphisms aren't even functions. $\endgroup$ – Mathematician 42 Feb 10 '17 at 15:12
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The way I think about category theory is a general setting to formalize the term "Mathematical object" as meaning "some object in some category". (this to some extent includes "categories of categories"...) You are free to choose what your objects (if you prefer keeping them around) and morphisms are. As long as they satisfy the axioms, category theory provides you methods to study those objects. Whether a certain category is worth studying is a matter of personal preference, or perhaps applicability of results about that category to problems to want so solve.

You seem primarily concerned with concrete categories where objects are sets with structure and morphisms are functions preserving that structure. In topological spaces we study continuous maps. Continuous maps are very general: The majority of continuous maps $\mathbb{R}\to\mathbb{R}$ are completely crazy, wildly fluctuating nonsense that we will never find applications of. Nevertheless they have nice properties, which you surely know many of. When we find a function we deem interesting for some reason AND it it continuous, we can use what we know about continuous functions to study the function we want to study.

The problem with defining "nice well-behaved" is difficult. Just about any definition of nice morphisms includes a lot of pathological cases. That is why we have so many spaces of functions to fit the needs in each application: Just talking about functions $\mathbb {R}\to\mathbb{R}$ you can consider continuous, bounded, of bounded variation, Lipschitz-continuous, uniformly continuous, measurable, integrable in some way... You can consider equivalence classes of functions almost everywhere w.r.t. some measure and pretend they're functions, define Sobolev spaces and the like... On the other hand distributions and measures can be thought of as generalized concepts of function. There is no one way to define "a function that is worth studying".

A different question is certainly how to get from "structure" to "morphism preserving that structure". The most common types of structure on a set X are selected elements (e.g. identity element), unary and binary maps (inversion, addition, multiplication) and subsets of the power set (open sets, measurable sets). In the algebraic setting it is easier to justify that definition. In more general settings it requires some though and the answer might not be unique and different definitions be suited for solving different problems. It would be unreasonable to expect that only those definitions of arbitrary mathematical objects make sense, that are obvious enough to be derived from a simple rule.

For the discussion of the algebraic setting see answers in the related topics: Especially What does Structure-Preserving mean?

and maybe What is a homomorphism and what does "structure preserving" mean?

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The definition of category requires things called objects, things called arrows (or morphisms), and some axioms defining the relationship between an arrow and objects and composition of arrows, and requiring associativity and identity arrows. This is a purely axiomatic definition. You can define a category by drawing some points, some arrows between them, and specifying composition in such a way that the axioms are satisfied. Wikibooks Category Theory has examples and A Slow Introduction to Categories has a more complicated example.

Nothing in the definition requires that objects be structures or that arrow preserve structure.

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