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Background: I have very little intuition about category theory but I’m trying to understand the motivation behind it.

Wikipedia states:

In mathematics, a topos [...] is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion of localization; they are in a sense a generalization of point-set topology.

Moreover, this video explains that a topos is a “mathematical universe” within which one can do math. I interpret this as saying that different topoi are “foundations of mathematics”. Perhaps the category of sets is an example of such a topos?

I am confused by this: topology is a specific mathematical theory, whereas set theory can be seen as a foundation of mathematics. It seems to me that set theory is far more general than topology, so I don’t understand why the structure of topology would be usable as a “mathematical universe” within which things like groups and linear algebra can be understood.

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  • $\begingroup$ It's more like set theory than topology (in my limited experience). Maybe someone more informed than I am would disagree. $\endgroup$ – Shaun Dec 29 '18 at 7:18
  • $\begingroup$ Since topology induces a Heyting algebra which is more general from logical point of view then Boolean algebra which are induced from set theory. $\endgroup$ – Evgeny Kuznetsov Dec 29 '18 at 7:37
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The two points of view are not in conflict. When one thinks of an ordinary set theory one has two truth values, so naively a statement is either true or false. In a topos there is an object of truth values. What would that look like? ... very like the open set lattice of a space. To improve one's imagination / intuition on this a thought experiment may help.

Imagine that a statement in the language of your set theory could have value an open set of a given topological space, $X$. In what way would the logic behind your set theory be different from the bog-standard 2 valued logic. For instance, an ordinary statement like 'the real valued function $f$ is non-zero', is either true or false, but we could also assign an open subset of $\mathbb{R}$ to $f$, namely the set of points at which $f(x)$ is not zero and that is a better measure of the truth of the statment that $f$ is not zero. In that case the set of truth values would quite naturally be $Open(\mathbb{R})$. (The topos here would be that of sheaves on $\mathbb{R}$.)

You might wonder how to do logic with such truth values but the union and intersection in $Open(\mathbb{R})$ allows on to form an elementary analogue of, for instance truth tables (or analogues of more abstract and powerful ways of viewing the logic). Have a look at Steve Vickers little book: Topology via logic for some very nice and intuitive aspects of this approach in a Computer Science setting. This is quite general provided you replace spaces by locales or Heyting algebras, which are an abstraction of the open set lattices of spaces.

If you want to look at toposes as generalisations of spaces, you can go the other way and, for instance, study versions of homotopy theory that apply to Grothendieck toposes. (But that would need another long answer and this one is long enough.)

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    $\begingroup$ I should have pointed out that 'set theory' is not a 'foundation for mathematics', by itself, as there are numerous different `set theories' each with its own axiom system. The progression one takes as one proceeds through an 'apprenticeship' as a mathematician / mathematics user is to use naive set theory to develop one's understanding of what maths is and how it is used. Then one may have to go back and make the set theory less naive. It all depends on the use you are going to make of it, e.g. in Computer Science based applications it can be very useful to have a good grip on formal logic. $\endgroup$ – Tim Porter Jan 1 at 7:16

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