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Different perspectives

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My favourite thing about studying topos theory in depth is that you get a feel for all of the different perspectives you can take to understand them. This undoubtedly was also the source of Grothendieck's appreciation for them.

Here's a fun example I've experienced this week. We can assemble the collection of toposes into a 2-category in a few different ways, but I want to consider the 2-category of all toposes with natural number objects and all geometric morphisms between them. There are some results about limits and colimits in this 2-category in Johnstone's Sketches of an Elephant, but we can also ask some more basic questions, like "which (1-)morphisms are faithful in this 2-category?"

It turns out to be easier to answer the 1-dual of this question, namely "which geometric morphisms are cofaithful in this category?" A 1-morphism g: \mathcal{F} \to \mathcal{E} is cofaithful if it induces a faithful functor \mathrm{Hom}(\mathcal{E},\mathcal{D}) \to Hom(\mathcal{F},\mathcal{D}) by composition for all objects \mathcal{D}. For geometric morphisms, since we work with natural transformations between inverse image functors, this corresponds to composition with the inverse image functor.

Why is that an easier question to answer? This brings us to the logical perspective. The category of geometric morphisms \mathrm{Hom}(\mathcal{E},\mathcal{D}) is equivalent to the category of models of any geometric theory classified by \mathcal{D}, and this equivalence in natural in \mathcal{E}. As it happens, one of the simplest geometric theories is the theory of objects, \mathbb{O}, whose models in a topos are precisely the objects of that topos. Over any topos \mathcal{S} with a natural number object, we can construct a classifying topos \mathcal{S}[\mathbb{O}] for the theory of objects. In particular, given g: \mathcal{F} \to \mathcal{E} as above, we can consider \mathcal{D} = \mathcal{E}[\mathbb{O}] and deduce that in order for g to be faithful as a 1-morphism, the inverse image functor of g must be a faithful functor, which is to same that g must be a surjection.

Shifting to the perspective of toposes as categories and examining the data of natural transformations, we can show, conversely, that the inverse image functor of g being faithful is sufficient to make g cofaithful as a 1-morphism in the 2-category of toposes. Hooray!

Exactly the same trick allows us to show that the cofully cofaithful morphisms (I'll let you work out the definition of those!) are precisely the connected geometric morphisms, whose inverse image functor is fully faithful. And how do we understand those? Using a geometric perspective on toposes! A localic geometric morphism is connected if and only if it corresponds to a connected internal locale in the codomain topos, hence the name "connected".

I'd love to hear what perspectives people personally find most useful or familiar on toposes. I haven't even mentioned the internal language perspective, the (related but subtly different) mathematical universe perspective, the groupoid perspective, the sites and sheaves perspective, the stacks perspective, the algebraic gadget perspective... You can see where Johnstone's allegory of the Elephant comes from!