lo. logic - What is a topos? - MathOverflow As Charley mentions, topoi have many nice properties, and since a topos is something which looks like sheaves of sets on a Grothendieck site, it should be clear why a topos theory would be useful In his book, Lurie develops, among other things, a theory of infinity-topoi, of which perhaps the main example of interest is sheaves of topological spaces on a Grothendieck site
Topos-theoretic Galois theory - MathOverflow So correct me if I'm wrong ) Is there more of "topos-theoretic Galois theory" in SGA 4 or are these the only two paragraphs about that topic? Concerning the definition of the fundamental group of a topos, there is a construction in Moerdijk's Classifying Spaces and Classifying Topoi, in which he nevertheless remarks:
Interview of Connes, Caramello, and L. Lafforgue about topos theory Around 45:30, the journalist claims that topos theory gets very bad press and asks why to Connes, Caramello and Lafforgue Connes says no, this is a completely external vision of reality Prompted by the journalist, he explains what is topos theory, where one studies a space not by looking at it directly, but by putting him behind the scenes
Major applications of the internal language of toposes applying Barr's theorem and passing to a Grothendieck topos that (is boolean and) satisfies the (external) axiom of choice (I mention homological algebra and homotopy theory here because, if I recall correctly, the former strategy was used – in the first instance – in the foundations of those topics, before the development of "direct" proofs )
Higher Topos Theory- whats the moral? - MathOverflow I've often seen Lurie's Higher Topos Theory praised as the next "great" mathematical book As someone who isn't particularly up-to-date on the state of modern homotopy theory, the book seems like a lot of abstract nonsense and the initial developments unmotivated
If I want to study Jacob Luries books Higher Topoi Theory, Derived . . . To read Higher Topos Theory, you'll need familiarity with ordinary category theory and with the homotopy theory of simplicial sets (Peter May's book "Simplicial Objects in Algebraic Topology" is a good place to learn the latter) Other topics (such as classical topos theory) will be helpful for motivation
What are Jacob Luries key insights? - MathOverflow People had looked at $\infty$-categories for years, and the idea of $(\infty,n)$-categories is not in itself new What was the key new idea which started "Higher Topos Theory", the proof of the Baez-Dolan cobordism hypothesis, "Derived Algebraic Geometry", etc ?
topos theory - Stone Spaces, Locales, and Topoi for the (relative . . . My first "Topos Theory" book was Johnstone (some title), was hard but page after page I assimilated this (I'm still alive more or less), but it was the only book in argument For me opinion the Borceaux's third volume is very good I indicate these text in progressive difficulty and depth : S Mac Lane, I Moerdijk, Sheaves in Geometry and Logic
Can a topos ever be an abelian category? - MathOverflow $\begingroup$ Reid's answer is clearly spot on, but I would just like to note that in a topos if an object has an arrow to an initial object, then it is itself initial First use the fact that the graph of that arrow is monic, and then use the subobject classifier pullback square to get the desired isomorphism
When is the category of small (pre)sheaves a(n elementary) topos? Certainly cartesian closedness is necessary to be a topos; I believe that a cocomplete topos is also complete, so that completeness is also necessary But I suspect that $\mathcal{P}C$ might only have a subobject classifier if $\mathcal{C}$ is essentially small