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Is there any application of topos theory in philosophy?

I am thinking that because topos theory is a sub-branch of category theory, and category theory is pretty much exclusively used in mathematics, I don't think there's any use of topos theory, but maybe I’m wrong.

In the broadest sense possible, is there any use for topos theory in philosophy? What’s the application of topos theory that's the most related to philosophy?

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There is a strand in philosophy that tries to mathematise philosophy, so it is in this strand you are likely to find topos theory applied to philosophy. Personally, I don't distinguish between category theory and set theory though they have different traditions and histories. As Marx turned Hegel upside down, likewise Category theory turned set theory upside down.

Badiou is one infamous example. He based an ontology on set and topos theory. Now one might think this might work on the basis of metaphor. But his work was rooted in Lacan. And Lacan said:

Topology is not ‘designed to guide us’ in structure. It is this structure.

Thus Badiou identifies ontology with set and topos theory. It is not a metaphor but it is this as such. This is baffling to me. Math has had a tangential relationship with philosophy and not its foundation. For example Plato eulogised math in philosophy but not as its foundation but as a step foward in the dialectic of understanding. With math, one becomes acquainted with neccessity and truth in a particularly pure form. But philosophy is far wider than math.

Basically, Badiou is led to this metaphysics by his materialist convictions: "the One is not".

But just as one should beware of sophistry in ordinary life, one ought to beware of sophistry in philosophy.

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You ask:

In the broadest sense possible, is there any use for topos theory in philosophy?

From the point of anyone in analytical philosophy who recognizes and advocates the linguistic turn and puts an emphasis on the philosophy of language, topoi are a potential foundation for building up descriptions of formal semantics. A philosopher who takes an interest in contemporary approaches to modeling thought using formal theories of meaning and formal systems, Richard Montague and his grammar (SEP) being the first, but certainly not the last to do so, will inevitably turn their eye to not the foundations of mathematics, but the foundations of logic.

There are logical systems aplenty, but there are key developments in logic, particularly with non-classical logics, that have spurred changes in various philosophical interpretations of topics. Consider Brouwer and his intuitionism as applied to his topology and math more broadly (SEP). Eventually, it inspired Per Martin-Löf and his various intuitionistic theories of types (SEP). If one is going to speculate as to how ontological thinking functions, why not use a formalism that allows one to construct the existential qualifier itself? Now, Quine and his advocacy of existential quantification for ontological purposes can be understood in terms of judgements and proof objects. Being is contingent upon proof.

But if being is contingent upon proof, and proof is contingent upon argument, then it would be nice to model how logic systems themselves might be constructed. Consider WP's article on topoi:

More recent work in category theory allows this foundation to be generalized using topoi; each topos completely defines its own mathematical framework. The category of sets forms a familiar topos, and working within this topos is equivalent to using traditional set-theoretic mathematics. But one could instead choose to work with many alternative topoi. A standard formulation of the axiom of choice makes sense in any topos, and there are topoi in which it is invalid. Constructivists will be interested to work in a topos without the law of excluded middle.

Robert Goldblatt's textbook Topoi: The Categorical Analysis of Logic (GB) presents a number of the applications of topoi for constructing otherwise recognizable formalisms used in the formalization of semantics in analytic philosophy. Chapter 6 is covers the construction of classical logic. Chapter 8 addresses intuitionistic logic. Chapter 12 produces a categorial set theory. Chapter 13 shows arithmetic and 14 includes examines the relationship to Kripke-Joyal semantics. This is the very heart of analytical philosophy itself as the project begun by Frege with his modern symbolization of logic; except now foundations is contagious, including the foundations of logic itself.

Thus, as formalisms grow in sophistication and serve to reinforce each other, we come to a more descriptive theory of what language and mind is. In fact, there is afoot a grand metaphysical project of sort to create a system to describe how the subject and objective relate in terms of formal semantics, much in the spirit of Kant's transcendentalism. Conifold's link to the book review of Badiou's Logics of Worlds and makes the same point:

Using this category-theoretical framework, Badiou can thus define the underlying structures determining the “logic” or relations of appearance determining what is treated as existent in each world, including various degrees of existence correspondent to the degrees of truth allowed by the world’s specific categorical architecture. He terms the specific structure determining these logical relationships and intensities of existence for a particular world its “transcendental”. Although the terminology echoes idealist theories from Kant to Husserl, Badiou emphasizes that in speaking of such structuring as “transcendental” he does not in any sense intend to give a theory of the transcendental subject. Instead, Badiou’s relationships of structuration of appearance are explicitly objective, determining without exception what can be understood to “exist” in a particular world, and what remains “inexistent” or invisible within its own particular way of structuring its phenomena (pp. 231-241)

Analytic philosophy has always emphasized the reliance of formal systems as a key to grounding philosophical theory. And it is by these formalisms by which analytic philosophy purports to reason philosophically about the world around us. Algebra and geometry became modern math, which stimulated proof and model theory. As mathematical logic, the systems were enriched with various non-classical logics and linguistic models. And these various mathematical logical structures can be built from topoi.

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