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ct.category theory - Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem? - MathOverflow

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Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The Lawvere fixed point theorem asserts that if X,Y are objects in a category with finite products such that the exponential YX exists, and if f:X→YX is a morphism which is surjective on points in the sense that the induced map Hom(1,X)→Hom(1,YX) is surjective, then Y has the fixed point property: for every morphism g:Y→Y there exists a point y:1→Y such that g∘y=y. The Brouwer fixed point theorem asserts that the closed n-disks, all of which I will denote by D for ease of notation, have the fixed point property as objects of Top. Seeing these two theorems together, it is tempting to try to prove the latter from the former by finding a topological space X such that the exponential DX exists, together with a surjective continuous map X→DX. Does there in fact exist such an X? Edit, 4/13/17: I'm

ct.category theory - Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem? - MathOverflow Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem? Ask Question Asked 13 years ago Modified 1 year, 3 months ago Viewed 6k times 83 $\begingroup$ The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective on points in the sense that the induced map $\text{Hom}(1, X) \to \text{Hom}(1, Y^X)$ is surjectiv

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