small fibration

A fibered category is a small fibration if it is equivalent to a fibered category which is obtained via a specific construction of the externalization of an internal category. The input for externalization is an internal category C=(C 0,C 1,s,t,c,i)C = (C_0,C_1,s,t,c,i) in a category EE with pullbacks, and the output is a fibered category P:FEP:F\to E, whose fiber over an object II (“index set”) in CC is the small category C I=hom E(I,C)C^I = \hom_E(I,C) whose object part is C 0 IC_0^I and morphism part is C 1 IC_1^I; and the structure maps of C IC^I are obtained by postcomposing with structure maps of CC.

Conventions for the internal category CC: the composition cc takes arguments in Leibniz convention, and the source map is the right leg of the span C 0C 1C 0)C_0\leftarrow C_1\to C_0); the projection p 1,p 2:C 1× C 0C 1p_1,p_2:C_1\times_{C_0} C_1 have tp 1=sp 2t\circ p_1 = s\circ p_2.

Consider objects m:IC 0m:I\to C_0 and n:JC 0n:J\to C_0 in fibers over II and JJ. Morphisms in hom F(m,n)\hom_F(m,n) are pairs (α,f):mn(\alpha,f):m\to n where α:IJ\alpha:I\to J and f:JC 1f:J\to C_1 are morphisms in EE satisfying sf=ms\circ f = m, tfα=nt\circ f\circ\alpha = n. The interesting part of the construction is the composition: given also p:KC 0p:K\to C_0 and m(α,f)n(β,g)pm\stackrel{(\alpha, f)}\to n\stackrel{(\beta,g)}\to p, their composition is (βα,ch)(\beta\circ\alpha,c\circ h) where c:C 0× C 1C 0C 0c:C_0\times_{C_1} C_0\to C_0 is the composition and h=(f,gα):IC 0× C 1C 0h=(f,g\circ\alpha):I\to C_0\times_{C_1} C_0 is the map given by the universal property of the fibered product and f=p 1hf =p_1\circ h and gα=p 2hg\circ\alpha = p_2\circ h. One checks that so defined composition is associative.

Finally, the projection P:FCP:F\to C is rather obvious: P(m:IC 0)=IP(m:I\to C_0) = I, and in notation from above, P(α,f)=αP(\alpha,f)=\alpha. One checks that PP is a fibered category with a canonical cleavage: the chosen cartesian morphism over α:IJ\alpha : I\to J with given target n:JC 0n:J\to C_0 is (α,inα):(I,nα)(J,n)(\alpha,i\circ n\circ\alpha):(I,n\circ\alpha)\to(J,n). Indeed, for any object l:LC 0l:L\to C_0 over LL and a morphism (λ,g):ln(\lambda,g):l\to n with λ=αβ\lambda = \alpha\circ\beta, one can decompose (λ,g)=(α,inα)(β,g)(\lambda,g)=(\alpha,i\circ n\circ\alpha)\circ(\beta,g) where (β,g)(\beta,g) is a morphism in FF with P(β,g)=βP(\beta,g)=\beta.

The basic example is the small fibration Set(C)Set\mathrm{Set}(C)\to\mathrm{Set} of set-indexed families of objects in CC, for a small category CC. The objects in the fiber C IC^I, that is the functions m:IC 0m:I\to C_0, m:im iC 0m:i\mapsto m_i\in C_0 are the II-indexed families (m i) iI(m_i)_{i\in I} of objects in CC. A map (α,f):mn(\alpha,f):m\to n is a map α:IJ\alpha:I\to J together with an II-indexed family (f i) iI(f_i)_{i\in I} of morphisms f i:m in α(i)f_i:m_i\to n_{\alpha(i)} in CC. The composition is (β,(g j) jJ)(α,(f i) iI)=(βα,(g α(i)f i) iI)(\beta,(g_j)_{j\in J})\circ(\alpha,(f_i)_{i\in I})= (\beta\circ\alpha,(g_{\alpha(i)}\circ f_i)_{i\in I}).

Last revised on February 26, 2018 at 15:30:31. See the history of this page for a list of all contributions to it.