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semi-left-exact reflection

Semi-left-exact reflections

Semi-left-exact reflections

Idea

A semi-left-exact reflection (also called a locally cartesian reflection) is a reflector into a reflective subcategory that preserves some pullbacks. It fits into a hierarchy of left-exactness properties for reflectors:

left exact? \Rightarrow stable units? \Rightarrow semi-left-exact \Rightarrow simple

In particular, semi-left-exactness is sufficient to imply that the corresponding reflective factorization system exists and can be constructed in one step, and that the reflective subcategory inherits local cartesian closure from the ambient category.

Definitions

Let CC be a category with finite limits, and BCB\subseteq C a full reflective subcategory with reflector L:CBL:C\to B. Let EE be the class of morphisms inverted by LL, and let MM be the class of morphisms right orthogonal to EE, i.e. (E,M)(E,M) is the associated reflective prefactorization system.

The following is a combination of Theorems 4.1 and 4.3 from CHK with Proposition 1.3 from GK.

Theorem

The following are equivalent:

  1. Every pullback of an EE-morphism along an MM-morphism is an EE-morphism.
  2. Every pullback of an EE-morphism along a morphism in BB is an EE-morphism.
  3. Every pullback of a reflection unit η x:xLx\eta_x : x \to L x along a morphism in BB is an EE-morphism.
  4. LL preserves pullbacks along MM-morphisms.
  5. LL preserves pullbacks along any morphism in BB.
  6. If f:xyf:x\to y is a morphism in BB, then f *:C/yC/xf^* : C/y \to C/x preserves EE-morphisms.
  7. (If CC is locally cartesian closed) If f:xyf:x\to y is a morphism in BB, then f *:C/xC/yf_* : C/x \to C/y maps B/xB/x into B/yB/y.

They all imply that (E,M)(E,M) is a factorization system and that the factorization of an arbitrary morphism f:xyf:x\to y can be constructed as the following pullback:

If these conditions hold, we say that the reflector is semi-left-exact.

Proof

Since all morphisms in BB are MM-morphisms, (1)\Rightarrow(2) and (4)\Rightarrow(5), while (2)\Rightarrow(3) since the reflection units are in EE. And since EE consists of the LL-inverted morphisms, (5)\Rightarrow(2) and (4)\Rightarrow(1). And clearly (6)\Rightarrow(2); while for fBf\in B the action of f *f^* on a morphism in C/yC/y is the latter’s pullback along a pullback of ff, and any pullback of ff lies in MM; thus (1)\Rightarrow(6). So to complete the equivalence of (1)-(6) it will suffice to show that (3) implies (4).

First we prove that (3) implies the factorization exists. For in the diagram above gg is a pullback of the unit yLyy\to L y along the morphism LxLyL x \to L y in BB, hence gEg\in E; so by 2-out-of-3 λ fE\lambda_f\in E, while ρ fM\rho_f\in M since MM is closed under pullback and LfML f \in M.

Now this construction implies that if fMf\in M then λ f\lambda_f is invertible, hence the naturality square for η\eta at ff is a pullback. Now if we want to pull back some h:ckh:c\to k along an MM-morphism f:akf:a\to k, we can paste two pullback squares to obtain a large pullback rectangle:

Since η kh=Lhη c\eta_k \circ h = L h \circ \eta_c by naturality, we can now factor this pullback rectangle through the pullback of ff along LhL h:

so that the left-hand square is also a pullback. But this is a pullback of the reflection unit η c\eta_c along the map eLce\to L c that lies in BB (since BB is closed under pullbacks). Thus by (3) the map ded \to e lies in EE, and hence exhibits ee as LdL d. The right-hand pullback square above is then LL applied to the given pullback square, so that LL indeed preserves this pullback, i.e. (4) holds.

Finally, if CC is locally cartesian closed, then (6) implies (7) by adjointness, while (7) implies (5) in the form L/xf *L/yf *L/x \circ f^* \cong L/y \circ f^* by passage to right-adjoint mates.

Of course, if LL preserves all pullbacks (i.e. it is left exact?), then it is semi-left-exact. The statement about the existence of the reflective factorization system implies that any semi-left-exact reflection is simple.

Remark

Note that for any xCx\in C, the functor L/x:C/xB/LxL/x\colon C/x \to B/L x has a right adjoint given by pullback along η x:xLx\eta_x : x \to L x. Condition (3) above then says that the counit of this adjunction is an isomorphism, which is to say that this right adjoint is fully faithful. In this form, semi-left-exactness is equivalent to (a particular case of) the notion of admissible reflection in categorical Galois theory.

References

  • Cassidy and Hébert and Kelly, “Reflective subcategories, localizations, and factorization systems”. J. Austral. Math Soc. (Series A) 38 (1985), 287–329 (pdf)

  • David Gepner and Joachim Kock, Univalence in locally cartesian closed categories, arxiv:1208.1749

Last revised on January 18, 2019 at 08:49:33. See the history of this page for a list of all contributions to it.