nLab semi-left-exact reflection

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 $C$ be a category with finite limits, and $B\subseteq C$ a full reflective subcategory with reflector $L:C\to B$. Let $E$ be the class of morphisms inverted by $L$, and let $M$ be the class of morphisms right orthogonal to $E$, i.e. $(E,M)$ 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 $E$-morphism along an $M$-morphism is an $E$-morphism.
2. Every pullback of an $E$-morphism along a morphism in $B$ is an $E$-morphism.
3. Every pullback of a reflection unit $\eta_x : x \to L x$ along a morphism in $B$ is an $E$-morphism.
4. $L$ preserves pullbacks along $M$-morphisms.
5. $L$ preserves pullbacks along any morphism in $B$.
6. If $f:x\to y$ is a morphism in $B$, then $f^* : C/y \to C/x$ preserves $E$-morphisms.
7. (If $C$ is locally cartesian closed) If $f:x\to y$ is a morphism in $B$, then $f_* : C/x \to C/y$ maps $B/x$ into $B/y$.

They all imply that $(E,M)$ is a factorization system and that the factorization of an arbitrary morphism $f:x\to y$ 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 $B$ are $M$-morphisms, (1)$\Rightarrow$(2) and (4)$\Rightarrow$(5), while (2)$\Rightarrow$(3) since the reflection units are in $E$. And since $E$ consists of the $L$-inverted morphisms, (5)$\Rightarrow$(2) and (4)$\Rightarrow$(1). And clearly (6)$\Rightarrow$(2); while for $f\in B$ the action of $f^*$ on a morphism in $C/y$ is the latter’s pullback along a pullback of $f$, and any pullback of $f$ lies in $M$; 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 $g$ is a pullback of the unit $y\to L y$ along the morphism $L x \to L y$ in $B$, hence $g\in E$; so by 2-out-of-3 $\lambda_f\in E$, while $\rho_f\in M$ since $M$ is closed under pullback and $L f \in M$.

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

Since $\eta_k \circ h = L h \circ \eta_c$ by naturality, we can now factor this pullback rectangle through the pullback of $f$ along $L h$:

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

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

Of course, if $L$ 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 $x\in C$, the functor $L/x\colon C/x \to B/L x$ has a right adjoint given by pullback along $\eta_x : x \to L x$. 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.

Related pages

• semi-left-exact left Bousfield localization

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