The Canonical Model Structure on Cat/Gpd

This follows by basic diagram chasing. If we have the following commuting triangle, with any two maps being equivalences, we can take pseudo-inverses to construct a pseudo-inverse to the third map.

Suppose we have the following commuting diagram where \(g\) is an equivalence. We show \(f\) is also an equivalence:

Since \(g\) is an equivalence, there exists a functor \(g'\) such that \(gg' \cong 1_V\) and \(g'g \cong 1_U\). The morphism \(pg'j\) is then easily seen to be a pseudo-inverse to \(f\), showing that it is also an equivalence.

Suppose the following diagram commutes and \(g\) is an isofibration. We need to show that \(f\) is an isofibration also:

Let \(x \in X\) and \(\alpha : fx \to y\) be an isomorphism in \(Y\). We apply \(j\) to get an isomorphism \(j\alpha : jfx=gix \to jy\). As \(g\) is an isofibration, there is an isomorphism \(\beta : ix \to u \in U\) such that \(g\beta = j\alpha\). Applying \(p\), we have an isomorphism \(p\beta : pix=x \to pu \in X\). If we apply \(f\) to this map, we have:

Hence, given an isomorphism \(\alpha\), there exists an isomorphism \(p\beta\) such that \(fp\beta = \alpha\), so \(f\) is also an isofibration.

Suppose we have the following diagram in \(\mathbf{Cat}\), where \(g\) is a functor injective on objects. We need to show that \(f\) is injective on objects also:

Suppose \(x\) and \(y\) are objects in \(X\) such that \(fx = fy\). We have the following chain of equalities:

Hence \(f\) is also injective on objects.

Suppose that \(F: \mathcal{A} \to \mathcal{B}\) is an isofibration and an equivalence. To see that it is surjective on objects, let \(b \in B\). As \(F\) is an equivalence, it is fully faithful, so there is some \(a \in \mathcal{A}\) and isomorphism \(\beta : Fa \to b \in \mathcal{B}\). As \(F\) is an isofibration, there is an isomorphism \(\alpha : a \to a' \in \mathcal{A}\) such that \(F\alpha = \beta\). Then, we must have that \(Fa' = b\), so \(F\) is surjective on objects.

If \(F\) is an equivalence that is surjective on objects, it is also an isofibration. Let \(a \in \mathcal{A}\) and \(\beta : Fa \to b\) be an isomorphism in \(\mathcal{B}\). As \(F\) is surjective on objects, \(b = Fa'\) for some \(a' \in \mathcal{A}\), so \(\beta\) is a map \(Fa \to Fa'\). By \(F\) being full, there is a morphism \(\alpha : a \to a'\) such that \(F\alpha = \beta\). By \(F\) being faithful, this \(\alpha\) must be an isomorphism, so \(F\) is an isofibration.

Suppose we have the following commuting diagram, where \(f\) is injective on objects and \(g\) is an equivalence and isofibration:

As this commutes, we have the following ``object square’’ commuting in \(\mathbf{Set}\):

In particular, \(f_0\) is injective and by Lemma 10, \(g_0\) is surjective. As \((\text{inj}, \text{surj})\) form a weak factorisation system on \(\mathbf{Set}\), there is a lift \(h_0\). We now aim to turn \(h_0\) into a functor. Let \(\alpha: y \to y'\) be a morphism in \(Y\). We then have a morphism \(j\alpha : jy = gh_0y \to gh_0y'=jy' \in V_0\). As \(g\) is fully faithful, there is a unique morphism \(\beta : h_0 y \to h_0 y' \in U\) such that \(g\beta = j\alpha\). We define \(h\alpha := \beta\). As \(j\) is a functor, it must preserve identity morphisms and as \(g\) is fully faithful, the lift of an identity morphism from \(v\) to \(u\) must be the identity, so \(h\) preserves identity morphisms. Similarly, by functorality of \(j\) and fully faithfulness of \(g\), we have that composites are mapped to composites, so \(h\) is a functor and makes the bottom triangle commute:

The top triangle commutes on objects, so we need to check it commutes on maps. Given \(\alpha : x \to x' \in X\),

So \(i \alpha = h f \alpha\) by fully faithfulness of \(g\), meaning that the top triangle commutes. This means that \(h\) is indeed a lift, which shows that functors injective on objects are in the left lifting class of fucntors that are equivalences and isofibrations.

Suppose we have the following diagram commuting, with \(f\) an equivalence that is injective on objects and \(g\) an isofibration:

To construct a functor \(h:Y \to U\), we use the axiom of choice to get some extra structure. As \(f\) is essentially surjective, for every object \(y \in Y\), we choose an isomorphism \(\alpha_y : fx_y \to y\), picking the identity morphism wherever possible, i.e \(\alpha_{fx} = 1_{fx}\) (we call the argument of \(f\) \(x_y\), which is well defined at \(f\) is injective on objects). As \(g\) is an isofibration, we choose for every \(y \in Y\) a morphism \(\beta_y : ix_y \to u_y \in U\) such that \(g\beta_y = j\alpha_y\), again, picking the identity morphism whenever possible, i.e \(\beta_{fx} = 1_{ix}\). On objects, we define \(h(y) = u_y\). For morphisms, the output is a little more complicated. Starting with a map \(\gamma : y \to y'\), we form the composite \(\alpha_{y'}^{-1} \gamma \alpha_y : fx_y \to fx_{y'}\). We let \(\overline{\alpha_{y'}^{-1} \gamma \alpha_y} : x_y \to x_{y'}\) be the unique map that \(f\) maps to \(\alpha_{y'}^{-1} \gamma \alpha_y\). Then, applying \(i\), we have a map \(i \overline{\alpha_{y'}^{-1} \gamma \alpha_y} : ix_y \to ix_{y'}\). We can then compose with maps \(\beta_y^{-1}\) and \(\beta_{y'}\) to get \(\beta_{y'} i \left(\overline{\alpha_{y'}^{-1} \gamma \alpha_y}\right) \beta_{y}^{-1} : u_y \to u_{y'}\). This is what we define \(h\gamma\) to be.

To see that \(h\) is a functor, we see that it sends the identity to the identity:

We also see that \(h\) preserves composites: let \(\gamma : y \to y'\) and \(\delta : y' \to y''\). Then:

We now show that it makes both the triangles commute:

Given \(x \in X\), we need \(ix = u_{fx}\), however this follows from our choice of maps being the identity wherever possible, which means \(\beta_{fx} = 1_{ix}\). Likewise, for maps \(\gamma: x \to x'\), our chosen isomorphisms are the identity, giving \(hf\gamma = i\gamma\). Now, if \(y \in Y\), we see by definition that \(gu_y = jy\), so the bottom triangle commutes on objects. Given a map \(\gamma : y \to y' \in Y\), by the properties of \(\beta_y\) we have:

Hence \(h\) is a lift, as required.

Given \(f:X \to Y\), we form a new category \(Z\) with objects given by \(X_0 \coprod Y_0\). We define the hom-sets as follows:

We then have a functor \(X \to Z\) which is the identity on objects, and \(f\) on maps, which, in particular, is injective on objects. We can also construct another functor \(Z \to Y\) which is formed by applying \(f\) to objects from \(X\) and the identity to objects from \(y\). It is then the identity morphism on maps, which means it is fully faithful. It is also surjective on objects as every object in \(Y\) is mapped to itself by this functor. The composite of these two morphisms is \(f\) on objects and \(f\) on maps, hence it is a factorisation of \(f\) into a functor injective on objects, followed by a surjective equivalence (which by Lemma 10) is an equivalence and isofibration), as required.

Given \(f:X \to Y\) we form the comma category \(F \downarrow 1_Y\), then take the full subcategory spanned by objects \((x, y, \phi : fx \to y)\) where \(\phi\) is an isomorphism in \(Y\). Denote this category as \(F \downarrow_{\cong} 1_Y\) Note that we have a functor \(X \to F \downarrow_{\cong} 1_Y\) defined by sending \(x \mapsto (x , fx, 1_{fx})\) and \(\alpha : x \to x' \mapsto (\alpha, f\alpha)\). This is clearly injective on objects and fully faithful. We also have that every \((x, y, \phi) \cong (x, fx, 1_{fx})\) by the following commuting square in \(Y\) (with horizontal maps being isomorphisms):

We have a functor \(f \downarrow_{\cong} 1_Y \to Y\) given by projecting out the second component. This functor is also seen to be an isofibration: given \((x, y, \phi)\) and an isomorphism \(\alpha : y \to y' \in Y\), the following square commutes, with horizontal maps isomorphisms, so is an isomorphism in \(f \downarrow_{\cong} 1_Y\):

Our projection functor sends this isomorphism to \(\alpha\), so it is an isofibration. The composite of the two functors defined is easily seen to be equal to \(f\), completing the factorisation of \(f\) as an equivalence that is injective on objects, followed by an isofibration.

Lemma 9 shows functors injective on objects are stable under retracts. Lemmas 9 and 9 show that equivalences that are isofibrations are also stable under retracts. Lemma 13 gives the desired lifting data and Lemma 14 gives the desired factorisation.

Lemmas 7 and 9 show that the class of equivalences that are injective on objects is stable under retracts. Lemma 8 shows the class of isofibrations are stable under retracts. Lemma 12 gives the desired lifting data and Lemma 14 gives the desired factorisation.

Lemma shows that \(W\) satisfies 2-out-of-3. Corollaries 15 and 16 show that \((C \cap W, F)\) and \((C, F \cap W)\) are weak factorisation systems.