Proposition [efr-ZRUV]

Let \mathcal {C} be a 2-category equipped with Lack's trivial model structure. Then a map f:X \to Y is a cofibration if and only if it satisfies any of the following equivalent conditions:

  1. For all tuples g: X \to A, h: Y \to A, \alpha : hf \simeq g, there exists a second map h' : Y \to A and isomorphism \beta : h \simeq h' so that h'f = g and \beta f = \alpha .
  2. For all A, the functor A^Y \to A^X is an isofibration

Furthermore f is a trivial cofibration if and only if there exists g: Y \to X so that gf = 1_X and fg \simeq 1_Y

Proof

It's not hard to see that the two conditions are equivalent. Let us prove that the first is equivalent to f being a cofibration.

For the "only if" part, consider the power of A by the walking isomorphism, A^I. Note that this exists because \mathcal {C} has finite limits. The map A^I \to A which picks out the first object is a trivial fibration.

Hence in the above diagram, a dashed arrow making it commute (strictly) exists, but under the adjunction identifying such maps with 2-cells in \mathcal {C}(Y,A), this is exactly the \beta we wanted.

Now suppose f has this lifting property. It's easy to see that this amounts to the claim that \mathcal {C}(Y,A) \to \mathcal {C}(X,A) is an isofibration for each A \in \mathcal {C} (in other words, it is a fibration in the trivial model structure on the opposite of \mathcal {C}).

Now let p: A \to B be a trivial fibration in \mathcal {C}. Then also A^I \to B^I is a trivial fibration, and hence the pullback A^I \times _{B^I} B \to B is a trivial fibration. This classifies "vertical" isomorphisms in A, i.e those that become identities in B.

Consider some lifting problem given by f: X \to Y, p, h_1: X \to A, h_2: Y \to B. Note that p: A \to B admits a strict section (as remarked by Lack), and so we can find \bar {h_2}: Y \to A, defined as the composite of h_2 with this section, which makes the lower triangle commute strictly, and the upper commute up to equivalence.

Hence the two maps \bar {h_2}f,h_1 and the equivalence between them describe a map X \to A^I \times _{B^I} B (a diagram chase shows that both of these maps, and the equivalence between them, goes to the same map X \to B, namely the diagonal composite in the square).

By construction this map admits an extension over Y up to equivalence, which by assumption means it admits a strict extension over Y. But after unwinding the definitions this is seen to be exactly a solution to the lifting problem (strictly).

Finally, if f is a weak equivalence, it admits a pseudoinverse g' so that fg' \simeq 1_Y, g'f \simeq 1_X. Applying the property above gives a g \simeq g' so that gf = 1_X.

Context

Notions of pseudocategory [efr-ZPPF]

Recall that a double category is a category internal to (the 1-category) \mathsf {Cat}. However, it has always been clear that this notion is often too strict, and that it is in some sense more natural to consider pseudo double categories, which satisfy the associativity and unitality law only up to a chosen (natural) family of globular 2-cells, which must in turn satisfy a coherence equation.

The defect of this definition, from our point of view, is that it is in some sense too algebraic. Just like the definition of internal category, it consists of objects equipped with operations and equations required to hold exactly, the only difference being that it operates on the 2-cells of \mathsf {Cat}, not merely the one-cells. In particular, it is not invariant under equivalence, in the sense that, given a pseudo double category \mathbb {C} with underlying graph of categories \mathbb {C}_1 \to \mathbb {C}_0, given equivalent categories \mathbb {C}_0',\mathbb {C}_1', there is not necessarily an induced (pseudo) double category structure on \mathbb {C}_1' \to \mathbb {C}_0'.

Example [efr-00A1]

Consider a double category with four objects A,B,B',C, two nonidentity vertical maps B \to B' \to B (which necessarily form an isomorphism), and two nonidentity horizontal maps A \nrightarrow B, B' \nrightarrow C (and no nonidentity squares). If \mathbb {C}_0 is the vertical category here, there is an obvious equivalence to the discrete category on three objects \mathbb {C}_0 \to \{A,B,C\} (identifying B,B'). But if we take any equivalence \mathbb {C}_1 \xrightarrow {\sim } \mathbb {C}_1' extending this diagram:

we find that the two horizontal arrows have become composable, and so there cannot be any pseudo double category structure on the right-hand side.

This problem prevents us from using certain "high-tech" methods to reason about pseudo double categories. While our eventual construction will have an "underlying" pseudo double category in the classical sense, we will introduce a different notion of "weak internal category" to enable a more conceptual approach. Especially once it comes time to construct a "symmetric monoidal weak triple category", this approach will be invaluable.

Segal objects [efr-00A0]

To motivate this approach, let us for a moment consider monoidal categories, rather than double categories. Note that, just as a monoid is essentially the same thing as a category with a single object, so a monoidal category is essentially the same thing as a pseudo double category with the vertical category equal to * (i.e, one object, one morphism).

Many monoidal categories are most naturally seen as representing an underlying promonoidal category. For example, the defining universal property of the vector product U \otimes V of vector spaces is that it represents bilinear functions on f: U \times V \to W. This only determines the tensor product up to isomorphism, but to make vector spaces into a monoidal category, we must choose a particular model for the tensor product for each U,V. The same phenomenon occurs for Cartesian products.

Hence, instead of considering a category equipped with a specific choice of tensor products, plus structural isomorphisms, it may be more natural to consider the family of all such choices, and the relationships between them. Of course, this system of data is, at least a priori, more complicated than the alternative, but we will see that it can be presented in a conceptually simple way, and that this approach has a number of advantages.

Definition Segal object [efr-001N]

Consider the simplicial category \Delta . For every n,m, the following square commutes:

A simplicial object X in a category \mathcal {C} is called a Segal object if each of these squares goes to a pullback square.

It is an old result that Segal objects in \mathsf {Set} are exactly those simplicial sets arising as nerves of categories, and simplicial maps between them are exactly functors. In fact more is true: Segal objects in any 1-category \mathcal {C} are equivalent to internal categories in \mathcal {C}, for exactly the same reason.

Proposition [efr-ZPPI]

Let \mathcal {C} be any 1-category, and let X: \Delta ^\mathrm {op} \to \mathcal {C} be a Segal object. Then there is an internal category with morphism object X[1], objects object X[0], domain and codomain given by d_1,d_0: X[1] \to X[0], identity map given by s_0: X[0] \to X[1], and composition given by X[1] \times _{X[0]} X[1] \cong X[2] \xrightarrow {d_1} X[1], where the isomorphism is the one induced by the Segal property.

This construction induces an equivalence of categories between the category \mathsf {Cat}(\mathcal {C}) of internal categories and internal functors, and [\Delta ^\mathrm {op},\mathcal {C}]_\mathrm {Segal} of Segal objects and natural transformations.

Proposition [efr-ZPPJ]

Let \mathcal {C} be any category with finite limits, and consider the full subcategory of \mathsf {Cat}(\mathcal {C}) spanned by those internal categories where \mathbb {C}_0 is terminal. This is equivalent to the category of internal monoids in \mathcal {C}.

Proposition [efr-ZPPI] and Proposition [efr-ZPPJ] together imply that a Segal object with X[0] terminal is equivalently an internal monoid.

Remark [efr-ZPPK]

It is of course perfectly meaningful to consider, for example, internal monoids in a category which does not admit all binary products---a monoid then is simply an object M and a choice of binary product M \times M and map m: M \times M \to M and so on, which simply means that only those objects which admit binary products with themselves admit monoid structures.

However, often an equivalence of categories between different "presentations", as the preceding ones, will depend on the existence of such limits---for example, given an internal category \mathbb {C}_1 \to \mathbb {C}_0, constructing the corresponding Segal object involves arbitrary iterated pullbacks \mathbb {C}_1 \times _{\mathbb {C}_0} \mathbb {C}_1 \cdots \times _{\mathbb {C}_0} \mathbb {C}_1, which may not exist even if the pullbacks implied by \mathbb {C} being an internal category do (but if they exist, they are essentially unique, so that we have the desired equivalence). Our categories here will certainly have all the limits we want, so this is only a minor point of warning.

Proposition The folk model structure on \mathsf {Cat} [efr-ZPPL]

There is a unique model structure on \mathsf {Cat} where the weak equivalences are the equivalences. It has the following characterization:

  1. The cofibrations are those functors which are injective on objects
  2. The fibrations are the isofibrations
  3. The acyclic fibrations are those Grothendieck fibrations where each fiber is a contractible groupoid
  4. Every category is fibrant and cofibrant

Proposition Trivial Model Structure [efr-MRQ3]

Let \mathcal {C} be a 2-category. Then the underlying category \mathcal {C}_0 carries a model structure defined as follows:

  1. A morphism A \to B is a weak equivalence, respectively a fibration if, for each X \in \mathcal {C}, the induced functor \mathcal {C}(X,A) \to \mathcal {C}(X,B) is a weak equivalence, respectively an isofibration
  2. A morphism is a cofibration if and only if it has the left lifting property against every morphism which is both a weak equivalence and a fibration

Moreover, when \mathsf {Cat} is equipped with its folk model structure, and \mathcal {C} is considered as enriched over \mathsf {Cat}, this is an enriched model structure in the usual sense.

This model structure is due to Lack Reference [lack-model-2-categories]. Since it is not exactly trivial, the name may be poorly chosen, but the idea is that it contains no "extra homotopy" except that implies by the \mathsf {Cat}-enrichment. For example, as Lack shows, the weak equivalences are exactly those morphisms which are invertible up to a 2-cell.

Note that, trivially, we have

Proposition [efr-OKKW]

The trivial model structure on \mathsf {Cat} coincides with the folk model structure.

We will now prove some new results about the trivial model structure

Proposition [efr-ZRUV]

Let \mathcal {C} be a 2-category equipped with Lack's trivial model structure. Then a map f:X \to Y is a cofibration if and only if it satisfies any of the following equivalent conditions:

  1. For all tuples g: X \to A, h: Y \to A, \alpha : hf \simeq g, there exists a second map h' : Y \to A and isomorphism \beta : h \simeq h' so that h'f = g and \beta f = \alpha .
  2. For all A, the functor A^Y \to A^X is an isofibration

Furthermore f is a trivial cofibration if and only if there exists g: Y \to X so that gf = 1_X and fg \simeq 1_Y

Proof

It's not hard to see that the two conditions are equivalent. Let us prove that the first is equivalent to f being a cofibration.

For the "only if" part, consider the power of A by the walking isomorphism, A^I. Note that this exists because \mathcal {C} has finite limits. The map A^I \to A which picks out the first object is a trivial fibration.

Hence in the above diagram, a dashed arrow making it commute (strictly) exists, but under the adjunction identifying such maps with 2-cells in \mathcal {C}(Y,A), this is exactly the \beta we wanted.

Now suppose f has this lifting property. It's easy to see that this amounts to the claim that \mathcal {C}(Y,A) \to \mathcal {C}(X,A) is an isofibration for each A \in \mathcal {C} (in other words, it is a fibration in the trivial model structure on the opposite of \mathcal {C}).

Now let p: A \to B be a trivial fibration in \mathcal {C}. Then also A^I \to B^I is a trivial fibration, and hence the pullback A^I \times _{B^I} B \to B is a trivial fibration. This classifies "vertical" isomorphisms in A, i.e those that become identities in B.

Consider some lifting problem given by f: X \to Y, p, h_1: X \to A, h_2: Y \to B. Note that p: A \to B admits a strict section (as remarked by Lack), and so we can find \bar {h_2}: Y \to A, defined as the composite of h_2 with this section, which makes the lower triangle commute strictly, and the upper commute up to equivalence.

Hence the two maps \bar {h_2}f,h_1 and the equivalence between them describe a map X \to A^I \times _{B^I} B (a diagram chase shows that both of these maps, and the equivalence between them, goes to the same map X \to B, namely the diagonal composite in the square).

By construction this map admits an extension over Y up to equivalence, which by assumption means it admits a strict extension over Y. But after unwinding the definitions this is seen to be exactly a solution to the lifting problem (strictly).

Finally, if f is a weak equivalence, it admits a pseudoinverse g' so that fg' \simeq 1_Y, g'f \simeq 1_X. Applying the property above gives a g \simeq g' so that gf = 1_X.

Corollary [efr-37LL]

Let \mathcal {C} be a 2-category. Then the cofibrations, weak equivalences, and fibrations of the trivial model structure on \mathcal {C} are exactly the fibrations, weak equivalences, and cofibrations, respectively, in the trivial model structure on \mathcal {C}^\mathrm {op}.

We may phrase this as saying the trivial model structure is self-dual.

If \mathcal {C} is a 2-category and I is a small category, the functor category [I,\mathcal {C}] acquires a \mathsf {Cat}-enrichment. Separately from that, there are a number of model structures we may put on the functor category---the main ones being the injective and projective model structures, and the Reedy model structure in case I comes with a Reedy structure. Since one can easily verify that all objects in the trivial model structure are both fibrant and cofibrant, it is clear that none of these model structures can be expected to coincide with the trivial model structure in general.

In fact, this difference is at the core of the problem with pseudo double categories noted above. Namely, in a functor 2-category [I,\mathcal {C}], a weak equivalence is not the same thing as a functor which is levelwise a weak equivalence. Returning to Example [efr-00A1], note that if we attempt to choose inverses \{A,B,C\} \to \{A,B,B',B\} and \mathbb {C}_1' \to \mathbb {C}_1 which respect the vertical maps, we will be forced to carry B to either B or B', and hence to carry the two maps which became composable in the codomain back into composable maps in the domain (but this is impossible). We can summarize by saying that the free pseudo double category monad on [\bullet \rightrightarrows \bullet , \mathsf {Cat}] does not preserve levelwise equivalences, but does preserve equivalences in the 2-category.

(Note that this is contrary to the case for 1-categories, where a natural transformation is a natural isomorphism if and only if it is levelwise an isomorphism.)

Lemma [efr-LSAI]

Let \mathcal {C} be a model category, and suppose there exists a family of functors F_i: \mathcal {C} \to \mathsf {Set}, which each preserve colimits, and so that a morphism f \in \mathcal {C} is a cofibration if and only if each F_i(f) is an injection. Then on the category of simplicial objects in \mathcal {C}, the injective and Reedy model structures coincide.

By definition, they have the same weak equivalences. Hence it suffices to prove they have the same cofibrations.

Observe that each F_i induces a functor from simplicial objects in \mathcal {C} to simplicial sets, which we also denote F_i. The latching object functors L_r commute with these F_i in that L_rF_i = F_iL_r, since the F_i preserve colimits. But recall that L_rX_\bullet for a simplicial set X is simply the subset of degenerate r-simplices.

Now suppose f: X_\bullet \to Y_\bullet is a morphism of simplicial objects in \mathcal {C}. Suppose it is an injective cofibration. This means each F_iX_n \to F_iY_n is an injection. We must show that, for each i, F_iX_n \coprod _{F_iL_nX} F_iL_nY \to F_iY_n is an injection. By assumption the map F_iX_n \to F_iY_n is injective, and the inclusion of the degenerate simplices is also an injection. So this is an injection as long as, whenever \sigma \in F_iX and \sigma ' \in F_iL_nY are sent to the same element, they are identified in the pushout. This amounts to the claim that if \sigma is sent to a degenerate simplex of F_iY, it is already degenerate in F_iX. This follows from the injectivity of the simplicial map F_iX \to F_iY assumed.

Conversely, suppose f is a Reedy cofibration. Suppose for induction that the maps F_iX_n \to F_iY_n are injective for all n<k and all i. Then the inclusion F_iX_k \coprod _{F_iL_kX} F_iL_kY is injective, since the map F_iL_kX \to F_iL_kY is the action of F_if on degenerate simplices, and hence is determined by those injective maps on lower-dimensional simplices. Hence as the composite of two injective maps, F_iX_k \to F_iY_k is also injective. (Note that L_0 = \emptyset so there is no problem starting the induction).

Proposition [efr-JD5V]

Let \mathcal {C} be any 2-category. Consider the 2-category \mathsf {PsCat}(\mathcal {C})_{\mathrm {ps},\mathrm {nm}} of pseudocategories, normal pseudofunctors and natural transformations in \mathcal {C}.

  1. For every X \in \mathcal {C}, there is a functor (-)^X: \mathsf {PsCat}(\mathcal {C}) \to \mathsf {PsCat}(\mathsf {Cat}) which simply applies (-)^X to all the data of the internal pseudocategory. This is contravariant functorial in X.
  2. If \mathcal {C} has finite strict 2-limits, for every finite pseudo double category K there is a functor (-)^K: \mathsf {PsCat}(\mathcal {C})_\mathrm {ps} \to \mathcal {C} defined by the universal property that \mathcal {C}(A,C^K) \cong \mathsf {PsCat}(\mathsf {Cat})(K,C^A)

Theorem [efr-2Q3G]

Let \mathcal {C} be any 2-category . Then the coherent nerve defines a functor N : \mathsf {PsCat}(\mathcal {C})_\mathrm {ps} \to [\Delta ^\mathrm {op}, \mathcal {C}]. Moreover,

  1. The coherent nerve carries fibrant pseudocategories to Reedy fibrant Segal objects.
  2. A pseudofunctor of pseudocategories is an equivalence of pseudocategories if and only if it is carried to an equivalence by N.
  3. Given two pseudocategories X,Y, the nerve induces a map \mathsf {PsCat}(X,Y) \to [\Delta ^\mathrm {op}, \mathcal {C}] (N(X),N(Y)) on the hom-objects in \mathcal {C}. For any fibrant X,Y, this is an equivalence. Moreover, there is a canonical strict retract. (Do we need fibrant here?)
  4. Up to weak equivalence, every Reedy fibrant Segal object is the nerve of a fibrant pseudocategory, whose underlying reflexive graph is given by the restriction of

In particular, the 2-functor N: \mathsf {PsCat}(\mathcal {C})_{\mathrm {ps},\fib } \to [\Delta ^\mathrm {op},\mathcal {C}]_\mathrm {Segal}^\circ is a biequivalence.

Definition Coherent Nerve [efr-FG0T]

Let \mathcal {C} be a 2-category with finite 2-limits and 2-colimits. Let X be a pseudocategory object in \mathcal {C}. Then, viewing the finite ordinals [n] as categories, we obtain for each n an object of \mathcal {C} classifying (normal) pseudofunctors [n] \to X. These assemble into a (strict) simplicial object which we denote N(X) \in [\Delta ^\mathrm {op},\mathcal {C}] and call the coherent nerve of X.