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.

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]

• June 25, 2024
• Eigil Fjeldgren Rischel

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

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

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.

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

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

• February 22, 2025
• Eigil Fjeldgren Rischel

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.

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}.

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).

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)

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.

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.