Add some lemmas to PathGroupoids
Add a file classifying the path space of precategories

It's rather slow.  Also currently ugly.

After     | File Name                                                          | Before   || Change
------------------------------------------------------------------------------------------------------
10m35.31s | Total                                                              | 9m25.53s || +1m09.78s
------------------------------------------------------------------------------------------------------
1m03.10s  | categories/Category/Paths                                          |    N/A   || +1m03.10s
0m26.36s  | categories/Grothendieck/PseudofunctorToCat                         | 0m25.31s || +0m01.05s
0m04.40s  | categories/DualFunctor                                             | 0m05.57s || -0m01.16s
1m45.60s  | categories/Comma/ProjectionFunctors                                | 1m45.22s || +0m00.37s
0m33.33s  | categories/Adjoint/Composition/AssociativityLaw                    | 0m32.79s || +0m00.53s
0m21.88s  | categories/Adjoint/HomCoercions                                    | 0m21.12s || +0m00.75s
0m16.64s  | categories/ExponentialLaws/Law4/Functors                           | 0m16.10s || +0m00.53s
0m16.20s  | categories/Adjoint/UnitCounitCoercions                             | 0m16.31s || -0m00.10s
0m16.11s  | categories/Comma/InducedFunctors                                   | 0m15.93s || +0m00.17s
0m15.80s  | categories/PseudonaturalTransformation/Core                        | 0m14.87s || +0m00.93s
0m15.65s  | categories/FunctorCategory/Dual                                    | 0m15.83s || -0m00.17s
0m12.82s  | categories/Pseudofunctor/Core                                      | 0m12.67s || +0m00.15s
0m12.75s  | hit/Flattening                                                     | 0m12.72s || +0m00.02s
0m12.27s  | categories/Functor/Pointwise/Properties                            | 0m12.91s || -0m00.64s
0m12.22s  | categories/Adjoint/Composition/Core                                | 0m12.44s || -0m00.21s
0m11.87s  | categories/Adjoint/Dual                                            | 0m11.80s || +0m00.06s
0m11.56s  | categories/Pseudofunctor/FromFunctor                               | 0m11.61s || -0m00.04s
0m10.76s  | categories/Adjoint/Composition/IdentityLaws                        | 0m10.62s || +0m00.14s
0m08.59s  | categories/ExponentialLaws/Law1/Law                                | 0m08.44s || +0m00.15s
0m08.26s  | categories/Yoneda                                                  | 0m08.96s || -0m00.70s
0m07.78s  | categories/Functor/Prod/Universal                                  | 0m07.75s || +0m00.03s
0m07.68s  | categories/GroupoidCategory/Morphisms                              | 0m07.52s || +0m00.16s
0m07.51s  | categories/UniversalProperties                                     | 0m07.98s || -0m00.47s
0m06.92s  | hit/Connectedness                                                  | 0m07.09s || -0m00.16s
0m06.79s  | categories/Comma/Dual                                              | 0m06.60s || +0m00.19s
0m06.68s  | categories/Adjoint/Paths                                           | 0m06.98s || -0m00.30s
0m06.49s  | categories/Structure/IdentityPrinciple                             | 0m06.32s || +0m00.16s
0m06.30s  | categories/Category/Morphisms                                      | 0m05.51s || +0m00.79s
0m06.28s  | categories/Adjoint/UnitCounit                                      | 0m05.73s || +0m00.54s
0m06.16s  | categories/Functor/Paths                                           | 0m06.24s || -0m00.08s
0m05.76s  | EquivalenceVarieties                                               | 0m05.53s || +0m00.22s
0m05.53s  | categories/Functor/Sum                                             | 0m05.48s || +0m00.04s
0m05.13s  | categories/ExponentialLaws/Law2/Functors                           | 0m04.83s || +0m00.29s
0m04.82s  | categories/Category/Sigma/OnMorphisms                              | 0m04.38s || +0m00.44s
0m03.73s  | categories/ProductLaws                                             | 0m03.76s || -0m00.02s
0m03.66s  | categories/Structure/Core                                          | 0m03.46s || +0m00.20s
0m03.29s  | types/ObjectClassifier                                             | 0m03.31s || -0m00.02s
0m03.24s  | HSet                                                               | 0m02.95s || +0m00.29s
0m03.14s  | categories/Functor/Composition/Functorial/Attributes               | 0m02.98s || +0m00.16s
0m03.10s  | types/Sigma                                                        | 0m02.94s || +0m00.16s
0m02.87s  | categories/Adjoint/UniversalMorphisms                              | 0m02.95s || -0m00.08s
0m02.79s  | categories/Functor/Prod/Functorial                                 | 0m03.00s || -0m00.20s
0m02.72s  | categories/FunctorCategory/Functorial                              | 0m02.47s || +0m00.25s
0m02.72s  | HProp                                                              | 0m02.69s || +0m00.03s
0m02.53s  | categories/Grothendieck/ToSet                                      | 0m02.39s || +0m00.13s
0m02.52s  | Equivalences                                                       | 0m02.53s || -0m00.00s
0m02.42s  | categories/Functor/Attributes                                      | 0m02.32s || +0m00.10s
0m02.28s  | categories/Category/Sum                                            | 0m02.27s || +0m00.00s
0m02.15s  | categories/NaturalTransformation/Isomorphisms                      | 0m02.12s || +0m00.02s
0m02.11s  | categories/Comma/Core                                              | 0m02.02s || +0m00.08s
0m02.05s  | categories/NaturalTransformation/Dual                              | 0m01.74s || +0m00.30s
0m02.04s  | categories/Category/Sigma/Core                                     | 0m01.94s || +0m00.10s
0m01.89s  | PathGroupoids                                                      | 0m01.83s || +0m00.05s
0m01.88s  | types/Record                                                       | 0m01.98s || -0m00.10s
0m01.85s  | categories/HomFunctor                                              | 0m01.74s || +0m00.11s
0m01.72s  | categories/FunctorCategory/Morphisms                               | 0m01.67s || +0m00.05s
0m01.61s  | categories/NaturalTransformation/Paths                             | 0m01.63s || -0m00.01s
0m01.40s  | categories/NaturalTransformation/Pointwise                         | 0m01.18s || +0m00.21s
0m01.33s  | categories/Category/Sigma/OnObjects                                | 0m01.20s || +0m00.13s
0m01.30s  | categories/Category/Prod                                           | 0m01.30s || +0m00.00s
0m01.26s  | types/Prod                                                         | 0m01.28s || -0m00.02s
0m01.25s  | categories/Limits/Core                                             | 0m01.10s || +0m00.14s
0m01.16s  | categories/NaturalTransformation/Composition/Laws                  | 0m01.16s || +0m00.00s
0m01.13s  | categories/Functor/Pointwise                                       | 0m01.08s || +0m00.04s
0m01.13s  | categories/Functor/Prod                                            | 0m00.93s || +0m00.19s
0m01.05s  | categories/Functor/Composition/Functorial/Core                     | 0m01.16s || -0m00.10s
0m01.05s  | categories/CategoryOfSections/Core                                 | 0m00.98s || +0m00.07s
0m01.03s  | categories/ExponentialLaws/Law1/Functors                           | 0m01.00s || +0m00.03s
0m01.01s  | categories/ExponentialLaws/Law   N/A                               | 0m00.97s || +0m00.04s
0m00.98s  | categories/NaturalTransformation/Prod                              | 0m00.97s || +0m00.01s
0m00.92s  | hit/Spheres                                                        | 0m00.88s || +0m00.04s
0m00.91s  | contrib/HoTTBookExercises                                          | 0m00.96s || -0m00.04s
0m00.88s  | types/Paths                                                        | 0m00.85s || +0m00.03s
0m00.86s  | hit/epi                                                            | 0m00.88s || -0m00.02s
0m00.77s  | categories/SetCategory/Morphisms                                   | 0m00.75s || +0m00.02s
0m00.74s  | contrib/HoTTBook                                                   | 0m00.76s || -0m00.02s
0m00.72s  | UnivalenceImpliesFunext                                            | 0m00.68s || +0m00.03s
0m00.67s  | hit/Suspension                                                     | 0m00.59s || +0m00.08s
0m00.66s  | types/Universe                                                     | 0m00.66s || +0m00.00s
0m00.65s  | categories/NaturalTransformation/Composition/Core                  | 0m00.65s || +0m00.00s
0m00.64s  | categories/categories                                              | 0m00.59s || +0m00.05s
0m00.63s  | types/Sum                                                          | 0m00.64s || -0m00.01s
0m00.62s  | categories/InitialTerminalCategory                                 | 0m00.64s || -0m00.02s
0m00.60s  | Trunc                                                              | 0m00.60s || +0m00.00s
0m00.57s  | Functorish                                                         | 0m00.50s || +0m00.06s
0m00.57s  | hit/quotient                                                       | 0m00.54s || +0m00.02s
0m00.54s  | categories/NaturalTransformation/Sum                               | 0m00.60s || -0m00.05s
0m00.53s  | categories/Limits/Functors                                         | 0m00.48s || +0m00.05s
0m00.53s  | hit/Circle                                                         | 0m00.56s || -0m00.03s
0m00.51s  | categories/Functor/Composition/Laws                                | 0m00.51s || +0m00.00s
0m00.50s  | categories/ExponentialLaws/Law3/Functors                           | 0m00.49s || +0m00.01s
0m00.48s  | categories/KanExtensions/Functors                                  | 0m00.49s || -0m00.01s
0m00.48s  | categories/Comma/Projection                                        | 0m00.54s || -0m00.06s
0m00.47s  | Tactics                                                            | 0m00.38s || +0m00.08s
0m00.46s  | types/Forall                                                       | 0m00.45s || +0m00.01s
0m00.43s  | categories/NaturalTransformation/Identity                          | 0m00.42s || +0m00.01s
0m00.40s  | categories/FundamentalPreGroupoidCategory                          | 0m00.40s || +0m00.00s
0m00.39s  | categories/Cat/Core                                                | 0m00.41s || -0m00.01s
0m00.38s  | TruncType                                                          | 0m00.34s || +0m00.03s
0m00.34s  | categories/ExponentialLaws/ExponentialLaws                         | 0m00.26s || +0m00.08s
0m00.34s  | Fibrations                                                         | 0m00.38s || -0m00.03s
0m00.33s  | categories/HomotopyPreCategory                                     | 0m00.32s || +0m00.01s
0m00.32s  | categories/Functor/Dual                                            | 0m00.35s || -0m00.02s
0m00.32s  | categories/NaturalTransformation/Composition/Functorial            | 0m00.33s || -0m00.01s
0m00.31s  | categories/KanExtensions/Core                                      | 0m00.34s || -0m00.03s
0m00.30s  | categories/NaturalTransformation/NaturalTransformation             | 0m00.25s || +0m00.04s
0m00.30s  | categories/IndiscreteCategory                                      | 0m00.27s || +0m00.02s
0m00.29s  | categories/NaturalTransformation/Core                              | 0m00.24s || +0m00.04s
0m00.29s  | categories/Cat/Morphisms                                           | 0m00.29s || +0m00.00s
0m00.28s  | categories/Functor/Prod/Prod                                       | 0m00.20s || +0m00.08s
0m00.28s  | categories/Grothendieck/Grothendieck                               | 0m00.26s || +0m00.02s
0m00.27s  | categories/Category/Objects                                        | 0m00.22s || +0m00.05s
0m00.27s  | categories/Functor/Composition/Core                                | 0m00.22s || +0m00.05s
0m00.27s  | categories/Notations                                               | 0m00.22s || +0m00.05s
0m00.27s  | categories/Functor/Functor                                         | 0m00.30s || -0m00.02s
0m00.26s  | categories/Utf8                                                    | 0m00.25s || +0m00.01s
0m00.25s  | hit/minus1Trunc                                                    | 0m00.27s || -0m00.02s
0m00.25s  | types/Arrow                                                        | 0m00.24s || +0m00.01s
0m00.24s  | categories/Adjoint/Hom                                             | 0m00.26s || -0m00.02s
0m00.24s  | categories/Adjoint/Adjoint                                         | 0m00.26s || -0m00.02s
0m00.23s  | categories/Grothendieck/ToCat                                      | 0m00.17s || +0m00.06s
0m00.22s  | categories/DependentProduct                                        | 0m00.20s || +0m00.01s
0m00.22s  | categories/Functor/Composition/Functorial                          | 0m00.22s || +0m00.00s
0m00.22s  | categories/Comma/Comma                                             | 0m00.21s || +0m00.01s
0m00.21s  | Conjugation                                                        | 0m00.18s || +0m00.03s
0m00.21s  | categories/NaturalTransformation/Utf8                              | 0m00.22s || -0m00.01s
0m00.20s  | categories/Limits/Limits                                           | 0m00.20s || +0m00.00s
0m00.20s  | hit/iso                                                            | 0m00.20s || +0m00.00s
0m00.20s  | categories/FunctorCategory/FunctorCategory                         | 0m00.20s || +0m00.00s
0m00.20s  | categories/Category/Category                                       | 0m00.12s || +0m00.08s
0m00.20s  | categories/NatCategory                                             | 0m00.18s || +0m00.02s
0m00.20s  | types/Bool                                                         | 0m00.18s || +0m00.02s
0m00.20s  | categories/FunctorCategory/Core                                    | 0m00.22s || -0m00.01s
0m00.19s  | categories/Comma/Utf8                                              | 0m00.22s || -0m00.03s
0m00.19s  | categories/Functor/Composition/Composition                         | 0m00.17s || +0m00.01s
0m00.18s  | categories/KanExtensions/KanExtensions                             | 0m00.18s || +0m00.00s
0m00.18s  | HoTT                                                               | 0m00.12s || +0m00.06s
0m00.18s  | categories/Functor/Notations                                       | 0m00.18s || +0m00.00s
0m00.18s  | Modality                                                           | 0m00.18s || +0m00.00s
0m00.18s  | categories/GroupoidCategory/Core                                   | 0m00.20s || -0m00.02s
0m00.18s  | categories/Pseudofunctor/Pseudofunctor                             | 0m00.18s || +0m00.00s
0m00.18s  | categories/Category/Pi                                             | 0m00.17s || +0m00.00s
0m00.17s  | categories/Profunctor/Representable                                | 0m00.12s || +0m00.05s
0m00.17s  | categories/PseudonaturalTransformation/PseudonaturalTransformation | 0m00.19s || -0m00.01s
0m00.17s  | categories/Functor/Utf8                                            | 0m00.20s || -0m00.03s
0m00.17s  | FunextVarieties                                                    | 0m00.20s || -0m00.03s
0m00.16s  | categories/Adjoint/Utf8                                            | 0m00.14s || +0m00.01s
0m00.16s  | categories/FunctorCategory/Utf8                                    | 0m00.19s || -0m00.03s
0m00.16s  | hit/Interval                                                       | 0m00.14s || +0m00.01s
0m00.16s  | categories/SetCategory/Functors/Functors                           | 0m00.19s || -0m00.03s
0m00.16s  | categories/Category/Univalent                                      | 0m00.14s || +0m00.01s
0m00.16s  | categories/Category/Core                                           | 0m00.14s || +0m00.01s
0m00.16s  | categories/ExponentialLaws/Law1/Law1                               | 0m00.15s || +0m00.01s
0m00.16s  | categories/Adjoint/Composition/LawsTactic                          | 0m00.12s || +0m00.04s
0m00.15s  | categories/Cat/Cat                                                 | 0m00.14s || +0m00.00s
0m00.15s  | categories/Adjoint/Identity                                        | 0m00.17s || -0m00.02s
0m00.15s  | categories/Adjoint/Notations                                       | 0m00.16s || -0m00.01s
0m00.15s  | categories/Adjoint/Composition/Composition                         | 0m00.18s || -0m00.03s
0m00.14s  | categories/Category/Dual                                           | 0m00.13s || +0m00.01s
0m00.14s  | categories/Comma/Notations                                         | 0m00.14s || +0m00.00s
0m00.14s  | categories/Category/Subcategory/Subcategory                        | 0m00.12s || +0m00.02s
0m00.14s  | hit/TwoSphere                                                      | 0m00.12s || +0m00.02s
0m00.14s  | Tests                                                              | 0m00.12s || +0m00.02s
0m00.14s  | categories/Profunctor/Identity                                     | 0m00.14s || +0m00.00s
0m00.14s  | categories/ExponentialLaws/Law2/Law2                               | 0m00.12s || +0m00.02s
0m00.14s  | Overture                                                           | 0m00.17s || -0m00.03s
0m00.13s  | categories/ExponentialLaws/Law3/Law3                               | 0m00.17s || -0m00.04s
0m00.13s  | categories/Profunctor/Core                                         | 0m00.15s || -0m00.01s
0m00.13s  | categories/NaturalTransformation/Composition/Composition           | 0m00.11s || +0m00.02s
0m00.13s  | categories/Category/Sigma/Sigma                                    | 0m00.09s || +0m00.04s
0m00.12s  | categories/DiscreteCategory                                        | 0m00.10s || +0m00.01s
0m00.12s  | categories/Category/Utf8                                           | 0m00.10s || +0m00.01s
0m00.12s  | categories/SetCategory/Core                                        | 0m00.14s || -0m00.02s
0m00.12s  | categories/Profunctor/Notations                                    | 0m00.13s || -0m00.01s
0m00.12s  | coq/Init/Peano                                                     | 0m00.13s || -0m00.01s
0m00.12s  | categories/Functor/Core                                            | 0m00.10s || +0m00.01s
0m00.12s  | hit/unique_choice                                                  | 0m00.14s || -0m00.02s
0m00.12s  | categories/NaturalTransformation/Notations                         | 0m00.13s || -0m00.01s
0m00.12s  | categories/SetCategory/SetCategory                                 | 0m00.13s || -0m00.01s
0m00.12s  | types/Unit                                                         | 0m00.11s || +0m00.00s
0m00.12s  | categories/Structure/Utf8                                          | 0m00.10s || +0m00.01s
0m00.12s  | categories/CategoryOfSections/CategoryOfSections                   | 0m00.11s || +0m00.00s
0m00.11s  | categories/ExponentialLaws/Law4/Law4                               | 0m00.11s || +0m00.00s
0m00.11s  | categories/FunctorCategory/Notations                               | 0m00.12s || -0m00.00s
0m00.11s  | hit/Truncations                                                    | 0m00.13s || -0m00.02s
0m00.11s  | categories/Profunctor/Utf8                                         | 0m00.16s || -0m00.05s
0m00.10s  | categories/GroupoidCategory/GroupoidCategory                       | 0m00.10s || +0m00.00s
0m00.10s  | categories/Structure/Structure                                     | 0m00.13s || -0m00.03s
0m00.10s  | Misc                                                               | 0m00.18s || -0m00.07s
0m00.10s  | categories/Profunctor/Profunctor                                   | 0m00.12s || -0m00.01s
0m00.10s  | categories/Structure/Notations                                     | 0m00.10s || +0m00.00s
0m00.10s  | categories/Category/Notations                                      | 0m00.10s || +0m00.00s
0m00.10s  | UnivalenceAxiom                                                    | 0m00.11s || -0m00.00s
0m00.10s  | categories/Functor/Pointwise/Pointwise                             | 0m00.10s || +0m00.00s
0m00.09s  | categories/Adjoint/Core                                            | 0m00.11s || -0m00.02s
0m00.09s  | Contractible                                                       | 0m00.10s || -0m00.01s
0m00.09s  | categories/Category/Subcategory/Wide                               | 0m00.12s || -0m00.03s
0m00.08s  | categories/Functor/Identity                                        | 0m00.08s || +0m00.00s
0m00.08s  | categories/Category/Strict                                         | 0m00.07s || +0m00.00s
0m00.08s  | coq/Init/Specif                                                    | 0m00.07s || +0m00.00s
0m00.08s  | categories/Category/Subcategory/Full                               | 0m00.10s || -0m00.02s
0m00.07s  | coq/Init/Datatypes                                                 | 0m00.06s || +0m00.01s
0m00.07s  | FunextAxiom                                                        | 0m00.05s || +0m00.02s
0m00.07s  | types/Empty                                                        | 0m00.08s || -0m00.00s
0m00.06s  | types/UniverseLevel                                                | 0m00.07s || -0m00.01s
0m00.06s  | Pointed                                                            | 0m00.09s || -0m00.03s
0m00.05s  | coq/Init/Logic_Type                                                | 0m00.06s || -0m00.00s
0m00.05s  | coq/Program/Tactics                                                | 0m00.06s || -0m00.00s
0m00.04s  | coq/Init/Logic                                                     | 0m00.02s || +0m00.02s
0m00.04s  | coq/Init/Prelude                                                   | 0m00.03s || +0m00.01s
0m00.03s  | coq/Init/Notations                                                 | 0m00.04s || -0m00.01s
0m00.02s  | coq/Bool/Bool                                                      | 0m00.03s || -0m00.00s
0m00.02s  | coq/Init/Tactics                                                   | 0m00.04s || -0m00.02s

WIP