1091 index index.xml Higher-categorical headcanon A collection of notes on the foundations of -category theory set up on top of HoTT. Based on Cisinski, Cnossen, Nguyen, and Walde's Formalization of Higher Categories, as well as Haugseng's Yet another introduction to -categories. These notes are not meant to be pedagogical or formalisation-ready. They are mostly self-contained but we assume familiarity with HoTT and the theory of wild categories therein. While the proofs are not complete, there should not be any big gaps, and the idea is that every claim should follow from the axioms listed. 260 000Q 000Q.xml The wild category of categories Axiom 2025 6 11 David Wärn We postulate a locally small -category object (in the external category of types), i.e. an infinitely coherent wild category, . Objects of are called categories and morphisms in are called functors. For , we denote the type of functors from to by . 262 000R 000R.xml On the coherences of Remark 2025 6 11 David Wärn The infinite tower of coherences of should be understood as an axiom schema. In type theory we cannot state all the axioms 'at once'. We will later see ways to internally get at such infinite tower of coherences, but this does not mean that we can get around postulating such an axiom schema in the beginning, since without it (the full schema) we see no way of going from the internal to the external. The expectation is that any concrete argument only invokes a finite number of coherences of . This can be compared with the length of contexts in type theory: the rules of type theory permit contexts of arbitrary size, while any concrete argument only ever uses contexts of some bounded size. This does not prevent type theory from dealing with infinite objects, such as infinite lists; in the same way we are not prevented from dealing with infinite towers of coherences internal to our development of higher category theory. 264 000S 000S.xml Univalence of Axiom 2025 6 11 David Wärn We postulate that is a univalent category. That is, we postulate that for any category , the subtype of consisting of invertible functors, is contractible. 266 000T 000T.xml The terminal category Axiom 2025 6 11 David Wärn We postulate the existence of a terminal category, i.e. a category such that for all categories , the type of functors is contractible. 268 0010 0010.xml There is only one terminal category Remark 2025 6 12 David Wärn Note that since is univalent, the existence of a terminal category is naturally a proposition. The same holds for other axioms about existence of categories with specified universal properties. 270 000V 000V.xml Objects of a category Definition 2025 6 11 David Wärn For a category , we denote by the type of functors from the terminal category to . 272 0013 0013.xml The action of a functor on objects Definition 2025 6 12 David Wärn Given categories and a functor , we have an induced map on objects which is simply post-composition with . 274 000U 000U.xml The walking arrow Axiom 2025 6 11 David Wärn We postulate a category together with a bijection between the type of objects of and the standard two-element type . 276 000W 000W.xml There is only one possible interval Remark 2025 6 12 David Wärn Every automorphism of fixes . Thus in the wild category there is only one object that can function as an interval. To see this, one can argue as follows. Suppose we are given an automorphism of that sends to some category . The automorphism necessarily fixes the terminal category and so the enumeration induces an enumeration of the objects of . One can then prove that the mapping spaces of are propositions (this is not entirely direct, since mapping spaces are defined in terms of functors out of , and we do not yet know that fixes ). One can show that the propositions and cannot both be true. It is a non-trivial exercise to show that the proposition is true (note that this is harder than showing it is not false!). From this we can build a functor which is easily seen to be an equivalence. 278 000X 000X.xml Morphisms in a category Definition 2025 6 12 David Wärn Given a category , functors are called morphisms (in ). Given a morphism , the object of given by the composite functor is called the domain of , written . The composition is called the codomain of , written . Given objects , the type of morphisms with domain and codomain is called a mapping space (from to in ) and is denoted . 280 001C 001C.xml Identity morphisms Construction 2025 6 13 David Wärn Given a category and an object , we obtain a morphism given by the composite functor . 282 001E 001E.xml The action of a functor on morphisms Construction 2025 6 13 David Wärn Given categories , a functor , and objects , we have an action of on morphisms given by postcomposing functors with . 284 000Y 000Y.xml has pullbacks Axiom 2025 6 12 David Wärn We postulate that has pullbacks. Explicitly, let be categories and let , be functors. Given a fourth category we can consider the type of 'cones from to '. We can compose a cone from to with a functor , to get a cone from to (this uses associativity of functor composition). The postulate is that we have a category with a cone from to such that for every category , composition induces an equivalence 286 000Z 000Z.xml has finite products Remark 2025 6 12 David Wärn From the facts that has a terminal object and pullbacks, it is easy to show that has binary products, and more generally arbitrary finite products. Explicitly, for any finite set (meaning, we have and ) and , the product exists. 288 0011 0011.xml has exponential objects Axiom 2025 6 12 David Wärn We postulate that has exponential objects. Explicitly, let be two categories. For any category we can consider the type of maps out of the product . Given a functor , we have an evident functor and so a map given by composition with . The postulate is that we have a category together with a functor which for all induces an equivalence We may also sometimes write instead of . 290 0015 0015.xml models simply typed lambda calculus Remark 2025 6 12 David Wärn Since has finite products and exponential objects, it is a Cartesian closed category, and so models simply typed lambda calculus, with all that it entails. For example we have an evident equivalence for all . Given an object , the corresponding functor is the composite . 292 001B 001B.xml preserves colimits Remark 2025 6 13 David Wärn For , the (wild) functor is a left adjoint, and so preserves the kinds of colimits one can talk about in wild categories, e.g. pushouts. 294 0012 0012.xml Functors into Axiom 2025 6 12 David Wärn For a category , the action of a functor on objects together with the enumeration of the objects of induces a map We postulate that the above map is an embedding whose image consists of those functions with the property that is monotone, in the sense that for all morphisms in , we have in . In other words, we postulate that is the cofree category of the poset . 296 0016 0016.xml The category Definition 2025 6 12 David Wärn We define by induction on a category , as follows. We take to be the terminal category, and to be the category of functors from to the interval . 298 0017 0017.xml Functors into Construction 2025 6 12 David Wärn For every , we have an enumeration and in this way, for all , we have that the map given by acting on objects is an embedding whose image consists of those functions such that for all morphisms in , we have in . In other words, is the cofree category on the poset . 300#253unstable-253.xmlProof2025612David Wärn By induction on . For there is nothing to explain. Thus assume the statement holds for some . We have which is equivalently the type of monotone maps from into . This is indeed easily seen to be . Now we have which corresponds to monotone maps from into . From here it is easy to get to the desired result. 302 0014 0014.xml The segal axiom Axiom 2025 6 12 David Wärn The square shown below is a pushout square in . 304 001D 001D.xml Composition of morphisms Construction 2025 6 13 David Wärn Given a category with objects and morphisms , , , say a witness of commutativity of the triangle consists of a functor whose action on objects is given by , and whose action on the morphisms is given by . We can thus interpret the Segal axiom as saying that for any such morphisms , the type of morphisms together with a witness of commutativity of is contractible. Let us denote by the unique morphism with a witness of commutativity of . Then by the fundamental theorem of identity types, a witness of commutativity of is equivalently an identification . 306 001F 001F.xml Unit laws Construction 2025 6 13 David Wärn Identity morphisms act as two-sided units for composition. That is, for any category with objects and a morphism , we have that and . Indeed this is witnessed by the composite functors and 308 0019 0019.xml The square has a triangulation Axiom 2025 6 13 David Wärn The square below is a pushout square in . In other words, a functor out of the walking square consists of two commutative triangles with common long edge. 310 0018 0018.xml Isomorphisms are closed under retracts Lemma 2025 6 13 David Wärn Let be objects of some wild category. Let , be morphisms with paths (). Suppose is a morphism with and . Then is invertible. 312#252unstable-252.xmlProof2025613David Wärn We claim that is a two-sided inverse to . Indeed we have and 314 001A 001A.xml is the colimit of its spine Lemma 2025 6 13 David Wärn For every , the category is freely generated by the family of objects together with the morphisms . Explicitly, this means that for any category , the evident map is an equivalence.In other words, is the colimit of a diagram of the form . 316#251unstable-251.xmlProof2025613David Wärn We induct on . For there is nothing to explain. Thus suppose for some , the category is the colimit of its spine in the sense of the lemma. Since preserves colimits, we get an analogous colimit description of , exhibiting it as the colimit of a diagram of the form . We also know that is the coreflection of the poset in . Fix an arbitrary retraction of this poset onto the linear poset . In this way we see that is a retract of . In particular, for any category this exhibits as a retract of . The colimit description of explained above together with the colimit descriptions of and give a description of the latter mapping space, . From this, and the chosen way of retracting onto , we see that is a retract of . The evident map from to the type above is a morphism of retracts of , and so is invertible. 318 001G 001G.xml Associators, pentagonators, and so on Construction 2025 6 13 David Wärn For any category , composition of morphisms in is associative (up to homotopy) and indeed has a whole (external) infinite tower of coherences. We explain only the construction of associators; the rest follows the same pattern. Given objects and morphisms , , , we would like to construct an identification . Since is the colimit of its spine, the morphisms induce a functor . Since we have a good understanding of functors , we see that the restrictions of along and are and , respectively. Hence the restriction of along equals both the desired ternary composites. 320 001H 001H.xml Natural transformations Definition 2025 6 13 David Wärn Given categories and two functors , we say that a natural transformation from to consists of a morphism in the functor category from (the object corresponding to) to . 322 001I 001I.xml The components of a natural transformation Construction 2025 6 13 David Wärn Given categories , functors a natural transformation , and an object , we have a morphism . Indeed this is simply the action on morphisms of the composite functor (It is an easy exercise to check that the action on objects of the above functor agrees with the action on of functors under the correspondence between functors and objects of .) 324 001J 001J.xml Natural transformations induce naturality squares Construction 2025 6 13 David Wärn Given categories , functors , a natural transformation , objects and a morphism , we have that . Indeed this is witnessed by the composite functor 326 001K 001K.xml The composition functor and whiskering Definition 2025 6 13 David Wärn Given categories , the operation of functor composition is captured by a functor between functor categories as is the case in any cartesian closed category. The action of the above functor on objects is called whiskering, or horizontal composition of natural transformations. The fact that the above functor respects composition gives an exchange law. 328 001L 001L.xml On coherences Remark 2025 6 13 David Wärn We have seen how to use our axioms to construct various laws relating composition of morphisms in categories, functors between categories, natural transformations between functors, and so on. There are many more similar laws (functors respect associators, naturality squares respect composition of morphisms and of functors, etc etc) which hold and are important to our development, but we will generally not spell out their constructions, with the hope that the reader has seen enough examples to understand the general strategy for obtaining such laws. It is worth noting that none of these laws are postulated: their witnesses are all constructed explicitly. The transparency of these constructions is important since it allows to extend the tower of coherences upward. The situation can be compared with that of coherences involving e.g. path composition and actions on paths in HoTT. 330 001O 001O.xml Fully faithful functors Definition 2025 6 13 David Wärn Given categories and a functor , we say that is fully faithful if for all , its action on morphisms is an equivalence of types. 332 001R 001R.xml Invertible morphisms Definition 2025 6 13 David Wärn Given a category , objects and a morphism , we say that is invertible if any of the following equivalent conditions hold: has a left inverse and a right inverse, i.e. with witnesses that and For all , precomposition with defines an equivalence of types For all , postcomposition with defines an equivalence of types 334 001S 001S.xml Fully faithful functors reflect isomorphisms Lemma 2025 6 13 David Wärn Let be categories and let be a fully faithful functor. Let be objects and a morphism. If is invertible then so is . 336#249unstable-249.xmlProof2025613David Wärn Immediate by either of the latter two characterisations of invertibility. 338 001Q 001Q.xml The Rezk axiom: categories are univalent Axiom 2025 6 13 David Wärn We postulate that for any category , the map of types that selects identity morphisms, is an embedding whose image contains all invertible morphisms of . 340 001P 001P.xml A fully faithful functor induces an embedding on objects Lemma 2025 6 13 David Wärn Let be categories and let be a fully faithful functor. Then is an embedding. 342#250unstable-250.xmlProof2025613David Wärn For objects we have to show that the action of on paths defines an equivalence By univalence, this is equivalent to the action of on isomorphisms: The above is a map of subtypes of , so it is at least an embedding. It is an equivalence since is conservative. 344 001M 001M.xml detects equivalences Axiom 2025 6 13 David Wärn We postulate that for categories and a functor , if the map of types given by postcomposition with is invertible, then so is the functor . 346 001N 001N.xml A functor is invertible if it is fully faithful and surjective on objects Remark 2025 6 13 David Wärn The axiom that detects equivalence has the following equivalent reformulation. Let be categories and a functor. First note that since is a retract of and equivalences are closed under retracts, if is invertible, then so is . Assuming that is invertible, we have that is invertible iff it is fibrewise invertible over . I.e. iff for every the action on morphisms is invertible, i.e. iff is fully faithful. Conversely, if is fully faithful, then is an embedding, so it is an equivalence iff it is surjective. Thus we have that is invertible if it is fully faithful and surjective on objects. 348 001T 001T.xml Full subcategories exist Axiom 2025 6 13 David Wärn For any category with a type family valued in small propositions, we postulate the existence of a category with a functor such that for all categories , postcomposition with induces an embedding whose image consists of precisely those functors such that for all , holds. It follows from the above that is fully faithful and that the image of is exactly the subtype determined by . 350 001U 001U.xml Fully faithful functors correspond to full subcategories Remark 2025 6 13 David Wärn Let be categories and let be a fully faithful functor. Then the action on objects has fibres that are small propositions. So there is a corresponding full subcategory inclusion . For any , clearly lies in the image of , so factors through via a functor . This is easily seen to be surjective and fully faithful and so an equivalence. In this way every fully faithful functor into a fixed category is uniquely determined by the corresponding subtype of . In particular, we have that for any category , the map given by postcomposition with is an embedding whose image consists precisely of those such that factors through . 352 001V 001V.xml Embeddings of functor categories Lemma 2025 6 14 David Wärn Let be categories and let be a fully faithful functor. Then for any category , the functor given by postcomposition with , is also fully faithful, and its image consists precisely of those such that factors through . 354#248unstable-248.xmlProof2025614David Wärn The claim that the action on objects is an embedding with image as specified follows from the fact that A has the universal property of a full subcategory. It remains to verify that the is fully faithful. By currying, arrows correspond to functors . Again by the universal property of , these form a subtype of functors . As needed 356 001X 001X.xml The full subcategory spanned by isomorphisms Lemma 2025 6 14 David Wärn Let be a category. Then the functor which is transpose to , is fully faithful and its image consists precisely of invertible morphisms in . 358#246unstable-246.xmlProof2025614David Wärn The fact that the action on objects is an embedding with image as specified is the content of the Rezk axiom. It remains to prove that is fully faithful (which has nothing to do with the Rezk axiom). Let be objects of . Then the action on morphisms corresponds, under the characterisation of morphisms in , to the map which sends to the data of , , and the composite of unit laws witnessing . We need only to observe that either of the projections from the latter type displayed above down to is an equivalence, since is equivalent to , and that the map displayed above is a section to this (invertible) projection. TODO: a better way to organise this proof is to first prove that is left (right) adjoint to () with invertible unit (counit). 360 001W 001W.xml Invertibility of natural transformations is deteced objectwise Lemma 2025 6 14 David Wärn Let be categories, functors, and a natural transformation. Suppose that for all objects , the component is invertible. Then itself is invertible. 362#247unstable-247.xmlProof2025614David Wärn The natural transformation corresponds, by flipping, to a functor The fact that is componentwise invertible means that it factors through the full subcategory . Thus corresponds to the identity morphism on a functor , and so is invertible. 364 0020 0020.xml Orthogonality Definition 2025 6 14 David Wärn Given morphisms , in some (wild) category, we say that and are orthogonal, written , if the following square is a pullback square of types. 366 001Y 001Y.xml Left and right fibrations Definition 2025 6 14 David Wärn A functor is said to be a left (resp. right) fibration if it is right orthogonal to the left (resp. right) endpoint inclusion . Explicitly, this means that is a left fibration if the map is an equivalence, where the maps and select the domain of a morphism. Equivalently, is a left fibration if for any objects , with a morphism , the type of lifts of with left endpoint , is contractible. 368 001Z 001Z.xml Orthogonality of an induced map on pushouts Lemma 2025 6 14 David Wärn Let be some morphism in a wild category, and consider a map of spans as follows. Suppose that both spans have pushouts, and . Assuming that are all left orthogonal against , then so is the induced map on pushouts . 370#245unstable-245.xmlProof2025614David Wärn This amounts to an interchange of pullbacks and pullbacks. We are tasked with proving that a certain square is a pullback square, where the square is induced by applying pullbacks objectwise to a square of cospans. This square of cospans is objectwise a pullback and Fubini gives the desired result. 372 0023 0023.xml 2-Out-of-3 for orthogonal morphisms Lemma 2025 6 14 David Wärn Let be a fixed morphism in some (wild) category. Let and be morphisms with left orthogonal to . Then is left orthogonal to if and only if is. 374#243unstable-243.xmlProof2025614David Wärn By pullback pasting / cancellation. 376 0022 0022.xml Cofinal functors Definition 2025 6 14 David Wärn A functor is said to be left cofinal (resp. right cofinal) if it is left orthogonal against every left fibration (resp. right fibration). 378 0021 0021.xml A left fibration induces a pullback square of arrow categories Lemma 2025 6 14 David Wärn Let be a left fibration. Then the following square is a pullback square in . 380#244unstable-244.xmlProof2025614David Wärn Since detects equivalences, it suffices to show that a certain square is a pullback square. This means showing that is left orthogonal to , i.e. that it is left cofinal. We use the colimit decomposition of and the condition for orthogonality of an induced map on pushouts. Namely, we have that is induced by the following map of spans. The component map is left cofinal by definition, and we have a factorisation which shows that the composite is left cofinal if the last map is. Finally we use that is a pushout of which is left cofinal. 382 0026 0026.xml Maps with left lifting closed under cobase change Lemma 2025 7 11 David Wärn Let be a fixed morphism in some wild category. Suppose given a pushout square. If is left orthogonal to , then so is . 384#242unstable-242.xmlProof2025711David Wärn By pullback pasting and cancellation. The top, bottom, and right faces in the cube below are cartesian, so the left face is too. TODO: abstract the above proof in terms of Leibniz calculus? 386 0027 0027.xml Externalising categories Remark 2025 7 14 David Wärn Categories have mapping spaces that are arbitrarily coherent and satisfy univalence, and functors between categories respect composition and so on. This means that the usual things one can do with wild categories and wild functors also apply to categories. For example we can talk about initial and terminal objects of categories, or pullback and pushout squares, and pastings thereof, or lifting properties. An important caveat here is that constructions in wild category theory do not a priori produce actual categories or functors (although they can produce objects and morphisms of given categories). 388 0028 0028.xml has pushouts Axiom 2025 7 14 David Wärn We postulate that has pushouts. Explicitly, let be categories and let , be functors. Given a fourth category we can consider the type of 'cocones from to '. We can compose a cone from to with a functor , to get a cone from to . The postulate is that we have a category with a cone from to such that for every category , composition induces an equivalence 390 002A 002A.xml Arrow categories Notation 2025 7 14 David Wärn Given a category , the functor category out of the walking arrow is sometimes denoted by . 392 002B 002B.xml Morphisms in arrow categories Remark 2025 7 14 David Wärn Let . Let be objects and let , be morphisms. We can the view and as objects of . By the colimit description of , the type of morphisms from to is equivalent to the following sigma-type. In other words, a morphism of arrows is a commutative square. 394 0029 0029.xml Slice and coslice categories Definition 2025 7 14 David Wärn Given a category with an object , the slice category over is the following pullback in . We may also denote by . The dual construction, where is replaced by , is denoted by and is called the coslice under . Note that an object of is given by an object together with a morphism . One also has the expected characterisation of morphisms in . 396 0025 0025.xml The gap map in a composite square Lemma 2025 6 15 David Wärn