2025-12-03T03:27:50Z https://yabu.me Evan Cavallo https://yabu.me/npub104k4jdh9t6qzsm20rc8j6hev94n04a7ppay7ysep4swxv4zrwa7qy4a3j0 https://media.mathstodon.xyz/accounts/avatars/109/370/069/239/586/021/original/7c05af7ae01f01c6.png https://media.mathstodon.xyz/accounts/avatars/109/370/069/239/586/021/original/7c05af7ae01f01c6.png https://yabu.me/nevent1qqstvhv0gkzqrq6e3ahe2h5jvr48h8nz5nwhgmy7zzqjjfqcqh8j9sgzyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhcp63pxl In reply to <a href='/nevent1qqs0z2n82jywy25u4fd43yjtjuv9xay2ttl5qd6myukc8a6za85ee7gpzemhxue69uhhyetvv9ujuerfw36x7tnsw43z79z23df'>nevent1q…23df</a><br/>_________________________<br/><br/><span itemprop="mentions" itemscope itemtype="https://schema.org/Person"><a itemprop="url" href="/nprofile1qyt8wumn8ghj7un9d3shjtnyd968gmewwp6kytcqyp2aucj8nm0d5n2ls2z3dy2z5udpsplzpjq60kur0q9qfax33370cyndand" class="bg-lavender dark:prose:text-neutral-50 dark:text-neutral-50 dark:bg-garnet px-1"><span>nprofile1qyt8wumn8ghj7un9d3shjtnyd968gmewwp6kytcqyp2aucj8nm0d5n2ls2z3dy2z5udpsplzpjq60kur0q9qfax33370cyndand</span> (<span class="italic">nprofile…dand</span>)</a></span> They have been at this for a while, scroll for example through <a href="https://arxiv.org/search/cs?searchtype=author&amp;query=de+Oliveira,+A+G">https://arxiv.org/search/cs?searchtype=author&amp;query=de+Oliveira,+A+G</a>. I read through one of them many years ago and recall not being convinced, but I don&#39;t remember anything specific now. 2025-11-27T13:27:37Z https://yabu.me/nevent1qqszc7nsf5wadjaapdgmxl2fmye2fs2fyze8fvnn62rrt48e6cd0wkqzyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhcyxv2t0 In reply to <a href='/nevent1qqswcwexwgq74akdnqkmz64f2mzlh2p2kpa39ap7vk5866kwha89umgpzemhxue69uhhyetvv9ujuerfw36x7tnsw43z7ruck79'>nevent1q…ck79</a><br/>_________________________<br/><br/><span itemprop="mentions" itemscope itemtype="https://schema.org/Person"><a itemprop="url" href="/nprofile1qyt8wumn8ghj7un9d3shjtnyd968gmewwp6kytcqyp2aucj8nm0d5n2ls2z3dy2z5udpsplzpjq60kur0q9qfax33370cyndand" class="bg-lavender dark:prose:text-neutral-50 dark:text-neutral-50 dark:bg-garnet px-1"><span>nprofile1qyt8wumn8ghj7un9d3shjtnyd968gmewwp6kytcqyp2aucj8nm0d5n2ls2z3dy2z5udpsplzpjq60kur0q9qfax33370cyndand</span> (<span class="italic">nprofile…dand</span>)</a></span> I see, thanks! 2025-10-21T05:45:42Z https://yabu.me/nevent1qqsrw60a90ul6q5525ncc8a3jhlrgak24yz6et8whgmnk45xsfyrs7szyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhcqpac6v In reply to <a href='/nevent1qqsgawm77nx3pjt0u7vl64fu7yewh9d82e3fhl2tsx9hy6vp6gq2apgpzemhxue69uhhyetvv9ujuerfw36x7tnsw43z7l7x2lh'>nevent1q…x2lh</a><br/>_________________________<br/><br/><span itemprop="mentions" itemscope itemtype="https://schema.org/Person"><a itemprop="url" href="/nprofile1qyt8wumn8ghj7un9d3shjtnyd968gmewwp6kytcqyp2aucj8nm0d5n2ls2z3dy2z5udpsplzpjq60kur0q9qfax33370cyndand" class="bg-lavender dark:prose:text-neutral-50 dark:text-neutral-50 dark:bg-garnet px-1"><span>nprofile1qyt8wumn8ghj7un9d3shjtnyd968gmewwp6kytcqyp2aucj8nm0d5n2ls2z3dy2z5udpsplzpjq60kur0q9qfax33370cyndand</span> (<span class="italic">nprofile…dand</span>)</a></span> What model is this? It may be relevant to me 🙃 2025-10-21T01:29:39Z https://yabu.me/nevent1qqsgxqcuje0z2eh0mlt4d4nw760gwq22968ulcaxe7a4yhngnu9dp9czyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhcf63c67 Type theory is weird 2025-10-20T08:59:08Z https://yabu.me/nevent1qqsfjnzsv320kf306yka7t725fsaupp6tw4zylffp5nj2qsgjvr4nwszyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhcrcudmz In reply to <a href='/nevent1qqsdh2sngxx0qt5q6ehm62y4nuetwywdg5mnkcyp27csqgzs32q2svqpz4mhxue69uhhyetvv9ujumt0wd68ytnsw43q9rppxv'>nevent1q…ppxv</a><br/>_________________________<br/><br/><span itemprop="mentions" itemscope itemtype="https://schema.org/Person"><a itemprop="url" href="/nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqjd874v63430gng67kpw5m597f34dr9vr0yhkp8dkmnma8208raysyd6kk0" class="bg-lavender dark:prose:text-neutral-50 dark:text-neutral-50 dark:bg-garnet px-1"><span>nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqjd874v63430gng67kpw5m597f34dr9vr0yhkp8dkmnma8208raysyd6kk0</span> (<span class="italic">nprofile…6kk0</span>)</a></span> <span itemprop="mentions" itemscope itemtype="https://schema.org/Person"><a itemprop="url" href="/nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpq3p50dwg48ec85vc22dr2u3xj5knm5jpn0p8kvvffcxvxl2djkgqqwd2pts" class="bg-lavender dark:prose:text-neutral-50 dark:text-neutral-50 dark:bg-garnet px-1"><span>nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpq3p50dwg48ec85vc22dr2u3xj5knm5jpn0p8kvvffcxvxl2djkgqqwd2pts</span> (<span class="italic">nprofile…2pts</span>)</a></span> The counterexample does apply. But the counterexample also applies to cartesian cubes, and as I wrote in <a href="https://mathstodon.xyz/@ecavallo/112262872461992336">https://mathstodon.xyz/@ecavallo/112262872461992336</a> I&#39;m optimistic that an equivariance condition can also save reversals.<br/><br/>Unfortunately it doesn&#39;t pay too well to write this up; one would rather take some time to figure out the right general setting under which these equivariant model constructions give the right answer, instead of doing one more special case. But at least some of the necessary components could now be taken off the shelf from the equivariant cartesian paper, so maybe once I clear my plate a little (🙃) I should sit down and bang it out... 2025-09-30T12:23:38Z https://yabu.me/nevent1qqsv6pamxf8lmddpllgwe5ygn5zktw9gkqyjj5lhytjg8kddrtu6qxgzyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhcgf59w2 My paper with Christian Sattler on cubes with one connection has been accepted to the Canadian Journal of Mathematics: <a href="https://doi.org/10.4153/S0008414X25101466">https://doi.org/10.4153/S0008414X25101466</a><br/><br/>The title buries the lede (oops), but it&#39;s about a constructive model of homotopy/cubical type theory and a (classical) proof that its associated homotopy theory agrees with the homotopy theory of topological spaces. The first such model to be found with this property was <a href="https://arxiv.org/abs/2406.18497">https://arxiv.org/abs/2406.18497</a>; this one is in some ways simpler, in other ways more complicated. The main technical work is in generalizing some combinatorial techniques from homotopy theory (Reedy category theory) to deal with a more exotic cube category (cartesian cubes with one connection) that came out of constructive semantics of homotopy type theory.<br/><br/>It&#39;s in copy-editing now, so this is your last chance to find us some typos ;) 2025-08-25T11:40:38Z https://yabu.me/nevent1qqsyeg89xz6hhugtyf3dqfgw3ek4fad954wu56ae3yk7pck2l23xduqzyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhc3gxd82 The usual proof of \((\lvert x\rvert_{n+1} = \lvert y \rvert_{n+1}) \simeq \lVert x = y\rVert_n\) uses a univalent universe. Anybody know a (non-univalent) model where it fails? 2025-08-16T09:55:23Z https://yabu.me/nevent1qqstp9wlk9h8hhpc8kxdun6q2d2y394qke405n7mn37ynm08wp3eq8gzyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhcskns3q (-1)-types (modest proposals) 2025-08-10T16:13:45Z https://yabu.me/nevent1qqswg2gt8we8hmtsguksl0saszz63gzv5raatd0u42q8mrfhrntgweszyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhcmg44qv Here&#39;s a puzzle (don&#39;t know an answer) that came up while I was thinking about <span itemprop="mentions" itemscope itemtype="https://schema.org/Person"><a itemprop="url" href="/nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpq67ehwdunn4l8upz54fmqammyrn7h0ptquhwd9jxxkzhqxd0k95sq209vc2" class="bg-lavender dark:prose:text-neutral-50 dark:text-neutral-50 dark:bg-garnet px-1"><span>nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpq67ehwdunn4l8upz54fmqammyrn7h0ptquhwd9jxxkzhqxd0k95sq209vc2</span> (<span class="italic">nprofile…9vc2</span>)</a></span>&#39;s &#34;type-theoretic axiom of replacement&#34;.<br/><br/>Let&#39;s say we&#39;re in intensional MLTT with funext and a univalent universe U (but you can drop funext and univalence and I&#39;m still interested in the answer).<br/><br/>Q: Are there any closed examples of a surjection f : A → B where A is U-small and B is locally U-small, but B cannot be proven to be U-small?<br/><br/>Here a type is U-small when it&#39;s equivalent to a type in U, and a type is locally U-small when its identity types are U-small. Since I&#39;m not assuming a propositional truncation, f being surjective means that for any B-indexed family of h-propositions P, precomposition (-) ∘ f : (Πb:B.P(b)) → (Πa:A.P(fa)) is an equivalence. 2025-08-01T15:35:08Z https://yabu.me/nevent1qqsds4upmu8wg0gcyy9guac4ve4nt90fymdvk434ll8mcxa0y68p6jgzyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhc5nlqgx In reply to <a href='/nevent1qqsquc30t34eyz02pzy9tcdf78rgnw4h29c26u94cn8tl70gwdup4pspz4mhxue69uhhyetvv9ujumt0wd68ytnsw43qktga06'>nevent1q…ga06</a><br/>_________________________<br/><br/><span itemprop="mentions" itemscope itemtype="https://schema.org/Person"><a itemprop="url" href="/nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpq73zugfcw5qemam49j598gfhr5nyctxun4mt3y8qf4ayj4pdnt7gqkfv2av" class="bg-lavender dark:prose:text-neutral-50 dark:text-neutral-50 dark:bg-garnet px-1"><span>nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpq73zugfcw5qemam49j598gfhr5nyctxun4mt3y8qf4ayj4pdnt7gqkfv2av</span> (<span class="italic">nprofile…v2av</span>)</a></span> For one thing I made a mistake, that should be S² → Type (corrected now). Thanks!<br/><br/>The universal property of S² as encoded in type theory is that a map S² → A is determined by a point a : A and a two-dimensional loop at a : A, i.e. an inhabitant of the interated identity type \(\mathsf{refl}_a =_{a =_A a} \mathsf{refl}_a\).<br/>In this case, we build a map H into a universe, which I called &#34;Type&#34;; to do so we choose a type in Type and a two-dimensional loop at that type.<br/>The type will be S¹.<br/><br/>For the two-dimensional loop, we use the univalence axiom for Type, which says that identities in it correspond to homotopy equivalences, and the reflexive identity corresponds to the identity equivalence.<br/>It also implies that identities between equivalences correspond to homotopies between the underlying functions.<br/>So giving an element of \(\mathsf{refl}_{S^1} =_{S^1 =_{\mathsf{Type}} S^1} \mathsf{refl}_{S^1}\) means giving an element of \(\mathsf{id}_{S^1} =_{S^1 ≃ S^1} \mathsf{id}_{S^1}\) means giving a family of identities \(x =_{S^1} x\) for all x : S¹.<br/>The Hopf fibration H : S² → Type is what we get when for each x : S¹ we take the loop \(x =_{S^1} x\) that goes around the circle once. 2025-07-26T13:45:22Z https://yabu.me/nevent1qqswze03a70xpp4ymch2fm4maj2c9h0ps579xfhru2fcpf5unmatzlqzyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhcef0a54 Doing synthetic homotopy theory in univalent type theory, we usually assume that we have a suspension type constructor from which we can define all the n-spheres. But a funny consequence of the Hopf fibration is that in univalent type theory with just a 1-sphere and a 2-sphere (but no other HITs assumed), we can already define a 3-sphere type, namely the Σ-type Σ(x:S²).H(x) with the Hopf fibration H : S¹ → Type definable by elimination into our univalent universe. Likewise, if we have 3- and 4-spheres we should be able to build the 7-sphere, and if we have 7- and 8-spheres the 15-sphere.<br/><br/>That exhausts the applications of this particular construction, by Adams&#39; theorem. Are there any other ways to get new spheres from existing spheres by standard type-theoretic operations? (I don&#39;t know and I&#39;m not the right person to ask. I also don&#39;t have any reason to ask; just wondering.) 2025-07-26T12:16:20Z https://yabu.me/nevent1qqs27g5g2um2mu4v8l4mm2yzp2qadq52arrq5fcu8nm0ks7fheh28xczyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhc0jkhg3 In reply to <a href='/nevent1qqsrp4p2razsz7ve4psledwppcf7qxmfjhg8k3k4ptfftssd7n6ng4gpz4mhxue69uhhyetvv9ujumt0wd68ytnsw43q5gp3tf'>nevent1q…p3tf</a><br/>_________________________<br/><br/><span itemprop="mentions" itemscope itemtype="https://schema.org/Person"><a itemprop="url" href="/nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqjd874v63430gng67kpw5m597f34dr9vr0yhkp8dkmnma8208raysyd6kk0" class="bg-lavender dark:prose:text-neutral-50 dark:text-neutral-50 dark:bg-garnet px-1"><span>nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqjd874v63430gng67kpw5m597f34dr9vr0yhkp8dkmnma8208raysyd6kk0</span> (<span class="italic">nprofile…6kk0</span>)</a></span> <span itemprop="mentions" itemscope itemtype="https://schema.org/Person"><a itemprop="url" href="/nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqxhmu64qvpl0efrty68zzwut2eg20dc354wz6mjd0jm9ml2crqusq8dtphs" class="bg-lavender dark:prose:text-neutral-50 dark:text-neutral-50 dark:bg-garnet px-1"><span>nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqxhmu64qvpl0efrty68zzwut2eg20dc354wz6mjd0jm9ml2crqusq8dtphs</span> (<span class="italic">nprofile…tphs</span>)</a></span> <span itemprop="mentions" itemscope itemtype="https://schema.org/Person"><a itemprop="url" href="/nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqjpjz2fr8cz3fc37qmta2tv32ndme82n5w72vyr2gyvzn4a4skk0suq9vrk" class="bg-lavender dark:prose:text-neutral-50 dark:text-neutral-50 dark:bg-garnet px-1"><span>nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqjpjz2fr8cz3fc37qmta2tv32ndme82n5w72vyr2gyvzn4a4skk0suq9vrk</span> (<span class="italic">nprofile…9vrk</span>)</a></span> Re the EDIT: the Smith paper that afaik introduces large elimination (<a href="https://www.cse.chalmers.se/~smith/Rek.pdf">https://www.cse.chalmers.se/~smith/Rek.pdf</a>) has the natural numbers as its main example, so imo it&#39;s not unclear what one needs to do; one just doesn&#39;t want to do it! 2025-07-02T13:29:49Z https://yabu.me/nevent1qqs8kzmfha0nedyr4y0tmdmd2p7nzqmdjhmwmktx6z9yt2neszgfnaszyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhc3dgqah Does anyone know of a model of intensional MLTT that satisfies LEM but not function extensionality? Preferably the strong version of LEM that says A or ¬A for all types A, not only h-propositions. 2025-06-20T21:21:59Z https://yabu.me/nevent1qqsgr07qxlx6mwkuuqtp65grp3l4mqrrk3z7ggj9kxxumeqr9wrpd8szyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhce7s5ga In reply to <a href='/nevent1qqsxa2tnkv6e70g5lswcd6yd96tp36kk0r6adpssmegfs7fxmm3l6ccpz4mhxue69uhhyetvv9ujumt0wd68ytnsw43qkrvc2v'>nevent1q…vc2v</a><br/>_________________________<br/><br/><span itemprop="mentions" itemscope itemtype="https://schema.org/Person"><a itemprop="url" href="/nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqez3yya8tpgge7lk4jq2zxz0cj3d2mcgev9mffd4ec7tfy84hre4sqchszg" class="bg-lavender dark:prose:text-neutral-50 dark:text-neutral-50 dark:bg-garnet px-1"><span>nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqez3yya8tpgge7lk4jq2zxz0cj3d2mcgev9mffd4ec7tfy84hre4sqchszg</span> (<span class="italic">nprofile…hszg</span>)</a></span> I think Bourke and Garner&#39;s paper <a href="https://arxiv.org/abs/1412.6559">https://arxiv.org/abs/1412.6559</a> contains a related result. I will not try to unfold everything here, but in Section 4.4 they show that for certain classes of monics M, there is an algebraic weak factorization system (L,R) where the L-coalgebras are the maps in M. They observe there that R sends a map to its partial map classifier. Then in Proposition 20 they prove that for general a.w.f.s, the R-algebras are maps with &#34;coherent&#34; lifting structure against a certain double category formed by the L-coalgebras. In this case, the two coherence conditions (in 5.1 and 6.1) are the pullback naturality and associativity you are looking at. 2025-06-09T10:35:09Z https://yabu.me/nevent1qqsyplvclnvffjcz9rrhc5yt76tlfxpa8ngtnnw0d9y7d5e37us8zuczyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhcuvehea <a href="https://arxiv.org/abs/2506.02759">https://arxiv.org/abs/2506.02759</a><br/><br/>Something on the arXiv from Christian Sattler and I today!<br/><br/>This paper is about small object arguments, which are used in homotopy theory to construct various factorizations of maps. A little more specifically, it&#39;s about Garner&#39;s *algebraic* small object argument, which is a more recent, more general, more frequently constructive variant of the original argument (due to Quillen). We start from a certain useful characterization of factorizations coming with Quillen&#39;s argument and hunt down an analogue of sorts for the algebraic case.<br/><br/>I hope the paper will be interesting to people who are interested in these things in general, but of course for me the motivation is in semantics of homotopy type theory. The algebraic small object argument is used in particular in the cubical model constructions, and the reason we ended up writing this paper is because we needed the result somewhere deep in an argument about these models. Someday you&#39;ll hear more about that :)<br/><br/>I talked about this work at HoTT/UF a month and a half ago, so you can check out some slides and video at <a href="https://hott-uf.github.io/2025/">https://hott-uf.github.io/2025/</a> 2025-06-04T04:13:41Z https://yabu.me/nevent1qqsp9r6dy6cpmxnkhnkvnqtlggcxqnrrvtkmh0xtzqh8t863wks3j7czyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhcc3cgkq Today&#39;s rediscovery of literature I should be citing more often: Raffael Stenzel&#39;s thesis <a href="https://etheses.whiterose.ac.uk/24342/">https://etheses.whiterose.ac.uk/24342/</a> 2025-03-31T22:02:33Z https://yabu.me/nevent1qqsfnkmtzlnt0w0yuy0wz9j2uc95uc9w9k02k80l9jx923ap7n0035szyp7k6kfku40gq2rdfu0q7t2l9skkd7hhcy85ncjryxkpcej5gdmhceh4uxc I see the contributed talks are now listed for HoTT/UF (<a href="https://hott-uf.github.io/2025/">https://hott-uf.github.io/2025/</a>) and the WG6 meeting (<a href="https://europroofnet.github.io/wg6-genoa/">https://europroofnet.github.io/wg6-genoa/</a>)! 2025-03-24T14:26:54Z