## Category Theory

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a place to discuss category theory, both applied and theoretical, and any other related things | part of the +mathematics:matrix.org community3 Servers

Timestamp Message
5 Sep 2019
15:15:52Joel Sjögren there is a map xy -> yx
15:17:07Joel Sjögren i think you could maybe use a "lax functor" to specify the permutations using this map
15:19:36Joel Sjögren intead of j(xy)=j(y)j(x) you would have j(xy)=j(x)j(y)->j(y)j(x)
15:19:46Joel Sjögren it's a bit foggy in my head
15:22:12Joel Sjögren i don't see where i was going with that
15:22:19Khinchin

First of all, for a contravariant functor, the functor maps a morphism f: A \to B where A and B both in C to a morphism Ff: FB -> FA in D where FB and FA both in D.

That is, there is a functor going reverse in a direction.

15:30:39Joel Sjögrenit seems like you should want the ability to specify a completely custom variance for the action of j on 1-morphisms (adjectives and nouns), but you still want the action on 2-morphisms (reductions) to be covariant
15:30:51Timyes, yes
15:31:58Joel Sjögren that is interesting input to my own (hobby) work on higher categories
15:32:35Timit's an interesting thing (to me, anyway) to think about in general, yeah
15:38:24Joel Sjögrenalternatively you can just say that the action of j on 1-morphisms is a function between sets, not necessarily functorial.
15:39:08Joel Sjögren * alternatively you can just say that the action of j on 1-morphisms is a function between the sets $J_E(\ast,\ast)$ and $J_S(\ast,\ast)$, not necessarily functorial.
15:39:36Joel Sjögren * alternatively you can just say that the action of j on 1-morphisms is a function between sets, not necessarily functorial.
15:41:41Joel Sjögren then you don't describe in an abstract way that this action is defined by an induction on the syntax.
15:43:27Joel Sjögren(oops, a typo above at 17:30, contravariant -> covariant)
15:43:35Joel Sjögren * it seems like you should want the ability to specify a completely custom variance for the action of j on 1-morphisms (adjectives and nouns), but you still want the action on 2-morphisms (reductions) to be covariant
15:43:46Joel Sjögren * (oops, a typo above at 17:30, contravariant -> covariant)
15:56:04Joel Sjögren and then j is captured entirely as its restriction as a functor between the 1-categories JE(.,.) and JS(.,.). it means that we forget about the monoidal structure for a moment.
15:57:03Timhm, that's an interesting way of seeing it actually
15:57:17Timmove the structure that isn't functorially respected to a higher level
15:59:20Joel Sjögren for j : JE -> JS, the nonfunctoriality lives at level 1, and the functoriality at level 2
16:05:01Timoh, that's the opposite of what i thought
16:10:54Joel Sjögren (now diverging from the topic by thinking about what it would mean be functorial at levels 1 and 3, but not 2 ...)
16:20:49Joel Sjögren even with a non-monoidal j you would still have (u : x -> x') => (j(uy) : j(xy) -> j(x'y)) which is reassuring i think
16:31:20Joel Sjögren the thing you don't get is j(uy) = j(u) id. to me this means that the translation could assign context-dependent (y-depdendent) meaning to the reduction u.
16:32:28Tim i don't see where j(uy) = j(u) id comes from?
16:33:23Joel Sjögren if j were monoidal then you would have j(uv) = j(u)j(v) wouldn't you?
16:34:01Joel Sjögren for u : x -> x' and v : y -> y' in JE
16:36:27Joel Sjögren i used (v := id a.k.a. y) : y -> (y' := y)
16:37:22Timoh, so y just means id_y