## 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 community2 Servers

Timestamp Message
5 Sep 2019
14:48:13Timsure
14:48:19Tim * sure
14:49:17Joel Sjögrenit seems to work in this case but if more reductions are defined maybe you want j(xy)=j(x)j(y) for those
14:49:41Joel Sjögrenso then you don't want j(xy)=j(y)j(x) for all x, y
14:50:02Timok, i think i see
14:50:44Timis there not some construction that lets us make the variance of the functor depend on e.g. the type of the variable?
14:51:24Timso, more generally, have a functor F:C->D with F contravariant on objects c:T and covariant on objects c:S
14:51:40Timsome sort of fibred construction
14:52:08Timor maybe embed the two subcategories (corresponding to each type) into C, and work with two functors separately
14:53:12Joel Sjögrenwhen you say variance, i take it that you mean that a monoidal category is a "2-category" (not sure about the terminology) with only one object. is this what you mean?
14:53:27Timyes, this
14:53:43Timi'm not fully certain of what i'm trying to say, at least, not formally
14:53:49Timjust have a vague idea
14:56:59Joel Sjögrendo we have (x->y) => (y*->x*)?
14:57:28Timwhat is * here?
14:57:35Timoh the dual
14:57:36Joel Sjögrendual
14:58:35Timwe should do, right? pretty sure you have this for pregroups anyway, but not certain here...
14:59:34Joel Sjögren i think so. i'm trying to work out the variance of *. is it in some formal sense more true that (xy)* = y*x* than (xy)* = x*y*?
14:59:55Joel Sjögreni need to make that code-style
15:00:13Joel Sjögren * i think so. i'm trying to work out the variance of *. is it in some formal sense more true that (xy)* = y*x* than (xy)* = x*y*?
15:00:16Tim(or you can disable markdown for an individual message with the toggle button)
15:00:33Tim(or use https://pigeon.digital 😉)
15:01:59Khinchin
15:02:23Timjust like latex: use enclose maths in $signs (e.g.$x^2$) 15:02:35Joel Sjögren$\pi\$