Timeline for Technical term for representing object of a presheaf determined by a left-adjoint?
Current License: CC BY-SA 3.0
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| Jun 28, 2017 at 12:51 | comment | added | Peter Heinig | Or, more precisely, ( constant functor $\mathsf{C}_0\rightarrow\mathsf{C}_1$ ) $=$ ( a functor of the form $\mathsf{C}_0\xrightarrow[]{!} 1 \xrightarrow[]{}\mathsf{C}_1$ ) . This is standard, and slightly off topic, of course---just mentioning it to round out this thread. | |
| Jun 28, 2017 at 12:40 | comment | added | Peter Heinig | One way to define constant functor without speaking of objects is to use "the terminal category 1" as some kind of logical primitive and say constant functor $=$ a functor of the form $1\rightarrow \mathsf{C}$. | |
| Jun 28, 2017 at 9:25 | comment | added | Peter Heinig | It is probably more useful to do this here, since there is not much to be explained: I was referring that one might be tempted (especially when weaned on classical analysis, with theorems like "locally-constant=>constant") to "define" a constant functor intrinsically (whatever that means in general; here it means: without ever mentioning the value of the functor), as a functor $F$ such that $F(o)=F(o')$ for all objects $o$ and $o'$. Someone coming from analysis might even consider this somehow better than mentioning a value explicitly. The latter is what was meant by "intrinsic definition". | |
| Jun 28, 2017 at 8:32 | comment | added | Todd Trimble | I don't understand your last sentence, but we can discuss this elsewhere. | |
| Jun 28, 2017 at 5:07 | comment | added | Peter Heinig | Thanks. The "an A consists not only of a B, but also of a specified C (as opposed to a D such that there exists a C)" pattern is useful. One could think of a list of such statements illustrating that, despite its focus on intrinsicality, category theory demands an additionally specified "part" of something. Basic example, modelled on your sentence: a constant functor consists not only of a functor such that any two of its values are equal, but also of a specified object giving its value. Here, there is another aspect: that the "intrinsic definition" would speak of equality of objects. | |
| Jun 28, 2017 at 4:52 | vote | accept | Peter Heinig | ||
| Jun 27, 2017 at 18:41 | history | edited | Todd Trimble | CC BY-SA 3.0 |
added a few variations on terminology
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| Jun 27, 2017 at 12:35 | history | answered | Todd Trimble | CC BY-SA 3.0 |