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Stream: learning: reading & references

Topic: Inversion of various morphisms

Elisha Goldman (Jan 09 2026 at 12:46):

Has there been anything written on the general case of how categories change when various categories of morphisms are inverted? Homotopical categories are my main example here, but those mainly come up within a particular context, same with the free group generated by a monoid. The passage from ZF to ZFC is also spiritually similar, since splitting an epi is just adding a one-sided inverse. What's the general theory linking all of these examples?

Federica Pasqualone 🦅 (Jan 09 2026 at 12:56):

Elisha Goldman said:

Has there been anything written on the general case of how categories change when various categories of morphisms are inverted? Homotopical categories are my main example here, but those mainly come up within a particular context, same with the free group generated by a monoid. The passage from ZF to ZFC is also spiritually similar, since splitting an epi is just adding a one-sided inverse. What's the general theory linking all of these examples?

Do you mean categories of fractions and localizations?

Elisha Goldman (Jan 09 2026 at 13:10):

Yes, thank you I forgot the words. Does there exist a similar concept of free splittings? (And can we do this in formal category theory? What structure does a 2-category need to have a good sense of localization?)

Elisha Goldman (Jan 09 2026 at 13:13):

For example, maybe a category with free splittings is given by some lax universal property

James Deikun (Jan 09 2026 at 13:38):

There is such a thing as freely adding splittings of idempotents (the [[Karoubi envelope]]), but things like this make me think there's probably not a nice way to freely (or cofreely) add splittings of epis to a general category.

James Deikun (Jan 09 2026 at 13:44):

Also, I think localization and splitting of epis are kind of inherently different, because localization becomes much more well-behaved in $\infty$-category theory, but even a single section-retraction pair turns into something that is a real pain to think about in that setting.

fosco (Jan 09 2026 at 13:49):

there's probably not a nice way to freely (or cofreely) add splittings of epis to a general category.

Yes there is!
http://www.tac.mta.ca/tac/volumes/34/46/34-46.pdf

James Deikun (Jan 09 2026 at 13:49):

(The 1-localizations of nice 1-category at a nice class of morphisms usually is not a nice 1-category, but the same $\infty$-localization usually is a nice $\infty$-category, and truncating it down to a 1-category is what destroys all the nice properties. Contrarily, the splitting of an idempotent is still an absolute limit and colimit in $\infty$-category theory, but not a finite one!)

Elisha Goldman (Jan 09 2026 at 13:51):

fosco said:

Yes there is!
http://www.tac.mta.ca/tac/volumes/34/46/34-46.pdf

Does that paper give an explicit construction in terms of equivalence classes of zigzags? if so, I got sniped with like one sentence left lmao

Elisha Goldman (Jan 09 2026 at 13:53):

James Deikun said:

Also, I think localization and splitting of epis are kind of inherently different, because localization becomes much more well-behaved in $\infty$-category theory, but even a single section-retraction pair turns into something that is a real pain to think about in that setting.

Maybe that's just because it has a more natural interpretation in $(\infty,\infty)$ categories which are just inherently more complicated?

fosco (Jan 09 2026 at 13:54):

iirc the paper gives a certain relation such that the coequalizer in Cat has the desired property (it does it for split mono, dually for split epi)

James Deikun (Jan 09 2026 at 13:59):

fosco said:

there's probably not a nice way to freely (or cofreely) add splittings of epis to a general category.

Yes there is!
http://www.tac.mta.ca/tac/volumes/34/46/34-46.pdf

Wow, color me surprised. However, even though the result is a 1-category, it seems not to be free as a 1-category, but only "weakly free" in some sense that is made precise as a universal property of a related poset-enriched category?

Elisha Goldman (Jan 09 2026 at 14:02):

Now I'm having the generalization impulse... Isomorphisms $x \cong y$ are equivalently isomorphisms $Y_x \cong Y_y$ and split epis are equivalently pointwise surjections $Y_x ↠ Y_y$, you could totally do this in enriched categories. The paper is from 2019 too, someone could probably enrich it right now

fosco (Jan 09 2026 at 16:02):

However, even though the result is a 1-category, it seems not to be free as a 1-category, but only "weakly free" in some sense that is made precise as a universal property of a related poset-enriched category?

Yes, this is what I understood as well. I think this goes well with the gut feeling that a truly universal way to "turn a map into a split epi" shouldn't exist...

Also related to this story, from a different angle: https://arxiv.org/abs/2507.12044 @Graham Manuell will tell you more about it

Oscar Cunningham (Jan 09 2026 at 16:35):

I asked a related question here: Is there a universal way to force the Axiom of Choice to be true?
If I have a topos an I freely split all the epimorphisms, is it still a topos?