The literature of an emerging subject can be difficult to track. As an effort to try and help us through the current works in spectral algebraic geometry, I decided to try and catalogue some properties of morphisms in SAG, similar to appendix C of Görtz--Wedhorn's book.
We take the (very biased) perspective that a stack is that as defined in [BDL25] and we work in their category of stacks Stk, although we try to find older references in similar contexts when available.
Double question marks means either I don't yet know or I haven't gotten around to checking. Any suggestions, additions, or updates are appreciated!
Let P be a property of morphisms stacks. We say that P satisfies:
BC (base change) if for each f:Y->X satisfying P and each g:X'->X, the pullback of f along g also satisfies P.
CANC (cancellation) if for each pair of morphisms f:Y->X and g:X->W, when their composite satisfies P, then f satisfies P.
COMP (composition) if for each pair of morphisms f:Y->X and g:X->W both satisfying P, their composite also satisfies P.
DESC (descent) if for each f:Y->X and each effective epimorphism g:X'->X such that the pullback of f along g satisfies P, then f satisfies P.
LOCS (local on the source) if for each f:Y->X and each effective epimorphism g:Y'->Y such that the composite satisfies P, then f satisfies P.
0-affine: The absolute definition is [MM15, Df.3.1] and the relative version is [BDL25, Df.2.2.1.1]; BC ??; CANC holds as long as g is 0-affine with [BCSY24, Pr.2.8] as the categorical reference and [BDL25, Pr.2.2.1.2(1)] in this context; COMP holds with [BCSY24, Pr.2.8] as the categorical reference and [BDL25, Pr.2.2.1.2(1)] in this context; DECS ?? []; LOCS ?? [].
0-semiaffine: The definition is [BDL25, Df.2.2.1.5]; BC ??; CANC holds as long as g is 0-affine [BDL25, Pr.2.2.1.2(1)]; COMP holds [BDL25, Pr.2.2.1.2(2)]; DECS ?? []; LOCS ?? [].
Adams flat: The definition is [BDL25, Df.2.3.2.3]; BC ?? []; CANC ?? []; COMP ?? []; DECS ?? []; LOCS ?? [].
Affine: The definition is obvious [BDL25, Df.2.2.2.4] or [SAG, Df.2.4.4.1] the Deligne--Mumford version; BC is clear [Dav26, Pr.A.3] or [Tok26, Pr.A.16]; CANC holds is g has affine diagonal [Dav26, Pr.A.3]; COMP is easy [Dav26, Pr.A.3] or [Tok26, Pr.A.16]; DECS holds [Dav26, Pr.A.3] or [Tok26, Pr.A.16]; LOCS is clearly false in general (consider Spec S/G).
Affine cover: The definition is [BDL25, Df.2.3.2.1]; BC ?? []; CANC ?? []; COMP ?? []; DECS ?? []; LOCS ?? [].
Affine étale: The definition is [Dav26, Df.A.2]; BC holds by definition [Dav26, Pr.A.3]; CANC ?? []; COMP is clear [Dav26, Pr.A.3]; DECS holds [Dav26, Pr.A.3]; LOCS ?? [].
Affine flat: The definition is [Dav26, Df.A.2]; BC holds by definition [Dav26, Pr.A.3]; CANC ?? []; COMP is clear [Dav26, Pr.A.3]; DECS holds [Dav26, Pr.A.3]; LOCS ?? [].
Closed immersion: The definition is [BDL26a, Df.5.5.3]; BC holds [Tok26, Lm.A.18]; CANC ?? []; COMP holds [Tok26, Lm.A.18]; DECS holds [Tok26, Lm.A.18]; LOCS ?? [].
Finite étale: The definition is [Dav26, Df.A.2]; BC holds by definition [Dav26, Pr.A.3]; CANC ?? []; COMP is clear [Dav26, Pr.A.3]; DECS holds [Dav26, Pr.A.3]; LOCS ?? [].
Flat map between geometric stacks: The definition is [BDL25, Df.3.4.2.1]; BC ?? []; CANC ?? []; COMP ?? []; DECS ?? []; LOCS ?? [].
Finite flat: The definition is [Dav26, Df.A.2]; BC holds by definition []; CANC ?? []; COMP is clear [Dav26, Pr.A.3]; DECS holds [Dav26, Pr.A.3]; LOCS ?? [].
Locally descendable: The definition is [BDL25, Df.2.3.3.3]; BC holds [BDL25, Pr.2.3.3.8(1)]; CANC ?? []; COMP ?? []; DECS ?? []; LOCS ?? [].
Monomorphisms: BC holds [Tok26, Lm.A.17]; CANC ?? []; COMP holds [Tok26, Lm.A.17]; DECS holds [Tok26, Lm.A.17]; LOCS ?? [].
Open immersion: The definition for Deligne--Mumford stacks is [SAG, Df.1.6.7.2] and for general stacks is [BDL25, Df.3.3.1.1] (just remove the word quasi-compact); BC hold [Tok26, Lm.A.30]; CANC ?? []; COMP holds [Tok26, Lm.A.30]; DECS holds [Tok26, Lm.A.30]; LOCS ?? [].
Quasi-affine: The is definition for Deligne--Mumford stacks is [SAG, Df.2.4.4.1] and for general stacks is [BDL25, Pr.3.3.1.1]; BC is clear [BDL25, Pr.3.3.1.5(1)]; BC ?? []; CANC ?? []; COMP ?? []; DECS ?? []; LOCS ?? [].
Quasi-compact open immersion: The is definition [BDL25, Pr.3.3.1.1]; BC ?? []; CANC ?? []; COMP ?? []; DECS ?? []; LOCS ?? [].
Perfect: The original definition is [BZFN10, Df.3.2] but we use the slight alternation of [DTL26, Df.??]/[Dav26, Df.??] here; BC ?? []; CANC ?? []; COMP ?? []; DECS ?? []; LOCS ?? [].
Relative nonconnective spectral Deligne--Mumford stack: The is definition [BDL25, Pr.3.3.1.4]; BC holds by definition; CANC ?? []; COMP ?? []; DECS ?? []; LOCS ?? [].
Universally 0-affine: The definition is [BDL25, Df.2.2.2.1]; BC holds [BDL25, Pr.2.2.2.5(1)]; CANC ??; COMP holds [BDL25, Pr.2.2.2.5(3)]; DESC holds [Tok26, Lm.A.22]; LOCS ??.
[BDL25]: Balderrama, Davies, Linskens, Affineness and reconstruction in complex-periodic geometry, arXiv (2025)
[BDL26a]: Balderrama, Davies, Linskens, Ambidextrous global spectra and tempered cohomology, arXiv (2026)
[BDL26b]: Balderrama, Davies, Linskens, Separable isogenies of oriented elliptic curves and geometric norms on equivariant elliptic cohomology, in preparation (2026)
[BCSY24]: Barthel, Carmeli, Schlank, Yanovski, The chromatic Fourier transform, arXiv (Pi, 2024)
[BZFN10]: Ben-Zvi, Francis, Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, arXiv (JAMS, 2010)
[Dav26]: Davies, On Galois extension of geometric fixed point spectra, in preparation (2026)
[DLT26]: Davies, Linskens, Tokic, On equivariant unipotence in elliptic cohomology, in preparation (2026)
[MM15]: Mathew, Meier, Affineness and chromatic homotopy theory, arXiv (JTop, 2015)
[Tok26]: Tokic, A family completion theorem for tempered cohomology, arXiv (2026)
[ECII]: Lurie, Spectral Algebraic Geometry, available on his website (2018)
[SAG]: Lurie, Spectral Algebraic Geometry, available on his website (2018)