Definition df-termc 49832

Description: Definition of the proper class (termcnex 49935) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 49845). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24.

As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 49874). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 49879). See also dftermc3 49890 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 49890, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 49850.

Followed directly from the definition, terminal categories are thin (termcthin 49836). The opposite category of a terminal category is "almost" itself (oppctermco 49864). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 49899).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 49860) and terminal objects (dftermo4 49861). The universal properties provide an alternate proof of initoeu1 17947, termoeu1 17954, initoeu2 17952, and termoeu2 49597. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 49875).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17623, 0thincg 49817, func0g 49448, 0funcg 49444, and initc 49450.

(Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
df-termc	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

Distinct variable group:   𝑥,𝑐

Detailed syntax breakdown of Definition df-termc

Step	Hyp	Ref	Expression
1		ctermc 49831	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1541	. . . . . 6 class 𝑐
4		cbs 17148	. . . . . 6 class Base
5	3, 4	cfv 6500	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1541	. . . . . 6 class 𝑥
8	7	csn 4582	. . . . 5 class {𝑥}
9	5, 8	wceq 1542	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1781	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 49776	. . 3 class ThinCat
12	10, 2, 11	crab 3401	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1542	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is referenced by:  istermc  49833