Definition df-termc 50251

Description: Definition of the proper class (termcnex 50354) 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 50264). 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 50293). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 50298). See also dftermc3 50309 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 50309, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 50269.

Followed directly from the definition, terminal categories are thin (termcthin 50255). The opposite category of a terminal category is "almost" itself (oppctermco 50283). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 50318).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 50279) and terminal objects (dftermo4 50280). The universal properties provide an alternate proof of initoeu1 18063, termoeu1 18070, initoeu2 18068, and termoeu2 50016. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 50294).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17739, 0thincg 50236, func0g 49867, 0funcg 49863, and initc 49869.

(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 50250	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1569	. . . . . 6 class 𝑐
4		cbs 17264	. . . . . 6 class Base
5	3, 4	cfv 6536	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1569	. . . . . 6 class 𝑥
8	7	csn 4589	. . . . 5 class {𝑥}
9	5, 8	wceq 1570	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1809	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 50195	. . 3 class ThinCat
12	10, 2, 11	crab 3416	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1570	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is referenced by:  istermc  50252